Determination of Tidal Response of the Moon with Fully Three

Home , Winslow (crater)

Determination of Tidal Response of the Moon with Fully Three-dimensional Elastic and Density Structures Using a Perturbation Method by Chuan Qin B.S., University of Science and Technology of China, 2008

M.S., University of Colorado at Boulder, 2011

A thesis submitted to the

Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirement for the degree of Doctor of Philosophy

Department of Physics

2015

This thesis entitled: Determination of Tidal Response of the Moon with Fully Three-dimensional Elastic and Density Structures Using a Perturbation Method has been approved for the Department of Physics

Shijie Zhong

John Wahr

Michael Ritzwoller

Date

The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii

Qin, Chuan (Ph.D., Physics, Department of Physics) Determination of Tidal Response of the Moon with Fully Three-dimensional Elastic and Density Structures Using a Perturbation Method Thesis directed by Professor Shijie Zhong

The Moon displays a number of hemispherically asymmetric (i.e., harmonic degree-1) features, including those in its topography, crustal thickness, mare volcanism, surface chemical compositions and deep seismicity distribution. These features suggest a long-wavelength (i.e., 3-

D) structure in the Moon’s interiors, which may have resulted from long-term lunar dynamic evolution. In recent years, high precision measurements of the Moon’s gravity field and topography have been made possible by the advancement of space geodetic techniques. New theoretical tools that can use those measurements to constrain the Moon’s 3-D interior structure are needed. In this thesis, I present a new semi-analytical method based on a perturbation theory that calculates the elastic response of a terrestrial planet with 3-D interior structures to time-varying body tides. The 3-D structure is represented by small lateral heterogeneities in the elastic properties and density of the planet. The presence of the 3-D structure excites response modes to tidal force that are added to those from the 1-D reference state. In a spherical harmonic representation, that every eigenstructure in the lateral heterogeneities is characterized by its induced modes and their relative responses holds the theoretical basis for inversion studies. The perturbation method is applied to solving the tidal response of the Moon, in which harmonic degree-1 lateral heterogeneities are assumed predominant. The perturbation solutions have been interpreted and compared with those from a finite element method, thus verifying the correct implementation of the perturbation method. My perturbation method is optimizable for inversions and promises to provide constraints on the Moon’s long-wavelength interior structures, with improved tidal response measurements from the GRAIL mission and future lunar missions.

Dedication

This dissertation is dedicated to my parents.

Acknowledgements

It has been a long journey and I am so pleased that it has come to a happy ending. I would like to acknowledge all those who helped and supported me along the way.

First and foremost, I would like to thank my advisor, Prof. Shijie Zhong, for offering me the opportunity to work in his geodynamics research group. It was my great honor to have worked with him in the past four and half years. Shijie always trusts my theoretical ability and technical skills, and let me work on many interesting geophysics projects. Without his consistent guidance and encouragement, I could not finish my PhD thesis with so many academic achievements. Shijie is a great mentor and has given me lots of good advice on how to actively pursue a successful career, from which I will benefit for my life.

I want to in particular acknowledge my co-advisor and committee member, Prof. John Wahr, who was both a scientific giant and a great mentor. Prof. John Wahr passed away on the next day of my thesis defense. I was in deep sorrow for the loss of him. I was so grateful to John for his trust on my ability and his guidance on the theoretical part of my thesis. It would be my forever regret not being able to present my thesis work in front of him and acknowledge him in person. I personally take my thesis to commemorate him. I want to thank Prof. Michael Ritzwoller, Prof. Michael Calkins and Prof. David Brain for their time to serve as my committee members. Their valuable comments helped me improve my thesis. I thank Prof. Michael Ritzwoller for recognizing my ability in theoretical work. I also thank him for his lecturing on the theory of seismology and the interesting topic of ambient noise tomography.

I would like to thank all my former and current colleagues in the geophysical groups on the 7th floor of Gamow Tower. It was great fun to discuss and work with them. Thank them all for helping me with my research programs and my thesis defense presentation. Next, I gratefully thank my parents and my family for their unconditional love and support. I owe to my parents for not being able to be with them for so many years. I love and miss them so much. I want to thank my girlfriend, Yang Gao. She is the most beautiful sunlight of my life, giving me so much energy to overcome the difficulties. She is so special to me and I cannot imagine my life without her. I love her. I also thank my best friends in the US and China,

Heming Zhen, Hao Tu and Jiwei Liu, for all their encouragements. I acknowledge NSF (US-NSF 1114168) and NASA (NNX11AP59G) for their financial supports on my thesis work. Finally, I thank the University of Colorado at Boulder and the beautiful city of Boulder for providing the best environments and resources. I have already taken Boulder as my second hometown. I will love and care about this place for my whole life.

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Contents

Chapter

1. Introduction ………………………………………………………………………………..1 1.1. Global asymmetries of the Earth’s Moon …………………………………………….2 1.2. Dynamic evolution of the Moon ……………………………………………………...7 1.3. Geodetic measurements from lunar missions ………………………………………...9 1.4. Elastic response to tidal forces………………………………………………………12 1.5. Motivation and purpose of the study………………………………….……………..15 1.6. Organization of the thesis…………………………………….……………………...17

2. Spatial correlation between deep moonquakes and mare basalts and timing of mare basalt eruptions: implications for lunar mantle structure and evolution ………………………..18 2.1. Spatial correlation of mare basalts and deep moonquakes ………………………….18 2.1.1. Data processing and methods………………………………………………...20

2.1.2. Results ………………………………………………………………………..23 2.2. Ages of mare basalt units ……………………………………………………………26 2.2.1. Age model of mare basalt units………………………………………………27

2.2.2. Results ………………………………………………………………………..29 2.3. Conclusion and discussion: a hypothetic process of lunar mantle evolution ……….34

3. A perturbation method and its application: elastic tidal response of a laterally heterogeneous planet……………………………………………………………………..38

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3.1. Introduction ………………………………………………………………………….39 3.2. Governing equations and perturbation theory… ……………………………………41 3.2.1. Governing equations…… ……………………………………………………41 3.2.2. Time-dependent body tides …………………………………………………..42 3.2.3. Perturbation theory…………………………………………………………...43 3.3. Methods... …………………………………………………………………………...46 3.3.1. Spherical harmonic and vector spherical harmonic expansions ……………..47 3.3.2. Equations in matrix form …………………………………………………….48

3.3.3. Spheroidal and toroidal modes……………………………………………….51 3.3.4. Mode coupling ……………………………………………………………….54 3.3.5. Semi-analytical approach and solutions……………………………………...56 3.4. Results……………………………………………………………………………….61 3.4.1. Finite element solutions ……………………………………………………...62 3.4.2. (1, 1) lateral heterogeneity throughout the mantle …………………………...64

3.4.3. (1, 1) lateral heterogeneity in either the bottom or the top half of the mantle…. ……………………………………………………………………………..…72 3.4.4. Other long-wavelength structures throughout the mantle ……………………75 3.5. Conclusion and discussion …………………………………………………………..78

4. Elastic tidal response of a laterally heterogeneous planet: a complete perturbation formulation ……………………………………………………………………………….81 4.1. Governing equations ………………………………………………………………...82 4.1.1. Background state ……………………………………………………………..83

4.1.2. Net force due to density anomaly ………...………………………………….84 4.1.3. Perturbation formulation ……………………………………………………..86 4.2. Solution method ……………………………………………………………………..90 4.3. Results……………………………………………………………………………….94 4.3.1. (1, 1) lateral heterogeneity in lambda ………………………………………..97

4.3.2. Horizontal displacement due to (1, 1) lateral heterogeneities in shear modulus and lambda……...……………………………………………..……………103 4.3.3. (1, 1) lateral heterogeneity in density…………………………………...…..107 4.3.4. Tidal effect of lunar crustal thickness variations and sensitivity kernel……110 4.4. Conclusion and discussion …………………………………………………………117

5. Summary and future plan ……………………………………………………………….120 5.1. Summary ...…………………………………………………………………………120

5.2. Future plan …………………………………………………………………………122

Bibliography...………………………………………………………………………………125

Appendix ……………………………………………………………………………………134

A. Harmonic expansion of mode coupling terms ……………………………………..134

Tables

Table

1.1 Degree 2 and 3 tidal Love number solutions from GRAIL primary mission.……..14 3.1 Lunar 1-D model parameters………………………………………………………61

3.2 Comparisons of h and k between the results from the perturbation method and those from the finite element methods……………...... 70

4.1 Relative differences between the responses from our perturbation method and those

from the finite element methods for both 0 and 0 cases at 10% lateral variabilities………………………………………………………………………..102

Figures

Figure

1.1 Lunar topographic map……....……………………………………………………...3 1.2 Crustal thickness model of the Moon……...………….……..…….………………..3

1.3 Thorium map on the lunar surface (top) and mare basalt distribution and ages of mare basalt units (bottom)…..……….……………………….……………………..4

1.4 Maps of (a) surface topography, (b) thorium concentration, (c) Bouguer gravity anomaly and (d) gravity gradient of the Moon……....…………….………………..6

1.5 GRAIL twin satellites (top) and free-air and Bouguer gravity anomaly maps from GRAIL lunar gravity model GL0420A………………………...…..………………11

1.6 Concept of lunar body tides...………....…….……………………………………..13

2.1 Distribution of the deep moonquake (DMQ) nests and FeO concentration in Mollweide projection centered on the nearside of the Moon (top) and lunar mare basalt distribution from geologic mapping (bottom)………………………………20

2.2 The percent area distribution versus FeO concentraion for the nearside and farside of the Moon……………………………………….………………………………..21

2.3 Statistical testing of spatial correlation between deep moonquake nests and mare basalt deposits….……...………………………….………………………………..24

2.4 Procedure in performing crater size-frequency distribution measurements..….…..28

2.5 Nearside impact basins for determination of ages of mare basalt units using crater size-frequency distribution measurements…………………………………..……..30

2.6 Number of nearside mare basalt units in each of 13 investigated mare basins verse model ages from crater size-frequency distribution measurements………………..31

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2.7 Age distribution of 346 mare basalt units in 13 investigated mare basins…………33

2.8 A schematic representation of the lunar interior structure and state during mare basalt emplacement ~3.9 Ga ago (left) and for the present-day (right)..……....…..36

3.1 Discretization of finite element mesh in CitcomSVE code………………...….…..63

3.2 Diagrams of mode couplings (up to second order in the perturbation) between a spherical harmonic (1, 1) laterally heterogeneous structure in shear modulus and the (2, 0) (top) and (2, 2) (bottom) tidal forces, respectively…...……………...….…..65

3.3 Log-log plots of the absolute values of the relative responses for a Moon with (1, 1) lateral heterogeneities in shear modulus, to (2, 0) and (2, 2) tidal forces………….67

3.4 Comparison of the gravitational responses of the self-coupling modes s2(2, 0) and s2(2, 2) with and without the contribution from toroidal mode t1(2, -1).………….73

3.5 The radial displacement responses of the Moon with (1, 1) shear modulus lateral heterogeneities in different depth ranges to (2, 0) and (2, 2) tidal forces.….……...74

3.6 High-order gravitational responses of the Moon to the entire degree-2 tidal forces for long-wavelength laterally heterogeneous structures of harmonic degrees from 1 to 3…………………………………………………………………………………76

4.1 Flowchart of solution procedure of the perturbation method…………………...…95

4.2 Hierarchies of mode couplings (up to the second order of perturbation) between the (2, 0) and (2, 2) tidal forces, respectively, and the (1, 1) laterally heterogeneities in shear modulus, first Lamé parameter or density…………………….……….…..98

4.3 The relative responses in radial displacement and gravitational potential of a Moon with (1, 1) lateral heterogeneity in , to (2, 0) and (2, 2) tidal forces……………..99

4.4 The relative responses in horizontal displacement of a Moon with (1, 1) lateral heterogeneities in  and , respectively, to (2, 0) and (2, 2) tidal forces………105

4.5 Comparisons of relative responses in gravitational potential, radial displacement and horizontal displacement between lateral heterogeneities in  and………109

4.6 Mapping of the lunar crustal thickness variations onto lateral variations in elastic properties……………………………………………………..…………………112

4.7 Lateral varying elastic structures in two near-surface layers of the Moon from conversion of the lunar crustal thickness variations……………………………113

4.8 Comparisons of gravitational tidal effects due to lunar crustal thickness variations and a presumed (1, 1) structure in the deep mantle……………………………..115

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4.9 Sensitivity kernels of first-order degree-3 gravitational responses to (2, 0) and (2, 2) tidal forces for (1, 1) lateral heterogeneity at different radii in the mantle……….116

Chapter 1

Introduction

The Moon, as the closest planetary object to the Earth, still withholds many of its mysteries. In the past decades starting from Apollo era, continuous space explorations to the Moon have gradually unveiled those mysteries and advanced human understanding of it. Seismic studies following the Apollo missions provide better constraints of the 1-D properties of the Moon’s interior (e.g., Khan et al., 2000; Nakamura, 1983; Weber et al., 2011). Remote sensing measurements help reveal significant variations in the geological structures and chemical compositions on the lunar surface (e.g., Lawrence et al., 1998, 2002; Lucey et al., 2000). Surface topography and gravity field measurements using lunar orbital satellites have also uncovered laterally varying (i.e., 3-D) structures in the Moon’s crust and upper mantle (e.g., Konopliv et al., 1998; Wieczorek et al., 2013; Zuber et al., 1994). However, 3-D structures and properties at greater depth, which contain important clues for the dynamic evolution of the Moon, are not yet understood. Recently, as space geodetic techniques advanced, global coverage high-precision measurements of the Moon’s gravity field and topography became possible (e.g., Smith et al.,

2010a; Zuber et al., 2013). New theoretical tools are needed to make use of those measurements to constrain long-wavelength components of potential 3-D structures in the Moon. These techniques, combined with the theoretical methods, may also apply to investigations of other planetary bodies in the solar system.

In this chapter, I review the observations of the Moon’s global asymmetries and models of lunar dynamic evolution that may lead to the asymmetries. I then introduce new geodetic observations that motivate development of new theoretical methods, which aim to find 3-D long- wavelength structures in a planet’s interior. This chapter ends by summarizing the organization of the thesis.

1.1 Global asymmetries of the Earth’s Moon

Lunar orbital observations discovered a number of globally asymmetric features on the Moon’s surface. Global altimetry mapping of the Moon’s surface shows an obvious hemispherical asymmetry in the lunar topography: the topography of the Moon’s nearside (i.e., the Earth-facing side) is on average several kilometers lower than that of the farside (i.e., the side facing away from the Earth) (e.g., Araki et al., 2009; Zuber et al., 1994). This asymmetry offsets the Moon’s center of figure from its center of mass by ~2 km (Araki et al., 2009; Smith et al., 1997). The latest lunar orbiters, LRO (Lunar Reconnaissance Orbiter) and SELENE, have mapped the lunar topography at unprecedentedly high precision and resolution using onboard laser altimeters (Smith et al., 2010a; Araki et al., 2009). These topographic maps display the surface geological structures of the Moon in great detail, with the topographic lows correlated with major impact basins on the Moon’s nearside and the South Pole-Aitken (SPA) basin and with the topographic highs represented by the lunar highlands on the farside (Fig. 1.1). Lunar topography, together with measurements of the Moon’s gravity field, is used to construct the lunar crustal thickness model. Significant thickness variations are present in the lunar crust (Fig.

1.2). The crust beneath the largest nearside impact basins (diameters from 200 to 2000 km) is less than 10 km thick and some impact basins even have near zero thicknesses, while crust below the farside Feldspathic Highlands Terrane (FHT) exceeds 50 km in thickness (Wieczorek et al., 2013). An over 10 km difference, on average, in crustal thickness between the nearside and the farside is another manifestation of the hemispherical asymmetry of the Moon, as is the surface

Figure 1.1. Lunar topographic map obtained from LALT altimetry data of SELENE (Kaguya) mission (figure credit Araki et al., 2009).

Figure 1.2. Crustal thickness of the Moon from GRAIL gravity and LRO topography (figure credit Wieczorek et al., 2013).

topography. The nearside-farside asymmetry of the Moon is also reflected from the surface chemical compositions. The Lunar Prospector (LP) was dedicated to mapping the compositional distributions on the lunar surface, such as that of iron, titanium, volatiles and radioactive elements (Binder, 1998). Mare basalts, formed by ancient volcanic eruptions, are spectrally defined by deposits of high-iron and high-titanium contents and are mostly distributed on the nearside of the Moon (Fig. 1.3b). While about 30% of the nearside hemisphere is resurfaced by

Figure 1.3. (a) Lunar Prospector surface thorium concentrations of Lawrence et al. (2003). The PKT is outlined in white and is defined by the 4 ppm thorium contour. (b) Mare basalt ages map of the nearside using data from Hiesinger et al. (2003) and the mare basalt map from the U.S. Geological Survey. Left and right are nearside and farside, respectively (figure credit Laneuville et al. (2013)).

basaltic flows, only 1% of the farside is covered by mare basalts (Fig. 1.3b). The cause of the biased distribution of mare basalts is still being debated, but topography and crustal thickness are unlikely to be the sole controlling factors (Parmentier et al., 2002). Although basaltic flows tend to intrude thinner crust and flood into low-elevation basins, the distribution of mare basalts is not necessarily correlated with low topography or thinned crust. In particular, the SPA basin, which holds the lowest elevation and very thin crust, possesses only modest amounts of mare basalts. Recent crater counting analyses of the remote sensing observations have also showed that, whereas most of the mare basalts were emplaced between ~3.9 to 3 Ga, subsequent eruptions as late as ~1.5 Ga were identified in some nearside basins, such as Oceanus Procellarum and Imbrium (Fig. 1.3b) (e.g., Hiesinger et al., 2003; Morota et al., 2011), implying a long-lasting mare volcanism that predominantly occurred on the nearside of the Moon. This suggests that endogenic processes maintain the long-term mare volcanism on one hemisphere and thus control the nearside emplacement of the mare basalts. Measurements by the Lunar Prospector (LP) gamma-ray spectrometer also revealed a vast geological province on the nearside of the Moon that contains much higher concentrations of KREEP (potassium (K), rare earth element (REE) and phosphorous (P)), incompatible and heat-producing elements, such as thorium and uranium, than the rest of the lunar surface (e.g. Lawrence et al., 1998, 2002). This province, called Procellarum KREEP Terrane (PKT), is geographically overlapped with or adjacent to the nearside basaltic regions (Fig. 1.3a). It has been suggested that KREEP may exist within the curst (Jolliff et al., 2000), while lack of KREEP within the crust of major basins outside the PKT region further suggests that KREEP is only localized to the nearside (e.g., Warren, 2001). Moreover, the latest analyses of the lunar gravity field data from the GRAIL (Gravity Recovery and Interior Laboratory) mission discovered a four-sided rifting structure that encompasses the PKT (Fig. 1.4). This implies that the heat- producing elements are also concentrated in the PKT region, such that the PKT may have once be heated more than its surrounding areas and consequently experienced faster cooling and more intense thermal contraction that caused the rifts. The strong spatial correlation between the

Figure 1.4. (a) Topography. (b) Th concentration. (c) Bouguer gravity anomaly. (d) gravity gradient. All maps are simple cylindrical projections centred on the nearside. The circular rim of the proposed Procellarum impact basin (black dashed line), the outline of the maria (white lines), and the extent of the PKT (red line, corresponding to a Th concentration of 3.5 ppm) are shown in (a). (figure credit Andrews-Hanna et al., 2014).

nearside mare basalt deposits and the PKT suggests a dynamic link between them.

1.2 Dynamic evolution of the Moon

There are two major magmatic episodes in the lunar geological history: 1) a global magma ocean and 2) the mare basalt emplacement (Wilhelm, 1977; Nyquist and Shih, 1992;

Shearer et al., 2006; Wieczorek et al., 2006). It has been widely accepted that the Moon originated from a giant impact ~4.5 Ga (Canup and Asphaug, 2001). After the impact, kinetic energy was transferred into a huge amount of thermal energy that caused a global magma ocean on the Moon (e.g., Elkins-Tanton et al., 2002). The magma ocean quickly crystallized due to cooling and differentiated into an anorthositic crust and an underlying olivine-pyroxene mantle

(e.g., Elkins-Tanton et al., 2002). As the magma ocean had mostly cooled, ilmenite-rich cumulates (IC), which contained high levels of incompatible and radioactive elements, accrued from the last liquid residue and would have formed a thin, dense layer (i.e., KREEP) at the base of the crust overlaying less dense mantle materials (Hess and Parmentier, 1995; Shearer and Papike, 1999). Geochemical and petrological evidence supports the derivation of the mare basalts from remelting of the IC (Ringwood, 1976; Shearer et al., 2006). Therefore, the evolution of the IC is crucial for understanding the asymmetric emplacement of the mare basalts (Hess and

Parmentier, 1995; Elkins-Tanton et al., 2002). There are two main classes of hypotheses for the origin of the mare basalts: deep versus shallow, as reviewed in Shearer et al. (2006). Wieczorek and Phillips (2000) proposed a purely conductive thermal model in which KREEP material is located in a 10 km thick spherical cap

with an angular radius of 40° at the base of the crust. The KREEP layer, rich in radioactive elements, heats itself and then melts the underlying mantle materials. At the time of mare basalt emplacement, melting would have reached about 200 km depth. This model may explain the genesis of some of the mare basalts as well as the connection between the one-hemisphere localization of the mare basalts and the PKT. Laneuville et al. (2013), using the similar setups of KREEP as in Wieczorek and Phillips (2000), developed a 3-D convection model in which mare basalt source material was melted due to KREEP heating and was transported to the surface by convective flows purely driven by the KREEP layer. This model, from a more dynamic point of view, partially explains why the mare volcanism preferentially occurred in the PKT region. However, the initial asymmetric placement of the KREEP layer did not necessarily induce a convective flow that led to mare volcanism only in one hemisphere; substantial melting also occurred on the antipodal hemisphere. This model also predicted mare volcanism that started too early and overestimated the melt volume. Models for a deep origin of the mare basalts assume that some portion of the IC sinks into the lunar mantle, possibly down to the top of the core, and is subsequently brought up to a depth of several hundred kilometers through fluid dynamic processes, leading to decompression melting (Hess and Parmentier, 1995; Zhong et al., 2000). Parmentier et al. (2002) also proposed that a single IC downwelling may have initially formed, if the viscosity of the KREEP layer is sufficiently smaller than the underlying mantle. The mixture (MIC) of IC and olivine orthopyroxene, formed during the sinking of IC in an olivine-pyroxene mantle, would have settled deep in the mantle because it is chemically denser (Hess and Parmentier, 1995; Elkins- Tanton et al., 2002; Zhong et al., 2000). The radioactive elements carried by IC will heat the

MIC until it becomes thermally buoyant to ascend and undergo decompression melting to cause mare basalt eruptions. Zhong et al. (2000) formulated this type of model through 2D thermochemical convection modelling with a MIC layer initially above the core. They found that if the core radius was relatively small (<300 km for simple models with homogeneous viscosity), the MIC might rise up to one hemisphere in a single plume, thus self-consistently explaining the

hemispheric asymmetry in mare volcanism and its onset timing. However, the orientation of the single MIC plume from this model was arbitrary, and any mechanism linking the mare basalts with the PKT was not investigated. Also, having focused on the onset of the MIC upwelling, its long-term evolution and dynamic effect on the mare volcanism was not discussed. Thinning of the nearside crust as well as thickening that of the farside may largely be attributed to early large impact events on the nearside, which ejected the nearside crustal materials onto the farside, causing the hemispherical asymmetry. Although the mechanism is not clear, the subsurface temperature anomalies that resulted from the impacts might have caused redistribution of the KREEP beneath the crust to the nearside (Ghods and Arkani-Hamed, 2007; Jutzi and Asphaug, 2011), thus explaining the nearside localization of PKT. The aggregated KREEP materials beneath the nearside crust may facilitate a single downwelling of IC and the subsequent single upwelling of MIC.

