Gravity Measurements in Bolivia and Their Implications for the Tectonic Development of the Central Andean Plateau

Gravity Measurements in Bolivia and their Implications for the Tectonic Development of the Central Andean Plateau

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Kevin M. Ahlgren

Graduate Program in Geodetic Science

The Ohio State University

2015

Dissertation Committee:

Michael Bevis, Advisor

Alan Saalfeld

Ralph von Frese

Kevin M. Ahlgren

2015

ABSTRACT

A geodetic network of gravity stations in Bolivia was established, reduced, adjusted, and analyzed. This network consists of more than 1300 relative stations with estimated precisions of less than ±0.15 mGal. The network is supported and adjusted using 15 absolute gravity stations. The motivation for observing a completely new and large gravity network in Bolivia is to contribute to the global geopotential model (GGM), which will eventually succeed EGM2008, to provide Bolivia with a much-improved height system, and to use these results to place new geophysical constraints on geodynamic models of the Central Andes.

The Central Andes comprise an area of South America associated with a widening of the Andean mountain belt. The evolution of this portion of the range is not completely understood. Traditionally, models with large amounts of steady crustal shortening and thickening over a period of time from 30 Ma - 10 Ma have been hypothesized. Recently, an alternative hypothesis involving a rapid rise of the Central Andes between 10 - 6.8 Ma has been proposed. A tectonic model of this rapid rise hypothesis is developed using locally-compensated isostatic assumptions and delamination of a dense layer of eclogite in the lithospheric mantle. The model is ultimately evaluated for consistency with the observed isostatic gravity anomaly field. Consistency between the model and the isostatic anomaly field is not supported though and fails to support the rapid rise delamination hypothesis. ii

DEDICATION

This document is dedicated to my daughters: Madeline and Ava

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ACKNOWLEDGMENTS

I wish to thank my advisor, Dr. Michael Bevis, for supporting this thesis, fruitful discussion, and guidance throughout my studies. I would also like to thank my committee members and instructors: Dr. Alan Saalfeld and Dr. Ralph von Frese. I also wish to acknowledge Dr. von Frese for the extensive use of his gravimeter. I also wish to acknowledge the National Geospatial-Intelligence Agency (NGA), who supported much of this project.

I also would like to thank a number of individuals related to this project. First,

Dana Caccamise and Eric Kendrick for their help in everything. Second, Colonel Arturo

Echalar Rivera for your leadership and guidance with the realization of this network and numerous other endeavors. Third, to all the sergeants, drivers, and other individuals who collected data that was used in this thesis. Finally, I also wish to thank fellow and former graduate students Abel Brown, Jacob Heck, and David Raleigh for data collection, comments, and suggestions.

My parents, John and Sally; sister, Erica; and brother, Keith have provided me so much throughout this endeavor, and I greatly appreciate their support, love, and guidance.

Finally, I wish to thank my wife, Christina, who has had to deal with my numerous trips to South America, weekends at work, long nights, and having to explain to people what her husband does.

VITA

2005...... B.C.E. Civil Engineering, University of

Minnesota – Twin Cities

2011...... M.S. Geodetic Science and Surveying, The

Ohio State University

2007 to 2012 ...... Graduate Research Associate, Department

of Earth Sciences, The Ohio State

University

2012 to present ...... Assistant Professor, Department of

Geography and Planning, St. Cloud State

University

Publications

Jekeli, C., Yang, H. J., & Ahlgren, K. (2013). Using isostatic gravity anomalies from

spherical harmonic models and elastic plate compensation to interpret the

lithosphere of the Bolivian Andes. Geophysics, 78(3), G41-G53.

Fields of Study

Major Field: Geodetic Science v

TABLE OF CONTENTS

ABSTRACT ...... ii

DEDICATION ...... iii

ACKNOWLEDGMENTS ...... iv

VITA ...... v

TABLE OF CONTENTS ...... vi

LIST OF TABLES ...... viii

LIST OF FIGURES ...... x

CHAPTER 1: INTRODUCTION ...... 1

CHAPTER 2: OPERATIONAL GRAVITY ...... 13

CHAPTER 3: GRAVITY ANOMALIES ...... 64

CHAPTER 4: DELAMINATION ...... 82

CHAPTER 5: DISCUSSION AND FUTURE WORK ...... 108

REFERENCES ...... 113

APPENDIX A: ADDITIONAL GRAVITY DATA IN BOLIVIA REGION ...... 119

APPENDIX B: LACOSTE AND ROMBERG MODEL G GRAVIMETERS ...... 129

APPENDIX C: NETWORK ADJUSTMENT WITH CONDITION EQUATIONS ..... 130

APPENDIX D: ISOSTATIC CORRECTIONS WITH SPHERICAL TESSEROIDS .. 133

APPENDIX E: LITHOSPHERIC MODEL PARAMETERS AND VALUES ...... 136

vii

LIST OF TABLES

Table 1: Gravity raw readings and corrections ...... 31

Table 2: Gravity observation 's with elapsed time for gravimeter unit: g142 and survey line: K016 - UYNI ...... 33

Table 3: Gravity observation 's with elapsed time for gravimeter unit: g1025 and survey line: K016 - UYNI ...... 34

Table 4: Absolute gravity differences – A10 in 2011 and FG5 in 1997 ...... 40

Table 5: Relative gravity misclosure statistics at absolute gravity stations ...... 55

Table 6: Relative gravity statistics - RMS of network adjusted residuals ...... 58

Table 7: Relative gravity statistics - maximum of network adjusted residuals ...... 59

Table 8: Relative gravity statistics - standard error of estimated gravity ...... 61

Table 9: Gravity statistics [mGal] ...... 70

Table 10: Isostatic Crustal Parameters ...... 80

Table 11: Ranges of Acceptable Outcome Values ...... 92

Table 12: Pop-up model critical parameters ...... 93

Table 13: Scenario parameters – See Figure 35 ...... 95

Table 14: Gravity amplitude [mGal] at varying distances. Tabulated values associated with Figure 38...... 98

Table 15: Observation statistics and high-velocity zone (HVZ) ...... 103

Table 16: Individual Gravimeter Observation Statistics ...... 129 viii

Table 17: Non-controlling model parameters ...... 136

LIST OF FIGURES

Figure 1: Geodetic Surfaces: Terrestrial, Geoid, and Ellipsoid ...... 2

Figure 2: Central Andes ...... 5

Figure 3: LaCoste and Romberg gravimeter ...... 14

Figure 4: Static GPS results - K169 – Height Residual [cm] computed from individual

CGPS stations as shown in map view in lower left plot. Plots on right hand side from top to bottom: Northing/Easting estimated position, height residual and baseline length, and height residual and station height difference...... 20

Figure 5: Single survey line / single gravimeter analysis overview ...... 21

Figure 6: Lacoste & Romberg Model G - Serial Number G801 Calibration Curve ...... 22

Figure 7: Survey line with Heaviside measured with 2 gravimeters (g1025 & g142).

Upper figure showing the mean gravity difference along the line. Lower figure showing the discrepancy between the two gravimeters’ observations...... 27

Figure 8: Re-observed survey line ...... 28

Figure 9: Simple observed gravity network ...... 29

Figure 10: g corrections due to linear drift estimates along a survey line for individual gravimeters with 2 error bars and percentage of data outside the 2 level. Colors and dashed vertical lines differentiate correction amounts for individual gravimeter units. ... 36

Figure 11: Bolivian absolute gravity network with elevation [m] ...... 39

Figure 12: Observational and network terminology ...... 44 x

Figure 13: Network adjustment overview...... 45

Figure 14: Example network ...... 50

Figure 15: Minimally constrained network misclosures at absolute gravity stations based on KIKA...... 55

Figure 16: Histogram of network adjusted residuals: N = 3470 ...... 56

Figure 17: Bolivian relative gravity - RMS of network adjusted residuals [mGal]...... 58

Figure 18: Bolivian relative gravity - maximum of network adjusted residuals [mGal] .. 59

Figure 19: Histogram of RMS of network adjusted residuals ...... 60

Figure 20: Bolivian relative gravity - standard error of estimated gravity ...... 61

Figure 21: Bolivian relative gravity - number of gravimeters ...... 62

Figure 22: Geoid and reference ellipsoid (Figure 2-12 Heiskanen & Moritz, 1967) ...... 65

Figure 23: Free-Air anomaly [mGal] ...... 69

Figure 24: Free-air anomaly histogram [mGal] ...... 70

Figure 25: Free-air anomaly and EGM2008 residual [mGal] ...... 71

Figure 26: Free-air anomaly and EGM2008 residual histogram ...... 72

Figure 27: Bouguer anomaly [mGal] ...... 73

Figure 28: Tesseroid geometry (Fig. 4 from Heck & Seitz, 2007) ...... 75

Figure 29: Isostatic geometry ...... 77

Figure 30: Terrain correction [mGal] determined using spherical tesseroids (Heck &

Seitz, 2007) based on a 333.4 km x 333.4 km grid centered at the individual stations. ... 79

Figure 31: Isostatic anomaly [mGal] ...... 81

Figure 32: High and low P-wave velocity zones with gravity observations (from Beck &

Zandt, 2002) ...... 84

Figure 33: A Simple Delamination Model. This model assumes delamination removes the lithospheric mantle but not the lower crust. The before (b) and after (a) configurations are isostatically balanced against a reference column (c), following Turcotte et al.’s

(1977) concept of the mantle manometer...... 86

Figure 34: The Modified Delamination Model. It is assumed that as a consequence of its unusual thickness, the lower part of the continental crust had transformed to eclogite.

Both this eclogite layer and the lithospheric mantle are removed by delamination...... 89

Figure 35: Partial Delamination Model Scenario ...... 91

Figure 36: Partial delamination Monte Carlo simulation (f = 0.5). Blue trials are successful outcomes; Cyan trials are unsuccessful outcomes. Scenarios I – III are highlighted with parameters in Table 13...... 94

Figure 37: Partial delamination Monte Carlo simulation (f = 0.75). Scenarios IV – VII are highlighted with parameters in Table 13...... 95

Figure 38: Generated gravity signal for tectonic Scenarios I - VII [mGal]. (Negative distances associated with non-delaminated situation; positive with delamination) ...... 97

Figure 39: Isostatic anomaly within the Altiplano [mGal]. White and black areas represent high velocity zones and low velocity zones (Beck & Zandt, 2002). White x’s are stations omitted due to being outside the range of elevations associated with the

Altiplano...... 100

Figure 40: Isostatic anomaly (with station height) distributions ...... 101

xii

Figure 41: Free-air anomaly distributions...... 102

Figure 42: Bouguer anomaly distributions ...... 102

Figure 43: Profiles across High-Velocity Zone ...... 104

Figure 44: Profile A – A’ ...... 105

Figure 45: Profile B – B’ ...... 105

Figure 46: Profile C – C’ ...... 106

Figure 47: YPFB and OSU gravity data with EGM2008 ...... 120

Figure 48: DoD/NGA and OSU gravity data with EGM2008 ...... 121

Figure 49: OSU - YPFB Free-Air Anomaly Residual [mGal] ...... 123

Figure 50: OSU - YPFB Free-Air Anomaly Residual Histogram [mGal] ...... 124

Figure 51: OSU - DoD Free-Air Anomaly Residual [mGal] ...... 125

Figure 52: OSU - DoD Free-Air Anomaly Residual Histogram [mGal] ...... 126

Figure 53: DoD Gravity Example I ...... 127

Figure 54: DoD Gravity Example II ...... 128

Figure 55: Scenario I Model with Densities [kg/m3] ...... 136

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CHAPTER 1: INTRODUCTION

1.1: Background

One of the central concerns of geodesy is the definition and realization of horizontal and vertical reference systems that can be used for scientific, engineering and societal purposes. Reference systems have progressed from spherical to ellipsoidal reference surfaces and incorporated or accommodated a wider range of geodynamic phenomena (earth rotation, plate motion, intraplate deformation, etc.). Vertical reference systems invoke a single vertical coordinate (i.e. elevation, height or altitude), which has historically been tied to a mean sea level surface of some kind. Presently, physical vertical reference systems usually invoke an equipotential surface, defined as a surface of constant gravity potential, W, which best approximates mean sea level (Heiskanen &

Moritz, 1967) in some global sense. That is, the particular surface with

푊 = 푊(푥, 푦, 푧) = V + Φ ≔ 푊0 = 푐표푛푠푡. ( 1 ) where V is the gravitational potential and Φ is the ‘centrifugal’ potential. This fundamental equipotential surface is called the geoid. Orthometric height is defined as the height about the geoid, as measured ‘along the plumb line’. In contrast, geometrical vertical reference systems, which gained great popularity with the rise of the Global

Positioning System (GPS), invoke a geometric or ellipsoidal height, h, measured perpendicular to a standard ellipsoid such as the WGS84 ellipsoid. The approximate (but 1

normally adequate) relationship between the orthometric height, H, and the ellipsoidal height, h, is given by:

ℎ = 퐻 + 푁 ( 2 ) where N is the geoid undulation. This relationship is illustrated in Figure 1:

Figure 1: Geodetic Surfaces: Terrestrial, Geoid, and Ellipsoid

The geoid is a crucial reference surface for many disciplines such as engineering, oceanography, and hydrology. All of which use the potential energy associated with the gravity field to determine the direction of water flow. The geoid surface is typically modeled and determined through geodetic measurements including gravity, topography, and artificial satellite orbits. In the past decades, the availability of data obtained through a number of gravity observing satellite missions has increased the precision of global geoid/gravitational models. Satellite missions including LAGEOS, CHAMP, GRACE, and GOCE have all been incorporated into various modern global geoid/gravitational

models (Lemoine, et al., 1996; Pail, et al., 2011; Pavlis, et al., 2008). One limitation with satellite gravity missions is that they sense gravity hundreds of kilometers above the

Earth's surface, rendering them incapable of resolving short wavelength features of the surface gravity field. As a result, satellite and aircraft measurements of gravity must be combined with surface gravity measurements in order to obtain the best possible geoid/gravity models. The current standard in global gravity/geoid models of this kind is the Earth Gravitational Model 2008 (EGM2008) (Pavlis, et al., 2008). This is a spherical harmonic model complete to degree and order 2159. It will be used for comparison purposes in subsequent sections of this dissertation. Our motivations for building a large gravity network in Bolivia were to contribute to the successor model to EGM2008, to provide Bolivia with a much-improved height system, and to use our results to place new geophysical constraints on geodynamic models of the Central Andes.

There are two categories of observing terrestrial gravity: absolute gravimetry measurements and relative gravimetry measurements. Absolute gravimetry measurements are much more accurate and precise than relative gravity measurements, and they provide a direct determination of the acceleration of gravity at each measurement station. Relative gravimetry measures only the change in gravity in either a spatial or temporal sense. Most often a relative gravity measurement determines the difference in the values of gravity between two geographically adjacent measurement stations. The major advantage of relative gravity measurement is that it can acquire a large number of observations in a relatively short period of time, and at modest cost, due to the portability, robustness, and ease of operation of the associated instrumentation.

Ideally both techniques are combined allowing the relative or differential gravity field measured with relative gravimeters to be converted or adjusted using estimates of absolute gravity.

Since this dissertation is focused on a specific region and has implications for the geophysical evolution of the region, a brief geographic and geological background of the region is relevant. Bolivia is located in the heart of South America occupying a region of the Andes mountain belt often described as the Central Andes (see Figure 2). The terrain is very dissimilar throughout the country with the Andes occupying the western portions and Amazon River basin and lowlands in the northern and eastern regions.

Figure 2: Central Andes

The higher (say > 3 km) portions of the Andean mountain belt, which runs all along the Pacific margin of South America are typically ~ 100 km wide, or less, but north of about 29° S, and south of about 13° S, the Andes are considerably wider than this, typically wider that 400 km. This very wide segment of the Andean chain is called the

Central Andes. In Bolivia and in adjacent parts of Chile, the Central Andes consist of the very high Western Cordillera (an active volcanic arc), the Altiplano, and the very high

Eastern Cordillera, which has progressively lower topography further east. The Altiplano is a relatively flat area characterized by internal drainage, about 350-400 km wide with an

average elevation close to 3.8 km (Isacks, 1988; Allmendinger et al., 1997). It is bordered to the south by a rougher and somewhat higher area called the Puna, which is almost entirely located in NW Argentina. There is a rather similar terrain at the northern end of the Altiplano in southern Peru.

The Altiplano and Puna are often referred to jointly as the Altiplano-Puna plateau

(Allmendinger, 1997). Another related morphological term is the ‘Central Andean

Plateau’, which has more than one definition, but which almost always refers to the high

Western and Eastern Cordillera and the intervening Altiplano-Puna. Some authors have suggested a particularly simple definition for the Central Andean Plateau, which we adopt: that portion of the Central Andes located at an elevation ≥ 3 km. Immediately to the east (or the northeast) of the Central Andean Plateau is the ‘East Flank’ of the plateau, which descends eastward from elevations of ~ 3 km to the edge of the lowlands whose elevation varies from 300-600 m. The eastern part of the East Flank is called the

Subandes or the Subandean Range. The Subandes have formed only in the last 10 Ma

(Isacks, 1988; Oncken et al., 2006). They may be of modest height, but this range is the locus of modern mountain building in the Central Andes. This uplift is thought to be driven by active underthrusting of the Brazilian craton beneath the high Central Andes

(Oncken et al., 2006, Brooks et al., 2011). The Subandes constitute the most rapidly deforming foreland fold and thrust belt in the western hemisphere.

