Model by Keven

High-quality constraints on the glacial isostatic adjustment process over North America: The ICE-7G_NA (VM7) model

Keven Roy

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto

High-quality constraints on the glacial isostatic adjustment process over North America: The

ICE-7G_NA (VM7) model

Keven Roy

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2017

The Glacial Isostatic Adjustment (GIA) process describes the response of the Earth’s surface to variations in land ice cover. Models of the phenomenon, which is dominated by the inﬂuence of the Late Pleistocene cycle of glaciation and deglaciation, depend on two fundamental inputs: a history of ice-sheet loading and a model of the radial variation of mantle viscosity. Various geophysical observables enable us to test and reﬁne these models.

In this work, the impact of the GIA process on the rotational state of the planet will be analyzed, and new estimates of the long-term secular trend associated with the GIA process will be provided. It will be demonstrated that it has undertaken a significant change since the mid-1990s. Other important observables include the vast amount of geological inferences of past sea level change that exist for all the main coasts of the world. The U.S. Atlantic coast is a region of particular interest in this regard, due to the fact that data from the length of this coast provides a transect of the forebulge associated with the former Laurentide ice sheet. High-quality relative sea level histories from this region will be employed to generate a new model of the GIA process that includes for the first time data from the forebulge region in its optimization process (the ICE-6G_C (VM6) model). Then, the series of analyses is extended to include space-geodetic observations of present-day vertical uplift of the crust. A solution reconciling all available data from the continent, named ICE-7G_NA (VM7), is obtained through modest further modifications of both the viscosity structure of the model and the North American component of the surface mass loading history. It provides an excellent fit to the constraining data related to the GIA process, including observations of the time-dependent de-levelling of the Great Lakes region. Finally, to test the global exportability of the new model, its predictions of relative sea level change are tested against observations from the Western Mediterranean region.

ii Pour mes parents, Chantal et Martin

iii Acknowledgements

First of all, I would like to thank my supervisor, Prof. W. R. Peltier, for his support and thoughtful mentoring throughout my project. Your guidance, advice and insight have helped me become a much better scientist and communicator. I am also grateful for the privilege you have given me to present my work and interact with other scientists at many conferences and workshops around the world during my time at the University of Toronto.

Special thanks go to Prof. Dylan Jones and Prof. Qinya Liu for their insightful comments and support while being on my Ph.D. committee. Rosemarie Drummond and Guido Vettoretti also deserve credit for the invaluable help they have provided throughout the project. I would also like to thank

Prof. Ben Horton and Dr. Donald Argus for sharing their expertise with me as I was learning about relative sea level data and space-geodetic observations. My external examiner, Prof. Geoﬀrey Blewitt, also deserves credit for his insightful comments on the ﬁnal version of this thesis. I would also like to thank the very hard-working people of the Physics Department, including Ana Sousa, Pierre Savaria,

Krystyna Biel and Teresa Baptista, who have helped me so many times. Special thanks also go to Dr.

Michel Bourqui. Your introductory course on atmospheric and oceanic physics at McGill University and your availability to discuss research in Earth sciences have helped me discover a passion for geophysics.

I would also like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC), the Ontario Ministry of Training, Colleges and Universities (through the Ontario Graduate Scholarship program) and the Fonds Nature et Technologies (FRQNT) for the funding I received in support of my project. I would also like to thank the Centre for Global Change Science (CGCS) for the funding I received to attend beneﬁcial and exciting conferences in Greenland and Australia.

Moreover, I am indebted to the amazing people I have met at the University of Toronto, and in particular the present and former members of the Atmospheric Physics and Geophysics groups. I have been lucky to be surrounded by such inspiring people and make lifelong friends in the process. In particular, special thanks go to Niall, Zen, Phil, Joseph, Ryan, Federico, Andrew, Alain, Ellen, Maria and Oliver. Your support and friendship have helped me carry this project through. I would also like to thank Simon: I was quite fortunate to have my best friend around for most of my time at the University of Toronto. Going rock climbing and spending time with you were always fun, and your friendship means a lot to me. The long discussions I have had with you and Jen have also helped me in the tougher times.

Cheers to both of you. Also, thanks to all my friends back in Québec and Montréal, who have helped me a great deal in staying balanced and motivated.

Finally, some very important people deserve recognition for the infallible support they have provided

iv me throughout my project. Betty, you deserve special credit for this project. Your love and constant support have helped me so much with ﬁnishing this project. J’aimerais ﬁnalement remercier ma famille, et particulièrement mes parents (Chantal et Martin), à qui cette thèse est dédiée, ainsi que mes soeurs

Mélissa et Karen. Votre support inconditionnel m’a permis de mener ce projet à terme. Je vous en suis extrêmement reconnaissant.

v Contents

1 Introduction 1 1.1 The orbital theory of the timing of the ice age cycle ...... 2

1.2 Ice age cycles in deep-sea sedimentary cores and ice cores ...... 7

1.3 Other geomorphological evidence and the Glacial Isotatic Adjustment (GIA) process . . . 11

1.4 Overview ...... 17

2 The Earth’s rotational state and its recent evolution 18 2.1 Fundamental dynamics ...... 19

2.1.1 A useful simpliﬁcation: the axisymmetric case ...... 21

2.1.2 A special case: the free Eulerian wobble ...... 22

2.1.3 The inﬂuence of the GIA process on the moment of inertia of the planet ...... 23

2.2 Observations of rotational variability and reference frame considerations ...... 25

2.2.1 Celestial reference systems and frames ...... 25

2.2.2 Terrestrial reference systems and Earth Orientation Parameters (EOPs) ...... 27

2.2.3 Rotational variability measurement techniques and the latest ITRF iteration . . . 32

2.3 The evolution of secular trends in the planetary rotational state: Methodology and results 34

2.3.1 Sources of variability in length-of-day measurements ...... 35

2.3.2 Sources of variability in polar wander measurements ...... 37

2.3.3 Polar wander and LOD evolution: Data set and methodology employed ...... 37

2.3.4 Secular trend pivot point determination: results and analysis ...... 41

2.4 Perspectives and future work ...... 45

3 GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 48 3.1 General approach and strategy ...... 50

3.2 Modelling the GIA process: Theoretical background ...... 51

vi 3.3 The ICE-NG models of ice sheet loading history ...... 55

3.4 Analysis of the performance of the ICE-6G_C ice loading history as a function of viscosity

model ...... 57

3.4.1 Relative sea-level history reconstructions ...... 57

3.4.2 The Engelhart and Horton (2012) data set of U.S. East coast RSL evolution . . . . 59

3.4.3 Analysis of the performance of the VM5a and VM5b viscosity structures ...... 59

3.4.4 The ﬁt to the Fennoscandian spectrum ...... 64

3.5 An exploration of alternative viscosity structures ...... 66

3.5.1 Basic assumptions and methodology employed ...... 67

3.5.2 Error analysis and model performance ...... 68

3.5.3 Alternative viscosity models: V1 and V2 ...... 69

3.5.4 Case study I: Mantle viscosity variations in the upper mantle ...... 74

3.5.5 Case study II: Viscosity changes in the upper part of the lower mantle ...... 77

3.5.6 Case study III: Viscosity variations in the transition zone ...... 77

3.5.7 Case study IV: Viscosity contrast variations between the upper mantle and lower

mantle ...... 80

3.5.8 Case study V: Lithosphere thickness variations ...... 82

3.5.9 Case study VI: Lower mantle viscosity variations and the Earth’s rotational state . 85

3.5.10 Other considerations and a summary of the insights gained through the sensitivity

analyses ...... 86

3.6 A preferred viscosity structure: the VM6 proﬁle ...... 88

3.6.1 The predictions for the U.S. East coast for ICE-6G_C (VM6) ...... 89

3.6.2 The ﬁt to the Fennoscandian relaxation spectrum ...... 93

3.6.3 Testing the viscosity structure against data from the North American West coast . 94

3.7 Conclusion ...... 97

4 Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 100

4.1 Geophysical observables related to the GIA process in North America and the performance

of current models ...... 101

4.1.1 Constraints on former ice sheet extent and thickness ...... 102

4.1.2 The importance of RSL data from the North American region of forebulge collapse 102

4.1.3 Space-geodetic uplift measurements over North America ...... 104

vii 4.1.4 Time-dependent gravity measurements from the Gravity Recovery and Climate

Experiment (GRACE) satellites ...... 108

4.1.5 Water level gauges in the Great Lakes region ...... 110

4.2 Characterization of the model misﬁts to present-day space-geodetic uplift rate observations110

4.2.1 The impact of elastic lithospheric thickness and mantle viscosity variations upon

vertical uplift rate predictions ...... 112

4.2.2 Impact of ice loading history variations on vertical uplift rate predictions . . . . . 114

4.2.3 The impact of ice loading history variations on relative sea level evolution . . . . . 118

4.2.4 The impact of mantle viscosity variations on the diﬀerential uplift of the Great

Lakes ...... 120

4.3 Parameter space search and model reﬁnement ...... 122

4.3.1 Model misﬁts and error determination ...... 124

4.3.2 Initial viscosity model generation and parameter space exploration ...... 125

4.3.3 Ice loading history variations and full model optimization ...... 128

4.3.4 The ICE-7G_NA (VM7) model of the GIA process ...... 130

4.3.5 The Upper Campbell (Lake Agassiz) strandline tilt ...... 141

4.3.6 The ﬁt to the time-dependent gravity measurements of the Gravity Recovery and

Climate Experiment ...... 145

4.3.7 Model reﬁnement considerations: margin evolution considerations of the south-

western edge of the Laurentide Ice Sheet ...... 148

4.4 Conclusion ...... 149

5 A regional test of the ICE-7G_NA (VM7) model in the Mediterranean basin 151 5.1 Some geological considerations in the Mediterranean basin ...... 152

5.2 The Vacchi et al. (2016) data set of Holocene RSL change ...... 152

5.3 Results for the ICE-7G_NA (VM7) model in the Mediterranean basin ...... 154

5.4 Evaluation of the ICE-7G_NA (VM7) model performance ...... 158

5.5 Revisiting the interpretation of Roman ﬁsh-tanks ...... 160

5.6 Perspectives ...... 162

6 Conclusion 164 6.1 Perspectives and future work ...... 165

Bibliography 168

viii A Copyright and contributions 191

ix List of Tables

2.1 Observed values of the principal moments of inertia of the Earth ...... 21

2.2 Inter-study comparison of secular trends in polar wander and in the J˙2 coeﬃcient . . . . . 42 2.3 Compilation of global mass balance estimates of mountain glaciers ...... 45

3.1 Relaxation times in Eastern James Bay ...... 74

3.2 Error comparison for the viscosity variations under consideration ...... 79

3.3 Radial viscosity structure of the VM6 viscosity proﬁle ...... 89

3.4 Change in error misﬁt between the ICE-6G_C (VM5a/VM6) models ...... 92

4.1 Parameter space exploration values considered ...... 127

4.2 Radial viscosity structure for the VM5a/VM6/VM7 proﬁles ...... 130

4.3 Values of Earth rotational observables predicted by ICE-6G_C and (VM5a) and ICE-

7G_NA (VM7) ...... 141

4.4 Relaxation times determined in Eastern James Bay ...... 141

5.1 Comparison of the performance of the ICE-6G_C (VM5a) and ICE-7G_NA (VM7) mod-

els in the Western Mediterranean ...... 160

5.2 Comparison of observational inference and GIA model predictions of Roman time RSL

along the Tyrrhenian coast ...... 162

x List of Figures

1.1 Examples of erratic boulders ...... 2

1.2 Orbital parameters of the Earth’s orbit ...... 3

1.3 Orbital parameter evolution over the last million years ...... 5

1.4 Examples of seasonal variations in latitudinal solar insolation ...... 6

1.5 δ18O record from the Panama Basin ...... 8

1.6 Evolution of atmospheric temperature and CO2 from the EPICA Dome C ice core . . . . 10

1.7 Idealized representation of the Glacial Isostatic Adjustment (GIA) process ...... 13

1.8 Relative sea level change at Barbados since the Last Glacial Maximum ...... 15

2.1 Location in the sky of the 295 deﬁning extralactic radio sources of the ICRF2 ...... 27

2.2 The ﬁve intermediate transformations linking the ICRF with the ITRF ...... 30

2.3 Correspondence between nutation and polar motion in various reference frames ...... 31

2.4 Polar wander values from the SPACE2008 data set ...... 38

2.5 Study of secular trends in the SPACE2008 data set of polar wander motion ...... 40

2.6 Fitted SPACE2008 data set of polar wander motion ...... 40

2.7 Study of secular trends in the J˙2 coeﬃcient of gravitational potential ...... 41

2.8 Total root-mean-squared error for J˙2 and polar wander ...... 42

2.9 Compilation of recent mass balance estimates of the Greenland and Antarctic ice sheets . 44

2.10 Impact of recent land ice melting on the J˙2 Stokes coeﬃcient ...... 46

3.1 Topographic height anomaly at LGM in the Northern hemisphere for ICE-5G and ICE-

6G_C...... 57

3.2 Schematic representation of a salt marsh environment and relative sea level indicators . . 60

3.3 Geographical location of the North American sites used in Roy and Peltier (2015) . . . . . 61

3.4 The VM5a, VM6 and VM2 viscosity proﬁles ...... 63

xi 3.5 Comparison of the Engelhart and Horton (2012) data set with the ICE-6G_C (VM5a/VM5b)

model predictions ...... 65

3.6 Comparison of the VM5a, VM5b, V1 (FM) and V2 (FM) viscosity proﬁles ...... 70

3.7 Comparison of the performance of selected viscosity models (VM5a, VM5b, V1, V2) along

the U.S. East coast ...... 71

3.8 Eastern James Bay isolation basin data of Pendea et al. (2010) ...... 73

3.9 Comparison of model performance along the U.S. East coast with upper mantle viscosity

variations ...... 76

3.10 Comparison of model performance along the U.S. East coast with lower mantle (upper

part) viscosity variations ...... 78

3.11 Comparison of model performance along the U.S. East coast with transition zone viscosity

variations ...... 81

3.12 Comparison of model performance along the U.S. East coast with viscosity contrast vari-

ations ...... 83

3.13 Comparison of model performance along the U.S. East coast with lithosphere thickness

variations ...... 84

3.14 Comparison of model performance along the U.S. East coast with lowermost mantle vis-

cosity variations ...... 87

3.15 ICE-6G_C (VM6) model performance when compared to U.S. East coast relative sea

level observations ...... 91

3.16 Observed and predicted rates of late Holocene sea level rise ...... 93

3.17 Comparison of the VM5a/VM5b/VM6 viscosity proﬁles ...... 94

3.18 Performance of the ICE-6G_C (VM6) model with respect to RSL observations along the

U.S. West coast ...... 96

4.1 Deglaciation isochrones for North America ...... 103

4.2 Space-geodetic measurements of vertical uplift rates over North America ...... 106

4.3 ICE-6G_C (VM5a/VM6) model predictions of vertical uplift rates over North America

and comparison with observational data ...... 107

4.4 GRACE observations of the time dependence of the gravitional ﬁeld of the Earth with

ICE-6G_C (VM5a/VM6) model predictions ...... 109

4.5 Observations of vertical uplift rate around the Great Lakes from water gauge observations

and ICE-6G_C (VM5a/VM6) model predictions ...... 111

xii 4.6 Comparison of the modern vertical uplift rate (mm/yr) over the North American continent

using model variations based upon the ICE-6G_C (VM5a) model ...... 113

4.7 Coral-based record of relative sea level evolution at Barbados with ICE-6G_C (VM5a)

model predictions (and a model variation based upon it) ...... 116

4.8 Model predictions of vertical uplift over North America for the ICE-6G_C (VM5a/VM6)

models ...... 117

4.9 Prediction of the vertical uplift rate ﬁeld over North American continent for modiﬁed ice

loading histories combined with the VM5a viscosity proﬁle ...... 119

4.10 Comparison of observations of relative sea level evolution along the U.S. East coast with

modiﬁed ice loading histories based upon the ICE-6G_C (VM5a) model ...... 121

4.11 Comparison of Great Lakes vertical uplift rate observations with model predictions based

upon ICE-6G_C (VM5a) with diﬀerent elastic lithosphere thickness ...... 123

4.12 Regions where modiﬁcations to the ICE-6G_C ice loading history were initially introduced128

4.13 Radial viscosity variations for the VM5a/VM6/VM7 proﬁles ...... 131

4.14 Snapshots of the thickness of the Laurentide and Greenland ice sheets for ICE-7G_NA

since LGM ...... 132

4.15 Diﬀerence in ice thickness between the ICE-7G_NA and ICE-6G_C ice loading histories

over North America ...... 133

4.16 Predicted topography over North America at LGM with respect to sea level ...... 133

4.17 Comparison between relative sea level observations along the U.S. East coast and ICE-

6G_NA (VM7) model predictions ...... 134

4.18 Background map of predicted present-day rate of vertical motion of the crust for the

ICE-7G_NA (VM7) model ...... 136

4.19 Comparison of observations of vertical uplift rates traverses of the North American con-

tinent with ICE-6G_C (VM5a/VM6) and ICE-7G_NA (VM7) model predictions . . . . . 137

4.20 Quality-of-ﬁt determination between space-geodetic measurements of present-day vertical

motion rates and model predictions ...... 139

4.21 Comparison of observed late Holocene relative sea level rise along U.S. East coast with

ICE-6G_C (VM5a/VM6) and ICE-7G_NA (VM7) model predictions ...... 140

4.22 Comparison of water gauge observations of present-day vertical uplift rates along the

Great Lakes with ICE-6G_NA (VM7) model observations ...... 140

4.23 Comparison of the modeled tilt of the Upper Campbell strandline associated with former

Lake Agassiz with the tilt measured at present ...... 142

xiii 4.24 Comparison of relative sea level evolution observations along the U.S. East coast with

ICE-7G_NA (VM7) model predictions ...... 144

4.25 GRACE observations of the time dependence of the Earth’s gravitational ﬁeld over North

America and ICE-7G_NA (VM7) model predictions ...... 146

4.26 Comparison between GLDAS-NOAH and WGHM hydrological contributions to the time

dependence of the Earth’s gravitational ﬁeld over North America ...... 147

5.1 Tectonic setting of the Western Mediterranean Sea and location of relative sea level ob-

servations ...... 153

5.2 Comparison of observations of relative sea level evolution in the Mediterranean Sea with

ICE-7G_NA (VM7) model predictions (part 1) ...... 155

5.3 Comparison of observations of relative sea level evolution in the Mediterranean Sea with

ICE-7G_NA (VM7) model predictions (part 2) ...... 157

5.4 Contour plot of relative sea level at 2 ka BP across the Mediterranean region predicted

by the ICE-6G_C (VM5a) and ICE-7G_NA (VM7) models ...... 161

6.1 Inferred GIA vertical uplift rates over Greenland and contributions of each drainage basin

to post-LGM eustatic sea level change ...... 166

xiv Chapter 1

Introduction

One of the first geological observations connected to past variability in climatic conditions came from the Alps, where the presence of erratic boulders, large rocks of a different rock type from their surround- ings, was seen as a major scientific puzzle (Figure 1.1). Initially portrayed as evidence of some of the catastrophic flooding events described in the Bible or in other ancient texts, it took until the end of the

18th century for works linking them to the action of ice, such as that of the famous early geologist James

Hutton (Krüger, 2013), to emerge, often on the account of folkloric tales of mountain glacier advance and retreat. It was only later, in the 19th century, that the notion was extended to the presence of large ice sheets over most of Europe, notably in the works of Jean de Charpentier (de Charpentier, 1841) or

Louis Agassiz (Agassiz, 1840), who observed most eloquently:

"The perched boulders which are found in the Alpine valleys [...] occupy at times positions so extraordinary that they excite in a high degree the curiosity of those who see them. For instance, when one sees an angular stone perched upon the top of an isolated pyramid, or resting in some way in a very steep locality, the ﬁrst inquiry of the mind is, When and how have these stones been placed in such positions, where the least shock would seem to turn them over?" (Agassiz, 1840).

These accounts sought to explain not only the presence of erratic boulders throughout the mountainous regions of central Europe, but also in numerous plains or marshes to the north. The obvious impact on oceanic depth of such large ice sheet growth was suggested a few years later (Maclaren, 1842), and the following decades saw an increasing number of further observations from many northern regions.

However, it was not until the middle of the 20th century that plausible mechanisms to explain their growth and decay were put forward, however limited these were by the lack of precisely dated geological evidence.

Amidst various propositions to explain the ice age phenomenon, the Serbian mathematician Milutin

1 Chapter 1. Introduction 2

Figure 1.1: (a) A drawing of an erratic boulder, called Pierre des Marmettes, in the Rhône valley. The 10-metre rock served as an inspiration to Jean de Charpentier, as its granitic composition indicated that it originated more than 30 kilometres up the valley. From de Charpentier (1841), and reproduced with permission. (b) Erratic boulders perched on the top of eroded accumulations of glacier material in Switzerland, similar to the ones that inspired Louis Agassiz in the 19th century. Picture from the Glacier Photograph Collection of the National Snow and Ice Data Center (NSIDC) (Reid, 1902), and reproduced with permission.

Milanković popularized the idea that cycles in the parameters describing the Earth’s orbit around the

Sun could induce variations in the insolation received by the Earth’s surface large enough to signiﬁcantly impact its climate and induce ice sheet growth and decay over the Northern hemisphere (Milanković,

1941). Although his original view has been modiﬁed as observations of past climate variability and models of the astronomical processes driving such cycles became available, he put emphasis on the cyclical nature of ice sheet formation that orbital pacing should induce, and the theory of orbitally-paced ice age cycles

(or Milanković cycles) still bears his name today.

1.1 The orbital theory of the timing of the ice age cycle

As it orbits the Sun, the Earth is subject to complex variations in the gravitational attraction it feels towards other bodies in the solar system, and these variations induce changes in the geometry of the

Earth’s orbit. This geometry is described by three main parameters, illustrated in the left panels of

Figure 1.2. The eccentricity of the orbit describes the departure of its shape from a circle through the ratio of the the semi-major axis of the orbit to its focus (with ratio values e ranging from 0 for a circle to

1 for an inﬁnitely elongated ellipse), while its obliquity angle refers to the tilt of the axis of rotation with respect to the ecliptic, or the plane with respect to which the Earth revolves around the Sun. The third parameter is the axial and apsidal precession of the orbit. Axial precession refers to the secular variation Chapter 1. Introduction 3

Figure 1.2: Presentation of the three main parameters describing the geometry of the Earth’s orbit around the Sun, namely (a) its eccentricity, (b) its axial obliquity, and (c) its precessional cycle (modulated by eccentricity). The left panels show graphical illustrations of each cycle, while the right panels show the variations in each parameter over the last 150,000 years, together with vertical lines representing the timing of the end of the Eemian interglacial (PEGI), the Last Glacial Maximum (LGM) and the start of the Holocene (HO). The left axes show the range in values for each quantity. For eccentricity-precession, it indicates the position of the northern hemisphere (NH) solstices with respect to perihelion, with the dashed lines showing the eccentricity modulation. When the full curve intersects the top dashed line, the NH winter solstice (WS) occurs at perihelion (P), and when it intersects the bottom dashed line, the NH summer solstice (SS) occurs at perihelion (P). The right axis show the main impact of each parameter on the insolation calculation. From Peltier (2007), reproduced with permission.

in the position of the rotation axis with respect to the stars caused by the diﬀerential gravitational pull on the equatorial bulge of the planet, a phenomenon which will be discussed in detail in a later section of this work. This eﬀect is compounded by orbital precession of the orbit itself, which sees the apside

(the line joining the perihelion and aphelion of the orbit) drifting with respect to the ﬁxed background of stars because of gravitational interactions with Jupiter and Saturn, and smaller contributions from relativistic eﬀects and the Sun’s oblateness (Fitzpatrick, 1970).

The evolution of the three parameters can be stably computed analytically (Berger and Loutre, 1991) or numerically (Laskar et al., 2004) for the past several million years, as an n-body problem including the planets, their satellites and the tidal interactions between them (corrected for relativistic eﬀects). A Chapter 1. Introduction 4 reconstruction of the eccentricity, the obliquity and the precession (modulated by eccentricity) over the past million years is presented in Figure 1.3, while a zoom over the last 150,000 years is provided in the right panels of Figure 1.2.

Careful examination of Figure 1.3 shows the periodic or quasi-periodic nature of the variations in orbital parameters. Obliquity is characterized by a 41,000-year cycle that brings the value of the axial tilt from 22° to 24.5° (the present value is 23° 44 ′), while eccentricity shows variations between e = 0 and e = 0.05 (present-day value of e = 0.0167) with periods of about 405,000 and 100,000 years.

Precession/eccentricity variations exhibit cycles with periods of 19,000 years, 22,000 years and 23,000 years, due to the combined effect of the 26,000-year cycle of axial precession with the 112,000-year cycle of orbital precession (which effectively shortens the effective cycle of the precession of the equinoxes as seen from Earth (Fitzpatrick, 1970)).

Changes in the geometry of the Earth’s orbit induce corresponding variations in the distribution of solar energy on the surface of the planet (in the case of obliquity and precession variations) or in the total amount of solar energy received by the planet (in the case of eccentricity), as well as changes in the phase of seasons as a function of the position of the Earth along its orbit. The impact of orbital geometry variations on the distribution of solar energy can be computed as a function of latitude and time using a geometrical expression based upon Kepler’s second law, and accounting for the various changes in eccentricity, obliquity and precession combined in the expression:

∆Q (θ,t)=∆RS (θ)∆ǫ (t)+ m (θ)∆[e (t) sin (˜ωt)] , (1.1) where ∆Q (θ,t) represents the anomaly in solar insolation at the top of the atmosphere, measured in

W/m2, as a function of time and latitude, with respect to a chosen background value. This quantity includes the variations in obliquity ∆ǫ (t), in eccentricity e (t) and in the precessional cycle sin (˜ωt), and takes into account the latitude dependence of this quantity via the parameters ∆RS (θ) and m (θ).

Figure 1.4 displays the present-day insolation at the top of the atmosphere as a function of latitude and time of year, together with the insolation anomaly for three other time periods. The mid-Holocene

(panel b) is the most recent manifestation of a major orbital forcing positive anomaly in the Northern hemisphere. Around the time of the Last Glacial Maximum (LGM), the insolation anomaly was minimal compared to present values (panel c), while there was a very strong negative summer anomaly over the

Northern hemisphere at the time of the onset of the last glaciation/deglaciation cycle, about 116,000 years ago (panel d).

The original Milankovi`ctheory states that changes in summer insolation received by mid-latitudes in Chapter 1. Introduction 5

Figure 1.3: Orbital parameter evolution over the last million years, for (A) eccentricity, (B) axial obliquity, and (C) precession (modulated by eccentricity). From Cronin (2010) (using data from Berger and Loutre (1991)), reproduced with permission. Chapter 1. Introduction 6

Figure 1.4: Latitudinal distribution of solar insolation at the top of the Earth’s atmosphere as a function of time of year, showing (a) present insolation values (W/m2), and anomalies with respect to present values (∆W/m2) at (b) the mid-Holocene (6,000 years ago), (c) the Last Glacial Maximum (21,000 years ago), and at (d) the onset of the last glacial cycle at the end of the Eemian interval (116,000 years ago). From Peltier (2007), reproduced with permission. Chapter 1. Introduction 7 the northern hemisphere should be key to the growth or decay of ice sheets, as this parameter is directly linked to the amount of summer melt an ice sheet will experience. He also postulated the existence of a positive feedback loop in which the increase in snow cover induced by low insolation values would increase the albedo and further cool these regions.

1.2 Ice age cycles in deep-sea sedimentary cores and ice cores

Evidence for the inﬂuence of orbital forcing on the climate system came to light in the 20th century with the development of paleoceanography and deep-sea sedimentary records, which enabled the recovery of long records of past climate conditions through the careful analysis and interpretation of isotopic proxies.

In particular, oxygen isotopic anomalies extracted from the shells of small sea-dwelling protists, called foraminifera, found in sedimentary cores can be used as a high-quality proxy for past continental ice sheet volume (Shackleton, 1967; Shackleton et al., 1990). This is due to the natural occurrence in sea water of three stable isotopes of oxygen (16O, 17O and 18O, with relative abundances of 99.75%, 0.04% and 0.2%, respectively (Bradley, 2015)), and the fact that evaporation is a kinetic mass fractionation

16 process that will preferentially use the lighter H2 O water molecules. As the evaporated water from the oceans eventually gets locked-up in the growth of continental ice sheets at high latitudes during

18 glacial eras, the oceanic water becomes anomalously rich in the heavier H2 O molecules. This anomaly is described by the δ18O ratio:

[18O/16O] δ18O = sample − 1 , (1.2) [18O/16O] standard where the relative concentration ratio [18O/16O] of a sample is compared to an oceanic average value. As the foraminifera shells use oxygen from oceanic water to make their calcium carbonate (CaCO3) shells, they will retain the oxygen isotopic composition of oceanic water during their lifetime as they eventually get deposited at the bottom of the ocean when they die. Records of δ18O can be obtained by associating the depth at which the foraminifera shells are found in the sedimentary core to a time of death: more positive values indicate a glacial era while more negative values are associated with interglacial periods.

This approach was ﬁrst used in a seminal paper by Hays et al. (1976) on two deep-sea sedimentary cores in the southern Indian ocean to recover a record of δ18O anomalies over the past 450,000 years as a proxy for continental ice cover. Beyond a clear ∼100,000-year glacial-interglacial cycle extending over the record, a spectral analysis revealed strong quasi-periodic ∼41,000-year and ∼23,000-year periodic signals, providing the ﬁrst support to the Milankovi`ctheory of orbital pacing of the climate. Figure 1.5 displays Chapter 1. Introduction 8

Figure 1.5: (a) Oxygen isotopic δ18O record from the Ocean Drilling Program Site 677 over the past two million years. (b) A power spectrum of the earlier and later million years of the record, showing a strong periodicity over the last million years in the record with a 100,000-year period, as well as smaller spectral power peaks at other frequencies associated with the cycles in the Earth’s orbital geometry parameters (periods of ≈41,000 and ≈22-23,000 years). From Peltier (2007), reproduced with permission. a two-million-year high-resolution record (Shackleton et al., 1990), extracted from the sedimentary core of the Ocean Drilling Program Site 677 oﬀ the coast of Panama and Colombia (Shackleton and Hall,

1989), together with a power spectrum analysis of the signal for the 0-1 Myr BP and 1-2 Myr BP periods.

The strong 100,000-year periodicity over the last million years is evident, as well as other spectral lines at periods of ∼41,000 and ∼22-23,000 years.

Evidence for climatic conditions appropriate for the growth and decay of such large continental ice cover can be found in numerous other paleoclimate records, most notably in ice cores recovered from major ice sheets. In particular, isotopic analysis of the gas trapped in small cavities found in the ice can be used as proxies to infer past atmospheric composition, temperature and circulation. Several ice cores have been recovered from Antarctica, including the 420-ka record from the Vostok ice core (Petit et al., 1999) and the 800-ka record from the EPICA ice core recovered from Dome C (Augustin et al.,

2004; Jouzel et al., 2007). Ice core records from Greenland, such as the GRIP (Greenland Ice Core

Project), GISP2 (Greenland Ice Sheet Project) (Grootes et al., 1993) and NGRIP (North Greenland Ice

Core Project) (Andersen et al., 2004) records, provide an inter-hemispheric connection to the records obtained in Antarctica, but they extend only into the last glacial cycle (or, in the case of the NGRIP Chapter 1. Introduction 9 core, into the last interglacial period) due to the younger age of the deepest part of the Greenland Ice

Sheet.

δ18O anomalies can also be recovered from ice cores. However, in this case, the recovered anomaly will provide information about the temperature of the ambient atmosphere from which the water molecule precipitated, with the δ18O anomaly roughly falling by ∼1‰ per 1.5° Celcius, although the relationship is also inﬂuenced by other factors, such as the source area of the precipitation or thermal diﬀusion during the transition from snow to ice (Cronin, 2010).

One of the most important direct measurements which can be obtained from ice cores is related to the past atmospheric concentration obtained from careful analysis of the gas trapped inside the ice core. In particular, records of past greenhouse gas concentrations in the atmosphere provide a unique, high-resolution window in the past state of the climate. Greenhouse gases are atmospheric constituents that absorb and emit radiation at infrared wavelengths of the electromagnetic spectrum that correspond to the thermal radiation spectrum emitted by the Earth’s surface, the atmosphere or clouds (IPCC,

2013). Although the lowermost atmosphere is dominated by latent and sensible heat transport, at upper altitude, where radiative effects dominate, greenhouse gases, together with cloud droplets and some aerosols, absorb the upward radiation and re-emit it in all directions, resulting in an effective warming of the lower atmosphere. This radiative effect is the so-called greenhouse effect. The main greenhouse gases are water vapour (H2O), carbon dioxide (CO2), methane (CH4), nitrous oxide (N2O) and ozone

(O3). An increase (decrease) in atmospheric concentration of these substances aﬀects the energy balance of the climate system (with the radiative forcing expressed in W/m2), provoking a warming (cooling) of the lower atmosphere and of the surface until the radiative equilibrium of the climate system is restored.

Although water vapor accounts for a signiﬁcant portion of the greenhouse eﬀect, it is generally considered as a positive feedback on a given episode of atmospheric warming rather than a "prime mover" of the warming itself, since its atmospheric concentration increases with temperature. A record of atmospheric

CO2 recovered from the EPICA Dome C ice core is shown in the lower panel of Figure 1.6. Both panels of the ﬁgure display a close correspondence and a similar 100,000-year periodicity to that found in the sedimentary core-based δ18O record shown in Figure 1.3(a).

The link between atmospheric composition, air temperature and ice sheet growth and decay is apparent from Figure 1.6, and provides the ﬁrst tangible indication that the study of past continental ice sheet extent must be considered within the broader framework of paleoclimate studies. Partly within the framework initially put forward by Milankovi`c,examination of paleorecords and of insolation model results reveals that strong positive feedbacks (from changes in greenhouse gas concentrations and surface albedo) are required to explain the large amplitude of the late Pleistocene glacial-interglacial cycles (e.g. Chapter 1. Introduction 10

Figure 1.6: Results from the EPICA Dome C ice core in Antarctica over the past 700,000 years. (a) Estimated local temperature anomaly from Dome C, inferred from oxygen isotopic δ18O anomalies, with respect to the average of the last 1000 years, and (b) Atmospheric CO2 concentration obtained from Vostok ice core for 0-413,000 years before present (BP) (open squares), and from EPICA Dome C for 413,000-650,000 years BP (open circles). From Cronin (2010) (using data from Jouzel et al. (2007) and Petit et al. (1999)), reproduced with permission. Chapter 1. Introduction 11

Abe-Ouchi et al., 2007, 2015; Tarasov and Peltier, 1997, 1999). A careful analysis of the large millenial- scale climate variability is also enabled by the high resolution of the proxy records, enabling the analysis of Heinrich events (e.g. Hemming, 2004), Dansgaard-Oeschger (D-O) oscillations (e.g. Peltier and Vet- toretti, 2014), the Younger Dryas cold interval (e.g. Tarasov and Peltier, 2005; Muscheler et al., 2008), and of the existence of any time lag or inter-hemispheric phase diﬀerence in the records (e.g. Shakun et al., 2012; Broecker, 1998).

1.3 Other geomorphological evidence and the Glacial Isotatic

Adjustment (GIA) process

Beyond the evidence of past ice sheet extent provided by erratic boulders and from proxies recovered from sedimentary and ice cores, direct geomorphological evidence can also be found over regions that were previously glaciated, including a large fraction of North America and Northern Europe. These hints include the presence of:

• glacial moraines, formed by debris (till) extracted by friction from the bedrock, carried by glaciers

and accumulated on their sides or at the foot of their maximal extent;

• drumlins, hills formed by deposition of till and moraine material or the abrasion of unconsolidated

bed material, elongated in the direction of motion of the ice ﬂow;

• grooves/striations formed by friction from ice movement over the bedrock;

• glacial valleys, long U-shaped troughs, such as fjords, carved by glaciers.

A spectacular, indirect observation of former glacial extent comes from the study of land movement over formerly glaciated areas. Historical observations of sea retreat were already made in Scandinavia at the end of the Middle Ages, where harbours on the Baltic Sea, such as Öregrund or Luleå, had to be abandoned and moved forward towards the shore to compensate for its eﬀect. For example, Stockholm was historically founded on a chain of islands in a deep and narrow bay giving on the Baltic Sea, which was progressively cut out as the sea retreated after the Viking age (Krüger, 2013). Today, the former bay is a freshwater lake (Lake Vätanen) and the city straddles the two water basins.

The potential strategic and economic consequences of such movements of the sea led to concern from the Swedish crown as early as the beginning of the 17th century, and the famous astronomer Anders

Celsius was commissioned in 1731 to investigate the possible reasons behind them. As part of his study, he inscribed a mark at the average position of the sea near the top of a large boulder in the Gulf of Chapter 1. Introduction 12

Bothnia used in his days by marine mammals as a resting spot due its near complete submergence, and decided to track the evolution of the sea at that location (the level of the mean sea was marked every following year on the so-called Celsius rock, and the boulder is now almost completely out of the water). He concluded that the sea was retreating throughout Sweden, and ascribed this change to the evaporation of water from the oceans (Ekman, 2009). After the publication of Louis Agassiz’s account of past large ice sheet cover (Agassiz, 1840), the issue of sea variations in Northern Europe was revisited and it was inferred that continental ice sheet formation would lead to the removal of a large volume of water from the oceans and aﬀect their depth (Maclaren, 1842).

The potential link between glacial cycles and land motion followed the introduction of the concept of isotasy in the middle of the 19th century, which refers to the idealized gravitational balance of the uppermost layer of the Earth (lithosphere) with the underlying, denser asthenosphere, developed in response to the discovery of negative gravity anomalies in the Himalayas by Louis Bouguer, a counter- intuitive result given the extra mass of the mountains over which the measurements were performed.

This result of apparent mountainous mass excess at the surface was explained using isostatic arguments as either due to the existence of a low-density "root" extending deeper in the mantle that balanced the mountainous rise (the Airy model in 1855), or due to a lateral gradient in density that resulted in mountains where the density of the crust was lower (the Pratt model in 1854). The potential role of

ﬂexure in the response of the crust was soon recognized by F.A. Vening Meinesz, and a hybrid solution using components of both the Airy and Pratt hypotheses was put forward by Aleksanteri Heiskanen.

Isostatic considerations of the Earth structure have played a fundamental role in the much later development of the theory of plate tectonics (Schubert et al., 2001), but it was not long before the potential link between isostatic considerations of the crust and the fast (geologically-speaking) growth and melt of large ice sheets was recognized by Jamieson (1882), who suggested that the melting of the large ice cover over Northern Europe should have induced an uplift of the crust (e.g. Peltier, 1998c). It was also an implicit realization that the rheology of the planet should have a viscous component, and not be completely elastic, due to the continuing adjustment of the Scandinavian surface to the long-gone ice age load.

Glacial Isostatic Adjustment (or GIA) refers to the deformation of the Earth’s surface in response to varying land ice surface loads, which induce corresponding changes in relative sea level and in the planetary gravitational ﬁeld. A schematic view of the process is presented in Figure 1.7. When a large ice sheet forms over a continent, global sea levels fall due to the removal of a large volume of water from the oceans. Also, as seen in the upper panel, the surface of the Earth under the ice sheet adjusts

(in an isostatic sense) to the extra weight put upon it, whereby mantle material ﬂows outward and the Chapter 1. Introduction 13

Figure 1.7: Idealized representation of the eﬀects of the Glacial Isostatic Adjustment (GIA) process on the surface of the planet in a region undergoing ice sheet growth during an ice age (upper panel), and after its removal (lower panel). Figure sourced from the Canadian Geodetic Survey (Natural Resources Canada).

lithosphere ﬂexes to create a peripheral bulge (or forebulge) in front of the ice sheet. The lower panel presents the response of the surface once the ice sheet has disintegrated. Global sea levels rise due to the redistribution of the water that was once locked in the former ice sheet, while the crust under it rises to recover isostatic equilibrium. The forebulge slowly collapses, which indicates that post-glacial isostatic adjustment will result in regions of uplift under former ice sheets and subsidence in their outskirts.