1.3 Geodetic measurements from lunar missions

In the past two decades, several lunar orbiters were used or dedicated to measure the topography and/or gravity field of the Moon. As geodetic techniques develop, topography and gravity measurements have been reaching increasingly high precision and resolution. The Clementine mission mapped the lunar topography in 60N to 60S latitude with a ~160 km surface resolution (Zuber et al., 1994). It also measured the gravity only for the nearside of the Moon with relatively poor resolution. Lunar Prospector (LP), designed mainly for mapping the global geochemical compositions on the Moon, was also the first orbiter to map the Moon’s gravity field with nearly global coverage, by measuring the S-band Doppler shift of the tracking signal in low-altitude polar orbit (e.g., Konopliv et al., 1998). Despite the increasing uncertainties in measurement of the gravity field on the farside, the nearside gravity was determined in much higher precision. The nearside gravity map was modeled up to spherical harmonic degree 180, corresponding to a surface resolution of ~30 km (Konopliv et al., 2001). This much improved

lunar gravity model helped identify density structures, such as three new mascons (i.e., mass concentrations) buried under the nearside crust (Konopliv et al., 1998). The operational LRO and recent GRAIL missions have pushed topography and gravity measurements, respectively, to a new limit. The LOLA (Lunar Orbiter Laser Altimeter) instrument onboard LRO produced the most accurate lunar topography map to date, with a global horizontal resolution down to ~100 m and a radial altimetric precision of ~10 m (Smith et al., 2010b). The GRAIL mission was composed of a twin-satellite orbiting system that inherited the same spacecraft-to-spacecraft tracking technique from the GRACE (Gravity Recovery and

Climate Experiment) mission, which was designed for measurement of the Earth’s gravity field (Tapley et al., 2004). The GRAIL twin satellites, tracked by the Earth-based facilities, measured small changes in the ranging rate between them caused by small gravitational variations through inter-satellite communications, thus determining the Moon’s gravity field to unprecedented precision (Fig. 1.5). Operated in a low polar orbit of 55 km on average, the GRAIL primary mission obtained the gravity field with a surface resolution of harmonic degree 420 (Zuber et al., 2013; Fig. 1.5). Lowering the orbital altitude to ~23 km above the surface, the extended mission was able to double the resolution to degree 900, corresponding to a spatial resolution of ~10 km (Lemoine et al., 2014). These measurements led to a number of new understandings of the Moon, including crustal thickness and density variations (Wieczorek et al., 2013) as well as new structures at both large and small scales (Andrews-Hanna et al., 2013, 2014; Melosh et al., 2013).

While joint analysis based on high-precision, high-resolution topography and static gravity data is a powerful method to discover new density structures as well as to infer the strain and stress states in the Moon’s crust and uppermost mantle, its power degrades with increasing depth as the gravitational effect due to deep density anomalies diminishes. Especially for the Moon, of which lateral heterogeneities near the surface are significant, effects caused by density anomalies in the deep interior will be buried in signals from the surface features and thus can hardly be reflected in surface gravity anomalies.

Figure 1.5. (top) GRAIL twin satellites. (bottom) (A) Free-air and (B) Bouguer gravity anomaly maps from GRAIL lunar gravity model GL0420A, to spherical harmonic degree and order 420 (figure credit Zuber et al., 2013).

The GRAIL satellites were also able to capture small time variations in the gravity field that are caused by the Moon’s time-varying body tide force. These time-varying signals, called the gravitational tidal response, can be separated from the static gravity field and contain important information of interior structures. The tidal response is usually determined spectrally in terms of spherical harmonics, by dividing the measured gravitational variation at a harmonic by the tidal forcing at the same harmonic. The GRAIL primary mission, for the first time, provided gravitational tidal solutions for individual degree-2 harmonics and even a preliminary degree-3 solution (Lemoine et al., 2013; Konopliv et al., 2013). Time-varying body tides also cause ground deformation on the Moon. Recently, degree-2 tidal response of the Moon in its surface radial displacement was determined using the LRO measurements, by applying an altemetric crossover strategy to reduce errors in the altimetry measurements (Mazarico et al., 2014).

1.4 Elastic response to tidal forces

The tidal force in a planetary body is a differential effect of the gravitational pull from another body. In the Earth-Moon-Sun system, the tidal force in the Moon is predominantly from the Earth. The Earth-raised gravitational potential in the Moon is given as (Agnew, 2008)

n  GM  r V(r , t )  Pn [cos ( t )] , (1.1) R()() tn0  R t

where M  is the mass of the Earth, r is the radius of an arbitrary point in the Moon, Rt() and

()t are distance and angle illustrated in Fig.1.6, respectively, where t is the time, Pn is the Legendre polynomial of degree n and can be further expanded into spherical harmonics. While the potential term of n  0 provides no force and the n 1 terms provides the orbital force of the Moon, the tidal force starts from terms of n  2 . Tidal forcing terms decay in magnitude rapidly with increasing n , since r/ R ( t ) 1/ 200 near the Moon’s surface. Therefore, the tidal force is

Figure 1.6. Concept of lunar body tides.

dominant at degree 2, whereas terms higher than degree 3 can hardly be detectable through modern techniques. Tidal forcing terms of each degree can be expanded into time-independent and time- dependent components, representing static and time-varying tides, respectively. For the Moon, the time-varying tidal force is caused by its orbital eccentricity and obliquity, and is a first-order modification to the static tide (discussed in Chapter 3). As a note, the Moon’s orbital eccentricity is 0.0549 and the obliquity is 0.0898.

A terrestrial planetary body, such as the Moon, will respond to a time-varying tidal force elastically in a relatively short time scale, experiencing deformation and deformation-induced gravitational change. The tidal response for each harmonic is fully described by three non- dimensional Love numbers (Love, 1911) h , l and k , which represent responses in radial displacement, horizontal displacement and gravitational potential, respectively, normalized by the tidal forcing magnitude. The theoretical framework for computing the Love numbers of a spherically symmetric (i.e. 1-D) planetary body has long been established (Longman, 1962, 1963;

Farrell, 1972). According to the 1-D theory, tidal forcing at a harmonic would only induce response at the same harmonic, and the Love number solutions are only functions of the 1-D profile of the structure (Love numbers for spherical harmonics with the same degree n have equal values). Therefore, Love numbers derived from the measurements can be used to constrain the 1-D structure of the planetary body.

The presence of a 3-D interior structure in a planetary body superposed onto the 1-D reference structure will modify the 1-D tidal response. Numerical studies have found that the tidal force acting on a 3-D planetary body would excite additional harmonics in the response that are different from those for a 1-D body, through the so-called mode coupling mechanism (Zhong et al., 2012). More specifically, a degree-2 tidal force will excite not only a response at the same degree-2 harmonic, as predicted by the 1-D tidal theory, but also other degree-2 as well as non- degree-2 terms in the response (Zhong et al., 2012). It has also been well understood from numerical calculations that a larger tidal force or a more heterogeneous 3-D structure can usually induce a larger response in those harmonics (Zhong et al., 2012). These findings suggest that measurement and analysis of those additional responses, although technically difficult, may provide significant constraints to the 3-D interior structures, especially at very long wavelengths. Recently, two GRAIL groups reported their degree-2 gravitational tidal Love number solutions for the Moon, i.e., those for harmonics (2, 0), (2, 1) and (2, 2) (Table 1.1), based on the

Table 1.1. Degree 2 and 3 tidal Love number solutions from GRAIL Primary Mission.

GSFC (Lemoine et al., 2013) JPL (Konopliv et al., 2013) k k 20 0.024615 ± 0.0000914 20 0.02408 ± 0.00045 k k 21 0.023915 ± 0.0000132 21 0.02414 ± 0.00025 k k 22 0.024852 ± 0.0000167 22 0.02394 ± 0.00028 k k 30 0.00734 ± 0.0015 3 0.0089 ± 0.0021

GRAIL primary mission data (Lemoine et al., 2013; Konopliv et al., 2013). Differences shown among the degree-2 Love numbers in each set, if they exist, can only be attributed to 3-D structures in the Moon, according to the numerical studies. Improvements on the degree-2 as well as degree-3 solutions are expected from the GRAIL extended mission.

1.5 Motivation and purpose of the study

Revealing and constraining laterally varying (3-D) structures of planetary interiors is a big leap forward in understanding both the composition and internal dynamics of a planet. Seismology, by investigating elastic wave propagation in planetary interiors, is undoubtedly the most powerful method for exploring planetary interior structures. In the past two decades, numerous 3D tomographic models of the Earth’s interior structures have been derived from seismological studies (e.g., Dziewonski, 1984; Van der Hilst et al., 1997), demonstrating that long-wavelength lateral heterogeneities exist in the Earth’s mantle. In particular, the Earth’s lower mantle is dominated by two antipodal structures below the central Pacific and Africa, called LLSVPs (Large Low Shear Velocity Provinces), which has revolutionized the understanding of the present-day state and thermochemical evolution of the Earth’s mantle. However, high-resolution 3-D seismic imaging of planetary interiors relies on a global network of seismometers, which are currently inapplicable on any of the planetary bodies other than the

Earth. Although seismological studies of the Moon were enabled by the Apollo missions (e.g., Nakamura et al., 1982), which led to a better constraint on the Moon’s 1-D properties, it has proved challenging to construct a tomographic model of the lunar interior due to limited seismic data.

Before seismometers may be planted onto the Moon in the future, advanced space geodetic techniques may become alternative and cost-effective tools to “image” the lunar interior structures. The method of tidal tomography (e.g., Latychev et al., 2009; Zhong et al., 2012) is one possibility, which is to use high-precision measurements of the elastic response of a planetary body (i.e., changes in gravitational potential and/or surface topography) to its time- varying body tide to constrain the 3-D interior structures, especially at very long wavelengths. If the hemispherical asymmetry (harmonic degree-1) of the Moon’s surface were also largely present in the interiors, it would be reflected in the degree-2 and degree-3 tidal responses. The

LRO and GRAIL observations, although not necessarily reaching that precision level, have offered this promise for future targeted lunar missions. Global asymmetry is not only the feature of the Moon. Space observations over the past decades have also revealed several other planetary bodies in the solar system that have apparent non-spherical long-wavelength features near their surfaces. Mars owns a crustal dichotomy between its northern and southern hemispheres (e.g., Zhong and Zuber, 2001). Mercury has a magnetic field with north-south asymmetry, for which the surface magnetic field at the northern hemisphere is about three times stronger than that at the southern hemisphere (e.g., Anderson et al., 2011). Saturn’s moon Enceladus features an ancient northern hemisphere but surprisingly young geology at the southern hemisphere, where an active geyser system and abnormally high surface temperature are observed at the south pole (e.g., Porco et al., 2006; Spencer et al., 2006). These features, like those on the Moon, also imply long-wavelength origins from the deep interiors. Therefore, a method that may prove effective for lunar exploration is also applicable to other planetary bodies with appropriate data. Implementation of tidal tomography relies highly on an analytical tool that can effectively and efficiently evaluate the effects of a 3-D interior structure on the elastic tidal response for inversion studies. The 3-D interior structure is usually represented by lateral heterogeneities in the elastic properties and density. Whereas the 1-D tidal response theory and the solution method are mature, developing an analytical method for the 3-D case is still challenging. More commonly, though, fully numerical approaches are used to solve for the response of a planetary body with 3-D structures (e.g., Kaufmann and Wu, 2002; Latychev et al., 2005, 2009; Zhong et al., 2003). One example is the finite element code CitcomSVE, which was originally developed to solve the post-glacial rebound problem for a 3-D incompressible viscoelastic Earth’s mantle (Zhong et al., 2003) and was later modified to include compressibility (A et al., 2013). However, grid-based numerical methods are much less efficient and flexible than a analytical method. Also, and more importantly, the accuracy of numerical solutions is limited by the grid resolution. Spherical harmonic analyses of them cannot retrieve

all the small harmonic components, thus limiting the capability of numerical methods in investigating the mode coupling. In this thesis, I develop and present a semi-analytic method based on the Love number formalism that solves the elastic tidal response of a planetary body with lateral heterogeneities in its elastic and density structures. I benchmark this method to validate its effectiveness. This method is used to predict the tidal response of the Moon, and the meantime, to understand the mechanism of mode coupling.

1.6 Organization of the thesis

This thesis is organized as follows. In Chapter 2, I first introduce the nearside-farside asymmetry in the distribution of the Moon’s deep seismicity, i.e., the deep moonquakes (DMQs). I establish a spatial correlation between the mare basalt deposits and the epicenters of the DMQs, based on robust statistical analyses. I then investigate the timing of mare basalt eruptions by using the ages of the nearside mare basalt units determined from latest remote sensing observations. Based on these two observations, I propose a hypothesis of dynamic process that links the Moon’s mare volcanism, the PKT and the occurrence of the DMQs, as well as testable predictions for the present-day state of the deep lunar mantle. In Chapters 3 and 4, I formulate the analytical method based on perturbation theory and develop an easy-to-apply solution method to calculate the elastic tidal response of a planetary body with lateral heterogeneities in its elastic and density structures. I apply the method to the Moon, in which long wavelength (primarily harmonic degree-1) structures are assumed to exist in the mantle. I compare the tidal response solutions with those from a finite element code, verifying the formulation and implementation of the perturbation method. Chapter 5 provides a summary of the thesis, as well as some plans to extend the perturbation method for future applications.

Chapter 2

Spatial correlation between deep moonquakes and mare basalts and timing of mare basalt eruptions: implications for lunar mantle structure and evolution1

The genesis of mare basalts and deep moonquakes are important events that have major implications for understanding the thermal evolution and interior dynamics of the Moon. The eruption of mare basalts predominantly from ~3.9 to 3 Ga ago represents one of the most important events in lunar geological history. Deep moonquakes recorded by the Apollo Seismic Network show the dynamic nature of the present-day lunar mantle.

In the first part of this chapter, I establish the spatial correlation of the presence of the mare basalts, using FeO concentration as a proxy, with the epicenters of well-located deep moonquake (DMQ) clusters, based on statistical analyses. In the second part, using crater counting measurement results, the ages of different mare basalt units are analyzed to show new phases of mare volcanism. These two observations, combined, suggest a dynamic process that governs the long-term evolution and present-day state of the lunar mantle.

2.1 Spatial correlation of mare basalts and deep moonquakes

1 The first part of this chapter was published in “Qin, C., Muirhead, A. C. & Zhong, S., 2012. Correlation of deep moonquakes and mare basalts: implications for lunar mantle structure and evolution, Icarus, 220, 100-105”. The second part of this chapter was presented in “Qin, C. & Zhong, S., 2014 December. Oscillatory thermochemical convection as a cause for the episodic mare basalt volcanism in the PKT region of the Moon. Poster session presented at the meeting of American Geophysical Union, San Francisco, CA”.

Mantle seismicity reflects the deformation and stress states and thermochemical structures within planetary interiors. The recognition of deep seismicity in the Earth’s mantle as subducting lithosphere plays an essential role in establishing the plate tectonics theory (e.g., Isacks and Molnar, 1969) and provides an important constraint on the Earth’s mantle dynamics. The Moon is the only other planetary body with known deep seismicity, known as deep moonquakes (DMQs), thanks to Apollo missions. DMQs are low magnitude events (generally less than magnitude 3) that occur predominantly at depths of 700-1200 km within about 300 nests and largely on the nearside of the Moon (Frohlich and Nakamura, 2009; Lammlein, 1977;

Nakamura, 2003, 2005; Toksoz et al., 1977) (Fig. 2.1). The DMQs exhibit periodicities that suggest an important role of tidal forces in the Earth-Moon-Sun system in causing them (Lammlein, 1977; Weber et al., 2009). However, it remains an open question as to the exact role of the tidal stresses (Frohlich and Nakamura, 2009; Weber et al., 2009); tidal stresses may only contribute 0.1 MPa (Minshull and Goulty, 1988). It has been suggested that tectonic stress and mechanical heterogeneities associated with water and volatiles play important roles in DMQs (Frohlich and Nakamura, 2009; Toksoz et al., 1977). This is important, considering recent studies that suggest that the Moon may contain significant amounts of water (Pieters et al., 2009), particularly in the source regions of mare basalts (Saal et al., 2008). It has long been recognized that DMQs occur mostly in a broad zone trending NE-SW below mare basalts (Fig. 2.1A) (Lammlein, 1977; Minshull and Goulty, 1988), suggesting a possible correlation of DMQs with the mare basalts (Lammlein, 1977). Understanding the cause of the DMQs will help illuminate the current state of the deep mantle, and determining the connection between the DMQs and mare basalts will further constrain the geological history of the Moon.

The goal of this study is to test the hypothesis of correlation between the DMQs and the mare basalts that was proposed based on visual inspection by Lammlein (1977), by quantifying the correlation using the most recent observations and more robust statistical methods.

Figure 2.1. (A) Distribution of the deep moonquake (DMQ) nests and FeO weight percents centered on the nearside. White contours outline 13wt% FeO and triangles represent locations of the DMQ nests. (B) Adapted from Wilhelms (1977) and Wieczorek et al. (2001), this map shows mare basalt distribution from geologic mapping. Visually correlating these two maps suggests that regions with FeO concentrations greater than 13wt% represent the mare basalts most accurately.

2.1.1 Data processing and methods

I used FeO weight percent (wt%) as a proxy for the mare basalts deposits (Head, 1976; Lawrence et al., 2002). The FeO wt% data that I used in my analyses are from Lunar Prospector’s gamma-ray and neutron spectrometers and have one-degree by one-degree

resolution (Fig. 2.1A). The data were previously analyzed and processed, and were described in Lawrence et al. (2002). I used the 106 DMQ nests in Nakamura (2005) for my analysis, but discarded the nests with unconstrained depths or with errors in longitude or latitude greater than 10 degrees. This leaves 52 well-located nests, 51 of which are on the nearside (Fig. 2.1A). In order to determine what FeO wt% is representative of the mare basalts, I analyzed the surface area covered by different weight percents (Fig. 2.2) as well as visually comparing maps

Figure 2.2. The percent area distribution versus FeO concentration in weight percents for the nearside (A) and farside (B) of the Moon. The dot-dash and dashed lines show the cumulative percent area and percent area distributions, respectively.

of mare basalt deposits from geologic mapping (Wilhelms, 1977; Wieczorek et al., 2001) (Fig. 2.1B). Fig. 2.2 shows the distribution of FeO deposits by plotting the percent surface area of FeO for a given concentration bin and the percent surface area of FeO equal to or greater than a given concentration. I determined that FeO weight percent of 13wt% and greater is the most representative of the mare basalts (Fig. 2.1), which is consistent with ~30% coverage of mare basalts on the nearside (Nyquist and Shih, 1992) (Fig. 2.2). To quantify the spatial correlation of the DMQs and the mare basalts (FeO), I calculated the arc distance from each DMQ epicenter to the closest deposit of FeO with concentration greater than 13wt%. When computing the arc distance, I used the latitude and longitude coordinates of the DMQ epicenters and FeO distributions and ignored the depth of the DMQs. I then defined the correlation as the percentage ratio of the number of DMQ clusters within either 1° or 5° from the FeO deposits >13wt% to the total number of clusters considered. In order to prove that my method is sound, I ran a calculation in which I saturated the nearside with 10,000 random points and found that about 30% of the random points were within mare basalt deposits (i.e., >13wt% of FeO), which was consistent with the values from Fig. 2.2.I then took into account the uncertainties in DMQ locations that may differ in longitudinal and latitudinal directions as documented by Nakamura (2005). I generated 1,000 sets of 52 random points in Gaussian distribution around each corresponding DMQ epicenter, by using the errors in longitude and latitude (Nakamura, 2005) as the standard deviations for the Gaussian distribution in corresponding directions. I then calculated the percentage correlation for each set of the 52 random DMQ nests, in order to show that the percentage correlation is robust against the DMQ location errors.

Contrary to the hypothesis of the correlation of the DMQs with mare basalts is a hypothesis that DMQs are randomly distributed with respect to mare basalts. To test the random distribution hypothesis, I generated 10,000 sets of 52 random DMQ events on a spherical cap on the nearside, centered at the sub-earth point of the Moon. The angular radius of the spherical cap is the largest arc distance between nearside DMQ epicenters and the sub-earth point, and is 66°. I

then calculated the percentage correlation between each random set of DMQs and the mare basalt deposits using 5-degree distance criteria. I compared the observed percentage correlation with the correlations from the random DMQs and made the randomness test.

2.1.2 Results

I now quantify the correlation between the DMQs and the mare basalts. Fig. 2.3A shows the frequency of the DMQs occurring within five-degree bins away from FeO deposits for FeO concentrations greater than 13wt%. My results show that out of 52 DMQ, 42 occur within 5° and 33 within 1° away from mare basalt deposits, which corresponds to the percentage correlation between the DMQs and the mare basalts of 81% and 63% for distances less than 5° and 1°, respectively. To test the robustness of the result, I generated 1,000 sets of random points in Gaussian distribution around each of the 52 epicenters with errors in longitudinal and latitudinal directions as the standard deviations and calculated the correlation as the ratio of the number of DMQ clusters within 5° away from the mare basalts deposits to 52 for each set. Fig. 2.3B shows the frequency of the correlations for these 1,000 sets of random DMQ events. The mean value of the percentage correlations is about 80%, while the standard deviation is about 3%. Assuming a Gaussian distribution for the percentage correlation, the probability that the correlation of one random set of DMQs relative to the mare basalts is larger than 70% was found to be over 99%.

This indicates that the uncertainties in DMQ locations do not affect the results of high correlation percentage significantly. I then calculated the distances between 52 randomly generated DMQ clusters in the spherical cap of 66° radius and the mare basalts deposits. Fig. 2.3C shows the frequency-distance distribution of one set of 52 random events. My calculations show that out of the 52 random events, 32 (or 62%) occur within 5° and 19 (or 37%) within 1° from the mare basalt deposits (Fig. 2.3C), which are much lower than that for the observed DMQs (Fig. 2.3A). The distribution of the 52 random events may not be uniform, and this may introduce statistical fluctuations. To

Figure 2.3. (A) The distribution of DMQ frequency versus the minimum arc distance from the FeO deposits greater than 13wt% in 5° bins for 52 well-located DMQ nests. (B) The distribution of the frequency of the percentage correlations for 1,000 sets of 52 random DMQ events in Gaussian distribution around each epicenter of the observed DMQs. The longitudinal and latitudinal errors of each epicenter from Nakamura (2005) are used as the standard deviations for Gaussian distribution in corresponding directions. The average, minimum and maximum values and the standard deviation of the percentage correlations are given in the inset. (C) The distribution of frequency versus 5° arc distance bins for one set of 52 random DMQ events generated on the spherical cap centered on the nearside, with angular radius 66°. (D) The distribution of the frequency of the percentage correlations for 10,000 sets of 52 random DMQ events generated as in (C). The solid and dashed curves in (D) are for the binomial distribution from equation (2) and Gaussian distribution with the average percentage correlation and the standard deviation given in the inset, which also shows minimum and maximum percentage correlations.

examine this effect, I ran 10,000 sets of calculations with 52 random events for each set and with different initial random seeds. For each random set, I determined the percentage of the random points occurring within 1° and 5° from FeO deposits greater than 13wt%. The frequency distribution of these correlations for distance less than 5° for 10,000 runs is shown in Fig. 2.3D. The average percentage correlation is 65% and the standard deviation is 6.6% (note that Fig. 2.3C is for one set of such calculations). Out of the 10,000 sets of 52 random DMQ events, only a small number of them produce the percentage correlations that are greater than 80%, suggesting that the observed 80% correlation is a small probability event. I may ask what the probability is for the observed 80% or greater correlation to happen if the DMQs are randomly distributed relative to mare basalts.

I now calculate this probability using two different approaches. First, since the percentage correlation shows a normal or Gaussian distribution with mean of xavg  65% and standard deviation of   6.6% (Fig. 2.3D), >80% correlation is outside of 2.4 with a probability given by

0.8 1 (xx )22 /(2 ) p1  eavg dx  0.012 . (2.1) 0 2 This small probability suggests that the hypothesis of a random distribution of DMQ relative to mare basalts can be rejected on a statistic ground. In the second approach, considering that the probability of a random event falls within 5° from the mare basalts (i.e., >13wt% of FeO) is q and that the random events follow a binomial distribution, the probability for n out of 52 events to fall within 5° from mare basalts is given by

52 nn52 pn  q(1 q ) , (2.2) n where q is the ratio of the area within 5° of the mare basalts overlapped with the spherical cap to the area of the spherical cap, and is determined numerically to be 0.655. Eq. (2.2) (i.e., the solid curve in Fig. 2.3D is 10000pn) agrees well with calculations from 10,000 sets of random cases (i.e., the bars in Fig. 2.3D). Also shown in Fig. 2.3D is an ideal Gaussian distribution (the dashed

curve) with xavg  65% and   6.6% . This demonstrates that the Gaussian distribution in Eq.