Note that, in map view (Figure 2), there is a ‘bend’ in the Central Andes, which is particularly sharp on its eastern side. This ‘corner’ of the mountain belt occurs near the city of Santa Cruz de la Sierra (or just Santa Cruz), and so is sometimes referred to as the

‘Santa Cruz corner’. The Subandean ranges located north and west of this corner are called the Northern Subandes, whereas the ranges to the south of the corner are called the

Southern Subandes.

The Central Andean Plateau is the second largest active tectonic plateau in the world after the Tibetan Plateau, which is considerably larger. Whereas the Tibetan

Plateau formed in the midst of a continental collision, the Central Andean Plateau did not.

Like the Andes as a whole, it formed adjacent to an active subduction zone. Why only the Central Andes (and not the rest of the Andes) evolved into a tectonic plateau is not well understood, and this is the subject of a great deal of speculation. Most traditional models for the Central Andes invoke unusually large degrees of crustal shortening and thickening which developed, more or less steadily, from about ~ 30 Ma to 6-10 Ma (e.g.

Isacks, 1988; Lamb & Davis, 2003; Oncken et al., 2006). A more recent and alternative hypothesis invokes a very rapid rise of the plateau from 10 Ma to 6.8 Ma (Garzione et al.,

2006) in response to abrupt ‘delamination’ and loss of dense lithospheric mantle and lower crustal material (Molnar & Garzione, 2007). In effect, the Central Andes ‘popped up’ in response to the sudden loss of a dense ‘keel’ that had been weighing them down as they floated in the underlying asthenosphere.

One of our motivations for collecting a new gravity dataset in Bolivia was to provide a geophysical test of the ‘pop-up’ hypothesis for the Altiplano and adjacent portions of the Central Andean Plateau. Other gravity datasets exist in this general region such as those described by Cady & Wise, (1992) and others. These datasets are a combination of data from the Defense Mapping Agency (DMA now NGA) and a local

exploration company, Yacimientos Petroliferos Fiscales Bolivianos (YPFB), which has organized much of the gravimetric work in this country over many decades (e.g. Lyon-

Caen, et al., 1985; Cady & Wise, 1992; Corchete et al., 2006, Borsa et al., 2008; etc.).

Another gravity dataset was collected by a German research group in an east-west transect near 20° S (Götze, et al., 1990 and Götze, et al., 1994). Most of these datasets either were not available to us or were deemed not very useful for our purposes because of the very large uncertainties attending the heights of the gravity stations. All of the gravity observations presented in this thesis were collected by Ohio State University and our partners, and our analysis of this data is completely independent of that associated with the DMA and YPFB datasets. A brief numerical comparison and analysis between the OSU data and the DMA/YPFB data is presented in Appendix A. Our intent, from the beginning, was to build a gravity network in which every station was very carefully located using GPS measurements and in which all relative gravity measurements were tied to and adjusted with absolute gravity measurements.

1.2: Objectives and Chapter Description

This dissertation has objectives associated with two broad categories: the observing, processing, and adjustment of a precise gravimetric network in Bolivia and testing the ‘pop-up’ model for the Altiplano using our gravity results. With respect to the first agenda, we shall describe the data collection process and associated logistical factors; data reduction, processing, and adjustment; and the determination of associated gravimetric quantities. The tectonic modeling has two main parts: development of a specific tectonic model associated with a proposed delimitation scenario, i.e. one that predicts the consequences for the overlying gravity field, and comparing the predictions of this model with observed gravity data.

We began building our gravity network in Bolivia in 2009. As of March 2015, this network consists of > 1300 gravity stations. Since this network will be used for national orthometric height control purposes in the future, we sought an accuracy of better than 1.0 mGal at all stations. When we began working in 2009, there were 3 already established absolute gravity stations in Bolivia. This was inadequate for our purposes so we collaborated with the absolute gravity group at the National Space

Institute of the Danish Technical University (DTU) so as to build 12 additional absolute gravity stations (as well as reobserving the original 3 stations). In addition to the gravity observations, the precise geodetic coordinates also needed to be established through the use of static GPS observations. The rugged terrain and other logistical factors in Bolivia

also impact how these observations are collected and an appropriate methodology taking these difficulties into consideration will be developed.

Secondly, a discussion will be presented regarding the reduction, processing, and adjustment of the GPS and gravity data. The most involved part of this discussion is with regard to the relative gravity data. The reduction for tidal and gravimeter calibration effects need to be considered; a linear drift rate for each gravimeter along a forward- reverse survey line (~36 hours) is estimated and removed from all observations; and the network must be adjusted using a least squares model of condition equations with absolute gravity stations acting as fixed conditions. This network adjustment is essential since there are multiple instruments providing redundant observations. Previous studies have utilized somewhat similar but not identical methodologies and adjustments in a relative gravimetric network. For example, the network adjustment presented is similar to the weighted constraint model from Hwang, et al (2002). However, the presented processing separates the adjustment into two separate stages: an initial stage to estimate and remove the instrument drift rate and then a network adjustment to introduce a gravity datum through the use of fixed constraints in the form of absolute gravity values. Since the drift rate changes throughout a gravity survey, it is easier and more appropriate to incrementally estimate and remove the effects due to instrument drift prior to network adjustment (Dias & Escobar, 2001).

Finally, a number of gravity anomaly fields are determined largely for the benefit of geophysical assessment. Both the free-air anomaly and Bouguer anomaly are presented - these are the ‘classical’ gravity anomalies. Due to the rugged terrain in the

region, and for use in subsequent geophysical analysis, we also compute an isostatic anomaly field.

Our main geophysical objective is to assess and analyze a proposed mechanism for the formation of the Altiplano using newly attained gravimetric results. Towards this objective, a simplified structural model of the lithosphere is needed at different epochs in the evolution of the Altiplano. This simplified model invokes local rather than regional isostasy. There are a number of material (i.e. density) and geometric variables (i.e. thicknesses) present in our version of the pop-up model which cannot be well-constrained a priori. Therefore, we use a Monte Carlo approach to search over all plausible combinations of these parameters and to characterize the range of gravitational signatures that would result. Finally, the predictions of the resulting family of tectonic models are compared with the observed gravity data.

This dissertation is presented with the following additional chapters: Chapter 2:

Operational Gravity, Chapter 3: Gravity Anomalies, Chapter 4: Delamination, and

Chapter 5: Conclusions and future work. Chapter 2 will present a synopsis of the gravimetric network established and observed in Bolivia. This will mainly focus on the relative gravimetry components of the network, but a brief section will be presented regarding the absolute gravimetry aspects, which constrain and support the entire network. This chapter will include the development and analysis of various reduction and adjustment methodologies. Chapter 3 will focus on the resultant and associated gravity anomalies including the free-air anomaly, simple Bouguer anomaly, and an isostatic anomaly. Chapter 4 will present a lithospheric model consistent with the current

understanding of the Altiplano region within the Central Andes. This model is developed and constructed using Airy-Heiskanen local isostasy based on a pre-delaminated state and after delamination occurs. A rapid rise of the Altiplano condition is invoked in the model, and its hypothetical gravimetric signal is then generated. Finally, the generated gravity signal is compared with the observed gravity anomaly data. Chapter 5 will present concluding statements regarding the gravity network and lithospheric delamination along with opportunities for future work related to these objectives.

CHAPTER 2: OPERATIONAL GRAVITY

2.1: Relative Gravity Field Observations

Relative gravity measurement refers to measuring the differences between gravity values at two or more stations (or two time epochs). The absolute values of gravity acceleration cannot be determined using relative gravity alone. But if an absolute gravimeter is used to determine the absolute value of gravity at one or more stations in a relative gravity network, it is then possible to adjust the entire relative network so as to estimate the absolute gravity amount at every station.

There are two different physical methods that can be utilized for observing the gravity difference - dynamic methods and static methods (Torge, 1989). Dynamic methods operate over extended periods of time to determine the period of pendulums and strings, whereas static methods utilize a test mass held in equilibrium with a counteracting force, which is measurable. Typically, this counteracting force is provided by an elastic spring. We do not use dynamic methods so our remaining discussion will focus on the static technique.

The LaCoste and Romberg (LCR) gravimeter developed in the 1930-40’s uses a

‘zero-length spring’, which provides a restoring force proportional to its length to support a test mass (see Figure 3). In moving the gravimeter from one location to another, the 13

change in the length of this spring manifests the change in the weight of the test mass and so gauges the change in the gravity. The LCR gravimeter is one of the most precise relative gravimeters ever produced with expected accuracies from ±0.01 − ±0.10 mGal

(Torge, 1989). The mechanical structure of the LCR gravimeter is shown in Figure 3

(Instruction Manual – Model G & D gravity meters, 2004).

Figure 3: LaCoste and Romberg gravimeter

The errors present in a relative gravity reading are typically categorized into gross, systematic, and random types. Gross errors are difficult to completely eliminate from the system but can be substantially remedied through careful and consistent survey methodology, procedures, observations, and processes. These gross errors often are in the form of operator recording errors like transposing two numerical digits, incorrect station identification, leveling errors, unstable voltage, etc. It is possible to detect gross errors through statistical outlier detection using statistical methods described by Baarda,

(1967); Koch, (1999); etc. However, outlier detection is currently not performed in the 14

current processing, but the removal of gross errors is still attempted through comparing two readings in a survey line, comparing the gravity differences from multiple gravimeters, and other etc. Systematic errors are typically due to the instrument drift,

Earth tides, air pressure changes, groundwater changes, etc. (Torge, 1989; Dias &

Escobar, 2001). The most significant of these are the instrument drift and Earth tides.

These two effects are treated in the initial stage of the reduction and adjustment as described in Chapter 2.2. The remaining error is assumed to be of the random type, which can be removed from the system through the network adjustment stage, which is described in Chapter 2.5.

All relative gravimeters have a distinct instrument drift causing the gravity measurement to change over time. The LCR gravimeters typically have drift rates on the order of 0.5 mGal per month (Instruction Manual – Model G & D gravity meters, 2004).

Like a fingerprint, individual gravimeters have differing drifts so differencing measurements with multiple gravimeters is not applicable. Additionally, the drift behaves differently depending on the transportation method (Hamilton & Brule, 1967).

As the amount of vibrations and bumps is increased, the irregularity of the drift is also increased (Instruction Manual – Model G & D gravity meters, 2004). The poor road and transportation conditions in Bolivia warrant the necessity to estimate the drift rate as often as possible. However, another factor must be considered. In a typical relative gravity survey, drift correction is achieved by making repeat observations at a base station where the gravity is assumed to be constant for a short time period (usually less than 4 - 6 hours) (Hinze, et al., 2013). This is more burdensome in Bolivia than in most

developed countries because the roads are often very poor, circuitous, and very slow to drive. It is not practical to re-observe a base station in 6 hour time intervals in Bolivia – one would never get very far from one’s starting point. Therefore, we utilize a similar philosophy of Dias & Escobar, (2001) to estimate the gravimeter drift as an unknown parameter for each survey line or interval with a survey design methodology as follows.

The closure of a survey line (either by completing a circuit or by ‘reversing’ the measurement traverse) was deemed to be within two working days (~36 hours) in order to estimate the unknown linear instrument drift. Typically, observations were started around dawn, the forward measurement line was driven, stops were made at increments of roughly 60 minutes to measure gravity at a station, and the forward line was terminated once the light began to fail. The route was then reversed the following day, revisiting every station observed during the forward leg. This is analogous to a ‘double- run’ style of survey in geodetic leveling or a ladder sequence in relative gravimetry field procedures (Federal Geodetic Control Committee [FGCC], 1984). Most importantly, the drift is not estimated using just the initial and final station readings, but at every re- observed station along the line. This serves two important purposes for the drift estimation: time differences are much shorter than the complete ~36 hour round trip and a reading error at the initial or final station won’t contaminate the readings of the entire survey line as all station readings are used in the drift estimation.

It is obvious that a third traverse is usually necessary in order to return to the end of the survey segment so as to be positioned to extend the survey into the next segment.

The amount of time lost searching for a monument, installing a destroyed monument, and

other logistical hindrances all results in a significant reduction in the productivity of a survey. It is not uncommon in Bolivia for a road to be open only for 3 - 4 hours per day because of construction purposes, landslides, cultural protests, and other environmental causes. So, we can take advantage of the need to have three passes by incorporating an initial reconnaissance pass where the gravity crew could scout the local roads, search out each survey marker, replace it if necessary, and understand the road conditions and closures. This recon pass takes place on day one. The first gravity measurement traverse is made the next day, the forward line. On the third day, the gravity measurements are made again in the reverse sequence. Finally, the GPS crew will come at a separate time and measure the 3D position of each gravity station. So, each survey segment is actually traversed four times and not twice as normally imagined. This ‘four pass’ approach proved to be very robust in Bolivia. The gravity crews worked quickly and efficiently but were never constantly working in very great haste and so very few logistical problems were encountered.

Another important concept in this environment is that it is necessary to have the utmost confidence in one’s survey measurements prior to leaving the survey area. For example, if a blunder is discovered a week after the observations were made, it may take

2 - 3 days of driving just to return to that survey area. The only way one can be really confident that the measurements are adequate is to provide additional redundancy by using two or more gravimeters. This allows two independent measurements of the gravity difference to be determined for each pair of gravity stations along the survey line, and if these estimates are in good agreement, then the survey field crew can leave the

area without any concern. Because it is necessary to have at least two gravimeters per field team and typically multiple teams are surveying simultaneously, a number of relative gravimeters were utilized over the duration of this study, all of which were the

LaCoste and Romberg Model G type. Additional information about specific gravimeters used, numbers of observations, reading interval amounts is provided in Appendix B.

The final component of the relative gravity field survey is the need for precise 3D positional information. This positional information was established at every gravity station using a static GPS occupation lasting ≥ 1 hour. At stations with poor sky view, this occupation time was extended to 2 hours. Because of these rather long occupation times, the gravity surveys and the GPS surveys are done completely independently of one another increasing efficiencies for both surveys.

The geodetic analysis of this “rapid static” GPS data was conducted using MIT’s

TRACK software suite (Herring, et al., 2006). It is advantageous in limiting positioning error to process the short occupation data in baseline mode using a single continuous GPS station (CGPS) to serve as a base station. The roving receiver is actually positioned, in independent analyses, relative to several CGPS stations (~ 6 - 8). These positioning results are compared epoch by epoch for each CGPS solution and a mean position is determined with an estimated vertical accuracy of less than ±0.20 m and horizontal accuracy of less than ±0.10 m. This is similar to methods utilized by the NGS for OPUS positioning results of static GPS observations (Eckl, et al., 2001 and Soler, et al., 2006).

A script to remove any CGPS solutions which exhibit a discrepant solution relative to the other CGPS solutions was developed. It takes into account the mean and standard

deviation of each set of results removing individual solutions that poorly fit the overall trend in an iterative process. A time series representing the residuals from the mean height for a single 1-hour occupation for station K169 is shown in the upper portion of

Figure 4. Reference stations shown in red in the various subplots (e.g. BLPZ) are removed from the relative results due to large discrepancies from the mean and/or large standard deviations. Other portions of Figure 4 are used for visual evaluation of the processing results including the map of CGPS sites used in processing, the local

Northing/Easting horizontal position residual, the height residual as a function of the

CGPS baseline distance, and the height residual as a function of the CGPS baseline height difference.

Figure 4: Static GPS results - K169 – Height Residual [cm] computed from individual CGPS stations as shown in map view in lower left plot. Plots on right hand side from top to bottom: Northing/Easting estimated position, height residual and baseline length, and height residual and station height difference.

2.2: Analyzing a Single Survey Line

Prior to any adjustment being performed, any obvious gross errors present in the observations are removed, and the remaining observations are ‘reduced’. The following section describes the procedure for reducing the readings of the gravimeter made in the field to an estimate of the gravity difference between adjacent stations. This reduction utilizes the individual gravimeter’s calibration curve provided by the manufacturer, a gravity tide correction, and a drift correction. The reduction, adjustment, and analysis were all performed using MATLAB software. The overall scheme for analyzing a single survey line is depicted in Figure 5.

Figure 5: Single survey line / single gravimeter analysis overview

Each relative gravimeter has an individually specific calibration curve provided by the manufacturer, which is considered the most practical procedure for obtaining gravity units from raw readings (Dias & Escobar, 2001). A representative calibration curve for a LCR Model G spring gravimeter is shown in Figure 6. These tables were digitized for data processing. The mean of three readings made at each station in the 21

field is used as the nominal reading. A spline interpolation converts this reading to gravity units. At this point, the gravity value is proportional to the actual gravity acceleration but offset by a fixed but unknown bias and contaminated by minor instrumental drift and random errors due to the environment, instrument, and operator.