The application of the isostatic adjustment theory to the problem of varying ice loading at its surface was originally described in a series of seminal papers in the 1970s (Peltier, 1974, 1976; Peltier and

Andrews, 1976; Farrell and Clark, 1976; Clark et al., 1978; Peltier et al., 1978), and its mathematical structure as a visco-elastic problem has been used since then to generate constraints on the eﬀective viscosity structure of the planetary mantle, a crucial ingredient in the design of models of the mantle convection process, and on reconstructions of the geographical distribution of land ice cover over the last glaciation-deglaciation cycle (e.g. Peltier, 1994, 2004; Argus et al., 2014; Peltier et al., 2015). This Chapter 1. Introduction 14 knowledge has direct impacts on many scientiﬁc domains, including the determination of boundary conditions on paleotopography and paleobathymetry that are required when studying ice-age climate conditions using modern coupled atmosphere-ocean climate models (e.g. Vettoretti and Peltier, 2013;

Peltier and Vettoretti, 2014; Abe-Ouchi et al., 2015). These boundary conditions are also important when studying the strong climate variability that has characterized some eras during the post-LGM deglaciation, such as the Younger Dryas period (Tarasov and Peltier (2005); also discussed in Chapter

4), or determining the origin and characteristics of post-LGM rapid sea level rise events inferred from tropical sea level records, such as the Meltwater pulses 1A and 1B (Peltier, 2005; Argus et al., 2014;

Peltier et al., 2015). It has also been recently suggested that they could be used to ﬁnd equivalents to present-day mass loss in Greenland (Khan, S. A. et al., 2016).

GIA models also provide a crucial way to determine coastline evolution and understand changes in sea level around the world that accompany the glaciation-deglaciation cycle. In particular, the large variations in ice cover that characterized the last ice age have also left an imprint on sea level worldwide.

One of the most famous examples of such changes comes from the long-term record of sea level history based upon U-Th dating of coral terraces on the island of Barbados in the Caribbean Sea (Fairbanks,

1989; Peltier and Fairbanks, 2006). This record, once corrected for tectonic uplift of the island, is presented in Figure 1.8 and reveals a large change in eustatic sea level of about 120 meters since the

Last Glacial Maximum, revealing the large volume of the ice sheets that once covered the Northern hemisphere. The importance of this crucial record will be discussed in further detail in Chapters 3 and

4 of this work.

Results from GIA models are important to understand relative sea level evolution around the world and to determine the present-day horizontal and vertical motion associated with the GIA process at every point on the planet. This information can then be used to understand and attribute present-day local land movements, or to interpret various archaeological sites, such as Neolithic sites in Scandinavia, the ﬁrst human settlements in the Americas and ﬁsh ponds used in former Roman settlements (Chapter

5) (Peltier, 1998c; Lambeck et al., 2004; Evelpidou et al., 2012). They can also be used to understand mass movements within the Earth system and interpret long-term changes in the rotational state of the planet (Chapter 2) and on the present evolution of its gravitational ﬁeld (Chapter 4).

Moreover, knowledge of the GIA process is important in the context of the present-day climate change primarily induced by anthropogenic emissions of greenhouse gases (IPCC, 2013), notably regarding the renewed episode of continental deglaciation that is occurring due to this warming of the lower troposphere. Our observations of the ongoing melting of both the great polar ice sheets on Antarctica and Greenland (as well as smaller ice catchments), which is responsible for an important fraction of the Chapter 1. Introduction 15

Figure 1.8: Relative sea level indicators at Barbados, indicating a very large change in relative sea level in meters (over 120 meters) as a function of time (in thousands of years before present) since the Last Glacial Maximum (LGM) about 21-26 thousand years ago. The data points are shown in green (from Fairbanks (1989)) or in purple (from Peltier and Fairbanks (2006)), with the horizontal bar showing the tectonic uplift-corrected depth of the indicator, and the vertical bar the range with respect to sea level in which the particular coral species from which the sample was recovered could be found. The eustatic sea level curve for a version of the precursor GIA model ICE-5G (VM2) of Peltier (2004) is shown as the black line. From Peltier and Fairbanks (2006), reproduced with permission. Chapter 1. Introduction 16 global rise of sea level occurring at the present time, are signiﬁcantly impacted by the remaining isostatic disequilibrium associated with the last ice-age cycle, the signal of which must be eliminated in order to more clearly identify the global warming component (e.g. Peltier and Tushingham, 1989; Peltier, 2009).

In particular, the GIA correction is central to the analysis of the time dependent gravity results being provided by the Gravity Recovery and Climate Experiment (GRACE) satellites (Chapter 4) and the determination of the recent melting of Greenland, Antarctica, and of other smaller ice catchments such as the Alaskan glaciers (Peltier and Luthcke, 2009; Velicogna and Wahr, 2013; Luthcke et al., 2013).

Also, the determination of global sea level rise and local relative sea level change since the mid-19th century relies heavily on the interpretation of the data collected by tide gauges worldwide, which must be corrected for GIA eﬀects originating from the lasting inﬂuence of the past ice age and from the recent increased melting of ice sheets and glaciers around the world (Hay et al., 2015).

The study of the GIA process relies on a suite of geophysical observables that provide crucial information on diﬀerent parameters of the models, which will be described in detail in this work. These include, but are not limited to:

• Observations of the rotational state of the planet (Chapter 2);

• Numerous indicators of past relative sea level (Chapters 3, 4 and 5);

• Space-geodetic observations of crustal motion (Chapters 4 and 5);

• Records of paleo-shorelines (Chapter 4);

• Space-based recovery of planetary gravitational anomalies (Chapter 4);

• Water level gauges from the North American Great Lakes (Chapter 4);

• Dated reconstructions of ice margins from geomorphological markers (Chapter 4).

In the past decade, the study of the GIA process has beneﬁted from a large increase in the number and quality of collected geophysical observables, most notably over North America. In fact, the advent of gravimetric measurements from the GRACE satellites, space-based geodetic observations made using the

Global Positioning System (GPS) and concerted eﬀorts to develop databases of relative sea level records with standardized error treatment (Engelhart and Horton, 2012; Vacchi et al., 2014) have generated a wide range of constraints for GIA models. Testing the most recently available GIA models against these new data sources has largely supported their validity, but it has nonetheless raised certain issues that might indicate systemic problems with some of the model inputs, notably regarding the mantle viscosity structure used (Engelhart et al., 2011). Chapter 1. Introduction 17

1.4 Overview

The overarching goal of this work is to construct a model of the GIA process that provides an acceptable

fit to the entire suite of geophysical observables related to the GIA phenomenon over North America and, in particular, that reconciles the model predictions with the latest generation of observables collected over the continent. The general procedure to be followed will be, (1) to test the ICE-6G_C (VM5a) model of Peltier et al. (2015) against the newly available geophysical observables for the North American continent; then, (2) to examine if the misfits that arise from these comparisons can be corrected by appropriate changes in the parameters of the model; and finally, (3) generate a cohesive GIA model that incorporates all of this knowledge and provides further constraints on the viscosity of the mantle and on the history of land ice loading.

In this context, a clear hierarchical consideration of the available geophysical constraints will be used. Chapter 2 will be based on the work of (Roy and Peltier, 2011) and focus on the rotational state of the planet. The latest observations of the main rotational anomalies and of their secular trends

(associated with the GIA process) will provide constraints for the following steps in the work presented here. Chapter 3 will then proceed to focus on the newly available database of high-quality relative sea level indicators for both coasts of the United States (Engelhart et al., 2011; Engelhart and Horton, 2012;

Vacchi et al., 2014), regions that bisect the forebulge associated with the former Laurentide ice sheet.

Following the work presented in Roy and Peltier (2015), it will see the introduction a new viscosity proﬁle

(VM6), obtained under the assumption all misﬁts between model predictions and the new dataset can be removed solely through appropriate mantle viscosity variations that enable a better capture of the behaviour of the forebulge. Chapter 4 will then relax this assumption, and following the work of Roy and

Peltier (2017), extend the methodology to the whole suite of the latest GIA-related observations over

North America to consider variations in both mantle viscosity and ice loading history. The expanded network of space-geodetic crustal uplift rate measurements and water level gauges from the Great Lakes will be used, in conjunction with the ice-margin reconstructions of the Laurentide ice sheet, to generate a new model of the GIA process, ICE-7G_NA (VM7) (Roy and Peltier, 2017). Its plausibility will then be tested (and conﬁrmed) against spaced-based gravimetric measurements of the GRACE satellites and paleo-shoreline records that have been used in the literature to generate alternate models of the GIA process. Finally, Chapter 5 will see the new model being successfully tested in another region of the world, namely in the western half of the Mediterranean basin. Chapter 2

The Earth’s rotational state and its recent evolution

Changes in the distribution of mass within the Earth system, whether in its interior or at its surface, result in variations in its gravitational ﬁeld (via Newton’s law of universal gravitation) and in its rotational state (via the conservation of angular momentum). In this chapter, the impact of the redistribution of mass associated with the previously described Late Pleistocene cycle of glaciation/deglaciation on the rotational state of the planet will be analyzed. In particular, secular trends in observables that capture the rotational variability of the planet have been associated with the GIA process since the 1980s (Peltier,

1982; Wu and Peltier, 1984), and the ability of GIA models to explain these trends is a major constraint of their plausibility (Peltier, 2007).

Here, the patterns of variability which have been identiﬁed in the rotational state of the planet on a wide range of time scales will be presented, alongside the observational data used to quantify and deﬁne them. The most recent observational data sets will be analyzed in order to (1) determine the recent evolution of the Earth’s rotational state, and (2) to update the rotational constraints to which GIA models should be tuned. This chapter follows the peer-reviewed contribution of Roy and Peltier (2011)

(Roy, K., and Peltier, W. R. (2011), ’GRACE era secular trends in Earth rotation parameters: A global scale impact of the global warming process?’, Geophysical Research Letters 38(10), L10306), with the addition of a more substantial introduction and overview of the information currently available on the rotational state of the planet. It will provide strong and original evidence that the Earth’s rotational state has undergone important changes in the last few decades, and hypothesize that modern climate change is the culprit behind this phenomenon.

18 Chapter 2. The Earth’s rotational state and its recent evolution 19

2.1 Fundamental dynamics

The Earth can be approximated as a quasi-rigid body (which can exhibit deformation and ﬂex over time scales much longer than the time scale associated with a full rotation), for which the three-dimensional rotational state can be described in a ﬁxed, inertial frame of reference by its angular velocity ~ω(t) and its moment of inertia tensor J . The angular momentum of the planet can be separated into contributions from the planetary moment of inertia, and from relative angular momentum ~h(t) that would result from any motion ~u(t) with respect to the axes of rotation. Using the center of mass as the origin, and using the Einstein summation convention, each component of the total angular momentum L~ will then take the form:

Li = Jij(t)ωj + hi(t), (2.1)

where Jij = V ρ (xkxkδij − xixj) dV are the elements of the moment of inertia tensor, ωi(t) is the i-th R component of the angular velocity vector, and the relative angular momentum hi(t)= V ρ (εijkxjuk) dV is due to relative motion ~u(t) about the principal axes of rotation. Without loss of generality,R a coordinate system coincident (momentarily) with the ﬁxed, inertial frame of reference, and rotating at angular velocity ~ω(t = 0) relative to it, is chosen. The Eulerian equations of motion describing the changes in angular momentum in the presence of external torque ~τ take the form:

dL~ ~τ = + ~ω × L~ (2.2) dt !

Equation (2.2) is called the Liouville equation of motion (Munk and MacDonald, 1960). Whereas it is valid for any group of rigid, rotating axes, it may be useful to restrict the chosen axes to a few special cases that take advantage of the inherent symmetry of the system, such as the so-called Tisserand’s mean- mantle axes, which are defined such that the relative motion of the crust and of the mantle with respect to the rotating frame of reference is null (~h(t)crust = 0 and ~h(t)mantle = 0) (e.g. Jeffreys, 1952; Tisserand, 1891). Since geodetic measurement techniques rely (at least partially) on ground-based measurements, it is also useful to include the mean motion of the crust in the definition of the Tisseyrand axes, such that the relative motion of both the mantle and the crust with respect to the frame of reference is null, an extension which is called the Tisserand mean-surface axes (Wahr, 1981). Also useful are the principal axes of rotation, with respect to which the moment of inertia tensor is diagonalized (Jij = 0 for i =6 j). However, although these options are attractive from a mathematical perspective, it is often preferable to determine the rotating coordinate system with respect to "fixed" celestial background objects (like a catalogue of stars or extra-galactic objects), as will be discussed in the following section. Considering a Chapter 2. The Earth’s rotational state and its recent evolution 20 case in which the zˆ-axis is approximately aligned with the rotation axis of the planet (see below), small perturbations to the mean state of rotation about the zˆ-axis ( ~ω0(t)=Ω0zˆ) are expressed as:

~ω(t)= ~ω0(t)+∆~ω(t)=Ω0zˆ + ~m(t). (2.3)

The moment of inertia tensor will be perturbed by introducing small variations around the diagonalized tensor J0 corresponding to the alignment of the Earth system with its principal axes of rotation:

A 0 0 ∆Ixx(t) ∆Ixy(t) ∆Ixz(t)     J (t)= J0 + ∆J (t)= 0 B 0 + ∆Ixy(t) ∆Iyy(t) ∆Iyz(t) (2.4)         0 0 C ∆Ixz(t) ∆Iyz(t) ∆Izz(t)         In Equation (2.4), A, B and C refer to the principal moments of inertia of the Earth, which are ordered such that A

2007):

dm B (C − B) 1/2 B 1/2 1 dβ (t) x + α m (t)= −α x − β (t) + τ (t)=Ψ (t) (2.5a) dt A (C − A) r y r A Ω dt y x x 0

dm A (C − A) 1/2 A 1/2 1 dβ (t) y − α m (t)= −α y + β (t) + τ (t)=Ψ (t) (2.5b) dt B (C − B) r x r B Ω dt x y y 0 dm dβ (t) z = − z + τ (t)=Ψ (t) (2.5c) dt dt z z

The above equations are the governing equations for the evolution of the angular velocity vector for the

Earth. Physically, changes recorded along the reference rotation axis (zˆ-axis) in Equation (2.5c) are perceived as variations in the length of the day, while the coupled diﬀerential equations (2.5a) and (2.5b) are perceived as a motion of the instantaneous axis of rotation of the Earth with respect to its reference rotation axis, a phenomenon which is also called polar wander. Above, the frequency of oscillation αr is:

(C − A)(C − B) 1/2 α = Ω , (2.6) r AB 0 and the Ψi(t) elements are the so-called excitation functions, in which βi(t) corresponds to the changes in moment of inertia: 1 1 βx(t)= ∆xz(t)+ hx(t) , (2.7a) − − Ω0 (C A)(C B) p Chapter 2. The Earth’s rotational state and its recent evolution 21

1 1 βy(t)= ∆yz(t)+ hy(t) , (2.7b) − − Ω0 (C A)(C B) p 1 1 β (t)= ∆ (t)+ h (t) . (2.7c) z C zz Ω z 0 Changes in the rotational state of the planet can be expected to impact both the Earth’s moment of inertia and relative angular momentum values. By using Tisseyrand’s mean-mantle frame, there will be, by deﬁnition, no change to the relative angular momentum of the planet caused by the motion of the mantle (Tisserand, 1891). This extension of Tisseyrand’s deﬁnition to include both the motion of the mantle and motion of the crust leaves only the contributions of the core, the oceans and of the atmosphere to contribute to the variations in relative angular momentum of the planet.

2.1.1 A useful simpliﬁcation: the axisymmetric case

A useful simpliﬁcation of Equations (2.5a) to (2.7c) can be obtained from the fact that the Earth is almost axisymmetric, suggesting that the two equatorial moments of inertia will be similar (A ≈ B, see

Table (2.1)).

Parameter Value (×1037 kg·m2) A 8.0101 B 8.0103 C 8.0365

Table 2.1: Observed values of the principal moments of inertia of the Earth (Groten, 2004)

′ A+B Using the average value of A and B (A = 2 ≈ A ≈ B) greatly simpliﬁes Equations (2.5a) to (2.7c) (Lambeck, 1980; Peltier, 2007), and the Liouville equations become

dm ′ ′ x + α m =Ψ (t), (2.8a) dt r y x

dm ′ ′ y − α m =Ψ (t), (2.8b) dt r x y

dm ′ z =Ψ (t), (2.8c) dt z with a natural frequency which takes the simpliﬁed form

′ ′ (C − A ) α = Ω (2.9) r A′ 0 Chapter 2. The Earth’s rotational state and its recent evolution 22

The excitation functions are also greatly simpliﬁed by the previous approximations and become

′ ′ ′ ′ ′ αr dβx(t) Ψx(t)= αrβy(t) − + τx(t), (2.10a) Ω0 dt

′ ′ ′ ′ ′ αr dβy(t) Ψy(t)= −αrβx(t) − + τy(t), (2.10b) Ω0 dt

′ dβ (t) Ψ (t)=Ψ (t)= − z + τ (t), (2.10c) z z dt z where the changes in moment of inertia are written:

′ 1 1 β (t)= ∆ (t)+ h (t) , (2.11a) x C − A′ xz Ω x 0

′ 1 1 β (t)= ∆ (t)+ h (t) , (2.11b) y C − A′ yz Ω y 0 1 1 β (t)= ∆ (t)+ h (t) . (2.11c) z C zz Ω z 0

2.1.2 A special case: the free Eulerian wobble

In the absence of excitation and external torques (βi(t) = 0 and τi(t) = 0, hence Ψi(t) = 0), the right- hand side of Equations (2.5a) to (2.5c) becomes null, and the coupled equations for polar wander are solved by:

mx(t)= M cos (αrt + ψ) (2.12) A(C − A) 1/2 m (t)= M sin(α t + ψ) (2.13) y B(C − B) r for a phase ψ determined by the initial conditions of the system, an amplitude M along the x-axis and a frequency of oscillation αr described by Equation (2.6). This motion, ﬁrst predicted by Euler (1765), is the free periodic motion that a rigid Earth should exhibit if the rotation axis were momentarily displaced from its principal axis, and should have a period of about 305 days using the values for the principal moments of inertia of the Earth from Table 2.1. Chapter 2. The Earth’s rotational state and its recent evolution 23

2.1.3 The inﬂuence of the GIA process on the moment of inertia of the planet

Mathematical models of the GIA process, which have been developed since the 1970s, have been successfully used to predict the impact of ice-age inﬂuence on the rotational state of the planet (e.g. Peltier,

1982; Wu and Peltier, 1984; Peltier, 2007). Although the detailed mathematical structure of these models will be discussed in the following sections, it is useful to present here the way in which moment of inertia perturbations that lead to polar wander and length-of-day variations can be generated by the large mass redistribution characteristic of the GIA process. As this relationship has been thoroughly reviewed in the literature (e.g. Peltier, 2007), only the key results will be shown here.

First, it should be noted that the length-of-day variations that are induced by changes in the axial component of the moment of inertia of the Earth can be equivalently expressed over long time scales as a change in the gravitational potential of the Earth. This effect is measured in terms of the evolution of the parameter J2, the dimensionless Stokes coefficient of degree 2 and order zero of the Earth’s gravitational field, which is directly related to the oblateness of figure of the Earth. This time evolution, denoted as

J˙ 2, is linked to the angular velocity of the Earth by a simple expression (Lambeck, 1980; Peltier, 2007):

˙ 3C ω˙ 3 J2 = 2 , (2.14) 2a me Ω0

where a represents the average radius of the Earth, me represents the mass of the Earth, ω3 is the angular velocity of the Earth with respect to its rotation axis, and Ω0 is the modern average angular velocity of the Earth.

With regards to the polar wander components of rotational variability, two sources of perturbation from the GIA process have been identified. The first relies on the direct effect of the viscoelastic nature of the modelled Earth undergoing isostatic adjustment. This is expressed through the so-called Love numbers (Love, 1911), which are dimensionless parameters that capture the elastic (or viscoelastic) response of a planetary object to tidal potential or surface mass loads (to be described in further detail in a following section of this work). In particular, Peltier (1982) has shown that the direct, loading-related

LOAD impact from the GIA process on the moment of inertia of the Earth, ∆Iij , takes the form:

LOAD L rigid ∆Iij = 1+ k2 (t) ⋆Iij (t), (2.15) L where k2 (t) is the degree 2 surface mass loading Love number, ⋆ represents a temporal convolution, and rigid ∆Iij (t) is the perturbations to the moment of inertia that the same surface load would have induced Chapter 2. The Earth’s rotational state and its recent evolution 24 on a totally rigid equivalent of the Earth.

The second contribution to the moment of inertia perturbations originates from the fact that the rotation itself is changing, and is described using the tidal potential loading Love number of degree 2,

T k2 (t), of Farrell (1972) in the so-called MacCullagh’s formula (Munk and MacDonald, 1960; Peltier, 2007): T ROT k2 ′ ∆Iiz = ⋆ mi (C − A ) , (2.16) kf

3G ′ where kf = 5 2 (C − A ), i can represent either the xˆ or yˆ component, the axisymmetric approxi- a Ω0 mation of section 2.1.1 has been used, and G is the universal gravitational constant.

Substitution of these moment of inertia perturbations in the Laplace-transformed versions of the polar wander components of Equations (2.8) and (2.9), while dropping the subscripts for clarity’s sake, leads to: kT (s) sm + α 1 − 2 m =Ψ (s), (2.17a) x r k y x f kT (s) sm + α 1 − 2 m =Ψ (s), (2.17b) y r k x y f where s is the transform variable.

Assuming that externally applied torques vanish, the excitation functions expressed in Equation

(2.10) take the form, in the Laplace domain:

Ω s Ψ (s)= 0 ∆I − ∆I , (2.18a) x A yz A xz

Ω s Ψ (s)= 0 ∆I − ∆I , (2.18b) y A xz A yz

L rigid where, based on Equation (2.15), ∆Iij(s)= 1+ k2 (s) ∆Iij (s). Thus, the impact of the GIA process on the polar wander component of rotational variability is obtained by solving for its two components mx and my:

Ω 1+ kL(s) m (s)= 0 2 ∆Irigid(s), (2.19a) x ′ kT (s) xz Aαr  1 − 2  kf  

Ω 1+ kL(s) m (s)= 0 2 ∆Irigid(s), (2.19b) y ′ kT (s) yz Aαr  1 − 2  kf   In Equations (2.19), the rigid equivalent to the moment of inertia perturbations induced by the surface load can be easily determined from the history of ice sheet loading used in the GIA model Chapter 2. The Earth’s rotational state and its recent evolution 25 realization (to be described in a following chapter), using the surface integral:

rigid 2 ∆Iij (t)= v (θ,φ,t) a δij − xij dS, (2.20) Z Z where v (θ,φ,t) is the surface mass loading at time t for latitude θ and longitude φ.

Further details for these calculations are provided in Peltier (2007), together with an in-depth discussion of the assumptions that are integral to their expression.

2.2 Observations of rotational variability and reference frame

considerations

Determining the evolution of the Earth’s rotation takes advantage of a wide range of terrestrially-based and space-based measurement techniques, which have considerably gained in precision over the last few decades. To achieve the precision required to understand the minute variations in the Earth’s rotation, all of the techniques discussed below require both a precise, uniform time scale and a suitable reference frame against which measurements can be established.

To determine the position of the Earth in space and the properties of its rotation, two rotation frames need to be defined: an Earth-fixed reference frame which follows the Earth’s mean rotation and is fixed to the Earth’s surface as best as possible, and a celestial, inertial reference frame that is based upon stellar, galactic or extra-galactic sources assumed to be fixed in space. In astronomy and geodesy, they are based upon a reference system, which specifies the rules concerning the origin and the axes to be used, which is then realized specifically for a given reference frame by giving the coordinates of a series of reference observations that follow the rules prescribed by the reference system. For celestial frames of reference, a list of specific astronomical sources to be located is prescribed, while the survey control stations, also called geodetic reference points, are the equivalent prescribed locations when defining a terrestrial reference frame (Boucher, 2001).

2.2.1 Celestial reference systems and frames

An inertial reference system should ideally be based upon a precise knowledge of all forces acting on the main bodies of a given system and their exact position, such that the equations of motion describing its evolution could be re-written in such a way that all rotational terms are removed. However, it is easier in practice to rely upon a kinematic deﬁnition for the reference system, which states that far objects Chapter 2. The Earth’s rotational state and its recent evolution 26 exhibiting no proper motion should remain ﬁxed and show no residual rotational or translational motion.

Locations of astronomical objects have traditionally been given using two angles: the declination of an object being its angular distance measured perpendicularly to the extension to inﬁnity of the

Earth’s equatorial plane, and its right ascension being the eastward angular distance measured along the equator between the object and a given reference point (that defines a zero meridian). This reference point is traditionally given by the vernal equinox, defined as the south to north (or upward) intersection of the Sun’s apparent yearly motion in the celestial sphere (the ecliptic) and of the equatorial plane of the Earth. Since both the ecliptic and the equatorial plane are not fixed, the coordinate systems that use them need to be associated with a specific time at which the origins were determined. Recent measurements are given with respect to the equinox at 12h (on the Terrestrial Time scale, measured by atomic clocks and uncorrected for irregularities in the rotation of the Earth) of January 1, 2000 (Julian), represented as J2000.0.

In order to develop the ﬁrst celestial reference frames, catalogues of star positions and proper motions were developed from the 19th century onwards using the timing of their lunar occultations, the precision of which was limited before the advent of atomic clocks. The ground-based catalogues of stars have been supplanted in the past decades by wide-ranging space-based surveys. In particular, the Hipparcos catalogue of star positions and proper motion (ESA, 1997) provides over 118,000 astrometric observations collected over the 1989-1993 period by the eponymous satellite of the European Space Agency (ESA).

The GAIA satellite, its ESA successor, has been in orbit since the end of 2013 and should provide a substantial update on the star catalogue, as it aims to make over a billion astrometric measurements in the galaxy (de Bruijne, 2012).

Since a given reference frame should be usable for any future observation, the accuracy of the proper motion of the reference objects used in its realization is a crucial limiting factor of its accuracy. This limitation is compounded by the fact that a large fraction of the astrometric observations used in the recent star catalogues are based upon relatively short time series, or ones for which the earliest measurements are unreliable (Feissel and Mignard, 1998). This implies that the quality of a given reference frame realization will deteriorate quickly with time if unaccounted systematic errors in the determination of proper motions exist. The issue was largely addressed with the advent of radio-astronomy and the discovery of powerful extragalactic compact radio sources such as quasars, highly energetic galactic nuclei located billions of light-years away, at such a distance that their proper motion is undetectable even when using the most precise techniques available today. The development of radio interferometry

(Very Long Baseline Interferometry (VLBI), described below) enabled the International Astronomical

Union (IAU) to locate these sources with a sub-milliarcsecond (mas) accuracy (Feissel and Mignard, Chapter 2. The Earth’s rotational state and its recent evolution 27

Figure 2.1: The location in the sky of the 295 deﬁning extragalactic radio sources of the Second Realiza- tion of the International Celestial Reference Frame (ICRF2). The celestial sphere is represented using an Aitoﬀ equal area projection. From Schuh and Behrend (2012), reproduced with permission.

1998).

The ﬁrst celestial reference system that used the newly gained precision of extragalactic radio sources is the International Celestial Reference System (ICRS) (Arias et al., 1995), adopted in 1998 (Feissel and

Mignard, 1998). Its origin is deﬁned at the barycenter of the solar system in order to facilitate the introduction of general relativistic corrections (Kovalevsky, 2001). The pole of the ICRS is given, for the sake of continuity from previous systems, in the J2000.0 direction, while the right ascension origin is aligned with a speciﬁc source from previous systems. The latest (second) version of the International

Celestial Reference Frame (ICRF) is an implementation of the ICRS that relies on a catalogue of 3414 extragalactic radio source positions determined by VLBI observations over two observational bands

(wavelengths of 3.6 cm and 13 cm), in which a subset of 295 sources are used as deﬁning sources that precisely align the frame with the J2000.0 equator and equinox (Feissel and Mignard, 1998).

2.2.2 Terrestrial reference systems and Earth Orientation Parameters (EOPs)

A terrestrial reference system is a spatial reference system that rotates together with the Earth’s surface in its diurnal cycle, in such a way that the ensemble of all deﬁning points on the surface displays no residual rotation (Kovalevsky, 2001; Petit and Luzum, 2010). Coordinates of points on the surface do not display variations beyond tidal and tectonic eﬀects.

After Euler’s suggestion that the Earth should exhibit wobbling around its rotation axis, the ﬁrst observational hints corroborating its existence came from observations of periodic variations in latitude by German astronomers in 1881, followed a few years later by the detection of their anti-phased counterpart in Hawaii (at roughly 180° of longitude away from Germany). In a seminal paper, H.C. Chandler discovered a 14-month phase in the signal (Chandler, 1891), which convinced the International Asso- Chapter 2. The Earth’s rotational state and its recent evolution 28 ciation of Geodesy (IAG) to set up the International Latitude Service (ILS) to study this motion and monitor its evolution. It oversaw the installation of six observatories placed at roughly the same latitude

(39° 8 ′) but at diﬀerent longitudes (three in the United States, one in Italy, one in Japan, and one in

Turkmenistan, later replaced by a station in Uzbekistan), which were tasked with the careful monitoring of the position of twelve groups of six pairs of stars (Yokoyama et al., 2000). The comparison of the collected measurements at each site, once corrected for refraction corrections, provided a means to record the evolution of the latitude measurements of the stars, while the spread of the observatories along the same latitude simpliﬁed the analysis.

The ﬁrst global terrestrial reference frame developed from the combination of site position measurements was performed by the Bureau International de l’Heure (BIH) in 1984. The rules deﬁning how realizations of terrestrial reference frames should be performed were laid down in 1988, with the creation of the International Terrestrial Reference System (ITRS):

• The frame should be geocentric (the origin of the system being at the center of mass of the entire

Earth system, which includes the atmosphere and the oceans);

• The time coordinate is determined by a geocentric local frame (accounting for general relativistic

eﬀects)

• The orientation of the poles is coincident with the BIH (1984) orientation;

• The time evolution of the frame is determined by the condition that the frame should undergo no

net rotation under the horizontal tectonic motion at the Earth’s surface.

Then, once the terrestrial reference system is realized into a particular frame of reference using a set of space-geodetic and/or astronomical observations, time-dependent coordinates of interest in the body-

ﬁxed, rotating terrestrial frame ~xT (t) can be transformed into the inertial celestial frame ( ~xC (t)), using a coordinate transformation of the type (e.g. Gross, 2007; Petit and Luzum, 2010):

~xC (t)= QRW ~xT (t) (2.21)

= PNRXY ~xT (t) (2.22) where Q = PN is associated with the celestial part of the motion of the pole in the celestial reference system caused by precession (P) and nutation (N), R represents the rotation of the Earth about this pole, and W = XY accounts for the remaining polar motion along two speciﬁed xˆ and yˆ directions. While only three time-dependent transformations should be necessary to perform a coordinate transformation Chapter 2. The Earth’s rotational state and its recent evolution 29 between the two frames, the International Astronomical Union (IAU) traditionally uses an intermediate frame of reference in order to separate celestial and terrestrial components of the rotation variability. In more detail, the celestial parameters under consideration in the transformation matrices linking the two frames are:

Axial precession (P) The precession of the Earth’s rotation axis was the ﬁrst observed source of variability in the rotational state of the planet in the 2nd century BC by the Greek astronomer

Hipparcus (however, anecdotal evidence suggests it might have been discovered earlier in India and

Ancient Egypt), who observed that the position of the Sun in the celestial sphere at the vernal

equinox was slowly precessing along the zodiac constellations. This slow motion, of about 50 ′′ per

year, is mostly due to the gravitational torque induced on the equatorial bulge of the planet by the

Moon and the Sun (the so-called lunisolar component of the precession, or precession of the equa-

tor). It causes the rotational axis of the planet to trace an imaginary cone around the ecliptic pole

axes with a period of about 28,500 years. Another precessional component comes from the move-

ment of the ecliptic itself due to the gravitational interactions between the planets (the so-called

planetary component of precession), although it occurs about 500 times slower than the lunisolar

component. Both the lunisolar and planetary components are included in the determination of

precession constants and transformation matrices, as well as dynamical processes that inﬂuence

the shape of the Earth and of its equatorial bulge (to be discussed later). The axial precession

transformation matrix links the geocentric version of the International Celestial Reference Frame

(ITRF) to the mean celestial equator system.

Nutation (N) Nutation originates from the same gravitational interaction of the Sun, of the Moon and of the other planets (to a lesser extent) with the non-spherical shape of the Earth that gives rise to

precession, but refers to the slight wobbles of the precessional motion caused by variations in the

orbits of these objects. Much smaller in amplitude than those caused by the precessional cycle of

the rotation axis, they also operate on shorter timescales. The Earth’s nutation is dominated by

the relative positions of the Moon and of the Sun (lunisolar nutation), which is characterized by

cycles ranging from a few days all the way to 18.6 years. The most important of these oscillations is

the 18.6-year cycle in the position of the lunar orbital node (intersection between the lunar orbital

plane and the ecliptic) caused by the 5° inclination of the orbital plane of the Moon with respect to

the ecliptic. An accounting of the various nutation eﬀects that impact the Earth’s precession are

generated using time-dependent harmonic series in precession-nutation models (such as the IAU

2006/2000A model of Mathews et al. (2002)), which include the direct gravitational eﬀect of other Chapter 2. The Earth’s rotational state and its recent evolution 30

Figure 2.2: The five intermediate transformations linking the space-fixed International Celestial Refer- ence Frame (ICRF) and the Earth-fixed International Terrestrial Frame (the W includes both xˆ and yˆ components). From Seitz and Schuh (2010), reproduced with permission.

planets on the Earth and their indirect eﬀect on the Sun, ocean tidal eﬀects, mantle anelasticity,

as well as core/mantle and inner/outer core electromagnetic coupling (Mathews et al., 2002). By

deﬁnition, these models are limited to motions with periods greater than two days as measured

from the celestial frame of reference. These models are used to determine the elements of the

precession and nutation matrices P and N and generate the true celestial equator system and the

Celestial Intermediate Pole (CIP) with respect to which the rotation rate of the planet and the

transient motion of the pole can be determined.

Diurnal spin (S) This transformation matrix represents the diurnal rotation of the planet around the Celestial Intermediate Pole.

Polar motion (X and Y) The transformation matrices record the position of the Celestial Interme- diate Pole in the frame obtained from the International Terrestrial Reference System in two di-

mensions, where the xˆ direction is in the direction of the Greenwich meridian and the positive yˆ

direction is along the 90° West meridian, following IERS conventions.

Figure (2.2) presents the link between the transformation matrices and the interim reference systems between the quasar position-based celestial frame (translated to its geocentric form instead of its barycentric form) and the crust and mantle-based terrestrial reference frame. Thus, for a particular frame of reference based on a set of space-geodetic and/or astronomical observations, only five variables are required to define the coordinate transformation between the space-fixed and Earth-fixed reference frames. These quantities, called Earth Orientation Parameters (EOPs), capture the variability of the

Earth’s rotation, and are deﬁned as:

Universal time (UT1) Length of day variations (∆LOD) are given as the diﬀerence between the measured rotation speed (UT1) and the uniform, reference atomic time (UTC). From it, a measure

of the diﬀerence in instantaneous rotation speed with respect to a nominal value (taken to be 86,400

seconds) can be inferred. Chapter 2. The Earth’s rotational state and its recent evolution 31

Figure 2.3: Conventions relating the frequency (in cycles per sidereal day) between nutation and polar motion as measured in the celestial (top) and terrestrial reference frames (bottom). Low-frequency motion is deﬁned as nutation in the celestial reference frame but as polar motion in the terrestrial reference frame. From Gross (2007), reproduced with permission.

Celestial pole offsets (dΨ, dǫ) (2) The celestial pole offsets represent the difference in longitude and obliquity between the measured celestial pole and the Celestial Intermediate Pole (CIP), with a

temporal resolution of 2 days as seen in the celestial frame of reference. They are the residuals

between the observed cycles in axis motion and the modelled predictions from precession-nutation

models for motions that are characterized by a frequency (prograde or retrograde) of less than

two days (as seen in Figure 2.3). The measured oﬀsets will include phenomena that are still not

modelled, such as the free nutation of the core (originating from the diﬀerence in axis direction

between the core and the rest of the planet).

Polar motion Complementary to nutation being deﬁned as motion with a frequency (prograde or retrograde) of less than two days as seen in the celestial reference frame, any motion of the Celestial

Intermediate Pole that falls out of this range is called ’polar motion’. To determine how the

frequency of any motion in the celestial frame (ωcel) will be observed in the rotating terrestrial frame, the diurnal rotational frequency of the planet needs to be removed from the celestial-based

frequency, as seen in the lower half of Figure 2.3. This implies that nutation as seen on Earth

encloses all retrograde motions that have a close-to-diurnal frequency, while polar motion includes

all other terms, including secular trends and high-frequency motions.

Another key consideration in determining rotational variability is to set up a time reference standard from which anomalies can be inferred. In this respect, Universal Time (UT1) is based upon the instantaneous rotation rate of the planet, determined with respect the International Celestial Reference

Frame (ICRF). However, since the advent of atomic clocks, an atomic time scale has been provided

(International Atomic Time, or TAI). Coordinated Universal Time (UTC), has been introduced to be an intermediary between TAI and UT1. Chapter 2. The Earth’s rotational state and its recent evolution 32

2.2.3 Rotational variability measurement techniques and the latest ITRF iteration

To infer a suitable terrestrial frame of reference, the variability of the Earth’s rotational state must be precisely monitored. Although optical methodologies (lunar occultations or meridian transit times) extend back well before the space era (into the 19th century), recent iterations of such terrestrial reference frames have increasingly relied on the much increased precision reached by space-geodetic observing systems. These methodologies include Very Long Baseline Interferometry (VLBI), Global Navigation

Satellite System (GNSS) measurements, satellite and lunar laser ranging (SLR and LLR, respectively), and Doppler-based orbitography (DORIS). Each methodology is brieﬂy summarized here:

Lunar occultation A lunar occultation refers to the passage of the Moon between an object and an observer on Earth. Before the atomic era, lunar occultations of stellar objects were the most

precise way by which to measure variations in the Earth’s rotational rate (Gross, 2007), and a

substantial number of such events have been collected since the beginning of the 19th century (for

instance, the Jordi et al. (1994) reduction contains over 53,000 events from 1830 to 1955). For

each collected occultation event, the stellar position is determined from star catalogues, a chosen

lunar ephemeris (giving the evolution of the position of the Moon in the sky over time) and a

correction for the uneven limb proﬁle of the Moon from photographic charts. Reductions then

compare the terrestrial time measurements (TT), based on astronomical measurements, to the

Universal Time measurements (UT1), based on the Earth’s rotation, to produce time series for

the length of day, albeit with signiﬁcant uncertainties originating from the ephemeris and limb

models used. Although the term can also be used to describe the passage of the Moon in front

of the Sun itself, otherwise known as a solar eclipse, the data sets of such events are not used in

the generation of the ITRF, due to the timing uncertainty associated to the large relative size of

both the source and screen. Solar (and lunar) eclipses provide valuable information, however, on

millenial variations in rotation speed.

Local meridian transit times With the advent of atomic clocks, it became more advantageous to measure star transits directly with respect to a given reference meridian, rather than with respect

to an intermediary like the Moon. Measurements of this type include the previously discussed

International Latitude Service (ILS), from 1899 to 1978 (Yumi and Yokoyama, 1980), with its

six stations located at about the same latitude in the Northern hemisphere, that recorded any

variations in the orientation of the planet’s rotation axis by monitoring the position of stellar

pairs. Other series include the later Bureau International de l’Heure (BIH) data set, taken from Chapter 2. The Earth’s rotational state and its recent evolution 33

136 stations, over the period 1962-1981 (Li and Feissel, 1986) and the more recent and precise

astrometric measurements from the Hipparcos system, which have been combined with the ILS

data to create a long uniform data set from 1899 to 1992 (Vondrák, 1999).

Very long baseline interferometry (VLBI) Very long baseline interferometry (VLBI) is based on the measurement of the diﬀerential arrival time of radio signals emitted by distant extragalactic

radio sources at diﬀerent radiotelescopes located around the globe. The methodology is sensitive

to any change in the relative position of the radiotelescopes with respect to the radio source,

which would be caused by a change in orientation of the planet or by the tectonic displacement of

the radiotelescopes on the surface. Measurements using more than two radiotelescopes (so-called

"multibaseline" measurements) enable the independent determination of all Earth orientation pa-

rameters (the only space-geodetic technique able to do so, as the other satellite-based methods

only record the rate of change of the length of day but not its instantaneous value) (Gross, 2007).