(2.1) and the binominal distribution in Eq. (2.2) describe accurately the numerical results. The probability to have >80% of the DMQs (i.e., more than 42 out of 52 events) to fall within 5° from the mare basalts is

52 pp n 0.009 , (2.3) n42 which is similar to that from the first approach in Eq. (2.1). With a relaxed selection criterion to include DMQs with epicenter errors up to 20°, the number of DMQs in my analysis is increased from 52 to 64. Applying the same analysis to 64

DMQs, I found that the percentage correlations between the DMQs and the mare basalts deposits are 75% and 59%, for 5° and 1° distance criteria, respectively, which are similar to those from analysis of 52 DMQs. Also, similar to preceding analysis to test the randomness of the DMQ distribution, I generated 10,000 sets of 64 random points in the spherical cap with arc radius of 72° (note that the larger cap is caused by the increased population of DMQs that spread out more, relative to 52 DMQs) and calculated the percentage correlations for each set. The correlations are also in Gaussian-like distribution with mean value 60% and standard deviation 6.2%. The probability that more than 75% of the 64 DMQs occur within 5° from the mare basalts deposits is ~0.01, thus rejecting the randomness of the DMQs relative to mare basalts in this scenario. Finally, if all the 106 DMQ clusters from Nakamura (2005) are considered, without constraints on depths and errors in longitude and latitude, I found that correlations between the DMQs and FeO deposits >13wt% were 68% and 50%, using 5° and 1° distance criteria, respectively, which are also significantly higher than those expected from random distributions of DMQs. My results therefore indicate that the DMQs are not randomly distributed relative to mare basalts and that the DMQs are correlated to mare basalts, supporting the proposal by Lammlein (1977).

2.2 Ages of mare basalt units

Determination of the timing of mare basalt eruptions is important in understanding the thermal evolution of the Moon. Through radiometric dating of the lunar samples from Apollo and Luna return missions, mare volcanism was estimated to be active from about ~3.9 to 3.1 Ga (e.g., Head, 1976; Nyquist and Shih, 1992). But since the basalt samples are limited to vicinity of the landing sites, majority of the lunar surface was not sampled. Thanks to remote sensing techniques, approaches such as crater degradation (e.g., Boyce, 1976), stratigraphic relationship (e.g., Wagner et al., 2002, 2010) and crater size-frequency distribution (e.g., Hiesinger et al., 2000; Neukum et al., 1975) can be used to estimate relative or absolute ages of mare basalts for a much larger area coverage, based on early image data. Prior analyses suggested that lunar mare volcanism may have extended far beyond 3.1 Ga, but the ages and spatial distribution of the latest eruptions were poorly constrained (Morota et al., 2011). Recent studies determined ages of the nearside mare basalt deposits using crater size- frequency distribution method, based on high-resolution Lunar Orbiter and SELENE images (Hiesinger et al., 2000, 2002, 2003, 2010; Morota et al., 2009, 2011). The studies covered all the major mare basins on the nearside, thus providing new and statistically more robust constraints on the timing of mare basalt eruptions.

2.2.1 Age model of mare basalt units

To perform crater size-frequency distribution measurements, mare basalt deposit units need to be defined in order to count craters in them. Galileo and Clementine mission data provided multispectral ratio images that were used to identify compositional variations between different mare basalt deposits (Hiesinger et al., 2000, 2003, 2010). Three channels of spectral ratios, which are sensitive to different composite contents, were employed to define spectrally homogeneous units. In this way, a single mare basin could be divided into a number of mare basalt units (Fig. 2.4a and b). These mare units were then mapped onto Lunar Orbiter (Hiesinger et al., 2000, 2003, 2010) or SELENE (Morota et al., 2009, 2011) images, which are of much

Figure 2.4. Procedure in performing crater size-frequency distribution measurements. (a) Terrain Camera mosaic image of the Oceanus Procellarum and Mare Imbrium regions. (b) Map of spectrally homogeneous mare basalt units defined by Hiesinger et al. (2000, 2003). (c) Sketch map of mare basalt units and count area (pink) in which crater size-frequency measurements were performed. (d) Map of model ages of mare basalts determined for central PKT region. (figure credit Morota et al., 2011)

higher quality to perform crater counting. In each mare unit, crater counting was performed only in a count area and the age determined for this area was used to represent the age for the entire mare unit (Fig. 2.4c and d).

Accurate measurements of the size of the count area and the crater diameters within the area are crucial for creating crater size-frequency distributions. For each unit, crater diameters D (usually in a discrete range between 1 and 10 km) were measured and cumulative densities (km-2) of craters of size equal to or larger than D, i.e. N(D), were determined, forming the observed size- frequency distribution. This distribution was fit against the empirical lunar production function (Morota et al., 2011)

11 n log10N ( D ) a 0 an (log 10 D ) , (2.4) n1 where a1 to a11 are coefficients given by Neukum (1983), to obtain a0 , which is also equal to N(1) . The absolute age of this unit can thus be determined by solving the cratering chronology function (e.g., Neukum, 1983)

N(1) A exp( Bt )  1  Ct , (2.5) where t is the age in Gyr, A, B and C are constant that were calibrated with radiometric ages of the lunar samples.

2.2.2 Results

I collected 346 mare units (including units that were resurfaced by basaltic flows) from 13 major mare basins on the nearside (Fig. 2.5). While all the 346 units were dated based on Lunar Orbiter images using crater size-frequency distribution measurements (Hiesinger et al., 2000, 2003, 2006, 2010), ages of 49 units distributed near PKT have been updated from SELENE images, including 36 in Oceanus Procellarum, 10 in Mare Imbrium, 2 in Mare

Insularum and 1 in Mare Nubrium (Morota et al., 2011). The ages of mare units in a single mare basin were plotted in individual histograms (Fig. 2.6). It was found that mare volcanism occurred for at least one episode in all of the mare basins: starting ~4Ga, reaching maxima ~3.6 Ga and then halting ~2.8 Ga (Fig. 2.6). 3 out of 13 mare

Figure 2.5. Nearside impact basins (yellow stars and blue circles) for determination of ages of mare basalt units using crater size-frequency distribution measurements. Yellow stars mark the mare basins with occurrence of late-stage mare volcanism.

basins that are located in the central region of the PKT, i.e. Oceanus Procellarum, Mare Imbrium and Mare Insularum, were found to have experienced secondary eruptions after 2.0 Ga (Fig. 2.6). Ages of 346 units were then stacked into one histogram to represent the age distribution of the nearside mare volcanism (Fig. 2.7A). The histogram shows three primary features (Fig. 2.7A): 1) the nearside mare basalts erupted predominantly from ~4.0 to 2.8 Ga, as suggested from the radiometric ages of lunar samples and previous remote sensing analyses. 2) The central PKT region, after its volcanic maxima, also experienced secondary eruptions of reduced intensities to as latest as ~1.5 Ga. 3) The volcanism in the PKT displays an episodic behavior with multiple eruptions in a ~2.5 Gyr duration. To provide a more robust analysis, two more plots were made: 1) surface areas of the units were summed into 0.1 Gyr time bins, indicating the total flooded area at different ages

Figure 2.6. Number of nearside mare basalt units in each of 13 investigated mare basins (see Fig. 2. 4) verse model ages from crater size-frequency distribution measurements (Hiesinger et al., 2000, 2003, 2006, 2010; Morota et al., 2011).

Figure 2.7. Age distribution of 346 mare basalt units in 13 investigated mare basins. (A) Total number of nearside mare basalt units, (B) total surface area of mare basalt units, and (C) total surface area of mare basalt units distributed in ±0.5 Gyr time slot verse model ages for every 0.1 Gyr time bin.

(Fig. 2.7B). 2) A smoothed surface area distributions verse ages (Fig. 2.7C). Surface area of each unit was re-distributed into 0.1 Gyr time bins for ±0.5 Gyr duration, by following a two-sided normal distribution with the standard deviation on each side to be the uncertainty in the age model for each unit. Fig. 2.7B shows very similar trend as Fig. 2.7A, with even a more obvious episodic pattern. Fig. 2.7C, although smoothed out, still displays clearly two phases of the mare volcanism in central PKT. The PKT, which is rich in heat-producing elements, may have slowed down cooling of the underlying mantle and maintained mare volcanism for a long time within its region.

2.3 Conclusion and discussion: a hypothetic process of lunar mantle evolution

I have correlated the presence of the mare basalts, using FeO concentration as a proxy, with the epicenters of 52 well-determined DMQs. I find FeO concentrations of 13wt% or higher to be representative of the mare basalt deposits which account for 30% of the surface area of the nearside. My analysis shows that over 63% of the DMQs occur within 1° from the mare basalt deposits, while over 80% of the DMQs are within 5° from the mare basalt deposits. My analysis also shows that for the same amount of randomly distributed DMQs within a spherical cap on the nearside that encompasses all the nearside DMQs, the probability of over 80% of the DMQs occurring within 5° from the mare basalt deposits is about 0.01, thus rejecting a random distribution of the DMQs with respect to the mare basalts. The DMQs are present-day processes that occur at large depths (average depth of the 52 nests used is 930 km), while most mare basalts erupted to the lunar surface over 3 Gyr ago. If they are correlated as I established here, then this result has a number of implications for the causes of the DMQs, the origin of mare basalts and the evolution of the lunar interior. This suggests that mare basalts may have a deep origin, because the same processes responsible for DMQs are also likely related to mare basalts. This is consistent with some petrological evidence (Delano 1986) and with previously presented arguments (Hess and Parmentier, 1995; Zhong et

al., 2000). These results also suggest that the asymmetrical distribution of the DMQs on the nearside may be real and not necessarily entirely due to seismic station bias, because mare basalts are predominately on the nearside as well. I have also investigated the timing of mare basalt eruptions using the latest remote sensing observations. I collect ages of 346 mare basalt units from crater size-frequency distribution measurement results (Hiesinger et al., 2000, 2003, 2006, 2010; Morota et al., 2011), which covers 13 major mare basins on the nearside. I build the distributions of ages in ~2.5 Gyr duration by counting 1) number of units, 2) total flooded area and 3) distributed flooded area within each 0.1 Gyr period, which represent the differential intensities of the mare volcanism in each time bin. I find that mare volcanism occurs in all 13 mare basins predominantly from ~4 to 2.8 Ga, close to what has been suggested from previous studies (Nyquist and Shih, 1992; Taylor et al., 1983). I have identified younger eruptions after ~2.5 Ga till ~1.5 Ga, all occurring in the central PKT region. The age distributions also show episodic occurrence of eruptions on the nearside. While mare volcanism stopped after its maximum in most areas that are distant from the PKT, the central PKT region experienced at least one more episode of eruption but with much reduced intensity. This, together with the latest finding from lunar gravity field analysis (Andrews-Hanna et al., 2014), suggests that PKT play an important role in keeping the underlying mantle from cooling and thus in controlling the mare volcanism. The strong spatial correlations between the mare basalts, the PKT and the present-day DMQs, as well as the potential thermal effect of the PKT on the mare basalt eruptions, suggest a long-term dynamic process that links them. I propose a new hypothesis, in which the link between mare basalts and DMQs is the

MIC material, i.e., the source material for mare basalts. Following Hess and Parmentier (1995), the MIC material initially resided at large depths, possibly just above the CMB, due to its larger compositional density than its ambient mantle materials. Being rich in heat-producing elements, the MIC would be gradually heated and become thermally buoyant to rise up. Under certain hydrodynamic conditions, the MIC material might rise in one hemisphere to a depth of ~400 km

through a single plume and results in the eruption of mare basalts mostly between ~3.9 and 3 Ga ago (Fig. 2.8A). The KREEP, which was partly trapped within or below the crust after the

Figure 2.8. A schematic representation of the lunar interior structure and state during mare basalt emplacement ~3.9 Ga ago (A) and for the present day (B). MIC is the mixture of ilmenite cumulates (IC) and olivine-orthopyroxene formed during the sinking of the IC in olivine and pyroxene mantle after magma ocean differentiation; MIC is rich in radioactive elements, water and other volatiles. The ascending MIC plume in (A) would undergo decompression melting to form mare basalts and is the source of mare basalts. The MIC material may then sink back toward the core due to its cooling and chemically greater density. The stars denote deep moonquakes (DMQ) that in our hypothesis may form preferentially in the MIC or mare basalt source regions where the water and other volatiles may weaken the materials locally and form cracks (denoted by lines in B) (e.g., Frohlich and Nakamura, 2009). A layer of partial melting may exist in the lunar mantle above the core (e.g., Nakamura, 2005; Weber et al., 2011).

sinking of MIC, might have been localized to the present-day PKT region. The PKT is also rich in heat-producing elements, which would cause a higher temperature anomaly and a reduced viscosity in the mantle beneath it to a few hundred kilometers depth (Lanueville et al., 2013; Wiezcorek and Phillips, 2000). This hotter region would be dynamically more mobile and might guide the upwelling plume of MIC to beneath the PKT (Fig. 2.8A).

After causing melting in the mare basalt source region, the MIC would be significantly cooled and become negatively buoyant again and sink back to the deep interior (Fig. 2.8B). Due to continuous but reduced radiogenic heating, the MIC might rise again and cause less amount of melting and mare basalt eruption. This rising-and-sinking process might occur repeatedly until it could not be sustained by heating, causing episodic mare volcanism as observed from the age distribution of the mare basalts. Due to cooling and increased viscosity of the MIC and lunar mantle, the sinking of the MIC might be sluggish, such that the MIC might remain largely on the nearside at or above the CMB today (Fig. 2.8B). The MIC material, as the source for mare basalts, contains significant amounts of water and volatiles (Saal et al., 2008). Under the Earth's tidal force, the water-rich MIC material may fail relatively easily, because water helps facilitate failure by reducing strength (Frohlich and Nakamura, 2009). Therefore, the locations of DMQs may delineate the regions in the lunar deep mantle where the MIC material may exist (Fig. 2.8B). High seismic attenuation was attributed to partial melting (Nakamura, 2005; Weber et al., 2011), and may also be related to the MIC materials. This is because the MIC material is more likely to contain partial melting due to the presence of water that helps reduce melting temperatures.

I suggest that DMQ may hold a key to understanding lunar mantle dynamics and evolution, similar to how deep seismicity in the Earth’s mantle illuminates Earth’s mantle dynamics. While the new hypothesis (Fig. 2.8) appears to reconcile most of the observations, it also makes specific testable predictions, including significantly reduced DMQ activities on the farside relative to the nearside, and distinct seismic structure at large depths on the nearside, all of which can be tested with future deployment of seismometers, particularly on the farside. For nowadays, constraining the deep interior structure of the Moon turns to precise geodetic measurements by lunar orbiters, such as the GRAIL satellites. If the MIC were largely in the nearside interior, it would effectively cause a nearside-farside asymmetry in the elastic properties and density of the deep mantle, e.g. a reduced shear modulus in the nearside due to lower elastic strength of MIC. This asymmetric structure in the Moon’s interior may be significant enough to be measured.

Chapter 3

A perturbation method and its application: elastic tidal response of a laterally heterogeneous planet2

Theory has been long established for computing the elastic response of a spherically symmetric terrestrial planetary body to both body tide and surface loading forces. However, for a planet with laterally heterogeneous mantle structure, the response is usually computed using a fully numerical approach. In this chapter, I develop a semi-analytic method based on a perturbation theory to solve for the elastic response of a planetary body with lateral heterogeneities particularly in the shear modulus of its mantle. I first present a derivation of the governing equations and their matrix form for the second-order perturbation method. I use the perturbation equations to study the mode coupling between degree-2 body tide forcing and the laterally heterogeneous elastic structure of the mantle, which leads to high-order tidal effects. I then develop a propagator matrix solution method, based on the fourth order Runge-Kutta numerical scheme, to obtain the tidal response solutions. I test the method by applying it to the Moon, in which long-wavelength lateral heterogeneities are assumed to exist in the shear modulus of the lunar mantle. The tidal response of the Moon is determined mode by mode, for lateral heterogeneities in different depth ranges within the mantle and different horizontal scales. I compare the perturbation solutions with the numerical results (i.e., benchmarks), showing

2 This chapter was published as “Qin, C., Zhong, S., & Wahr, J., 2014. A perturbation method and its application: elastic tidal response of a laterally heterogeneous planet, Geophys. J. Int., 199(2), 631-647”.

remarkable agreement between the two methods. I then summarize the results and discuss some potential future applications of the perturbation method.

3.1 Introduction

Some knowledge of the interior structure of a planetary body may be obtained by analyzing the elastic and viscoelastic response to other forces, such as tidal forces. This requires highly accurate observations of tidal deformation (often done using space geodetic methods) and efficient analytical or numerical techniques for determining the dynamic response of planetary bodies. The theoretical framework for computing the elastic response of a homogenous or spherically symmetric planetary body (i.e., a 1-D model) has long been established to help understand the dynamic response of a spherically symmetric planet to tidal forces at short time scales (Longman, 1962, 1963; Farrell, 1972). Later, this formulation was extended to include viscoelasticity in spherically symmetric models for surface mass loading problem (e.g., Wu and Peltier, 1982). However, finding a general analytic framework for computing the response of a laterally heterogeneous planetary body is still a challenge. For a planetary body with a 3-D structure consisting of small lateral heterogeneities, a perturbation theory is generally used. Wang (1990) developed a perturbation method to study the tidal deformation of the Earth with lateral heterogeneities in the mantle. This method, however, was not checked for its validity in solving real 3-D problems. Tromp & Mitrovica (1999a, 1999b,

2000) later develop a normal-mode perturbation formulation for solving the surface loading response of a viscoelastic aspherical Earth. Recently, stimulated by the more advanced geodetic measurements, two theoretical methods in solving 3-D tidal response problems have been newly developed. Besides my method that will be presented in the following chapters, Lau et al. (2015) adopted the theoretical framework from Tromp & Mitrovica (1999a, 1999b, 2000) and developed normal mode method that can compute the tidal response of a laterally heterogeneous, rotating and anelastic Earth in the semi-diurnal band. Their ultimate goal is to make use their

method to inverse the GPS measurements of the ground deformation in constraining the Earth’s deep mantle structure. The space observations of the Moon’s hemispherical asymmetries (Chapter 1.1) and the recent statistical study of the spatial correlation between the DMQ and the mare basalts (Chapter 2) support the hypothesis that the hemispherically asymmetric features of the early lunar mantle may have survived to the present day (Qin et al., 2012). By assuming that a spherical harmonic (1, 1) structure can best represent such hemispherical asymmetry, Zhong et al. (2012) used the finite element code CitcomSVE (Zhong et al., 2003; A et al., 2013) to solve for the response of the lunar mantle to degree-2 tidal forcing, and explicitly determined the amplitudes of what appeared to be the largest spherical harmonic coefficients in the solution. This numerical study (Zhong et al., 2003) provided the quantitative basis for the idea of lunar tidal tomography: the harmonic degree-2 tidal force, when applied to the Moon with laterally heterogeneous structure in the lunar mantle, will excite not only degree-2 terms, but also detectable non-degree-2 terms, in the gravitational response. Until future seismological investigations can be made, studies of the lunar tidal response utilizing precise geodetic measurements (e.g., the GRAIL gravity measurements (Konopliv et al., 2013; Lemoine et al., 2013; Zuber et al., 2013)) may hold the promise for constraining the long-wavelength structure and material properties of the lunar mantle (Zhong et al., 2012). The main goal of this study is to develop a new perturbation method that can determine the elastic response of a planetary body with 3-D interior structure. Although the method is generally applicable to different planetary bodies, I limit my analyses to the Moon’s tidal response. Compared to numerical methods, the perturbation method can calculate the tidal solutions much more efficiently and accurately. The perturbation method is also better for understanding the mode coupling effects between the tidal forcing and a laterally heterogeneous (i.e., 3-D) structure. Those coupling effects tend to be weak, and can be obscured by noise in numerical analyses. Compared to other semi-analytical methods (e.g., Lau et al., 2015; Wang, 1990), the perturbation method features a solution method that is much easier for implementation.

More importantly, the perturbation method is carefully benchmarked for 3-D cases against the numerical method. The perturbation analysis can model the effects of lateral heterogeneities in a planet’s solid mantle, but not in its fluid core. Because of the high mobility of a fluid, the core is likely to have much smaller lateral heterogeneities than the mantle. This, combined with the large depth of the core for most planetary bodies, especially for the Moon, implies that in most cases it is reasonable to ignore the impact of the lateral heterogeneities in the core on the tidal response. Lateral heterogeneities in the mantle are likely to exist both in the density and in the elastic moduli, i.e. the first Lamé parameter and the shear modulus. To avoid mathematical complications, I do not include lateral heterogeneities in the density or the Lamé parameter in this chapter, but focus my analyses on the effects of lateral heterogeneities only in the shear modulus. I also focus on descriptions of the mode coupling mechanism and the solution techniques. Lateral heterogeneities in density and the Lamé parameter will be incorporated into the formulation in the next chapter.

3.2 Governing equations and perturbation theory

3.2.1 Governing equations

I start from the governing equations for the tidal response of an elastic, compressible, and spherically symmetric planet. In this study, I define the entire mantle and crustal materials overlying the fluid core as the solid “mantle”, and restrict the solution region to the mantle. The equation of motion in the mantle (Farrell, 1972; Wahr et al., 2009) is

 0     1g 0 rˆ (  0 g 0 ur ) f td  0 , (3.1)

where  is the incremental stress tensor,  is the incremental gravitational potential, 0 is the reference density of the mantle, g0 is the unperturbed gravitational acceleration, ur is the radial component of displacement, 10 ()u is the Eulerian density perturbation (Wu and Peltier,

1982) due to tidal deformation, where u is the displacement vector, and ftd is the body tide force and it is given by

ftd  0 V td , (3.2)

where Vtd is the tidal potential. The elastic constitutive relation that relates the stress to the displacement is

T  00()(()) uI   u   u , (3.3)

where 0 ()r and 0 ()r are the first Lamé parameter and the shear modulus, respectively, from the spherical reference model. The gravitational potential  is governed by Poisson’s equation

2 4 G 1 . (3.4)

3.2.2 Time-dependent body tides

In the Earth-Moon-Sun system, the solar tidal forcing in the Moon, which has entirely different time dependence from the Earth’s forcing, is ~200 times weaker and is thus neglected from the modelling. The tidal potential in the Moon, caused by the Earth, can be expressed as an infinite series of spherical harmonics of degree l and order m (Agnew 2008; eq. (1.1)), with by far the largest terms occurring at degree l = 2. Here, I only consider the degree-2 terms caused by the monthly variation in the Earth-Moon distance due to the Moon’s eccentric orbit; that is the terms in Wahr et al. (2009) that have a cos(nt +0 ) temporal signature:

2 2 3Gma r 22 Vtd( r , ,  , t )3  [(1  3cos  )  3sin  cos(2  )]cos( nt +  0 ) , (3.5) 4Ra where r, , , and tare the spherical coordinates and the time, respectively, is the lunar orbital eccentricity, a is the Moon’s radius, R is the semi-major axis, m is the mass of the Earth, nT 2/ , where T is the orbital period, and 0 is the initial phase angle. Note that for an

elastic body, the tidal response must have the same cos(nt +0 ) time dependence as the applied tidal force. This tidal potential in eq. (3.5) does not include contributions from the librational tide: i.e. tidal forcing caused by the fact that for an eccentric orbit the Moon does not keep exactly the same face pointed towards the Earth, but rocks back and forth relative to the Earth-Moon vector.

The librational tide has a sin(nt +0 ) time dependence, and is of the same order as the terms shown in eq. (3.5). The purpose here, though, is simply to describe and assess a new semi- analytic method of computing the tidal effects of laterally heterogeneous structure, and this does require I use a complete description of the tidal potential. So, I have omitted the librational tidal forcing, for simplicity.

Hereafter, I ignore the time-dependent coefficient cos(nt 0 ) and eq. (3.5) is simplified to

2 Vtd(,,)() r  r Z 1 Y 20 Z 2 Y 22 , (3.6)

where Y20 and Y22 are the real-form spherical harmonic functions of degree l = 2, order m = 0 and m = 2 (i.e., (2, 0) and (2, 2)), respectively (see Appendix A for the definition of Ylm ), and where 9Gm Z  and ZZ 3 are constant coefficients. Below, I sometimes consider the (2, 1 5 R3 21

0) or (2, 2) tidal forcing cases separately.