Figure 6: Lacoste & Romberg Model G - Serial Number G801 Calibration Curve

The second aspect of the reduction is the removal of the gravity tide using the formulas of Longman (1959) and based on the FORTRAN code GRREDU developed by

Rene Forsberg for use within the GRAVSOFT software package (Tscherning, et al.,

1994). Our results were compared to a similar MATLAB function written at the

Proudman Oceanographic Institute in England and produced identical results.

After successful reduction of the raw readings using the gravimeter calibration curve and tidal corrections, a least squares adjustment is performed to estimate the

gravimeter drift rate and all station pair gravity differences along the survey line. The gravimeter drift, D(t), can be represented with a polynomial of the form (Torge, 1989):

푎 푝 퐷(푡) = ∑ 푑푝(푡 − 푡0) ( 3 ) 푝=1

However, only the time-dependent linear component of the instrument drift is estimated, as estimating higher order term (i.e. a ≥ 2) produces very little significant improvement to the overall results. This adjustment of the survey data is characterized as a ‘local’ adjustment since it is confined to a specific segment (the forward and reverse measurements), and one such adjustment is made using the observations acquired by each gravimeter.

The derivation of this local adjustment procedure is described below. A gravity observation, mQ is made at station, Q. This observation consists of the ‘true’ value of gravity, gQ, an error associated with the instrument drift, D(t), and some random error, eQ, assumed to have zero mean and a normal distribution.

푚푄 = 푔푄 + 퐷(푡푄) + 푒푄 ( 4 )

The difference in observed gravity between adjacent stations measured by a single gravimeter in a line is calculated thus:

푑푚푃푄 = 푚푃 − 푚푄 ( 5 )

푑퐷 Combining equations ( 4 ) and ( 5 ) and assuming a constant drift rate, 퐷̇ = : 푑푡

푑푚푃푄 = 푔푃 − 푔푄 + (퐷(푡푃) − 퐷(푡푄)) + (푒푃 − 푒푄) = ( 6 )

푑푔푃푄 + 퐷(푡푃 − 푡푄) + (푒푃 − 푒푄) 23

In addition, the reverse line produces a similar result with only a change of sign as the difference:

푑푚푄푃 = 푚푄 − 푚푃 = 푔푄 − 푔푃 + (퐷(푡푄) − 퐷(푡푃)) + (푒푄 − 푒푃) ( 7 ) = 푑푔푄푃 + 퐷(푡푄 − 푡푃) + (푒푄 − 푒푃)

The error terms shown in parenthesis in ( 6 ) & ( 7 ) are now the difference of the errors between the two independent observations. There may be a small amount of correlation in these error terms as the measurements were made by the same gravimeter and operator, but these errors are assumed to be uncorrelated. The drift rate, 퐷̇ is an additional unknown that needs to be estimated and removed. Rewriting the time difference slightly from ( 6 ) & ( 7 ), results in the following:

퐷(푡푃 − 푡푄) = 퐷̇ ∗ 푑푡푃푄 ( 8 )

This is the common practice of estimating an unknown drift rate from an individual dg based on the time elapsed between measurements, dtPQ. The estimated difference in gravity is then determined by the following two equations for a pair of consecutive stations (Note: the ∆푔 term is not a gravity anomaly as is common with the notation; it is a difference in gravity (g) between consecutive stations):

∆푔푃푄 = 푑푔푃푄 + 퐷̇ ∗ 푑푡푃푄 + 푒푃푄 ( 9 ) ∆푔푄푃 = 푑푔푄푃 + 퐷̇ ∗ 푑푡푄푃 + 푒푄푃

The assumption in this model is that the value of the drift rate is constant for a given forward and reverse gravity line pair. Depending on the network geometry and local situations, this assumption might degrade slightly especially with ‘tares’ that can

occur in gravimetric surveys. For the full forward and reverse line, the constant drift rate is estimated along with the ∆푔 terms from the left hand side of equation ( 9 ); however, the relationship between the left hand sides is known and shown in ( 10 ) along with the rearranged model from ( 9 ):

∆푔 ≔ ∆푔푃푄 = −∆푔푄푃 ( 10 )

푑푔푃푄 = ∆푔 − 퐷̇ ∗ 푑푡푃푄 ( 11 ) 푑푔푄푃 = −∆푔 − 퐷̇ ∗ 푑푡푃푄

The set of equations in ( 11 ) can be expanded into matrix form for the complete survey line with a set of stations, Sn = {P, Q, ..., Sn-1, Sn}, and forward and reverse segments:

푦 = 퐴푥 = 푑푔푃푄 −푑푡푃푄 1 … 0 퐷̇ 푑푔 −푑푡 −1 … 0 푄푃 푄푃 ∆푔 ( 12 ) ⋮ = ⋮ ⋮ 푃푄 ⋱ ⋮ 푑푔 −푑푡 0 ⋮ 푆푛−1,푆푛 푆푛−1푆푛 … 1 [∆푔푆푛−1,푆푛] [푑푔푆푛,푆푛−1] [−푑푡푆푛푆푛−1 0 … −1]

The least squares solution to the linear system of equations in ( 12 ) for the unknown parameters is then solved with a unit-weight covariance matrix as follows:

푥̂ = (퐴푇퐴)−1퐴푇푦 ( 13 )

There are n resultant unknowns in 푥̂: the n-1 ∆푔 terms, and the single drift rate for this forward and reverse gravity line and gravimeter. There are 2n observations in the y vector. This arrangement provides a high level of redundant observations, n degrees of freedom, for the system.

The previous derivations in terms of the ∆푔 elements are applied to each line individually to adjust the line and determine the best values for the vector of ∆푔 values.

The vector of ∆푔 values is now adjusted within a local frame, i.e. that of the survey line.

The typical survey line in Bolivia has been observed using 1 - 10 gravimeters (see

Figure 21). Each gravimeter produces a solution for the ∆푔 values for each station pair encountered along the survey line. We can compare the results from two or more gravimeters initially to identify problems. But ultimately, all gravimeters used in the segment enter the network analysis as stand-alone results. However, the use of observations from multiple gravimeters can cause significant improvement in blunder detection such as operator reading errors and transcribing errors. For example, if the level of consistency between ≥ 2 gravimeter ∆푔 values along a survey line is not adequate (typically a RMS ≥ 0.5 mGal), the following procedure is undertaken. First, the individual raw readings are investigated, compared, and repaired, if possible. Secondly, the station pairs that are not consistent will be re-observed. This re-observation obviously consists of a tremendous time and labor cost and should be avoided unless absolutely necessary.

The following describes a first-hand experience requiring re-observation. When a gravimeter is strongly jolted (usually in transport), a loss of measurement continuity may result. That is, the reading after the jolt would be systematically different from that immediately before it. This is often described as a tare. The survey line shown in Figure

7 implies such a “Heaviside jump” in the observations since the residuals are relatively small on either side of the jump and a large discrepancy between stations where the jump

occurred. The lower portion of Figure 7 is the residual from the mean for the individual

∆푔 values between consecutive station pairs for all gravimeters that observed the segment. In this case, there are only two gravimeters. The jump occurred at some time between stations K022 and UP03, but it is impossible to determine which gravimeter is in error since only 2 are used.

Figure 7: Survey line with Heaviside measured with 2 gravimeters (g1025 & g142). Upper figure showing the mean gravity difference along the line. Lower figure showing the discrepancy between the two gravimeters’ observations.

This phenomenon cannot be completely removed without removing the entire line segment for one of the gravimeters. Since there are only two gravimeters, it is impossible to determine which one contains the error and which one does not. The redundancy in

the network design though enables large errors (≥0.5 mGal) to be isolated so that they can be removed by re-measuring the pair of stations, in this case, K022-UP03.

The gravity between the station pairs in error was re-measured along with the adjacent stations on each side and the results are shown in Figure 8.

Figure 8: Re-observed survey line

The agreement between the measurements is significantly improved with residual magnitudes less than 0.04 mGal versus the 1.85 mGal residual previously obtained.

The following provides a brief explanation of the relative gravity reduction and adjustment methodology with actual numeric observations made in Bolivia. The goal of which is to utilize redundant observations to estimate the value of gravity (~978000 mGal) at all observed stations. To illustrate these reductions and adjustments, a small 28

subset of the relative gravity network is isolated and shown in Figure 9, which exhibits many of the situations encountered throughout the network.

Figure 9: Simple observed gravity network

Overall, this sub-network consists of two absolute gravity stations (AGUY and

AGPT). The absolute gravity difference from AGUY to AGPT was observed as

-111.175 ± 0.011 mGal (Nielsen, 2013). These two absolute stations were established after the relative gravity observations were made in this area; therefore, relative gravity observations were also made connecting the absolute stations to a nearby relative station

using the same methodology described. The relative and absolute gravity stations are not collocated but usually only separated by a few hundred meters. For example, AGUY and

UYNI are approximately 15 m apart.

The observational data from the forward and reverse segments of one survey line are shown in Table 1. This line between stations K016 and UYNI was measured using two gravimeters (serial numbers: g142 & g1025) in October 2009. Prior to measurements being taken, the gravimeter power is started approximately 24 hours before the start of the survey and kept up throughout the survey using batteries during transport. This ensures that the internal temperature is adequate throughout the entire survey, and the effects due to temperature variation are minimized. In both the forward and reverse segment, each and every station is observed in both directions (i.e. A-B-C-B-

A) using three readings per gravimeter. The gravimeter dial is turned a small amount in between sequential readings, which results in additional time for the unit to relax after long stretches of transportation in a vehicle. The three readings are then averaged.

Additional readings are made if field operators deem initial results unacceptable

(differences ≥ 0.05 mGal). The averaged reading is then converted to mGals using the calibration table provided by LaCoste & Romberg for each gravimeter. Finally, a correction is applied to remove the tidal components present in the observation. This correction follows a method developed by Longman (1959) and is dependent on the station latitude/longitude and time of observation. At this stage of the process, the readings are now considered reduced. These reduced readings are those shown in the last column of Table 1.

Table 1: Gravity raw readings and corrections

Tidal Unit Calibrated Reduced Readings Mean UTC – Model Index Station Serial Reading Reading (n = 3) Reading Date/Time Correction Number [mGal] [mGal] [mGal] 1476.81 10/9/09 1 K016 g142 1476.81 1476.81 1560.097 11:30 -0.05859 1560.03841 1476.81 1415.221 10/9/09 2 K016 g1025 1415.221 1415.221 1435.8457 11:32 -0.059 1435.7867 1415.221 1198.3 10/9/09 3 K017 g142 1198.3 1198.3 1266.0451 13:39 -0.0553 1265.9898 1198.3 1126.528 10/9/09 4 K017 g1025 1126.528 1126.528 1142.9148 13:40 -0.05506 1142.85974 1126.528 1269.469 10/9/09 5 K060 g142 1269.468 1269.468333 1341.1851 14:43 -0.03054 1341.15456 1269.468 1200.365 10/9/09 6 K060 g1025 1200.365 1200.365 1217.83034 14:45 -0.029542 1217.800798 1200.365 1223.659 10/9/09 7 K018 g142 1223.66 1223.659667 1292.8204 17:22 0.064 1292.8844 1223.66 1152.704 10/9/09 8 K018 g1025 1152.704 1152.704 1169.4725 17:25 0.0657 1169.5382 1152.704 1271.97 10/9/09 9 K059 g142 1271.97 1271.97 1343.8263 18:18 0.09141 1343.91771 1271.97 1202.958 10/9/09 10 K059 g1025 1202.958 1202.958 1220.4613 18:22 0.0930377 1220.554338 1202.958 1280.435 10/9/09 11 K019 g142 1280.435 1280.435 1352.7635 19:37 0.11193 1352.87543 1280.435 1211.788 10/9/09 12 K019 g1025 1211.788 1211.788 1229.42086 19:38 0.112 1229.53286 1211.788 1278.71 10/9/09 13 UYNI g142 1278.71 1278.71 1350.9423 21:58 0.0873 1351.0296 1278.71 1210.1 10/9/09 14 UYNI g1025 1210.1 1210.1 1227.7081 21:59 0.08693 1227.79503 1210.1 1278.7 10/8/09 15 UYNI g142 1278.7 1278.7 1350.9317 10:50 -0.07009 1350.86161 1278.7 1210.062 10/8/09 16 UYNI g1025 1210.062 1210.062 1227.6695 10:53 -0.071 1227.5985 1210.062 Continued

Table 1 continued

1280.498 10/8/09 17 K019 g142 1280.498 1280.498 1352.83 13:11 -0.0646 1352.7654 1280.498 1211.142 10/8/09 18 K019 g1025 1211.142 1211.142 1228.7654 13:15 -0.06291 1228.70249 1211.142 1271.988 10/8/09 19 K059 g142 1271.988 1271.988 1343.8453 14:30 -0.01718 1343.82812 1271.988 1202.91 10/8/09 20 K059 g1025 1202.91 1202.91 1220.4126 14:33 -0.01497 1220.39763 1202.91 1223.678 10/8/09 21 K018 g142 1223.678 1223.678 1292.8398 15:26 0.02705 1292.86685 1223.678 1152.761 10/8/09 22 K018 g1025 1152.761 1152.761 1169.5303 15:28 0.02865 1169.55895 1152.761 1269.489 10/8/09 23 K060 g142 1269.489 1269.489 1341.2069 18:50 0.13168 1341.33858 1269.489 1200.405 10/8/09 24 K060 g1025 1200.405 1200.405 1217.8709 18:07 0.1255 1217.9964 1200.405 1198.359 10/8/09 25 K017 g142 1198.36 1198.359667 1266.1081 19:19 0.1299 1266.238 1198.36 1126.505 10/8/09 26 K017 g1025 1126.505 1126.505 1142.8914 19:23 0.12927 1143.02067 1126.505 1476.862 10/8/09 27 K016 g142 1476.862 1476.862 1560.1519 21:28 0.07376 1560.22566 1476.862 1415.481 10/8/09 28 K016 g1025 1415.481 1415.481 1436.10955 21:35 0.06925 1436.1788 1415.481

A reduced gravity reading at a station is of no use by itself, though. Only the difference or relative amount between two locations or epochs can produce the desired value of gravity. So, we are only concerned with the differences in consecutive gravity readings i.e. the difference between station i and station i+1. The differences in reduced readings between consecutive stations are shown in Table 2 and Table 3 for g142 and g1025, respectively. To obtain the first difference value in Table 2, the readings using g142 from K016 to K017 are differenced (from Table 1, line index 3 minus line index 1) 32

as shown in ( 14 ). The reverse from K017 to K016 is found from Table 1 (index 27 minus index 25) as shown in ( 15 ).

∆푔푔142, 푘016→푘017 = 1265.9890 − 1560.03841 = −294.04861 ( 14 )

∆푔푔142, 푘017→푘016 = 1560.22566 − 1266.2380 = 293.98766 ( 15 )

Table 2: Gravity observation 's with elapsed time for gravimeter unit: g142 and survey line: K016 - UYNI

From Station To Station g [mGal] t [days] K016 K017 -294.04861 0.089583333 K017 K016 293.98766 0.089583333 K017 K060 75.16476 0.044444444 K060 K017 -75.10058 0.020138889 K060 K018 -48.27016 0.110416667 K018 K060 48.47173 0.141666667 K018 K059 51.03331 0.038888889 K059 K018 -50.96127 0.038888889 K059 K019 8.95772 0.054861111 K019 K059 -8.93728 0.054861111 K019 UYNI -1.84583 0.097916667 UYNI K019 1.90379 0.097916666

Table 3: Gravity observation 's with elapsed time for gravimeter unit: g1025 and survey line: K016 - UYNI

From Station To Station g [mGal] t [days] K016 K017 -292.92696 0.088888889 K017 K016 293.15813 0.091666666 K017 K060 74.941058 0.045138889 K060 K017 -74.97573 0.052777778 K060 K018 -48.262598 0.111111111 K018 K060 48.43745 0.110416667 K018 K059 51.0161377 0.039583333 K059 K018 -50.83868 0.038194444 K059 K019 8.9785223 0.052777778 K019 K059 -8.30486 0.054166667 K019 UYNI -1.73783 0.097916666 UYNI K019 1.10399 0.098611111

A major concern with relative gravity observations is controlling the instrument drift. This drift can be treated as a systematic error, which is both time and instrument dependent, and can be mostly removed from the reduced observations. This is done by including a linear drift rate term in the least squares adjustment model and estimating it

(see ( 4 ) - ( 13 )). The time between observations at consecutive stations is shown in the last column of Table 2 and Table 3.

In this procedure, a gravimeter forward and reverse survey segment provides two observations of the gravity difference between each pair of consecutive stations.

However, only one is needed to determine a solution, but both are used in the adjustment resulting in additional redundancy. The least squares results of this adjustment are illustrated in equations ( 16 ) – ( 19 ) below. After this local-level adjustment, an estimate of the gravity difference between a pair of stations is obtained for each

gravimeter along with the estimated linear drift rate for that forward and reverse survey line.