Global navigation satellite system (GNSS) Global navigation satellite systems consist of a network of satellites with on-board atomic clocks broadcasting orbital data and precise emission time,

with a ground-based multi-channel receiver combining satellite signals to orient itself precisely with

respect to the satellite network. The American Global Positioning System (GPS) is the most ex-

tensive (as of October 2016, 32 satellites (U.S. Coast Guard Navigation Center, 2016)) and oldest

truly global system (worldwide coverage since 1995), while the Russian Global Navigation Satellite

System (GLONASS) network has also reached full global operational capacity in 2011 (after a

brief stint in the mid-1990s). The Galileo (European Union) and Compass (China) systems are

planned to go in fully global operational mode around 2020. Trilateration is used to generate the

position of the ground receiver (using four or more signals), taking into account a wide range of

error sources (relativistic eﬀects, ionospheric eﬀects, etc.) (Gross, 2007; Leick, 2015).

Satellite laser ranging and lunar laser ranging Laser ranging relies on the timing of a laser pulse trip from an observing station towards and back from a reﬂector installed on a satellite orbiting the

Earth (satellite laser ranging) or on the surface of the Moon during the Apollo and Luna missions

(lunar laser ranging). Although many artiﬁcial satellites carry reﬂectors to aid with their tracking,

the geodetic community mostly relies on the specialized LAGEOS-1 and LAGEOS-2 satellites (from

LAser GEOdynamics Satellite) for geodetic measurements, because of their high orbital stability

(spherical shape) and highly reﬂective surface (ease of tracking) (Tapley et al., 1985). Range

measurements are realized at many tracking stations around the globe, and the orbit of each

satellite ﬁtted to them, while taking into account the geophysical parameters which may inﬂuence Chapter 2. The Earth’s rotational state and its recent evolution 34

the position of each tracking station (tectonics, tidal motion, Earth orientation changes). From this

ﬁtting process, most Earth orientation parameters, except absolute length-of-day measurements,

can be obtained (Tapley et al., 1985). Lunar ranging is more challenging version of the technique

because of the much weaker signal to be detected, and only two observatories actively perform the

measurements. Earth orientation parameters are obtained from analysis of the range measurement

residuals, once the Moon’s orbit and the other ground eﬀects have been taken into account (Gross,

2007).

Doppler orbitography The Doppler Orbitography and Radio positioning Integrated by Satellite (DORIS) system is a French network based on the Doppler shift analysis of dual-frequency signals emitted

from ground stations towards an orbiting receiver. Once corrected for ionospheric eﬀects, the orbit

of the satellite can be inferred from the measurements, as well as changes in the orientation of the

planet (Willis et al., 2006).

Each methodology described above has its strengths and limitations, which are the topic of much literature and analysis by the geodesy community (Gross, 2007; Petit and Luzum, 2010). In particular, the information gathered obtained from each of these methodologies can then be combined together, given appropriate weight, to minimize the individual uncertainty associated with each methodology. Thus, the International Earth Rotation Service (IERS) has aimed to provide periodic realizations of the most accurate reference frame. There have been thirteen iterations of the International Terrestrial Reference

Frame (ITRF), each of them providing a diﬀerent long-term solution to the Earth’s orientation axes and their motion based on diﬀerent observational data sets. They are denoted by the ITRF acronym followed a number representing the last year for which observational data was used to constrain the computed solution (starting from ITRF88 to ITRF2014), with each iteration provided with full variance-covariance information. Since ITRF2000, the relative motion of all geodetic sites on the surface of the planet are accounted for by aligning the orientation rates of each site with a plate tectonic motion model (the

NNR-NUVEL-1A model of Argus and Gordon (1991)) (Altamimi et al., 2001).

2.3 The evolution of secular trends in the planetary rotational

state: Methodology and results

The large gains in precision obtained in the space era using geodetic measurement techniques, combined with the development of suitable reference frames to which measurements can be compared, has enabled us to build upon the previous optical methods and study precisely the variability characteristic of the Chapter 2. The Earth’s rotational state and its recent evolution 35

Earth’s rotational state. The impact of various physical processes has now been characterized and compared to model realizations (Gross, 2007). The study of planetary rotation follows the evolution of two particular observables, namely the wander of the pole with respect to the geographic background

(polar wander) and changes in the length of the day (LOD) (or equivalently, in the Earth’s dynamic oblateness of ﬁgure, also known as the J2 coeﬃcient).

Predictions of the impact of the GIA process on these observables have been realized and tested against observations since the 1980s (Peltier, 1982; Wu and Peltier, 1984), and have explained secular trends in these observables as being mostly due to the GIA process (Peltier, 2007). However, recent changes in these observables have been noted. For polar motion, a possible kink in its secular trend was suggested to have begun in the mid-1990s (Gross and Poutanen, 2009), while studies of the Earth’s dynamic oblateness (J2) have also pointed towards a recent change in its secular trend (Cox and Chao, 2002). However, no simultaneous study of those two observables had ever been attempted before the study of (Roy and Peltier, 2011), which is presented here. We ﬁrst start by brieﬂy reviewing known sources of variability in both observables, and then show in what follows that they both began to depart from their previously established, Late Quaternary ice-age related, GIA-controlled trends, at a somewhat earlier time. The simultaneity of the onset of this new regime of behaviour is suggestive of a common cause, which is discussed thereafter.

2.3.1 Sources of variability in length-of-day measurements

The existence of anomalies in the Earth’s rate of rotation was ﬁrst suggested in the context of tidal dissipation and angular momentum conservation in the Earth-Moon system, resulting in the transfer of rotational angular momentum to its orbital counterpart (Halley, 1695; Spencer Jones, 1939). The study of this phenomenon over historical periods can be performed through the careful analysis of numerous records of the timing of past lunar and solar eclipses, as performed by ancient Babylonian, Chinese, Arab and Greek astronomers (Stephenson and Morrison, 1995; Morrison and Stephenson, 2001). These studies have inferred, from the diﬀerence between historical records the "expected" timing of eclipses based on constant, modern rotational rates and historical records, the average acceleration in the Earth’s rate of rotation since then. However, the rate of planetary rotation can be impacted through both variations in the relative angular momentum of the Earth-Moon system and by variations in the moment of inertia of the planet, as demonstrated in Equation (2.7). This becomes apparent as the component of the total acceleration resulting from the tidal braking of the Earth’s spin can be precisely known from laser ranging observations of the recession of the Moon. Using these values, Stephenson and Morrison (1995) inferred Chapter 2. The Earth’s rotational state and its recent evolution 36 the existence of a non-tidal component acceleration of 1.6(±0.4)·10−22 rad·s−2, primarily associated with the change in oblateness of the Earth resulting from the ongoing glacial isostatic recovery of the planet.

Following the direct correspondence between changes in the J2 zonal component of the gravitational ﬁeld of the planet (the oblateness of ﬁgure of the Earth) and length-of-day variations, this non-tidal acceleration corresponds to a rate of change in the oblateness of the Earth of −3.5(±0.8) · 10−11 yr−1.

The advent of the space age also enabled the study of the Earth’s gravitational ﬁeld from the precise monitoring of satellite orbital parameters, which are in turn connected to the rotational rate of the planet

(Yoder et al., 1983). From orbital tracking of the orbits of geodetic satellites using the satellite laser ranging (SLR) technique, a measurement of the time rate of change of the oblateness of the Earth can be obtained. From a constellation of seven satellites, Cheng and Tapley (2004) studied the variation of

−11 −1 the J2 component over a 28-year period, and found a secular trend of −2.75 · 10 yr , consistent with the Stephenson and Morrison (1995) inference based upon the analysis of the timing of ancient eclipses.

The existence of strong variability in the J2 component of the Earth’s gravitational field and in length-of-day measurements has been established at various time scales. On the daily scale, the periodic displacement of fluid at the surface of the planet associated with tides is responsible for large cyclical signals in LOD measurements at tidal frequencies, which are easily distinguishable in high-accuracy VLBI measurements (Gross, 2007). Effects from longer-period ocean tidal effects can be accounted for using assimilated models of tide influence from radar altimetry data sets, such as the TOPEX-POSEIDON

(T/P) sea surface height record (Kantha et al., 1998; Gross, 2007). Long-period solid-Earth tidal eﬀects have also been identiﬁed at the biannual, annual and multi-year (9.3 years and 18.6 years) time scales

(Yoder et al., 1981). On the annual and seasonal scales, measurable cycles in LOD measurements have been associated with meteorological processes, such as annual and semiannual cycles in the angular momentum carried by zonal winds (Munk and MacDonald, 1960; Lambeck, 1980; Gross et al., 2004), the Madden-Julian oscillation (Gross et al., 2004) or by ocean currents (Gross et al., 2004). Strong

ENSO-induced interannual variations over timescales of 4 to 6 years have also been identiﬁed, and have been associated with the transfer of angular momentum between the atmosphere and the solid Earth induced by the collapse of tropical easterlies during an ENSO event. Finally, the existence of strong decadal variations in the length of the day has also been revealed by the study of the 400-year lunar occultation record (Munk and MacDonald, 1960; Lambeck, 1980; Gross, 2007). Chapter 2. The Earth’s rotational state and its recent evolution 37

2.3.2 Sources of variability in polar wander measurements

The study of polar wander variability beneﬁts from the existence of a long time series extending back to the end of the 19th century with the previously described record of the International Latitude Ser- vice (ILS), which has collected data until 1980 (Yumi and Yokoyama, 1980). The development of precise space-era star catalogues, such as Hipparcos, has greatly increased the precision of transit time techniques, and space-geodetic techniques have provided supplementary means to infer changes in the position of the poles.

Studies of the long-term ILS record have revealed the dominating presence of the free Chandler wobble with a period of about 433 days (amplitude of 100-200 milliarcseconds, or mas), caused by a combination of atmospheric, oceanic and hydrological processes (Gross, 2007). An annual cycle in polar motion has also been detected. At the decadal time scale, the Markowitz wobble, with a period of 24 years and an amplitude of about 30 mas, dominates the variability seen in the ILS data (Markowitz, 1961;

Gross and Vondrák, 1999; Gross, 2007). Beyond the existence of these periodic signals in polar motion, secular trends in the ILS and Hipparcos polar motion series have been identiﬁed (Gross and Vondrák,

1999), with an estimate of 3.51(±0.01) mas/year towards the meridian 79.2(±0.20)°W. If adjusted for a diﬀerent terrestrial frame of reference (in the so-called hotspot frame), the rate is increased to 4.03 mas/year towards the meridian 68.4°W (Argus and Gross, 2004).

2.3.3 Polar wander and LOD evolution: Data set and methodology employed

The recent evolution of polar wander is determined through a study of the previously described Earth

Orientation Parameters (EOPs), which provide the transformation between the International Terrestrial

Reference System (ITRS) and its celestial counterpart. They are determined from a set of independent space-geodetic techniques, which includes very long baseline interferometry (VLBI), global positioning system (GPS) measurements, as well as lunar and satellite laser ranging (SLR) techniques. Once a speciﬁc terrestrial frame of reference is determined, keeping track of any anomalies recovered from these techniques provides useful supplementary information on the evolution of the rotational state of the planet. Moreover, combining the information gained from the strengths and weaknesses of each methodology has resulted in the generation of an unbiased measure of polar motion variations since the mid-1970s (Ratcliﬀ and Gross, 2010).

The analysis of the recent evolution of rotational parameters presented here takes advantage of the recent SPACE2008 data set of Ratcliﬀ and Gross (2010), which is part of a family of Earth orientation parameters generated annually from a combination of space-geodetic measurements by the Geodynamics Chapter 2. The Earth’s rotational state and its recent evolution 38

Figure 2.4: Polar wander values from the SPACE2008 data set (Ratcliff and Gross, 2010), covering the 1976-2009 time period, with the raw xˆ-component (a) and yˆ-component of polar wander shown in the upper panels, with the corresponding 1-σ uncertainty presented in the lower panels, for the xˆ-component (c) and yˆ-component (d), respectively. From Ratcliff and Gross (2010), reproduced with permission. and Space Geodesy Group of the Jet Propulsion Laboratory (JPL), and obtained via anonymous ftp at . Once the time components of the measurement series were corrected for solid-Earth tide effects (Yoder et al., 1981), and for the impact of the roughly fortnightly (Mf ) and monthly (Mm) tides, the individual data sets are combined by iterative comparison to a combination of all the other data sets, using a Kalman filter (Gross et al., 1998). Bias-rate corrections and uncertainty scale factors are provided for each data set, and a final combined time series is generated with its corresponding uncertainty range. The particular set used in this study, denoted SPACE2008, spans the period from 28 September 1976 to 2 July 2009. The values of the polar motion components are presented in Figure 2.6 at daily intervals, with components extending from the Conventional International

Origin (CIO) along an x-axis aligned with the Greenwich meridian (panel (a) of Figure 2.6), and along a y-axis aligned with the 90° W meridian (panel (c) of Figure 2.6) (Ratcliﬀ and Gross, 2010).

Linear trends in the polar motion series are studied by removing the large high-frequency variations that dominate the signal. In this study, a zero-phase Butterworth-type low-pass ﬁlter is applied to the polar motion series (Butterworth, 1930), with a period cut-oﬀ set at 6 years, a value that is consistent Chapter 2. The Earth’s rotational state and its recent evolution 39 with previous studies of polar motion trends (Gross and Vondrák, 1999) and chosen to eliminate the signal associated with the beat period between the annual wobble and the Chandler wobble. The Butterworth

filter is preferred to simpler, idealized digital filters such as the boxcar filter, as it has a maximally

flat frequency response and phase shift effects that are simple to compensate for. The smoothed series, shown in panels (a) and (c) of Figure 2.5 (for the xˆ- and yˆ-components, respectively), serve as the basis for the analysis of quasi-periodic low-frequency residual variability, using a methodology similar to that employed in Gross and Vondrák (1999). Low-frequency and quasi-periodic terms are determined by performing a simultaneous weighted least-squares fit for a mean, a linear trend and low-frequency periodic terms that correspond to the most prominent peaks in the amplitude spectrum of the time series, including terms with periods of 8.3 years, 10.7 years and 18.8 years. Changes in the linear trend are then found by dividing the time series in two segments. The position of the "pivot-time" that separates the two segments is varied systematically, and the linear trend and mean are determined for each segment in order to find the pivot-time that minimizes the overall root-mean-square error of the fit. The quality of the best two-segment fit is then compared to the best single-segment fit. To be preferred, the best two- segment fit must be characterized by a substantially lower root-mean-square error. This simple analysis provides an approximation to any broad changes having occurred in the linear trend, but it should be remembered that such changes may not be as abrupt as approximated by the methodology used here.

The resulting two-segment ﬁts to the low pass ﬁltered polar wander observations are presented in panels

(b) and (d) of Figure 2.5 for the xˆ- and yˆ-components, respectively. Figure 2.6 demonstrates the quality of the fit provided by the final fitted function for both components.

Recent variations in the linear trends for the non-tidal acceleration of planetary rotation rate are analysed using a single data set, namely the satellite laser ranging (SLR) data set of Cheng and Tapley

(2004), which uses a subset of seven geodetic satellites (Starlette, Ajisai, Stella, LAGEOS 1 and 2,

Etalon-1 and -2, and BE-C, from 1999 onwards). The data used for this analysis is derived from the acceleration of the precession rate of the orbital node of each satellite. It was obtained via anonymous ftp from the Center for Space Research at the University of Texas at Austin, spans January 1976 to the end of 2009, and is presented in panel (a) of Figure 2.7. In order to study variations in the linear trend over this period, low-frequency terms are also eliminated using a low-pass Butterworth filter. The cut-off for the analysis of this data is fixed at 20 years, which is sufficient to remove the suggested 18.6-year tidal and decadal variations, and is similar to the cut-off frequency employed by Cheng and Tapley

(2004). The smoothed series, plotted in panel (a) of Figure 2.7, reveals a marked change approximately in the mid-1990s. Linear trends are then determined in the smoothed J˙2 series, by separating the time series into two parts and ﬁtting the trend separately for each part, as for the polar motion. The pivot- Chapter 2. The Earth’s rotational state and its recent evolution 40

400 60 a) b) 300 50 200 40 100 0 30 −100 20 (x−component) (x−component) −200 10 Polar Motion (mas) Polar Motion (mas) −300 −400 0 1975 1980 1985 1990 1995 2000 2005 2010 1975 1980 1985 1990 1995 2000 2005 2010 Year Year 600 360 c) d) 500 340

400 320

300 300

200 280 (y−component) (y−component)

Polar Motion (mas) 260 Polar Motion (mas) 100

0 240 1975 1980 1985 1990 1995 2000 2005 2010 1975 1980 1985 1990 1995 2000 2005 2010 Year Year

Figure 2.5: Study of the polar wander from the SPACE2008 data set (Ratcliff and Gross, 2010). (a)-(b) Raw xˆ-component of polar wander (dark grey), its subjection to a 6-yr, low-pass, Butterworth filter (red), and the two linear fits for the time periods 1976-1992 (rate: 1.7 mas/yr), and 1992-2009 (rate: 0.9 mas/yr). (c)-(d) Raw yˆ-component of polar wander (dark grey), its subjection to a 6-yr, low-pass, Butterworth filter (red), and the two linear fits for the time periods 1976-1992 (rate: 4.1 mas/yr), and 1992-2009 (rate: 1.5 mas/yr). From Roy and Peltier (2011).

Figure 2.6: Representation of the SPACE2008 polar wander data set (Ratcliff and Gross, 2010) subjected to a 6-yr, low-pass, Butterworth filter (red), together with the fitted signal used in the determination of the secular trends (black). This fit function includes a constant term, a linear trend and the periodic signals described in the text (black), for the xˆ-component (a) and the yˆ-component (b) of the signal. Chapter 2. The Earth’s rotational state and its recent evolution 41

8 4 a) b)

6 3 ) ) −10 −10 2 4 1 2 0 0 Component (x10 Component (x 10 2

2 −1 −2 −2 −4 −3 Variation in J Variation in J −6 −4

−8 −5 1975 1980 1985 1990 1995 2000 2005 2010 1975 1980 1985 1990 1995 2000 2005 2010 Year Year

−10 Figure 2.7: Changes in the J2 Stokes coefficient (×10 ), for the period 1976-2009, from the orbital 10 parameters of seven geodetic satellites (Cheng and Tapley, 2004). (a) Raw variations of J2 (×10 ) (dark grey), and subjected to a 20-year, low-pass filter (green). (b) Two linear fits for the time period 1976-1992 [rate: −0.37(±0.01) · 10−10 yr−1], and 1992-2009 [rate: −0.09(±0.02) · 10−10 yr−1]. From Roy and Peltier (2011). time is again shifted along the series to find the position of the knot in time that minimizes the overall root-mean-square error of fit. The final result of this process is presented in panel (b) of Figure 2.7.

2.3.4 Secular trend pivot point determination: results and analysis

Changes in the linear trends of the secular drift of the pole relative to the surface geography and of the non-tidal acceleration of the rate of rotation have been clearly identiﬁed. It is most revealing, however, to compare the variation of the total root-mean-square error for the two rotation-related time series as the position of the transition in the linear trend is varied systematically. These results are presented in Figure 2.8 and demonstrate that the transition in the linear trends for both anomalies occurs in approximately 1992.

Our results are compared to historical inferences in Table 2.2. Our analysis of the polar motion time series demonstrates that its secular rate of drift remained approximately ﬁxed, prior to 1992, at 4.5(±0.1) mas/yr (or 1.3° Myr−1), along the 68(±8) °W meridian, while it slowed to 1.8(±0.4) mas/yr (or 0.5°

Myr−1), along the 58(±9) °W meridian subsequently. Although not corrected for the hotspot frame of reference, as done by Argus and Gross (2004), the current analysis demonstrates that dividing the time series into two distinct epochs increases the magnitude of the established secular rates of change prior to 1992, but decreases them for modern times subsequent to this. As no uncertainties were provided in previous inferences of these quantities, it is hard to state if the observed differences between our inferred values and previous ones are significant or not, but different data processing methodologies and data set time spans might explain part of the differences. Concerning the corresponding changes in J˙2, our Chapter 2. The Earth’s rotational state and its recent evolution 42

14 0.6

12 0.4

10 0.2 Root−mean Squared Error (rms) Root−mean Squared Error (rms)

8 0 1980 1985 1990 1995 2000 2005 Year

Figure 2.8: The total root-mean-squared error for the J˙2 Stokes coefficient fit (dark green) and for the polar wander fit (red), as a function of the position of the pivot-time, with a common minimum observed around 1992. From Roy and Peltier (2011).

Table 2.2: Inter-study comparison of secular trends in polar wander and in the J˙2 coeﬃcient. From Roy and Peltier (2011)

analyses estimate a rate of change of −3.7(±0.1) · 10−11 yr−1 before 1992, and −0.9(±0.2) · 10−11 yr−1 after 1992. Splitting the time series fully reconciles the observed rate of change for J˙2 presented in Cheng and Tapley (2004) with the estimate from historical records of Stephenson and Morrison (1995)

(−3.5(±0.8) · 10−11 yr−1). No uncertainty is provided for the Cheng and Tapley (2004) inference, which complicates any comparison to our values, it would be expected to lie between our pre-1992 and post-1992 inferences for J˙2 as it straddles both time periods.

The identification of significant shifts in the secular rates of change of the two primary Earth rotation observables, which become evident in the same epoch of time, is highly significant. This is especially the case as the variation in J˙2 and polar wander are dependent upon completely independent elements of the Earth’s moment of inertia tensor (e.g. Munk and MacDonald, 1960; Peltier and Luthcke, 2009). Chapter 2. The Earth’s rotational state and its recent evolution 43

In models of the GIA process, which are able to simultaneously ﬁt those two rotational anomalies, variations in the elements of the moment of inertia tensor of the planet are determined by a speciﬁc model of time-dependent continental ice sheet loading and of the depth-dependence of mantle viscosity.

It would therefore be very surprising if the observed recent changes in the secular trends inferred to have occurred since 1992 were not caused by recent changes in surface ice sheet loading.

Our hypothesis is that these simultaneous shifts in Earth’s rotational state are caused by additional melting of continental ice cover induced by recent global climate change. The existence of a link between the timing of the changes in the linear trends for the rotational state anomalies and recent changes in the ice sheet loading on the planet is supported by studies of the year-to-year melting rates of major continental ice sheets and of small ice sheets and glaciers distributed globally. For instance, Thomas et al.

(2006) compared various estimates of the time dependence of elevation changes and mass balance for the

Greenland Ice Sheet and found that snow-accumulation rates and airborne laser altimetry measurements support the idea of a sharp decrease in ice sheet mass beginning in the mid-1990s. The mass decrease is inferred to come predominantly from the coastal areas of the ice sheet, as seen in Figure 2.9(a). Figure

2.9(b) shows mass balance estimates for the Greenland Ice Sheet for the 1958-2007 period resulting from aircraft radar campaigns, estimates of ice discharge, and positive-degree-day estimates (Box et al., 2006), which indicate that the ice sheet was close to mass equilibrium during the 1970’s and 1980’s, with a rapid melting onset during the 1990’s and onwards (Rignot et al., 2008). Other studies of uplift rates measured from GPS ground stations located around the Greenland ice sheet (Jiang et al., 2010; Khan, S. A. et al.,

2015a, 2016), have found regions peripheral to the Greenland ice sheet to be uplifting at an accelerating rate, also indicating an accelerated melting of land ice having also occurred since the mid-1990s. These results, combined with radar altimetry, laser altimetry and gravimetric measurements (for the post-2002

GRACE era) are seen in Figure 2.9(c) (Khan, S. A. et al., 2015a). Other studies also indicate a similar behaviour for the West Antarctic Ice Sheet, albeit with a later onset of the melting phase (Velicogna and

Wahr, 2013; IPCC, 2013). Mass balance results for the ice sheets covering Antarctica and Greenland, expressed in Sea Level Equivalent (SLE), are summarized in Figure 2.9(d).

The results indicating an accelerated melting of large ice sheets are robust, but extending these results to the smaller mountain glaciers that cover high-altitude areas is not straightforward. These smaller ice catchments are widely distributed, and can be found notably in northern and Arctic regions (Canada,

Russia, Svalbard, Iceland, Scandinavia and Alaska), the western United States and Canada, the southern

Andes (including Patagonia), Central Europe, New Zealand and the Central Asian mountains. Estimates of the mass balance of these mountainous regions over the 20th century indicate that there are strong regional diﬀerences between the response of mountain glaciers, with some showing an accelerated negative Chapter 2. The Earth’s rotational state and its recent evolution 44

Figure 2.9: Representation of the best estimates of ice sheet mass balance values or indicators from the Greenland and Antarctic ice sheets. (a) Rates of surface elevation change (dS/dt) for the central (red) and coastal (blue) regions of the Greenland Ice Sheet, coming from snow-accumulation estimates (box a), solely airborne radar measurements (boxes b and c), or from a combination of airborne and satellite radar measurements (box d). Figure from Thomas et al. (2006), reproduced with permission. (b) Total mass balance (TMB) and surface mass balance (SMB) estimates for the Greenland ice sheet over 1958-2007, using interpolated (light blue diamonds) and observed (dark blue circles) values (uncertainty of ± 60 Gt/yr). The other lines show the anomalies comprising the total values (SMB anomalies in green circles, interpolated ice discharge (D) in red squares and observed ice discharge (D) in pink triangles). Figure from Rignot et al. (2008), reproduced with permission. (c) Total mass balance estimates of the Greenland ice sheet over 1992-2012, using GRACE observations (red), laser altimetry (green), radar altimetry (blue), so-called "input-output" methods (precipitation and ice discharge estimates, in black), and the IMBIE (Ice-sheet Mass Balance Inter-Comparison Exercise) combination of the previous observations (light blue). Figure from (Khan, S. A. et al., 2015a), reproduced with permission. (d) Rate of ice-sheet loss equivalent averaged over 5-year periods over Antarctica (red), Greenland (blue), and a combination of the two (green), with corresponding uncertainties, coming a large combination of estimates over the two regions collected by IPCC (2013). Figure from IPCC (2013), reproduced with permission. Chapter 2. The Earth’s rotational state and its recent evolution 45 mass balance since the mid-century (in particular in Patagonia and Western North America), and other regions showing an inconclusive signal. Overall, the studies of Cogley (2009) and Marzeion et al. (2012) suggest a likely acceleration in the melting rates of the ice cover in most of these mountainous regions

(although not conclusive, given the uncertainty level), but it should be noted that Gardner et al. (2013), who focused solely on the 2003-2009 time period, have found no significant acceleration in mountain glacier mass loss. Whether this inconsistency is indicative of any long-term mass balance signal that applies to the earlier time periods not covered by the study or simply a short-term discrepancy is not clear (IPCC, 2013; Gardner et al., 2013; Cogley, 2009; Dyurgerov, 2010). In the context of the work presented in this chapter, the regional variability in the mass balance estimate would need to be taken into account, as ice catchments at lower latitudes would be expected to contribute more significantly to changes in polar wander. The summarized results of these analyses were presented in IPCC (2013) for different time periods are shown in Table 2.3.

Table 2.3: Average annual rates of global mass change in Gt yr−1 for diﬀerent time periods for all glaciers globally, except those peripheral to the Antarctica and Greenland ice sheets, with 90% uncertainty range, with data sources. The data was originally compiled in IPCC (2013). Time period Reference Mass balance for all glaciers (Gt yr−1) 1901-1990 Marzeion et al. (2012) -197 ± 24 1971-2009 Cogley (2009); Marzeion et al. (2012) -226 ± 135 1993-2009 Cogley (2009); Marzeion et al. (2012) -275 ± 135 2005-2009 Cogley (2009); Marzeion et al. (2012) -301 ± 135 2003-2009 Gardner et al. (2013) -215 ± 26

2.4 Perspectives and future work

We infer that these melting events would induce, depending on their origin, changes in either or both of the rotational observables under consideration, with the melting events closer to the rotation axis (i.e. the poles) inducing changes in the oblateness of ﬁgure of the planet, and hence of the LOD, while melting events occurring further from the axis of rotation aﬀecting predominantly polar wander observations.

It is also probable that these melting events have an impact on any comparison between observations of the time derivatives of the degree two and order one Stokes coeﬃcients observed by the Gravity

Recovery and Climate Experiment (GRACE) satellites and the values predicted by GIA models, such as the earlier ICE-5G (VM2) model of the GIA process (Peltier and Luthcke, 2009). In their analysis, the authors suggested that the misﬁt could be due to the fact that the ICE-5G (VM2) model of the

GIA process is designed to fit only data that is unambiguously associated with the Late Quaternary ice-age. The model specifically does not include the influence of the additional changes in continental Chapter 2. The Earth’s rotational state and its recent evolution 46

Figure 2.10: Simple test of the hypothesis of the impact of recent land ice melting on inferences of the rate of change of the J2 coefficient of the Earth’s gravitational field, prior and after the break in the J˙2 time series identified in Roy and Peltier (2011). The post-1992 change is roughly explained by the progressive addition of additional land ice melting from the polar regions of Greenland (GRN), West Antarctica (WAN) and Alaska (AK) in the model. The rates of melting for these regions (in global sea- level rise equivalent, in mm/yr) were inferred by Peltier (2009) from the Gravity Recovery and Climate Experiment (GRACE) satellite observations of the time dependence of the Earth’s gravitational field. Reproduced from Peltier et al. (2012), with permission from the American Geophysical Union. Chapter 2. The Earth’s rotational state and its recent evolution 47 ice-cover associated with the modern global warming process. The validation of the Peltier and Luthcke

(2009) hypothesis concerning the origins of the inferred misfit between the ICE-5G (VM2) prediction of the time derivatives of the degree two and order one Stokes coefficients and the GRACE inferences of the same properties of Earth’s gravitational field requires the development of a forward model that is in accord with the known characteristics of modern continental ice-sheet disintegration and global sea-level observations.

Initial work on this front has indicated that a relatively good ﬁt to the post-1992 value of J˙2 inferred in Roy and Peltier (2011) can be recovered using a simple linear melting rate over the Alaskan mountain ranges, Greenland and Antarctica (Peltier et al., 2012), presented in Figure 2.10. Recently, Adhikari and Ivins (2016) have shown that, using a simple model considering only the elastic response of the solid

Earth to recent changes in terrestrial water storage and land ice cover can explain a good fraction of the change in polar wander and J˙2. However, to fully reconcile recent observations with the full theory describing the dynamic response of the Earth system to the recent melting of ice sheets and mountain glaciers induced by modern climate change, future work will require the augmentation of the latest GIA model structure (presented in the following chapters of this work) to include this additional source of rotational forcing. In the absence of the demonstration herein of the occurrence of a marked change in rotational state prior to the launch of the GRACE satellite, work of this kind would not be warranted. Chapter 3

GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model

The study of the Glacial Isostatic Adjustment (GIA) process, in which the solid Earth responds visco- elastically to varying ice and water loads at its surface which are associated with the Late Quaternary ice- age cycle, has signiﬁcantly contributed to our understanding of both paleoclimatological phenomenology and solid Earth geophysics. A contribution in the latter area has involved the provision of robust constraints on the eﬀective viscosity of the planetary mantle, a crucial ingredient in the design of models of the mantle convection process. Furthermore, comparisons of the relative sea-level histories predicted by

GIA models to the large and globally distributed database of geologically-derived records of such histories, such as that originally assembled at the University of Toronto and described in Tushingham and Peltier

(1992), has led to signiﬁcant advances in our understanding of the most recent cycle of Late Quaternary glaciation and deglaciation. As previously mentioned, understanding the geographical distribution and temporal variability of land ice cover over the approximately 100,000-year period of this cycle provides the detailed boundary conditions of paleotopography and paleobathymetry that are required as a basis for the reconstruction of ice-age climate conditions using modern coupled atmosphere-ocean climate models (Peltier, 1994, 2004; Vettoretti and Peltier, 2013; Abe-Ouchi et al., 2015; Ivanovic et al., 2016).

Because the quality of the Relative Sea Level (RSL) database is especially high in the period since LGM

(which occurred between 26,000 and 21,000 years ago), the focus upon this period, both geophysically and climatologically, has been especially intense. This has led to the development of regional RSL databases with exceptional quality control such as that for the United Kingdom (Shennan et al., 2002), for the Canadian land mass that was once covered by the vast Laurentide/Cordilleran/Innuitian ice

48 Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 49 sheet complex (Dyke et al., 2002; Dyke, 2004), the Caribbean region (Khan, N. S. et al., 2015b), the

Mediterranean region (Vacchi et al., 2014, 2016) and for both the East and West coasts of the continental

United States (Engelhart et al., 2011; Engelhart and Horton, 2012; Engelhart et al., 2015). These two last data sets will be of special interest for the purpose of the present section.

It is also important to appreciate the role played by GIA modelling in the context of the current renewed episode of continental deglaciation caused by the rapidly increasing atmospheric concentration of greenhouse gases. Our observations of the ongoing melting of the great polar ice sheets on Antarctica and Greenland, and of many small ice catchments and glaciers (responsible for an important fraction of the global rise of sea level occurring today) is signiﬁcantly impacted by the remaining isostatic disequilibrium associated with the last ice-age cycle, the signal of which must be accounted for in order to more clearly identify the global warming component (e.g. Peltier and Tushingham, 1989; Peltier,

2009; Velicogna and Wahr, 2013). In particular, the GIA correction is central to the analysis of the time-dependent gravitational measurements provided by the Gravity Recovery and Climate Experiment

(GRACE) satellites, which provide constraints on the the recent melting rate of ice sheets and other smaller ice catchments (e.g. Peltier, 2009; Peltier and Luthcke, 2009; Velicogna and Wahr, 2013; IPCC,

2013), but which can show some degree of variability depending on the GIA correction (Ivins et al., 2013;

Argus et al., 2014). Given the large potential economic and social impacts of future sea level change, it becomes crucial to evaluate models of the GIA process against all available geophysical observables, at all stages of the post-LGM deglaciation, and to consider them from a global perspective.

As will be discussed, the Atlantic coast of the continental United States is a region of special interest for the study of the GIA process, given that data from this coast provides a transect of RSL history associated with the collapse of the forebulge induced outboard of the former Laurentide ice sheet that once covered all of the Canadian landmass to the north. Given the availability of a newly compiled database of 14C-dated records of sea-level evolution of very high quality for the Holocene period (Engelhart and

Horton, 2012), it has become possible to more stringently test the accuracy of recently constructed

GIA models than had previously been possible given the modest quality control to which previous such compilations were subject. Testing the performance of these models and determining whether further adjustments are necessary to their current iteration to recover a best ﬁt to this new high-quality data for the forebulge region are primary goals of the work presented in this chapter. The results presented here follow the peer-reviewed contribution of Roy and Peltier (2015) (Roy, K., and Peltier, W. R. (2015),

’Glacial isostatic adjustment, relative sea level history and mantle viscosity: reconciling relative sea level model predictions for the U.S. East coast with geological constraints’, Geophysical Journal International

201(2), 1156-1181), with a slightly expanded theoretical background section providing more information Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 50 about the mathematical structure of the model (section 3.2), as well an additional section to introduce the relative sea level records used in the study (section 3.4.1).

3.1 General approach and strategy

Models of the GIA process depend on two fundamental inputs: a history of ice-sheet loading and a model of the radial variation of mantle viscosity. The methodology traditionally employed to address the complex interaction between the depth-dependent viscosity of the mantle and ice loading history inputs to GIA models has taken advantage of the fact that certain characteristics of some of the geological observables are relatively insensitive to either the exact nature of the ice loading history or to the viscosity in certain ranges of depth, thereby enabling the development of an iterative approach that maximizes the information content gathered from all available data (an overview of these considerations is presented in

Peltier (1998b)). This has involved the determination of a spherically-symmetric viscosity structure for the Earth’s mantle from certain speciﬁc constraints that are understood to be suﬃciently independent of the deglaciation history, such as the relaxation times that are characteristic of the "rebound" of the surface of the solid Earth in regions that were previously covered by thick continental ice sheets at the

Last Glacial Maximum in central parts of North America and Fennoscandia (Mitrovica and Peltier, 1991,

1993a,b, 1995; Wieczerkowski et al., 1999), or from rotational state constraints that provide information about the viscosity of the lower mantle (Wu and Peltier, 1984; Peltier and Jiang, 1996).

GIA models may be tested and reﬁned by comparing their local predictions of relative sea-level history to geological inferences based upon appropriate sea level indicators. The U.S. Atlantic coast is a region of particular interest in this regard, due to the fact that data from the length of this coast provides a transect of the forebulge associated with the former Laurentide ice sheet. High-quality relative sea level histories from this region are employed herein to explore the ability of current models of mantle viscosity to explain the inferred evolution of relative sea level that have accompanied forebulge collapse following deglaciation. Existing misﬁts are characterized, and alternatives are explored for their reconciliation.

Then, it will be demonstrated that a new model of mantle viscosity (referred to herein as VM6) is able to eliminate the majority of these misﬁts when coupled with the latest model of deglaciation history

ICE-6G_C, while continuing to reconcile a wide range of other important geophysical observables, as well as additional relative sea-level data from the U.S. West coast which also record the collapse of the forebulge but which have not been employed in tuning the viscosity proﬁle to enable ICE-6G_C (VM6) to ﬁt the East coast data set.

In this work, the global ice-sheet loading history is ﬁxed to that of the most recently constructed Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 51 model, known as ICE-6G_C (Peltier et al., 2015), while the radial proﬁle of mantle viscosity is initially

ﬁxed to the VM5a proﬁle of Peltier and Drummond (2008) or to the variant upon it labelled VM5b

(Engelhart et al., 2011). Characterizing these misﬁts and attempting to remove them solely through appropriate variations in mantle viscosity will lead to a further reﬁned model of mantle viscosity referred to as VM6.

The systematic series of sensitivity analyses that have led to this reﬁned model, which involves a search through the space of plausible parametrization of viscosity depth dependence, will in many ways be ad hoc rather than based upon the application of the formal Bayesian methodology that led to the original VM2 model of Peltier (1996b, 1998b,c). The reason this approach has been followed is that there does not appear to be an equally simple parametrization of the RSL histories from the forebulge region as that possible for histories from the regions previously covered by thick accumulations of land ice, where sea level histories have a simple exponential form described entirely by an amplitude and a characteristic relaxation time. Prior to implementing a brute force statistical search based upon

Monte Carlo methods, a restricted search will be pursued, during which a physical understanding of the sensitivity of RSL histories from sites along the U.S. East coast to variations of viscosity over various depth intervals will be discussed.

3.2 Modelling the GIA process: Theoretical background

The mathematical structure of the global GIA process was ﬁrst developed in the 1970s in a series of seminal papers (Peltier, 1974, 1976; Farrell and Clark, 1976; Peltier and Andrews, 1976; Clark et al.,

1978; Peltier et al., 1978). It has since been extended and further reﬁned from this earliest form. The current and most elaborate version of the theory (and the one that is employed herein) takes the form of a Fredholm equation of the second kind, an integral equation whose solution provides a prediction of the time varying level of the sea with respect to the continuously deforming (visco-elastically) surface of the solid Earth (Peltier et al., 2015). The explicit form of this Sea Level Equation (SLE) is as follows:

t ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ S(θ,λ,t)= C(θ,λ,t) L(θ ,λ ,t )GL(γ,t − t )+ΨR(θ ,λ ,t )GT (γ,t − t ) dΩ dt +  R  −∞Z ZZ Ω (3.1) △Φ(t)  C(θ,λ,t) , g where S(θ,λ,t) represents the local level of the sea relative to the local surface of the solid Earth at time t and at the geographical location with co-latitude θ and longitude λ. The function L(θ′,λ′,t′) is Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 52 the surface mass loading history per unit area which consists of both grounded land ice and ocean water components. It may be written in the composite form:

L(θ,λ,t)= ρI I(θ,λ,t)+ ρW S(θ,λ,t), (3.2)

where ρI and I(θ,λ,t) represent ice density and ice thickness history, whereas ρW and S(θ,λ,t) respectively represent water density and relative sea level. The integral nature of Equation (3.1) is thus clear on the basis that the unknown ﬁeld S(θ,λ,t) appears both in the integrand of Equation (3.1) and on its left-hand side. The function ΨR(θ′,λ′,t′) represents the change in centrifugal potential associated with the GIA process itself. These rotational changes are induced by the large mass movements associated with the glaciation and deglaciation processes. The solution of the Sea-Level Equation to be employed herein includes the renormalized representation of the rotational feedback terms discussed explicitly in

Peltier et al. (2012) and Peltier et al. (2015). This renormalization (see equation (3b) of Peltier et al.

(2015)) is necessary to ensure that changes in the centrifugal potential at any single location do not impact the centrifugal potential at any other location (such that they only exert "local" inﬂuence).