3.2.3 Perturbation theory

Eqs (3.1) – (3.4), when accompanied by appropriate continuity conditions, can be solved semi-analytically to determine the body-tide-induced displacement u0 and gravitational potential

0 for a spherically symmetric planetary mantle (e.g. Farrell, 1972; Tromp and Mitrovica,

1999b). I express that spherically symmetric solution as (,)u00 for brevity. When there are small lateral heterogeneities in the elastic moduli in the planet’s mantle, eqs (3.1) – (3.4) are modified and solving them becomes more complicated. Using a perturbation method, I

reformulate the modified eqs (3.1) – (3.4) by treating the lateral heterogeneities in the elastic properties as a perturbation, and reorganize the differential equations in terms of different orders of the perturbation. In this chapter, I consider effects of lateral heterogeneities only in the shear modulus, and I assume they have a laterally varying pattern described by a specific spherical harmonic of degree l and order m, as

(,,)()()(,)r    r 0 r Ylm   , (3.7)

where 0 ()r is the reference shear modulus, Ylm is the real-form spherical harmonic function at (l, m), and  ()r is the amplitude of the lateral variability and denotes the asymptotically small expansion parameter, which is chosen below to be either constant through the entire mantle, or constant through an individual shell of the mantle and zero outside that shell. Including this lateral heterogeneity in the shear modulus, the constitutive relation, eq. (3.3), becomes

T  00( u )I  (    )(  u  (  u ) ) . (3.8)

Replacing  in eq. (3.3) with that in eq. (3.8) leads to a set of slightly modified differential equations, and I denote the corresponding tidal response solution as (,)u  . From the perturbation theory point of view (e.g., Richards and Hager, 1989), if the perturbation is small

(i.e.  1), (,)u00 is slightly modified to (,)u  , where (,)(,)u u00  u    and (,)u represents the small modification to (,)u00 due to . I call (,)u00 the zeroth-order solution and (,)u the high-order residual. I further write (,)(,)(,)(,)u  u1  1  u 2  2  u    , where

(,)u11 and (,)u22 are the first- and second-order (in the amplitude,  , of the lateral heterogeneity) corrections to (,)u00 , respectively, and (,)u  is the combined correction from all higher orders. This expansion to (,)u could be extended to third- and higher-orders, but a truncation at second order is sufficient for the range of values of considered below. I re-write eqs (3.1) and (3.4), combined with eq. (3.8), into three separate groups of differential equations, representing the equations for the zeroth, first, and second orders of the

perturbation, respectively. These three sets of equations have the same general form, which can be expressed as follows,

T [()(())]0uDI   0 u D u D   0  D ()  0 u Dg 0 rˆ  (  0 g 0 u Dr ) F D , (3.9)

2 DD  4 G  ( 0u ) , (3.10)

where the subscript D denotes the order of the perturbation and D = 0, 1, or 2, FD is the effective forcing vector and has different forms for different orders of the perturbation.

Specifically, when D = 0, F0 is just the tidal force and F0 f td . When D = 1 or 2,

T FDDD ( [ uu11  (  ) ]), and the equations are forced by the coupling between  and the lower-order displacement field uD1 . To solve eqs (3.9) and (3.10) for each order of the perturbation, appropriate continuity conditions are needed at the core-mantle boundary (CMB), the outer surface, and any internal boundary in the mantle. At any solid-solid spherical internal boundary, the normal traction rˆD , the displacement uD , the incremental potential D , and the normal component of the adjusted incremental gravitational acceleration rGˆ( DD  4  0u ) are continuous, expressed as

rˆ  uu     rˆ (    4  G  )   0 , (3.11)  DDDDD     0  

 where [] denotes the jump of the enclosed quantity from the lower () to the upper () side of an interface, rˆ is the unit vector in the radial direction (it enters into these equations because it is the normal to spherical boundaries), and  D is the stress tensor for order D of the perturbation. The radial traction is

T rrˆDDDDD ˆ 00()[()] uI   u   u  B , (3.12)

T where BDDD (Dr )ˆ  [ ( uu11  (  ) )], and (D ) 0 when D = 0 and (D ) 1 when D =

1 or 2 (i.e., a non-vanishing BD exists in the radial traction, only in the first- and second-order equations). The boundary conditions at the CMB and the outer surface, however, cannot be fully described by eq. (3.11), and are given in matrix form in eqs (3.45), (3.46), (3.51) and (3.52).

By comparing the zeroth-order equations and the associated boundary conditions with those of the high-order equations, the only differences are the two terms, FD and BD . The zeroth-order motion is driven solely by the tidal force, while the first- and second-order motions are both driven by an effective force, together with an inhomogeneous term in the radial traction continuity condition. I call and in the high-order equations the high-order forcing terms or, later, the coupling terms. The solution to the zeroth-order equations (D = 0) is the tidal response of a spherically symmetric planet. The spherical response to any given force is described with the same spherical harmonics that are present in that force. Since in this case the tidal force ftd consists solely of the spherical harmonics (2, 0) and (2, 2) (see eq. (3.6)), those are the harmonics that enter into the zeroth-order response. I refer to this zeroth-order tidal response as the primary response. The solution to the high-order equations represents the response of a spherically symmetric planet to the high-order forcing terms, and (D = 1 or 2). Thus, these solutions are described with the same harmonics that enter into those forcing terms. The first-order response includes harmonics other than (2, 0) and (2, 2), since the forcing terms couple the (2, 0) and (2, 2) harmonics in u0 with the harmonics in . Similarly, the second-order response results from the coupling of u1 and and spans an even larger set of harmonics than those in the first-order response. I refer to the first- and second-order responses as the high-order (or secondary) response.

3.3 Methods

In this section, I convert eqs (3.9) and (3.10) and the boundary conditions eq. (3.11) into matrix forms, illustrate the mode coupling process, and present solution strategies for the matrix equations to obtain the zeroth, first, and second order responses.

3.3.1 Spherical harmonic and vector spherical harmonic expansions

In order to solve the partial differential equations eqs (3.9) and (3.10) semi-analytically, an effective procedure is to first convert them into ordinary differential equations, which is generally done by performing eigenfunction expansions to the unknown variables. In the spherical coordinates, a scalar and a vector are naturally expanded by spherical harmonic (SH) functions and vector spherical harmonic (VSH) functions, respectively. The real-form spherical harmonic function is defined as

 2N Pm (cos )cos m m  0  lm l  m Ylm( ,  ) N lm P l (cos  ) m 0 , (3.13)  2N Pm (cos )sin m m  0  lm l

m where Pl (cos ) is the associated Legendre polynomial of non-negative m, Nlm is the (2l 1) ( l m )! normalization constant of non-negative m and N  , and l and m are called lm 4 (lm )! spherical harmonic degree and order, respectively, and l  0, l  m  l . The associated derivatives of Ylm are defined as

 YYlm lm , (3.14) 1 YY  , (3.15) lmsin  lm

  YYlm lm , (3.16) 1 YY  , (3.17) lmsin  lm 1 YY  , (3.18) lmsin  lm respectively. Any 3-D vector can be expanded in terms of vector spherical harmonics (VSH). The three orthogonal bases of the VSH can be constructed from SH and are given by (Dahlen and Tromp, 1998)

PBClmrYˆ lm,,, lm  11 Y lm lm  rˆ  Y lm (3.19)

respectively, where Plm is in the radial direction, Blm and Clm span in horizontal directions, and 1  ˆ  ˆ is the reduced gradient operator. An arbitrary scalar S can be expanded 1 sin   into SH as

 l S(,,)()(,) r   slm r Y lm   , (3.20) l0 m  l while an arbitrary vector V can be expanded into VSH as

 l VPBC(,,)[()()()]r  plm r lm  b lm r lm  c lm r lm , (3.21) l0 m  l where slm , plm , blm , and clm are the expansion coefficients and are only functions of the radius. The expansion coefficients at a harmonic mode (l, m) can be obtained through

s()(,) r S  Y d  (3.22) lm lm  and

p( r ), L b ( r ), L  c ( r ) VPBC  , , d  , (3.23)  lm lm lm   lm lm lm   respectively, where L l( l 1). A nonzero set of expansion coefficients represent a modal component at harmonic (l, m) of or .

3.3.2 Equations in matrix form

To solve for the tidal response, it is convenient to reformulate eqs (9) and (10) into a matrix form (Tromp and Mitrovica, 1999b; Wu and Peltier, 1982) that permits propagator matrix solution techniques. The mathematical procedure of converting the zeroth-order equations into matrix form is well developed (e.g., Dahlen and Tromp, 1998). A similar procedure is applied here to the high-order equations. I expand the incremental potential D into spherical harmonics,

while perform VSH expansions on the displacement vector uD and the non-coupling terms in the radial traction rˆD , as follows:

DDD uD()UVW lm P lm  lm B lm  lm C lm , (3.24) lm,

D D  KY lm lm , (3.25) lm,

ˆ TDDD r00()[()]() uDI   u D   u D  R lm P lm  S lm B lm  T lm C lm , (3.26) lm, where all the expansion coefficients are functions of radius, while the angular dependence is

D D D D present in Ylm , Plm , Blm , and Clm . Ulm , Vlm , Wlm and Klm fully describe the order D solutions of the tidal response. The expansion coefficients in eq. (3.26) are given respectively by (Dahlen and Tromp, 1998; Tromp and Mitrovica, 1999b)  RDDDD(  2 ) U 0 (2 U  l ( l  1) V ) , (3.27) lm00 lmr lm lm

VUDD SVDD () lm  lm , (3.28) lm0 lm rr W D TWDD ()lm , (3.29) lm0 lm r where a dot denotes d/ dr . I introduce another auxiliary variable

l 1 QKKGUDDDD   4 , (3.30) lm lmr lm0 lm which is related to the incremental gravitational acceleration.

The forcing terms FD in eq. (3.9) and BD in eq. (3.12) also need to be expanded into

VSH. For the zeroth-order equations, B0  0 and F0 f td . The VSH expansion of ftd is straightforward, and is given in Appendix A. For the high-order equations (D = 1 or 2), and refer to the coupling terms. Physically, these coupling terms demonstrate the mode coupling between the lateral heterogeneities in shear modulus and the lower-order (i.e. order D 1) tide- induced deformation field, and lead to responses of order D at predictable harmonic modes. The

VSH expansions of the coupling terms are derived in Appendix A, while their general forms are given here as,

p,,, D b D c D FD()FFF lmPBC lm  lm lm  lm lm (3.31) lm, and

cp,DDD cp, cp, BD ()()DRST lmPBC lm  lm lm  lm lm . (3.32) lm, Substituting eqs (3.24) – (3.32) into eqs (3.9) and (3.12) for each order D yields two decoupled sets of ordinary differential equations along with boundary conditions. The two sets of equations govern two individual modes of motion, the spheroidal (s) and the toroidal (t) modes, which can be investigated separately. The two sets of equations have different dimensions and forms, but can be written in the same general matrix form for each harmonic (l, m), as

dX D lm AXFDD , (3.33) dr l lm lm

D where X lm is the solution vector for harmonic (l, m), Al is a square matrix and depends on the

D harmonic degree l but not on the order m, and Flm is the vectorized form of the VSH expansion of the coupling term FD . Note that the original partial differential equations eqs. (3.9) and (3.12) have been transformed into a ordinary differential equation system in matrix form with respect to radius only. I usually non-dimensionalize eq. (3.33) and the corresponding boundary conditions before solving it, using the following normalization scalings:

    r ar, 0  cmb 0 , 0 = cmb 0 , 0 = cmb 0 , g 0  4  G cmb ag 0 , (3.34)

DDDDDD   (Ulm , V lm , W lm ) a ( U lm , V lm , W lm ), (3.35)

DDDDDD   (RSTRSTlm , lm, lm ) cmb ( lm , lm , lm ), (3.36)

DDDD2  Klm4 G cmb a K lm , Q lm 4  G  cmb aQ lm , (3.37)

cp,DDDDDD cp, cp, cp, cp,  cp,  (RSTRSTlm , lm , lm ) cmb ( lm , lm , lm ), (3.38)

 (FFFFFFpD,,,,,, , bD , cD ) cmb ( pD , bD  , cD  ), (3.39) lm lm lma lm lm lm where the primed variables are non-dimensional, a is the Moon’s radius, g0 is the gravitational acceleration, cmb and cmb are the density and the shear modulus on the mantle side of the CMB. The non-dimensional form of eq. (3.33) is thus given by

dX D lm AF X DD    . (3.40) dr l lm lm

Hereafter, all the variables are non-dimensional and I omit the primes for simplicity. The explicit

D D forms of X lm , Al , and Flm will be demonstrated in the next subsection.

3.3.3 Spheroidal and toroidal modes

The explicit expressions of , , , the associated boundary conditions at the CMB and the outer surface, and the continuity conditions at internal boundaries depend on whether eq. (3.33) describes a spheroidal or a toroidal motion that is excited by the coupling terms. The spheroidal and toroidal modes describe two non-overlapping sets of components in the response: the spheroidal (s) mode contains all the vector components that span in Plm and

Blm , plus potential related quantities, while the toroidal mode has no radial components and only contains vector components in Clm . Specifically,

1. Spheroidal mode

For a spheroidal mode, eq. (3.33) is a 6-by-6 system and thus has 6 components, as

DDDDDDDT Xlm (,,,,,)UVRSKQ lm lm lm lm lm lm , (3.41)

2ll ( 1) 1  00 0 0 0 rr    1 1 1  0 0 0 rr  0

4l (1)2 l 40 l (1)(1) l   l   ()()     r r r r r r g00 r g Al   , (3.42)  1 2 10 3 ( ) 2  2 00 ll (  1)(    )   0 r r r r r rg0 l 1 0 0 0 0 1 r l1 l ( l  1) l  1  0 0 0 00r r r

0(3  0 2  0 ) where  002  ,   ,  00g ,  is a non-dimensional coefficient and is 00 2 22 4Gacmb D given by   . The vectorized forcing vector Flm has the form of cmb

D p,,T D b D Flm (0,0,FF lm , lm ,0,0) , (3.43)

pD, bD, where Flm and Flm are VSH expansion coefficients and all the other entries are zeros. In my calculations, I start the solution process from the CMB and propagate up through the mantle. To find a unique solution to eq. (3.33), the boundary conditions at the CMB and the outer surface, as well as the internal continuity conditions are needed. The vectorized continuity conditions in the solid mantle are

DDDDDDDD   cp,  cp,  [][][][][UVKQRRSSlm lm   lm   lm   lm  lm ][   lm  lm ]0   , (3.44) which ensure the continuity of the displacement, the gravitational potential/acceleration, and the radial traction, everywhere in the mantle. The CMB and surface boundary conditions are

21l  X DDDDDDDDT()(,,(+r U V K g U ),0,, K K  U ) (3.45) lmcmb lm lm 0c lm 0 lm lm lm 0c lm r rcmb rc and

DDDDT Xlm()a ( U lm , V lm ,0,0, K lm ,0) r a , (3.46)

D D respectively, where Xlm ()rcmb and Xlm ()a are also the solutions at the CMB and the outer D surface, respectively, 0c is the density of the fluid core. Solving for the six unknowns, Urlm ()cmb ,

D D D D D Vrlm ()cmb , Krlm ()cmb , Ualm (), Valm () and Kalm () uniquely determines the spheroidal response at harmonic (l, m) of order D of perturbation.

D D The last components of both Xlm ()rcmb and Xlm ()a are different from the classical forms of the boundary conditions used in tidal forcing problems (Tromp and Mitrovica, 2000). The last component in has been mathematically simplified by assuming a uniform fluid core below the CMB instead of a solid inner core overlain by a fluid outer core. This simplification has negligible effect on the results. For , which represents the boundary conditions at a free outer surface, the last component is zero instead of the nonzero constant usually used for tidal forcing in the classical form. This classical nonzero constant comes from re-defining the gravitational potential variable so that it includes the tidal potential. In that case the effects of the tidal force appear in the outer surface boundary condition instead of in the differential equations. But, in this study, I do not include the tidal potential in the gravitational potential variable, and so the effects of the tidal force appear in the differential equations rather than in the boundary condition.

2. Toroidal mode

D For a toroidal mode, eq. (3.3) is a 2-by-2 system and X lm has 2 components, as

DDDT Xlm (,)WT lm lm , (3.47)

11  r 0 Al   . (3.48) (ll 2)( 1) 3 0  rr2

D The vectorized forcing vector Flm is

D c, D T Flm=F(0, lm ) . (3.49)

The associated internal continuity conditions and boundary conditions are (Tromp and Mitrovica, 2000), respectively,

DDDcp, [WTTlm ] [ lm  lm ]  0 , (3.50)

X DDT(rW ) ( ,0) , (3.51) lmcmb lm r rcmb and

DDT Xlm(aW ) ( lm ,0) r a , (3.52)

D D Solving Wrlm ()cmb and Walm () determines the toroidal response at harmonic (l, m) of order D of perturbation.

3.3.4 Mode coupling

Mode coupling causes the high-order (or secondary) responses. Based on eqs (3.9) and

(3.12), the coupling of the lateral heterogeneity in the shear modulus (i.e.,  ) and the degree-2 primary response gives rise to the first-order responses, while further coupling of with the first-order responses leads to the second-order responses. Here, I call the lower-order (i.e. order

D 1) mode (response) the parent mode (response) and the higher-order (i.e. order D) mode (response) that results from the coupling as the child mode (response).

The VSH expansions of the coupling terms F and B make it possible to determine the harmonic modes that are present in the first- and second-order solutions, before actually solving the differential equations (see Appendix A). For the following, I suppose that is at harmonic

(,)lm11 and the parent response (either spheroidal or toroidal) is at harmonic (,)lm00. These combine into F and B , and separate into individual spheroidal (s) and toroidal (t) coupling terms at a few harmonics (,)lm. Because the high-order equations are still spherically symmetric, each spheroidal (or toroidal) coupling term will induce exactly the same spheroidal (or toroidal) mode in the response, denoted here as s(,) l m (or t(,) l m ).

I categorize the mode coupling processes into four types of parent-child mode pairings: s to s, s to t, t to s, and t to t, and I summarize these processes into generalized selection rules that can be described by

if sign ( m01 ) sign ( m ) 0

 p(l , m ), ( l , m )LM11 p(,)(,)l0 m 0 l 1 m 1   p(l , m ), ( l , m )LM22

if sign ( m01 ) sign ( m ) 0

 p(l , m ), ( l , m )LM12 p(,)(,)l0 m 0 l 1 m 1   p(l , m ), ( l , m )LM21  10x  where sign ( x )  , and , (3.53) 10x 1 L : l l  l  2 i , 0  i  ( l  l  l  l ), i is integer 1 0 12 0 1 0 1 1 L : llli 2 1, 0 illll ( ) 1, i is integer ( ll  0, 0) 2 0 12 0 1 0 1 0 1

M1: m m 0  m 1 or m 0  m 1 , and m  l

M2: m  m 0  m 1 or  m 0  m 1 , m  l , and m  0 where p denotes the s (or t) mode and p denotes the other mode, i.e. the t (or s) mode,

LLMM1, 2 , 1 , and 2 are the sets of the harmonic degrees and orders that are allowed by the selection rules. While the selection rules predict the harmonic modes caused by the coupling, the

pD, bD, cp,D radially-dependent expansion coefficients from the coupling terms, i.e., Flm , Flm , Rlm and

cp,D cD, cp,D Slm for spheroidal mode and Flm and Tlm for toroidal mode, govern the strength of the coupling (see Appendix A). Only when at least one of all the coefficients is nonzero, a specific spheroidal or toroidal mode is indicated to exist. The amplitude of the tidal response for each predicted mode depends on the amplitude, , of the lateral variability Specifically, the amplitude of the first-order response is linearly proportional to  while the amplitude of the second-order response is a quadratic function of .

3.3.5 Semi-analytical approach and solutions

Given the matrix forms of eq. (3.33) and the associated boundary conditions, the tidal response for each order of the perturbation can be solved mode by mode using the propagator matrix method (e.g., Hager and O’Connell, 1981) together with a Runge-Kutta numerical scheme. For numerical implementation of the propagator matrix method, I set up a 1-D radial grid that has a fine-enough resolution to ensure convergence of the Runge-Kutta solution. I make sure that there are two grid points at each radial discontinuity in material properties. I also make sure that there are two grid points at each radial discontinuity of , so that I can apply the appropriate boundary conditions at those discontinuities. The basic idea of the propagator matrix method is that, knowing the explicit matrix form of eq. (3.33) (i.e., for a specific spheroidal or toroidal mode at harmonic (l, m) and order of perturbation D, determined directly from the mode coupling) and a starting solution X()r0 (here

I do not show the subscript ‘lm’ or superscript ‘D’ for brevity) at radius r0 in the mantle, the solution X()r at any radius r can be determined by successively propagating X ()r0 from r0 to r using pre-determined propagator matrix. To simplify the solution method, I start the propagation from the CMB (i.e. rr cmb ) instead of from the center of the inner core (i.e. r  0 ), and the starting solution X ()rc is partly unknown (see eqs (3.45) and (3.51)). Such simplification proves to produce little inaccuracy to the calculations. Only a ~0.1% level difference is found when comparing the solutions with those calculated using the classical methods that include a core, for 1-D planetary models with varying core sizes. To fully determine , I propagate it to the outer surface (i.e. ra ), imposing the appropriate boundary conditions as it passes across internal discontinuities, and equate the resulting undetermined solution to the outer surface boundary value X()a (see eqs (3.46) and (3.52)). By solving a linear equation relating the unknowns in X()rcmb and X()a , I am able to solve for

and , and determine the solution, X()ri , at any grid point ri . Once I have solved

the order D equations, I use these solutions to construct the order D coupling terms and solve the resulting order D equations using the same procedure. It is conceivable that some planetary bodies, including possibly even the Moon, might not possess a fluid core, in which case there would be no CMB to start the propagation from. For such a body, however, I can circumvent this problem by inserting a very small fluid core at the center of the planet, and starting the solutions at the surface of that small sphere. For implementation of the propagator matrix method for the perturbation solution, special treatment has to be made in setting up the 1-D radial grid from the CMB to the outer surface of the mantle. Since continuity conditions must be applied at every internal boundary, I define the grid so that there are two grid points at each boundary: one just inside and one just outside. The internal boundaries occur wherever there is a discontinuity, due either to I) a discontinuity in material properties of the spherical background structure, which causes the matrix A to be discontinuous, or II) a radial discontinuity in or the lateral variability . In the latter case, although there is continuity of the radial traction components (see eqs (3.44) and (3.50)), X and F are discontinuous due to the jump of in the coupling terms. The entire mantle is divided into n layers by identifying all the non-overlapping discontinuity boundaries (Type I plus Type II), and subsequently, each layer L (1Ln) is equally divided into fine layers with step size hL . I number all the grid points in order from the CMB to the outer surface. I make use of the fourth order Runge-Kutta numerical method to construct the propagator matrix. According to the Runge-Kutta method, an unknown solution vector X ()ri1 can be evaluated from knowledge of X()ri , as

h XXKKKK(rr ) ( ) L (  2  2  ) , (3.54) ii16 1 2 3 4 where ri and ri1 are two neighboring grid points that belong to layer L, and ri1  r i h L . K1 , K 2 ,

K3 and K4 are defined as

KMAXNFj j( ) (rj i )  j ( ),  1,2,3 or 4 , (3.55)

where MAj () and NFj () are given, respectively, as

MAA1()() ri , (3.56) h MAAIMA( ) (r )[L ( )] , (3.57) 2i 1/22 1 h MAAIMA( ) (r )[L ( )] , (3.58) 3i 1/22 2

MAAIMA4( ) (rhiL 1 )[ 3 ( )] , (3.59) and

NFF1()() ri , (3.60) h NFFANF()()()()rrL , (3.61) 2ii 1/22 1/2 1 h NFFANF()()()()rrL , (3.62) 3ii 1/22 1/2 2

NFFANF4()()()()ri 1 h L r i 1 3 , (3.63)

where ri1/2( r i r i 1 ) / 2 , A and F are the matrix and forcing vector from eq. (3.33), and I is the identity matrix. The dimensions of and are the same as those of and , respectively, which depend on whether eq. (3.33) is spheroidal or toroidal. Eq. (3.54) can thus be re-written as

XPXG()(,)()(,)ri1 r i r i  1 r i r i r i  1 , (3.64) where

h PIMAMAMAMA(,)rr  L (()2()2()    ()) , (3.65) ii16 1 2 3 4 and

h GNFNFNFNF(,)rr L (()2()2()    ()) . (3.66) ii16 1 2 3 4

Eqs (3.64) – (3.66) are the fundamental equations for the propagator matrix method. P(,)rrii1 is the differential form of the propagator matrix, and G(,)rrii1 is the differential contribution of the vector F from ri to ri1 . Note that when rrii 1, PI(,)rrii+1  and G(,)rrii+1  0 . The differential propagation can be easily generalized to perform the propagation of X from ri to rik (k > 1) in the same layer L as

XPXG()(,)()(,)ri k r i r i  k r i r i r i  k , (3.67) where

PPPP(,)(,)(,)(,)riik r r ikik  1 r   r ikik   2 r   1   r ii r  1 , (3.68) and

GGPGPG(,)(,)(,)(,)(,)(,)rriik rr ikik 1  rr ikik  1  rr ikik  2 1   rr iik  1  rr ii  1 . (3.69)

However, when the propagation reaches a Type II discontinuity boundary at rb (here, includes the CMB at rcmb and the outer surface at a), a correction needs to be made to include the discontinuity of X. This correction, derived from eqs (3.44) and (3.50), is

 XX()()rb r b  r b  , (3.70)

  where  rb  is the correction term that relates the solution vectors X at rb and rb . Specifically, for a spheroidal mode,

cp cp  cp  cp  T (rb ) (0,0, R ( r b )  R ( r b ), S ( r b )  S ( r b ),0,0) , (3.71) while for a toroidal mode

cp cp T (rb ) (0, T ( r b ) T ( r b )) , (3.72) where the subscript ‘lm’ and superscript ‘D’ for Rcp , S cp and T cp are ignored for brevity. Note

   that if rb is equal to rcmb (or a), , and on the rcmb (or a ) side are zero since rcmb (or ) is out of the solution region. Once across a Type II discontinuity boundary, successive propagation of X continues until reaching another such boundary.