1 0 0 0 0 0 −0.0896

−1 0 0 0 0 0 −0.0896 0 1 0 0 0 0 −0.0444

0 −1 0 0 0 0 −0.0201 0 0 1 0 0 0 −0.1104 0 0 −1 0 0 0 −0.1417 퐴 = ( 16 ) 0 0 0 1 0 0 −0.0389 0 0 0 −1 0 0 −0.0389 0 0 0 0 1 0 −0.0549 0 0 0 0 −1 0 −0.0549 0 0 0 0 0 1 −0.0979 [ 0 0 0 0 0 −1 −0.0979] −294.0486

293.9877 75.1648

−75.1006 −48.2702 48.4717 ( 17 ) 푦푔142 = 51.0333 −50.9613 8.9577 −8.9373 −1.8458 [ 1.9038 ] ( 푇 −1)−1 푇 푥푔142 = 퐴 퐴 퐴 푦142 −294.0181 퐾016 − 퐾017 75.1278 퐾017 − 퐾060

−48.3646 퐾060 − 퐾018 ( 18 ) = 50.9973 퐾018 − 퐾059 8.9475 퐾059 − 퐾019 −1.8748 퐾019 − 푈푌푁퐼 [ −0.4047 ] (퐷푟𝑖푓푡 푟푎푡푒 [푚퐺푎푙⁄푑푎푦]) ( 푇 −1)−1 푇 푥푔1025 = 퐴 퐴 퐴 푦푔1025 −293.0422 퐾016 − 퐾017 74.9594 퐾017 − 퐾060

−48.3501 퐾060 − 퐾018 ( 19 ) = 50.9272 퐾018 − 퐾059 8.6419 퐾059 − 퐾019 −1.4208 퐾019 − 푈푌푁퐼 [ −0.2603 ] (퐷푟𝑖푓푡 푟푎푡푒 [푚퐺푎푙⁄푑푎푦]) 35

The amount of correction due to drift that is applied to the reduced g values throughout the network is illustrated in Figure 10. The absolute maximum linear drift correction is 1.07 mGal, but the vast majority of adjustment amounts (91.6%) are less than ±0.10 mGal. The small magnitudes present in these corrections demonstrate the high level of observational accuracy as even the extreme changes are only approximately

1 mGal.

Analysis at the survey line is a crucial preliminary step in identifying blunders as well as estimating and removing instrument drift from the observations. This analysis at the survey line level generates estimates of the ∆푔 values between consecutive station 36

pairs from ( 13 ), which will be used in the network based adjustment described in

Chapter 2.5. The survey line processing takes into consideration the instrument calibration functions provided by the manufacturer, tidal corrections, and an estimated linear drift rate for each individual gravimeter and survey line. Further validation of the observational accuracy is found in the 90% + of drift corrections that are at insignificant magnitudes (< 0.1 mGal).

2.3: Absolute Gravity Network

From a geodetic perspective, the observations obtained with relative gravimeters are of somewhat limited value, in a larger context, without the inclusion of stations with known gravity values. The network of relative gravity allows one to predict the change in gravity between one station and another, but not the absolute value of gravity at any station. The values obtained from a relative gravity survey are all biased by a constant, an unknown systematic offset or bias. In order to account for this offset, it is necessary to measure the absolute value of gravity at one or more stations in the network. This is done using an absolute gravimeter typically measuring the acceleration of a freely-falling test mass using laser interferometry. Extensive discussion of absolute gravimetry techniques and instrument development can be found in the geodetic and geophysical literatures (e.g.

Torge, 2001; Faller, 2002; etc.)

The Bolivian gravimetric network was originally based on 3 absolute gravity stations established by the National Imagery and Mapping Agency (NIMA) in 1997 using an FG5 absolute gravimeter. This original sparse configuration is illustrated in Figure 11.

The original network configuration is too concentrated in the central portion of the country and additional absolute stations are needed to adequately support relative gravity surveys for the entire country. In a global collaboration involving OSU, DTU, and IGM-

Bolivia, the absolute gravity network in Bolivia was expanded from its 3 original stations to 15 stations (Nielsen, 2013). The 3 original stations at La Paz, Santa Cruz, and

Trinidad were re-observed, and 12 new stations were measured throughout the country

providing a more expansive network for future gravity work. Figure 11 shows the newly observed absolute gravity network configuration consisting of 15 absolute gravity stations.

Figure 11: Bolivian absolute gravity network with elevation [m]

The absolute gravity network was observed with a Micro-g LaCoste A10 (SN-

019) operated by J. Emil Nielsen from DTU with an accuracy of 10 μGal and precision of

10 μGal in 10 minutes (Micro-g LaCoste, 2008). The differences between the FG5 and

A10 measurements are shown in Table 4 as evidence of the consistency between the two absolute gravity surveys and instruments.

Table 4: Absolute gravity differences – A10 in 2011 and FG5 in 1997

Gravity Station Difference [Gal] La Paz -19 Santa Cruz -8 Trinidad -4

Absolute gravity stations often have special requirements (e.g. sheltered from wind, stable platforms, etc.) so we do not normally occupy an existing relative gravity point. Instead, a nearby point (typically indoors) is occupied, and this point is tied to a nearby relative station (and so to the entire network) using a relative gravimeter.

Typically, this relative gravity difference is observed repeatedly between the two stations and the average difference is used in subsequent adjustments so as to shift the relative gravity network values so as to be consistent with the absolute gravity measurements.

Procedurally, the absolute gravity network provides additional constraints for the realization and adjustment of the relative gravity network. However, the geodetic position of the absolute gravity stations is typically unknown as they are inside or near buildings with solid foundations. Therefore, the absolute gravity stations provide constraints for the relative network but will not contribute any gravimetric quantity or anomaly requiring a known station height, unless extraordinary measures are taken to precisely determine the height of the absolute gravity station. In the future, these sites could be included in the anomaly fields by determining the height of each absolute 40

gravity station through traditional terrestrial positioning, like geodetic leveling, that

‘connects’ the absolute station to a GPS station located nearby.

2.4: Building Networks

The following section provides a description of the terminology used for global analysis of a gravity network consisting of many relative gravity stations and a much smaller number of absolute gravity stations. This terminology also will promote a cleaner understanding of the mathematical framework used for the adjustment presented in Chapter 2.5.

From the point of field operations, a survey ‘segment’ is a group or ‘line’ of consecutive stations that are observed in both forward and reverse directions in two consecutive days. These segment surveys are subjected to a local adjustment analysis as described in Chapter 2.2. For a segment containing n stations, this adjustment provides n-1 estimates of the true gravity difference between each station pair for each gravimeter used in the survey. The results from the entire history of segment surveys are then used in the network adjustment.

The network analyst has a different point of view. Gravity lines comprised of many segments often form closed loops. Most stations are part of a single line or ‘edge’, and as such, are connected to just one or two other stations (see Figure 12). But some stations are connected to three or more other stations and are designated as ‘nodes’. In effect, nodes are where survey lines or edges meet.

An edge is then a line of stations that begins at one node and may end at another node. Nodes never occur within an edge only at one or both ends of an edge. All nodes are stations, but most stations are not nodes. For our Bolivia network, there are typically

3-8 stations in an edge. The exception is the tie lines joining absolute gravity stations

(AGS) to the corresponding nearest relative gravity station (RGS). This tie line is an edge containing one node (the RGS) and one other station (the AGS). Some edges consist of entirely of RGS, have a node at one end, but not at the other. Such hanging lines or hanging edges do not form part of a loop. Edges that form part of a loop are described as 1st Order, whereas RGS edges that do not are referred to as 2nd Order. This is because the solutions for RGS in a hanging edge do not benefit from the constraint imposed by a loop. The final gravity estimates for a station at the end of a long hanging edge typically have a higher formal uncertainty than a station, which lies within a loop.

There are two general classes of a network adjustment model (i.e. two ways of formulating the inverse problem for a final network solution). One formulation uses closure constraints explicitly, and the other does not though the constraints are still present, implicitly. This network adjustment model will be described and derived in more detail in Chapter 2.5.

Note that the survey segments may, but usually do not constitute network edges.

As a network is constructed, a station once connected to two neighbor stations may form a junction with a newly constructed segment and change to a node. The relationship between survey segments and network edges evolves as the network expands. This is why the network analyst prefers a terminology divorced from that associated with field operations.

Figure 12: Observational and network terminology 1st Order Stations = Part of a closed loop

2nd Order Stations = Not part of a closed loop (i.e. a “hanging” edge)

2.5: Network Adjustment

The two classes of adjustment associated with gravity networks can be characterized as (1) a local or segment level adjustment to remove the drift rate and (2) a network level adjustment to utilize multiple gravimeters, absolute gravity stations, geometric requirements (i.e. implicit loop closures), and minimize random errors. In this section, we discuss (2) network level adjustment, which uses all results previously produced from a large number of segment adjustments (See Chapter 2.2 for the segment level adjustment). Just as horizontal coordinates in a horizontal control network can be adjusted within a local frame or adjusted to align with a global frame, a similar, though not exact, representation is used here. The network adjustment scheme is illustrated in .

Figure 13: Network adjustment overview

There are two adjustment techniques that can be adopted for network adjustment: adjustment by condition equations and the Gauss-Markov model with fixed constraints.

However, they are duals of each other and produce identical results. We present here the

Gauss-Markov model with fixed constraints, which we prefer. (See Appendix C for a

discussion of the alternative approach). As noted in Strang & Borre, (1999): ‘It is far more difficult to program a computer to set up the conditions than just to read the observation equations’. The automatic detection and identification of loops within a network is an intensive algorithm to develop. The Gauss-Markov model with fixed constraints is a much easier model to fully implement with confidence in a software package. The main drawback of the Gauss-Markov model with fixed constraints is a slightly more complicated description of the error distribution throughout the network.

There are simply no numerical misclosures present after the adjustment; they all sum exactly to 0. The Gauss-Markov model with fixed constraints will be derived in the following section and illustrated with reference to a small example network.

This network adjustment uses the Gauss-Markov model with fixed constraints or a model of observation equations with fixed constraints. The observations can be modeled using the following (Koch, 1999 and Schaffrin, 2007):

푦 = 퐴휉 + 푒 ( 20 ) where: y = n x 1 vector of observations (g values between station pairs)

A = n x m matrix of coefficients representing the observation equations

휉 = m x 1 vector of unknown parameters (g values at stations)

2 −1 e = n x 1 error vector with statistical expectation = 0 and dispersion = 휎0 푃 = Σ

P = n x n weight matrix

The second portion of the model consists of the constraints imposed on the observation equations. These constraints are modeled using the following equation:

휅 = K휉 ( 21 ) where:

휅 = l x 1 vector of constraints

K = l x m matrix of coefficients representing the constraint equations

The least squares solution minimizes the quantity, 푒푇푃푒 subject to the requirements imposed by the constraints. Thus, the function to minimize can be described as:

푒푇푃푒 = (푦 − 퐴휉)푇푃(푦 − 퐴휉) ( 22 ) subject to:

휅 − K휉 = 0 ( 23 )

To estimate the solution, the use of a Lagrange function is needed. The Lagrange function for the Gauss-Markov model with fixed constraints is shown in the following equation:

Φ(휉, 휆) = (푦 − 퐴휉)푇푃(푦 − 퐴휉) − 2휆푇(휅 − K휉) ( 24 )

A little bit of linear algebra and substitution can be performed on this Lagrange function to simplify the equation resulting in:

Φ(휉, 휆) = 푦푇푃푦 − 2푐푇휉 + 휉푇푁휉 − 2휆푇(휅 − K휉) ( 25 ) where the system of normal equations is defined by:

[푁, 푐] = 퐴푇푃[퐴, 푦] ( 26 )

Taking the partial derivatives of ( 25 ) with respect to 휉 and 휆 and setting them equal to 0 results in the following:

휕Φ = 2(−푐 + 푁휉 + K푇휆) = 0 ( 27 ) 휕휉

휕Φ ( 28 ) = 2(Κ휉 − 휅) = 0 휕휆 The resultant matrix equation for the partial derivatives can be expressed as follows where the parameters with a ‘hat’ now represent the estimates of the original parameters:

푇 휉̂ 푐 [푁 K ] [ ] = [ ] ( 29 ) Κ 0 휆̂ 휅

When the rank of A is equal to m, N is invertible and solving the matrix equation for the parameters results in the following:

휉̂ = 푁−1푐 − 푁−1Κ푇휆̂ ( 30 )

휆̂ = −(퐾푁−1Κ푇)−1(휅 − 퐾푁−1푐) ( 31 )

Substituting the result from ( 31 ) into ( 30 ) results in:

휉̂ = 푁−1푐 + 푁−1K푇(K푁−1K푇)−1(휅 − K푁−1푐) ( 32 )

A more typical case encountered in gravimetric networks is an overdetermined system, where the rank of A is some arbitrary value q which is less than m. However, the following information about the rank of A and K is known:

푟푎푛푘 [퐴푇 K푇] = 푚 ( 33 )

In this case, an invertible quantity must be substituted into the matrix equation shown in

( 29 ). This is done by adding 퐾푇퐾휉 to both sides of the first row of ( 29 ). This results in the updated matrix equation:

푇 푇 휉̂ 푇 [푁 + 퐾 퐾 퐾 ] [ ] = [푐 + 퐾 휅] ( 34 ) 퐾 0 휆̂ 휅

Again, isolating the two parameters in ( 34 ) results in the following two equations:

휉̂ = (푁 + 퐾푇퐾)−1(푐 + 퐾푇휅) − (푁 + 퐾푇퐾)−1Κ푇휆̂ ( 35 )

휆̂ = −[퐾(푁 + 퐾푇퐾)−1Κ푇]−1[휅 − 퐾(푁 + 퐾푇퐾)−1(푐 + 퐾푇휅)] ( 36 )

The error estimates can be determined by rearranging ( 20 ) and incorporating a tilde to signify that they are estimated values, which results in the following from ( 20 ):

푒̃ = 푦 − 퐴휉̂ ( 37 )

For completeness, the estimate of the variance of unit weight can also be determined from the following equation:

푒̃ 푇푃푒̃ 휎̂2 = ( 38 ) 0 푛 − 푚 + 푙

A very simple, example gravimetric network is shown in Figure 14 with two absolute gravity stations (A and E) with known gravity and four connected relative gravity stations.

Figure 14: Example network

The network above would consist of the following Δ푔 values, where the subscripts are indices corresponding to the direction of the observation based on a from station and a to station which makes up the vector of observations, y:

Δ푔 퐶,퐴 350.6 Δ푔 퐷,퐵 −125.8 Δ푔퐶,퐷 62.3 푦 = Δ푔퐸,퐶 = 148.2 ( 39 )

Δ푔퐶,퐹 47.9 −14.6 Δ푔퐷,퐹 [−195.4] [Δ푔퐹,퐸 ]

The Gauss-Markov model with fixed constraints uses a design matrix shown in

( 40 ) with each row corresponding to an observation (the difference in consecutive station values) and each column corresponding to an individual station. Additionally, all observations are adjusted with equal weight, thus P = In.

1 0 −1 0 0 0 0 1 0 −1 0 0 0 0 −1 1 0 0

퐴 = 0 0 1 0 −1 0 ( 40 ) 0 0 −1 0 0 1 0 0 0 −1 0 1 [0 0 0 0 1 −1]

As the title of the adjustment model suggests, one or more fixed constraints must be included in the model. In this example network, stations A and E are given as known or fixed values. A constraint matrix (K) associated with these constraints is constructed with a similar structure as the design matrix, A where each row contains a single coefficient of 1 corresponding to the location of the constraint station. This constraint matrix is shown in ( 41 ). The values of these constraints, which will stabilize the network, are contained in the 휅 vector in ( 42 ).

1 0 0 0 0 0 퐾 = [ ] ( 41 ) 0 0 0 0 1 0

978500 휅 = [ ] ( 42 ) 978000 The system of normal equations can then be solved using ( 26 ) with results shown as:

1 0 −1 0 0 0 0 1 0 −1 0 0

푁 = −1 0 4 −1 −1 −1 ( 43 ) 0 −1 −1 3 0 −1 0 0 −1 0 2 −1 [0 0 −1 −1 −1 3 ]

350.6 −125.8

푐 = −312.6 ( 44 ) 202.7 −343.6 [ 228.7 ]

The rank of the design matrix in ( 40 ) needs to be verified to determine if the matrix has full column rank. In this example, the rank of A is equal to 5, which is less than m = 6.

Thus, the equations in ( 35 ) & ( 36 ) will be used utilizing the matrices shown in ( 40 ) –

( 44 ). The results for the vector of Lagrange multipliers is shown in ( 45 ) followed by the estimated parameters in ( 46 ).

휆̂ = −[퐾(푁 + 퐾푇퐾)−1Κ푇]−1[휅 − 퐾(푁 + 퐾푇퐾)−1(푐 + 퐾푇휅)] −0.884615 ( 45 ) = [ ] 0.884615

978500 978084.892

휉̂ = (푁 + 퐾푇퐾)−1(푐 + 퐾푇휅) − (푁 + 퐾푇퐾)−1Κ푇휆̂ = 978148.515 ( 46 ) 978210.692 978000 [978195.969]

The network adjustment estimates for the unknown parameters in ( 46 ) indeed ‘fix’ the

̂ ̂ value at the constraints (휉1 and 휉5), which correspond to the absolute gravity stations (and

1’s in the K matrix). Finally, the estimated error vector can be determined using ( 37 ) with results as:

−0.8846 0 0.1231 푒̃ = 푦 − 퐴휉̂ = −0.3154 ( 47 ) 0.4462 0.1231 [ 0.5692 ]

2.6: Adjustment Results

The results of the network adjustment of the relative gravity network are presented in the following section. There are two possible models that can be used in the adjustment: the model of condition equations and the Gauss-Markov model with fixed constraints. These two models appear on the surface to be different as they use different equations to solve and different forms of the observables. However, they are duals of one another, so only the results from Gauss-Markov model with fixed constraints will be presented. This section only describes the quantities relative to the gravimetric network

(i.e. residual values, RMS of the residuals, etc.) and not the underlying gravity anomalies

(see Chapter 3 for gravity anomaly determination).