The function C(θ,λ,t) in Equation (3.1) is the so-called ocean function (Munk and MacDonald,

△Φ(t) 1960), which has a value of unity over oceans and zero elsewhere. The time-dependent function g is constructed so as to ensure that there is a precise match between the mass of water generated by melting land ice and that which appears in the ocean basins (see Peltier (1998c); Peltier et al. (2012) for thorough discussions of this term). The complete Sea-Level Equation (3.1) is solved in such a way as to simultaneously compute the time dependence of the ocean function using the iterative methodology introduced in Peltier (1994, 1998a) and Peltier and Fairbanks (2006).

ΨR(θ,λ,t), the time and space dependent change in the centrifugal potential associated with the

GIA process, can be expressed as a ﬁrst-order perturbation, following Dahlen (1976):

1 R Ψ (θ,λ)=Ψ00Y00(θ,λ)+ Ψ2mY2m(θ,λ), (3.3) m=−1 X

2 2 2 where Ψ00 = 3 mz(t)Ω0a , 2 2 Ψ20 = − 3 mz(t)Ω0 1/5, 2 2 p Ω0a Ψ2,+1 =(ω1 − iω2) 2 2/15, 2 2 Ω0a − p Ψ2, 1 = −(ω1 + iω2) 2 2/15, ωx,y,z p mx,y,z = Ω . Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 53

L ′ T ′ In Equation (3.1), the functions G (γ,t − t ) and GR(γ,t − t ) are Green functions representing the impulse response of the surface of the Earth which, when convolved with the surface loading history and the centrifugal potential loading, predict the evolving separation between the surface of the solid Earth and the surface of the sea. The angle γ represents the separation between the source point and the ﬁeld point at which the response is to be determined. The impulse function GL was ﬁrst described in Peltier

(1974), subsequently reﬁned in Peltier and Andrews (1976) and Peltier (1985), and takes the form:

∞ L a L L G (γ,t)= 1+ kl (t) − hl (t) Pl(cos γ), (3.4) Me l=0 X where a and Me refer to the Earth’s mean radius and mass, while the Pl(cos γ) terms are the standard

T Legendre polynomials evaluated at angle γ. A similar formulation exists for the Green function GR(γ,t) associated with the change in centrifugal potential ΨR(θ′,λ′,t′), the form of which has recently been reviewed in Peltier et al. (2012):

∞ 1 2l + 1 GT (γ,t)= 1+ kT (t) − hT (t) P (cos γ). (3.5) R g 4π l l l l=0 X

L L In Equation (3.5), kl (t) and hl (t) are so-called surface load Love numbers (Love, 1911), dimensionless quantities that capture the nature of the visco-elastic response of the Earth’s surface to surface loading.

They are time-dependent, visco-elastic extensions of the equivalent elastic surface load Love numbers of

Farrell (1972), and are related to the radial displacement of the surface induced by the load (h) and the induced gravitational potential perturbation (k). The exact form of the impulse functions is determined using a linear Maxwell rheology for the Earth’s interior and can be expressed, following Peltier (1976,

1985), at spherical harmonic expansion degree l, as:

m − l L L,E l sj t hl (t)= hl δ(t)+ rje , (3.6a) j=1 X

m − l L L,E l sj t kl (t)= kl δ(t)+ qje , (3.6b) j=1 X L,E L,E l l l where hl and kl are the elastic surface load Love numbers of Farrell (1972), while rj, qj and sj are the amplitudes and relaxation times of the m exponential decay responses that are required to deﬁne the time-dependent behaviour of the Love numbers, determined as the zeros of an appropriate secular function (Peltier, 1985) or using a collocation method (Peltier, 1974, 1976) while using a speciﬁc model of the elastic properties of the Earth’s interior (using here the Preliminary Reference Earth Model Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 54

(PREM) of Dziewonski and Anderson (1981)) and of the radial variations in viscosity. Another suite of

L Love numbers, ll (t), describes the tangential displacement resulting from an applied surface load, and takes the form: m − l L L,E l sj t ll (t)= ll δ(t)+ tje , (3.7) j=1 X L,E l where ll is the equivalent elastic surface load Love number and the tj coeﬃcients are the amplitudes of the normal mode expansion. The Love numbers at expansion degree l (hl, kl and ll) are linked to the induced radial displacement Ul, tangential displacement Vl and perturbation in the gravitational potential Φ3,l:

hl Ul g     =Φ ll , (3.8) Vl 2,l g         Φ3,l kl          where Φ2,l refers to the gravitational potential perturbation of Legendre degree l due to a point mass load applied to the surface of the Earth to determine its impulse response, the exact expression of which is derived in Peltier (1974).

T In Equation (3.6), corresponding time-dependent visco-elastic tidal loading Love numbers kl (t) and T hl (t) are included, and can be expressed as:

m ′ − l T T,E l sj t hl (t)= hl δ(t)+ rj e , (3.9a) j=1 X

m ′ − l T T,E l sj t kl (t)= kl δ(t)+ qj e , (3.9b) j=1 X m ′ − l T T,E l sj t ll (t)= ll δ(t)+ tj e . (3.9c) j=1 X The required inputs to Equation (3.1) consist of two geophysical ﬁelds, namely a global ice loading history I(θ,λ,t), and a spherically-symmetric radial viscosity structure (on the basis of which the exact

L L T forms of the time-dependent visco-elastic Love numbers included in the equation, kl (t), hl (t), kl (t) T and hl (t), are determined). Using this information, Equation (3.1) is solved iteratively to determine the rotational feedback component and the time dependence of the ocean function C(θ,λ,t) (the land- ocean interface) induced by the changing ice cover, with the first step neglecting the rotational feedback component and fixing the ocean function C(θ,λ,t) to its present value. The time dependence of the ocean function is determined following the steps of Peltier (1994), in which topographical self-consistency is obtained by recognizing the arbitrariness of the point with respect to which S(θ,λ,t) is defined. In other Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 55

words, the difference between the predicted present sea level, S(θ,λ,tpresent), and modern topography with respect to sea level, (Tpresent), is calculated (using Tdiff (θ,λ)= T (θ,λ,tpresent) − S(θ,λ,tpresent)). Past topography at any time t is then determined by adjusting the sea level history with respect to the observed topographic difference at present (Tdiff ), such that T (θ,λ,t) = S(θ,λ,t)+ Tdiff (θ,λ). The ocean function is then updated taking into account this new topographic field T (θ,λ,t), the ice sheet loading history I(θ,λ,t), and effects related to grounded ice and the inundation that results when regions initially covered by ice become inundated when it melts (also known as "implicit ice") (Peltier, 1998a).

For the rotational feedback component (Equation 3.3), an initial estimate of its value is determined from ice and ocean loading histories obtained from solving the Sea Level Equation (3.1) without the rotational feedback. Then, Equation (3.1) is solved again, but this time incorporating the inﬂuence of this ﬁrst estimate of the rotational feedback ΨR(θ,λ,t). The iterative process is repeated until convergence is achieved. For both the rotational feedback component and the time dependence of the ocean function, this convergence occurs after only a few iterations, because of the relative low amplitude of the rotational and topographic corrections (Peltier, 1998c). It results in a solution for the sea level history that is both gravitationally and topographically self-consistent.

3.3 The ICE-NG models of ice sheet loading history

The ﬁrst input required to infer a global evolution of relative sea level is a representation of the history of ice sheet loading associated with the last glaciation/deglaciation process. Such models are largely constrained on the basis of surface geomorphological evidence, such as carbon-datable material from terminal moraines, which record the location of the margins of a retreating ice sheet. In the recent past, cosmogenic exposure age dating, which may be invoked to determine the time since a surface outcrop or glacial erratic boulder has been exposed following the retreat of an ice sheet that once covered the location, has also become a widely employed tool (e.g. Dyke et al., 2002; Dyke, 2004; Whitehouse et al.

(2012); Argus et al. (2014)). These methods are particularly suited to infer the evolution of the margins of large concentrations of ice, but most often only weakly constrain their former thickness. Signiﬁcant progress in the estimation of the evolving thickness of ice was primarily realized through the application of GIA analyses. Because of the simple exponential relaxation form of relative sea level records from regions that were previously covered by thick ice sheets (e.g. Northern Canada, Fennoscandia), one may employ the amplitude of the exponential rebound curves to "weigh" the thickness of the ice that must have been removed to induce the amplitude of the rebound observed. This requires, however, that the viscosity structure of the Earth’s deep interior be known. It is possible to separate both problems by ﬁrst Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 56 determining the viscosity of the Earth on the basis of observations of the relaxation times characteristic of the RSL time series in previously glaciated regions, before using the amplitude of these rebound time series to focus on ice sheet thickness (of course, granted that such additional constraints from the time of load removal exist) (e.g. Peltier, 1982, 1998a, 2007).

However, the early attempts to determine the history of ice sheet loading during the last ice age cycle

(e.g. Peltier and Andrews, 1976; Wu and Peltier, 1983) benefited subsequently from the development of large databases of relative sea level histories spanning the whole globe, which led to the development of the ICE-3G structure, presented in Tushingham and Peltier (1991). Subsequent changes, which included refinements in the analysis methodology and the use of further observational constraints (such as the high-quality Barbados RSL history based upon U-Th dating of coral terraces (Fairbanks, 1989)) resulted in the improved ICE-4G model (Peltier, 1994). Later, the ICE-5G model (Peltier, 2004) was obtained from a careful analysis of geodetic and gravity-based data, most notably over North America (Argus et al., 1999), and extended the ice loading history to the entirety of the most recent glacial-interglacial cycle. Other improvements to the model included the application of refined GIA data over the British

Isles (Peltier et al., 2002; Shennan et al., 2002) and new observations of far-ﬁeld sea-level histories

(Peltier, 2004). The ICE-5G model has been widely used in conjunction with the VM2 viscosity proﬁle of the mantle to provide a global theory of relative sea-level adjustment due to the glaciation/deglaciation process.

Further refinements to the ICE-5G loading history over North America have more recently been introduced, in which the misfits to Global Positioning System (GPS) observations of vertical motion of the crust documented in Argus and Peltier (2010) have been eliminated by appropriate modifications of the ice thickness history over North America, Northwestern Eurasia and Antarctica. This process has led to the ICE-6G_C model (Argus et al., 2014; Peltier et al., 2015). Figure 3.1 shows a comparison of the LGM topographic height anomaly generated by this most recent model with respect to LGM sea level over North America with that of the precursor ICE-5G model. Inspection will show that the primary modifications of the ice thickness distribution consist of a thinning of the Laurentide ice sheet in central Canada and a thickening over northern Quebec and Labrador, as well as over the northern border region between the Canadian provinces of Alberta and British Columbia. The Antarctic component of the global model is discussed in Argus et al. (2014). The ICE-6G_C model will be employed for the purpose of the analyses presented in this chapter. Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 57

Figure 3.1: (a) Geographical extent and topographic height anomaly at LGM in the Northern hemisphere for the ICE-6G_C ice loading history. (b) Diﬀerence in ice thickness between the ICE-6G_C loading history and its ICE-5G predecessor (Peltier, 2004). From (Roy and Peltier, 2015), adapted from Vettoretti and Peltier (2013) and used with permission.

3.4 Analysis of the performance of the ICE-6G_C ice loading

history as a function of viscosity model

Among the regions for which high quality records of past sea level based on the recovery of biological material from past coastal environments now exist, the eastern seaboard of the United States is of particular interest, as it straddles the glacial forebulge consisting of an upwarping of the crust induced by the viscous ﬂow of material from the Earth’s interior beneath the previously ice sheet covered region into its periphery. The regional extent of this forebulge covers the entirety of the coast, from just south of the LGM margin of Laurentian ice in New England, through the peak of present-day subsidence around

New Jersey and Chesapeake Bay, and into the Caribbean Sea, with its trailing edge being located in the near vicinity of the critical island of Barbados.

3.4.1 Relative sea-level history reconstructions

Reconstructions of past sea level rely on diﬀerent types of information sources, which include, in order of increasing reach in the past, (1) instrumental records of sea level (tide gauges, satellite altimetry), (2) archaeological evidence, (3) biological indicators and (4) geological evidence of past shoreline features

(Kearney, 2009). Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 58

One of the primary observational data sets providing constraining information on GIA model reconstructions of past sea level evolution is the isotopic dating of relevant biological markers interpreted in the context of their possible elevation range with respect to relative sea level at the time of their growth (e.g. van de Plassche, 1986). Examples of such indicators include shells of intertidal or marine mollusc species (providing a sea level range or a marine limit, respectively), high/low marsh plants and salt marsh micro-fossils indicating a freshwater limit, or coral samples that indicate a possible range of relative sea level based on the living depth range of speciﬁc species recovered in situ. Today, an extensive number of such records exist at a wide range of locations around the globe, and their use has greatly beneﬁted from the availability of precise geolocation techniques based on the Global Positioning System

(GPS). These measurements also take advantage of precise dating techniques, and in particular of the

14C and U/Th methods of radioactive dating. 14C dating (Arnold and Libby, 1949; Libby, 1952) relies on the natural presence of 14C radionuclides in the atmosphere, which are produced by the constant bombardment of the Earth’s atmosphere by natural cosmic rays and eventually get absorbed by living organisms after mixing in the atmosphere. After their death, 14C atoms progressively undergo β-decay to become 14N, allowing the determination of the moment of death of a biological sample if 14C atoms can be properly counted (using accelerator mass spectrometry) and if a proper transfer function from

14C age to sidereal age is used (e.g. Shennan and Horton, 2002). This mapping is not linear due to changes in cosmic ray bombardment rates, and depends on cross-calibration with other methodologies, such as dendrochronology or U/Th methods (van de Plassche, 1986; Kearney, 2009). Information on the indicative range of sea level of each species from which biological samples are recovered is crucial in the inference of past sea level (e.g. van de Plassche, 1986). Various organisms have been used to determine past sea level extent, including numerous shell species living in shallow waters close to tidal mean sea level, tree stumps or whale bones.

One of the most promising means of recovering past sea level ranges relies on the careful study of intertidal deposits in past salt marsh environments. Sediments from these environments, common in most coastal areas around the world, provide information about the separation between high (freshwater) and low (intertidal) marshes, when correctly interpreted in light of reference water levels, mean tide levels and highest astronomical tides and the cross-referencing of indicative altitude ranges for the available vegetation and intertidal microfossil assemblages (van de Plassche, 1986; Engelhart and Horton, 2012).

Figure 3.2 shows a schematic representation of a coastal salt marsh environment and how the information recovered from sediments can be interpreted and turned into a relative sea level curve. Beyond providing a direct inference of past sea level height with respect to present (so-called index points), biostratigraphic data can also indicate deposition in a purely terrestrial or marine environment. In this case, it will only Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 59 provide limiting information (either marine or terrestrial) about past sea level. Once an elevation and an age are determined, a careful error analysis must be performed in order to correctly account for altitude errors, extrusion errors and past tidal range inferences (Engelhart and Horton, 2012). Another potential source of error is related to the potential natural compaction of the sediment column. In this respect, so-called base of basal index points are of particular interest, as they are recovered from the sediments that are located within a few centimeters of an incompressible substrate, such that they are held to be unaﬀected by compaction (Engelhart and Horton, 2012; Horton et al., 2013).

3.4.2 The Engelhart and Horton (2012) data set of U.S. East coast RSL evolution

In this analysis, the quality-controlled database presented by (Engelhart and Horton, 2012) (with a preliminary iteration ﬁrst used in Engelhart et al. (2011)) will be employed. The data set consists of

686 individual sea-level indicators from locations along the coast ranging from northern Maine to South

Carolina. These data may be assigned to 16 distinct locations, which are indicated on Figure 3.3.

A substantial advantage of this particular data set over the previously employed tools used for the same region (e.g. Tushingham and Peltier, 1992; Peltier, 1996a) derives from the fact that the methodology employed for its construction was consistent throughout the coast and signiﬁcant eﬀorts have been invested in assigning meaningful, consistent error estimates to each of the sea level indicators employed, especially with regards to the indicative range of habitat of each sample, altitude, extrusion errors and tidal level errors Engelhart and Horton (2012). It should be noted, however, that compaction errors were not included in the original data set, but were subsequently added for the New Jersey site

(Horton et al., 2013). The data used here for the New Jersey site includes the eﬀect of compaction.

3.4.3 Analysis of the performance of the VM5a and VM5b viscosity structures

The availability of this new, high-quality data set of relative sea level index points and limiting indicators provides an opportunity to test the performance of current GIA models and their inference of the viscosity structure of the Earth’s interior. Early models of its radial structure, such as the VM1 model (Peltier,

1986), were simple two-layer models that attempted to distinguish the viscosity between the average values appropriate for the upper and lower mantle with the boundary between these regions deﬁned by the phase transition interface at 660 km depth (Peltier et al., 1986; Tushingham and Peltier, 1992).

Using a theoretical framework based on the formal Bayesian methodology (Mitrovica and Peltier, 1991, Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 60

Figure 3.2: Schematic representation of a salt marsh environment and how local RSL evolution can be eventually reconstructed from radiocarbon-dated biological markers in sediment cores. (a) Typical vegetation zones in the typical tidal frame for a mid-latitude coastal environment, such as the U.S. East coast. (b) Depth and altitude of samples in a typical core showing a transition from freshwater to marine environment. (c) Conversion of the recovered information into a collection of dated sea level index points (estimate of past sea level height), or limiting data. The ﬁgure notes Reference Water Levels (RWL), Indicative Range (IR), Altitude (A), Mean Tide Level (MTL), Mean High Water (MHW), Mean Higher High Water (MHHW) and Highest Astronomical Tide (HAT). Figure from Engelhart and Horton (2012), reproduced with permission. Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 61

Figure 3.3: (a) Geographical location of the 16 composite sites identiﬁed in Engelhart and Horton (2012) along the U.S. East coast and used in the current work. (b) Geographical extent of the 14 composite sites used in the current work for the U.S. West coast (Engelhart et al., 2015). From Roy and Peltier (2015). Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 62

1993a, 1995), various observables were considered in an effort to better understand their resolving power for the inference of viscosity depth dependence. Thereafter, Peltier (1996a) used the totality of the data, each type having its own resolving power insofar as viscosity depth dependence is concerned, to demonstrate that when all such data were employed one could obtain estimates of mantle viscosity from the surface to the core-mantle boundary. These data included the Fennoscandian relaxation spectrum of relaxation times (described below), a suite of site-specific relaxation times in regions heavily influenced by the former Laurentide and Fennoscandian ice sheets, as well as two anomalies of the Earth’s rotational state: the speed and direction of the true polar wander phenomenon and the rate of nontidal acceleration of the Earth’s axial rotation rate (Peltier and Jiang, 1996). The resulting profile of radial variations in viscosity, capped by a 90-kilometre elastic lithosphere, is referred to in the literature as VM2 and is presented in Figure 3.4(a). Despite the high-quality fits that the VM2 viscosity profile combined to the ICE-5G history of ice sheet loading provided when compared to a large number of globally distributed 14C-dated relative sea level histories (Peltier, 2004), discrepancies were subsequently found to exist between predicted and observed modern-day horizontal motion of the crust of the solid Earth over North America (Argus et al., 1999; Sella et al., 2007). These results were obtained from an extensive network of GPS observations over the continent. Peltier and Drummond (2008) demonstrated that the horizontal velocity misfits could be eliminated by a simple modification of the shallow structure that was characteristic of the VM2 viscosity profile by introducing radial viscosity stratification of the near-surface lithosphere, a feature that is expected based on the exponential temperature dependence of the creep resistance of a solid. Their preferred viscosity profile, referred to as VM5a, is a 5-layer approximation to the VM2 viscosity profile. It incorporates at the surface a 60-km elastic lithosphere overlaying a 40-km region of high viscosity (1022 Pa·s), and becomes a multi-layer average of the VM2 model at greater depths.

Engelhart et al. (2011) compared predictions of relative sea-level history at locations along the U.S.

East coast for the VM5a viscosity structure when combined with either the ICE-5G or an early version of the ICE-6G ice loading history with their observational database and found persistent misfits between them, irrespective of which of these loading histories was employed. In an initial simple attempt to eliminate these misfits, they introduced the VM5b viscosity profile, a modified version of the VM5a profile for which the viscosity was halved in the upper mantle and transition zone. Such a modification was found to remove some of the identified discrepancies in the mid-Atlantic region, although significant misfits remained for the southernmost part of the region covered by the data set. The VM5a and VM5b profiles are presented in Figure 3.4(a), together with the parent VM2 profile.

The model predictions of relative sea level histories based upon the use of the VM5a and VM5b Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 63

Figure 3.4: (a) Radial variations of the viscosity in depth for the viscosity profiles VM5a (green), VM5b (blue) (note that VM5a and VM5b are identical except in the upper mantle), and VM2 (blue/green), for which VM5a is a five-layer approximation with added stratification (high viscosity layer between 60 and 100 km depths) just below the elastic lithosphere (0-60 km depth). (b) Inverse relaxation time as a function of spherical harmonic degree obtained from observations of the glacial isostatic adjustment of Fennoscandia, with the relaxation spectrum (black dashed line) and corresponding 1-σ uncertainties (dark gray area) calculated by Wieczerkowski et al. (1999), with the inferred 2-σ uncertainties extending to the light gray area. The predicted spectra for the VM5a model (green) and for the VM5b model (blue) are superimposed. (c) Fit of the ICE-6G_C model predictions to the Barbados record of coral-based sea level indicators when combined to the VM5a viscosity structure. From Roy and Peltier (2015). Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 64 viscosity profiles and of the ICE-6G_C ice loading history are first revisited and compared to the

Engelhart and Horton (2012) data set, using the most recent version of the University of Toronto model.

All model predictions include the impact of rotational feedback. The resulting comparisons are shown in Figure 3.5 for the 16 distinct locations for which data is included in the compilation.

For the northernmost locations (Southern Maine, Massachusetts), inspection of the fits to the observational data shows that the VM5a profile performs better than VM5b, as the softer model of the upper mantle and transition zone predicts the existence of a highstand, which is not observed in the observational records, and significantly under-predicts relative sea level changes over the past few millennia. It is worthwhile to note that for Eastern Maine, neither VM5a nor VM5b succeeds in reconciling model predictions and observations. In fact, the predictions of the two models bracket the observations. Pro- gressing further south along the coast, the performance of the VM5a model degrades noticeably, while the VM5b model is either the equal of VM5a or an improvement, depending on the region of interest. For coastal Connecticut and New York, neither of the models adequately reproduces the observed relative sea level evolution, but they rather bracket the observations. Continuing further south, the VM5b profile begins to perform better than VM5a for coastal locations (in Long Island, New Jersey, Outer Delaware and East Virginia), although misfits remain for observations older than 6.5 ka. VM5a and VM5b bracket the available observations for the regions further inland (Inner Delaware, Inner Chesapeake). However, progressing into North Carolina, the misfits between the observations and the model predictions for both VM5a and VM5b become larger for the earliest part of the records. Model predictions become very similar for both VM5a and VM5b, and a clear misfit can be identified prior to 5 ka for the locations in

North Carolina and South Carolina. Thus, although the VM5b proﬁle does indeed provide an improvement in performance for mid-Atlantic locations (Engelhart et al., 2011), important misﬁts remain for the southernmost sites.

3.4.4 The ﬁt to the Fennoscandian spectrum

An important caveat needs to be addressed concerning the marginal increase in performance of the

VM5b viscosity profile, which concerns its fit to one of the most important observational data sets relating to the GIA process, namely the fit to the so-called Fennoscandian spectrum of relaxation times,

ﬁrst introduced by McConnell (1968).

Fennoscandia is a region of considerable interest with regards to the study of the GIA process, due to the large ongoing surface adjustments due to the inﬂuence of the former ice sheet which covered it during the last glaciation-deglaciation cycle (e.g. Steﬀen and Wu, 2011 for a comprehensive review). Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 65

Figure 3.5: Comparison of the Engelhart and Horton (2012) data set of relative sea-level histories along the U.S. Atlantic coast for the 16 composite regions with the predicted relative sea-level history at those locations for the ICE-6G_C model of ice-loading history combined to the VM5a (blue) and VM5b (black) radial viscosity proﬁles. Green data points represent sea-level index points, whereas blue crosses represent marine-limiting data and orange crosses represent terrestrial-limiting data. From Roy and Peltier (2015). Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 66

The Fennoscandian spectrum provides, as a function of deformation wavelength, a series of constraints on the relaxation time associated with the GIA process, which have been inferred using a database of six former strandlines from diﬀerent locations along the coast of the Gulf of Bothnia and of the Gulf of

Finland. The analysis of McConnell (1968) has been shown to provide a constraint on Earth rheology to a depth of approximately 1,500 km, with the considerable advantage of being relatively insensitive to the temporal and geographical evolution of the thickness of the Fennoscandian ice sheet. This important property has led to considerable interest being devoted to the area and to the data set employed (e.g.

Mitrovica and Peltier, 1993a; Peltier, 1998a; Wieczerkowski et al., 1999; Steffen and Wu, 2011). In particular, Wieczerkowski et al. (1999) applied a damped least-squares solution methodology to the strandline data of Donner (1995) to revisit the McConnell (1968) constraints. Their result provided a somewhat modified inference of the relaxation time dependence on deformation wavelength, complete with a 1-σ uncertainty range, which is reproduced in Figure 3.4(b) and extended to 2-σ, together with the prediction of the spectrum for the VM5a and VM5b viscosity profiles.

While the VM5a profile succeeds in producing a relaxation time spectrum consistent with the inferences of McConnell (1968) and Wieczerkowski et al. (1999) (unsurprisingly so, one might say, given that it is a 5-layer approximation to the VM2 profile, which is heavily constrained by an earlier analysis of the spectrum (Peltier, 1996a)), the VM5b profile fails to reproduce the Fennoscandian spectrum, especially at longer wavelengths (lower spherical harmonic degree). The inability of the VM5b model to reproduce this constraint indicates that the gain in performance from the VM5b profile in the mid-Atlantic might not be acceptable from a global GIA perspective, as accepting it would require one to entertain models with significant lateral heterogeneity of viscosity and our goal in the first instance is to attempt to define the best possible spherically symmetric model of the Earth’s viscosity structure.

The focus of the further analyses to follow is therefore to attempt to optimally reconcile observations of relative sea level evolution on the U.S. East coast with predictions of a spherically symmetric viscosity model, while using other key global constraints related to the GIA process (such as the Fennoscandian relaxation spectrum), as further independent checks of the ﬁnal solution.

3.5 An exploration of alternative viscosity structures

The nature of the misfits identified between observational data and model predictions of relative sea level evolution on the U.S. East coast is investigated in what follows. Whether they can be eliminated using suitable variations of the radial mantle viscosity structure, while maintaining a good fit to other geophysical constraints, such as the Fennoscandian spectrum of relaxation times, will be investigated. Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 67

3.5.1 Basic assumptions and methodology employed

In this analysis, a two-pronged approach will be employed, in which the reasonableness of some alternative viscosity structures that have been suggested in the literature, and which diﬀer signiﬁcantly from the

VM2/VM5a proﬁles, will be explored. In particular, the V1 and V2 structures, based on the work of

Forte and Mitrovica (1996), will be analyzed. Following this initial analysis, and having demonstrated that these alternatives are not in fact acceptable substitutes for the Toronto models, our second step will be to perform a direct analysis of the sensitivity of model predictions of sea-level evolution on U.S.

East coast to radial viscosity variations at diﬀerent depths. These tests are performed using the ICE-

6G_C ice loading history (Argus et al., 2014; Peltier et al., 2015), since it is a self-consistent global model constrained, among other observables, by the total eustatic sea level change that has occurred since the Last Glacial Maximum at the far-ﬁeld Barbados location Peltier and Fairbanks (2006) (the

fit of the precursor ICE-6G_C (VM5a) model is provided in Figure 3.4(c)). The final step involves merging the knowledge gained from all of the sensitivity analyses into a single model of radial viscosity structure that minimizes the misfits between observations of past RSL evolution along the U.S. East coast and model inferences of the same quantities, which will lead to a new viscosity structure named

VM6. The high quality of the ﬁt of this model to the totality of the geophysical observables will then be demonstrated, including not only the relative sea level evolution along the U.S. East coast to which the model is tuned (as the current work is especially focused on recovering a ﬁt to this data set), but also other independent tests that are not included in the optimization process, including the Fennoscandian spectrum of relaxation times, a comparison to relative sea level evolution data along the U.S. West coast and the overall shape of the late Holocene forebulge.

In this analysis, it is important to note that the focus will be solely on radially symmetric models of mantle viscosity. Even though lateral heterogeneity in mantle viscosity is fully expected to exist based on our knowledge of the temperature dependence of the creep resistance of mantle material, our goal is to develop a model of minimal complexity. It is therefore seen as important to explain as many geophysical observables as possible using a simple one-dimensional viscosity model, as this is intended to provide a suitable background structure onto which accurately inferred lateral viscosity variations may be superimposed.

Another caveat is related to the ice model being fixed in all cases explored to ICE-6G_C. As models of the GIA process have as inputs both the viscosity structure of the planet’s interior and a suitable history of ice sheet and ocean loading on its surface, it should be noted that this work is thus part of an iterative process during which those two input fields should be allowed to vary in turn. The goal of such Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 68 an approach is to reduce the overall misfit between GIA model predictions and all available relevant geophysical observables, using the knowledge that the misfits to different observables are sensitive to different model inputs in order to eventually reach convergence. The work presented in this section focuses solely on the impact of the viscosity structure on these misfits, while follow-up work will also focus on the ice loading history, and will enable us to address the issue of potential non-uniqueness in a more thorough manner.

3.5.2 Error analysis and model performance

The performance of the viscosity structure variations considered in this study is first determined by a visual inspection of the difference in relative sea level evolution at all locations along the U.S. East coast, which provides an invaluable source of information about the complex variations in the temporal and geographical responses of the forebulge. This analysis is complemented by a quantitative analysis of the misfit between the relative sea level index and limiting data points and the model predictions, which is described by a χ2-like relationship defined as:

1 N ∆ 2 χ2 = i . (3.10) N σ i=1 i X In this relationship, N represents the number of historical data points (index points and limiting data points) and σi is the uncertainty in the sea level height of the i-th data point (95% conﬁdence level).

Also, ∆i represents the diﬀerence between an observed sea level indicator and the model prediction. For sea level index points, it takes the simple form:

∆i = Si,obs − Si,pred, (3.11)

where Si,obs represents the observed sea level height and Si,pred is the model prediction. Limiting data points are treated diﬀerently, as they are given as an upper or lower restriction on relative sea level. Their contribution is considered to be non-zero only if the predicted sea level falls above a maximal terrestrial- limiting data height or below a minimum marine-limiting data height. In the terrestrial-limiting case, the expression for the diﬀerence ∆i is written as:

Si,pred − hi,obs + σi, if Si,pred ≥ (hi,obs − σi) ∆i =  , (3.12)  0, otherwise

 where hi,obs is the height of the observed limiting observational data point. In the case of a marine- Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 69

limiting data point, ∆i takes the form:

hi,obs − Si,pred + σi, if Si,pred ≥ (hi,obs − σi) ∆i =  . (3.13)  0, otherwise  It should be noted that while this treatment of the limiting data induces some limited departure from a pure χ2 methodology, it accurately captures the physical meaning of each observation. The uncertainty in the age of the observational data points is taken into account in a simple manner: if the predicted sea level falls below the 95% confidence height interval of an observation, the difference ∆i is calculated for the time at the upper edge of the age uncertainty range, while it is calculated at the lower edge of the age uncertainty range if the predicted sea level is above the 95% confidence height interval of an observation.

In all cases, the χ2-like values are deﬁned for each site along the U.S. East coast as well as for the entire coast and a sub-ensemble of the four southernmost locations, in order to focus speciﬁcally on the regional model response to depth-dependent mantle viscosity perturbations.

3.5.3 Alternative viscosity models: V1 and V2

An alternative family of viscosity models under consideration in this study is derived from a methodology that involved a joint inversion of data that are related both to the mantle convection process and to glacial isostatic adjustment (Forte and Mitrovica, 1996; Mitrovica and Forte, 1997, 2004). The convection data sets used in those two studies include a map of the free-air gravity ﬁeld, the non-hydrostatic ellipticity of the core-mantle boundary from measurements of the Earth’s free-core nutation, the horizontal divergence of tectonic plates (e.g. forte.etal.1991) and a map of dynamic topography (Forte et al. (1993); see also

Pari and Peltier (1996, 2000)). The GIA data includes postglacial relaxation times for the Fennoscandian region and for Hudson Bay (two separate locations in James Bay and at Richmond Gulf). Their analysis was performed using a non-linear, iterative procedure, and was further reﬁned in the upper mantle using site-speciﬁc decay times (Forte and Mitrovica, 1996; Mitrovica and Forte, 2004). In their analysis, a soft layer was included just above the 660 km discontinuity (as had been earlier suggested by Forte et al.

(1991) and Pari and Peltier (1995)), based on the idea that it should be a consequence of transformational superplasticity due to grain size reduction during the transformation of mineral phase that occurs as material ﬂows across the transition interface because of mantle convection. The resulting preferred model of Mitrovica and Forte (2004), referenced herein as V1 (FM), is presented in Figure 3.6. The second model shown in the same ﬁgure, referenced herein as V2 (FM), was presented in Moucha et al. Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 70

Figure 3.6: Comparison of the viscosity profiles VM5a (black) and VM5b (red) with the V1 (FM) profile of Mitrovica and Forte (2004) (green), and the V2 (FM) variation profile of Forte et al. (2009) (blue). From Roy and Peltier (2015).

(2008) and Forte et al. (2009), and is a variation based upon the same methodology, but where the thin low-viscosity layer is practically absent and where the lower mantle is more viscous.

The relative sea-level history predictions for these two viscosity structures when combined with the

ICE-6G_C ice loading history are shown in Figure 3.7, where they are compared to the Engelhart and

Horton (2012) data set for a subset of four locations (Southern Maine, New York, Inner Delaware and

Southern South Carolina). For the northernmost part of the Atlantic coast of the United States, the two

FM profiles display predictions that are very similar to those of VM5a. However, for locations further south along the East coast, into regions located within the remains of the proglacial forebulge associated with the maximal extent of the former Laurentide Ice Sheet, the two FM profiles become less and less able to explain the observations of past relative sea level, with a systematic decrease of the quality of the fit as a function of latitude, which becomes particularly acute in the Carolinas. In particular, in

South Carolina, the misﬁt characterizing the use of the VM5a and VM5b proﬁles with ICE-6G_C is notably worsened by a switch to the alternative FM models (V1 and V2). This could be explained by their overall higher viscosity in the lower mantle, which would further extend the region of proglacial forebulge collapse associated with deglaciation following the Last Glacial Maximum.

One possible limitation of this simple comparison might come from the fact the ICE-6G_C ice loading history is optimized to be used with the VM5a model, and might thus be expected to fail when combined with viscosity structures that are very diﬀerent. One way to explore this possibility is to look at predictions that do not depend signiﬁcantly on the ice loading history, such as the relaxation time Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 71

Figure 3.7: Comparison of a subset of regions of the Engelhart and Horton (2012) data set of relative sea-level histories along the U.S. Atlantic coast with the predicted relative sea-level history at those locations for the ICE-6G_C model of ice-loading history combined to the V1 (FM) (green) and V2 (FM) (blue) radial viscosity proﬁles. Green data points represent sea-level index points, whereas blue crosses represent marine-limiting data and orange crosses represent terrestrial-limiting data. From Roy and Peltier (2015). Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 72 experienced by the solid Earth under formerly heavily glaciated areas like Hudson Bay (at the center of the former Laurentide Ice Sheet) or Fennoscandia. As these regions were under considerable ice cover at

Last Glacial Maximum (LGM), model-predicted relaxation times at locations near the centre of the ice loads are relatively protected from load history uncertainties near the ice sheet margins. For the Hudson

Bay area, analyses performed by Peltier (1998a) on a complete set of 14C-dated RSL constraints suggested a best estimate for the relaxation time in Hudson Bay (Richmond Gulf, Quebec) of approximately 3.4 ka, whereas the ICE-4G (VM2) model predictions suggested a relaxation time of approximately 4.1 ka.

A subsequent analysis of this data set by Mitrovica et al. (2000) led them to revisit the Peltier (1998a) inference, on the premise that close sites in the area might have largely diﬀerent relaxation times which would be reﬂected on the individual curves for each site and that only a single type of RSL data should be employed. They suggested that the relaxation time at the Richmond Gulf site is approximately

5.0 ka (with a permissible range of 4.0-6.6 ka) and that it should be significantly lower at James Bay, at around 2.5 ka (with a permissible range of 2.0-2.8 ka) (Mitrovica et al., 2000). This difference is important, because whereas a single relaxation time is used to represent the whole Hudson Bay region in the formal Bayesian inversion leading to the VM2 radial viscosity model (of which VM5a is a five-layer approximation) (Mitrovica and Peltier, 1995; Peltier, 1998a), the inversion performed by Mitrovica and

Forte (2004) that leads to the FM family of viscosity proﬁles uses two diﬀerent relaxation times for the area. Given that these region are in such geographical proximity this would appear to be unphysical.

A further contribution to this debate was provided by Dyke and Peltier (2000), who revisited the exponential model RSL curve to qualitatively assess the impact of having various types of sea-level indicators, in particular with respect to 14C-dated mollusc shells whose habitat could extend (for many species) to a significant depth beneath mean sea level. However, the revised relaxation times produced by this analysis, which were around 2.5 ka, further exacerbated the misfit between the observed relaxation time and model predictions based upon the ICE-4G loading history coupled to the VM2 viscosity profile.

An important recent additional perspective is provided by the work of Pendea et al. (2010), who have published the ﬁrst marine-to-freshwater transition shoreline displacement model for the James Bay area using Accelerator Mass Spectrometry (AMS) dating of organic material in paleo-tidal wetlands as emergence indicators. As pointed out by Pendea et al. (2010), the isolation basin methodology had been used in the past for individual lakes in the Hudson Bay region (Saulnier-Talbot and Pienitz, 2001;

Miousse et al., 2003), but not using a sequence of distinct basins, and never reaching as far south as

James Bay. A similar methodology had previously been applied to great eﬀect by Shennan et al. (1994) in constructing the RSL history record for the Arisaig location and other sites in western Scotland. The

Pendea et al. (2010) data set provides an independent take on the debate concerning relaxation times Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 73

Figure 3.8: High quality sea-level index points from the Pendea et al. (2010) analysis in the eastern part of James Bay, and the exponential decay fit that best represents the relaxation of the area after its deglaciation. The locations used in the Pendea et al. (2010) analysis are shown in the inset. From Roy and Peltier (2015). in the James Bay area, and is significant given the high accuracy of the methodology. The location of the individual sites included in their analysis is shown in Figure 3.8. Following Peltier (1998a), a simple exponential decay curve was fitted to the Pendea et al. (2010) data, while constraining it to zero present-day change in relative sea level S(t), following Equation (3.14):

S(t)= A expt/τ −1 , (3.14) h i where τ is the inferred relaxation time and A is the amplitude characterizing the post-glacial RSL history.

This best ﬁt, along with the Pendea et al. (2010) data, is presented in Figure 3.9. A relaxation time of

3.9(±0.8) ka is obtained for this site. This result is noteworthy, as it calls into question the validity of the lower estimates for relaxation times for the area, such as the estimates of Dyke and Peltier (2000), as well as that of Mitrovica et al. (2000).

A comparison of the various a posteriori predictions of relaxation time for the various radial viscosity proﬁles under discussion in this analysis is presented in Table 3.1. The relaxation time for the Hudson

Bay region predicted by VM2 and that of its related 5-layer approximation, VM5a, is approximately

4.1 ka. Understandably, the VM5b proﬁle, which has an upper mantle half as viscous as VM5a does, exhibits a much lower relaxation time than the other stiﬀer models. All previous Toronto models have relaxation times that are within the new observational inference obtained from the Pendea et al. (2010) data. On the other hand, the FM family of models exhibits relaxation times that are much higher than Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 74 the other models, with values of approximately 5.7 ka for V1 (FM) (a similar value to Mitrovica and

Forte (2004)), and 6.4 ka for V2 (FM). These high values are due to the excessively high lower mantle viscosity that is delivered by the inversion procedure when a high relaxation time is assumed to govern the rebound process in the region near the centre of the Laurentide ice sheet. The FM family of models fails to predict relaxation times that fall within the error on the relaxation time estimated from the highly accurate isolation basin record of Pendea et al. (2010) as well as the previous estimates in Peltier

(1998a).

Table 3.1: Relaxation times determined in Eastern James Bay for various viscosity proﬁles coupled with the ICE-6G_C ice loading history

Initial observations strongly suggest that neither of the FM models (V1 and V2) are plausible insofar as the understanding of the GIA process is concerned, as comparisons to the relative sea level data along the U.S. East coast (when combined with the realistic ICE-6G_C loading history) has revealed important misﬁts between the model predictions and observations, especially along the southernmost portion of this coast. This view is now reinforced by the fact that these viscosity models deliver excessively high relaxation times in the James Bay area when compared to the latest data set of high-quality information from this region. This observation puts into question the assumptions made regarding relaxation times in the Hudson Bay and James Bay areas in the design of these models. Although the claim was that the

V1 and V2 models were constrained to ﬁt both GIA and convection related constraints, it seems clear that they are not in fact able to explain GIA observables.