Starting from the CMB boundary value, i.e. X()rcmb , the generalized expression of the solution X()ri at any grid point can be obtained through

XPXGE()(,)()(,)()ri rcmb r i  r cmb  r cmb r i  r i , (3.73)

where P(,)rrcmb i and G(,)rrcmb i are the propagator matrix and the propagation of F from rcmb to ri that are defined in eqs (3.68) and (3.69), respectively. E()ri is the total contribution of the correction terms at from all the Type II boundaries below , and

EP()(,)()()r r r  r  r , (3.74) i b i b rii , r1 i rrb  i where  1 if rr and zero otherwise. rrii, 1 ii1

Let in eq. (3.73) be a, and solve the resulting linear equations eq. (3.73), with X()rcmb and X()a given by eqs (3.45) and (3.46) for the spheroidal case and eqs (3.51) and (3.52) for the toroidal case. After and are fully determined, eq. (3.33) can be solved at every grid point by applying eq. (3.73) repetitively. For the special case of the high-order degree-1 spheroidal mode, an unconstrained translational motion associated with a rigid shift of the planet is contained in the solution to eq. (3.33). This translational motion does not cause any deformation and thus needs to be subtracted from the solution to obtain the real degree-1 response (Farrell 1972). The translational mode always satisfies the homogeneous form of eq. (3.33), and its unit form can be expressed as

X tr (r ) (1,1,0,0, g ( r ),0)T , (3.75) where gr() is the gravitational acceleration at radius r. Note that in the center of mass reference frame I consider here, the degree-1 response must have zero gravitational potential at the outer surface. This condition is applied to determine the degree-1 response by adding a multiple of eq. (3.75) to the solution to eq. (3.33), to obtain

XXXnet()()()r r   tr r , (3.76)

where   K()/() a g a , and X net ()r is the degree-1 physical response in the center of mass reference frame (Farrell, 1972).

3.4 Results

I apply the perturbation method described above to solve for the tidal response of the Moon. The Moon is modeled with a solid inner core, a fluid outer core and seven mantle and crust layers of different properties, according to a recent lunar seismic study (Table 3.1). In my

Table 3.1. Lunar 1-D model parameters (Weber et al., 2011)

3 Depths (km) 0 (kg/m ) Vp (km/s) Vs (km/s) 0  15 2700 3.2 1.8 15 40 2800 5.5 3.2 40 238 3300 7.7 4.4 238 488 3400 7.8 4.4 488 738 3400 7.6 4.4 738 1257 3400 8.5 4.5 1257 1407 3400 7.5 3.2 1407 1497 (outer core) 5100 4.1 0.0 1497 1740 (inner core) 8000 4.3 2.3 Semi-major axis, a 3.844108 m Eccentricity, 0.0549 Earth’s mass, m 5.97 1024 kg 6 Moon’s radius, Rs 1.74 10 m

calculations, I adopt this seismic model, and refer to the top seven layers as the lunar “mantle”. I determine the reference elastic properties in each material layer using the seismic velocities from the 1-D model. Specifically,

2 00 VS (3.77) and

2 0  0VP 2  0 (3.78)

where VP and VS are P-wave and S-wave velocities (Table 3.1), respectively. I assume for most of my calculations that there are long-wavelength lateral heterogeneities in the shear modulus throughout some prescribed depth range of the lunar mantle. For most, though not all, of these calculations I assume that the heterogeneities can be represented by a (1, 1) harmonic, since that best represents a nearside-farside asymmetric structure. I calculate the tidal response of the Moon for three scenarios regarding the choice of lateral heterogeneities in the shear modulus: 1) a (1, 1) harmonic pattern throughout the entire mantle, 2) a (1, 1) harmonic pattern in either the bottom or the top half of the mantle, 3) other long-wavelength patterns of harmonic (,)lm11 throughout the mantle. I apply the (2, 0) and (2, 2) tidal force components separately for the first two scenarios, but apply the sum of those two components (i.e., eq. (3.6) for the total tidal potential) for the third scenario. I calculate the full set of first- and second-order responses for both the spheroidal and torodial modes. However, I present the results primarily for the spheroidal modes, since only the spheroidal modes contribute to the radial displacement and the gravitational potential, which are the quantities that can potentially be inferred using observations from current space missions. Note that toroidal modes, when coupled with laterally heterogeneous structure, can induce higher-order spheroidal motions. For each high-order spheroidal response s(l, m) at order D, I define the relative tidal response

DD 0 using the normalized radial displacement hlm U lm ()/() a Utd a and the normalized gravitational

DD 0 0 0 potential klm K lm ()/() a Ktd a at the lunar surface, where Uatd () and Katd () are the associated primary responses at degree 2.

3.4.1 Finite element solutions

I also compute the tidal response for the same model using the finite element code CitcomSVE (A et al., 2013; Zhong et al., 2003, 2012), by solving eqs (3.1), (3.4) and (3.8). I then compare the results from the perturbation method with those from the finite element method. In CitcomSVE, the mantle is modeled as a series of spherical shells overlying a fluid core, from the CMB to the surface. The whole domain is horizontally divided into 12 spherical caps and each cap is further divided into elements in both radial and horizontal directions (Fig. 3.1).

Figure 3.1. Discretization of finite element mesh in CitcomSVE code (Zhong et al., 2003). (a) global view of finite element mesh. The mantle shell is horizontally divided into 12 spherical caps. (b) Each cap is further divided in both horizontal and radial directions into elements.

In my calculations, each cap is divided into 483 elements and the entire domain is comprised of

12 483 (~1.3 millions) elements in total. The same 1-D model of the lunar mantle is used in CitcomSVE as in the perturbation method. Lateral variations in the elastic moduli as well as the tidal forces are mapped onto each element. The tidal response solutions from CitcomSVE are quantified by hlm and klm , which are the spherical harmonic expansion coefficients of the surface radial displacement field and surface gravitational potential field, respectively. In this study, and are determined for harmonic degrees and orders of 04l and 0 ml, and are normalized (i.e., relative response) in the

same way as in my perturbation calculations, such that solutions from the two methods can be compared directly. Note that, when lateral heterogeneities are added into the finite element

model, hlm and klm are obtained by subtracting the 1-D response (i.e., without lateral heterogeneities) from the 3-D response (i.e., with lateral heterogeneities), in order to suppress the numerical errors that may be significant at very small perturbations.

3.4.2 (1, 1) lateral heterogeneity throughout the mantle

I define the harmonic (1, 1) lateral heterogeneity in  as

   0 Y 11    0 sin  cos  , (3.79)

3 where  is a nonzero constant throughout the mantle, and   is the peak amplitude of 4

the lateral variability in the shear modulus. The value of is chosen to vary from 2.5% to 40%,  so that it spans a large range of perturbations. (The upper values of this range are larger than what I would expect for the real Moon.)

Using this perturbation method, I first determine the high-order response modes from the mode coupling analysis, for (2, 0) and (2, 2) tidal forcing separately (see Fig. 3.2). I define the spheroidal (toroidal) mode at harmonic (,)lm of the order of perturbation D as sD (,) l m (tD (,) l m ). A (2, 0) tidal force induces a s0(2, 0) primary response. Based on the selection rules

in eq. (3.53), the coupling of s0(2, 0) with the (1, 1) lateral heterogeneity gives rise to a first- order response consisting of two spheroidal modes s1(1, 1) and s1(3, 1) and one toroidal mode t1(2, -1). These first-order modes couple with the (1, 1) structure to excite second-order

responses that span harmonic degrees 2 to 4. In particular, all three first-order modes excite s2(2, 0) and s2(2, 2), while s1(3, 1) generates additional degree-4 spheroidal modes, which are s2(4, 0) and s2(4, 2). For (2, 2) tidal forcing, the same three first-order modes are generated as for (2, 0) tidal forcing, along with an additional s1(3, 3) mode. Thus, the harmonic content of the second-

Figure 3.2. Diagrams of the mode couplings (up to second order in the perturbation) between a spherical harmonic (1, 1) laterally heterogeneous structure in shear modulus and the (a) (2, 0) and (b) (2, 2) tidal forces, respectively. In both diagrams, s and t represent spheroidal and toroidal modes, respectively. The superscript 0, 1, and 2 denote the zeroth-order, first-order, and second-order modes, respectively.

order response to (2, 2) forcing includes all the same harmonics as for (2, 0) forcing, plus the harmonics that derive from the s1(3, 3) mode. Those latter harmonics include s2(4, 4) and an additional s2(2, 2) contribution (note that s1(3, 3) does not induce a s2(2, 0) term). Though the responses to (2, 0) and (2, 2) forcing have nearly the same high-order mode content, the amplitude of each mode differs in the two cases. Note that the amplitude of the same mode from different couplings need to be summed together to obtain the total amplitude of that mode.

I calculate the relative response of the radial displacement hlm and the gravitational potential klm , of all the high-order spheroidal modes for different values of , and show the absolute values of the results in Figs 3.3a and b for (2, 0) forcing, and in Figs 3.3c and d for (2, 2) forcing. The response amplitudes of the first-order modes, i.e. s1(1, 1) and s1(3, 1) for the (2, 0) tidal forcing, and s1(1, 1), s1(3, 1), and s1(3, 3) for the (2, 2) tidal forcing, increase linearly with , while the amplitudes of the second-order modes, i.e. s2(2, 0), s2(2, 2), s2(4, 0), and s2(4, 2) for the (2, 0) tidal forcing, and s2(2, 0), s2(2, 2), s2(4, 0), s2(4, 2), and s2(4, 4) for the (2, 2) tidal forcing, are quadratic functions of . Note that the degree-1 responses in the gravitational potential are zero in the center of mass reference (see eq. (3.76)). For relatively weak perturbations (i.e. small ), the first-order responses are generally orders of magnitude stronger than those of second-order (Fig. 3.3). However, due to their quadratic dependence on , the second-order responses become increasingly significant with increasing  and some of them approach or exceed the first-order responses at largeFig. 3.3) I take s2(2, 0) induced by the (2, 0) tidal forcing and s2(2, 2) induced by the (2, 2) tidal forcing (i.e. the “self-coupling” modes) as examples. When is as large as

2 2 1 40%, the s (2, 0) gravitational potential response, k2,0 , is 70% of the associated s (3, 1) response,

2 2 1 while the s (2, 2) gravitational potential response, k2,2 , is 53% of the associated s (3, 3) response and is 113% greater than the s1(3, 1) response (Fig. 3.3). I also find that these self-coupling responses are always one order of magnitude stronger than the other second-order responses, on average. This happens because these second-order self-coupling responses are generated through a forward-and-backward mode coupling process. More specifically, if a child mode sD (,) l m

Figure 3.3. Log-log plots of the absolute values of the relative responses for a Moon with (1, 1) lateral heterogeneities in shear modulus, to (2, 0) (a and b) and (2, 2) (c and d) tidal forces. The amplitude of the lateral variability in the shear modulus is represented by  and the results are shown for  from 2.5% to 40%. The relative responses in the surface radial displacement (a and c) and the surface gravitational potential (b and d) are shown, respectively, for each spheroidal harmonic mode s(l, m) from the mode couplings. The straight solid lines represent the response modes predicted by our second-order perturbation method (denoted by P). The triangles (first order), inverted triangles (second order), and open circles (higher than second order), represent results computed using CitcomSVE, that we believe to be correct (denoted by C). The dashed lines are also results from CitcomSVE calculations for all the other harmonics of 04l and

0 ml, and are believed to be numerical noise (denoted by N_err). Note that k11， from CitcomSVE (b and d) are highlighted.

DD1 from the forward coupling of s(,)(,)(,) l0 m 0 l 1 m 1 s l m has the same (or opposite) sign as

D1 DD1 its parent mode s(,) l00 m , then the backward coupling s(,)(,)(,) l m l1 m 1 s l 0 m 0 must

D1 D generate a s(,) l00 m mode that also has the same (or opposite) sign as s(,) l m . Therefore, the self-coupling responses always have the same sign as their primary response, which may not be true for the other second-order responses. Since it is guaranteed that the contributions from all the first-order modes to the self-coupling modes add constructively, the self-coupling response is always the largest among the second-order responses. I also compute the tidal responses numerically using the finite element code CitcomSVE (A et al. 2013; Zhong et al. 2003). By comparing the CitcomSVE results with those from the perturbation method, I categorize the numerical results into three types: 1) those that are correct and are predicted by the second-order perturbation method (triangles in Fig. 3.3), 2) those that are probably correct but that are not predicted by the perturbation method (open circles in Fig.

3.3), and 3) those that are probably caused by numerical noise (dashed lines in Fig. 3.3). I find that the most significant responses from CitcomSVE are all predicted by the perturbation method, and that they agree remarkably well with those from the perturbation method (both in their amplitude and in their linear or quadratic dependence on , see Fig. 3.3) for small and moderate values of . As shown in Table 3.2, when  = 10%, the relative

Table 3.2. Comparisons of h and k between the results from the perturbation method and those from the finite element methods, for =10%.

Ia II III b c Mode h (%) k (%) 1 s (1, 1)(2,0) 5.56 N/A 10.1 N/A 1.46 N/A 1 s (3, 1)(2,0) 0.51 0.51 0.31 0.33 0.36 0.33 2 s (2, 0)(2,0) 0.48 0.48 0.10 0.10 0.06 0.04 2 s (2, 2)(2,0) 0.59 0.65 0.19 0.26 0.65 0.57 2 s (4, 0)(2,0) 0.58 0.19 0.23 0.51 0.13 0.15 2 s (4, 2)(2,0) 0.74 0.70 0.12 0.12 0.63 0.51 1 s (1, 1)(2,2) 8.82 N/A 9.14 N/A 0.94 N/A 1 s (3, 1)(2,2) 0.28 0.26 0.34 0.50 0.24 0.17 1 s (3, 3)(2,2) 0.49 0.45 0.28 0.32 0.41 0.34 2 s (2, 0)(2,2) 0.64 0.51 0.46 0.81 0.26 0.28 2 s (2, 2)(2,2) 0.41 0.42 0.33 0.35 0.62 0.53 2 s (4, 0)(2,2) 0.46 0.51 0.34 0.49 0.35 0.21 2 s (4, 2)(2,2) 0.44 0.26 0.66 0.70 0.48 0.27 2 s (4, 4)(2,2) 0.77 0.82 0.33 0.44 0.66 0.64 a Lateral heterogeneity in I. the entire mantle II. the bottom half of mantle III. the top half of mantle b Spheroidal response of harmonic (l, m), to the (2, 0) and (2, 2) tidal forcing, respectively c Relative difference  X ()/XXXCitcom pert pert

difference between the hlm (or klm ) values of the two methods is no greater than 1% for all non- degree-1 responses. The degree-1 responses h11， from CitcomSVE show a ~10% deviation from

1 those computed using the perturbation method, although h1,1 is small. This deviation may be attributed to numerical errors in CitcomSVE. I notice that although the gravitational potential

1 response k1,1 should be zero by definition, the k11， results from CitcomSVE display a linear dependence on , and so behave like a first-order response (see Figs 3.3b and d). The CitcomSVE values for are ~10-6 for  = 10%. Considering that the primary response is 1, a ~10-6 numerical error is small and not unexpected.

The CitcomSVE results gradually deviate from the semi-analytic results as  increases, as the perturbation theory suggests. As  = 20%, the difference between the numerical results and the semi-analytic results is at the 3% level, for all the first- and second-order modes (i.e. triangles and solid lines in Fig. 3.3). When  is as large as 40%, the differences grow to larger than 10%. This is because for large perturbations, contributions from effects higher than the second order become significant in the numerical results, and the second-order perturbation method is unable to predict them. According to the selection rules, the higher-order contributions to the first- and second-order responses start from the third and the fourth orders in the perturbation, respectively, so they grow slowly as  increases and are not obvious in Fig. 3.3 even for large values of . Clear evidence of higher-order (D > 2) effects can be seen in Figs 3.3a and b (i.e. for (2, 0) forcing), which show significant s(3, 3) and s(4, 4) (open circles) responses in the CitcomSVE results that are not predicted by the second-order theory. As 10% , these s(3, 3) and s(4, 4) responses display approximately cubic and quartic dependence on , implying that they are third- and fourth-order effects, respectively. More specifically, s3(3, 3) is from the coupling between s2(2, 2) and (1, 1) structure, while s4(4, 4) is from further coupling between s3(3, 3) and (1, 1) structure. However, no such individual higher-order responses are observed for (2, 2) tidal forcing (e.g. Figs 3.3c and d). This is because all the higher-order effects in that case are coincidently buried in the first- and second-order responses, which are dominant.

Of all the other modes from CitcomSVE (dashed lines in Fig.3.3), the hlm and klm are at a level of 10-6 or less and display pseudo-random behavior with respect to increasing . Considering that they are six orders of magnitude smaller than the primary response, and that they are not predicted to exist from the perturbation method, I believe them to be numerical noise. Additionally for (2, 0) tidal forcing, both the s3(3, 3) and s4(4, 4) responses are at the level of ~10-6 when 10% and contain significant errors (Figs 3.3a and b). I believe that a relative amplitude of ~10-6 for both and is approximately the limit of accuracy that CitcomSVE can reach, given that the numerical models use 12 483 elements (or ~40 km surface resolution)

and a 10-3 convergence tolerance. However, it is still surprising to see the high accuracy (six orders of magnitude) of CitcomSVE. Having shown that the perturbation method can determine the spheroidal responses accurately, its accuracy in solving for the toroidal responses is not examined directly in this chapter. Here, I consider the t1(2, -1) mode and its contributions to s2(2, 0) and s2(2, 2) responses for both (2, 0) and (2, 2) tidal forcing. s2(2, 0) and s2(2, 2) are also excited from the degree-1 and 3 spheroidal modes (Fig. 3.2) and this portion of the responses from s to s couplings are believed to be calculated accurately. With the total responses of s2(2, 0) and s2(2, 2) being accurate (Table

3.2), I argue that the t1(2, -1) solutions should also be accurate. Fig. 3.4 compares the gravitational responses of both the s2(2, 0) and s2(2, 2) modes with (solid lines) and without (dashed lines) the contribution from t1(2, -1), for 10%   40% . Without t1(2, -1), the s2(2, 0) and s2(2, 2) responses deviate significantly from the CitcomSVE results. Adding the contribution by t1(2, -1) eliminates this deviation in the response, which implies that the perturbation method can also determine the toroidal responses accurately.

3.4.3 (1, 1) lateral heterogeneity in either the bottom or the top half of the mantle

I have shown that the second-order perturbation method accurately solves for the tidal response of a laterally varying Moon with (1, 1) lateral heterogeneities throughout the lunar mantle (case I). To investigate how a Moon with lateral heterogeneities localized to different depth ranges within the mantle would respond to the tidal force, and to test the robustness of the perturbation method, I explore two additional cases that assume the same (1, 1) lateral structure but restrict it to either the bottom (case II) or the top (case III) half of the mantle, respectively. For both cases II and III, the tidal responses to the (2, 0) and (2, 2) tidal forces are calculated for from 2.5to 40%using both the perturbation method and CitcomSVE. Results for both cases II and III continue to show an overall < 1% relative difference in hlm and klm between these two methods (see Table 3.2 for = 10).In Fig. 3.5, I show the displacement

Figure 3.4. Comparison of the gravitational responses of the self-coupling modes s2(2, 0) (red) and s2(2, 2) (blue) with (solid lines) and without (dashed lines) the contribution from t1(2, -1), for from 10% to 40%. (a) and (b) are for (2, 0) and (2, 2) tidal forcing, respectively. The triangles and inverted triangles are the s2(2, 0) and s2(2, 2) responses from the CitcomSVE calculations.

Figure 3.5. The radial displacement responses of the Moon with (1, 1) shear modulus lateral heterogeneities I) throughout the entire mantle (blue), II) in the bottom half of the mantle (green), and III) in the top half of the mantle (red), for , to (a) (2, 0) and (b) (2, 2) tidal forces, respectivelyThe x-axis lists all the high-order spheroidal response modes. The responses of the same harmonic from cases II and III are summed into one column for comparison with those from case I.

responses, hlm , of each high-order mode for all three cases for both (2, 0) and (2, 2) tidal forcing as = 10. for cases II and III are added to compare with those in case I. Since case I includes lateral heterogeneities throughout the entire mantle, the high-order responses in case I are generally greater in amplitude than those in cases II and III separately (Fig. 3.5). However,

1 exceptions exist. The responses h1,1 for both (2, 0) and (2, 2) tidal forcing in case III are greater than those in cases I and II. However, since cases II and III have non-overlapping depth ranges, and since the sum of their ranges equals the depth range for case I, I expect the sum of the first- order responses from cases II and III to be equal to those from case I, due to linearity of the first- order responses. This is consistent with what I have found in the first-order solutions (Fig. 3.5).

1 Therefore, it is evident that the responses h1,1 in cases II and III have opposite signs and thus

1 partially offset one another, explaining the reduced h1,1 in case I. However, such simple addition cannot be applied to the second-order responses (e.g., self-coupling responses), due to non-linear effects (Fig. 3.5).

3.4.4 Other long-wavelength structures throughout the mantle

I also use the perturbation method to calculate the tidal response of the Moon for long- wavelength structures other than (1, 1). I consider lateral heterogeneities in the shear modulus throughout the mantle for nine long-wavelength structures, i.e. harmonics (,)lm11 for 13l1 and 0 ml11. Here, I use  (given in eq. (3.7)), instead of as the measure of the lateral variability of the shear modulus. Note that I use the total tidal potential that includes both (2, 0) and (2, 2) terms in this scenario.