Another property of the network adjustment based on two network constraint configurations will be investigated: the minimally constrained solution and the fully constrained solution. The addition of a single fixed value (or constraint) to the network will allow the unknown gravity values to be known relative to a known level, (i.e. a datum). Typically, a few to many known values are present and support a relative gravity network; however, many essential elements of the observations, network configuration, and existence of errors or blunders can be excavated with the adjustment based on a minimal set of known constraints (i.e. use only a single absolute gravity station as a constraint). These constraints are the known absolute gravity station observations. The inclusion of all known absolute gravity station values produces the statistically most probable and best solution. This type of fully constrained solution will also be discussed.

The minimally constrained solution provides an essential understanding of the internal consistency of the network and observations. However, the minimally constrained solution is not statistically the ‘best’ solution for the network. However, this solution provides an excellent understanding on how errors are being propagated radially away from the fixed constraint location. The unused known constraints will be estimated in the adjustment and estimates can be compared to the known values. By selecting a centrally-located constraint, the distance and number of observations required to traverse the network are assumed to be kept at a minimum. Generally, this assumption is correct; however, there are situations where it would not be met, e.g. higher density sampling intervals and circuitous routes in the network. The minimally constrained network based on a centrally-located station at roughly (-17.3o, -66.2o) is shown in Figure 15.

Figure 15: Minimally constrained network misclosures at absolute gravity stations based on KIKA

Table 5: Relative gravity misclosure statistics at absolute gravity stations

N = 15 Gravity [mGal] Mean 0.04 Minimum -0.35 Maximum 0.55 Standard Deviation (  ) 0.25

The fully constrained adjustment solution will utilize all constraints, and this comparison between estimated and known gravity value is useless as the difference is zero at all constraints. However, the estimated errors can be utilized to better sense the distribution and propagation of the estimates. These estimated errors or residuals in the observations are propagated throughout the network and provide estimates at the unknown gravity stations. The distribution of network adjusted residuals is shown in

Figure 16. Note that the inter-quartile range (IQR) is very small compared with the root- mean squared value. These residuals are very tightly restricted especially when compared against a normal distribution.

Figure 16: Histogram of network adjusted residuals: N = 3470

Since there are multiple gravimeters observing the forward-reverse gravity station pair, there is an observation for each in the adjustment. However, this proves difficult to graphically visualize on the network. We use two statistics to summarize the residuals between a given station pair: the root mean squared (RMS) of all the residuals for a gravity station pair and the maximum residual of these residuals. The RMS of the network adjusted residual and the maximum network adjusted residual are shown in

Figure 17 and Figure 18.

Figure 17: Bolivian relative gravity - RMS of network adjusted residuals [mGal]

Table 6: Relative gravity statistics - RMS of network adjusted residuals

N = 1506 RMS of residuals [mGal] Mean 0.05 Minimum 0 Maximum 0.89 Standard Deviation (  ) 0.07

Figure 18: Bolivian relative gravity - maximum of network adjusted residuals [mGal]

Table 7: Relative gravity statistics - maximum of network adjusted residuals

N = 1506 Maximum of residuals [mGal] Mean 0.07 Minimum 0 Maximum 0.90 Standard Deviation (  ) 0.09

Figure 19: Histogram of RMS of network adjusted residuals

Ultimately, the desired quantity is the unknown values of gravity at all stations.

The estimates of the unknown gravity quantities is found using ( 35 ). The standard error of the estimated gravity stations provides another sense of overall quality in the network adjustment. The standard error for the estimated gravity stations is shown in Figure 20.

The magnitude of the standard error is mainly dependent on two factors in this network: the degree to which the station is connected to the network and the number of gravimeters observing any particular station pair.

Figure 20: Bolivian relative gravity - standard error of estimated gravity

Table 8: Relative gravity statistics - standard error of estimated gravity

N = 1395 Standard Error [mGal] Mean 0.057 Minimum 0.022 Maximum 0.119 Standard Deviation (  ) 0.010

Figure 21: Bolivian relative gravity - number of gravimeters

The preceding section investigated a number of network related results and misclosures present within the Bolivian relative gravimetric network. No geodetic network is error-free; however, the errors that are present and how they are related to distance and elevation are essential and important aspects to understand when designing and observing a similar new gravimetric network or densifying and expanding an existing network. The minimally constrained solution provides a sense of the internal consistency within the network though not the best estimate of the unknown station values. This most 62

probable estimate is achieved using the fully constrained adjustment. For either of these two network adjustment configurations, a case can be made to use either the Gauss-

Markov model with fixed constraints or the model of condition equations.

CHAPTER 3: GRAVITY ANOMALIES

3.1: Relevant Gravity Anomalies

The results of the relative gravity network have been discussed previously in a network-based context. The goal of which is to provide an understanding of the network errors present, isolate those errors, and if necessary re-observe any number of stations.

However, typically the adjusted observational data is investigated in a geophysical sense with the classical free-air and Bouguer anomalies. These anomalies will be presented in this section. As with any observed quantity, the true value is unknown. However, there are a number of satellite and terrestrial global gravity models that have been developed in the past few years, and the estimated gravity quantity can be compared with the modeled value at the same location. The global model that will be used as a comparison is the

Earth Gravitational Model 2008 (EGM2008) (Pavlis et al., 2008). The modeled values were determined using averaged 2.5 minute x 2.5 minute free-air anomaly values determined by the International Gravimetric Bureau (BGI). This 2.5’ x 2.5’ grid was interpolated to the actual station coordinates using a bilinear spline function.

A number of gravity anomalies are relevant to discuss for different applications and analysis. For the complete derivation and description of the anomalous gravity field and its relative components, see (Heiskanen & Moritz, 1967). A point on the geoid, P

can be projected onto a point Q on an ellipsoid surface through the ellipsoid normal direction as illustrated in Figure 22.

Figure 22: Geoid and reference ellipsoid (Figure 2-12 Heiskanen & Moritz, 1967)

The gravity anomaly vector is defined as the difference between the gravity vector g at P on the geoid and the normal gravity vector g at Q on the reference ellipsoid:

Δ̅̅̅푔̅ = 푔̅̅푃̅ − 훾̅̅푄̅ ( 48 )

The difference in magnitude is defined as the gravity anomaly:

Δ푔 = 푔푃 − 훾푄 ( 49 )

The gravity observations are not associated with either of these two surfaces; they are related to the terrestrial surface. To determine the gravity anomaly for a station, the surface observations need to be reduced to the geoid surface and the normal gravity needs to be determined.

The normal gravity on a reference ellipsoid with semi-major axis length = a and semi-minor axis length = b can be calculated in closed form from Somigliana’s formula

(Heiskanen & Moritz, 1967):

2 2 푎훾푎 cos 휙 + 푏훾푏 sin 휙 훾 = ( 50 ) √푎2 cos2 휙 + 푏2 sin2 휙 where:

훾푎 = normal gravity at the equator

훾푏 = normal gravity at the pole

The reduction of gravity observed on the surface to the geoid surface requires the

휕푔 knowledge of the vertical gradient of gravity, . Typically, the vertical gradient of 휕퐻 gravity is unknown but is approximated with the normal gradient of gravity:

휕푔 휕훾 ≅ = −0.3086 ( 51 ) 휕퐻 휕ℎ

The value of gravity at the geoid can then be determined from a first-order Taylor expansion and the height of the observation point above the geoid:

휕푔 휕훾 푔 = 푔 − 퐻 … ≅ 푔 − 퐻 = 푔 + 0.3086퐻 ( 52 ) 퐹퐴 휕퐻 휕ℎ

The free-air anomaly is then found by subtracting the normal gravity on the reference ellipsoid:

Δ푔퐹퐴 = 푔퐹퐴 − 훾 = 푔 − 훾 + 0.3086퐻 ( 53 )

An estimate of the accuracy of the free-air anomaly can be made by propagating the errors of the individual components in ( 53 ). From the static GPS positioning results, the horizontal accuracy and vertical accuracy is ±0.20 m and ±0.10 m, respectively. This manifests into ±0.062 mGal for the gradient term and ±0.010 mGal for . The maximum

standard error for estimated g values is ±0.119 mGal (see Table 8). Finally, the maximum estimated standard error for gFA is calculated as ±0.135 mGal.

The simple Bouguer anomaly is found by removing the gravity effect of the topographic masses outside the geoid and the free-air correction. The attraction of the infinite Bouguer plate can be found from the following:

퐴퐵 = 2휋퐺휌퐻 ( 54 )

This can be simplified to the following using a standard density for the crust, 휌 =

푔 2.67 : 푐푚3

퐴퐵 = 0.1119퐻 ( 55 )

The simple Bouguer correction can then be found from the following:

휕푔 푔 = 푔 − 퐴 − 퐻 = 푔 − 0.1119퐻 + 0.3086퐻 = 푔 + 0.1967퐻 ( 56 ) 퐵 퐵 휕퐻

By subtracting the normal gravity on the reference ellipsoid, the simple Bouguer anomaly is obtained:

Δ푔퐵 = 푔퐵 − 훾 = 푔 − 훾 + 0.1967퐻 ( 57 )

An estimate of the maximum standard error of the simple Bouguer anomaly can be found in a similar fashion as the free-air anomaly standard error. The maximum estimated standard error for gB is ±0.126 mGal.

|3.2: Free-Air Anomaly

The free-air anomaly map and its corresponding histogram for the Bolivian gravimetric network are shown in Figure 23 and Figure 24. This is found from the first order approximation shown in ( 52 ). The orthometric height (H) needed in this equation is not known for the network stations, so the geoid undulation (N) is approximated from

EGM2008 along with the observed ellipsoid height (h) to estimate H. The free-air anomaly statistics are also shown in Table 9 along with other relevant gravimetric quantities discussed in subsequent sections. Note there is a discrepancy between the number of stations described in Table 9 and Table 8 of ~30 stations. This difference is typically due to incomplete or missing geodetic positional information for a station. An approximate geodetic position can be used for reduction and network adjustment purposes, but only stations with accurate, geodetic-grade positions derived from GPS are suitable for determination of anomalous fields. A large peak is present in the lower values present in Figure 24, which represents the large number of observations at a smaller sampling interval near {-16.5o, -65.5o}.

Figure 23: Free-Air anomaly [mGal]

Figure 24: Free-air anomaly histogram [mGal]

Table 9: Gravity statistics [mGal]

Standard N = 1366 Mean Minimum Maximum Deviation Free-Air 29.645 83.495 -171.580 334.838 Anomaly EGM2008 -8.257 28.353 -182.950 117.640 Residual Bouguer -186.447 146.541 -436.286 57.276 Anomaly Isostatic -8.531 41.957 -126.402 105.479 Anomaly

The free-air anomaly values are then compared with the interpolated model values. The residual free-air anomaly values when compared with the EGM2008 model values amounts are shown in Figure 25 along with the corresponding histogram in Figure

26. Like the free-air anomaly data, the residuals do not exhibit any abnormalities apart 70

from rather large magnitudes. There are no regional nor local areas of significant bias. A few isolated areas along the Eastern flank do not agree well with the model as can be expected in a region with rapidly changing terrain. In general, a station having a large residual will likely reside on the flank, but not every station on the flank has a high residual.

Figure 25: Free-air anomaly and EGM2008 residual [mGal]

Figure 26: Free-air anomaly and EGM2008 residual histogram

Investigating the residual statistics a bit further, a cumulative distribution of the free-air anomaly residual is shown in the inset of Figure 26. Aside from the extremes present in the residuals, the vast majority of the stations still do not agree well with the model. Only ~ 60% of the stations are within 20 mGals of their corresponding modeled values. Considering the gravimetric requirements for a geoid model, these observation residuals are troubling regarding the use of EGM2008 in this region. The likely cause of these large discrepancies is the EGM2008 model’s spatial resolution of 2.5’. This resolution is approximately 4.6 km in the latitude ranges present for the given network, and at these distances, significant elevation changes can occur resulting in vastly different gravitational observations.

3.3: Bouguer Anomaly

The simple Bouguer anomaly map is shown in Figure 27. On a regional scale, the simple Bouguer anomaly exhibits correlations with the local topography, as is expected.

Since the terrain in Bolivia is vastly different across the country, the simple Bouguer anomaly is only presented for completeness and not used in further analysis.

Figure 27: Bouguer anomaly [mGal]

3.4: Isostatic Anomaly

The calculation of the gravitational effect due to the surrounding terrain has historically been a very tedious task for geodesists and geophysicists. In 1909, J.F.

Hayford and colleagues with the United States Coast and Geodetic Survey needed an average time of 17 hours to compute the topographic and isostatic corrections for a single station (Watts, 2001)! The computing power available today along with mathematical methods (i.e. FFT) have greatly increased the efficiency in computing topographic and isostatic corrections since 1909; however, it is still arduous. There are a number of algorithmic methods to determine the gravitational attraction of masses (Forsberg, 1984;

Forsberg & Tscherning 1997; etc.), and we utilize a method developed by Heck & Seitz,

2007. This method uses geometric shapes in the form of spherical tesseroids bounded by spherical coordinates, which results in a spherical rather than a planar approximation.

Figure 28: Tesseroid geometry (Fig. 4 from Heck & Seitz, 2007)

This makes it very convenient for geodetic or geographic coordinates present in a digital terrain model as there is no need for additional coordinate transformations, conversion to a grid based system, etc. Two separate but similar corrections are of interest: the terrain correction due to the surrounding topography and the isostatic reduction due to the ‘roots’ supporting or compensating the topography. Both of these corrections are used in the complete isostatic anomaly:

Δ푔푖푠표 = 푔 + 0.1967퐻 + 푇퐶 + 푑푔푖푠표 − 훾 ( 58 )

These two effects while different in geometry and magnitude can be implemented quite efficiently using the spherical tesseroid model. The general form of the gravity reduction of a spherical tesseroid can be determined from:

훿푔(푟, 휑, 휆) = 퐺휌Δ푟Δ휑Δ휆 ∗

1 ( 59 ) [퐿 + (퐿 Δ푟2 + 퐿 Δ휑2 + 퐿 Δ휆2)] 000 24 200 020 002 where G is Newton’s constant of gravitation;  is the mass density; ∆푟, ∆휑, & Δ휆 are the geometric bounds of the tesseroid; and the L terms are series coefficients shown in

Appendix D.

The terrain correction (TC) and isostatic reduction (dgiso) in ( 58 ) are only evaluated differently with ( 59 ) in terms of the radial bounding coordinates: r1 and r2. In computing the isostatic reduction due to compensating masses in the lower lithosphere, the radial coordinates are defined as (see Figure 29):

푟1 = 푅퐸 − 푇 − 푡 ( 60 )

푟2 = 푅퐸 − 푇 ( 61 ) where:

푅퐸 = 6371 푘푚 ( 62 )

푇 = 30 푘푚 ( 63 )

휌 2670 푘푔/푚3 푡 = 퐻 = 퐻 = 4.45퐻 ( 64 ) Δ휌 600 푘푔/푚3

Figure 29: Isostatic geometry

Additionally, the gravitational effect due to the visible and surrounding topography is based on r1 and r2 values from:

푟1 = 푅퐸 + 퐻푃 ( 65 )

푟2 = 푅퐸 + 퐻′ ( 66 )

The total reduction amount applied to an observed gravity value is then determined based on the individual components of a grid of spherical tesseroids, which covers a rectangular area of 333.4 km by 333.4 km (2 x 166.7 km). This radial distance of 166.7 km is based on Zone O (Hayford and Bowie, 1912; Hammer, 1939; Heiskanen

& Vening Meinesz, 1958; and Hinze, et al., 2005). The grid is centered on the station that is receiving the correction, and all grid values are totaled to arrive at the terrain correction and/or root compensation. 77

Both the terrain and isostatic root effects were evaluated based on topography from the ~90-meter SRTM dataset (Jarvis, et al., 2008). It is difficult to evaluate the accuracy of the SRTM data as stated accuracies are typically for global use, but more significant errors are present in mountainous terrain due to steep slopes, shadowing, etc.

(Gamache, 2004), which is clearly evident in Bolivia. Preliminary comparisons between station heights and SRTM interpolated heights produces maximum differences of ±30 m, but this only impacts terrain corrections and isostatic corrections for stations in rugged terrain. Due to the irregular nature of the errors in the SRTM dataset and their spatial distance relationship with gravity observations, the standard error is next to impossible to determine appropriately. There are additional DTM’s available (~30-meter / 1 arc-sec.), but these do not significantly impact the overall results while increasing the computational time immensely. For most topography and many of the stations in

Bolivia, this terrain correction is minuscule. However, along the Eastern Cordillera, this terrain correction is quite significant and approaches 50 mGals. Along the most westerly portions of the Bolivian Altiplano, some minor effects of the Western Cordillera are apparent in the terrain correction but are much less significant compared with those in the

Eastern Cordillera. In the central Altiplano and in the Eastern lowlands, there is essentially no residual terrain effect. The map showing the terrain correction is shown in

Figure 30.