3.5.4 Case study I: Mantle viscosity variations in the upper mantle

Rather than further mining the existing literature for additional alternative models of mantle viscosity that might be employed to reconcile observations of relative sea level history with GIA model predictions along the U.S. East coast, a systematic exploration of the sensitivity of model predictions to mantle viscosity variations at diﬀerent depths is preferred. It is well-known that, for regions previously located beneath a former ice sheet and therefore undergoing uplift today, the sensitivity of the characteristic Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 75 relaxation time for uplift of the crust depends strongly on the horizontal scale of the previously ice- covered region (e.g. Peltier, 1998a), but the sensitivity to mantle viscosity variations is more complex for regions located in the surrounding glacial forebulge region, as the variation of viscosity with depth strongly impacts both the extent and amplitude of the forebulge itself (Tushingham and Peltier, 1991).

The viscosity of the upper mantle and transition zone is known to have a profound impact on RSL predictions associated with ice sheets of moderate horizontal extent, such as the Fennoscandian ice sheet (Shennan et al., 2002). In the context of the larger Laurentide ice sheet, the impact is expected to be focused on the region in the ice sheet periphery. A wide range of upper mantle and transition zone viscosity structures has been investigated, and a representative subset of this family of models is presented here in Figure 3.9, where upper mantle and transition zone viscosity values are varied from

0.25 · 1021 Pa·s (the VM5b model of Engelhart et al. (2011)) to 0.75 · 1021 Pa·s, in a layer between the

660-km discontinuity and the base of the rheologically-stratified lithosphere (100 km depth), while the remainder of the radial viscosity structure is assumed to be identical to that of the VM5a profile. The range of viscosities considered here is representative of various viscosity values for this depth range in the literature. The RSL evolution predictions resulting from these models when combined to the ICE-6G_C loading history are presented in Figure 3.9 for representative locations on the U.S. East coast, while a quantitative estimate of the error in the fit to the observational data is provided in Table 3.2.

Varying progressively the value of the viscosity in the upper mantle and transition zone from 0.25 ·

1021 Pa·s to 0.75 · 1021 Pa·s results in a broad but consistent range of predictions. Focusing ﬁrst on the northernmost locations, the viscosity of the target region strongly impacts the predictions of RSL evolution. Models characterized by a more viscous upper mantle have reduced forebulge amplitudes which develop later in time, while models with less viscous upper mantles recover more quickly towards equilibrium. Progressing further south into the forebulge, softer models such as VM5b perform better at reconciling GIA predictions and observations, especially for the most recent 4 ka. However, beginning in

North Carolina, such variations in mantle viscosity fail to explain the older data, and by South Carolina, relative sea level history predictions become insensitive to variations of viscosity in the target region.

However, it should be noted that the spectrum of Fennoscandian relaxation times is highly sensitive to the viscosity in the region above the 660 km seismic discontinuity (Peltier, 1998a), so much so that the

ﬁt to this spectrum constitutes an important test of any spherically symmetric viscosity model. Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 76

Figure 3.9: Comparison of a subset of regions of the Engelhart and Horton (2012) data set of relative sea- level histories along the U.S. Atlantic coast with the predicted relative sea-level history at those locations for the ICE-6G_C model of ice-loading history combined to viscosity proﬁles where the viscosity of the upper mantle (UM) is allowed to vary (as discussed in the text). (a) Viscosity variations for the upper mantle (values are shown in the legend). The region of the mantle in which the variations occur is highlighted in the inset. (b) For the models shown in (a), comparison of RSL history predictions with observational data at four locations of the Engelhart and Horton (2012) data set, namely Southern Maine (2), New York (6), Inner Delaware (9), and Southern South Carolina (16). Green data points represent sea-level index points, whereas blue crosses represent marine-limiting data and orange crosses represent terrestrial-limiting data. From Roy and Peltier (2015). Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 77

3.5.5 Case study II: Viscosity changes in the upper part of the lower mantle

Given the large horizontal scale of the Laurentide Ice sheet, it is known that the predictions of relative sea level evolution under the former Laurentide ice sheet are sensitive to mantle viscosity variations in the upper part of the lower mantle (as determined from the site-speciﬁc Fréchet kernels, or sensitivity kernels, which link arbitrary perturbations in the radial proﬁle of mantle viscosity to resulting variations in the post-glacial recovery of the surface (e.g. Peltier and Andrews, 1976; Peltier and Jiang, 1996;

Peltier, 1998a; Wu et al., 2010)). This sensitivity can be expected to extend into the peripheral bulge region associated with the former Laurentide ice sheet, and to impact not only the geographical extent of the forebulge, but also its temporal evolution. The sensitivity of RSL predictions along the U.S.

East coast to a wide range of variations in the viscosity of the upper part of the lower mantle has been investigated, and in a representative subset of those models, the viscosity of the upper part of the lower mantle has been taken to vary over the range from 0.518·1021 Pa·s (one third of the value of the viscosity of the VM5a model at this depth), through the sequence 0.78 · 1021 Pa·s, 1.57 · 1021 Pa·s (the VM5a profile), and 3.14 · 1021 Pa·s, while the viscosity structure of the rest of the mantle is kept fixed to that of the VM5a profile. In this initial treatment, the range of values is selected only to provide a broad overview of the dependence of RSL predictions on viscosity variations at this range of depths.

Figure 3.10 compares the RSL predictions resulting from these variations of the VM5a structure when combined to the ICE-6G_C ice loading history for the same subset of locations from Engelhart and Horton (2012) that were previously employed, with a quantitative estimate of the error in the fit to the observational data being provided in Table 3.2. In the northernmost regions of the coast, the softer models produce a peripheral bulge of larger amplitude that reaches its maximum earlier following initiation of the deglaciation process. Moving further south along the coast reveals that this difference increases in magnitude, and leads to the presence of a mid-Holocene highstand south of Virginia for the softest of the models analysed, a feature that is incompatible with the RSL data from the region. From these observations, when compared to the predictions for the VM5a viscosity structure, a lower viscosity in the upper part of the lower mantle would be expected to partly eliminate the misfits observed along the southern part of the U.S. East Coast, although the optimal viscosity must be high enough to prevent the formation of highstands in the regions south of Virginia and at Barbados.

3.5.6 Case study III: Viscosity variations in the transition zone

When characterizing the sensitivity of RSL history predictions to mantle viscosity, a further avenue that warrants explicit study concerns the possibility of the existence of a harder transition zone between Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 78

Figure 3.10: Comparison of a subset of regions of the Engelhart and Horton (2012) data set of relative sea- level histories along the U.S. Atlantic coast with the predicted relative sea-level history at those locations for the ICE-6G_C model of ice-loading history combined to viscosity proﬁles where the viscosity of the upper part of the lower mantle (ULM) is allowed to vary (as discussed in the text). (a) Viscosity variations for the upper part of the lower mantle (values are shown in the legend). The region of the mantle in which the variations occur is highlighted in the inset. (b) For the models shown in (a), comparison of RSL history predictions with observational data at four locations of the Engelhart and Horton (2012) data set, namely Southern Maine (2), New York (6), Inner Delaware (9), and Southern South Carolina (16). Green data points represent sea-level index points, whereas blue crosses represent marine-limiting data and orange crosses represent terrestrial-limiting data. From Roy and Peltier (2015). Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 79 depths of 410 kilometers and 660 kilometers, as has been previously shown to trade oﬀ perfectly with a softening of viscosity in the vicinity of the deeper endothermic phase transition itself (e.g. Peltier,

1998a). Because the mineral assemblage present in the transition zone is especially rich in garnet, there are a priori reasons to believe that a feature of this kind in the depth dependence of mantle viscosity could exist. However, the recent discovery of ringwoodite, a high-pressure polymorph of olivine thought to originate in the transition zone, embedded in a diamond from a Brazilian diamond pipe (Pearson et al., 2014), suggests that viscosity of the transition zone might instead be lower than the viscosity of the upper mantle, since ringwoodite is a hydrous phase and this might be interpreted to suggest that the transition zone is rich in water. In the context of the GIA response to the melting of the Laurentide ice sheet over the East coast of North America, modifying the viscosity structure in this region alone and not in the upper mantle could result in a change in the characteristics of the forebulge, while also modifying somewhat the ability of the model to ﬁt the Fennoscandian relaxation spectrum.

Table 3.2: Error comparison for the viscosity variation cases considered

To investigate the impact of viscosity variations in the transition zone on sea level dynamics in the glacial forebulge region, a sequence of modifications of the VM5a profile was once again looked at. These new models are identical to the VM5a profile, except in the channel between the base of the transition Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 80 zone at 660 km depth and the top of the transition zone at 410 km depth. The RSL predictions at a few selected sites along the U.S. East coast are presented in Figure 3.11 for a subset of the tests, covering a viscosity range from 0.30 · 1021 Pa·s to 1.50 · 1021 Pa·s. A quantitative estimate of the error in the fit to the observational data is provided in Table 3.2. In the northern and central regions of the coast, increasing the viscosity in the channel between the depths of 410 and 660 kilometers creates a less pronounced forebulge that achieves its maximum displacement at a later time. In these regions, a softer transition zone seems to be favoured when comparing to sea level index point data, although stiffer models enable a slightly better fit to the marine-limiting observational data at some locations (e.g.

Massachusetts). The difference between the predictions resulting from the stiffer viscosity profiles is quite limited for sites in the mid-Atlantic regions (such as Connecticut and New York), but the soft transition zone models perform notably better in those regions. Indeed, for New Jersey, Delaware, Virginia and the index points of the Chesapeake Bay record, the softer transition zone models provide a better fit to the observational data provided by the Engelhart and Horton (2012) database. For regions further south, the inter-model comparison differences become less important. One explanation for this behaviour could be that, as the viscosity under the lithosphere is decreased, the geographical extent of the forebulge decreases, a feature that is accompanied by a larger amplitude of response at locations closer to the former Laurentide ice sheet. Correspondingly, for models with a stiffer transition zone, the impact of the forebulge is felt further south along the eastern U.S. seaboard, and the predicted RSL curves for those models are shifted to shallower depth compared to the original VM5a model and with less curvature in the recovery phase. Hence, increasing the viscosity in the transition zone provides a poorer fit to the RSL observations for most locations along the U.S. East coast. Thus, large changes in mantle viscosity in the transition zone would have to be incorporated simultaneously with large viscosity changes in either the upper mantle or the upper part of the lower mantle to account for the RSL evolution misfits introduced in the northernmost and southernmost parts of the U.S. East coast.

3.5.7 Case study IV: Viscosity contrast variations between the upper mantle and lower mantle

Most simple models of mantle viscosity include a discontinuity in mantle viscosity associated with the presence of the 660-km seismic discontinuity (e.g. Peltier, 1989), and the issue of determining an appropriate contrast between the upper and lower mantle is explored here in the context of relative sea level evolution predictions for the Eastern seaboard of the United States. Results for a representative subset of viscosity structures are shown here, where two diﬀerent viscosity values are explored for both Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 81

Figure 3.11: Comparison of a subset of regions of the Engelhart and Horton (2012) data set of relative sea- level histories along the U.S. Atlantic coast with the predicted relative sea-level history at those locations for the ICE-6G_C model of ice-loading history combined to viscosity proﬁles where the viscosity of the transition zone (TZ) is allowed to vary (as discussed in the text). (a) Viscosity variations for the transition zone (values are shown in the legend). The region of the mantle in which the variations occur is highlighted in the inset. (b) For the models shown in (a), comparison of RSL history predictions with observational data at four locations of the Engelhart and Horton (2012) data set, namely Southern Maine (2), New York (6), Inner Delaware (9), and Southern South Carolina (16). Green data points represent sea-level index points, whereas blue crosses represent marine-limiting data and orange crosses represent terrestrial-limiting data. From Roy and Peltier (2015). Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 82 the upper part of the lower mantle (1.57 · 1021 Pa·s and 0.785 · 1021 Pa·s) and for the upper mantle and transition zone (0.75 · 1021 Pa·s and 0.37 · 1021 Pa·s), which results in four diﬀerent viscosity structure combinations.

Figure 3.12 presents the relative sea-level history predictions resulting from these structures for the subset of representative locations on the U.S. East coast, while a quantitative estimate of the error in the fit to the observational data is provided in Table 3.2. Beginning with the northernmost locations, it is noticeable that for the lowest upper mantle and transition zone viscosity (0.37 · 1021 Pa·s), increasing the contrast between the upper and lower mantle (by increasing the value of the lower mantle viscosity) leads to a less prominent forebulge that reaches its maximum amplitude at a slightly later time, a direct consequence of the slower mantle material flow caused by its higher overall viscosity. In terms of relative sea level evolution at the locations of interest on the U.S. East coast, this increase of the lower mantle viscosity results in flatter sea level curves and slower recovery to today’s observed sea level. Repeating the analysis for a higher upper mantle viscosity (0.75 · 1021 Pa·s), a similar behaviour is observed for the northernmost sites. However, for mid-Atlantic sites (Massachusetts to Maryland), the higher upper mantle viscosity masks any impact arising from a higher lower mantle viscosity. In

Virginia, North Carolina and South Carolina, as was the case for the models with the lower upper mantle and transition zone viscosity, relative sea level history predictions are highly dependent on the value of lower mantle viscosity employed. Models with softer lower mantles tend to reproduce the observational data of Engelhart and Horton (2012) more accurately, although in the two sets of experiments shown here, the overall sea level evolution curvature is not captured accurately.

3.5.8 Case study V: Lithosphere thickness variations

Another important sensitivity analysis concerns the thickness of the elastic lithosphere. Here, lithospheric thickness is varied from 60 kilometers to 195 kilometers (through the sequence of 60, 90, 130 and 195 kilometers), while the rest of the mantle viscosity structure is fixed to the VM5a profile. The rheological stratification of the lithosphere present in the VM5a and VM5b viscosity profiles is ignored in this test.

Relative sea level evolution predictions for the U.S. East coast are compared to the sea level indicators in four regions of the Engelhart and Horton (2012) database in Figure 3.13. A quantitative estimate of the error in the ﬁt to the observational data is provided in Table 3.2.

The model predictions for the northernmost sites (in particular Eastern and Southern Maine) are particularly interesting. While the forebulge is almost absent when using models with a very thin lithosphere, as lithospheric thickness is increased from 60 to 130 kilometers, the forebulge gains in Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 83

Figure 3.12: Comparison of a subset of regions of the Engelhart and Horton (2012) data set of relative sea- level histories along the U.S. Atlantic coast with the predicted relative sea-level history at those locations for the ICE-6G_C model of ice-loading history combined to viscosity proﬁles where the viscosity contrast between the upper and lower mantles is allowed to vary (as discussed in the text). (a) Upper/lower mantle viscosity contrast variations (values are shown in the legend). The region of the mantle in which the variations occur is highlighted in the inset. (b) For the models shown in (a), comparison of RSL history predictions with observational data at four locations of the Engelhart and Horton (2012) data set, namely Southern Maine (2), New York (6), Inner Delaware (9), and Southern South Carolina (16). Green data points represent sea-level index points, whereas blue crosses represent marine-limiting data and orange crosses represent terrestrial-limiting data. From Roy and Peltier (2015). Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 84

Figure 3.13: Comparison of a subset of regions of the Engelhart and Horton (2012) data set of relative sea-level histories along the U.S. Atlantic coast (Southern Maine (2), New York (6), Inner Delaware (9), and Southern South Carolina (16)) with the predicted relative sea-level history at those locations for the ICE-6G_C model of ice-loading history combined to viscosity proﬁles where the thickness of the elastic lithosphere is allowed to vary (as discussed in the text). Green data points represent sea- level index points, whereas blue crosses represent marine-limiting data and orange crosses represent terrestrial-limiting data. From Roy and Peltier (2015). Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 85 maximal amplitude and reaches it at earlier times. However, the behaviour becomes less marked for a thicker lithosphere (or even slightly reverses for the highest values considered), which implies that there might be a lithospheric thickness value that maximizes the amplitude of the local forebulge for the northernmost sites. Similarly, progressing further south along the coast, models with a thinner lithosphere demonstrate a wide range of predictions, but models with very thick elastic lithospheres (130 km and above) display a similar RSL evolution. In the mid-Atlantic region (e.g. New York and Inner

Delaware), this situation manifests itself in the form of a forebulge of lower amplitude for the thickest lithosphere under consideration. Conversely, in the southernmost regions, models with a thin lithosphere exhibit a similar behaviour, while the viscosity structures with a thicker lithosphere display a broader range of predictions with a more prominent forebulge, a feature which is not desirable given the small change in RSL history over the past 6 ka captured by the sea level indicators in the region.

This series of features can be explained in terms of the physical nature of the movement of mass in the mantle associated with the former Laurentide ice sheet and the corresponding flexure of the lithosphere it induces. For a thicker lithosphere, the greater stiffness of the Earth’s solid surface tends to spread the forebulge over a larger geographical area, while diminishing its maximal amplitude. Viscosity models with thinner lithospheres result in a more pronounced but localized shape. A simple variation of this parameter does not provide a satisfactory improvement in the ability of the VM5a viscosity profile to explain the RSL observational data over all regions along the coast.

3.5.9 Case study VI: Lower mantle viscosity variations and the Earth’s rotational state

It has been long established that the Earth’s rotational state continues to be inﬂuenced by the isostatic adjustment of the planet resulting from the deglaciation process that followed the Last Glacial Maximum

(e.g. Peltier, 1982; Wu and Peltier, 1984; Peltier, 1996a), in particular concerning the two main rotational anomalies observed today and previously discussed, namely the non-tidal acceleration of the rate of planetary rotation and the secular drift of the pole of rotation with respect to the surface of the planet

(true polar wander). Modern observations of these anomalies are numerous, and recent observations of changes in the non-tidal acceleration of the planetary rotation rate and in the secular drift of the poles have been described earlier in this work (based upon measurements of satellite orbital parameters, historical records of lunar and solar eclipses and space-geodetic techniques (Yoder et al., 1983; Stephenson and Morrison, 1995; Gross and Vondrák, 1999; Cheng and Tapley, 2004; Roy and Peltier, 2011; Cheng et al., 2013)). Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 86

Predictions of these GIA-induced anomalies are mostly sensitive to the viscosity of the lowermost mantle (Peltier, 1982; Wu and Peltier, 1984). Although predictions of RSL evolution following the deglaciation were found to be sensitive to the viscosity of the upper part of the lower mantle but only weakly sensitive, if at all, to changes in the viscosity of the deepest portion of the lower mantle (Peltier,

1998a), it is important to investigate the impact of such viscosity changes in the deepest mantle on RSL predictions along the U.S. East coast. Such modifications might be required to maintain the quality of fit to Earth rotation constraints when an appropriately softened shallower structure is introduced to improve the quality of GIA predicted RSL histories along its southernmost parts. In Figure 3.14, predictions of RSL evolution for four key sites along the U.S. East coast are presented when the viscosity of the lowermost lower mantle is allowed to vary between three different values (chosen for illustrative purposes) in panel (a), while a quantitative estimate of the error in the fit to the observational data is provided in Table 3.2. The impact of constraining a potential viscosity increase to the lowermost section of the lower mantle was also investigated. The panels shown in Figure 3.14(b) demonstrate the minimal impact induced on RSL evolution predictions by slight changes in lower mantle viscosity, with the only notable difference seen at the very edge of the former Laurentide ice sheet in Maine. Furthermore, by concentrating the increase in lower mantle viscosity to the very deepest part of the mantle and keeping the rest of the viscosity structure fixed to VM5a, it should be noted that the RSL predictions are identical to the ones made using the VM5a structure.

3.5.10 Other considerations and a summary of the insights gained through the sensitivity analyses

Synthesizing the results of these calculations, it appears that, for the ICE-6G_C ice loading model, combinations of viscosity changes in different layers of the mantle could potentially lead to a good fit to RSL observations in the northernmost locations along the U.S. East coast. Indeed, for these regions, the thickness of the elastic lithosphere and the viscosity of the upper mantle and transition zone have an important influence on the RSL predictions, while the viscosity of the upper part of the lower mantle has a significant impact on the shape of the forebulge and its decay. However, the RSL data from the southern part of the U.S. East coast is only better represented with a viscosity in the upper part of the lower mantle that is slightly lower than that characteristic of the VM2 and VM5a profiles, which reduces the ambiguity as to which of the possible combinations one should first apply to better fit the data from the northernmost points along the coast. For instance, an increase in lithospheric thickness decreases the quality of the fit in the southernmost locations while a reduction of its thickness does not Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 87

Figure 3.14: Comparison of a subset of regions of the Engelhart and Horton (2012) data set of relative sea- level histories along the U.S. Atlantic coast with the predicted relative sea-level history at those locations for the ICE-6G_C model of ice-loading history combined to viscosity profiles where the viscosity of the lower part of the lower mantle is allowed to vary (as discussed in the text). (a) Viscosity variations in the lower part of the lower mantle (values are shown in the legend). (b) For the models shown in (a), comparison of RSL history predictions with observational data at four locations of the Engelhart and Horton (2012) data set, namely Southern Maine (2), New York (6), Inner Delaware (9), and Southern South Carolina (16). Green data points represent sea-level index points, whereas blue crosses represent marine-limiting data and orange crosses represent terrestrial-limiting data. From Roy and Peltier (2015). Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 88 reconcile RSL observations with the model predictions, which restricts the range of possibilities in the quest for an improved model that will enable us to fit the majority of the observational data. Changes in the viscosity of the transition zone have some impact on RSL predictions at northern locations along the U.S. East coast. Also, a softer transition zone fits the mid-Atlantic data points better, but as the viscosity of the upper part of the lower mantle must be decreased to fit the observational data from the southernmost locations along the U.S. East coast, it is interesting to note that the viscosity in the transition zone could potentially be used as a fine-tuning parameter in balancing the quality of the fit to the geologically derived RSL data from the southernmost locations of the Engelhart and Horton (2012) data set and to other global geophysical constraints (such as the Fennoscandian spectrum of spherical harmonic degree-dependent relaxation times).

3.6 A preferred viscosity structure: the VM6 proﬁle

Here, using the previous analyses and the most recent ICE-6G_C globally consistent ice loading history

(Argus et al., 2014; Peltier et al., 2015), a new radially-symmetric viscosity profile for the Earth’s mantle that addresses the shortcomings of the VM5a and VM5b profiles along the U.S. East coast is found. In each layer listed in the previous section, the viscosity was perturbed to minimize the misfit between the

GIA prediction of RSL along the U.S. East cost and the Engelhart and Horton (2012) data set, starting with the thickness of the lithosphere, progressing with the viscosity of the upper mantle and of the layers beneath it until reaching the upper part of the lower mantle (RSL predictions along the U.S. East coast have no meaningful sensitivity to viscosity variations in the lower part of the lower mantle). The process was repeated for the whole mantle until convergence was achieved. In this process, a rheological stratification of the lithosphere, with a 30-km thick layer with higher than upper mantle viscosity located right under the elastic lithosphere, was used, following its introduction in the VM5a and VM5b models in order to reconcile GIA model predictions of present-day horizontal motion of the solid Earth surface with space-geodetic observations (Peltier and Drummond, 2008). The resulting new viscosity structure, which is referred to as VM6, is illustrated in Figure 3.17(a) and its properties are listed in Table 3.3. It contains a slightly thicker elastic lithosphere (90 km thickness) superimposed upon a 30-km thick layer with higher than upper mantle viscosity representing the rheological stratification of the lithosphere. In the VM6 profile, the viscosity of the upper part of the mantle, located between the base of the lithosphere and the base of the transition zone at 660 km depth (0.45·1021 Pa·s), lies between the viscosity of VM5a and VM5b. No additional structure in the transition zone (between the 420-km and 660-km seismic discontinuities) was found to provide a consistent decrease in the error function values obtained from Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 89 comparisons to the Engelhart and Horton (2012) data set, but it was preferable to reduce the viscosity in the upper part of the lower mantle, where it is set to 0.90 · 1021 Pa·s, or about half the viscosity characteristic of the VM5a structure in this depth range. This decrease is key to addressing the misfits identified between the observational data in the Carolinas and the model predictions for the VM5a and

VM5b viscosity profiles. The lower part of the lower mantle requires a slight decrease in viscosity. These modifications do enable the revised model to address decisively the shortcomings of the VM5a viscosity profile regarding the reconstruction of past relative sea level from the Engelhart and Horton (2012) data set.

Table 3.3: Radial viscosity structure of the VM6 viscosity proﬁle down to the core-mantle boundary

The new ICE-6G_C (VM6) model is tested here not only against the RSL data from the U.S. East coast of Engelhart and Horton (2012), but also against geophysical observables that were not used in the optimization process that led to it. This enables us to test the plausibility of the new VM6 viscosity structure and its potential global exportability, but it should be remembered that these extra constraints have not been used in the tuning process itself.

3.6.1 The predictions for the U.S. East coast for ICE-6G_C (VM6)

Comparisons between the complete Engelhart and Horton (2012) data set of relative sea level indicators and the ICE-6G_C (VM6) predictions of RSL history are shown in Figure 3.15, while a summary of the changes in reduced error function performance introduced by the new model when compared to the fit provided by the ICE-6G_C (VM5a) model is shown in Table 3.4. For indicative purposes, the change in performance resulting from a transition from the VM5a viscosity profile to the VM5b viscosity profile of Engelhart et al. (2011) is also included in Table 3.4. However, as demonstrated earlier, it should be noted that VM5b is not as favoured as a spherically-symmetric model of mantle viscosity, as it provides a worse fit to the Fennoscandian spectrum of relaxation times than that provided by VM6.

For the northernmost sections of the U.S. East coast, VM6 provides some improvement to the fit to the observations over that of the VM5a profile over the latest 4 ka of the observational record, but a Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 90 decrease in the quality of the fit at earlier times. Since the region was co-located with the margin of the

Laurentide ice sheet during the initial phases of the last deglaciation and is relatively close to a region of complex margin evolution (e.g. Borns et al., 2004), slight modifications to the ice loading history would be able to resolve this localized misfit. It should be noted, however, that VM6 provides an improved fit compared to VM5a for most of the sea-level index points south of Connecticut. In particular, in New

York (site 6), VM6 provides a fit to the observational data much superior to both VM5a and VM5b, especially for the more recent sea-level index points. For Long Island and New Jersey (sites 7 and 8), the change in viscosity structure from VM5a to VM6 improves the fit to the more recent index points, but does not provide a substantial improvement for indicators older than 4 ka. The performance difference observed between VM5a and VM6 (or between VM5b and VM6) in Long Island (site 7) is strongly affected by a few outlying data points (mostly marine-limiting) that none of the models can explain, and which dominate the percentage change shown in Table 3.4 for these locations. Progressing further south, the VM6 model is found to be much superior to VM5a, increasing substantially the quality of the

ﬁt of the GIA model to the Engelhart and Horton (2012) data set in Maryland, Virginia and Delaware

(sites 9 to 12). In the Carolinas (sites 13 to 16), particularly in South Carolina, VM6 provides a noted improvement over both VM5a and VM5b. As the misﬁts associated with the use of these models for the southern part of the Atlantic coast of the United States was an outstanding problem (Engelhart et al.,

2011), the introduction of this new viscosity structure is important. However, it should be noted that the error function is strongly impacted by a limited number of relatively old marine-limiting data points in the Chesapeake Bay region (site 11) and in the northern part of North Carolina (site 13), where some misﬁts remain.

Another key result is related to the age dependence of the performance of the ﬁt with latitude.

As indicated by Table 3.4, for sites south of Delaware, the increase in performance provided by VM6 is particularly notable for older RSL geological data points (older than 4 ka). In particular, for the southern part of the coast, using the new VM6 viscosity structure removes practically all the misfits with respect to older geological RSL data that were identified when using the VM5a model, with some misfit reductions of more than 95% at some locations. This result is important, as the ability to model older geological data points is crucial if the behaviour of the forebulge collapse observed along the U.S.

East coast is to be explained correctly, a feature which we now turn our attention to.

Looking in detail at the form of the ongoing collapse of the forebulge predicted to extend over much of the U.S. East coast is an important test of the new VM6 viscosity structure, as the accurate description of the properties of the forebulge is a key component of the study of relative sea-level changes observed over the later part of the Holocene and during the 20th century along the U.S. East coast (e.g. Engelhart Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 91

Figure 3.15: Comparison of the Engelhart and Horton (2012) data set of relative sea-level histories along the U.S. Atlantic coast for the 16 composite regions with the predicted relative sea-level history at those locations for the ICE-6G_C model of ice-loading history combined to the VM5a (green), VM5b (blue) and VM6 (black) radial viscosity proﬁles. Green data points represent sea-level index points, whereas blue crosses represent marine-limiting data and orange crosses represent terrestrial-limiting data. From Roy and Peltier (2015). Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 92

Table 3.4: Percentage change in the reduced error misfit for the transition from ICE-6G_C (VM5a) to ICE-6G_C (VM6) along the U.S. East coast (negative changes indicate a better fit to the observational data, with -100% representing a complete removal of the misfits observed with ICE-6G_C (VM5a)). The change in error misfit performance resulting from the transition from the VM5a viscosity profile to the VM5b viscosity profile of Engelhart et al. (2011) is also shown for indicative purposes. From Roy and Peltier (2015).

et al., 2009). The shape of the forebulge inferred from the geological records described in Engelhart and

Horton (2012) is presented in Figure 3.16 by considering the rates of relative sea-level rise over the late Holocene. Following Engelhart et al. (2009), these late Holocene rates of relative sea-level rise are shown in Figure 16 for a series of locations along the U.S. East coast as a function of distance from one of the main centers of glaciation of the Laurentide ice sheet (Keewatin dome), approximated to be in modern-day Churchill, Manitoba, on the western shore of Hudson Bay. These values were determined in Engelhart et al. (2009) by running a linear regression over the geological data covering the past 4,000 years at each site under consideration (with 2-σ error bars). The results obtained for the ICE-6G_C

(VM5a) and ICE-6G_C (VM6) models at the same locations are superimposed on Figure 3.16.

As shown in Figure 3.16, using the VM5a mantle viscosity proﬁle over-estimates the late Holocene rates of relative sea-level rise for most locations along the U.S. East coast, most notably for locations in the mid-Atlantic states of New Jersey and Delaware (around 2,500 kilometers away from the former centre of glaciation), where the most signiﬁcant forebulge collapse occurs. This results in a forebulge collapse shape (illustrated in Figure 3.16 by the slope of uplift rate change as a function of distance Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 93

Figure 3.16: Comparison of observed Late Holocene relative sea level rise (mm/a) for locations along the U.S. East coast (dark grey diamonds) (with 2-σ uncertainty ranges in light gray) (Engelhart et al., 2009), with predicted values for the ICE-6G_C (VM5a) (green dots) and ICE-6G_C (VM6) models (red dots). The data points are plotted as a function of distance from the city of Churchill, Manitoba, on the western shore of Hudson Bay (km). From Roy and Peltier (2015). from the center of former glaciation) that compares unfavourably with the geological inference. The

VM6 proﬁle captures much better the amplitude of forebulge collapse in most locations, in particular its maximal range, which results in a forebulge collapse shape that is very close to that geologically inferred.

The ability of the ICE-6G_C (VM6) model to capture the geographical extent and the amplitude of forebulge collapse is very signiﬁcant, since this data was not used in the development of the model.

It should also enable an improvement in the characterization of the relative sea level rise observed during the late Holocene and during the 20th century (see Engelhart et al. (2009); Kemp et al. (2013)).

Discrepancies remain for the northernmost locations (around Maine), but as these locations are very close to the margin of the former Laurentide ice sheet, these misfits could be eliminated by slight changes in the timing of margin retreat or in the thickness of ice cover in the margin regions. The slight remaining misfit observed in the southernmost locations (around South Carolina) at the trailing edge of the forebulge can be eliminated by a minor change in the stratification of the elastic lithosphere, which will be demonstrated elsewhere.

3.6.2 The ﬁt to the Fennoscandian relaxation spectrum

A global constraint to which the new model may be tested, and which is directly linked to the properties of the viscosity structure itself, is the ﬁt to the Fennoscandian spectrum, provided in Figure 3.17(b) Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 94

Figure 3.17: (a) Comparison of the radial variations of the viscosity in depth for the new VM6 viscosity profile (black) with the VM5a (green) and VM5b (blue) profiles. (b) Inverse relaxation time as a function of spherical harmonic degree obtained from observations of the glacial isostatic adjustment of Fennoscandia, with the relaxation spectrum of Wieczerkowski et al. (1999) (black dashed line) and corresponding 1-σ and 2-σ uncertainties (dark gray and light gray areas, respectively). The predicted spectra for the new VM6 viscosity structure is shown in red, while the predictions for the VM5a (green) and VM5b (blue) models are superimposed. Adapted from Roy and Peltier (2015). for the VM6 viscosity profile. The new model provides a generally adequate fit to this constraint. The relaxation times inferred for the largest deformation scales are somewhat lower than for the VM5a profile

(also shown), largely owing to the slight decrease in eﬀective viscosity across much of the upper mantle

(including the slight reduction in the transition zone) and the upper part of the lower mantle. However, they still remain within the 2-σ uncertainty range of the Fennoscandian spectrum suggested by the formal analysis of Wieczerkowski et al. (1999). For high spherical harmonic degrees (e.g. smaller deformation scales), the relaxation times for VM6 lie well within the 1-σ uncertainty range. As the optimization process leading to VM6 was focused on the U.S. East coast, modifications to the VM6 profile which may have improved the fit to the Fennoscandian spectrum at lower spherical harmonic degree (such as a higher viscosity in the transition zone) were not included, as these variations would have decreased the quality of the fit to RSL observations for a substantial number of sites along the U.S. East coast, most notably in the mid-Atlantic region. A slight reduction in mantle viscosity in the transition zone from the VM5a model, similar to the change applied to the rest of the upper mantle, was found to fit the observational data at these sites optimally. This conclusion may have to be revisited in future analyses that expand the methodology employed here to include a broader suite of global constraints.

3.6.3 Testing the viscosity structure against data from the North American West coast

The new viscosity structure can also be tested by comparing its relative sea level evolution predictions to other regional data sets. Such a region is the western coast of North America, where a large data set Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 95 of calibrated 14C-dated sea level indicators has also recently become available (Engelhart et al., 2015).

The locations of the sites under consideration in this analysis were shown in Figure 3.3(b) and span the entire West coast of Canada and the United States. Since the area is another region undergoing postglacial forebulge collapse, it will provide another important test of the quality of the new model, given that it has not been employed in its development. It is important to note, however, that unlike the Atlantic coast of North America, where the continental margin is relatively "passive", the Paciﬁc coast is much more active tectonically, as it is notably host in the south to the San Andreas strike slip fault and in the north to the subduction zone that marks the eastern ﬂank of the Juan De Fuca plate.

For example, the West coast is known to have been host to the great Cascadia earthquake of 1700AD

(magnitude 8-9), which was accompanied by signiﬁcant vertical motion (Hawkes et al., 2010, 2011). The area is also aﬀected by uplift/subsidence associated with the subduction of the Juan de Fuca plate, especially over Vancouver Island (James et al., 2000). Thus, when comparing our results for relative sea level history predictions with the observational constraints, the existence of a potential tectonic overprint should be considered. However, as the database of 14C-dated sea level indicators covers only a few millennia, this impact should be relatively minor compared to the other uncertainties involved in the study of the GIA problem (viscosity structure of the mantle, ice loading history uncertainty, large spontaneous subsidence/uplift events, impact of sediment compaction on geological records, etc.).

The performance of the VM6 viscosity proﬁle is evaluated along the North American West coast by comparing its RSL history predictions at the sites shown on Figure 3.3(b) with the data presented in

Figure 3.18, starting with the northernmost locations. The model provides a good ﬁt to the available sea level indicators for Queen Charlotte Strait (site 1), as well as the eastern coast of Vancouver Island

(site 3). However, on the west coast of Vancouver Island (region 2), a region strongly aﬀected by tectonic impact associated with the subduction of the Juan De Fuca plate, the model not surprisingly fails to reproduce the relative sea level evolution inferred from observational constraints, even though it performs marginally better than VM5a in this regard. For the southeastern part of Georgia strait

(site 4), a region located directly on the mainland, the model fails to predict the subsidence revealed by the geological record for all viscosity structures employed. Instead, it predicts a complex RSL history probably impacted by the rebound associated with the melting of the adjacent Cordilleran ice sheet, the collapse of the forebulge associated with the more remote but much larger Laurentide ice sheet to the east, and the global rise in sea level associated with the last deglaciation. These discrepancies could be explained by various phenomena. In fact, not only is the response in this region heavily impacted by tectonic activity, but the site is also very close to the advancing front of Cordilleran ice during the Last Ice

Age and is thus highly sensitive to variations in the ice thickness in its immediate vicinity, most notably Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 96

Figure 3.18: Comparison of a data set of sea-level indicators along the U.S. Pacific coast (Engelhart et al., 2015) with the predicted relative sea-level history at those locations for the ICE-6G_C model of ice- loading history combined to the VM5a (green), VM5b (blue) and VM6 (black) radial viscosity profiles. Green data points represent sea-level index points, whereas blue crosses represent marine-limiting data and orange crosses represent terrestrial-limiting data. From Roy and Peltier (2015). Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 97 in the earlier part of the sea level record. Furthermore, there might be subsidence effects associated with sediment loading caused by glacier meltwater-fed (and thus sediment-rich) water outflows towards

Georgia Strait (such as the Fraser and Columbia rivers), which could have impacted certain locations on the eastern shore of Georgia Strait.

For the southern coast of Vancouver Island (site 5), the new ICE-6G_C (VM6) model performs adequately with regards to the limiting data points, although ICE-6G_C (VM5a) ﬁts marginally better the index points. Progressing further south to the Juan de Fuca Strait (site 6) and onwards, the quality of the ﬁt provided by the new model improves substantially. Most notably, along the western coast of

Washington State and Oregon, the ICE-6G_C (VM6) model is able to ﬁt most of the observational constraints derived from both sea level index points and terrestrial- or marine-limiting data at all locations.

Finally, for the southernmost part of the data set, in California, the models struggle to reproduce the

flattening of relative sea level observed between 7.5 ka and 4 ka. However, it should be noted that the northern part of the Californian coast could be affected by the complex tectonic setting of the region, close to the Mendocino Triple Junction (e.g. Merritts and Bull, 1989), while the central coast site, being located within the Sacramento River basin, could also be affected by sediment compaction effects.

Although the new viscosity structure VM6 provides a notably improved fit to the observational data of relative sea evolution on both the East and West coasts of North America, small misfits persist for some locations and time periods. For example, Eastern Maine is a region where a misfit persists, but as mentioned earlier, the issue should not be significant given its location at the margin of the former

Laurentide ice sheet. In fact, as relative sea level evolution predictions close to the edges of a former ice sheet are highly dependent on the exact melting history and geometry of the ice sheet, it is expected that these misﬁts could be removed by modifying the ice loading history in the margins of the former ice sheets. A similar situation applies to the two sites on the eastern shore of Georgia strait on the West coast of the continent. Also, some misﬁts remain for the U.S. East coast locations situated right at the center of the forebulge associated with the former Laurentide ice sheet, namely Long Island and New

Jersey, regions that could be perhaps aﬀected by compaction issues (Horton et al., 2013).

3.7 Conclusion

The East coast of the continental United States is a very important region in the study of the glacial isostatic adjustment process. The availability of a new, high-quality data set that covers the entire coast provides an opportunity to test current GIA models, most notably concerning models of the radial variation of mantle viscosity. Here, misﬁts resulting from the use of either the VM5a model (Peltier and Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 98

Drummond, 2008) or the VM5b model (Engelhart et al., 2011) have been analysed in detail and shown to be eﬀectively eliminated by the new VM6 model when coupled to the new ICE-6G_C model of global planetary glaciation and deglaciation. Alternative models in the literature (the V1 and V2 models of

Mitrovica and Forte (2004) and Forte et al. (2009)) were also tested and rejected on the basis that the misﬁts to the data associated with them were found to be even greater than those characteristic of the

VM5a and VM5b models.