Fig. 3.6 shows the high-order gravitational tidal responses of the Moon for all nine laterally heterogeneous structures for . All the responses are normalized by the primary s0(2, 0) response. Figs 3.6a – i show global maps of the lunar surface gravitational potential anomaly contributed by all the first- and second-order responses, with the sub-Earth point at the center of the map. Fig. 3.6j demonstrates the spectrum of the gravitational responses that spans

Figure 3.6. High-order gravitational responses of the Moon to the entire degree-2 tidal forces (i.e., the (2, 0) forcing plus the (2, 2) forcing) for long-wavelength laterally heterogeneous structures of harmonic degrees from 1 to 3, with  lm 10% in each case. (a) – (i) are gravitational anomaly maps in Mollweide projection for the lateral heterogeneities of harmonic (1, 0), (1, 1), (2, 0), (2, 1), (2, 2), (3, 0), (3, 1), (3, 2), and (3, 3), respectively. The center of the map is the sub-Earth point of the Moon. (j) shows the response spectrum for each harmonic structure. The x-axis represents the high-order spheroidal modes that span from (0, 0), (1, 0), (1, 1), (2, 0), … , to (8, 8) in harmonics. The y-axis lists the nine long-wavelength harmonics of the structure. The maps (a) – (i) and the spectrum plot (j) use different color scales.

harmonic degrees 0 to 8 for each associated structure. I make the following observations from my calculations. First, consistent with the selection rules (eq. (3.53)), a degree-1 harmonic structure generates degree-3 first-order and degree-2 and 4 second-order responses; a degree-2 harmonic structure generates degree-2 and 4 first-order and degree-2, 4, and 6 second-order responses; and a degree-3 harmonic structure generates degree-3 and 5 first-order and degree-2, 4, 6, 8 second-order responses. Second, all nine gravitational potential anomaly patterns are dominated mostly by first-order responses with amplitudes ranging from ~0.2% to less than 2%, relative to the s0(2, 0) response. Third, the second-order self-coupling modes s2(2, 0) and s2(2, 2) show up in all cases, with relative amplitude ranging from 0.04% to 0.07% for s2(2, 0) and from

0.09% to 0.14% for s2(2, 2), which are approximately one order of magnitude weaker than most of the first-order modes. However, for some structures, the s2(2, 0) and s2(2, 2) responses are comparable with first-order responses that are relatively weak (e.g., s1(3, 1), 0.2%, in Fig. 3.6g; s1(5, 0), 0.14%, in Fig. 3.6h; s1(5, 1), 0.04%, and s1(5, 3), 0.19%, in Fig. 3.6i). The other second- order responses are two to three orders of magnitude smaller than the first-order responses. Therefore, for lateral variability ofor less, I can neglect the second-order non-self-coupling responses. For much larger values of , all the second-order responses become significant and should all be taken into account.

3.5 Conclusion and discussion

I have developed a semi-analytic method based on perturbation theory, to solve for the tidal response of a planetary body with 3-D elastic structure in the mantle. The method can be applied to a planetary body where there are weak lateral heterogeneities in the elastic moduli (in this chapter, in the shear modulus only) superimposed on a radially stratified reference model.

The second-order perturbation method predicts all the spherical harmonic modes resulting from first- and second-order couplings between the body tide force and the 3-D elastic structure, and determines the responses at these modes. I apply the second-order perturbation method to the Moon to calculate the responses to degree-2 tidal forcing for a variety of laterally heterogeneous lunar mantle structures, including harmonic (1, 1) structure restricted to different depth ranges in the mantle, and other long-wavelength structures with harmonic degrees up to 3. I also compute the tidal responses for the same models using the 3-D finite element model CitcomSVE (A et al., 2013; Zhong et al., 2003). I find remarkable agreement between the perturbation method and the finite element method for small and moderate magnitudes of the perturbation, suggesting that the perturbation method provides accurate solutions to the tidal response problem. Furthermore, the perturbation method provides the first true 3-D benchmark for the finite element code CitcomSVE. The benchmark results show that the finite element

model, which has a resolution of ~40 km at the outer surface with 12 483 elements, achieves a relative error of ~10-6. Since the primary responses (i.e., the tidal love numbers) are generally at the order of 10-2, the high-order responses can be determined accurately down to the level of ~

108 , which is remarkable. Such a surprisingly small relative error deserves more study from a computational science point of view. The perturbation method, which can be used as an alternative to a fully numerical approach, has advantages for solving a tidal response problem. First, the perturbation method is computationally much more efficient than CitcomSVE in doing forward calculations. For the same problem setup with single harmonic lateral heterogeneity, the perturbation calculation only costs less than 1/10 of the time cost by a CitcomSVE run with 12 483 elements. Second, the perturbation method is able to directly predict all the spherical harmonic modes in the tidal solution, and the effects of different laterally heterogeneous model parameterizations on each modal coefficient can be investigated individually. This type of mode coupling analysis cannot be easily done using grid-based numerical methods. Third, given observational measurements of the tidal response, inverse modeling to constrain the laterally varying mantle structure would be more efficient using the perturbation method than using a fully numerical approach. Monte Carlo sampling in the parameter space could be easily done to look for plausible long-wavelength pattern(s) and depth range(s) for the mantle lateral heterogeneities that could explain the observations.

A potential application of the perturbation method is to constrain the elastic structure of the present-day lunar mantle, by inverting the gravitational tidal solutions from the GRAIL mission (Zhong et al., 2012). Due to the unprecedented accuracy of the GRAIL satellites in measuring the near-surface gravity field, long-wavelength gravitational tidal terms can be determined accurately enough that small tidal variations among different harmonics can be observed and evaluated (Konopliv et al., 2013; Lemoine et al., 2013). The gravitational tidal signal can be interpreted as a surface representation of the lateral heterogeneities in the lunar interior, and an inversion analysis can thus be performed to determine possible laterally varying

structures. It is also important to note out that the degree-2 self-coupling response may cause significant variations in degree-2 Love numbers, thus complicating the usage of degree-2 tidal response to constrain the lunar core size (e.g. Williams, 2007). The perturbation method can be improved and expanded in the following ways. First, lateral heterogeneities not only in the shear modulus (as considered in this chapter), but also in the first Lamé parameter and density need to be incorporated into the formulation. Second, 3- D surface mass loading problems can be solved using the perturbation method by changing only the outer surface boundary condition. Third, the perturbation method can be adapted for tidal or surface loading problems for a planet with an elastic shell of variable thickness (A et al., 2014).

Chapter 4

Elastic tidal response of a laterally heterogeneous planet: a complete perturbation formulation3

In Chapter 3, I develop a new perturbation method that can effectively calculate the elastic tidal response of a terrestrial planetary body with lateral heterogeneity in the shear modulus of its mantle to very high accuracy. The validity of the perturbation method has been checked by implementing it on the Moon for varieties of presumed 3-D cases. I take this chapter as a continuation of Chapter 3. I complete the perturbation method by incorporating lateral heterogeneities not only in the shear modulus but also in the first Lamé parameter and the density, into one formulation. Based on similar procedures to those described in Chapter 3, the new perturbation equations are transformed into a matrix form in a spherical harmonic representation, such that the total response is split into multiple single-harmonic responses through mode coupling and can be determined semi-analytically. I then continue to apply the perturbation method to a hemispherically asymmetric Moon for benchmarks. The benchmark tests also include comparison of horizontal displacement in addition to radial displacement and gravitational potential, to verify the correctness of the perturbation method. This chapter is organized as follows. In Section 1, I derive the governing equations and the perturbation formulation. The 3-D density anomaly, unlike any 3-D elastic structures that

3 This chapter has been prepared as a manuscript for submission to Geophysical Journal International. The title of the manuscript is “elastic tidal response of a laterally heterogeneous planet: a complete perturbation formulation”.

have no effects on the background state, would break the 1-D hydrostatic equilibrium of the system. This physical effect caused by the density anomaly is discussed. In addition, special treatment on the harmonic degree-1 mode that can potentially be induced by the density anomaly is illustrated on a physical basis. In Section 2, I proceed to transform the perturbation equations into a matrix form through spherical harmonic (SH) and vector spherical harmonic (VSH) expansion processes. I then demonstrate the programming flow of the whole solution process. In Section 3, I test the perturbation method by calculating the tidal response of a hemispherically asymmetric Moon for individual sources of lateral heterogeneity. The perturbation solutions are then compared with their numerical companions (not available for density perturbation), as done in Chapter 3. In Section 4, remarks and future prospects are provided.

4.1 Governing equations

I seek to derive governing equations for tidal response of a spherically stratified planetary body superposed with small lateral heterogeneities in its rocky mantle and crust. The mantle is assumed to be compressible and respond to time-varying tidal force elastically with self- gravitational effect. I include all sources of lateral heterogeneities, which can exist in the two elastic moduli (the shear modulus and the first Lamé parameter ) and the density . The equation of motion originates from the conservation of momentum for a non-inertial and non- rotating system, which is given as (Dahlen and Tromp, 1998; Tromp and Mitrovica, 1999b)

EEE rr T    0 , (4.1) where r denotes a point in the deformed system (while x denotes the same point in the undeformed system) ,  E ,  E and TE are Eulerian density, gravitational potential and stress field at r , respectively.

4.1.1 Background state

For a spherically symmetric (1-D) planetary body with no external force, the system is in hydrostatic equilibrium with no deformation, such that rx= . From eq. (4.1), I have

T0  0  0 , (4.2)

2 004 G  , (4.3)

where 0 , 0 and T0 are the reference density profile, the gravitational potential and the stress field, respectively, and are all functions of radius. TI00p and p0 is the hydrostatic pressure.

When there exist small lateral heterogeneities, i.e. 0 , 0 and 0 (00/1  ,

00/1  and 00/1  , where 0 , 0 and 0 are the shear modulus, the first Lamé parameter and the density of the 1-D reference state, respectively), although 0 and 0 alone have no direct effect, the density anomaly 0 would provide a buoyancy field that breaks the hydrostatic equilibrium and drives the mantle flow and deformation. I assume that 0 will cause the system to deform by s ( is the displacement vector and srx). Considering s is small, quantities at r can be mapped linearly onto x as follows (Sec. 3.3 of Dahlen and Tromp, 1998),

E   00 s   , (4.4)

E   00 s   , (4.5)

E  TTT00 s  , (4.6)

r   () s  , (4.7)

   where 0  0  0 , 0  0  0 and T0p 0 I + 0 are defined at x , and   x . Notice that I consider 0 , 0 and  0 already contain any incremental effect induced by the deformation. Substituting eqs (4.4) – (4.7) into eq. (4.1) result in

   T0  0  0  O()s , (4.8)

   where 0  0  0 , 0  0  0 , T0p 0 I + 0 , respectively,  0 is the stress induced by the buoyancy (i.e., the pre-stress), 0 is the change in gravitational potential caused by 0 , s is the displacement vector and is assumed to be very small, and O()s represents all the terms that explicitly contain s . The Poisson’s equation is

2  004 G  . (4.9)

Eqs (4.8) and (4.9) describe the new background state without time-varying body tidal force being applied. In fact, eqs (4.8) and (4.9) can be generalized to describe a system that is pre- stressed by a combined effect of not only internal buoyancy but also other sources, such as static body tide, rotational potential, surface loading and etc., as long as they act in an entirely different frequency band from that of the time-varying tidal response I consider. In such generalized form,  0 needs to be updated to include all those potential components while any surface loading should be appropriately reflected through boundary conditions. In this study, I assume that the system is strained and pre-stressed only by the internal density anomaly 0 .

4.1.2 Net force due to density anomaly

The time-varying body tide may be induced by the orbital eccentricity or obliquity of the planetary body in question. For example, the Mercury’s tide is almost solely contributed by the Sun, and its time-dependent part is caused by the fairly large orbital eccentricity. Body tide in the moons of the gas giants is predominantly raised by their planets, while the effect from the Sun and other planets is negligible. For the Earth’s Moon, the tidal potential from the Sun is about

200 times smaller than that due to the Earth, and thus is excluded from my analyses. Also, I only consider the leading terms in the Moon’s tidal potential in my analyses, i.e. the harmonic degree-

2 terms caused by its eccentric orbit. I denote the tidal potential as Vttd () [or Vttd (,)r ], and the tidal force can be expressed as ()()0   0 Vt td . Since Vttd () is at degree-2 spherical harmonics, the response of a spherically symmetric (1-D) body would be at the same harmonics.

However, when coupled with non-spherical structures, the degree-2 tidal forcing also excites non-degree-2 harmonics in the response. Tidal response solution must be solved in a static system, on which the total external force or the net force is zero. On a spherically symmetric body, the tidal force 0Vt td () causes zero net force. However, when density anomaly 0 is present, the coupling between and the tidal force, i.e. 0Vt td (), may resonate at degree-1 harmonics and result in a nonzero net force on the system. This net force tends to accelerate the system and thus break the force balance in the equation of motion. To resolve this problem, I solve the degree-1 tidal response in the reference frame that is accelerating with the system. The rationale is that, physically, the net force would not cause any deformation and would be balanced out instantaneously by the orbital centripetal force; only the degree-1 internal force causes the deformation. I remove this net force by adding an inertial force to the system, keeping only degree-1 internal force that causes the deformation in the equation of motion.

Due to mode coupling, the forcing term fr(,)()t 0  V td t can be decomposed components in different harmonics, as f(,)(,) rtt  flm, r . It proves that only degree-1 lm, spheroidal force component can cause a net force on the system. Assuming that the degree-1 spheroidal force be expressed as

PB f1,m(,)(,)(,) rt f 1,m r t P 1, m f 1, m r t B 1, m , (4.10)

where m can be 1, 0 or -1. I project fr1,m (,)t on to the Cartesian coordinates, as

P fx(r , t )   sin cos  cos  cos  sin   f1, m ( r , t ) Y 1, m    B  f(r , t ) sin sin  cos  sin  cos  f ( r , t ) Y , (4.11) y   1, m 1, m    B  fz(r , t )   cos sin  0  f1, m ( r , t ) Y 1, m / sin 

where ftx (,)r , fty (,)r and ftz (,)r are x , y and z components of fr1,m (,)t , respectively. I compute the degree-1 net force by integrating fr1,m (,)t throughout the whole volume of the mantle, which is

F()t f (,) r txfˆ  (,) r tyfˆ  (,) r tzrˆ 2 sin drdd   . (4.12) net  x y z

Fnet ()t depends on the choice of m and

FPBnet(t ) f 0 ( t )( 1,mm 1, ) , (4.13) where

4 a ft( ) frtPB ( , ) 2 frtrdr ( , ) 2 , (4.14) 03   1,mm 1, rcmb

The net force accelerates the system, and the acceleration is

F ()t a ()t  net , (4.15) tot M where M is the total mass of the system. Solving degree-1 deformation in the accelerating reference requires the inertial force be added to every portion of the system. The inertial force, fint ()t , is in the opposite direction to the acceleration and is equal to

faint()()()tt  0   0 tot . (4.16)

4.1.3 Perturbation formulation

The time-varying tidal force induces displacement u()t in the pre-stressed system in addition to the 0 induced displacement s (see Sec. 4.1.1). Thus, the total deformation becomes su ()t . The physical variables in eq. (4.1) are now given by the linearized relations, as

E E1 E1  00  (u )  ( s  u )  [    ( u )] , (4.17)

E E1 E1  0 VV td  (u )  ( s  u )  [  0  td   ( u )] , (4.18)

E E1 E1 TTTTT00 (u )  ( s  u )  [  ( u )] , (4.19)

r   ()su   , (4.20) where  E1 ,  E1 and TE1 are Eulerian perturbations in density, gravitational potential and stress, respectively, due to the incremental deformation u()t . I substitute eqs (4.17) – (4.20) into eq.  (4.1), and introduce the inertial force term faint 0 tot to cancel the degree-1net force on the system, if there were. Ignoring all the terms in the second and higher order of displacement, i.e.

O(|u |2 ) and O(|su || |) (I assume s is reasonably small), decouples u()t from the background deformation (eqs (4.8) and (4.9)) into a linearized equation of motion, which is

TTTE1+  (uu )  (  )   (  E1 V )   E1    0 0 0 td 0 . (4.21) a   (  u   )    ( u   )  ( u   )    0 0 tot 0 0 0 0 0 0

Here, the Eulerian incremental stress is transformed into Lagrangian form through the relation,

L1 E1  TTT u  0 , (4.22)

  Also, the terms u () T0 and 00()u  in eq. (4.21) cancel each other in that

      0(u  0 )  u ()TT 0  u ( 0   0   0 )(||||)O s u , (4.23) considering O(|su || |) is a small second order term.

Eqs (4.22) and (4.23) remove the unknown pre-stress term  0 from eq. (4.21), leading to the final equation of motion,

TL1 (  E1 V )   E1    a  ( u   )     ( u   )  0 , (4.24) 0 td 00tot 000 0 where  E1 is the Eulerian density perturbation and

E1   ()0 u , (4.25)

TL1 is the incremental Lagrangian Cauchy stress, which is related to u through the constitutive relation

L1 L1    T T T(,,)()()0  0u   0  u I +  0  u  u , (4.26)

  where 0  0  0 , 0  0  0 . The Poisson’s equation is

2 E14 G  E1 . (4.27)

E1 L1 Note that u ,  , T and atot have the same time dependence as the tidal force Vttd (), given that the response is elastic. It is also worth noting that I do not need to determine the unknown pre-stress  0 in solving the tidal response, since those containing terms are cancelled through eqs (4.22) and (4.23). Compared with the governing equations in eqs (3.8) – (3.10), eqs (4.24) – (4.27) here include the impact of 0 and 0 , together with that of 0 , on the response. When , and vanish, eqs (4.24) – (4.27) reduce to the equations for the 1-D case and reconcile eqs (3.1) – (3.4). Hereafter, I omit the superscript indicator for an Eulerian or Lagrangain quantity.

I solve eqs (4.24) – (4.27) using a perturbation method, similar to in Chapter 3 that considered only 3-D structure in the shear modulus. I regard any differential tidal effect that is caused by the lateral heterogeneities as a small perturbation to the tidal response of its 1-D reference state. I denote the tidal solution of the 1-D reference system as (,)u00 and that of the 3-D system as (,)u  . The difference between the two solutions is consequence of the lateral heterogeneities in the mantle, denoted as (,)u . can be further divided into solutions that represent different levels of accuracy in terms of the order of perturbation in the lateral heterogeneities. Naming as the zeroth order solutions, includes the first and second order effects and beyond, i.e. (,)u11 , (,)u22 and (,)u  , respectively. Dropping

and replacing u and  in eqs (4.24) – (4.27) with u0 u 1 u 2 and 0  1  2 group different orders of solutions into individual sets of perturbation equations. These equations, up to the second order of perturbation, can be expressed into a generalized form as

TDDDDD ()()0uu   0   0    0   0 F , (4.28)

2 DDD 4 G  ( 0u )  G , (4.29)

T TDDDD00()() u I +  u  u , (4.30)

where FD and GD are the coupling terms, the subscript D indicates the order of the perturbation and can be 0, 1 or 2, i.e. the zeroth, the first or the second order. Eqs (4.28) – (4.30) have non- trivial solutions only when the coupling term FD or GD is nonzero.

When D  0 (i.e. the zeroth order), F0  0 V td and G0  0 , and eqs (4.28) – (4.30) solves the tidal response of the spherically symmetric (1-D) body. The technique in solving this equation was well developed decades ago (Farrell 1972; Longman 1962, 1963). Difficulties lie in solving the higher order equations, that is for D 1 or 2, due to involvement of the coupling

      terms. At higher orders, FFFFDDDD   , where FD , FD and FD represent contributions to FD from the coupling between the lower order ( D 1) solutions and the lateral heterogeneities 0 , 0 and 0 , respectively. , and are mutually independent and are given, respectively, as

 T , (4.31) FDDD 0() uu 1  1

 FDD ()01 u  , (4.32)

 FDDDD0()() uu 1   0   1   0    1  , (4.33)  0 ( uDDD 1 )   0  ( u 1   0 )  f

   Note that FD and FD keep the same form in the first and second order equations, while FD

 differs by the extra term fD between different orders. Specifically,

 fa1  0 V td   0 tot , (4.34)

 f2  00 ()() u   0   00  u   0   0tot a . (4.35)

At higher orders,

GDD4G  ( 01u  ) , (4.36)

and GD is nonzero only when 0  0 . Solving eqs (4.28) – (4.30) requires appropriate continuity conditions as well as boundary conditions at the outer surface and the core-mantle boundary (CMB). Across any concentric interface in the spherically layered mantle, uD , D , the normal traction (rˆTDD +B ) and rGˆ(   4   u ) H must be continuous, which can be expressed as DDD0  uu  rˆ T +   rˆ  (    4  G  )   0 , (4.37)  DDDDDDD   B   0 H 

 which is modified from eq. (3.11), where [] symbolizes the jump of the enclosed quantity across an interface, and rˆ is the normal vector in the radial direction. BD is caused by 0 or

0 while H D results from 0 , both of which are nonzero only in the higher order equations (i.e. D 1 or 2). Dividing into terms that represent and contributions, respectively,

  as BBBDDD, BD , BD and are

 ˆ T BDDDr 0() uu 1  1 , (4.38)

 BDDrˆ 01() u  I , (4.39)

H DDrGˆ 4 01u  , (4.40) respectively.

4.2 Solution method

I solve the perturbation equations, i.e. eqs (4.28) – (4.30), through a two-step procedure, as described in Chapter 3. Step one: I perform spherical harmonic (SH) and vector spherical harmonic (VSH) expansions respectively for any 3-D scalars and vectors that enter in eqs (4.28) – (4.30). In this way, a complete set of modes in the higher order response can be determined from the coupling terms. Separating radial and angular dependence reduces the partial

differential equations (PDEs, eqs (4.28) – (4.30)) to the ordinary differential equations (ODEs) with respect to the radius, which are then converted into a matrix form (Dahlen and Tromp, 1998; Tromp and Mitrovica, 1999b). Step two: I apply the propagator matrix method to solve the resulting matrix equation for each response mode. Eqs (4.28) – (4.30) and the corresponding continuity and boundary conditions are non- dimensionalized by eqs (3.34) – (3.39), and additionally,

0( 0 )  cmb0  ( 0 ),  0 ( 0 )   cmb0   ( 0  ),  0 ( 0 )   cmb0   ( 0 ), (4.41)

where the prime denotes a non-dimensional quantity, cmb and cmb are the reference density and shear modulus right above the CMB, respectively. Hereafter, all the following quantities are non- dimensional and I omit the primes for brevity. I expand 3-D scalars and vectors in the spherical coordinates (,,)r  into SH and VSH, respectively, of which the basis functions are given by eqs (3.13) and (3.19), respectively. Specifically, assuming that the lateral heterogeneities are composed of different eigen-structures in terms of spherical harmonics, they can be expanded into

 ()()(,)r  r Y   , (4.42) 00 l1 m 1 l 1 m 1 lm11,

 ()()(,)r  r Y   , (4.43) 00 l1 m 1 l 1 m 1 lm11,

 ()()(,)r  r Y   , (4.44) 00 l1 m 1 l 1 m 1 lm11, respectively, where  ()r measures the lateral variability of the heterogeneities and lm11  (r ) 1 . lm11

T uD , D and rˆTD (note that here TIDDDD00()[()] u   u   u ) are expanded respectively into eqs (3.24) – (3.30). Changes are made to expansions of the high-order ( D  0 ) coupling terms FD and GD , and the continuity conditions BD and H D . The expansion coefficients of the coupling terms have complicated forms, and solving them at order D requires

the response at order D 1 be fully determined. Here, I only present the symbolic forms as follows,

p,,, D b D c D FD()FFF lmPBC lm  lm lm  lm lm , (4.45) lm,

D GD GY lm lm , (4.46) lm,

p,,, D b D c D BD()BBB lmPBC lm  lm lm  lm lm , (4.47) lm,

D H D  HY lm lm , (4.48) lm, where D 1 or 2, while the complete expressions are given in Appendix A. The coupling between a single eigen-structure in the lateral heterogeneities and a single response mode at order gives rise to multiple response modes at order D, which are indicated by each non- trivial set of expansion coefficients from FD , BD , H D and GD . Therefore, the tidal solutions are limited to a finite number of pre-determined modes rather than occur in infinite number of modes. The SH and VSH expansion forms help turn the perturbation equations at order D (eqs

(4.28) – (4.30)) into equations for individual response modes, and these equations can be further reduced to ODEs with respect to radius, by dropping angular dependence in the harmonics. The ODEs of a single mode belong to either one of the two categories, i.e. spheroidal (s) and toroidal (t), I convert these ODEs into a matrix form, and the spheroidal and toroidal equations share the same general form, as eq. (3.40). By including 0 and 0 in addition to 0 , the only changes are made to the forcing vector and the continuity conditions. Specifically, for a spheroidal mode,

D p,,T D b D D Flm (0,0,FFG lm , lm ,0, lm ) , (4.49)

D D  D  DpD,,  DbD  DD  [][][][UVKRBSBQHlm lm   lm   lm  lm ][   lm  lm ][   lm  lm ]0   . (4.50)

For a toroidal mode

D c, D T Flm=F(0, lm ) , (4.51)

D D c, D [WTBlm ] [ lm  lm ]  0 . (4.52)

pD, bD, cD, Note that, here, Flm , Flm and Flm , which include the contributions from 0 and 0 , are different from those in eqs (3.43) and (3.49). The surface and the CMB boundary conditions remain the same, as in eqs (3.45), (3.46), (3.51) and (3.52).