Figure 30: Terrain correction [mGal] determined using spherical tesseroids (Heck & Seitz, 2007) based on a 333.4 km x 333.4 km grid centered at the individual stations.

The isostatic anomaly is also presented in Figure 31 below. This isostatic anomaly is based on the simple, local compensation model of Airy-Heiskanen

(Heiskanen & Moritz, 1967). The depth of compensation at sea level, T is assumed to be

30 km with a density contrast, Δ휌 = 600 kg/m3.

Many previous studies involving the isostatic anomaly have used slightly different crustal parameters (e.g. Lyon-Caen, et al., 1985; Gotze, et al., 1990; Chapin, 1996; and

Whitman, 1999).

Table 10 illustrates these differences. However, the choice of these parameters does not significantly affect the isostatic anomaly results and the values from Heiskanen & Moritz,

(1967) will be used in subsequent analysis. The resulting map of the isostatic anomaly is shown in Figure 31.

Table 10: Isostatic Crustal Parameters

T [km] Δ휌 [kg/m3] Heiskanen & Moritz, 1967 30 600 Whitman, 1999 35 350 Gotze, et al., 1990 40 500

Figure 31: Isostatic anomaly [mGal]

CHAPTER 4: DELAMINATION

4.1: Geological Background

The modern Andean Cordillera of South America is associated with the subduction of the oceanic Nazca plate beneath the western edge of the South American continent (Dewey & Bird, 1970; Barazangi & Isacks, 1976; Dewey & Lamb, 1992). The highest and widest portion of the Andes, known as the Central Andes or the Andean

Plateau, consist of the high and flat Altiplano surrounded by the even higher Eastern and

Western Cordillera (Allmendinger et al., 1997; Isacks, 1988). This is one of just two large and active tectonic plateaus on earth, the other being the Tibetan Plateau. The traditional model for the evolution of the Andean Plateau is that it formed gradually over a period of more than 20 MY, in response to shortening and thickening of the continental crust (e.g. Isacks, 1988; Lamb & Davis, 2003; Oncken et al., 2006). However, Garzione et al., (2006) suggested an alternative to this scenario, later expanded upon by Ghosh et al., (2006); Molnar & Garzione, (2007); and Garzione et al., (2008), in which the Andean plateau ‘popped up’ in late Miocene time, between about 10 and 6.8 Ma. This extremely rapid rise can only be explained by rapid removal of the dense lithospheric mantle root of the continent, either by delamination (Bird, 1979) or by means of a convective instability

(England & Houseman, 1989) in which the cold and dense lithospheric mantle ‘drips’ off 82

the South American lithosphere. In order to invoke > 2 km of sudden uplift, removal of cold (and therefore dense) lithospheric mantle is not enough. It is also necessary to assume that a dense eclogite layer, comprising the lower crust, was removed along with the lithospheric mantle (Molnar & Garzione, 2007). However, Beck & Zandt, (2002) concluded that the Altiplano has no lithospheric mantle in some places, and an apparently normal (high velocity) lithospheric mantle in others. If only partial removal of the lithospheric mantle and lower crust could produce two or more kilometers of uplift, as the pop-up hypothesis suggests, then there should be very large gravity changes within the

Altiplano as one moves from areas underlain by lithospheric mantle to areas where no significant amount of lithospheric mantle remains.

The following chapter will describe a means of assessing this pop-up hypothesis by assuming that local isostatic equilibrium prevailed both before and after delamination.

We use a Monte Carlo simulation to identify the ensemble of model instances capable of producing uplift amounts of 3 ± 1 km as proposed by Garzione, et al., (2006), and producing the current (final) topography. This suite of viable delamination models is then used to predict a suite of gravity signatures expected as one crosses between delaminated and non-delaminated parts of the Altiplano (see Figure 32). The goal is to identify the ‘typical’ gravity contrast expected if the delamination hypothesis is true.

This is compared to the actual gravity variations within the Altiplano, providing us with a test of the delamination model.

Figure 32: High and low P-wave velocity zones with gravity observations (from Beck & Zandt, 2002)

From Figure 32, high P-wave velocity zones are equated with the presence of lithospheric mantle, meaning delamination has not occurred. The velocities are estimated to be 8.3 km/s within these high velocity zones at a depth of 90 km (Myers, et al., 1998).

Areas in which the uppermost mantle has a low P-wave velocity are assumed to have been delaminated with approximate P-wave velocities of 7.8 km/s.

4.2: Lithospheric Delamination Model

The model of the lithosphere used in this analysis is developed using Airy-

Heiskanen isostatic theory based on local compensation (Heiskanen & Moritz, 1967).

Local isostatic compensation is a very crude model for the complexities present in the lithosphere. However, a number of other investigations have determined to varying degrees that much of the Altiplano is more or less locally compensated (e.g. Lyon-Caen, et al., 1985, Gotze, et al., 1990, Chapin 1996, and Whitman, 1999). The model is depicted in Figure 33, which includes the current state (i.e. after delamination), the before delamination state, and a reference column.

With reference to the initial, pre-delamination state (Figure 33(b)), assume that the temperature in the lithospheric mantle at some depth z beneath the base of the crust is given by:

푧 푇 = 푇 + ∆푇 ( 67 ) 퐿푀 퐶 퐿 where: ∆푇 = 푇퐴 − 푇퐶, 푇퐴 is the temperature of the asthenosphere at the base of the lithospheric mantle (z = L), and 푇퐶 is the temperature at the base of the crust (z = 0).

Note that L is the initial thickness of the lithospheric mantle, and ∆푇/퐿 is the vertical temperature gradient within the lithospheric mantle.

Because the lithospheric mantle is cooler than the underlying asthenosphere, the depth dependent density of the lithospheric mantle, 휌퐿푀 is always greater than the density of the asthenosphere, 휌퐴 except at their interface. Since:

휌퐿푀(푧) − 휌퐴 = 휌퐴훼푉(푇퐴 − 푇퐿푀(푧)) ( 68 ) where: 훼푉 is the volumetric coefficient of thermal expansion for mantle rock (Turcotte &

Schubert, 2002, Eq. 4-205), the average density of the lithospheric mantle is given by:

∆푇 휌̅ = 휌 (1 + 훼 ) ( 69 ) 퐿푀 퐴 푉 2

We now model the uplift of the Altiplano produced by delamination beginning with the first and simplest scenario (Figure 33(b)) in which the entire ‘lid’ of lithospheric mantle was removed, and the asthenospheric mantle takes its place. The asthenospheric temperature (푇퐴), the crustal thickness (H), and the crustal density, (휌퐶) all remain the same. Under the assumption of local isostasy, we can balance mass columns depicted by

Figure 33(a) and (b), implying:

퐻휌푈 + (퐿 + ∆ℎ)휌퐴 = 퐻휌푈 + 퐿휌̅퐿푀 ( 70 ) and therefore:

∆푇 ∆ℎ휌 = 퐿(휌̅ − 휌 ) = 휌 훼 퐿 ( 71 ) 퐴 퐿푀 퐴 퐴 푉 2 which simplifies to:

1 ∆ℎ = 훼 ∆푇퐿 ( 72 ) 2 푉 87

or equivalently:

2∆ℎ ∆푇 = ( 73 ) 훼푉퐿

-5 -1 If we assume that 훼푉 = 3 x 10 K , ∆푇 = 600 K, and L = 100 km, then ( 72 ) yields an uplift estimate of 900 m, very far short of the 2500 - 3500 m of abrupt uplift invoked by Ghosh et al., (2006) and Garzione et al., (2006). This calculation suggests that a late Miocene ‘pop-up’ of the Altiplano cannot plausibly be explained by simple delamination of the lithospheric mantle alone.

4.3: Lithospheric Delamination Model with an Eclogite Layer

In order to explain > 2 km of uplift, it is necessary to invoke a dense layer of eclogite, at the base of the crust, prior to delamination, and assume that both the lithospheric mantle and this eclogite layer are removed during the delamination event

(Garzione et al., 2006 and Molnar & Garzione, 2007). The partial or complete removal of this dense eclogite layer has the potential to produce the 3±1 km of uplift claimed by the ‘pop-up’ theory. The modified delamination model is illustrated in Figure 34.

Figure 34: The Modified Delamination Model. It is assumed that as a consequence of its unusual thickness, the lower part of the continental crust had transformed to eclogite. Both this eclogite layer and the lithospheric mantle are removed by delamination.

Using the layer thicknesses and densities as shown in Figure 34(b) and (c), the pre-delamination height (h) can be found from:

푊휌푊 + 퐵휌퐵 + 푋휌퐴 = 퐻휌푈 + 퐸휌퐸 + 퐿휌̅퐿푀 ( 74 )

푊휌푊 + 퐵휌퐵 − 퐸휌퐸 − 퐿휌퐿푀̅ ℎ = + (퐻 + 퐸 + 퐿 − 푊 − 퐵) ( 75 ) 휌퐴

Additionally, the change in height (∆h) can be found from balancing the columns in Figure 34(a) and (b).

퐻휌푈 + (퐿 + 퐸 + ∆ℎ)휌퐴 = 퐻휌푈 + 퐸휌퐸 + 퐿휌̅퐿푀 ( 76 )

퐸(휌 − 휌 ) + 퐿(휌̅ − 휌 ) ∆ℎ = 퐸 퐴 퐿푀 퐴 ( 77 ) 휌퐴

Using the results from ( 75 ) and ( 77 ), the current (post-delamination) height of the surface (hf) can be found from:

ℎ푓 = ℎ + ∆ℎ ( 78 )

The delamination invoked using ( 74 ) - ( 78 ) assumes complete delamination of the Altiplano. Yet according to seismic observations (Beck & Zandt, 2002), the

Altiplano is likely in a state of partial delamination. This scenario can be represented as shown in Figure 35 by assuming that some areal fraction, f, of the Altiplano is delaminated, whereas the fraction (1 - f ) is not. In this case we multiply the right hand side of ( 77 ) by f in order to obtain the correct uplift:

퐸(휌 − 휌 ) + 퐿(휌̅ − 휌 ) ( 79 ) ∆ℎ = 푓 ∗ [ 퐸 퐴 퐿푀 퐴 ] 휌퐴

Figure 35: Partial Delamination Model Scenario

4.4: Monte Carlo Simulations

All of the density and depth parameters in this model have uncertain values but they are known within a particular range of values. This analysis uses a Monte Carlo simulation (Metropolis & Ulam, 1949), which is based on a large number of trials (~105) or combinations of parameter values. Each trial uses a parameter value selected randomly from a pre-determined range. Each trial calculates a value of hf and ∆h, which are then compared to the current elevation of the Altiplano (hf) and the amplitude of rapid uplift according to the pop-up hypothesis (∆h). The current topography is rather straightforward with mean elevations of the Altiplano known to be ~3800 m. The uplift amount, which results in a successful outcome in the simulation, is found from a range invoked using results from Ghosh et al., (2006) and Garzione et al., (2006). The particular ranges of ‘acceptable’ output values are shown in Table 11.

Table 11: Ranges of Acceptable Outcome Values

Parameter Range [m]

hf 3700 – 3900 ∆ℎ 2500 - 3500

The great majority of Monte Carlo trials or instances produces results in which either hf or ∆h, falls outside of their acceptable ranges, and the instance is rejected.

Otherwise it is retained as an acceptable or plausible delamination scenario.

The input density and depth parameters can be separated into two subsets: non- critical and critical. There are a number of non-critical input parameters that will produce acceptable output values over the entire range of acceptable input values. These parameters are assigned a plausible, average value and do not fluctuate in the Monte

Carlo simulation. The non-critical assigned values used in the simulation are shown in

Appendix E. A couple of input parameters do largely control the successful trial outcomes: E is the thickness of the delaminated eclogite layer, E is the density of the eclogite, and U is the density of the upper crust. The values of these critical parameters will either produce acceptable or rejected outcomes with respect to hf and ∆h calculated from ( 78 ) and ( 79 ). Only a very narrow range of values will result in an acceptable outcome. These parameters are shown in Table 12 with their permissible ranges used in the simulation.

Table 12: Pop-up model critical parameters

Parameter Range E [km] 10 – 40

3 휌퐸 [kg/m ] 3350 - 3650 3 휌푈[kg/m ] 2600 - 2900

Two Monte Carlo simulations were conducted based on an appropriate amount of partial delamination: f = 0.50 and f = 0.75 using n = 100,000 trials for each simulation.

Acceptable combinations for the three critical parameters are shown in Figures 5 and 6 for f = 0.50 and f = 0.75, respectively. The dark blue trials represent acceptable outcomes

with regards to hf and Δh. The cyan trials represent rejected values. (Note: the cyan

‘background’ is all rejected outcomes).

Based on the Monte Carlo simulation results, seven plausible model scenarios were selected (colored squares in Figure 36 and Figure 37 with Roman numerals I - IV).

(See Figure 55 in Appendix E for the geometry and density distribution of Scenario I of the delamination model). The parameters associated with each scenario are shown in

Table 13.

Figure 37: Partial delamination Monte Carlo simulation (f = 0.75). Scenarios IV – VII are highlighted with parameters in Table 13.

Table 13: Scenario parameters – See Figure 35

휌푈 f 3 3 E [km] 휌 [kg/m ] [kg/m ] hf [m] ∆ℎ [m] Fractional 퐸 Scenario Eclogite Density of Density of Final Uplift Amount of Thickness Eclogite Upper Height Amount Delamination Crust I 0.50 25 3616 2710 3756 2518 II 0.50 30 3550 2700 3885 2580 III 0.50 35 3600 2670 3854 3183 IV 0.75 15 3585 2795 3806 2533 V 0.75 20 3500 2795 3764 2656 VI 0.75 25 3470 2790 3777 2905 VII 0.75 35 3350 2790 3850 2686

4.5: Gravity Signal Response

Each of these delamination scenarios will generate a particular gravity signal that should be present in the observed gravity anomalies if the scenario is likely to have occurred. The synthetic gravity signals were generated using a script developed from the equations provided by Won & Bevis, (1987). The seven synthetic gravity signals associated with the selected model scenarios are shown in Figure 38. Gravity should increase in moving from delaminated to non-delaminated zones, since the latter are characterized by higher densities at depth. (Think: higher velocity, higher density, excess mass, and higher gravity)

Figure 38: Generated gravity signal for tectonic Scenarios I - VII [mGal]. (Negative distances associated with non-delaminated situation; positive with delamination)

In Table 14, we characterize the surface gravity change as a function of the distance travelled across (perpendicular to) the edge of the boundary between non- delaminated and delaminated portions of the Altiplano.

Table 14: Gravity amplitude [mGal] at varying distances. Tabulated values associated with Figure 38.

Distance [km] Scenario 25 50 100 200 I 66 128 235 373 II 65 128 235 376 III 80 156 287 462 IV 45 87 159 252 V 46 89 163 260 VI 49 96 175 281 VII 42 82 152 248

The gravity signals associated with the seven selected scenarios provide a model gravity signal that can be compared with the observed gravity anomalies. Note that the surface gravity signature of the delamination/non-delamination boundary below the

Moho has a gradient, or rate of accumulation, that varies from about 2 mGal/km to more than 2.8 mGal/km. The modeled gravity and the observed gravity will be compared within a focused area of the Altiplano where seismic observations sense differences in subcrustal material i.e. areas of high-velocity mantle versus ‘normal’ or low mantle velocities (see Figure 32). Many of these areas are quite small relative to the gravity observation sampling density and cannot be investigated very well. However, the high- velocity zone bounded by ~ -17o to -21o and -67o to -68o has a number of observations within and across it a shown in Figure 39. As noted above, we would expect surface gravity to rise as one moves into the high mantle velocity zones. However, some care is required because the isostatic corrections related to topography, particularly for the

Eastern Cordillera bordering the Altiplano, are only approximate, or incomplete. No single class of isostatic model can encompass both the Altiplano and the Eastern

Cordillera as the Eastern Cordillera is likely in a state of regional compensation or flexure rather than locally compensated (Lyon-Caen, et al., 1985; Jekeli, et al., 2013; etc.).

Before considering specific gravity profiles, we attempt a holistic comparison of gravity inside and outside of the mantle high velocity zone. To a certain extent, one would expect the distributions of gravity observations within each zone to be significantly different. The distributions of the isostatic gravity anomaly and the station heights for both areas are shown in Figure 40. The area outside the high-velocity zone has some considerably larger values compared to inside the high-velocity zone, which is the opposite of what the delamination model predicts (see Table 15). Moreover, there is not a significant difference in the means of the two sets of gravity anomalies. The two sample means are within 10 mGal of each other. This is much less than even the minimum ~40 mGal difference expected from the delamination model scenarios.