Through a detailed series of sensitivity tests designed to investigate the response of U.S. East coast

RSL histories to variations in the radial variation of viscosity over specific ranges of mantle depth, a new model of the depth dependence of viscosity in an assumed spherically symmetric model of the Earth’s interior was constructed using a simple, iterative methodology. This new model is referred to as VM6, the sixth in the series of models that continue to be refined using the iterative methodology developed to converge upon an optimal structure of this kind. Although the predecessor model VM5a is a perfectly acceptable model insofar as the reconciliation of relative sea level histories and geodetic data from the ice covered regions is concerned, the model fails in the region of forebulge collapse along the U.S. East coast, especially in its southern section. Demonstrating this has required the availability of the new high quality data base that has been assembled in Engelhart and Horton (2012). The new viscosity model, when combined with the ICE-6G_C loading history, has been shown to eliminate most misfits that otherwise exist at sites along the U.S. East coast which record the process of forebulge collapse. Some misfits still remain in isolated regions, most notably in the northernmost locations of the U.S. East coast in the data set of Engelhart and Horton (2012), and in some locations near the crest of the forebulge, but they are relatively minor when compared to those associated with the predictions of the previous

ICE-6G_C (VM5a,b) models. These remaining misfits appear to be resilient, however, in the sense that they cannot be eliminated without sacrificing the fit the model provides to the Fennoscandian spectrum of scale-dependent relaxation times (those tests were not explicitly shown here). Such persistent misfits could be an indication of lateral heterogeneity in the viscosity structure of the Earth, which is expected to exist based on the nature of the mantle convection process, where upwellings and downwellings are respectively associated with hotter (and less viscous) and colder (and more viscous) material. However, these misfits could also be related to compaction effects in some areas (Horton et al., 2013), and will be the subject of further work. Independent additional tests of the new model have also been discussed in this paper. The first such independent test involved the analysis of RSL data from the U.S. West coast which is also undergoing the process of glacial forebulge collapse but the data from which were not employed to constrain the new VM6 viscosity structure. With the exception of misfits associated with tectonic influence at a few sites along this coast and with the close proximity of the glacial advance of Chapter 3. GIA constraints from the U.S. East coast: the ICE-6G_C (VM6) model 99 the Western Cordilleran ice sheet, the new model was shown to reconcile equally well the data from this region as it does along the U.S. East coast.

Another independent test of the new model in the North American domain would verify the impact upon the misfit to the voluminous data base of GPS observations of vertical motion of the crust to which the ICE-6G_C (VM5a) model has also been tuned. An analysis of this nature will be presented in the following section, and will identify remaining regions of misfit. This presents a potentially major issue that arises when one proceeds using the methodology employed in the analyses presented in this chapter, as it focuses solely on changes to the viscosity structure to eliminate any existing misfits. As the loading history was fixed here to the ICE-6G_C model, which tuned to enable a good fit to a rich database of

GPS observations of vertical motion of the crust and to a large number of carbon-dated RSL histories from both Northern Canada and Fennoscandia (Peltier et al., 2015) when used with the VM5a viscosity model. Therefore, whether these critical data sets will remain well reconciled by the new ICE-6G_C

(VM6) model of the global GIA process remains an important question. If significant misfits to these data were found to exist even when variations to the ice loading history were allowed, then the iterative process we are employing to refine the model might not be convergent. This iterative process consists of alternately fixing the viscosity and the loading history and then refining the other. By returning to investigate the fit to GPS observations over the North American continent, the circle can be closed to test convergence. Such a process, using the advocated modifications to the radial profile of mantle viscosity presented herein, will be detailed in the following chapter. Chapter 4

Full GIA constraints over N. America: the ICE-7G_NA (VM7) model

The increasing availability of large global and regional databases of relative sea level history records

(Tushingham and Peltier, 1992; Engelhart and Horton, 2012; Engelhart et al., 2015; Khan, N. S. et al.,

2015b; Vacchi et al., 2016) has greatly beneﬁted the study of the GIA process. Together with the con- current development of space-geodetic constraints on the rotational state of the planet (Peltier, 1982;

Wu and Peltier, 1984; Peltier and Jiang, 1996), they have led to the successful development of comprehensive models such as the ICE-5G (VM2) model of Peltier (2004). Further signiﬁcant improvements to

GIA models have been enabled by the availability of large networks of space-geodetic measurements of crustal motion (Argus et al., 1999; Milne et al., 2001; Argus and Peltier, 2010; Argus et al., 2014), and of satellite-based time-dependent gravity observations from the Gravity Recovery and Climate Experi- ment (GRACE) satellites (Peltier, 2004; Peltier et al., 2015), which have been used most recently in the development of the ICE-6G_C (VM5a) global model of Peltier et al. (2015). In the previous section, a detailed analysis of the performance of this model in terms of its ability to reconcile data from the region of forebulge collapse outboard of the Laurentide ice sheet has been invoked to further tighten the constraint on mantle viscosity (beyond that provided by RSL data from the region that was previously ice covered). It led to the ICE-6G_C (VM6) model of Roy and Peltier (2015), which reconciled observations of past RSL along the U.S. East coast with model predictions, while producing a good or excellent ﬁt to a suite of other observables linked to RSL evolution or the post-deglaciation relaxation of the centers of the former ice sheets covering North America and Fennoscandia. However, modiﬁcations to the viscosity structure of the mantle may also be expected to impact the ability of the new model to

100 Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 101 maintain its ﬁt to additional constraints related to the present-day crustal uplift rate of the continent that had led to the precursor ICE-6G_C (VM5a) model. This situation is addressed in the present chapter, where the optimization methodology used in Roy and Peltier (2015) is expanded to encompass not only the U.S. East coast relative sea level records, but also those space-geodetic constraints on the uplift rate of the crust. Also, whereas the previous analysis was only concerned with mantle viscosity variations, variations in continental ice cover over the region will also be allowed. The plausibility of the

Roy and Peltier (2015) assumption that the ICE-6G_C loading history may be considered converged will thus be directly addressed.

It will demonstrate that updating the most recently published reﬁnement of the global model (ICE-

6G_C (VM5a)) with the VM6 viscosity structure leads to significant misfits with the space-geodetic uplift rate observations which the original ICE-6G_C (VM5a) model fit with high accuracy. This raises the issue of the convergence of the iterative methodology being employed in the process of model construction. Through modest further modifications of both the viscosity structure of the model and the

North American component of the surface mass loading history, a locally optimized solution which eliminates these misfits and maintains an excellent fit to the RSL constraints from the U.S. East coast is obtained. The new model presented herein, ICE-7G_NA (VM7), is then tested against other independent constraints to which it was not tuned, including the time dependent de-levelling of the Great Lakes region, the time dependent gravity results being provided by the GRACE satellites and the shape of the late Holocene forebulge. In this chapter, the observables used in the study will first be described. Then, the predictions of the same observables resulting from the latest ICE-6G_C (VM5a) and ICE-6G_C

(VM6) models (Peltier et al., 2015; Roy and Peltier, 2015) will be assessed, the sensitivity of any resulting misﬁts to appropriate variations in mantle viscosity and ice cover determined, and a new model adjustment, ICE-7G_NA (VM7), which removes those misﬁts, will be presented. It follows closely the contribution of Roy and Peltier (2017), under review for publication (Roy, K., and Peltier, W. R. (2017),

’Space-geodetic and water level gauge constraints on continental uplift and tilting over North America:

Regional convergence of the ICE-6G_C (VM5a/VM6) models’, Geophysical Journal International).

4.1 Geophysical observables related to the GIA process in North

America and the performance of current models

Models of the GIA process can be constrained, reﬁned and tested by comparing their predictions of various observables to geophysical and geological inferences of the same quantities. An overview of the Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 102 main data sets used in this analysis is provided here.

4.1.1 Constraints on former ice sheet extent and thickness

In the construction of the ICE-5G (VM2) and ICE-6G_C (VM5a) models of the GIA process, one of the main parameters to be constrained is the evolution of ice sheet cover over the continents during the last glaciation-deglaciation cycle (Peltier, 2004; Peltier et al., 2015). The determination of the time dependence of the area covered by grounded continental ice may be constrained on the basis of radiocarbon dated biological material from ice sheet margin locations (end moraines, former pro-glacial lakes, etc.) and exposure dating from cosmogenic nuclide abundances (10Be, 14C,26Al, 36Cl, 3He and 21Ne) which are based on the inhibition of their natural formation on the bedrock when the surface is covered by a thick ice sheet. These clues have been successfully used to reconstruct past ice sheet margins over

North America for the Laurentide, Cordilleran and Innuitian ice sheets (Dyke et al., 2002; Dyke, 2004), shown in Figure 4.1, and over Fennoscandia (Gyllencreutz et al., 2007; Hughes et al., 2016).

Estimates of ice sheet thickness variations within the evolving margins are provided by the amplitude of the emergence of the land with respect to evolving sea level provided by radio-carbon dated relative sea level histories or from the application of fully coupled models of ice-Earth-ocean interactions in conjunction with them (Tarasov et al., 2012; Stuhne and Peltier, 2015). Appropriate records of sea level evolution at far-ﬁeld sites also provide strong constraints on the time dependent total volume of water locked in grounded land ice: in particular, the high-quality record of relative sea level change at Barbados provides an excellent approximation to the eustatic (globally averaged) changes in sea level owing to its location at the trailing edge of the forebulge associated with the former Laurentide ice sheet (Fairbanks,

1989; Fairbanks et al., 2005; Peltier and Fairbanks, 2006). This record, shown earlier in Figure 1.8, will play an important role in the current discussion.

4.1.2 The importance of RSL data from the North American region of forebulge collapse

Regional databases of RSL history, such as that for the British Isles (Peltier et al., 2002; Shennan et al., 2002), the U.S. East coast (Engelhart et al., 2011; Engelhart and Horton, 2012), the U.S. West coast Engelhart et al. (2015), the Caribbean Sea (Khan, N. S. et al., 2015b) or the western half of the

Mediterranean Sea (Vacchi et al., 2016), can be used to provide further constraints on the GIA process, but great care must be taken in the interpretation of any misﬁts between observations and GIA model predictions, given the potential additional contribution of other local geophysical eﬀects such as tectonics, Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 103

Figure 4.1: Deglaciation isochrones for North America, based on the work of Dyke et al. (2002) and Dyke (2004), which are the margin constraints applied to the ICE-5G (Peltier, 2004) and ICE-6G_C (Peltier et al., 2015) ice sheet loading histories. From Peltier et al. (2015). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 104 sediment compaction or tidal range change (e.g. Horton et al., 2013; Engelhart et al., 2015; Roy and

Peltier, 2015). The high-quality database for the U.S. Atlantic coast of Engelhart and Horton (2012)

(see also Engelhart et al. (2011)) was employed by Roy and Peltier (2015) to further constrain the depth dependence of viscosity, and the individual time series of which the database is comprised were employed to demonstrate that by slightly modifying the viscosity model VM5a to VM6, a marked improvement in the fit to RSL data along the U.S. East coast could be achieved, particularly in the southernmost part of the coast, where systematic misfits had been previously identified (Engelhart et al., 2011). This work will use the same crucial data set to constrain the evolution of relative sea level along the forebulge associated with the former Laurentide ice sheet.

4.1.3 Space-geodetic uplift measurements over North America

A further set of geophysical observables which is critical for present purposes is the large number of space- geodetic measurements of vertical crustal motion currently available for the North American continent.

The data set to be employed for the purpose of the discussion herein is a reﬁnement and updated version of the initial database presented in Argus and Peltier (2010), and recently employed in Argus et al. (2014) and Peltier et al. (2015). It includes GPS records from the most recent JPL solution,

SLR measurements, VLBI records, Doppler Orbitography and Integrated Radio-Positioning by Satellite

(DORIS)-based measurements, as well as GPS measurements from the Canadian Base network, with appropriate reference frame and tectonic plate motion considerations in order to measure accurately long-term ground motions (Argus, 2007; Argus et al., 2010). It covers the time period from 1994 to

2012. A suitable transformation from the GPS measurements to a realization of the 2008 International

Terrestrial Reference System (ITRF2008) (Altamimi et al., 2011) is realized using a subset of the GPS data, and using appropriate tidal standards (Petit and Luzum, 2010), from which estimates of the time evolution of each site is determined (Argus et al., 2010, 2014). For the purpose of this work, the linear trends in vertical motion (together with its 2-σ conﬁdence range) found in each record are used. The total suite of measurements includes not only 509 JPL GPS sites, but also 52 VLBI, 20 SLR and 37

DORIS sites, as well as 142 GPS sites from the Canadian Base Network (see Peltier et al. (2015)). An in-depth discussion of the methodology employed can be found in Peltier et al. (2015), Argus et al. (2014) and Argus and Peltier (2010). In the reference frame construction (and velocity of the Earth’s center determination), estimated parameters include the velocity (rotational and translational components) between the reference frames of each of the four space-geodetic techniques, the angular velocities of the major plates, and the velocities of sites on sites that are not currently undergoing signiﬁcant post-glacial Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 105 recovery or ice loss.

In Peltier et al. (2015), it was shown that the ICE-6G_C (VM5a) model provides an excellent fit to almost all such available measurements. The locations of the vertical motion measurements upon which we will focus for present purposes are shown on Figure 4.2, together with a graphical representation of the values of the measurements and their associated error bars. Clearly evident is the well defined boundary between the region of present day uplift in the region that was once covered by the Laurentide ice sheet and the forebulge region south of the U.S.-Canada border in which present day crustal sinking is occurring. Figure 4.3 presents maps of the predicted present-day uplift rate by the ICE-6G_C (VM5a) and ICE-6G_C (VM6) models over North America (in panels (a) and (b), respectively). Inspection of the results in this figure demonstrates that the predictions of both models are characterized by the existence of a clearly evident double "bull’s eye" pattern of uplift which straddles Hudson Bay. Switching the viscosity structure from VM5a to VM6 dramatically reduces the amplitude of predicted present-day vertical crustal motion, as the reduced viscosity in the upper part of the lower mantle leads to much more rapid relaxation, thereby diminishing the uplift rate extrema seen over both former Laurentide ice domes (by as much as 5 mm/yr for the Keewatin Dome located west of Hudson Bay, and by 3 mm/yr for the dome that was located over present-day Quebec and Labrador) and over all the regions that were once covered by the Laurentide ice sheet. The forebulge region is also strongly impacted both in extent and amplitude from a switch to the VM6 profile, with strong extrema occurring on the oceanic floor of the Labrador Sea and in the American Midwest. Along the U.S. East coast, a reduction in subsidence is also observed (reaching 1.4 mm/yr in the mid-Atlantic portion of the coast). The demarcation line that separates regions of uplift and subsidence is, however, not affected by the change in viscosity structure.

The large diﬀerence in vertical motion predicted by these models has consequences for their ability to explain the space-geodetic measurements recording the crustal vertical uplift rate over the continent.

The lower panels of Figure 4.3 provide a closer look at the ability of both the ICE-6G_C (VM5a) and ICE-6G_C (VM6) models (in Figs 4.3(d) and 4.3(e), respectively) to explain them within their observational uncertainties. Clearly, the fit to the space-geodetic data is significantly degraded when the VM5a viscosity structure is simply replaced with VM6. A primary issue addressed here is whether there exists or not a modified version of the ICE-6G_C (VM6) GIA model that is simultaneously able to reconcile both the RSL data from the region of forebulge collapse along the U.S. East coast and the space-geodetic observations of crustal uplift rates, while also continuing to reconcile all of the other geophysical observables (global and regional) successfully explained by these models (including RSL history observations from the ice covered regions of the planet).

Finally, an examination of the quality of the ﬁt provided by the ICE-6G_C (VM5a) and ICE-6G_C Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 106

Figure 4.2: Measurements of vertical motion at sites from which space-geodetic measurements are available for the central and eastern parts of the North American continent in the Jet Propulsion Laboratory (JPL) data set, following the reduction methodology of Peltier et al. (2015). The color of each circle represents the vertical uplift inferred at each site, whereas its radius is inversely proportional to the standard error of the individual measurements, following the scale provided at the bottom left of the map. From Roy and Peltier (2017). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 107

Figure 4.3: Model predictions of vertical uplift rates over North America for the (a) ICE-6G_C (VM5a) and (b) ICE-6G_C (VM6) models in mm/yr. The diﬀerence between the predictions of those two models over North America is shown in panel (c) (in mm/yr). The bottom panels show a comparison of any existing misﬁts between space-geodetic measurements of present-day vertical motion rates (mm/yr) with GIA predictions of those same rates (taking into account observational uncertainties) for the ICE-6G_C (VM5a) (panel (d)) and ICE-6G_C (VM6) (panel (e)) models at the site of each uplift rate measurement. From Roy and Peltier (2017). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 108

(VM6) models to the (mostly) GPS-based observations of modern-day vertical uplift rates reveals the existence of two regions of persistent misﬁt in south-western Canada and in the vicinity of the Gulf of

St. Lawrence. A recent analysis of three-dimensional crustal velocities over North America by Snay et al. (2016) also found residual misfits in these regions. This work will attempt to address whether any changes in ice margin evolution in this region are required (and sufficient) to remove these misfits.

4.1.4 Time-dependent gravity measurements from the Gravity Recovery and Climate Experiment (GRACE) satellites

Another independent test of the quality of the ICE-6G_C (VM5a) and ICE-6G_C (VM6) models is provided by measurements of time-dependent gravity being provided by the GRACE satellites, which have not been employed to tune the model parameters.

The GRACE system relies on two identical satellites that orbit on the same near-circular low Earth orbit (altitude of ∼500 km), but separated by about 220 km (Tapley et al., 2004). A microwave ranging system is used, together with GPS receivers, attitude sensors and accelerometers, to monitor variations in the separation between the two satellites. These variations are converted into gravity anomalies using a precise sequence of data manipulations, together with a suitable background gravity model to which the anomalies can be determined and a reference frame. These "release products" are currently generated by the U.S. Center for Space Research (CSR) and the GeoForschungsZentrum (GFZ) in Potsdam, Germany.

By monitoring the evolution of these anomalies over many orbits, a sequence of gravity estimates can be obtained from the measurements, from which gravity anomaly maps and time series can be generated.

The signal observed by GRACE over North America, obtained using the Release 5 product from the

U.S. Center for Space Research, is shown in Figure 4.4(a) in terms of the rate of thickness change of an equivalent water layer at Earth’s surface, and clearly reveals the double "bull’s eye" pattern (with extrema existing on either side of Hudson Bay, as described in the recent analysis of Peltier et al. (2015)).

The only processing applied to the Release 5 data before analyzing the linear rate of change has involved use of a Gaussian ﬁlter with a 300-km half-width. In Figure 4.4(b), the prediction of the time-dependent gravitational changes over North America from the ICE-6G_C (VM5a) model is presented, while Figure

4.4(c) shows the residual when the GIA prediction is subtracted from the GRACE observation so as to clearly reveal the remaining signal over Greenland and Alaska, which has been associated with the modern global warming induced rate of loss of grounded ice from these regions (for example, in Luthcke et al. (2013)). Figure 4.4(d) further corrects the GRACE inference for the gravitational inﬂuence of changes in surface hydrology based upon data from the Global Land Data Assimilation System (GLDAS- Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 109

Figure 4.4: (a) Gravity Recovery and Climate Experiment (GRACE) observations of the time dependence of the gravitational field over North America from the Release 5 data from the Center for Space Research (CSR) for the period covering January 2003 to October 2013, on which no correlated filter was applied but a spatial filter of Gaussian half-width of 300 km was applied; (b) ICE-6G_C (VM5a) model predictions of this field; (c) Difference between the raw GRACE field (a) and the ICE-6G_C (VM5a) prediction (b); (d) Difference between the GRACE field (a), corrected for hydrological effects using the GLDAS correction of Rodell et al. (2004), and the ICE-6G_C (VM5a) prediction; (bottom panels) Same figures as (b) to (d), with using the ICE-6G_C (VM6) model predictions instead. All results are presented in equivalent water layer thickness change (cm/yr). From Roy and Peltier (2017).

NOAH) model (Rodell et al., 2004). Application of this correction ampliﬁes the strength of the "bull’s eye" pattern. Figs 4.4(e) to 4.4(g) show the same information as Figs 4.4(b) to 4.4(d), except using

ICE-6G_C (VM6). As described in Peltier et al. (2015), ICE-6G_C (VM5a) is able to reproduce the double "bull’s eye" pattern extremely well while the application of the hydrology correction slightly degrades the quality of the fit provided by the model over formerly glaciated areas. The use of the VM6 viscosity profile, however, results in a notable degradation of the quality of the model fit to the GRACE observations, especially with respect to the strength of the bull’s eye over southern Hudson Bay, a result that is fully consistent with the misfit to the GPS observations introduced when the VM6 viscosity model is employed in place of VM5a. This misfit becomes even more prominent if a hydrological correction is applied to the GRACE data. These questions, in particular with respect to the characteristics of the hydrology correction provided by the GLDAS-NOAH model (and also by competing representations), will be explored in a subsequent section of this chapter. Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 110

4.1.5 Water level gauges in the Great Lakes region

An additional data set that will be employed for present purposes is related to the evolving tilt of the

North American Great Lakes basin associated with the diﬀerential uplift of the crust between the regions north and south of the lakes and which is recorded in the time dependent warping of their shorelines.

This differential uplift is accurately recorded by long-term records from water level gauges. Analyses of such data were initially published in Peltier (1986) but higher quality and longer records have since become available (Mainville and Craymer, 2005), based on water level measurements at 55 lake shore locations (some of which extending back to 1860). In their methodology, the seasonal and systemic water level variations that are common to all locations within each lake are taken into account by pair-wise comparisons of water level change, and a least-squares fit is performed to remove any inconsistencies between each pair of sites (Mainville and Craymer, 2005). Long-term uplift measurements are provided with respect to the outlet of each lake. The differential uplift measurements obtained by Mainville and

Craymer (2005) for each of the Great Lakes are presented in Figure 4.5(a) (with respect to each lake outlet, shown in white). They are then compared to the ICE-6G_C (VM5a) and ICE-6G_C (VM6) model predictions in panels (b) and (c) of the same figure, where the observations in each lake are adjusted uniformly by the predicted uplift at each lake outlet. Panels (d) and (e) show the difference between the model predictions of differential uplift for each lake and the Mainville and Craymer (2005) inference when accounting for the 2-σ uncertainty in the observations. Both models can adequately represent the differential uplift observed in Lakes Ontario and Erie, while misfits exist for the other Great Lakes. In particular, both ICE-6G_C (VM5a) and ICE-6G_C (VM6) do not accurately capture the differential uplift seen across the northern and southern shores of Lake Superior and in the southern half of Lake

Michigan, while ICE-6G_C (VM5a) also predicts misﬁts in the vicinity of the Upper Peninsula of Lake

Michigan.

4.2 Characterization of the model misﬁts to present-day space-

geodetic uplift rate observations

Since a primary goal of this chapter is to demonstrate that the degradation of the ﬁt to the space-geodetic measurements of vertical crustal motion may be simply eliminated without sacriﬁcing the quality of the

ﬁt to the U.S. East coast data that constrain the time dependence of forebulge collapse, the nature of the misﬁts to these data induced by the switch from VM5a to VM6 is explored. A simple sensitivity analysis is performed, in which the impact of a systematic scan of depth-dependent changes in the viscosity Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 111

Figure 4.5: (a) Mainville and Craymer (2005) observations of vertical uplift for sites along the shores of the Great Lakes (mm/yr), compared to the reference point of each individual lake outlet (indicated as a white circle); (b) Predicted vertical uplift rate field from the ICE-6G_C (VM5a) model superimposed by the Mainville and Craymer (2005) measurements of vertical uplift (the data of each lake being corrected by the amount of uplift predicted by the model at each lake outlet); (c) Same field for the ICE-6G_C (VM6) model; (d) Difference between the modeled vertical uplift rate and the adjusted observations of Mainville and Craymer (2005) at each outlet, accounting for the 2-σ uncertainty in the inferences, for the ICE-6G_C (VM5a) model (with its predicted uplift rate field in the background); (e) Same as (d), but for the ICE-6G_C (VM6) model. From Roy and Peltier (2017). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 112 structure on vertical motion predictions is assessed. Following this analysis, the strong constraint of a loading history to be fixed to that of ICE-6G_C (as in Roy and Peltier (2015)) is relaxed.

4.2.1 The impact of elastic lithospheric thickness and mantle viscosity variations upon vertical uplift rate predictions

Variations in the viscosity structure of the mantle are known to signiﬁcantly aﬀect the sea level evolution predictions obtained from GIA models (e.g. Roy and Peltier, 2015), but the response of the solid Earth to ice loading and the range of mantle depth over which the response to deglaciation will be sensitive to viscosity is also highly dependent on the horizontal scale of the former ice sheets (Peltier, 1974; Wu and

Peltier, 1984; Mitrovica and Peltier, 1993b; Peltier, 1996b, 1998a). In Fennoscandia and in the British

Isles, due to the relatively small horizontal extent of the ice sheets that once covered these regions, the relative sea level evolution predictions are mostly sensitive to the viscosity structure at shallower depths (upper mantle and lithosphere) (Peltier et al., 2002; Shennan et al., 2002). For North America, it was demonstrated that, due to the larger scale of the former Laurentide ice sheet, the evolution of the forebulge associated with it is sensitive primarily to the viscosity at a greater depth in the mantle, namely the lower part of the upper mantle and the upper part of the lower mantle (Roy and Peltier,

2015). However, the initial shape of the forebulge (namely, its maximum amplitude and geographical extent) was found to be signiﬁcantly impacted by the thickness of the elastic lithosphere and the viscosity contrast between the upper and lower mantles (e.g. Tushingham and Peltier, 1991).

It is expected that this degree of sensitivity to variations in mantle viscosity and elastic lithosphere thickness should extend to predictions of vertical uplift rates. To this end, the same simple approach employed by Roy and Peltier (2015) is now applied to the case of vertical uplift rate predictions. A series of experiments is performed in which the thickness of the elastic lithosphere and the viscosity of the mantle are allowed to vary independently at various depth ranges in the mantle, while the rest of the profile remains identical to the VM5a profile of Peltier and Drummond (2008). Throughout this initial process, the ice loading is fixed to the ICE-6G_C model.

A summary of the most important results of this sensitivity study is presented in Figure 4.6. The impact of variations in the viscosity of the upper mantle (UM) is shown in the ﬁrst row. The resulting uplift rate ﬁeld for two VM5a variations is presented (upper mantle viscosities of 0.2 · 1021 Pa·s (panel

(a)) and 0.8 · 1021 Pa·s (panel (b)), together with the diﬀerence between each of these signals and the original VM5a-based prediction. An increase in upper mantle viscosity leads to a stronger uplift signal over the formerly glaciated area and more subsidence over the inner forebulge. On the eastern seaboard, Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 113

Figure 4.6: Comparison of the modern vertical uplift rate (mm/yr) over the North American continent using model variations based upon the ICE-6G_C (VM5a) model, where the ICE-6G_C loading history is kept fixed, but a single feature of the VM5a viscosity model is modified. (Upper row) Predicted vertical uplift rate field over North America, together with the difference between this field and the original ICE-6G_C (VM5a) model, for the cases in which the viscosity of the upper mantle (UM) is changed to (a) 0.2·1021 Pa·s, and (c) 0.8·1021 Pa·s; (2nd row) Same as the upper row, except for modified versions of the VM5a structure in which only the viscosity of the upper part of the lower mantle (ULM) is changed to (e) 0.8 · 1021 Pa·s, and (g) 2.4 · 1021 Pa·s. From Roy and Peltier (2017). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 114 the impact of viscosity variations in the upper mantle upon present-day uplift rates is negligible south of New Jersey. The second row of Figure 4.6 shows the results of a similar analysis, but for viscosity variations in the upper part of the lower mantle (ULM: 420 to 1260 km depth), for two examples (one below and one above the VM5a value), namely 0.8 · 1021 Pa·s (panel c) and 2.4 · 1021 Pa·s (panel d), together with the difference between these fields and the initial ICE-6G_C (VM5a) result. A visual comparison of the UM and ULM results shows that the impact of viscosity variations on the measured uplift rate under previously glaciated regions becomes more geographically uniform with increasing depth of the viscosity perturbation. As the geographical extent of the region of the forebulge sensitive displays a strong dependence to variations in viscosity in the upper part of the lower mantle, whether a significant gain in the quality of the fit could be obtained by increasing the resolution of the viscosity profile in this region will be considered later in this analysis.

4.2.2 Impact of ice loading history variations on vertical uplift rate predictions

To evaluate the sensitivity of model predictions of present-day vertical uplift rates to changes in ice loading history, two families of scenarios will be considered. The ﬁrst involves changes in ice sheet melting rates, notably in the vicinity of the Younger Dryas cold interval (that extended from approximately

12,800 to 11,500 years Before Present (BP) (Muscheler et al., 2008)), and the second will consider mass-conserving changes in the geographical distribution of ice thickness. The ﬁrst family of sensitivity analyses focuses upon the global eustatic change of sea level that has occurred since LGM, recorded by the Barbados record of relative sea-level evolution (Fairbanks, 1989; Peltier and Fairbanks, 2006). The

fit provided by the ICE-6G_C (VM5a) model of Peltier et al. (2015) to this crucial record of past sea level evolution is shown in Figure 4.7. The model provides an excellent fit through most of the period around LGM and during the deglaciation phase, but an important misfit to this coral based record is evident in the Younger Dryas period, which ends at approximately 11.5 ka. These errors of fit, which were not present in the precursor ICE-5G (VM2) model (Peltier, 2004; Peltier and Fairbanks, 2006), could conceivably be due to the slight difference in total ice melt (and its timing) in Antarctica and

North America between the two models. It may be most natural to seek an improvement of ﬁt to the

Barbados record in the Younger Dryas interval of time in terms of a modification to the ice melting history of the Laurentide ice sheet, since it has been argued that the return to near-glacial conditions itself was triggered by a meltwater injection into the Arctic Ocean from proglacial lakes associated with the Laurentide deglaciation (Tarasov and Peltier, 2005). In a simple sensitivity experiment, variations Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 115 in the history of the Laurentide Ice Sheet complex are introduced, with successively larger reductions in ice sheet melt rates over the entire ice sheet during the Younger Dryas period, until a best fit to the Barbados record is achieved. The reductions in melt rate are introduced abruptly at the start of the Younger Dryas (reflecting the abrupt nature of its onset), while progressively returning to the original ICE-6G_C ice loading model at the end of the Younger Dryas (with the difference between the original and the perturbed model following a decaying exponential curve). The fit provided by the best-fitting model variation based upon this simple methodology is presented in Figure 4.7. In this version, called ICE-6G_C_BARB, the melt rate falls to zero during the Younger Dryas period. It is clear that this new loading model eliminates the misfit to the Barbados record through the interval from 12.8 ka to 11.5 ka, which was an evident flaw in the ICE-6G_C (VM5a) model of Peltier et al.

(2015) (who commented explicitly on the issue). This modiﬁcation is an expected consequence of the return to extreme glacial conditions during this interval of time. Finally, in this context, it should be noted that some recent analyses of the living depth range of various coral species (Hibbert et al., 2016) have suggested an uncertainty depth range for the Acropora Palmata species of 3 meters (1-σ error) or

9.4 meters (2-σ error), indicating the very shallow depth at which most individuals can be expected to reside. In the analysis of Peltier and Fairbanks (2006) (and subsequent studies based upon this record, including here), an uncertainty of 5 meters is used (Fairbanks, 1989), which effectively falls between the two new interpretations of the biological record. It should be noted that the misfit identified around the

Younger Dryas period would remain even if one were to consider the larger depth range value of Hibbert et al. (2016).

Figure 4.8 presents the vertical uplift rate predictions over North America provided by this modified version of the ICE-6G_C ice loading history when it is employed in conjunction with the VM5a (upper row) and VM6 (lower row) profiles of radial variations in viscosity. The first figure of each row presents the predicted field of vertical uplift rate predicted for each viscosity structure using the ICE-6G_C model (panels (a) and (d), respectively), while the central figure of each row presents the same field, but using the ICE-6G_C_BARB test scenario (panels (b) and (e) for VM5a and VM6, respectively).

The difference between the test ICE-6G_C_BARB scenario and the ICE-6G_C model is shown on the rightmost panels of Figure 4.8 (panels (c) and (f)). The modifications in melting rate that lead to the ICE-6G_C_BARB model not only recover a better fit to the Barbados sea-level record, but also substantially increase the amount of predicted present-day uplift under the former Laurentide ice sheet that is obtained with both viscosity structures, with the maximum occurring in the core of the former

Laurentide ice sheet, and minor maxima located west of Hudson Bay and over Baﬃn Island. For both viscosity structures, given the short timescale over which the modiﬁed melting scenario has been applied, Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 116

Figure 4.7: The coral-based record of relative sea level evolution at the Caribbean island of Barbados of Peltier and Fairbanks (2006), with the vertical bars representing the depth range of the corresponding coral species (including 5 meters for Acropora palmata and 20 meters for Monastera annularis). The U/Th age uncertainties are small and close to the thickness of the plotted lines, so are not shown. Comparison of the ﬁt provided by the ICE-6G_C (VM5a) (green), together with the modiﬁed ICE- 6G_C_BARB (VM6) model described in the text (black). From Roy and Peltier (2017). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 117

Figure 4.8: Model predictions of vertical uplift over North America for the (a) ICE-6G_C (VM5a) and (d) ICE-6G_C (VM6) models in mm/yr. The predictions for the models using a modified version of the ICE-6G_C model that corrects the fit to the Barbados data is shown for the (b) VM5a structure, and (e) VM6 structure. The difference between the ICE-6G_C (VM5a) and ICE-6G_C_BARB (VM5a) models is shown in (c), whereas the difference between the ICE-6G_C (VM6) and ICE-6G_C_BARB (VM6) models is shown in (f) (in mm/yr). From Roy and Peltier (2017). the impact of the modified ice loading history is not felt in the forebulge, except in the Labrador Sea region. This slight shift in the melting history through the Younger Dryas interval significantly increases the predicted present-day rates of vertical motion by delaying the timing of deglaciation, an effect that could partly cancel the decrease in predicted uplift rate caused by a shift towards slightly less viscous model (in particular for the upper part of the lower mantle, as in the transition from VM5a to VM6).

These results support the idea that there could potentially be a model of the GIA process that is able reconcile both RSL records from the region of forebulge collapse and the present day measurements of vertical motion of the crust.

The second series of sensitivity analyses concerning glaciation history will consider the thickness distribution of the ice at LGM and during the deglaciation phase, and explore the impact that variations in Laurentide ice sheet "shape" have on predictions of present-day vertical uplift. Here, only the impact of a simple redistribution of the existing ice between the northern and the southern sectors of the ice Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 118 sheet are considered. The goal of this simple experiment is to determine whether simple variations in the distribution of ice could also be involved in understanding the marked reduction in predicted uplift rate delivered by the ICE-6G_C (VM6) model. The result of these experiments is shown in

Figure 4.9. Two families of variations are considered: one in which mass from the northern half of the Laurentide ice sheet is redistributed to the southern half between 19 ka and 13 ka (the "Early" case, upper-row panels of Figure 4.9), and one in which the same variation is introduced between 15 ka and 9 ka (the "Late" case, lower-row panels of Figure 4.9). Two amplitudes are considered (thickness redistributions that reach a maximal amplitude of 200 and 400 meters) and, in each case, the amplitude of the perturbation is smoothed at the beginning and at the end of the time period under consideration to ensure continuity in the local thickness of the ice sheet. It should be noted that the amplitudes of the redistributions remain small compared to the overall thickness of the ice sheet at the majority of time steps under consideration (except perhaps for the youngest time under consideration in the "Late" case, but it should be remembered that this simple test is, at this early stage in the methodology, focused on obtaining a physical understanding of the overall sensitivity of the predictions to such redistributions and that any ﬁnal modiﬁcations of this kind to the model would be required to be fully physically consistent).

The vertical uplift rate ﬁeld predicted by the model and its diﬀerence with respect to the original ICE-

6G_C (VM5a) prediction are investigated. This simple experiment results in a reduction of the uplift predicted over northern Canada and a corresponding increase in uplift predicted in southern Canada, but the uplift in the forebulge is once more unaffected. Increasing the magnitude of the modification in ice thickness from 200 to 400 meters simply increases the magnitude of the uplift change, without impacting its geographical extent. For a given change in ice thickness, an earlier application results in a less pronounced and smoother variation across the entire region over which the modification was applied.

4.2.3 The impact of ice loading history variations on relative sea level evolution

The sensitivity of the evolution of relative sea level in the forebulge to the simple variations in ice melting history and ice redistribution described above is now considered. In Figure 4.10, the inferences of relative sea level evolution from the Engelhart and Horton (2012) compilation are presented for a subset of sites along the U.S. East coast (Southern Maine, New York, Inner Chesapeake Bay and South Carolina), and are compared to predictions provided by the variations of the ICE-6G_C model discussed above (all of which are used in combination with the VM6 viscosity proﬁle). The cases considered here all include the simple modiﬁcations to the ice sheet melting rate around the Younger Dryas cold event and the Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 119

Figure 4.9: Prediction of the vertical uplift rate field over the North American continent from modified ICE-6G_C (VM5a) models (mm/yr), in which the ice loading history is modified following the description in the text, for the case of (a) an early application of a 200 m change; (c) an early application of a 400 m redistribution; (e) a late application of a 200 m redistribution, and (g) a late application of a 400 m redistribution. The difference between each field and the original ICE-6G_C (VM5a) prediction is shown in (b), (d), (f), and (h) for each respective case (mm/yr). From Roy and Peltier (2017). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 120 simple "Early" (19 ka to 13 ka) and "Late" (15 ka to 9 ka) ice redistribution scenarios (amplitude of 200 meters).

The impact of ice thickness redistributions within the Laurentide ice sheet on sea level history determination along the U.S. East coast is small, except for the northernmost sites where edge eﬀects are to be expected. Changing the ice melting rate to recover a ﬁt to the Barbados record of sea level history around the Younger Dryas cold event has a small impact on the sea level history predicted along the

U.S. East coast. The eﬀect is stronger for the northernmost sites, and fades as one moves south into the forebulge. This seems to indicate that the impact on RSL evolution along the U.S. East coast will be small, though not entirely negligible, at most sites if such a modiﬁcation to the ice melting history is introduced, due to the limited memory of the system to these relatively small mass loading variations.

Nevertheless, since VM6 was optimized to be used in conjunction with the ICE-6G_C loading history, it will be necessary to determine if another viscosity proﬁle is necessary to maintain (or improve upon) the high quality of the ﬁt provided by the VM6 model.

4.2.4 The impact of mantle viscosity variations on the diﬀerential uplift of the Great Lakes

The final data set considered in this sequence of preliminary sensitivity analyses will focus upon the differential uplift recorded by water gauge measurements in the Great Lakes that record the ongoing post-glacial tilting of the region and the transition from the region of deglaciation-induced uplift to that of forebulge collapse. The impact of shallow Earth structure variations on the ability of the model to reproduce the Great Lakes coastal water gauge uplift rate compilation of Mainville and Craymer (2005) is investigated by considering changes in elastic lithosphere thickness and upper mantle viscosity, while the rest of the viscosity structure is kept identical to that of the VM5a profile and the ice loading history is fixed to ICE-6G_C. In this analysis, as the observed uplift data is presented for each of the Great

Lakes with respect to its outlet, the observations need to be adjusted by an amount corresponding to the model prediction at the outlet of the Great Lake in which it is located, in order to capture the uplift gradient across each one of the Great Lakes. Results for a subset of two elastic lithosphere thicknesses are considered in the upper panels of Figure 4.12 (40 km and 220 km). Inspection of this

ﬁgure demonstrates that increasing the thickness of the lithosphere results in a much smoother transition between the uplifting regions of the continent and the forebulge collapse region. These variations do not signiﬁcantly impact the ability of the model to reproduce the uplift observed in most of the Great Lakes area, except for Lake Erie, where the observed gradient in vertical motion is not compatible with the Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 121

Figure 4.10: Comparison of the relative sea level history predicted at four key sites along the U.S. East coast from the (Engelhart and Horton, 2012) data set, namely Southern Maine, New York, the Inner Chesapeake Bay and Southern South Carolina. The ICE-6G_C (VM6) prediction (black) is compared to the ICE-6G_C_BARB (VM6) model (red), and the two cases where a 200 meter redistribution of ice is performed "early" (green) and "late" (blue) on the ICE-6G_C model (description in the text). Green data points represent sea-level index points, whereas blue crosses represent marine-limiting data and orange crosses stand for terrestrial-limiting data. From Roy and Peltier (2017). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 122 smooth gradient predicted when choosing models with a thick elastic lithosphere. All models are equally able to reproduce the gradient observed in Lakes Huron and Superior.