Solving eqs (5) – (7) for the total tidal response (,)u  , up to the second order accuracy in the perturbation, is now equivalent to adding the responses of all the individual modes from different orders of perturbation together. Response at each mode is a combined effect of the three lateral heterogeneities and its relative magnitude is only a function of the lateral variabilities in the heterogeneities (see Appendix A). As a direct consequence of the perturbation formulation, the first-order response is simply the sum of the first-order responses on three single structures, in which only one source of lateral heterogeneity appears while the other two are zeroed out. This can be expressed as

1 1, 1,  1,  XXXXlm lm  lm  lm , (4.53)

1, 1, 1, 2, 2, 2, where X lm , X lm and X lm ( X lm , X lm and X lm for later) are first order (second order) solutions due to lateral heterogeneity only in and , respectively. Such simple rule, however,

1 does not hold for the second order responses. The first order solution X lm enters in the second-

2, 2, order coupling terms (eqs (4.31) – (4.36) and eqs (4.38) – (4.40)) and results in X lm , X lm ,

2, X lm and, additionally, cross effect between the lateral heterogeneities into the second order

2 solution X lm , i.e.,

2 2, 2,  2,          XXXXlm lm  lm  lm  lm   lm   lm  , (4.54)

where lm ’s represent the cross effect between two lateral heterogeneities out of three. The whole solution procedure of the perturbation method has been coded up and can be summarized as a flowchart in Fig. 4.1. At the initial step, the mode of the tidal force (i.e.,

(,)lm00), the 1-D reference properties of the planetary body and the 3-D elastic and desity structures (i.e., lateral heterogeneities in form of eigenstructures, eqs (4.42) – (4.44)) are set up.

All quantities are non-dimensionalized. Then, an outer loop solves the mode couplings up to the second order of perturbation for each eigenstructure in the lateral heterogeneities (eq. (3.53)), and all the high-order response modes and their associated matrix equations are determined, depending on different ‘parent’ – ‘child’ pairings. An inner loop then goes through all the response modes and solves the tidal responses, mode by mode, from zeroth to second order of perturbation. I implement a propagator matrix method based on a fourth-order Runge-Kutta numerical scheme to solve the matrix equations. The description of the Runge-Kutta implementation is provided in Subsection 3.3.5. The total tidal response is the sum of responses of each mode.

4.3 Results

I test the perturbation method by calculating the response of a laterally heterogeneous planetary body to a pre-determined tidal force. As I did in Chapter 3, I continue to apply the method to the Earth’s moon, for which the 1-D structure (see Table 3.1) is better constrained through seismological studies, compared to other planetary bodies. As the Moon’s long- wavelength structure in depth is still unknown, I propose a hypothetic structure in the Moon’s elastic properties and density that can be described predominantly by a (1, 1) spherical harmonic in certain depth range, in the chosen coordinate system (i.e., the origin of the coordinate system is fixed at the center of the Moon, with x-axis pointing towards the sub-Earth point and z-axis perpendicular to the Moon’s orbital plane). The lateral heterogeneities in the two elastic moduli and the density can thus be expressed as

 0  1,1()r  0 () r Y 1,1    1,1 () r  0 ()sincos r   , (4.55)

 0  1,1()r  0 () r Y 1,1    1,1 () r  0 ()sincos r   , (4.56)

 0  1,1()r  0 () r Y 1,1    1,1 () r  0 ()sincos r   , (4.57)

Figure 4.1. Flowchart of solution procedure of the perturbation method.

3 where Y  sin cos ,  reflects the peak value of the lateral variability and 1,1 4 1,1 3  . In general, the lateral variability  ()r is piecewise constant as a function of 1,14 1,1 lm11

   the radius. In the following calculations, however, I set 1,1() 1,1 , 1,1() 1,1 and 1,1() 1,1 to be constant throughout the mantle for test purposes. I select the time-varying part of the tidal potential that is due to the eccentricity of the Moon’s orbit, and only consider its harmonic degree-2 terms in my calculations (eq. (3.5)). In a simplified form, the tidal potential is given by

Vtd(,)()()()r t  2,0 r Y 2,0   2,2 r Y 2,2 T t , (4.58) which contains (2, 0) and (2, 2) harmonics, respectively. Tt() represents the time dependence of the tidal potential, which can be dropped in static tidal response calculations. The tidal response can be fully described by non-dimensional Love numbers (Love 1911). For a 1-D planetary body, the tidal response must be spheroidal and the solution can be expressed by

0 Ul m()()/ r   h l r g0  0 0   0  V0()()()/ r td r l r g , (4.59) l0 m 0  l 0 m 0  l 0 0      K0 ()() r k r l0 m 0   l 0  where h , l and k are tidal Love numbers, which depend only on the 1-D structure and the l0 l0 l0 harmonic degree of the tidal force (Love 1911; Farrell 1972). For a planetary body with 3-D structures, response due to mode coupling occurs in multiple harmonics of either spheroidal or toroidal mode. For each response mode at order D of the perturbation, I convert the tidal solution into non-dimensional form using eq. (4.59), which is then further normalized by the three Love numbers. As a result,

UDDDD()()()() r V r K r W r hDDDD() rlm , l () r  lm , k () r  lm , w () r  lm , (4.60) lmU0()()()() r lm V 0 r lm K 0 r lm V 0 r l0 m 0 l 0 m 0 l 0 m 0 l 0 m 0

D D D D where hlm , llm and klm ( wlm ) are called relative responses of spheroidal (toroidal) mode D D D D D D s(,) l m [t(,) l m ]. Note that hlm , llm , klm and wlm can be positive or negative values. At the zeroth order, h0 l 0   k 0  1, and w0  does not exist. l0 m 0 l 0 m 0 l 0 m 0 lm00

4.3.1 (1, 1) Lateral heterogeneity in lambda

In Chapter 3, benchmark tests were performed for the cases in which lateral heterogeneity is purely in the shear modulus. Here, I conduct the same procedure for lateral heterogeneity in lambda, by comparing the results from the perturbation method and the finite element method ( A et al., 2013; Zhong et al., 2003). For test purposes, the response to (2, 0) and (2, 2) components of the tidal potential is computed separately, so the relative amplitude of each

 forcing component is not a concern. I vary the lateral variability 1,1 in the range from 2.5% to 40%, within which nine different values are picked for numerical calculations. Figs 4.2a and b show the diagrams of mode coupling to the second order of perturbation for (2, 0) and (2, 2) tidal forcing components, respectively. The modal selection rule (see eq.

(3.53)) holds the same for both 0 and 0 (and 0 ) cases, except that does not induce any toroidal modes as 0 (or 0 ) does. This being said, I obtain the same set of spheroidal modes in the higher order response, but the toroidal modes t1(2, 1) and t 2 (3, 2) that occur in the case and the mode couplings that are associated with them (marked by dashed frames and lines in Fig. 4.2) do not exist in the case. Fig. 4.3 shows the absolute values of the relative tidal response of all the spheroidal modes for different values of , computed from both the perturbation method and the finite element method. The whole mantle shell is divided into 12 483 elements in the finite element models. SH expansion is done for the grid-based results, and the expansion coefficients can then be directly compared with results from the perturbation method. The top and bottom panels of Fig. 4.3 show the responses at different modes to (2, 0) and (2, 2) tidal forcing components, respectively. For each tidal forcing component, the left and right panels show relative responses

Figure 4.2. Diagrams (a) and (b) show the hierarchy of the mode couplings (up to the second order of perturbation) between the (2, 0) and (2, 2) tidal forces, respectively, and the spherical harmonic (1, 1) laterally heterogeneities in shear modulus, lambda or density. s or t represents a spheroidal or toroidal mode, and the superscript denotes the order of perturbation (0, 1 or 2). The dashed boxes and lines are toroidal modes and their associated couplings, which are prohibited in the response of the lateral heterogeneity in lambda.

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Figure 4.3. The relative responses (absolute values) in the radial displacement (a and c) and the gravitational potential (b and d) of a Moon with (1, 1) lateral heterogeneity in , to (2, 0) (top) and (2, 2) (bottom) tidal forces. The plot is in a log-log scale. x-axis is the lateral variability of  the lambda 1,1 , where is varied from 2.5 to 40 per cent. Each response mode is indicated by a solid straight line and the triangles in the same color, which represent the results from our second-order perturbation method (P) and the finite element method (C), respectively. The open circles are responses that are beyond the second order of perturbation. The dashed lines are all the other harmonics in 04l and 0 ml, which are numerical artifacts from the finite element calculations.

D in the surface radial displacement (i.e. |halm ( ) |) and incremental gravitational potential (i.e. D |kalm ( ) | ), respectively. Fig. 4.3 demonstrates a remarkably good agreement between the results from these two different methods for the 0 case (also Table 4.1). All the significant modes from the finite element models are exactly the first and second order spheroidal modes predicted by the perturbation method, and they are clearly distinguishable from the numerical noise (see Fig. 4.3). Furthermore, the finite element results for each first (second) order modes follow very

 closely to the linear (quadratic) function of 1,1 (solid lines in Fig. 4.3), as expected from the perturbation formulation. By comparing Fig. 4.3 with Fig. 3.3 in Chapter 3, which showed the responses due to

0 , I find that the responses due to in general appear to be much weaker than the corresponding responses due to for the same level of lateral variability. I have also observed that the gravitational response is about one order of magnitude weaker than the corresponding radial displacement response for the case (Fig. 4.3). This is different from the case where the relative gravitational and displacement responses are comparable (Fig. 3.3). For

 1  1  example, when 1,1 10 per cent, the first order displacement responses |ha3,1 ( ) | and |ha3,3 ( ) | from the (2, 0) and (2, 2) forcing, respectively, are at the level of 103 , while the corresponding gravitational responses |ka1  ( ) | and |ka1  ( ) | are both at 104 level. In comparison, when 3,1 3,3  1  1  1  1  1,1 10 per cent, |ha3,1 ( ) | and |ka3,1 ( ) | from the (2, 0) forcing and |ha3,3 ( ) | and |ka3,3 ( ) | from the (2, 2) forcing are all greater than 102 . The mode s1(1,1) is the only exception. The

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Table 4.1. Relative differences between the responses from our perturbation method and those  from the finite element methods for both 0 and 0 cases at the lateral variability of 1,1 1,1 10 per cent.

Modea b  k (%)  h (%)  l / w (%) / (%) 1 s (1, 1)(2,0) — 0.62 0.21 0.12 1 s (3, 1)(2,0) 5.07 0.48 0.04 1.42 1 t (2, -1)(2,0) — — — 0.41 2 s (2, 0)(2,0) 3.31 2.37 1.44 0.52 2 s (2, 2)(2,0) 0.82 0.37 0.87 0.97 2 s (4, 0)(2,0) 1.84 0.19 0.36 0.46 2 s (4, 2)(2,0) 1.61 0.07 0.57 2.88 2 t (3, -2)(2,0) — — — 0.63 1 s (1, 1)(2,2) — 0.66 0.04 0.31 1 s (3, 1)(2,2) 6.86 0.93 0.42 0.91 1 s (3, 3)(2,2) 5.34 0.50 0.14 0.73 1 t (2, -1)(2,2) — — — 0.44 2 s (2, 0)(2,2) 1.29 0.38 1.00 0.69 2 s (2, 2)(2,2) 0.95 0.49 0.91 0.53 2 s (4, 0)(2,2) 1.96 0.18 1.14 0.23 2 s (4, 2)(2,2) 21.2 1.95 1.49 0.63 2 s (4, 4)(2,2) 8.36 0.33 0.26 1.65 2 t (3, -2)(2,2) — — — 0.69 a Spheroidal or toroidal mode of harmonic (l, m) from (2, 0) and (2, 2) tidal forcing, respectively b Percentage difference in the relative response X,  X ()/XXXCitcom pert pert , where X can be k (potential), h (radial displacement), l and w (horizontal displacement of spheroidal and toroidal, respectively) c 0 results for  k and  h are in Table. 3.2

1  response |ha1,1 ( ) | to either (2, 0) or (2, 2) forcing is even larger than that of the 0 case, making 1  |ha1,1 ( ) | comparable with the first order degree-3 responses (see Figs 4.3a and c; note that 1  1  |ha1,1 ( ) | coincides with |ha3,3 ( ) | in Fig. 4.3c). Because the responses due to 0 are relatively

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small, especially for the gravitational responses, they reach the numerical noise level of ~106 (i.e., those dashed lines. Note that in Chapter 3, I estimated this empirical noise level per the

 numerical resolution) when 1,1 is small. This explains why the agreement on the gravitational responses between the two methods for the 0 case degrades, and some results from the finite element calculations fluctuate around the predictions from the perturbation method even by more than 10 per cent, like s2 (4,2) from the (2, 2) tidal forcing as one of the examples (Fig. 4.3 and Table 4.1). I find that the significant difference in the tidal response amplitude between the and

0 cases occurs not only for (1, 1) lateral heterogeneity, it also exists for lateral heterogeneity in other harmonics. Thus, in general, when similar level of lateral heterogeneity exists in  and, tidal response due to , especially in the gravitational potential, would be significantly smaller than that due to . Finally, like in Chapter 3, some weak modes from the finite element solution (e.g., s(3,3) and s(4,4) marked by circles in Figs 4.3a and b) do not appear in the perturbation solution. This is because these modes are only expected from third or higher order perturbation theorem, which I do not attempt here.

4.3.2 Horizontal displacement due to (1, 1) lateral heterogeneities in shear modulus and lambda

All the tests have so far been performed for tidal response in the radial displacement and the gravitational potential, as in Figs 3.3 and 4.3. No direct comparison has been made for tide- induced horizontal displacement, although Fig. 3.4 indirectly showes a good agreement for the toroidal component of the horizontal deformation. Here, I check the tidal response solution in horizontal displacement due to and directly, as complementary validation of the perturbation method. I add into the finite element code a post-processing functionality to perform the VSH expansion of the horizontal displacement field uh at the surface, where

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DD uhh u(,,)()()a  Vlm a B lm  W lm a C lm , (4.61) lm, and it contains both spheroidal and toroidal components. According to the selection rules for degree-2 tidal forcing and (1, 1) lateral heterogeneity, only certain spheroidal and toroidal modes need to be considered in my tests (i.e., 14l and 0 ml ( lm   0 ) for spheroidal

D D (toroidal) modes). I compute |lalm ( ) | and |walm ( ) | for the spheroidal and toroidal modes, respectively, and plot them together with the perturbation predictions for comparison. Results are shown in Fig. 4.4 for four individual cases: (2, 0) and (2, 2) tidal forcing on

0 (a and b, respectively) and on 0 (c and d, respectively) cases, respectively. Again, all the first and second order spheroidal and toroidal modes (triangles and squares in Fig. 4.4, respectively) predicted by the perturbation method are also the most significant modes from the finite element results, and the analytic results (solid lines in Fig. 4.4) match the corresponding

D D |lalm ( ) | and |walm ( ) | from the finite element models remarkably well (Fig. 4.4 and Table 4.1).

Also, the responses in horizontal displacement due to are on average larger by an order of magnitude than those due to (Figs 4.4a and b verse 4.4c and d), similar to what I found in the radial displacement responses in the last subsection. The relative amplitudes of the responses in horizontal displacement (Fig. 4.4) are similar to those in radial displacement (Fig. 4.3 for the s3 (3,3) case and Fig. 3.3 of Chapter 3 for the case), including those of the modes and s4 (4,4) (i.e., open circles in those figures), which are beyond the second order perturbation s1(1,1) theorem but are obtained from the finite element models. However, the mode in the 1  case is different in that its horizontal displacement response |la1,1 ( ) | , as one of the largest responses for horizontal displacement, is about two orders of magnitude larger than its

1  corresponding radial displacement response |ha1,1 ( ) | (Figs 4.4a and b verse Fig. 3.3 of Chapter 3). For the cases, both (2, 0) and (2, 2) tides induce the same set of toroidal modes in the

1 2 1  response, which are t (2, 1) and t (3, 2) . The toroidal response |wa2, 1 ( ) | is surprisingly

 large, especially for that induced by the (2, 0) tide, and when 1,1 is sufficiently large (e.g.

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Figure 4.4. The relative responses (absolute values) in the horizontal displacement of a Moon with (1, 1) lateral heterogeneities in  (top)and (bottom), respectively, to (2, 0) (left) and (2, 2) (right) tidal forces. The plot is in a log-log scale. x-axis is the lateral variability of the shear   modulus 1,1 or the lambda 1,1 , where both and vary from 2.5 to 40 per cent. Each response mode that arises from our second-order perturbation method (P) is indicated by a solid straight line. The response of the same mode acquired from the finite element results (N) is marked by a triangle or a square, which denotes the occurrence of a spheroidal or toroidal mode. The open circles and diamonds represent the spheroidal and toroidal responses that are higher than the second order of perturbation, respectively. The dotted and dashed lines are all the other spheroidal and toroidal components for 04l , respectively, that are prohibited by the selection rule, and are numerical artifacts from the finite element calculations.

 1  1,1  30 per cent), |wa2, 1 ( ) | is even close to 1. According to the selection rules, two third order toroidal modes t3 (4, 1) and t3 (4, 3) are predicted to exist, and with responses at the same level as those second order degree-4 responses, they can be clearly identified in the finite element results (diamonds in Figs 4.4a and b). Note that 0 does not cause any toroidal modes 1  2  in the response, which is evident in Figs 4.4c and d: |wa2, 1 ( ) | and |wa3, 2 ( ) | in the 0 case are both at the noise level (dashed lines in corresponding colors). The dotted and dashed lines represent numerical errors in the spheroidal and toroidal components, respectively, which are not predicted from the perturbation theorem. However, the finite element models show large amplitude but random pattern for |wa ( ) | (yellow dashed line in Fig. 4.4), although the 1, 1 perturbation theorem asserts its non-existence. I think that this is caused by the inaccuracy of the finite element code in performing VSH expansion for degree-1 toroidal component.

4.3.3 (1, 1) lateral heterogeneity in density

In this subsection, I calculate the tidal response to (1, 1) lateral heterogeneity in density

(0 ) using the perturbation method. For now, since the finite element code is unable to include a 3-D density structure, similar benchmark tests as those done for 0 and cannot be

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fulfilled for 0 . Instead, I only show the perturbation solutions for the case at a specific lateral variability and compare it with that for 0 . I first determine a reasonable lateral variability for the 3-D density structure. In the Earth system, the largest heterogeneities are present in the uppermost (top 200 km, including the crust) and the lowermost (bottom 300 km) mantle. Acknowledging that the shear velocity variation

(VVss/ ) in the Earth can reach ~10 per cent in the uppermost mantle (e.g., Shapiro and Ritzwoller, 2002) and ~2.5 per cent at long wavelengths in the deep mantle (e.g., Su and Dziewonski, 1997), a 10 per cent lateral variability may be taken as a global upper bound for lateral heterogeneities in the elastic structure (note that the shear modulus variation is approximately twice as much as the shear velocity variation), which may also apply to the Moon. In comparison, the density variation (/  ) in the Earth’s mantle is on average less than 0.5 per cent (e.g., Forte and Mitrovica, 2001). For the lunar mantle, I roughly estimate the bulk density variation from the thermal or thermochemical convection modelling. For examples, in Zhong et al. (2000), the onset of a degree-1 thermochemical upwelling ~4 Ga introduces less than 1 per cent density variation for different models; in Laneuville et al. (2013), long-term thermal convections predict on average a less than 2 per cent density variation in the present-day lunar

 mantle. I thus choose 1,1  2.5 per cent for in the lunar model, and compare its impact on

 the tidal response with that of at 1,1 10 level. and induce the same set of high order modes (Fig. 4.5), and I compare the

D D D D relative responses (i.e., klm , hlm , llm and wlm ) between the corresponding modes from and to (2, 0) and (2, 2) tides, respectively, in Fig. 4.5. Each colored bar represents the absolute value of the response for each mode from or (the response without a bar is smaller than

105 ). The diamond on the bar means that response has a negative value. I find that the amplitudes of most first order responses due to and are comparable, while the second order responses due to are on average two orders of magnitude weaker than those due to . Also, the responses due and for some modes have opposite signs, such as the first order degree-3 spheroidal modes and degree-2 toroidal modes. Because a first order response is

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Figure 4.5. Comparisons of the relative responses (absolute values) in gravitational potential (left region, k), radial displacement (center region, h) and horizontal displacement (right region, l   and w) between lateral heterogeneities in  and, for 1,1 10 % and 1,1  2.5 %. Panels (a) and (b) are for (2, 0) and (2, 2) tidal forces, respectively. The y-axis is in log scale and the x-axis indicates all the modes that occur in a specific type of response (harmonics without an initial letter indicate spheroidal modes). The height of a bar means the amplitude of a response, and those responses without a bar are smaller than 105 . The grey diamond in a bar indicates that response has a negative value.

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proportional to  while a second order response is proportional to  2 , the sign of the first order response will be controlled by the choice of  . In Qin et al. (2012), I proposed a (1, 1) lateral heterogeneous structure in the lunar mantle that depicts a hotter but chemically denser region at

 depth, which implies a reduced shear modulus (i.e. 1,1  0 ) and a positive density anomaly (i.e.

 1,1  0) within that region. As a result, the degree-3 spheroidal and degree-2 toroidal responses induced by 0 and 0 , respectively, would add to each other, enhancing the total responses. On the other hand, if both density and elastic heterogeneities are caused by thermal effects (i.e.,

  for hot regions, 1,1  0 and 1,1  0 ), then their combined effects may reduce those responses. In Zhong et al. (2012), I asserted that the induced degree-3 tidal response in the Moon’s gravitational potential was a promising measurable to constrain the (1, 1) structure in the lunar mantle. Now, the combined effects of and potentially make detection of the degree-3 spheroidal responses (maybe also the degree-2 toroidal responses), if there were, more plausible.

4.3.4 Tidal effect of lunar crustal thickness variations and sensitivity kernel

Significant thickness variations are present in the lunar crust (e.g., Wieczorek et al., 2013). Here, I use the perturbation method to explore the effect of lunar crust on the tidal response, based on which I answer two questions concerning the tidal response measurements: 1) would lateral heterogeneities in the lunar crust (top tens of kilometers) have significant impact on the tidal response, and 2) how significant would such impact be compared with that due to a 3-D structure in the deep mantle? I first calculate the gravitational tidal response caused by the lunar crust. Since a non- spherical topography of an interface cannot be directly incorporated into the perturbation formulation, lunar crustal thickness variations are modelled into lateral heterogeneities in the material properties in my calculations. Here, I assume full compensation in the lunar crust and upper mantle at long wavelengths, and thus ignore any density variations caused by crustal thickness variations. I use a simple strategy to map the crustal thickness onto the variations in the

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shear modulus and lambda, which is illustrated in Fig. 4.6. I adopt the crustal thickness model from Wiezcorek et al. (2013), of which the mean crustal thickness is 34 km. I take the crustal layer (top 34 km) as Layer 1 (L1) and define the layer between the deepest point of Moho and the 34 km depth as Layer 2 (L2) (Fig. 4.6). Material properties below L2 are considered purely 1-D or spherically symmetric. I use the following conversions to obstain the 3-D elastic structures in these two pre-defined layers (shear modulus is shown only and lambda is mapped in the same way):

L1:

[(hs h m ) c  h m m ] / D 1 if h m  0 1  (,)     hs c/ D 1 if h m  0 , (4.62) L2: [(h )  ( h  D ) ] / D if h  0  m c m 2 m 2 m 2 (,)     m if hm  0

where hhss (,) and hhmm (,) are heights of surface and Moho relative to the mean

Moho at every point, D1 and D2 are the thickness of L1 and L2, respectively, c and m are shear modulus of the crust and upper mantle, respectively. Fig. 4.7a shows the maps of percentage differences in the shear modulus (top) and the first Lamé parameter (bottom) from reference values for L1 (left) and L2 (right), respectively. The thin crust on the nearside leads to increased values of elastic moduli in L1, since mantle material with larger elastic moduli intrudes into L1 (Fig. 4.7a). The farside has excess crustal material above the mean surface and below the Moho, thus causing elastic moduli increased in L1 and reduced in L2 (Fig. 4.7a). The laterally varying (3-D) structures in the elastic moduli, before being included in the perturbation formulation, are expanded into spherical harmonics, which are truncated to degree 5 in my calculations (because degree-5 harmonics are of the highest degree to induce degree-3 first-order responses when coupled with degree-2 tidal force). Fig. 4.7b shows that L1 and L2 are dominated by (2, 1) and (1, 1) harmonics, respectively, and the lateral variabilities in the shear modulus and first Lamé parameter are close. Since the tidal

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Figure 4.6. Mapping of the lunar crustal thickness variations onto lateral variations in elastic properties. (top) lunar crustal thickness model from Wieczorek et al. (2013). The mean crustal thickness is 34 km. (bottom) Mapping strategy that converts crustal thickness to 3-D structure in shear modulus and first Lamé parameter.