Figure 40: Isostatic anomaly (with station height) distributions

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Figure 41: Free-air anomaly distributions

Figure 42: Bouguer anomaly distributions

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Table 15: Observation statistics and high-velocity zone (HVZ)

Area Zone N Mean h Min. h Max. h Mean Min. Max. [103 km2] Inside 45 120 3771 3697 3990 -23.8 -51.7 14.2 HVZ Outside 100 195 3884 3699 4099 -15.0 -56.1 19.2 HVZ

Aside from comparing the distributions of the two subsets, which might be obscured by other sources of variability (such as proximity to the high Eastern

Cordillera), a number of cross sections can be determined which cross some or even the full width of the high-velocity zone and then cross over parts of the Altiplano underlain by ‘average’ or low velocity mantle. These profiles are modest in total number (because our network is still rather sparse, even in the Altiplano). A general overview of the three profiles along with the high-velocity zone is shown in Figure 43. The individual three profiles (A, B, & C) are shown in Figure 44 - Figure 46 and briefly described below.

Notice the vertical magenta bar, which represents the assumed boundary of the high- velocity zone.

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Figure 43: Profiles across High-Velocity Zone 104

Figure 44: Profile A – A’

Figure 45: Profile B – B’ 105

Figure 46: Profile C – C’

Profile A has the appearance of a promising jump in the observed isostatic anomaly at roughly the boundary of the high-velocity zone, but the magnitude is not quite large enough at only approximately 30 mGals of difference over a 25 km distance compared with the minimum increase of approximately 40 mGal associated with the model scenarios. Profile B also has no evidence of a ≥ 40 mGal jump correlated with the boundary of the high-velocity zone. Profile B does have a noticeable signal at a distance from ~75 km - 100 km where the gravity swings from -10 to -50 and back to -10 all over the course of ~25 km. However, this effect is due to the more recently deforming Corque syncline (see Figure 45) rather than the larger scale delamination and deformation of the

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Altiplano. This syncline is also causing the minimum in Profile A at approximately the

90 km distance. Profile C represents a cross section in the southern high-velocity zone much of which is over the Salar de Uyuni and associated with the flat elevation shown in

Figure 46. While the profile shows some significant changes in the anomalous fields over a completely flat terrain with ~25 mGal swings, the observed anomaly data shows no remnant of a 40 mGal jump in the vicinity of the boundary of the zone.

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CHAPTER 5: DISCUSSION AND FUTURE WORK

5.1: Discussion

The ‘pop-up’ scenario for the rise of the Altiplano and adjacent portions of the

Central Andean Plateau in ~ 3 Ma requires that a rapid delamination event occurred between ~10 Ma and ~6.8 Ma, but seismological data requires any such delamination affected only part, not all, of the Altiplano. Since the final, present-day Altiplano is nearly level as well as remarkably flat, the delaminated areas must be held down by the non-delaminated areas, and the non-delaminated areas must be pulled up by the delaminated areas. That is, there should be excess mass beneath the non-delaminated areas characterized by a high velocity mantle ‘lid’ and a mass deficit beneath the zones of normal and low mantle velocity. Accordingly, in crossing from delaminated to non- delaminated portions of the Altiplano, it should be possible to detect quite strong lateral variations in surface gravity.

In our Monte Carlo analysis, we assume two values for the areal fraction (f) that have undergone delamination: 50% and 75%. With only 50% delamination, many of the model parameters (Figure 35) have to take rather extreme values in order to produce 2.5

± 0.5 km of uplift. This is not so much the case when we assume 75% delamination, but the seismological findings suggest that 75% is already pushing the scenario to the extremes of plausibility relative to seismic results from Beck & Zandt, 2002. Indeed, if

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we invoke 75% delamination, much of it has to occur outside of the Altiplano proper.

Our Monte Carlo analysis highlights three parameters that strongly impact the model outcomes: the thickness of the eclogite layer (E), the density of the eclogite layer (ρE), and the average density of the crust (ρU). Additionally, it becomes apparent that the model has the most sensitivity to changes in the thickness of the eclogite layer. The crux with this eclogite layer is that a small amount of eclogite (E = 5-15 km) is the most plausible, but does not result in uplifts consistent with the pop-up hypothesis.

Alternatively, larger thicknesses of eclogite (E = 30 km +) can produce the necessary uplift but aren’t very likely from a geological perspective. Ultimately, seven models were selected along with their corresponding parameters so as to represent the broader suite of acceptable model instances. The gravity response signal from each of these models was calculated. These gravity response functions were then compared with observed lateral gravity changes in the Altiplano along profiles that crossed from high velocity zones to normal or even low velocity zones.

The observed gravity data within this high-velocity zone does not exhibit characteristics that mimic those associated with the models’ gravity responses. The statistical distribution of observed gravity inside the high-velocity zone does not exhibit uniquely identifying behavior compared with the rest of the Altiplano. Based on these results and observations, there is essentially no gravimetric observational evidence in support of the pop-up hypothesis. Furthermore, the synthetic gravity signal associated with the tectonic scenarios is at a minimum different by 10 mGal from the observed gravity. It might be possible to arrive at extremely elaborate delamination scenarios, with

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non-uniform crustal thicknesses, eclogite thicknesses, and lithospheric mantle thicknesses prior to delamination, that allow both for 2.5 ± 0.5 km of uplift, and yet produce little in the way of a visible surface gravity signature. But we suspect that such scenarios would constitute ‘special pleading’. Clearly the lack of strong and sustained lateral gravity gradients in the Altiplano will rule out most, if not all, plausible models for a delamination-induced uplift of ~ 2.5 km.

Overall, the Central Andes region provides an interesting test laboratory for gravimetric network investigations. With the diverse terrain and geography, the observation of a gravimetric network simply requires different methodologies than would be utilized in other parts of the world ranging from station sampling intervals, line closure requirements, absolute gravity ties, etc. The estimated standard error of gravity at all stations is quite good with a maximum of approximately ±0.12 mGal (2휎 = ±0.24).

We also find gravity anomaly magnitudes such as the free-air anomaly that are matched at only a few other places on earth. Most importantly, the observed gravity compares at a somewhat questionable level with modeled anomaly values from EGM2008 generally.

The mean of the residuals is less than 10 mGal, but there are no systematic discrepancies in a geographic sense as illustrated in Figure 25 and Figure 26. However, there are numerous locations both in mountainous regions and in more subdued terrain, which exhibit significant deviations from modeled values mostly in the range of approximately

±50 mGals but some locations are approaching ±180 mGals.

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5.2: Future Work

A variety of additional tasks are in various stages of implementation regarding the gravimetric network in Bolivia and the region as well as implications for the state of the lithosphere. Currently, additional network densification and expansion in targeted areas is being undertaken; the development of a geoid model for use within the country is being assessed; and the possibility of using existing GPS/leveling data as further model verification.

Network densification and expansion is currently being implemented with regards to a couple goals. First, expansion and connection to neighboring countries in South

America is a major goal. This is a very logistically challenging endeavor as most geodetic partners within their respective countries are in the military, and there are 5 neighboring countries to coordinate with: Peru, Brazil, Paraguay, Argentina, and Chile.

Nonetheless, initial conversations have occurred with geodetic partners in Brazil and

Chile and fieldwork is expected in the next year. Secondly, there are a couple of areas within Bolivia where additional gravimetric observations would provide further clarification regarding the possible delamination beneath the Altiplano. These areas include further observations around the Salar de Uyuni (-20o, -67.5o) where both high- velocity and low-velocity zones are lacking in data coverage and additional coverage along the Eastern Cordillera where gaps currently exist in the network.

Another logical step is to construct a gravimetric geoid model for use in the region. A gravimetric geoid model has been determined for the extent of the Altiplano

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region (15oS - 23oS, 65oW – 70oW) by Corchete, et al., (2006). The additional gravity data presented in this dissertation along with advances in global gravity models like

EGM2008 and GOCE warrant the determination of a new geoid model that could be put to use in Bolivia. This new model will likely require the inclusion of additional terrestrial gravity datasets as well as satellite gravity models. This will also likely warrant additional corrections and higher-order approximations to be included like an atmospheric effect, a second-order height correction, etc. Further development of various terrain corrections could also benefit a geoid model.

Finally, there are leveling datasets present in Bolivia that have been measured in the last decade as roads have been reconstructed and built. New leveling surveys are probably out of the scope of additional work, but utilizing some of the previously collected data would provide additional constraints and verification to the work that has already been completed and towards a new geoid model.

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REFERENCES

Allmendinger, R. W., Jordan, T. E., Kay, S. M., & Isacks, B. L. (1997). The evolution of the Altiplano-Puna plateau of the Central Andes. Annual Review of Earth and Planetary Sciences, 25(1), 139-174.

Baarda, W., (1967). Statistical concepts in geodesy. Netherlands Geodetic Commission, Publications on Geodesy, Delft, Vol. 2(4).

Barazangi, M., & Isacks, B. L. (1976). Spatial distribution of earthquakes and subduction of the Nazca plate beneath South America. Geology, 4(11), 686-692.

Beck, S. L., & Zandt, G. (2002). The nature of orogenic crust in the Central Andes. Journal of Geophysical Research: Solid Earth, 107(B10), 2230. doi:10.1029/2000JB000124.

Bird, P. (1979). Continental delamination and the Colorado Plateau. Journal of Geophysical Research: Solid Earth (1978–2012), 84(B13), 7561-7571.

Borsa, A. A., Bills, B. G., & Minster, J. (2008). Modeling the topography of the Salar de Uyuni, Bolivia, as an equipotential surface of earth's gravity field. Journal of Geophysical Research: Solid Earth (1978–2012), 113(B10).

Brooks, B. A., Bevis, M., Whipple, K., Arrowsmith, J. R., Foster, J., Zapata, T., Kendrick, E., Minaya, E., Echalar, A., Blanco, M., Euillades, P., Sandoval, M., & Smalley Jr, R. J. (2011). Orogenic-wedge deformation and potential for great earthquakes in the central Andean backarc. Nature Geoscience, 4(6), 380-383.

Cady, J., & Wise, R. (1992). Gravity and magnetic studies. US Geological Survey and Geobol (Servicio Geológico De Bolivia), Geology and Mineral Resources of the Altiplano and Cordillera Occidental, Bolivia: US Geological Survey Bulletin, 1975, 56-62.

Chapin, D. A. (1996). A deterministic approach toward isostatic gravity residuals-A case study from South America. Geophysics, 61(4), 1022-1033.

113

Corchete, V., Flores, D., & Oviedo, F. (2006). The first high-resolution gravimetric geoid for the Bolivian tableland: BOLGEO. Physics of the Earth and Planetary Interiors, 157(3), 250-256.

DeMets, C., Gordon, R. G., & Argus, D. F. (2010). Geologically current plate motions. Geophysical Journal International, 181(1), 1-80. doi:10.1111/j.1365- 246X.2009.04491.x.

Dewey, J., & Lamb, S. (1992). Active tectonics of the Andes. Tectonophysics, 205(1), 79-95.

Dewey, J. F., & Bird, J. M. (1970). Mountain belts and the new global tectonics. Journal of Geophysical Research, 75(14), 2625-2647.

Dias, F. J. S. S., & Escobar, I. P. (2001). A model for adjustment of differential gravity measurements with simultaneous gravimeter calibration. Journal of Geodesy, 75(2-3), 151-156.

Eckl, M. C., Snay, R. A., Soler, T., Cline, M. W., & Mader, G. L. (2001). Accuracy of GPS-derived relative positions as a function of interstation distance and observing-session duration. Journal of Geodesy, 75(12), 633-640.

England, P., & Houseman, G. (1989). Extension during continental convergence, with application to the Tibetan Plateau. Journal of Geophysical Research: Solid Earth (1978–2012), 94(B12), 17561-17579.

Faller, J. (2002). Thirty years of progress in absolute gravimetry: A scientific capability implemented by technological advances. Metrologia, 39(5), 425.

Federal Geodetic Control Committee (FGCC). (1984). Standards and Specifications for Geodetic Control Networks. Federal Geodetic Control Committee. Rockville, MD.

Forsberg, R., & Tscherning, C. (1997). Topographic effects in gravity field modelling for BVP. Geodetic boundary value problems in view of the one centimeter geoid (pp. 239-272) Springer.

Forsberg, R. (1984). A study of terrain reductions, density anomalies and geophysical inversion methods in gravity modelling. (No. 355). Columbus, Ohio: OSU Reports.

Forsberg, R. (1985). Gravity field terrain effect computations by FFT. Bulletin Géodésique, 59(4), 342-360.

114

Gamache, M. (2004). Free and low cost datasets for international mountain cartography. http://www.terrainmap.com/downloads/Gamache_final_web.pdf. Accessed: 7/22/2015.

Garzione, C. N., Hoke, G. D., Libarkin, J. C., Withers, S., MacFadden, B., Eiler, J., Ghosh, P., & Mulch, A. (2008). Rise of the Andes. Science, 320(5881), 1304- 1307. doi:10.1126/science.1148615.

Garzione, C. N., Molnar, P., Libarkin, J. C., & MacFadden, B. J. (2006). Rapid late miocene rise of the Bolivian Altiplano: Evidence for removal of mantle lithosphere. Earth and Planetary Science Letters, 241(3–4), 543-556. doi:http://dx.doi.org/10.1016/j.epsl.2005.11.026.

Ghosh, P., Garzione, C. N., & Eiler, J. M. (2006). Rapid uplift of the Altiplano revealed through 13C-18O bonds in paleosol carbonates. Science, 311(5760), 511-515. doi:10.1126/science.1119365.

Götze, H.-J., B. Lahmeyer, S. Schmidt, S. Strunk, and M. Araneda (1990), Central Andes Gravity Data Base, Eos Trans. AGU, 71(16), 401–407, doi:10.1029/90EO00148.

Götze, H., Lahmeyer, B., Schmidt, S., & Strunk, S. (1994). The lithospheric structure of the Central Andes (20–26 S) as inferred from interpretation of regional gravity. Tectonics of the southern Central Andes (pp. 7-21) Springer.

Götze, H., & Li, X. (1996). Topography and geoid effects on gravity anomalies in mountainous areas as inferred from the gravity field of the Central Andes. Physics and Chemistry of the Earth, 21(4), 295-297.

Hamilton, A. C., and B. G. Brulé (1967), Vibration-induced drift in LaCoste and Romberg Geodetic Gravimeters, J. Geophys. Res., 72(8), 2187–2197, doi:10.1029/JZ072i008p02187.

Hammer, S. (1939). Terrain corrections for gravimeter stations. Geophysics, 4(3), 184- 194.

Hayford, J. F., & Bowie, W. (1912). Geodesy: the effect of topography and isostatic compensation upon the intensity of gravity. US Govt. Print. Off.

Heck, B., & Seitz, K. (2007). A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling. Journal of Geodesy, 81(2), 121-136.

Heiskanen, W. A., & Moritz, H. (1967). Physical Geodesy. San Francisco: W.H. Freeman and Company. 115

Heiskanen, W. A., & Vening Meinesz, F. A. (1958). The Earth and its gravity ﬁeld. New York, McGraw-Hill.

Herring, T. A., King, R. W., & McClusky, S. C. (2006). GAMIT reference manual. Department of Earth, Atmospheric, and Planetary Sciences.

Hinze, W. J., Aiken, C., Brozena, J., Coakley, B., Dater, D., Flanagan, G., Forsberg, R., Hildenbrand, T., Keller, G., Kellogg, J., Kucks, R., Li, X., Mainville, A., Morin, R., Pilkington, M., Plouff, D., Ravat, D., Roman, D., Urrutia-Fucugauchi, J., Veronneau, M., Webring, M., Winester, D. (2005). New standards for reducing gravity data: The North American gravity database. Geophysics, 70(4), J25-J32.

Hinze, W. J., von Frese, R. R., & Saad, A. H. (2013). Gravity and magnetic exploration: principles, practices, and applications. Cambridge University Press.

Hofmann-Wellenhof, B., & Moritz, H. (2006). Physical Geodesy (2nd ed.). New York: SpringerWien.

Hwang, C., Wang, C. G., & Lee, L. H. (2002). Adjustment of relative gravity measurements using weighted and datum-free constraints. Computers & Geosciences, 28(9), 1005-1015.

Instruction Manual - Model G & D Gravity Meters. (2004). (Manual). Lacoste & Romberg.

Isacks, B. L. (1988). Uplift of the Central Andean plateau and bending of the Bolivian Orocline. Journal of Geophysical Research: Solid Earth (1978–2012), 93(B4), 3211-3231.

Jarvis, A., Reuter, H. I., Nelson, A., & Guevara, E. (2008). Hole-filled SRTM for the globe version 4. Available from the CGIAR-CSI SRTM 90m Database (http://srtm.csi.cgiar.org).

Jekeli, C., Yang, H. J., & Ahlgren, K. (2013). Using isostatic gravity anomalies from spherical harmonic models and elastic plate compensation to interpret the lithosphere of the Bolivian Andes. Geophysics, 78(3), G41-G53.

Koch, K. (1999). Parameter estimation and hypothesis testing in linear models. Springer Science & Business Media.

Lamb, S., & Davis, P. (2003). Cenozoic climate change as a possible cause for the rise of the Andes. Nature, 425(6960), 792-797.