The equivalent experiment, but performed with viscosity changes in the upper mantle, is presented in the lower panels of Figure 4.12 for two different values (0.2 · 1021 Pa·s and 0.8 · 1021 Pa·s). In this experiment, the rest of the viscosity structure is kept identical to that of VM5a. It can be seen that changing the viscosity of the upper mantle strongly impacts the amplitude of the observed uplift/subsidence, but does not affect the shape of the contours (and in particular of the zero uplift transition). A much stronger gradient is observed when using a stiffer upper mantle. The stiffest model (0.8 · 1021 Pa·s) overestimates the north/south gradient that is observed across the majority of the lakes (in particular for Lake Huron), while the softest model considered (0.2 · 1021 Pa·s) underestimates it.

4.3 Parameter space search and model reﬁnement

Using the knowledge gained from the analysis of the sensitivity of the ﬁt to the suite of geophysical observables that may be invoked to constrain the GIA process provided by variations of the ICE-6G_C

(VM5a/VM6) models (both in mantle viscosity and ice loading history over North America), we investigate whether a single best-ﬁtting model can be identiﬁed. In particular, we wish to answer the question as to whether the iterative process employed in GIA model construction actually converges to an optimal solution, and to address the meaning of such convergence (or otherwise). This work may also provide insight into the issue of the uniqueness of the global solution to the GIA problem.

In this analysis, we will continue to focus on spherically-symmetric models of mantle viscosity. The reason for this approach, even if the temperature dependence of the creep resistance of mantle material would lead one to expect the existence of lateral viscosity heterogeneity to be characteristic of the convecting mantle on a priori grounds, is to fully explore the explanatory capabilities of a model of minimum complexity. The best-ﬁtting spherically symmetric model thereby determined is expected to provide a suitable background viscosity structure upon which lateral variability can be superimposed and additional degrees of freedom added (if necessary). In the absence of a well-constrained background model of this kind, it is our view that the ad hoc addition of new degrees of freedom embodied in models characterized by signiﬁcant lateral heterogeneity of viscoelastic structure would not be meaningful.

The strategy to be employed here will expand the "brute force" methods originally employed in the iterative process, where the ice loading history is first kept intact to focus on viscosity refinements, and then fix viscosity to refine loading history. A stage has now been reach where the refinement of both components of the model structure will be required. In broad terms, a voluminous suite of over 380 Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 123

Figure 4.11: Comparison between the Mainville and Craymer (2005) observations of vertical uplift for sites along the shores of the Great Lakes presented with respect to the value modelled at the reference point of each individual lake outlet (filled circles representing the value in mm/yr) with a modified ICE- 6G_C (VM5a) scenario where only the thickness of the elastic lithosphere is changed to (a) 40 km, and (b) 220 km, or where only the viscosity of the upper mantle (UM) is modified to (c) 0.2 · 1021 Pa·s and (d) 0.8 · 1021 Pa·s. From Roy and Peltier (2017). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 124 mantle viscosity profiles is first generated, covering a plausible range of parameter space in each viscosity layer of the mantle determined by the sensitivity analyses discussed above and in Roy and Peltier (2015).

The ability of each model to recover a good fit to both the vertical uplift rate data and the U.S. East coast relative sea level data of Engelhart and Horton (2012) is then carefully determined. Then, the 25% top performing viscosity structures are kept, and perturbations in the ICE-6G_C ice loading history are introduced upon them. These include both the introduction of a perturbation in the time evolution of the Laurentide ice sheet mass to recover a fit to the Barbados record during the Younger Dryas, as well as perturbations in the geographical distribution of the ice melt during the deglaciation phase across various regions of the Laurentide ice sheet. These modifications are applied successively, with the idea that the misfits should be reduced as significantly as possible with each iteration. Finally, predictions from the best performing model resulting from the parameter space exploration will be tested against a series of additional important geophysical observables that were not employed in the optimization process, such as observations of modern-day differential tilt of the Great Lakes inferred on the basis of water level gauges, and time-dependent gravity measurements from the GRACE mission.

4.3.1 Model misﬁts and error determination

The performance of the GIA models considered in this study will be evaluated using both quantitative and qualitative approaches. A quantitative analysis of the ﬁt of a given model to the data will be performed by comparing two sets of important geophysical observables and corresponding model predictions of the same quantity, namely for the extensive database of relative sea level history observations of Engelhart and Horton (2012) for the U.S. East coast and for the observed present-day vertical uplift rate of the

North American continent. Although the fit of the model to the totality of the relative sea level records available from the previously ice covered region is also, of course, of critical importance, they are not included in this further stage of model iteration, as we are already sufficiently close to a converged solution that the fit of the current set of model variations to these data remains of the same high quality as that provided in the Supplementary Materials section of Stokes et al. (2015), such that no useful purpose would be served by reproducing the fits to these data herein.

The determination of the quality of the ﬁt provided by the predictions of the model to geological records of relative sea level index and limiting data points is evaluated using the same error function presented in the previous chapter (Roy and Peltier, 2015). The contribution of limiting data points to the error function is only non-zero if the predicted sea level falls above a terrestrial-limiting data height or below a marine-limiting height, as these data provide only an upper or lower bound on relative sea Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 125 level at a given time in the past. The uncertainty related to the accuracy of the dating of the geological inferences of past sea level is taken into account following the methodology of Roy and Peltier (2015), which takes into account the uncertainty range associated with the age determination of each sample.

The misﬁts are calculated for each site along the U.S. East coast, and also combined in a single composite value for the entire coast.

For the misﬁts between present-day continental uplift rates observed from space-geodetic techniques and corresponding model inferences, a similar methodology is employed, with the total misﬁt between observational data and the prediction of a GIA model at the given location for the present-day vertical uplift rate presented as:

N 1 G ∆ 2 χ2 = G,i , (4.1) G N σ G i=1 G,i X where NG represents the number of present-day uplift observations in the data set as compiled and tabulated in Peltier et al. (2015), σG,i is the uncertainty in each uplift rate observation (95% conﬁdence level), and ∆G,i represents the diﬀerence between each measurement and the GIA model prediction for the same location.

The misfits for each major data set are combined into a single composite value. For relative sea-level evolution along the U.S. East coast, the performance indicator of a given GIA model is given as a ratio of the χ2-like error it generates to that provided by the ICE-6G_C (VM6) model of Roy and Peltier (2015), a model which will be taken as a benchmark for present purposes. To ensure that the focus of the fitting process is on determining the best fit to the shape of the forebulge, the ratio is weighted for the density of constraining data at each site (an average of all regional ratios is used). It should be noted that, as opposed to Roy and Peltier (2015), where the stated goal was to remove specific misfits pertaining to the earlier part of the relative sea level record, the performance indicator used here encompasses the entirety of the geological record. With regards to the fit to the space-geodetic constraints on vertical uplift rates, a similar ratio is generated with respect to the fit provided by the ICE-6G_C (VM5a) model. Then, a

ﬁnal, synthesized value is generated as a simple average of the two ratios generated.

4.3.2 Initial viscosity model generation and parameter space exploration

First, a wide family of initial viscosity structures covering a large region of parameter space is generated, using the VM6 proﬁle as an initial model upon which modiﬁcations are introduced. However, based on the previous sensitivity analysis, mantle viscosity variations in the upper part of the lower mantle

(or slightly below) appear to play a fundamental role in reconciling the fit to both important data sets Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 126 at hand, but in opposite directions, as a lower viscosity at this depth, favored by the RSL constraints, exacerbates the misfits to the space-geodetic observations. To account for this observation, the upper part of the lower mantle is further subdivided in two sections to account for potential differences in peak sensitivity with depth between the two data sets. Also, the transition to the lower part of the lower mantle is pushed slightly deeper in the mantle to account for the residual sensitivity of forebulge RSL observations observed in the lower part of the lower mantle in Roy and Peltier (2017). Thus, the layers under consideration are the elastic lithosphere (of variable thickness), a high-viscosity stratification layer beneath the elastic lithosphere, the upper mantle and transition zone (down to a depth of 670 km), the upper part of the lower mantle (depth range of 670-980 km), the middle part of the lower mantle (depth range of 980-1470 km), and the lower part of the lower mantle (from 1460 km depth to the core-mantle boundary). The thickness of the high-viscosity stratification layer beneath the elastic lithosphere is

ﬁxed to 40 km, the value found by Peltier and Drummond (2008) to reconcile observations of horizontal crustal displacement with model predictions (model predictions of North American RSL and vertical uplift rates were found to be quite insensitive to changes in its thickness (not shown here)).

The range of viscosity values and elastic lithosphere thicknesses considered in each of these layers is shown in Table 4.1. The greater number of values considered in the parameter space exploration for lithospheric thickness, upper mantle viscosity and for the upper and middle parts of the lower mantle reﬂects the high sensitivity of model predictions of RSL along the U.S. East coast to even small variations in mantle viscosity in the upper mantle and in the upper part of the lower mantle (Roy and Peltier, 2015) and the high sensitivity of model predictions of vertical uplift rates to changes in lithospheric thickness.

The values of mantle viscosity considered in each layer or lithospheric thickness are above and below that of the initial VM6 viscosity proﬁle, and their range is determined following a simple methodology.

In the lithosphere, the upper mantle and the upper and middle parts of the lower mantle, an initial test is performed in which the value in each layer is varied independently (while the rest of the mantle is

ﬁxed to VM6), until the error function increases by ≈50%, determining the upper and lower ranges to be considered. To reduce computational costs, and given the lower sensitivity of the predictions to a change in its viscosity, in the lower part of the lower mantle, only two values (VM5a-like and VM6-like) are considered for the initial run.

A ﬁrst range of 384 spherically-symmetric viscosity models is thus generated, and their misﬁts to the uplift rate observations over North America and to the U.S. East coast relative sea level data computed.

Each viscosity structure is ﬁrst run over the last glacial cycle in conjunction with the global ICE-

6G_C ice loading history of Peltier et al. (2015). The misﬁts between model predictions and the geological record of RSL evolution of Engelhart and Horton (2012) and the space-geodetic measurements Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 127

Table 4.1: Elastic lithosphere thickness and mantle viscosity values considered in the parameter space exploration, with the values in each layer varied independently. From Roy and Peltier (2017).

of vertical uplift rates of Peltier et al. (2015) are generated using the methodology outlined in the previous section.

Several important inferences follow on the basis of this initial analysis. Concerning the performance of the viscosity structure variations with respect to relative sea level evolution along the U.S. East coast, our results support the Roy and Peltier (2015) observation that models with a thick elastic lithosphere are rejected, especially for the southernmost sites, where they systematically exacerbate the misﬁts identiﬁed in the region when using the ICE-6G_C (VM5a) model. However, these thick-lithosphere models tend to perform better than their thin-lithosphere counterparts with regards to the vertical uplift predictions over North America. This apparent conundrum can be resolved only by an increase in viscosity in the deeper mantle, a region to which the present-day uplift measured near the center of the former Laurentide ice sheet still displays weak, though non-negligible, sensitivity.

Models based upon viscosity structures featuring the further subdivision of the lower mantle into three sections provide a fit that is significantly better than the equivalent models that do not have this feature, as the changes in viscosity can be focused at a greater depth in the lower mantle. This is beneficial, since RSL predictions for the Atlantic coast are not nearly as sensitive to the viscosity at this depth as are vertical uplift rates over the center of the former Laurentide ice sheet.

After an exploration of the errors generated by this scan of parameter space and a careful analysis of the results, the best-ﬁtting mantle viscosity variations that are consistent with these observations favour a viscosity structure that is somewhat similar to the VM6 proﬁle of Roy and Peltier (2015).

Indeed, among all the alternatives considered in this analysis, only a few are able to provide a signiﬁcant improvement over ICE-6G_C (VM5a) and ICE-6G_C (VM6). These models are all found to have a thin elastic lithosphere (60-80 km), and an upper mantle viscosity equal to that of the VM6 viscosity proﬁle of Roy and Peltier (2015). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 128

Figure 4.12: Map of the regions in which modiﬁcations to the ICE-6G_C ice loading history were initially introduced. From Roy and Peltier (2017).

4.3.3 Ice loading history variations and full model optimization

The second step of the analysis involves the introduction of appropriate ice loading history perturbations. The ﬁrst ice loading history variations to be introduced is the modiﬁcation that led to the

ICE-6G_C_BARB model introduced in the earlier sensitivity analysis, with its Laurentide ice sheet component modified from ICE-6G_C by ice melt rate changes in the vicinity of the Younger Dryas event (between 12.8 ka and 11.5 ka). This modification, while helping the model recover a good fit to the extensive coral-based record of sea level evolution at the island of Barbados, also reduces the misfits observed between observations of vertical uplift rates over the center of North America and ICE-6G_C

(VM6) predictions of the same quantity.

Then, mass-conserving redistribution of ice between diﬀerent regions of the Laurentide ice sheet during the initial phase of deglaciation are introduced, smoothed at the beginning and at the end of each time period, which eﬀectively acts as a change in the regional melt rate of the ice sheet. The regions of equal area for which such redistribution scenarios are investigated are shown in Figure 4.12, and include the western (Keewatin) dome of the Laurentide ice sheet, the south-western part of the ice sheet over

Alberta, Nunavut, the southern shore of Hudson Bay, the northern and southern halves of the Great

Lakes region, central Quebec, southern Quebec, and the Gulf of St Lawrence.

The modiﬁcations were allowed to occur in the early part of the deglaciation phase (from 26 ka to Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 129

16 ka), or in its late phase (from 16 ka to 8 ka, or until the ice sheet had retreated from the region, whichever comes ﬁrst). The location of the ice sheet margins are kept identical to those of the ICE-

6G_C model at all times, which are based upon the best estimate isochrone compilation of Dyke et al.

(2002) and Dyke (2004). Each step of the optimization procedure allows mass conserving ice load redistribution to occur between any two of the regions described above in the early or late phase of the deglaciation (for various amplitudes, including a smooth merger of the modification introduced for each region with the surrounding areas, and including ice sheet smoothing inside each region if necessary), and the redistribution resulting in the most significant reduction of the misfit function is chosen from an extensive list of such trials.

Subsequent to this step, slight perturbations to the viscosity structure or to the elastic lithosphere thickness are introduced, and these further modifications to the viscosity structure are kept if any such change results in a significant further reduction of the misfit function. The process is repeated until convergence is achieved (i.e. no further ice loading redistribution or viscosity perturbation results in a significant reduction of the misfit function).

Then, as a final step, the viscosity in the lowermost part of the lower mantle is considered. The viscosity layer found at great depth in the VM6 model is similarly introduced here, and the viscosity of the lowermost mantle is adjusted to fit the constraints on the rotational state of the planet provided by the recent analyses of Roy and Peltier (2011) (namely, on the speed and direction of the true polar wander phenomenon and on the rate of non tidal acceleration of the Earth’s axial rate of rotation). As the fit to the space-geodetic uplift rate constraints has only limited sensitivity to viscosity changes at the top of the lowermost mantle, and none at greater depths, the best-fitting viscosity model is one that shows a gradual increase in mantle viscosity throughout the layer.

To check the final result of this process, the best-fitting ice loading history resulting from the optimization procedure is re-run with the entirety of the viscosity profiles considered in the initial parameter space exploration. Also, small perturbations to the mantle viscosity structure are introduced in each layer (with changes of ±0.1×1021 Pa ·s in the upper mantle and the upper part of the lower mantle, and

±0.5 × 1021 Pa ·s in the deeper layers of the mantle), but no change in our best-fitting viscosity profile provided notable improvements to the quality of the fit, suggesting that our methodology is converging towards a best, locally optimal spherically symmetric viscosity solution (within the range of parameter space considered). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 130

4.3.4 The ICE-7G_NA (VM7) model of the GIA process

The ﬁne-tuning process performed in this analysis results in a ﬁnal model, named ICE-7G_NA (VM7), which will be shown to satisfy all of the major geophysical constraints related to the GIA process over the North American continent. In the model, the ice loading history for all regions outside of the North

American continent is kept fixed to that of the ICE-6G_C model of Peltier et al. (2015). The final viscosity profile, called VM7, is shown in Figure 4.13, where it is seen to be very similar to the VM6 profile of Roy and Peltier (2015) in the upper mantle and upper part of the lower mantle. The profile is characterized by a much more systematic increase in viscosity in the lower mantle, and it has the same viscosity in the upper mantle as VM6, but a slightly thinner elastic lithosphere thickness (75 km). A detailed comparison of the viscosity variations with depth found in the VM5a, VM6 and VM7 viscosity profiles is provided in Table 4.2. The modification in lower mantle viscosity from VM6 to VM7 is consistent with the idea that the tests employed in the Roy and Peltier (2015) study, which focused exclusively on reconciling the model predictions with the geological records of past sea level along the

U.S. East coast, did not provide sufficient resolution in the lower mantle to justify a further refinement of the viscosity structure in this depth range. However, this increase of viscosity with depth in the lower mantle is fully expected on a priori grounds based upon a homologous temperature argument, given the greater rate of increase with depth of the solidus as compared to that of a mantle adiabat. Because the mantle convective circulation operates at very high Rayleigh number (of O(107)), the depth dependence of mantle viscosity is expected to be adiabatic and therefore to depend only on depth throughout the vast majority of the mantle volume (e.g. Peltier and Solheim, 1992; Butler and Peltier, 2000; Shahnas and Peltier, 2010, for specific examples based upon mantle convection model results).

Table 4.2: Radial viscosity structure of the VM5a, VM6 and VM7 viscosity proﬁles down to the core- mantle boundary. From Roy and Peltier (2017).

The ICE-7G_NA model of ice loading history includes slight redistributions of ice from the region Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 131

Figure 4.13: Comparison of the radial variations of the viscosity in depth for the VM7 viscosity proﬁle (blue/green) with the VM5a (black) and VM6 (red) proﬁles. From Roy and Peltier (2017). south of the western (Keewatin) dome to the southern and eastern shores of Hudson Bay and in southern

Quebec, and from the southern part of the Great Lakes region to the Gulf of St. Lawrence region. The thickness of the Laurentide ice sheet in the ICE-7G_NA model is shown in Figure 4.14 for several time steps during the deglaciation phase, while diﬀerences in Laurentide ice sheet thickness between the initial

ICE-6G_C model and the new ICE-7G_NA model are shown in Figure 4.15 for a selection of time steps.

In Figure 4.16, the predicted topography over North America at LGM for both the ICE-7G_NA (VM7) and the ICE-6G_C (VM5a) models are shown, together with the diﬀerence between the two.

The fit provided by the new ICE-7G_NA (VM7) model to the database of relative sea-level histories of Engelhart and Horton (2012) is shown in Figure 4.17. The model provides a better fit to the sea- level index points in the northernmost sites than did ICE-6G_C (VM6), improving one of the minor drawbacks of that model, and even provides a slight improvement over the ICE-6G_C (VM5a) model predictions in this region. For the rest of the U.S. East coast, the ICE-7G_NA (VM7) model provides an equally good fit to the observational data as did ICE-6G_C (VM6).

Figure 4.18 presents the vertical uplift rate predicted by the ICE-7G_NA (VM7) model over North

America in its upper panel, upon which a number of transects have been superimposed. Explicit point- by-point comparisons between model predictions and observations along these transects are performed for the new ICE-7G_NA (VM7) model and its precursors ICE-6G_C (VM5a) and ICE-6G_C (VM6).

These transects are labelled A-A’ to P-P’. The comparisons along each transect are presented in groups Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 132

Figure 4.14: Snapshots of the thickness of the Laurentide Ice Sheet (m) and of the Greenland Ice Sheet in the ICE-7G_NA ice loading history since the Last Glacial Maximum. The state of the ice sheets in each panel is given at the time indicated at the bottom left corner, in ka. From Roy and Peltier (2017). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 133

Figure 4.15: Diﬀerence in ice thickness between the ICE-7G_NA and ICE-6G_C ice loading histories (meters) over the Laurentide Ice Sheet (m) and of the Greenland Ice Sheet for a selection of time steps (given in the corner of each panel, in ka) in the reconstructions since the Last Glacial Maximum. From Roy and Peltier (2017).

Figure 4.16: Predicted topography over North America (meters) with respect to sea level at LGM for the (a) ICE-6G_C and for the (b) ICE-7G_NA ice loading histories, with (c) the diﬀerence between the ICE-7G_NA and ICE-6G_C topography at LGM. From Roy and Peltier (2017). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 134

Figure 4.17: Comparison of the Engelhart and Horton (2012) data set of relative sea-level histories along the U.S. Atlantic coast for the 16 composite regions with the predicted relative sea-level history at those locations for the ICE-6G_C model of ice-loading history combined to the VM5a (green) and VM6 (blue) radial viscosity proﬁles, together with the prediction of the new ICE-7G_NA (VM7) model (black). Green data points represent sea-level index points, whereas blue crosses represent marine-limiting data and orange crosses represent terrestrial-limiting data. From Roy and Peltier (2017). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 135 of four in Figure 4.19, together with a schematic location of each transect on the continent (note that all data points within 200 kilometers of a given transect are considered to be "on" the transect following a projection normal to it). The transects A-A’ to K-K’ are the same as those employed in Peltier et al.

(2015) for their evaluation of the performance of the ICE-6G_C (VM5a) model, while the other eight transects provide new insight on additional features of the uplift ﬁeld over North America: the L-L’ transect, in particular, follows the southern forebulge associated with the former Laurentide ice sheet west to east; the M-M’ transect follows the southern shore of Hudson Bay, through Quebec and into the Gulf of St. Lawrence; the N-N’ and O-O’ transects provide information on the uplift along the

U.S. Atlantic coast and along the St. Lawrence River, respectively, while the P-P’ transect cuts across

Hudson Bay between the former ice domes that existed on either side of it. In general, VM6 predicts signiﬁcantly lower uplift rates than VM5a in regions that were once under signiﬁcant Laurentide ice cover. VM6 also predicts lower subsidence rates in the forebulge (L-L’, N-N’ and O-O’) and fails to reproduce the gradient in observed uplift along the eastern shore of Hudson Bay (P-P’), but performs well in the transition between the uplifting and subsiding regions and for some western transects (I-

I’ and J-J’). However, the vertical uplift rate ﬁeld provided by the new model over North America is much closer to that of the ICE-6G_C (VM5a) model than ICE-6G_C (VM6), in fact performing better than both models along a few transects. The maximum uplift rate value predicted in central Quebec is 12 mm/yr, similar to the ICE-6G_C (VM5a) prediction and still within the constraints provided by the GPS observations of uplift in the region. West of Hudson Bay, the amplitude reached by the uplift rate is slightly lower than that predicted by ICE-6G_C (VM5a) while, in the mid-Atlantic, the predicted subsidence by the new model is lower and much closer to that of the ICE-6G_C (VM6) model, maintaining one of the main advantages of that model over the original model of Peltier et al. (2015).

The new model also performs slightly better in the region surrounding the Gulf of St. Lawrence, as visual inspection of the B-B’ transect will demonstrate.

The fit provided by this model to the observations of present-day vertical uplift rate (considering their observational uncertainty) is presented in Figure 4.20(b), where it is seen that most misfits that existed in comparison to the predictions of the ICE-6G_C (VM6) model (shown in Figure 4.20(a)) are now eliminated. The more sophisticated iterative process that has been employed in this further step of analysis leading to the ICE-7G_NA (VM7) model enables a fit to the uplift rate observations that is at least as good as the fit provided by the parent ICE-6G_C (VM5a) model (previously shown in Figure

4.3(d)). The two areas of misﬁt previously identiﬁed for the ICE-6G_C (VM5a) model in the Gulf of St. Lawrence and in south-western Canada deserve special attention. Using the new ICE-7G_NA

(VM7) model reduces the misﬁts found in the former region, but it does not reduce the signiﬁcant Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 136

Figure 4.18: Background map of predicted present-day rate of vertical motion of the crust for the ICE- 7G_NA (VM7) model of the GIA process following the scale at the bottom of the panel (mm/yr), superimposed by the traverses along which comparisons of the model to the observations of crustal uplift rate are performed in Figure 4.19, and for which the end points are labelled. From Roy and Peltier (2017). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 137

Figure 4.19: Comparison of the predicted uplift rate along the traverses identified and labelled in Figure 4.18 for the new model (green), the ICE-6G_C (VM5a) model (red) and ICE-6G_C (VM6) models (blue) with the observations measured at sites located within 200 km of each traverse, and which are projected onto it. Schematic maps for each subgroup of four traverses show their location on the map. From Roy and Peltier (2017). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 138 misfits observed in the latter. This persistent misfit will be addressed in a subsequent section. Following the methodology of Argus et al. (2010), Argus et al. (2014), and Peltier et al. (2015), Figure 4.20(c) presents the normalized quality of the fit provided by the new ICE-7G_NA (VM7) model with regards to the space-geodetically observed vertical uplift rates, together with the same quantity determined for precursor models. It can be seen that the excellent gain in quality of fit to the North American uplift rates that the ICE-6G_C (VM5a) model provided over the ICE-5G (VM2) model was partially lost when switching to the VM6 viscosity model. Introducing the vertical uplift rate predictions in our methodology

(which led to the VM7 viscosity profile) substantially reduced the misfit compared to VM6. The rest of the gain in quality of fit is provided by the slight ice loading melting rate variations introduced in the passage from ICE-6G_C to ICE-7G_NA.

The ﬁt provided by the ICE-7G_NA (VM7) model to the history of sea level change observed along the U.S. East coast during the late Holocene (since 4ka) as a function of distance from western Hudson

Bay (following Engelhart et al. (2009)) is shown in Figure 4.21, together with the same model predictions for the ICE-6G_C (VM5a) and ICE-6G_C (VM6) models. It was shown in Roy and Peltier (2015) that one of the main strengths of the new VM6 viscosity profile was its ability to reproduce the shape of the forebulge associated with the former Laurentide Ice Sheet much more accurately than was possible with the VM5a viscosity structure when this was employed with the ICE-6G_C loading history. Not only does the new model fit the data from the mid-coast region as well as ICE-6G_C (VM6) did, but it also better captures the location of the maximum subsidence of the forebulge and its amplitude, as well as the behaviour of the tail of the forebulge (in South Carolina), which is a strong indication of the improvement that this further refined model represents.

The ﬁt provided by the ICE-7G_NA (VM7) model to the uplift data based upon water gauges in the

Great Lakes is shown in Figure 4.22, where it is demonstrated that the ICE-7G_NA ice loading history with its slightly modified deglaciation rates compared to ICE-6G_C enable the full model to recover a much better fit to the observational data in the southern region of the Great Lakes than that provided by ICE-6G_C (VM6) or ICE-6G_C (VM5a). The only remaining misfit is in the southern half of Lake

Michigan.

It should be also noted that ICE-7G_NA has less ice than the Laur16 model of Simon et al. (2016) over Manitoba and more ice over the southern shores of Hudson Bay at LGM. While the two models generally seem to be qualitatively similar in other regions at LGM (an entirely expected consequence of the fact that both are variants of ICE-6G_C), a detailed comparison of these models will be described elsewhere.

Moreover, the ﬁt provided by the new ICE-7G_NA (VM7) model to the Roy and Peltier (2011) Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 139

Figure 4.20: Quality-of-fit determination between the JPL space-geodetic measurements of present- day vertical motion rates (mm/yr) and the GIA predictions of those same rates (taking into account observational uncertainties) for the ICE-6G_C (VM5a) (panel (a)) and the new ICE-7G_NA (VM7) (panel (b)) models at the site of each uplift rate measurement; (c) Misfit between GIA model predictions of vertical uplift rates and observations over North America, with the vertical bars representing the normalized quality of the fit, and the values at the top the square root of the reduced chi-square value (also called the Normalized Sample Standard Deviation, or NSSD), for the ICE-5G (VM2) (red), ICE- 6G_C (VM5a) (cyan), ICE-6G_C (VM6) (green), ICE-6G_C (VM7) (purple), and ICE-7G_NA (VM7) (dark blue) models. From Roy and Peltier (2017). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 140

Figure 4.21: Comparison of observed late Holocene relative sea-level rise (mm/yr) for locations along U.S. East coast (dark grey diamonds) (with 2-σ uncertainty ranges in light gray) (Engelhart et al., 2009), with predicted values for the same quantity from the ICE-6G_C (VM5a) (green dots), ICE-6G_C (VM6) (blue dots), and from the new ICE-7G_NA (VM7) (red dots) models. The data points are plotted as a function of distance from the city of Churchill, Manitoba, on the western shore of Hudson Bay (km). From Roy and Peltier (2017).

Figure 4.22: (a) Mainville and Craymer (2005) inferences of vertical uplift rate for sites along the shores of the Great Lakes presented with respect to the reference point of each individual lake outlet (filled circles representing the value in mm/yr); (b) Map of the ICE-7G_NA (VM7) model prediction of vertical uplift rates in the Great Lakes region, together with the Mainville and Craymer (2005) observations of the same quantity adjusted to the modeled uplift rates at each reference outlet point; (c) Difference between the modeled ICE-7G_NA (VM7) vertical uplift rates and the adjusted observations of Mainville and Craymer (2005) (b), accounting for the 2-σ uncertainty in the inferences. From Roy and Peltier (2017). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 141 constraints on true polar wander (speed and direction) and on the change of oblateness of figure of the planet (the change in the J2 zonal harmonic of the gravitational field of the Earth) is provided in Table 4.3.

Table 4.3: Values of Earth rotational observables predicted by ICE-6G_C and (VM5a) and ICE-7G_NA (VM7)

Finally, the relaxation time at the heart of the former Laurentide ice sheet, namely along the southeastern coast of Hudson Bay, provided by the new ICE-7G_NA (VM7) model is provided in Table 4.4, where its excellent ﬁt to the relaxation time inferred from high-quality isolation basin data from the region can be observed.

Table 4.4: Relaxation times determined in Eastern James Bay

4.3.5 The Upper Campbell (Lake Agassiz) strandline tilt

A further data set to which GIA model predictions can be compared, and which was not used in the optimization process leading to the ICE-7G_NA (VM7) model, is the observed present-day tilt of ancient strandlines associated with former proglacial lakes that existed during the deglaciation phase of the Laurentide ice sheet. In particular, a recent attempt at reconstructing the ice thickness history of the western part of the Laurentide ice sheet by Gowan et al. (2016) was explicitly tuned to the amount of observed tilt along strandlines in the region. A central constraint of this optimization was the

Upper Campbell strandline, the longest such feature in the vicinity spanning more than 1000 kilometers from Saskatchewan to the tri-border area between North Dakota, South Dakota and Minnesota. The ice loading history for the western Laurentide ice sheet of Gowan et al. (2016) is characterized by a signiﬁcant reduction of ice volume in the area as compared to that of the ICE-6G_C model (a decrease in ice volume at LGM of about 30% has been suggested to be appropriate), and the authors suggest that the observed tilt along the Upper Campbell strandline of Glacial Lake Agassiz "provides strong support Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 142

Figure 4.23: Comparison of the modeled tilt of the Upper Campbell strandline associated with former Lake Agassiz with the tilt measured at present along the strandline with respect to the point of lowest elevation along its length (Gowan et al., 2016). The black dashed line represents the line of best ﬁt. Results for the ICE-6G_C (VM5a) (blue) and ICE-7G_NA (VM7) (red) models are shown, together with the linear ﬁt best characterizing the modeled values (unconstrained at the origin) and the value of its slope. The inner panel shows the location of the Upper Campbell strandline in central North America. From Roy and Peltier (2017).

for thin ice cover" in the region (Gowan et al., 2016). We revisit this statement here in the context of the new ICE-7G_NA model, which embodies slight modiﬁcations to the ice loading history over the region of interest.

Figure 4.23 compares model predictions of the amount of tilting that has occurred along the Upper

Campbell strandline of former Lake Agassiz since its formation approximately 10,500±300 years BP with present-day elevation diﬀerences observed along the strandline. The methodology used in this analysis follows that of Gowan et al. (2016) to favour inter-model comparisons with their NAICE model, and uses the same conversion of elevation values to orthometric height diﬀerences along the strandline as compared to the point of present-day lowest elevation (Gowan et al., 2016) . In Figure 4.23, modeled values of tilt at locations for which elevation values are available are plotted against the observed values

(thus, observations follow a perfect one-to-one slope in the ﬁgure). Model predictions for the ICE-6G_C Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 143

(VM5a) and ICE-7G_NA (VM7) models are shown in Figure 4.23, with the uncertainty provided by using the range of ages allowed for the age of the strandline. Gowan et al. (2016) observed that the ICE-

6G_C (VM5a) predicted slightly too much uplift in the northern part of the strandline, thus lowering the quality of the ﬁt it provides to the data, but it should be noted that this inference is somewhat dependent on the predicted uplift in the southern part of the strandline and on potential errors in elevation measurements (suggested to be ±5 m by the authors). The ﬁt provided by the ICE-7G_NA

(VM7) model to this observational data set is also provided in Figure 4.23, and shows that the new model entirely corrects the misfit predicted by the ICE-6G_C (VM5a) model. This is due to the slight adjustment of the ice loading history introduced over central Canada that was necessary to recover a good fit to the network of space-geodetic uplift rate measurements over the continent. It is clearly important to understand that our new model was not tuned to fit the strandline tilt observation but rather to fit the space-geodetic observations.

This result is highly significant because it demonstrates the ability of the ICE-7G_NA (VM7) model to fit observations to which it was not tuned. Also, it is important to note that the ICE-7G_NA model is only marginally thinner than the original ICE-6G_C ice loading history over the region over which the NAICE model is defined. This means that the total ice volume found in ICE-7G_NA at LGM over this area of the western Laurentide ice sheet is very close to that of ICE-6G_C, and thus far above that of the NAICE model of Gowan et al. (2016). The fact that a fit to the Upper Campbell strandline tilt observations can be obtained with the "thicker" ICE-7G_NA model contradicts the assertion of Gowan et al. (2016) that the Upper Campbell strandline tilt observations should necessarily support a "thin ice cover" in the region. The slight misfits identified by Gowan et al. (2016) for the ICE-6G_C ice loading history can be corrected by slight variations in ice thickness over only a small fraction of central Canada that corresponds to the northern portion of the strandline.

As the strandline is found in a region that was completely ice-covered at LGM, it is also important to recognize that the quality of the fit provided by a given ice sheet reconstruction to strandline tilt observations is sensitive only to the differential ice thickness over the region, rather than to the thickness of ice removed. Moreover, the preference expressed by Gowan et al. (2016) for thin ice cover over the region is a consequence of their preference for a viscosity structure that is characterized by a very high viscosity in the lower mantle (1.0 · 1022 Pa·s) below 670 km depth, which requires a significant reduction in ice thickness of the Laurentide ice sheet to maintain an acceptable fit to the space-geodetic constraints on present-day vertical uplift rates. In fact a viscosity model of this kind is entirely ruled out by the characteristic exponential relaxation times that are characteristic of every relative sea level curve that has ever been measured from coastlines of the region that was once covered by Laurentide ice (e.g. Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 144

Figure 4.24: Comparison of the relative sea level history predicted at four key sites along the U.S. East coast from the Engelhart and Horton (2012) data set, namely Southern Maine, Southern Massachusetts, Northern South Carolina and Southern South Carolina, between the ICE-7G_NA (VM7) model (black) and the viscosity model used by Gowan et al. (2016), when coupled with the ICE-7G_NA ice loading history (red). Green data points represent sea-level index points, whereas blue crosses represent marine- limiting data and orange crosses represent terrestrial-limiting data. From Roy and Peltier (2017). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 145

Peltier (1998b) and the more recent analysis based upon isolation basin data in Roy and Peltier (2015)).

Furthermore, it should be noted that in the present analysis we have shown that using a viscosity structure with a very high lower mantle viscosity (and a thick elastic lithosphere as also suggested by these authors) has a significant impact on the ability of the model to fit constraints from the forebulge region. Figure 4.24 compares the fit at some sites in the U.S. East coast RSL database of Engelhart and Horton (2012) between VM7 and the viscosity model used by Gowan et al. (2016) when combined to the ICE-7G_NA ice loading history. Significant misfits are introduced in the ability of the model to reproduce the sea level evolution along the coast, especially at the southernmost sites. It should also be noted that, even though Gowan et al. (2016) suggest that supplementary ice might perhaps be required in other parts of the Laurentide ice sheet to maintain an adequate sea level budget over the last glaciation-deglaciation cycle, the misfit to the RSL data in the southernmost parts of the U.S. East coast remains even if a significant thinning is applied to the ICE-7G_NA model. Finally, it should be noted that inspection of the results of their parameter space exploration does not in fact preclude the validity, in the first place, of a viscosity structure with a lower mantle viscosity that is much closer to that of VM7 (see figure 6 of Gowan et al. (2016)).

4.3.6 The ﬁt to the time-dependent gravity measurements of the Gravity Recovery and Climate Experiment

Another important data set in terms of which the quality of a given model of the GIA process may be assessed is that provided by the time-dependent gravity measurements of the GRACE satellites. The

GRACE inference of this field, presented in terms of the time rate of thickness change of an equivalent water layer placed upon the surface of the Earth (cm/yr) is provided in Figure 4.25(a). As before, the Release 5 GRACE product from the U.S. Center for Space Research (CSR) is used without the application of a correlated error filter, but with a 300-km half-width Gaussian filter applied to the raw data. The equivalent field predicted by the new ICE-7G_NA (VM7) model is presented in panel (b) of the same figure. The new model displays the same double "bull’s eye" pattern that GRACE observes in the change in the gravitational field over North America, and predicts a higher maximum amplitude of this field than did ICE-6G_C (VM6). This difference enables the model to provide a much better fit to the time-dependent gravity signal over central Quebec than that provided by ICE-6G_C (VM6), as can be observed in panel (c) where the residual between the new model’s predicted field and the GRACE observations is presented.

Variations in surface hydrology impact the gravitational signal seen by the GRACE satellites, and Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 146

Figure 4.25: (a) Gravity Recovery and Climate Experiment (GRACE) observations of the time dependence of the gravitational field over North America from the Release 5 data from the Center for Space Research (CSR) for the period covering January 2003 to October 2013, on which no correlated filter was applied but a spatial filter of Gaussian half-width of 300 km was applied; (b) ICE-7G_NA (VM7) model predictions of this field; (c) Difference between the raw GRACE field (a) and the ICE-7G_NA (VM7) model predictions (b); (d) Difference between the GRACE field (a), corrected for hydrological effects by the application of the GLDAS correction of Rodell et al. (2004) and the ICE-7G_NA (VM7) model predictions. All results are presented in equivalent water layer thickness change (cm/yr). From Roy and Peltier (2017). the time-dependent gravity signal provided by the mission has been successfully used to determine continental water storage trends for many regions of the world (e.g. Ramillien et al., 2008; Landerer and

Swenson, 2012; Longuevergne et al., 2013). Due to the background signal that the GIA process imprints upon the gravity signal, these eﬀorts have mostly focused on regions where the GIA signal is expected to be small, although some eﬀort has been expended in an attempt to separate the GIA and hydrological components over North America and Scandinavia (Wang et al., 2012; Lambert et al., 2013; Wang et al.,

2015). In spite of these diﬃculties and the limited nature of the surface hydrological observations, hydrological models can be used to estimate their contribution to the gravity signal and may potentially be employed to evaluate the quality of GIA models (for example, see Peltier et al. (2015)). As in the case of ICE-6G_C (VM5a) and ICE-6G_C (VM6), adding the hydrological correction of GLDAS-

NOAH (panel (d)) results in a degradation of the quality of the ﬁt provided by the model, especially in

Quebec and Ontario, but the degree of degradation that occurs is not as marked as was the case for the

ICE-6G_C (VM6) model (see Figure 4.4).

It is useful to investigate in this context the diﬀerent impact that the application of diﬀerent hydrological models can have upon the portion of the GRACE signal that should be explicable as a consequence of the ongoing GIA process. The two most prominent surface hydrology models that may be employed for the purposes of such analyses are the Global Land Data Assimilation System model (GLDAS-NOAH)

(Rodell et al., 2004) and the WaterGAP hydrology model (WGHM) (Döll et al., 2014). The diﬀerent Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 147

Figure 4.26: (a) GLDAS-NOAH model prediction of the time dependence of the gravitational field over North America induced from hydrological effects from Rodell et al. (2004); (b) Residual from the GRACE observation of the time dependence of the gravitational field corrected for hydrological effects using the GLDAS-NOAH model results and the ICE-7G_NA (VM7) prediction of this field; (c) WGHM model prediction of the time dependence of the gravitational field over North America induced from hydrological effects from Döll et al. (2014); (b) Residual from the GRACE observation of the time dependence of the gravitational field corrected for hydrological effects using the WGHM model results and the ICE-7G_NA (VM7) prediction of this field. All results are presented in equivalent water layer thickness change (cm/yr). From Roy and Peltier (2017). Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 148 hydrology corrections for time dependent gravity provided by these models over North America, when the same filtering is applied to them as to the GRACE signal, is shown in Figs 4.26(a) (GLDAS-NOAH) and 4.26(c) (WGHM). Comparing the two panels demonstrates the existence of clear differences between the two models over North America. The difference in model design and input data accounts for the most prominent signal difference found over Greenland. The residual between the GRACE time-dependent gravitational signal corrected for hydrology and the GIA signal predicted by the ICE-7G_NA (VM7) model is presented in Figure 4.26(b) when using GLDAS-NOAH and Figure 4.26(d) when using WGHM.