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Figure 4.7. Lateral varying (3-D) elastic structures in two near-surface layers of the Moon from conversion of the lunar crustal thickness variations. a) Maps of percentage lateral variabilities in shear modulus (top) and first Lamé parameter (bottom) in layers L1 (left) and L2 (right), respectively. b) lateral variabilities for harmonics up to (5, 5) in lateral variations in shear modulus (solid lines) and first Lamé parameter (dashed lines), in layers L1 (red) and L2 (navy) (defined in the main text), respectively.

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effect due to first Lamé parameter is much weaker than that due to shear modulus, I therefore ignore the effect of the first Lamé parameter in my calculations. The gravitational tidal responses up to degree 4 are calculated for (1, 1) lateral heterogeneities in the shear modulus in L1 and L2, as shown by the dark grey bars in Fig. 4.8 for (2, 0) (top) and (2, 2) (bottom) tidal forces, respectively. The response at each mode is simply doubled as to take into account any laterally varying effects that are missed to be considered in the modelling (shown by the light grey bars in Fig. 4.8). It shows that the tidal response caused by the crust is only at a 104 level and is dominated by some degree-2 and degree-3 responses (Fig. 4.8). I also calculate the tidal response caused by a (1, 1) structure in the shear modulus at great depth of the mantle for three cases: 1) 5% lateral variability in bottom half of the mantle (red squares in Fig. 4.8), 2) 2.5% lateral variability in bottom half of the mantle (navy squares in Fig. 4.8) and 3) 5% lateral variability in bottom quarter of the mantle (green squares in Fig. 4.8). I find that whereas all the second-order responses caused by the (1, 1) deep structure are small and are buried by the responses caused by the crust, the first-order degree-3 responses are much stronger than those caused by the crust, for all three cases (Fig. 4.8). It suggests that the tidal effect of the lunar crustal thickness variations is not significant and it would not affect the idea of using measurements of degree-3 tidal responses to constrain the degree-1 interior structure of the Moon, even if the degree-1 strucutre is relatively deep and weak. I then calculate the sensitivity kernel of degree-3 gravitational response to degree-2 tidal forces with respect to different radii of a (1, 1) structure in shear modulus, in order to determine at what depth would a degree-1 structure have relatively large impact to the tidal response. I equally divide the lunar mantle into 60 layers (23.5 km thick) in the radial direction, and calculate the first-order degree-3 gravitational responses (relative response as described before) due to 10% (i.e., lateral variability) (1, 1) lateral heterogeneity in the shear modulus of each layer for (2, 0) and (2, 2) tidal forces separately. The gravitational responses as functions of the radius of the lateral heterogeneity, i.e., sensitivity kernels, are shown in Fig. 4.9. Based on the kernel functions, I find that the degree-3 gravitational responses (positive or negative) do not change

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(2, 0)

(2, 2)

Figure 4.8. Comparisons of gravitational tidal effects due to lunar crustal thickness variations and a presumed (1, 1) structure in the deep mantle. Gravitational tidal responses are solved using the perturbation method for (2, 0) (top) and (2, 2) (bottom) tidal forces, separately. x-axis shows the harmonic modes up to (4, 4). Dark and light grey bars are responses and doubled responses caused only by the lunar crust. Red, navy and green squares show tidal responses caused by a (1, 1) structure in shear modulus for different depths and lateral variabilities, respectively, as shown in the inset.

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Figure 4.9. Sensitivity kernels of first-order degree-3 gravitational responses to (2, 0) and (2, 2) tidal forces for (1, 1) lateral heterogeneity at different radii in the mantle. Red curve is (3, 1) response to (2, 0) tidal force, and navy and green curves are (3, 1) and (3, 3) responses to (2, 2) tidal force, respectively. The responses are normalized by the maximum absolute value of the responses. The lateral variability of shear modulus is set to a constant of 10%.

signs with changes in depth of the lateral heterogeneity, which means the first-order tidal responses of (1, 1) lateral heterogeneity at different depths will add to the todal response without cancellation effect. I also find that lateral heterogeneity at intermediate depths have much larger

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impact on the tidal response than that near the surface or the CMB. This conclusion has also been generalized to non-degree-1 eigenstructures, suggesting that tidal response of the Moon is more sensitive to a 3-D structure at intermediate depth of the mantle while it is less likely to be measured if the 3-D structure is closer to the surface or the CMB.

4.4 Conclusion and discussion

I have developed a complete perturbation method in calculating the elastic response of a laterally heterogeneous (3-D) planetary body to time-varying body tides, by incorporating both 3-D elastic and density structures of the mantle into one formulation. As done in the previous study, I apply the perturbation method to a laterally varying Moon with small (1, 1) heterogeneity in its elastic or density structure, and solve its response to degree-2 tides to the second order accuracy in the perturbation. I have also benchmarked the perturbation method by comparing its solution of each modal response with that from a finite element code, in not only radial displacement and gravitational potential but also, for the first time, horizontal displacement. For now, I am only able to perform the benchmarks for elastic heterogeneities in the shear modulus and the first Lamé parameter (lambda), not for the density perturbation. The great match between the perturbation and numerical solutions verifies the correctness of the perturbation method as well as the finite element method.

In deriving the perturbation equations, density anomaly is treated differently from lateral heterogeneities of the two elastic moduli, due to its involvement of more complex physical process. First, density anomaly acts as a buoyancy force that drives the mantle flow, and thus causes a pre-stressed and deformed background state before tidal force is applied; the larger the density anomaly is, the larger the pre-stress is and the more the system gets deformed. A reasonably small density anomaly is a prerequisite for the perturbation formulation, otherwise significantly large deformation in the background state would couple with the tidal effect (i.e. a non-negligible O(|su || |) terms), making the equations too complicated to be solved.

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Fortunately, because large long-wavelength density variations cannot be dynamically maintained in long-term evolution, they are small (<1 per cent) in a real planetary system, for which the perturbation method is applicable. Second, tidal force acting on a 3-D density structure may induce a net force on the system, which will break the force balance in the equation of motion. To resolve this problem, I introduce an inertial force onto the system to zero out the effect of the net force, keeping the internal force that causes the deformation in the equation. The perturbation method enables investigation of the impact of individual source of lateral heterogeneities on the tidal response as well as their combined effect. I have compared the tidal responses on the lunar model due to those three (1, 1) lateral heterogeneities, with the lateral variabilities in the elastic and density variations set to be 10 per cent and 2.5 per cent, respectively. I find that the lateral heterogeneity in the shear modulus has the greatest impact on the tides; the first order effect on the response from the density heterogeneity is comparable to that from the shear modulus heterogeneity; the lambda heterogeneity induces much weaker response, especially for that in the gravitational potential, which is on average two orders of magnitude smaller than that from the shear modulus or the density. These findings are also true for lateral heterogeneities of different eigen-structures (harmonics). Therefore, as a generalized remark, lateral heterogeneities in the shear modulus and the density are primary contributions to the tidal response, at a similar mutual level of lateral variability. Using detectable degree-3 tidal responses to constrain the longest-wavelength (i.e. degree-1) structure in the Moon is the motivation of my studies. In Zhong et al. (2012) and Qin et al. (2014), I found that (1, 1) shear modulus variations at a 10 per cent level throughout the lunar mantle can cause >2 per cent degree-3 (relative) responses, which is pretty significant. By including the density variations, I find that a reduced (enhanced) shear modulus accompanied by an enhanced (reduced) density anomaly in one hemisphere always reinforces the degree-3 responses, which is true not only for (1, 1) structure but also for other degree-1 structures (i.e., (1, 0) or (1, -1)). For the model proposed in Chapter 2, if a residual pile of the hotter and chemically denser MIC material (reduced shear modulus and larger density than the ambient mantle material)

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remained in one hemisphere of the Moon, the degree-3 tidal responses would approximately be twice as much as those by only considering the shear modulus effect. If, for simplicity, assuming such degree-1 structure were present throughout the mantle, the degree-3 responses would reach a 5 per cent level. In contrary, if the lunar mantle convection were thermally driven, as proposed by Laneuville et al. (2013), the degree-1 component of the hotter and lighter upwelling (reduced shear modulus and lower density than the surrounding code material) would cancel out the degree-3 responses, making detecting them more difficult. Although tested only for (1, 1) lateral heterogeneities, the perturbation method can apply to a more realistic model, in which the three lateral heterogeneities have more complicated angular and radial dependence: the angular dependence can be expanded into spherical harmonics, while the expansion coefficients can effectively be a piecewise constant function of the radius. The seismic tomographic model is a direct representation of this type of model, which is currently only available for the Earth. For other planetary bodies, the 3-D elastic and density structures are unknown and thus can only be inferred from 3-D mantle convection modelling (e.g., Laneuville et al., 2013; Zhong et al., 2012), through conversion of the modelled 3-D temperature and pressure fields. Prospects of the applications of the perturbation method rely on more advanced geodetic techniques in future missions. The perturbation method may be adapted and optimized for efficient 3-D inversions, provided that highly precise tidal response measurements as well as abundant geological constraints are available. Taking the Earth’s moon as one example, Monte Carlo inversions can be done to locate the most plausible parameter regimes of the 3-D structure, by minimizing the misfit on the tidal response between the measurements and the simulations. An optimal constraint of long-wavelength structures of the present-day lunar mantle can help build more reliable convection models, say, one can tweak the initial conditions by filtering out the unlikely final states of the convection.

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Chapter 5

Summary and future plan

5.1 Summary

In this thesis, I have developed a new semi-analytical method based on perturbation theory to calculate the elastic tidal response of a terrestrial planetary body with 3-D interior structures in the mantle. This theoretical study was motivated by: 1) the Earth’s Moon possesses a hemispherical asymmetry near its surface, and recent studies implied that such asymmetry may also be present in the lunar deep interiors; 2) numerical studies suggested that long-wavelength (primarily harmonic degree-1) 3-D structures of the Moon’s interiors may be constrained by its response to the time-varying body tides (i.e., tidal tomography); and 3) advancement of modern geodetic techniques would make lunar tidal tomography possible in future missions, thus requiring effective theoretical tools be developed to analyze the data. The perturbation method is complete in formulation in that it incorporates all isotropic sources of lateral heterogeneities in a 3-D structure, including those in the two elastic moduli and the density. The validity of the perturbation method has been checked through benchmarks against a finite element code. The perturbation method is computationally more efficient than a numerical method and is advantageous for performing spectral analysis on the tidal response. In Chapter 2, I propose that the present-day Moon may have a long-wavelength 3-D structure in its deep interior, based on statistical analyses on the recent seismological and remote

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sensing data. I first establish a spatial correlation between the mare basalt deposits and the epicenters of well-determined deep moonquake events based on robust statistical tests, which links the Moon’s ancient volcanic eruptions with its current state in the deep mantle. Then, I analyze the timing of lunar mare volcanism using the latest age model of the nearside mare basalt deposits that were derived from crater counting measurements. I find that mare volcanism may have experienced different episodes of eruptions in a long duration, and the PKT may have played an important role in causing that. I propose a long-term thermodynamic process in the lunar mantle that connects the nearside localization of PKT, the nearside emplacement of mare basalts and the nearside occurrence of DMQs, in a dynamic and self-consistently fashion. This hypothetical process suggests that the present-day Moon may have a nearside-farside asymmetry in its deep interior structure, similar to that from surface manifestations. In Chapter 3, I formulate the perturbation method for 3-D structure only in the shear modulus, and present a semi-analytical solution method to calculate its effect on the tidal response. I take the lateral heterogeneity in the shear modulus as a small perturbation to the 1-D reference structure, which allows splitting of the governing equations into the equations of different orders of perturbation. The perturbation equations, up to the second order in this study, are transformed into a matrix form through SH and VSH expansions. The tidal response is then solved in a two-stage process: 1) determine the high-order harmonic modes in the response from the mode coupling terms, and 2) solve the tidal response for each individual mode by implementation of the propagator matrix method. The perturbation method is tested on the Moon for various cases, especially for a Moon with a hemispherically asymmetric (i.e., harmonic (1, 1)) structure in the shear modulus. The tidal response solutions in radial displacement and gravitational potential agree remarkably well with those from the finite element code, meaning the perturbation method has been correctly formulated and implemented. Chapter 4 is a continuation of Chapter 3. I complete the perturbation formulation by including lateral heterogeneities in the first Lamé parameter (lambda) and the density, in addition to that in the shear modulus. Introduction of density anomalies causes both mathematical

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complications in the formulation and extra physical effects on the system, including a pre- stressed background state and a nonzero net force induced by tidal forces. As done in Chapter 3, I test the perturbation method on the (1, 1) Moon for the tidal effects of each source of lateral heterogeneity, for responses not only in radial displacement and gravitational potential but also in horizontal displacement. Within a reasonable upper bound for lateral variability of each source of heterogeneity, I find that 1) lateral heterogeneity in the shear modulus has the largest effect on the tidal response, 2) lateral heterogeneity in density has a comparable effect on the response with that of shear modulus and thus may reinforce or cancel the shear modulus effect in the response, and 3) the tidal effect due to lateral heterogeneity in lambda is generally 1 to 2 orders of magnitude weaker than that due to lateral heterogeneity in shear modulus or density. The perturbation method is then used to calculate the gravitational tidal response caused purely by the lunar crust, which is compared with that caused by a (1, 1) structure in deep mantle. Based on these calculations, I conclude that 1) the tidal effect due to lunar crustal thickness variations is not significant and 2) the degree-3 response due to a moderate degree-1 structure at depth is more significant than that due to the lunar crust. Therefore, the large thickness variations in the lunar crust would not affect the idea of using degree-3 tidal response to constrain the degree-1 interior structure of the Moon.

5.2 Future plan

With a complete formulation, the perturbation method presented here can be further improved or adapted for future applications.

The perturbation formulation can be adapted to include viscoelasticity to solve for the surface loading response of a self-gravitating planetary body in Maxwell time scale. Different from an elastic solid, the constitutive relation for a viscoelastic medium is time dependent, in which the stress is related to the strain rate through not only elastic moduli but also viscosity (e.g., Peltier, 1974). However, under a Laplace transform, the governing equations and the

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constitutive relation for a viscoelastic body will turn into the identical form as for an elastic body, with the two ‘effective’ elastic moduli being functions of the viscosity. Thus, solving the viscoelastic response problem is equivalent to solving its corresponding elastic problem in the transform domain, to which the perturbation method applies, followed by an inverse transform back to the time domain (Wu and Peltier, 1982). With the same idea as tidal tomography, the perturbation method I present here may be used to infer laterally varying structures in viscosity from surface loading responses. The perturbation method may be used to investigate the effect of crustal thickness variations of a planetary body on the tidal response, knowing that tidal response is more sensitive to structures near the surface than those at depth. For the Moon, the lunar crustal models (top ~60 km) have been well constructed from the joint analyses on the topography and gravity field measurements (Wieczorek et al., 2013), showing significant lateral variations in the thickness. Since the crustal thickness variations cannot be modelled within the current perturbation formulation directly, they may be converted into ‘effective’ lateral heterogeneities in the elastic properties and the density of the crust, through appropriate modelling. Tidal response due to the crust can be calculated separately and compared with that due to deep structure. I thus can make an important assessment of whether the lunar crustal signal in the tidal response would overwhelm that from depth, or the deep signal could still stand out, for future lunar missions. Similar calculations can be performed to estimate the tidal effect of the (potentially) temperature- induced crustal thickness variations in the icy moons of gas giants, such as Jupiter’s moons Europa and Ganymede and Saturn’s moon Enceladus (A et al., 2014). In these cases, a fluid layer needs to be included in the formulation to represent a liquid ocean that may underlay the icy shell (crust). My calculations may help determine the existence of the liquid ocean as well as provide implications on the characteristics of the icy shell, when future measurements are available. The perturbation method is optimizable for inversion studies. Preliminary analyses have been done to invert for the longest-wavelength (harmonic degree-1) in the elastic structure of the

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lunar mantle by using GRAIL degree-2 gravitational tidal Love number solutions (Qin et al., 2014b). My results suggest that, in order to provide reliable constraints on the degree-1 structures, degree-2 Love numbers alone are insufficient, while precise determination of the degree-3 tidal response may be required. For now, instead, inversion analyses on synthetic models can be useful in determining the minimum tidal measurements needed to reconstruct a prescribed structure. For example, relatively strong degree-1 or (and) degree-2 (i.e., very long wavelengths) structures may be introduced as lateral heterogeneities in shear modulus and density (the effect of lambda is much weaker), while other degrees are set to be noise. Response to degree-2 tides for this structure is calculated using the perturbation method, and the solutions are taken as the ‘measurements’. Monte Carlo sampling is performed using the perturbation method to determine a minimum suite of ‘measurements’, starting from the lowest degree (i.e., degree-2) to higher degrees, which can robustly reconstruct the prescribed structure with a relatively small misfit. It would also be important to examine if low-degree gravity ‘measurements’ are sufficient to constrain a structure or both gravity and uplift (radial displacement) ‘measurements’ are needed.

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Appendix A

Harmonic expansion of mode coupling terms

The coupling terms FD , BD , GD and H D defined in Chapter 3 and Chapter 4 (note that and in Chapter 3 is only the contribution from the lateral heterogeneity in shear modulus,

  which is FD and BD ), which involve mode coupling between the lateral heterogeneities in the elastic and density structures and the lower order ( D 1) response, only exist in the first and second order perturbation equations. In order to implement the propagator matrix method (e.g., Hager and O’Connell, 1981), I expand those coupling terms into different harmonics that separate into either spheroidal (s) or toroidal (t) mode. Here, I call the lower order response the “parent”, and the modes that result from the coupling are the “children”. A “child” at a specific mode is indicated by a non-trivial set of expansion coefficients of the coupling terms at that harmonic. Assuming the “parent”, a single eigen-structure of the lateral heterogeneities and a single “child” are at harmonics of (,)lm00, (,)lm11 and (,)lm, respectively, the expansion of the coupling terms can be generalized into four categories, depending on whether it is spheroidal or toroidal that the “parent” and the “child” belong to.

xD, xD, Following eqs (4.45) and (4.47) , I divide Flm and Blm further into 0 , 0 and 0 induced terms, as

x,,,, D x D x D x D FFFFlm,,, lm   lm   lm (A1) and

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x,,, D x D x D BBBlm,, lm lm , (A2) where x denotes p , b or c component. Note that a spheroidal (toroidal) mode does not have c

( p or b ) component(s). The four categories of the expansion coefficients of FD , BD , GD and

H D are thus expressed, respectively, as

I) s(,)(,)(,) l0 m 0 l 1 m 1 s l m

i. 0 induced coupling:

D1 22UUVS FpD,    a() U    a  a , (A3) ,lm l11 m 0 122 2 3 r r r0 r lm00

D1 SSUV3 FbD,    a()   a  a , (A4) ,lm l11 m 0 4 522 6 00r r r lm00

D1 BpD,    aU , (A5) ,lm l11 m 0 1 lm00

bD,  D1 B,4lm l m  a S , (A6) 11 lm00

where []D1 means the enclosed r-dependent quantities belong to mode (,)lm at order D 1 of lm00 00 the perturbation.

ii. 0 induced coupling:

D1 FpD,    b X , (A7) ,lm l11 m 0 1 lm00

D1 X FbbD,   , (A8) ,lm l11 m 0 2 r lm00

pD,  D1 B,lm l m  0 b 1 X  , (A9) 11 lm00

bD, B,lm  0 , (A10)

D1 2U l( l 1) V where XUD1    00 . lm00 rr lm00

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iii. 0 induced coupling:

VP D1 p,, D ddlm11 p D F,lm 0  l m cgX 1[ 0  ( gUKcXP 0  )]  1 l m  ( cUPc 1 l m  2 )  f , lm , (A11) 1 1dr 1 1 dr 1 1 r lm00

D1 XP UP VP b,, Dl1 m 1 l 1 m 1 l 1 m 1  gU0 K b D F,lm 0 c 4  c 1  c 22   l m c 3()   f , lm , (A12) r r r11 r r lm00 where P is the SH expansion coefficient of  due to  , and is at the same harmonics as lm11 0 0  . Specifically,   PY . f pD, and f bD, are from the additional terms that depend on 0 0  l1 m 1 l 1 m 1 ,lm ,lm lm11, the order of perturbation D . When D 1, f fp,1   c  td  ( l  1, m  m )  0 , (A13) ,lm l11 m 0 1 L , M 0 M

td f fb,1    cLM,  ( l  1, m  m )  0 , (A14) ,lm l11 m 0 3rM 0

td where I assume the tidal potential is at harmonic (,)LM and thus VYtdLMLM , , . Here,  (xy , ) 1 if and only if xy0 , otherwise  (xy , ) 0 . The second terms on the right-hand side of eqs (A29) and (A30) are non-vanishing only when degree-1 mode exists with a proper f choice of m (see eq. (4.13) for the meaning of 0 ). When D  2, M

0 VP p,2  d lm11 f0 f,lm  l m 0 cXP 5 l m ()(,) cUPc 5 l m  6  llmmc   1 , (A15) 1 1 1 1dr 1 1 r M LM,

0 XPl m UP l m VP l m f b,2  1 1 1 1 1 1 0 f,lm  l m  0 c 9  c 7  c 82 (,) l  l m  m c 3 . (A16) 11 r r r M LM,

In addition, GD and H D are caused by 0 only, where

D1 V GD    c X  c (A17) lm l11 m 0 1 2 r lm00

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and

D  D1 Hlm l m 01 cU  (A18) 11 lm00

respectively. a1,,,,,,,,, a 2 a 6 b 1 b 2 c 1 c 2 c 9 are coupling constants and they vary for different couplings.

II) s(,)(,)(,) l0 m 0 l 1 m 1 t l m

i. 0 induced coupling:

D1 SSV3 FcD,    a()   a , (A19) ,lm l11 m 0 1 2 2 00rr lm00

cD,  D1 B,1lm l m  a S , (A20) 11 lm00

c,, D c D ii. 0 does not induce toroidal mode, therefore FB,,lm lm 0.

iii. 0 induced coupling:

D1 g U K XP Fc,, D   c0  f c D , (A21) ,lm l11 m 0 1r , lm lm00 where when D 1

td fcc,1    LM, , (A22) ,lm l11 m 0 1 r when D  2

0 XPl m UP l m VP l m f c,2  1 1 1 1 1 1 0 f,lm  l m  0 c 2  c 32 (,) l  l m  m c 1 . (A23) 11 r r M LM,

Note that GD or H D does not appear in the toroidal equations.

III) t(,)(,)(,) l0 m 0 l 1 m 1 s l m

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i. 0 induced coupling:

D1 T FapD,   , (A24) ,lm l11 m 0 1 0r lm00

D1 TTW3 FbD,    a()   a , (A25) ,lm l11 m 0 2 3 2 00rr lm00

pD, B,lm  0 , (A26)

bD,  D1 B,2lm l m  a T  . (A27) 11 lm00

ii. 0 induced coupling:

WP D1 pD, d lm11 Fc,lm  0 1 () , (A28) dr r lm00

WP D1 bD, lm11 Fc,lm  0 1 2 . (A29) r lm00

Note that eqs (A28) and (A29) contain no extra terms.

D1 W GcD   , (A30) lm l11 m 01r lm00

D Hlm  0 . (A31)

A toroidal mode will never couple with 0 to induce a “child” mode.

IV) t(,)(,)(,) l0 m 0 l 1 m 1 t l m

i. 0 induced coupling:

D1 TTW3 FcD,    a()   a , (A32) ,lm l11 m 0 1 2 2 00rr lm00

cD,  D1 B,1lm l m  a T  . (A33) 11 lm00

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I find that a toroidal mode coupling with 0 does not further induce any toroidal mode, meaning

c, D D D FGH,lm lm  lm  0 .