Given the limited availability of hydrological constraints and the large differences of approach between the two models, it is hard to determine on a priori grounds which one is more plausible, but the fact that there are such significant differences underscores the need to develop a sufficiently robust model of surface hydrological change that can be confidently used to facilitate the interpretation of GRACE measurements.

4.3.7 Model reﬁnement considerations: margin evolution considerations of the south-western edge of the Laurentide Ice Sheet

The analysis of the ability of the ICE-6G_C (VM5a) and ICE-6G_C (VM6) models to match the observations of present-day vertical uplift provided by the extended network of space-geodetic observations over North America (see Figure 4.3) reveals a region of persistent misﬁt in the southern part of the Canadian Prairies, centered in Saskatchewan, which is also found in the comparison to the GRACE time-dependent gravity signal (Figure 4.4), even with the inclusion of a hydrological correction (GLDAS-

NOAH) to the GRACE data (see also Peltier et al. (2015)). These misfits were also mentioned by Snay et al. (2016), who found in their analysis of observed crustal motion that large residuals in vertical velocities remained in the region even if GIA effects from the ICE-6G_C (VM5a) model were taken into account. The persisting misfit is also visible along the L-L’ transect of Figure 4.19. From Figure 4.3, it is clear that this region of misfit remains highly localized within the borders of the Canadian provinces of

Saskatchewan and Manitoba. This misfit is insensitive to large variations in elastic lithosphere thickness and lower mantle viscosity, while it is relatively sensitive to changes in upper mantle viscosity. However, the best-fitting models would provide a highly significant reduction in the quality of the fit provided to

RSL data along the U.S. East coast, and are thus excluded.

The possibility that this regional misfit could be influenced by the position and timing of the retreating ice sheet margin or by the past thickness of the ice sheet covering the region is addressed by revisiting the ICE-6G_C and ICE-7G_NA models in the area. Slight changes in the configuration of the ice sheet Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 149 margin in the southern Prairies were considered within the acceptable range permitted by the limiting data provided by the Dyke (2004) (also Dyke et al. (2002)) reconstruction of ice sheet margin evolution, taking into account the relative paucity of constraints at the onset of deglaciation in this region. Also, changes in the ice thickness covering the region at LGM and during the earlier part of the deglaciation were introduced, with variations of up to 30% in total ice thickness around LGM and during the onset of the deglaciation (from 26 ka to 14 ka). These variations were accompanied by slight changes in ice thickness elsewhere on the ice sheet to keep its total mass similar to that of the ICE-7G_NA model.

It was found that the misﬁts to the GPS observations of vertical uplift were insensitive to even large increases in ice sheet thickness of the region, or to the slight changes in ice margin location that would be consistent with the constraints provided in Dyke (2004) and Dyke et al. (2002).

Misfits between GIA model inferences of vertical uplift and gravity change and both GPS observations of vertical uplift rate and GRACE observations of time-dependent gravity in the Canadian Prairies were identified in previous studies (Wang et al., 2012, 2015). Wang et al. (2012) suggested a methodology that might be employed to separate the signal from hydrology to that of GIA in the GRACE observations of time-dependent gravity, in a way that should be GIA model-independent (making the assumption that a generated uplift field from the GPS observations would, within the first 60 degrees of a spherical harmonic decomposition except the first, show the GIA signal alone), and the subsequent analysis of

Wang et al. (2015) found that a positive water storage trend over the Prairies and the Western Great

Lakes could explain the remaining misﬁt. This possibility is consistent with our determination of a positive residual in the GRACE observations of time-dependent gravity over the region, which cannot be explained by GIA inﬂuence.

4.4 Conclusion

The current analysis has introduced the voluminous quantity of space-geodetic measurements of crustal motion available over North America into the methodology that has led to the recent development of the ICE-6G_C (VM6) model of Roy and Peltier (2015). The resulting parameter space exploration and careful characterization of the misﬁts between the observational data sets and the GIA model predictions has led to the ICE-7G_NA (VM7) model of the GIA process, which is able not only to reproduce the sea level history database of Engelhart and Horton (2012), but also explain the present-day crustal uplift and subsidence over the North American continent as observed by the GPS network. Furthermore, it also explains the crustal tilting over the Great Lakes region that has occurred following deglaciation and provides a good ﬁt to the geological inference of the shape and extent of the Late Holocene forebulge Chapter 4. Full GIA constraints over N. America: the ICE-7G_NA (VM7) model 150 evolution along the U.S. East coast.

Our goal has been to provide a model that enables a further improvement of ﬁt to the totality of the available geophysical observables, while continuing to employ the same basic iterative methodology that has been so successfully employed in the previous steps in global GIA model development. Given the selected region of parameter space that was explored, and the strong focus put on the importance of global constraints, we suggest the ICE-7G_NA (VM7) model to be robust and the preferred solution within the region of parameter space that has been explored. The work performed here also establishes a framework in terms of which an explicit measure of the uncertainty associated with the ICE-7G_NA

(VM7) model can be provided. However, given how interconnected the ice loading history and the mantle viscosity structure are, it is important to consider carefully how this uncertainty should be quantiﬁed.

Work to quantitatively deﬁne such a measure of uncertainty will be presented elsewhere. Finally, given the inherently global nature of the problem we are addressing, the methodology used in this analysis will have to be expanded to other regions that have undergone large changes in ice cover since LGM to fully understand this uncertainty within the global framework in which this analysis must be applied.

As the added constraints were heavily focusing upon the North American component of the model, it remains important to test the global exportability of the adjustments that were made to the viscosity structure and to the Laurentide ice sheet loading history. The next chapter of this work presents the

ﬁrst such test, focusing on the Mediterranean basin, a region where a large number of past relative sea level observations have been collected. Chapter 5

A regional test of the ICE-7G_NA

(VM7) model in the Mediterranean basin

The study of past relative sea level change in the Mediterranean Sea has been an active ﬁeld of research, which has greatly beneﬁted from the collection of a large number of biological, geological and archaeological indicators (e.g. Lambeck et al., 2004; Antonioli et al., 2001, 2003, 2006, 2009; Vacchi et al., 2016, among others). Among the regions from which such knowledge of past relative sea level is available, the

Mediterranean basin is of particular interest, given the large quantity of archaeological evidence that exists along its coasts (owed to the long human presence there), as well as its location south of the former ice sheets that covered Northern Europe and the British Isles at the height of the last glacial cycle. This wealth of past sea level information can be used in the context of the development and testing of the global exportability of the ICE-7G_NA (VM7) model, as constraining information from the Mediterranean basin was not included in the optimization process that led to it.

Using a recent database of relative sea level indicators gathered by Vacchi et al. (2016), it will be shown that the new model performs very well in this new regional comparison test. It provides strong support for the global exportability of the ICE-7G_NA (VM7) model, while identifying isolated regions of remaining misﬁt which would beneﬁt from further study, and providing further clues towards the correct interpretation of certain archaeological evidence.

This chapter follows closely a publication in preparation (Roy, K., and Peltier, W. R. (2017), ’Relative

151 Chapter 5. A regional test of the ICE-7G_NA (VM7) model in the Mediterranean basin152 sea level in the Western Mediterranean basin: A regional test of the ICE-7G_NA (VM7) model’, in preparation).

5.1 Some geological considerations in the Mediterranean basin

The western part of the Mediterranean basin is a tectonically complex region, marked by the existence of a large convergent margin (Nubia-Eurasia), two small oceanic basins (the Tyrrhenian basin under the homonymous regional sea extent, and the Liguro-Provencal basin along its northern coast) and a continental block under the islands of Sardinia and Corsica (Faccenna et al., 2014). There is evidence suggesting the existence of an Adriatic micro-plate, extending in an arc covering the Po Valley, the

Adriatic Sea, the Ionian Sea and the southern part of Sicily, which is thought to move independently from the main Eurasian plate and to be responsible for considerable tectonic activity along its margins

(Devoti et al., 2002; Faccenna et al., 2014). Some parts of the basin margins are undergoing compressional tectonics (Billi et al., 2011), with signiﬁcant tectonic activity exists along the southern Mediterranean and southern Tyrrhenian margins, while contractional activity characterizes the Adriatic region. The eﬀect of long-term tectonic activity on geological records is also compounded by numerous co-seismic events that can result in rapid local sea level change after volcanic activity or earthquakes of large magnitude (e.g.

Pirazzoli et al., 1994; Mastronuzzi and Sansò, 2002; Antonioli et al., 2007). The existence of numerous inter-glacial shorelines enables a comparison of tectonic uplift rates over diﬀerent time scales, and has been used to argue for the long-term tectonic stability of certain regions in the Mediterranean basin when interpreting relative sea level data (e.g. Lambeck et al., 2004; Antonioli et al., 2015). Figure 5.1 shows the tectonic setting of the basin, with known tectonic faults, present-day GPS uplift rates, and the elevation of MIS 5e shorelines recovered from ancient beaches. This complex regional geological setting is very diﬀerent from the geological setting of the eastern coast of the United States, where the relative sea level records upon which the ICE-7G_NA (VM7) model of Roy and Peltier (2017) was calibrated were originating.

5.2 The Vacchi et al. (2016) data set of Holocene RSL change

The Vacchi et al. (2016) data set of relative sea level change is formed of 469 sea-level index points

(capturing past local mean sea level value) and 177 terrestrial-limiting and marine-limiting constraints

(which provide upper and lower limits for the past level of the sea), gathered around the western part of the Mediterranean basin and separated in 22 individual regions (Vacchi et al., 2016). These include Chapter 5. A regional test of the ICE-7G_NA (VM7) model in the Mediterranean basin153

Figure 5.1: Tectonic setting of the Western Mediterranean Sea and precise location of the 22 regional subdivisions of the Vacchi et al. (2016) data set, presenting tectonic faults as red lines (Faccenna et al., 2014), the color-coded elevation of MIS 5e shorelines as squares (Ferranti et al., 2006; Pedoja et al., 2014), and modern GPS vertical motion as color-coded circles (Serpelloni et al., 2013). The 22 regions are shown as numbered black squares. Figure from (Vacchi et al., 2016), reproduced with permission.

a large number of re-interpreted 14C-dated biological material, and a large number of archaeological inferences from historical coastal infrastructure that takes advantage of the long human presence around the Mediterranean Sea. Several other data points were excluded from the analysis in regions potentially affected by large co-seismic or tectonic uplift/subsidence. However, whereas most past sea level indicators are based in other data sets on salt marsh environments (Shennan and Horton, 2002; Engelhart and Horton, 2012), the small tidal range observed in the Mediterranean basin has limited the development of salt-marsh areas (except near large deltas). Most sea level index points are thus based on the interpretation of past coastal lagoonal environments, most notably from the transition from open marine- influenced lagoons to semi-enclosed, brackish environments (Vacchi et al., 2016). Index points are also provided through the study of beachrocks, sedimentary rock formations cemented by carbonate minerals in coastal environments that will often imprison shell fragments (Mauz et al., 2015). Terrestrial limiting data points are provided by biological markers deposited in freshwater marshes or alluvial plains, while marine limiting points will be obtained from open lagoonal settings or in situ marine benthos. Coastal archaeological evidence recovered along the coastlines include former harbour structures, coastal quarries and Roman fish tanks and can provide limiting or index points for the past few thousand years, given Chapter 5. A regional test of the ICE-7G_NA (VM7) model in the Mediterranean basin154 the long occupation of the Mediterranean basin (Auriemma and Solinas, 2009). Errors on each analyzed data point are described in detail in Vacchi et al. (2016), but follow largely the methodology of Shennan and Horton (2002). Compaction-free and basal index points comprise 28% and 14% of the data points, while the rest are intercalated data points (Vacchi et al., 2016), which could potentially be impacted by compaction effects.

The separation of the data set into 22 diﬀerent regions by Vacchi et al. (2016) is based upon the geographical proximity of the data points and the regional tectonic setting, which is derived from the height of the last interglacial (MIS 5e) shorelines and from modern GPS-derived vertical uplift or subsidence rates (Vacchi et al., 2016). These results, and the precise location of the regional subdivisions of the data set, are presented in Figure 5.1. The RSL data for each of the 22 sub-divisions of the Vacchi et al. (2016) data set is presented in Figs. 5.2 and 5.3.

5.3 Results for the ICE-7G_NA (VM7) model in the Mediter-

ranean basin

While the ICE-7G_NA (VM7) model of Roy and Peltier (2017) was able to reconcile the latest space- geodetic observations over the North American continent and the extensive observations of RSL evolution along the U.S. East coast of Engelhart and Horton (2012) with GIA model predictions of the same quantities, the extensive data set of past sea level evolution in the western half of the Mediterranean

Sea of Vacchi et al. (2016) provides a new testing ground for the model and a crucial veriﬁcation of its global exportability.

The predictions of the ICE-7G_NA (VM7) model for the 22 regional subdivisions are provided in

Figs. 5.2 and 5.3, together with the predictions provided by the precursor ICE-6G_C (VM5a) model of

Peltier et al. (2015). Along the north-western coast of the Mediterranean Sea, the new model improves on the performance of the precursor along the southern coast of France (site 3), where the data set is comprised of a large fraction of basal index points in the mid-Holocene that are less likely to be impacted by compaction (Vacchi et al., 2016). The new model does not perform as well in Central Spain (site 1), however, as it misses some of the older index points in the earlier part of the Holocene.

In the Ligurian Sea (sites 4 and 5), the new ICE-7G_NA (VM7) model performs very well. It is able to fit all index points on its western flank, while the lone basal index point on its eastern shore (at 6 ka) is also fit by the model. There are discrepancies with the older intercalated data points on the eastern

Ligurian shore (site 5), but this might be due to compaction. The sea level index points recovered from Chapter 5. A regional test of the ICE-7G_NA (VM7) model in the Mediterranean basin155

Figure 5.2: Comparison of the Vacchi et al. (2016) data set of relative sea-level histories in the Western Mediterranean Sea with the predicted relative sea-level history at those locations for the ICE-6G_C (VM5a) (green) and the ICE-7G_NA (VM7) (black) models of the GIA process, for the ﬁrst 14 locations in the data set (out of 22). Green data points represent sea-level index points, whereas blue crosses represent marine-limiting data and orange crosses represent terrestrial-limiting data. The top map shows the location of each site in the Mediterranean basin. Chapter 5. A regional test of the ICE-7G_NA (VM7) model in the Mediterranean basin156

Corsica (site 6) are entirely based on biological markers and archaeological inferences from Roman ruins on Pianosa Island, located between Elba and Corsica, and only extend to a bit more than 4 ka. Both models perform equally well in the area, given the uncertainty range associated with each relative sea level inference. In Southern Corsica (site 7), the ICE-7G_NA (VM7) model enables a better ﬁt to the terrestrial limiting data points, and it generally ﬁts the expected RSL evolution in the region. In Sardinia

(site 8), the new model provides an excellent ﬁt to the basal index points slightly older than 9 ka, but suggests that the younger intercalated data points might be impacted by compaction. Between 4.6 ka and 4.0 ka, Vacchi et al. (2016) identiﬁed considerable scatter between the two intercalated data points from Cagliari (−6.8 ± 1.0 m) and Is Mistras (−2.2 ± 1.0 m). The new model suggests an intermediate

RSL value around this time, and its suggested RSL evolution is consistent with the younger index points.

In central Latium (site 9), there is signiﬁcant scatter between the various index points used to constrain RSL evolution, which were associated by Vacchi et al. (2016) to the potential inﬂuence of long-term tectonic uplift (also seen on the large elevation of the last interglacial beaches in the area from

Figure 5.1) and sedimentary deposition from the Tiber Delta. In this context, it is hard to evaluate the performance of each GIA model, but it should be noted that the new model tends to overestimate past relative sea level changes in the area. It remains to be seen whether any tectonic correction (consistent with the MIS 5e shoreline elevation) would reduce this misﬁt. In the Gulf of Gaeta (site 10), the new model presents a signiﬁcant performance upgrade over the precursor ICE-6G_C (VM5a) in the late

Holocene. However, for the earlier part of the Holocene, a misfit remains between the terrestrial-limiting data points and the predictions offered by both GIA models. In Salerno Bay (site 11), the new model provides a slight degradation in the quality of the fit it provides to the data.

In Northwestern Sicily (site 12), a region of considerable interest because of its low local tectonic activity (Antonioli et al., 2002), the new model provides an excellent ﬁt to the available RSL data, including a pair of limiting data points, marine and terrestrial, of similar age and elevation. In mid- eastern Sicily (site 13), there is considerable scatter among the late Holocene index points (Vacchi et al.,

2016), and the earlier part of the record is limited to three intercalated and a single marine-limiting data points. In this context, the predictions of both GIA models are consistent with the limiting data points, but the index data points do not enable a clear evaluation of their relative performance, except that they both seem to underestimate relative sea level change since the mid-Holocene (albeit with respect to intercalated data points that might be aﬀected by compaction). In Malta and Southern Sicily (site

14), the new model provides a signiﬁcant improvement over the precursor ICE-6G_C (VM5a) model.

The data from Southern Tunisia (site 15 on Figure 5.3), collected around the Gulf of Gabes, is of considerable interest as it suggests the existence of a late Holocene highstand in this part of the Chapter 5. A regional test of the ICE-7G_NA (VM7) model in the Mediterranean basin157

Figure 5.3: Comparison of the Vacchi et al. (2016) data set of relative sea-level histories in the Western Mediterranean Sea with the predicted relative sea-level history at those locations for the ICE-6G_C (VM5a) (green) and the ICE-7G_NA (VM7) (black) models of the GIA process, for the last 8 locations in the data set (out of 22). Green data points represent sea-level index points, whereas blue crosses represent marine-limiting data and orange crosses represent terrestrial-limiting data. The top map shows the location of each site in the Mediterranean basin. Chapter 5. A regional test of the ICE-7G_NA (VM7) model in the Mediterranean basin158

Mediterranean Sea, which was inferred to provide information about the late post-LGM melting of the

Antarctic ice sheet (Stocchi et al., 2009). Although the ICE-6G_C (VM5a) model supports the existence of a strong highstand in the region, similar to that predicted by Stocchi and Spada (2009), the new ICE-

7G_NA (VM7) model reduces its amplitude. Further study is required to characterize and determine the cause and signiﬁcance of this diﬀerence, especially in the context of the sensitivity analysis performed by Stocchi et al. (2009).

In the Adriatic Sea, where the observational data set is able to provide useful constraints, the new

ICE-7G_NA (VM7) model generally performs better than the precursor ICE-6G_C (VM5a) model.

At the northern end of the sea, close to Venice and its lagoon (site 16), there is signiﬁcant scatter among the mid-Holocene and late Holocene data, most likely caused by compaction (Vacchi et al.,

2016). Nevertheless, the new model provides a marked improvement in the qualitative shape of the

RSL curve, as the precursor ICE-6G_C (VM5a) model suggests the existence of a highstand, a feature that is absent from the observational data. This observation extends to the other sites in the Adriatic

Sea (sites 17 to 20), where the new ICE-7G_NA (VM7) provides a higher rate of late Holocene RSL change, consistent with the observational record for the region. The ﬁt provided by the new model is of particularly high quality in the middle part of the Adriatic Sea, including the Italian mainland and the islands of Vis and Bisevo (sites 19 and 20, respectively). In Istria and northern Dalmatia (North-East

Adriatic, site 17), the new model provides an excellent ﬁt to the high-quality RSL data points sourced from marshes and lagoons.

In the north-western extent of the Adriatic Sea, in a region corresponding to the Romagna and

Commachio coastal plains (site 18), both models are unable to reproduce all characteristics of the inferred sea level history. Although they are able to ﬁt the older basal data points. they are unable to reproduce the rapid rise in relative sea level suggested by the younger data. Compaction eﬀects may be important when studying the younger data points, however Vacchi et al. (2016).

Finally, in the southern extent of the Adriatic Sea, the new model provides a signiﬁcant improvement over its precursor. In both Northern Apulia (site 21) and Southern Apulia (site 22), the new model is able to reproduce the entirety of the index points, although there are some inconsistencies with the younger terrestrial limiting data points along the Salento coast (site 22).

5.4 Evaluation of the ICE-7G_NA (VM7) model performance

The gain in performance provided by the new model over its ICE-6G_C (VM5a) precursor is shown in

Table 5.1. It is evaluated following the χ2-like error function introduced in the previous chapters (see Chapter 5. A regional test of the ICE-7G_NA (VM7) model in the Mediterranean basin159

Roy and Peltier (2015) and Roy and Peltier (2017)). The increase in performance obtained by the new

ICE-7G_NA (VM7) model for most locations is evident. This is a highly signiﬁcant result, given how the evolutionary step between each model was driven by constraints from North America.

In particular, an important gain in the ability of the model to explain the observational constraints is seen at most sites in the Western Mediterranean, especially in the Adriatic Sea and along the Ligurian

Sea. The two GIA models perform similarly well at most locations in Sicily. A performance decrease from the use of the ICE-7G_NA (VM7) model is observed at only two locations out of the 22 locations included in the Vacchi et al. (2016) data set: in Central Spain (site 1) and along the Salerno Bay (site 11). The existence of significant misfits between GIA model predictions of past sea level evolution and geological inferences in Central Spain has been identified as an open issue (Vacchi et al., 2016), and the ICE-7G_NA

(VM7) is unable to reconcile model predictions with observations at this specific site. As the present-day vertical uplift rates, inferred from GPS observations (Serpelloni et al., 2013), are not sufficient to easily ascribe this difference to tectonic effects, further investigation of this situation is warranted. In Salerno

Bay (site 11), the ICE-7G_NA (VM7) displays a significant decrease in performance when compared to the predictions of its ICE-6G_C (VM5a) precursor. However, the southern Italian coast is very complex tectonically (Faccenna et al., 2014), and although several MIS-5e shorelines in the area have been used to infer little to no long-term uplift (Vacchi et al., 2016), our model results suggest that it would be hard to fit both the RSL data for the Salerno Bay area (site 11) and for the Gulf of Gaeta (site 10), which is located only 100 kilometers to the north, without accounting for some local effects (perhaps due to the geographical proximity of the Naples volcanic area).

However, these local issues are heavily compensated by the strength of the model at all other sites, where important gains in model ﬁt are obtained at most locations. The new ICE-7G_NA (VM7) model performs very well in France, around the Ligurian Sea and in Corsica (sites 3 to 7). In Northern Corsica

(site 6), the increase in model performance is entirely due to the ability of the new model to ﬁt the lone terrestrial-limiting data point at about 2 ka BP, while remaining within the uncertainty range of the other index points. In the Gulf of Gabes, in Southern Tunisia (site 15), it is important to note that both models perform equally well, due to the uncertainties associated with the geological record. The ICE-

6G_C (VM5a) and ICE-7G_NA (VM7) models perform equally well in Sicily, with the remaining misﬁt in Mid-Eastern Sicily (site 12) due to the scatter in late Holocene data points, associated by Vacchi et al.

(2016) to the probable uplift trend in the area. In the Adriatic Sea, the new model significantly reduces the misfits observed between the ICE-6G_C (VM5a) model predictions and the geological inferences of past relative sea level. However, while the ICE-7G_NA (VM7) model is able to eliminate almost entirely the misfits for the mid-Adriatic Sea locations (sites 19 to 22), some misfits remain in the northern section Chapter 5. A regional test of the ICE-7G_NA (VM7) model in the Mediterranean basin160 of the Adriatic Sea (sites 16 to 18), probably due to the general subsidence of the area (Antonioli et al.,

2009).

Table 5.1: Comparison of the performance of the ICE-6G_C (VM5a) and ICE-7G_NA (VM7) models in the Western Mediterranean basin in absolute and relative terms. Gains (drops) in the χ2-like error greater than 20% are displayed with a green (red) background, with changes of less than 20% are shown with a yellow background.

5.5 Revisiting the interpretation of Roman ﬁsh-tanks

One of the main archaeological tools used to infer relative sea level during Antiquity is based on the interpretation of Roman-era piscinae (fish tanks), structures built close to sea level that acted as storage pens for live fish, since their architecture and age was thought to be well constrained (e.g. Lambeck et al., 2004; Auriemma and Solinas, 2009; Morhange and Marriner, 2015). However, the interpretation of the mechanism controlling the water inflow and outflow, based on field studies and study of original descriptions in Latin, has recently been the subject of much debate in the community (e.g. Lambeck et al., 2004; Auriemma and Solinas, 2009; Evelpidou et al., 2012; Morhange et al., 2013). The main features from which sea level can be inferred include the sluice gate that enabled the flow of water while keeping the fish inside the tank, water exchange channels and foot-walks that delimited the extent of the basins and enabled workers to move between them (Lambeck et al., 2004; Antonioli et al., 2007; Vacchi et al., 2016). While some authors (Lambeck et al., 2004; Antonioli et al., 2007; Auriemma and Solinas, Chapter 5. A regional test of the ICE-7G_NA (VM7) model in the Mediterranean basin161

Figure 5.4: Contour plot of relative sea level at 2 ka BP across the Mediterranean region, as predicted by (a) the ICE-6G_C (VM5a) model of Peltier et al. (2015), and (b) the ICE-7G_NA (VM7) model of Roy and Peltier (2017). A full contour line represents a relative sea level rise while a dashed contour line represents a relative sea level fall, with each contour line signifying a RSL increment of 20 cm from the bolder zero contour line.

2009) have suggested that the upper RSL limit was 0.2 meters below the lowest foot-walks and that water inflow was tidally controlled, others (Evelpidou et al., 2012; Morhange et al., 2013) have suggested that this interpretation would not have allowed a sufficient water flow and instead that the fixed gates were built in a sub-tidal position.

These diﬀerent interpretations lead to diﬀerent estimates of relative sea level during Roman times.

Along the Italian coast of the Tyrrhenian Sea, Lambeck et al. (2004) place Roman RSL in the range of

-1.37 m to -1.20 m with respect to present, while Evelpidou et al. (2012) place it at -0.58 m to -0.32 m with respect to present. This diﬀerence has strong implications in the determination of late Holocene rates of relative sea level change, indispensable for the correct interpretation of modern tide gauge rates of sea level change in the context of anthropogenic climate change (e.g. Engelhart et al., 2009; Church and White, 2011; Hay et al., 2015; Vacchi et al., 2016).

In this context, it is interesting to determine whether the new ICE-7G_NA (VM7) model can shed some light on the disagreement. Figure 5.4 displays the predicted values of relative sea level around the time at which the ﬁsh tanks were constructed (at 2 ka BP) across the Mediterranean region for the

ICE-6G_C (VM5a) and ICE-7G_NA (VM7) models. Table 5.2 compares the observational inference Chapter 5. A regional test of the ICE-7G_NA (VM7) model in the Mediterranean basin162 of relative sea level at 2 ka BP along the Italian coast of the Tyrrhenian Sea, obtained from the interpretation of Roman ﬁsh-tanks of Lambeck et al. (2004) and Evelpidou et al. (2012), together with the predictions from GIA models of the same quantity at the location where the main ﬁsh tanks are located

(Civitavecchia). The new ICE-7G_NA (VM7) model predicts a sea level rise of 0.54 m since Roman times at that location, a value that is consistent with the upper margin of the Evelpidou et al. (2012) result. The ICE-6G_C (VM5a) model of Peltier et al. (2015) predicts a sea level change since Roman times that cannot be easily reconciled with the two interpretations of Roman ﬁsh tank indicative meaning. As the available geological data points are not able to diﬀerentiate between the two interpretations once altitude errors are accounted for (Vacchi et al., 2016), the predictions of GIA models (which were not tuned to this data source) are an important tool by which the interpretation of archaeological information can be assessed. This preliminary result supports, albeit at the limit of its uncertainty range, the Evelpidou et al. (2012) analysis.

Table 5.2: Comparison of observational inference and GIA model predictions of Roman time RSL along the Tyrrhenian coast

5.6 Perspectives

It has been shown herein that the new ICE-7G_NA (VM7) model of Roy and Peltier (2017) is able to reproduce the available observations of past relative sea level in the western part of the Mediterranean basin gathered by Vacchi et al. (2016). Observations from this region were not used in the optimization process that led to the ICE-7G_NA (VM7) model, which focused on constraints from the North

American continent. Thus, the current analysis is highly significant, as it is the first supplementary regional test of the model. Some misfits remain in specific regions of interest, most notably in Central

Spain and Southwestern Italy. Further work will be required to analyze carefully these misfits in the context of the complex geological setting of the Mediterranean basin. Also, some further assessment of the model predictions in the Gulf of Gabes will be required, in particular to revisit the sensitivity Chapter 5. A regional test of the ICE-7G_NA (VM7) model in the Mediterranean basin163 analyses performed by Stocchi et al. (2009) regarding the far-field influence of the melting history of the

Antarctic ice sheet on the small highstand observed in the geological record. Finally, it will be necessary to extend similar tests of the global exportability of the ICE-7G_NA (VM7) model to other regions, including the eastern half of the Mediterranean basin, for which geological inferences of past relative sea level also exist (among others, Vacchi et al. (2014) for the Aegean Sea). Chapter 6

Conclusion

This work has focused on the study of the Glacial Isostatic Adjustment (GIA) process, and has presented the clear opportunities that high-quality data sets of geophysical observables related to the process can provide to further constrain our models of the phenomenon.

First, the impact of the GIA process on the rotational state of the planet, caused by the large redistribution of mass associated with the Late Pleistocene cycles of glaciation/deglaciation, was analyzed.

An inference of the impact of this cycle on the most recent observational data sets of the J˙2 coeﬃcient and of polar wander was presented, together with evidence of the impact of present-day anthropogenic climate change on the secular trends inferred in those observables (Roy and Peltier, 2011).

Then, the importance of the Atlantic coast of the continental United States for the study of the GIA process and the availability of a newly compiled database of 14C-dated records of sea-level evolution of very high quality for the Holocene period in the region (Engelhart and Horton, 2012) were discussed.

It was shown that misfits identified between observations of Holocene relative sea level change for the southern part of the coast and model inferences of the same quantities (Engelhart et al., 2011) could be eliminated by carefully considered changes in the structure of the mantle viscosity profile used in the

ICE-6G_C (VM5a) model of Peltier et al. (2015). This work, which was the ﬁrst time that constraints from regions of forebulge collapse were actively used in the optimization process leading to a global GIA model, has led to the new VM6 viscosity proﬁle (Roy and Peltier, 2015), which was also tested against data for the West coast of the United States.

The following step led to the expansion of the methodology used in Roy and Peltier (2015) to include the voluminous quantity of space-geodetic measurements of crustal motion available over North America

(Argus et al., 2010; Peltier et al., 2015), and an exploration of the impact of ice sheet loading history

164 Chapter 6. Conclusion 165 variations on the resulting quality of the fit to the combined suite of observables that also included the relative sea level history database of Engelhart and Horton (2012). The resulting parameter space exploration and careful characterization of the misfits between the observational data sets and the GIA model predictions has led to the ICE-7G_NA (VM7) model of the GIA process, which is able not only to reproduce the sea level history database of Engelhart and Horton (2012), but also explain the present-day crustal uplift and subsidence over the North American continent as observed by the GPS network. The model also explains the crustal tilting over the Great Lakes region that has occurred following deglaciation and provides a good fit to the geological inference of the shape and extent of the

Late Holocene forebulge evolution along the U.S. East coast.

The resulting ICE-7G_NA (VM7) model enables a further improvement of ﬁt to the totality of the available geophysical observables over the North American continent, while continuing to employ the same basic iterative methodology that has been so successfully employed in the previous steps in global

GIA model development. It takes advantage of the high quality of the new observational data sets related to the GIA process, particularly in regions of forebulge collapse.

Finally, the performance of the ICE-7G_NA (VM7) model in the Mediterranean basin, a region to which the model was not tuned, was presented. This work, to be published imminently, shows the increased quality of ﬁt provided by the viscosity structure variations that led to the VM7 viscosity proﬁle.

6.1 Perspectives and future work

Several avenues could be explored to further exploit and enhance the advances described in this work.

In particular, our knowledge of the post-LGM evolution of the ice sheets covering Antarctica has beneﬁted from the rapid development of an extended GPS observation network over the region and of an increase in suitable geological inferences of ice sheet retreat (notably bedrock exposure age estimates), which have recently been used in generating new models of Antarctic ice sheet evolution (Argus et al.,

2014; Whitehouse et al., 2012). In light of the slight adjustments that led to the ICE-7G_NA (VM7) model, and furthering the exportability process initiated in the present work, predictions of geophysical observables related to the GIA process over Antarctica will be addressed. However, as most of the sensitivity of the GIA model predictions of geophysical observables over West Antarctica to changes in mantle viscosity lies within the upper mantle (Argus et al., 2014), it is expected that the impact of the switch to the new ICE-7G_NA (VM7) model on the regional GIA predictions over Antarctica will be minor, since the new VM7 viscosity structure is very similar to its precursors VM6 and VM5a in the upper mantle. This work would also enable a revisit of the Stocchi et al. (2009) hypothesis of the Chapter 6. Conclusion 166

Figure 6.1: (a) Inferred GIA vertical displacement rates (mm/yr) inferred over the GPS sites of the Greenland Global Positioning System Network (GNET) by Khan, S. A. et al. (2016), together with estimates of the contribution of each drainage basin of the ice sheet to total eustatic sea level change since LGM. (b) GIA vertical displacement rates inferred from the simple GIA model of Khan, S. A. et al. (2016), that shows the GIA uplit rates over Greenland that would be required to explain the GNET observations. The path of the Iceland hot spot, inferred by the authors to be a source of potential lateral heterogeneity in mantle viscosity that could explain this misﬁt, is shown as a dashed line. From Khan, S. A. et al. (2016). Reprinted with permission from AAAS. impact of Antarctica post-LGM deglaciation on relative sea level evolution in some isolated sections of the Mediterranean Sea.

Similar analyses will be performed to carefully assess the quality of the new GIA model predictions over Fennoscandia (Steﬀen and Wu, 2011) and Greenland (Wake et al., 2016). Although these analyses are also expected to lead only to minor adjustments, if any, to the model, they are nonetheless necessary in order to turn ICE-7G_NA (VM7) into a truly global model. The model could then be used to revisit the present-day mass balance estimate of the Greenland and Antarctic ice sheets and estimates of 21st century sea level rise in the context of anthropogenic climate change.

A further topic of interest is related to Greenland, where space-geodetic measurements from the

Greenland Global Positioning System Network (GNET), maintained by the Danish National Space

Institute and other partners, have revealed the existence of regional misﬁts between observed vertical uplift rates and GIA model predictions of the same quantities (Khan, S. A. et al., 2016). Although these Chapter 6. Conclusion 167 misﬁts have been associated by the authors to potential lateral heterogeneity in the viscosity of the mantle associated with the Iceland hot spot (Khan, S. A. et al., 2016), it would be enlightening to see if the methodology applied in Roy and Peltier (2015) and Roy and Peltier (2017) could introduce suitable ice loading history variations that would explain this regional discrepancy (shown for indicative purposes in Figure 6.1) with a simple 1D model of the viscosity of the mantle. If they cannot be reconciled using the current model structure, an interesting research question would be to address the extent to which the GIA-related predictions and the optimization method that led to the ICE-7G_NA (VM7) model might be impacted by lateral heterogeneity in mantle viscosity. Various techniques, including coupled

Laplace ﬁnite-element methods (Wang and Wu, 2006), have been used in previous studies, and have also relied on further constraints provided by modern global seismic tomography models (such as the one of Ritsema et al. (2011)). As the Greenland Ice Sheet is already showing evidence of accelerated decay (Khan, S. A. et al., 2015a), it is imperative to better understand how it responded to past climate change, in order to better constrain its current response and predict its future behaviour.

Finally, as the ICE-7G_NA (VM7) model is able to explain available GIA-related geophysical observables over North America, it will be interesting to use this model in the study of recent relative sea level change associated with the global warming process, and recorded over the past century or so in a large number of tide gauge measurements that cover the globe. As an accurate inference of modern-day changes in relative sea level from these instruments depends on an appropriate correction for GIA eﬀects

(e.g. Douglas and Peltier, 2002; Church and White, 2011; Hay et al., 2015), and since a large fraction of the longest North American tide gauge records are found along the United States East coast (e.g. Hay et al., 2015), it might be insightful to revisit the GIA-related assumptions of these studies in the context of the new ICE-7G_NA (VM7) model. This work looks especially interesting in light of the extension of the Jet Propulsion Laboratory network of GPS receivers along the U.S. East coast (unpublished work).

As sea level changes are one of the most striking manifestations of past and present climate change, it is crucial to develop our understanding of the processes that inﬂuence its characteristics. Their breadth is very extensive, and includes processes that control the growth and disappearance of large continental ice sheet cover and the feedbacks that relate these components. It also relies on a knowledge of the

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This section contains the copyright agreements for the previously published material in this work.

The following ﬁgures are shown in the current work under a License Agreement between the author,

Keven Roy, and the publisher (in parentheses for each listed ﬁgure), as provided by the Copyright

Clearance Center for the reuse in a thesis. Figure 1.2: (Peltier, 2007) (Elsevier). Figure 1.3: (Cronin,

2010) (Columbia University Press). Figure 1.4: (Peltier, 2007) (Elsevier). Figure 1.5: (Peltier,

2007) (Elsevier). Figure 1.6: (Cronin, 2010) (Columbia University Press). Figure 1.8: (Peltier and

Fairbanks, 2006) (Elsevier). Figure 2.1: (Schuh and Behrend, 2012) (Elsevier). Figure 2.2: (Seitz and Schuh, 2010) (Springer). Figure 2.3: (Gross, 2007) (Elsevier). Figure 2.9(a): (Thomas et al.,

2006) (John Wiley and Sons). Figure 2.9(b): (Rignot et al., 2008) (John Wiley and Sons). Figure

5.1: (Vacchi et al., 2016) (Elsevier). Figure 6.1: (Khan, S. A. et al., 2016) (American Association for the Advancement of Science).

Figure 1.1(a) is from de Charpentier (1841), is in the public domain and was obtained from the Open Internet Archive (Open Library ID - ia:essaisurlesglac02chargoog; Internet Archive ID - essaisurles- glac02chargoog). The permission to use Figure 1.1(b) was obtained from the National Snow and Ice

Data Center, in Boulder (CO), USA. Figure 1.7 is modiﬁed from a ﬁgure provided courtesy of the Canadian Geodetic Survey (Natural Resources Canada), covered under a Creative Commons Attribution

Share-Alike 3.0 License (CC BY-SA 3.0). The rights for Figure 6.1 are also covered under a Creative Commons Attribution NonCommercial 4.0 License (CC BY-NC) linked to the American Association for the Advancement of Science. Figure 2.6 (Ratcliﬀ and Gross, 2010) was reproduced with permission, courtesy of NASA/JPL-Caltech. Figure 2.9(c) (IPCC, 2013) was reproduced with permission from the IPCC Secretariat (World Meteorological Organization).

191 Appendix A. Copyright and contributions 192

The work presented in Chapter 2 contains material from Roy and Peltier (2011) (Roy, K., and Peltier,

W. R. (2011), ’GRACE era secular trends in Earth rotation parameters: A global scale impact of the global warming process?’, Geophysical Research Letters 38(10), L10306) and Figure 2.10 from Peltier et al. (2012) (Peltier, W. R., Drummond, R., and Roy, K. (2012), Comment on "Ocean mass from

GRACE and glacial isostatic adjustment" by D.P. Chambers et al., Journal of Geophysical Research:

Solid Earth, 117, B11403). All necessary permissions were obtained from the publishers. The work in Chapter 3 contains material from Roy and Peltier (2015) (Roy, K., and Peltier, W. R. (2015), ’Glacial isostatic adjustment, relative sea level history and mantle viscosity: reconciling relative sea level model predictions for the U.S. East coast with geological constraints’, Geophysical Journal International 201(2), 1156-1181). All necessary permissions were obtained from the publishers. The work in Chapter 4 is under review at the time of printing (accepted with minor modiﬁcations) under Roy and Peltier (2017) (Roy,

K., and Peltier, W. R. (2017), ’Space-geodetic and water level gauge constraints on continental uplift and tilting over North America: Regional convergence of the ICE-6G_C (VM5a/VM6) models’, Geophysical

Journal International). The work in Chapter 5 will be submitted to Geophysical Research Letters as

(Roy, K., and Peltier, W. R., ’Relative sea level in the Western Mediterranean basin: A regional test of the ICE-7G_NA (VM7) model’).