116

Lemoine, F. G., Kenyon, S. C., Factor, J. K., Trimmer, R. G., Pavlis, N. K., Chinn, D. S., . . . Torrence, M. H. (1998). The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96.

Li, X., & Götze, H. (2001). Ellipsoid, geoid, gravity, geodesy, and geophysics. Geophysics, 66(6), 1660-1668.

Longman, I. M. (1959). Formulas for computing the tidal accelerations due to the moon and the sun. Journal of Geophysical Research, 64(12), 2351-2355.

Lyon-Caen, H., Molnar, P., & Suárez, G. (1985). Gravity anomalies and flexure of the Brazilian shield beneath the Bolivian Andes. Earth and Planetary Science Letters, 75(1), 81-92.

McQuarrie, N. (2002). The kinematic history of the Central Andean fold-thrust belt, Bolivia: Implications for building a high plateau. Geological Society of America Bulletin, 114(8), 950-963.

Metropolis, N., & Ulam, S. (1949). The Monte Carlo method. Journal of the American Statistical Association, 44(247), 335-341.

Micro-g LaCoste. (2008). A-10 Portable Gravimeter User’s Manual. Micro-g LaCoste. Lafayette, CO.

Molnar, P., & Garzione, C. N. (2007). Bounds on the viscosity coefficient of continental lithosphere from removal of mantle lithosphere beneath the Altiplano and Eastern Cordillera. Tectonics, 26(2), - TC2013. doi:10.1029/2006TC001964.

Myers, S. C., Beck, S., Zandt, G., & Wallace, T. (1998). Lithospheric-scale structure across the Bolivian Andes from tomographic images of velocity and attenuation for P and S waves. Journal of Geophysical Research: Solid Earth, 103(B9), 21233-21252. doi:10.1029/98JB00956.

Nielsen, J. E. (2013). Absolute gravimetry - for monitoring climate change and geodynamics in Greenland. (PhD dissertation). Technical University of Denmark, Kgs. Lyngby.

NIMA. (1997). Department of Defense World Geodetic System 1984, Its Definition and Relationships with Local Geodetic Systems. Technical Report TR8350.2, 3rd Ed. National Imagery and Mapping Agency, Washington. D.C.

Oncken, O., Hindle, D., Kley, J., Elger, K., Victor, P., & Schemmann, K. (2006). Deformation of the Central Andean upper plate system—Facts, fiction, and constraints for plateau models. The Andes (pp. 3-27) Springer. 117

Pavlis, N. K., Holmes, S. A., Kenyon, S. C., & Factor, J. K. (2012). The development and evaluation of the earth gravitational model 2008 (EGM2008). Journal of Geophysical Research: Solid Earth, 117(B4), - B04406. doi:10.1029/2011JB008916.

Prezzi, C. B., Götze, H., & Schmidt, S. (2009). 3D density model of the Central Andes. Physics of the Earth and Planetary Interiors, 177(3–4), 217-234. doi:http://dx.doi.org/10.1016/j.pepi.2009.09.004.

Schaffrin, B. (2007). Geodetic Science 650 Course Notes - The Ohio State University. Columbus, Ohio.

Soler, T., Michalak, P., Weston, N. D., Snay, R. A., & Foote, R. H. (2006). Accuracy of OPUS solutions for 1-to 4-h observing sessions. GPS solutions, 10(1), 45-55.

Strang, G. and Borre, K. (1997). Linear algebra geodesy and GPS. Wellesley-Cambridge Press, USA.

Torge, W. (1989). Gravimetry. Berlin; New York: de Gruyter.

Torge, W. (2001). Geodesy (3rd ed.). Berlin; New York: de Gruyter.

Tscherning, C. C., Knudsen, P., Forsberg, R. (1994). Description of the GRAVSOFT package. Geophysical Institute of Copenhagen, Technical Report, 1991, 2 Ed. 1992, 3 Ed. 1993, 4 Ed. 1994

Turcotte, D. L., Haxby, W. F., & Ockendon, J. R. (1977). Lithospheric instabilities. Island arcs, deep sea trenches and back-arc basins, 63-69.

Turcotte, D. L., & Schubert, G. (2002). Geodynamics 2nd Edition. Cambridge, UK: Cambridge University Press.

Watts, A. B. (2001). Isostasy and flexure of the lithosphere Cambridge University Press.

Whitman, D. (1999). The isostatic residual gravity anomaly of the Central Andes, 12 to 29 S: A guide to interpreting crustal structure and deeper lithospheric processes. International Geology Review, 41(5), 457-475.

Won, I., & Bevis, M. (1987). Computing the gravitational and magnetic anomalies due to a polygon: Algorithms and FORTRAN subroutines. Geophysics, 52(2), 232-238.

Yuan, X., Sobolev, S. V., & Kind, R. (2002). Moho topography in the Central Andes and its geodynamic implications. Earth and Planetary Science Letters, 199(3–4), 389- 402. doi:http://dx.doi.org/10.1016/S0012-821X(02)00589-7. 118

APPENDIX A: ADDITIONAL GRAVITY DATA IN BOLIVIA REGION

The datasets described in Cady & Wise, (1992) consist of two separate databases available from National Centers for Environmental Information (NERI) (formerly

NGDC). The first is a compilation of surveys from the DoD-NGA (formerly DMA)

Gravity Library; the second is from a local oil and gas exploration company, Yacimientos

Petroliferos Fiscales Bolivianos (YPFB). Both of these databases have quite different characteristics and should be considered separately from one another for any investigation. The YPFB data (Figure 47) is quite dense with points spaced at < 1 km intervals and lines spaced at distances of approximately 10-20 km. There are 55,146 points located exclusively in the Altiplano region. The DoD-NGA data (Figure 48) has points spaced at 1-5 km intervals and lines separated by 50-250 km. The distribution of points is more evenly distributed throughout Bolivia and surrounding countries with

8,162 points. Most importantly, the original survey metadata such as base stations, types of gravimeters, and other technical information that was used as a basis for the gravity data is not known (Cady & Wise, 1992). We are only provided with a digital file with positional information and gravity values. For this reason alone, it is quite difficult to merge the existing datasets with unknown positional and gravity accuracies with the newly observed OSU data. In the following section, the OSU gravity data will be compared relative to the DoD-NGA data and the YPFB data. Additionally, two scenarios

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are presented at the end of this section to illustrate the lingering problems with the existing data.

Figure 47: YPFB and OSU gravity data with EGM2008

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Figure 48: DoD/NGA and OSU gravity data with EGM2008

Other studies have utilized some of the NGA and YPFB data in mostly localized areas such as Lyon-Caen, et al., (1985); Corchete et al., (2006); Borsa et al., (2008); etc.

Lyon-Caen, et al., (1985) evaluated two gravity profiles for investigating the lithospheric flexure: one across the entire Andean range at -19.5o and another shorter profile at approximately {-18o, -60o}. Cochete et al., (2006) and Borsa et al., (2008) employ data from mostly the Altiplano and mountainous regions in southwestern Bolivia for geoid determination and geopotential surface determination, respectively.

In order to provide even a brief comparison between these databases and the OSU gravity data, some processing of the existing gravity data must take place and can be separated into three categories: the horizontal position (latitude/longitude), the vertical position, and the gravity value. The horizontal coordinates are rather straightforward to 121

arrive at; however, it is highly likely that these values were originally determined from scaled positions based on topographic maps and other less accurate techniques than available today. The NGA data is assumed to be in a WGS84 (or similar and insignificantly different) reference frame. The YPFB data is transformed from a local horizontal datum, Provisional South American 1956 (PRP-A) to WGS84 (NIMA, 1997 and Borsa, et al., 2008). The most widely discrepant component is the vertical coordinate, which also has the most influence on gravity values and anomalies. For this comparison, we simply use a standard reference model, SRTM90 (Jarvis, et al., 2008) to base all gravity values in both datasets on. The SRTM90 data isn’t without limitations but removes the inconsistencies and inaccuracies from the varied original vertical positions. Finally, the gravity values need to be transformed slightly. The YPFB dataset was originally referenced to an older gravity datum, and a constant bias of 14.0 mGal was subtracted from this data to correspond with the NGA and OSU datasets (Cady & Wise,

1992). The free-air anomaly was then calculated using ( 53 ) with the elevation based on

SRTM90.

With the scarcity of gravity values, significantly different sampling distributions, and introduction of additional approximations, it is inappropriate to grid all the data and compare the computed grids. Instead, we compute grids for the NGA and YPFB datasets individually and evaluate at the OSU point locations within the corresponding grids. In order to limit the effect of distant data points, the extrapolation of the grid is limited to 10 km from a gravity value. The results of this comparison are shown in Figure 49 - Figure

52. The YPFB data agrees quite well with the OSU results with maximum residuals at

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the ±10 mGal level. There is a slight systematic discrepancy of approximately 3.5 mGal, which is unexpected considering the overall agreement between the datasets. The NGA data is completely dissimilar to the OSU results with only half the data within approximately ±13 mGal. Even more concerning is the huge residual values obtained ranging from -360 to +260 mGal. There are scenarios where lines are essentially collocated, and the residual value is greater than ±100 mGal.

Figure 49: OSU - YPFB Free-Air Anomaly Residual [mGal]

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Figure 50: OSU - YPFB Free-Air Anomaly Residual Histogram [mGal]

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Figure 51: OSU - DoD Free-Air Anomaly Residual [mGal]

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Figure 52: OSU - DoD Free-Air Anomaly Residual Histogram [mGal]

Two example situations are further presented to illustrate the potential problems encountered with the NGA dataset (Figure 53 and Figure 54). Figure 53 shows a profile of the EGM2008 free-air anomaly model along with the long wavelength characteristic modeled with a quadratic function. While local, high frequency effects are expected; there is a significant departure from the long wavelength gravity field of approximately

50 mGals at the 140 km and 170 km distance along the profile. These two gravity values are included in the EGM2008 model but are in complete disagreement with the long wavelength gravity field. Another very common problem encountered with the DoD-

NGA dataset is illustrated in Figure 54. While the OSU gravity data align quite well with digital road maps, the NGA data is often not aligned with the existing roads. The line of

DoD gravity stations in the northeast corner of the map aligns with an existing road; the

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stations in the central portion of the map do not though. But, there is a general similarity between the features of the existing road and the line of gravity stations. By scaling and moving the existing road to approximately fit the gravity data locations, the correlation between the gravity stations and the existing road is clear. In this situation, there is clearly a scaling problem in determining the horizontal position of the gravity stations.

Figure 53: DoD Gravity Example I

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Figure 54: DoD Gravity Example II

Based on the numerical comparisons and example situations presented in the preceding discussion, there is a strong argument to not include these datasets in this investigation. The YPFB data might warrant inclusion in future work; however, some additional verification will likely be conducted. From the onset, the present investigation was meant to be based on the accurately positioned OSU gravity data. The DoD-NGA data simply has too many inconsistencies and unknowns for use in this investigation. It is not trivial to visually inspect more than 8,000 points and determine the seemingly random positioning inconsistencies.

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APPENDIX B: LACOSTE AND ROMBERG MODEL G GRAVIMETERS

A number of relative gravimeters owned by a variety of institutions have been used in the Bolivian gravimetric network. The gravimeters are summarized in Table 16 below:

Table 16: Individual Gravimeter Observation Statistics

Observed absolute g’s LCR Stations Num. 25 – 50 – 100 – 200 – Serial Institution < 25 > 300 Max. Observed g 50 100 200 300 No. g42 NGA 376 333 222 59 37 13 2 0 262.90 IGM- g131 108 97 36 13 24 15 6 3 370.65 Chile IGM- g142 168 147 78 28 25 10 3 3 577.47 Chile The Ohio g710 State 964 847 584 113 87 48 11 4 576.83 University g801 NGA 983 833 613 102 72 37 7 2 320.43 IGM- g1024 774 663 502 81 53 23 3 1 380.18 Bolivia IGM- g1025 639 550 344 77 71 47 8 3 370.02 Bolivia

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APPENDIX C: NETWORK ADJUSTMENT WITH CONDITION EQUATIONS

The network adjustment based on a model of condition equations categorizes two types of condition equations in this gravimetric network: (1) the g values making up an individual loop within the network must total zero and (2) the g values connecting two absolute gravity nodes must sum to the difference in the absolute gravity observations.

The observations are assumed to fit the model of condition equations, which has a vector of discrepancies or misclosures defined as:

푤 = 퐵푦 ( 80 ) where B is a matrix consisting of the coefficients of the condition equations and y is defined in ( 81 ) based on the model of direct or repeated observations with random and non-random components:

푦 = 휏휇 + 푒 ( 81 ) where is the non-random coefficient of the observables,  is the non-random observable, and e is the random error vector. Combining ( 80 ) and ( 81 ), results in the following:

푤 = 퐵푦 = (퐵휏)휇 + 퐵푒 ( 82 )

The B term shown in parenthesis in ( 82 ) is equal to 0 and the vector of misclosures can be described simply as: 130

푤 = 퐵푒 ( 83 )

The model also assumes that the error vector has an expectation of 0 and dispersion, Σ =

2 −1 휎0 푃 . The least squares solution is determined by minimizing the sum of squared errors times their corresponding weights subject to the conditions of ( 83 ):

푒푇푃푒 → 푚𝑖푛𝑖푚𝑖푧푒푑 ( 84 )

In order to achieve this minimization, the Lagrange target function,  is defined as:

Φ(푒, 휆) = 푒푇푃푒 + 2휆푇(푤 − 퐵푒) ( 85 )

Partial derivatives are taken of the Lagrange target function with respect to e and . The results are then set equal to 0 to find potential minima/maxima values.

휕훷 ( 86 ) = 2푃푒̃ − 2퐵푇휆̂ 휕푒

휕훷 ( 87 ) = 2(푤 − 퐵푒̃) = 0 휕휆 where 푒̃ and 휆̂ are the estimated residual and Lagrange multiplier vectors, respectively.

Solving ( 86 ) for 푒̃ results in the following:

푒̃ = 푃−1퐵푇휆̂ ( 88 )

Substituting the result from ( 88 ) into ( 83 ) results in the following:

푤 = 퐵푃−1퐵푇휆̂ ( 89 )

Solving for 휆̂ in ( 89 ) and substituting into ( 88 ) results in the vector of residuals:

푒̃ = 푃−1퐵푇(퐵푃−1퐵푇)−1푤 ( 90 )

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The vector of adjusted observations (푦̂) can be determined from ( 90 ) and the vector of observations (y):

푦̂ = 푦 − 푒̃ = 푦 − 푃−1퐵푇(퐵푃−1퐵푇)−1푤 = 푦 − 푃−1퐵푇(퐵푃−1퐵푇)−1퐵푦 ( 91 )

The estimated variance component can subsequently be used to aid in verification of the model. The derivation for the estimated variance component is as follows using ( 88 ):

2 푇 ̂푇 −1 −1 푇 ̂ 휎̂0 (푛 − 1) = 푒̃ 푃푒̃ = 휆 퐵푃 푃푃 퐵 휆 ( 92 )

Substituting the solution from ( 89 ) results in the following result for the estimated variance component:

휆̂푇푤 ( 93 ) 휎̂2 = 0 푛 − 1

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APPENDIX D: ISOSTATIC CORRECTIONS WITH SPHERICAL TESSEROIDS

The gravitational potential of a spherical tesseroid can be generally developed as the following integral based on Newton’s law of gravitation:

휆2 휑2 푟2 푟′2 cos 휑′ 푑푟′푑휑′푑휆′ ν(푟, 휑, 휆) = 퐺휌 ∫ ∫ ∫ ( 94 ) 푙 휆1 휑1 푟1 where the spherical distance between the gravity point P and integration point Q is:

푙 = √푟2 + 푟′2 − 2푟푟′ cos 휓 ( 95 ) and:

cos 휓 = sin 휑 sin 휑′ + cos 휑 cos 휑′ cos(휆′ − 휆) ( 96 )

Since only the vertical component of the gravitational potential is sensed in observations, the partial derivative with respect to the vertical component, r, is determined as:

휆2 휑2 푟2 휕휈(푟, 휑, 휆) 푟′2(푟 − 푟′ cos 휓) cos 휑′ 푑푟′푑휑′푑휆′ 훿푔 = − = 퐺휌 ∫ ∫ ∫ ( 97 ) 휕푟 푙3 휆1 휑1 푟1

The volume integral in ( 97 ) can be calculated using a series expansion to second-order in ( 98 ) (See Heck & Seitz, 2007 for complete derivation).

훿푔(푟, 휑, 휆) = 퐺휌Δ푟Δ휑Δ휆 ∗ ( 98 ) 1 [퐿 + (퐿 Δ푟2 + 퐿 Δ휑2 + 퐿 Δ휆2)] 000 24 200 020 002 133

where the zero-order coefficient (L000) and second order coefficients (L200, L020, L002) are:

2 푟0 (푟 − 푟0 cos 휓0) cos 휑0 퐿000 = 3 ( 99 ) 푙0

푟 cos 휑0 3푟0 퐿200 = 3 {2 − 2 [5푟0 − (2푟 + 3푟0 cos 휓0) cos 휓0] 푙0 푙0 ( 100 3 15푟0 2 ) + 4 sin 휓0 (푟0 − 푟 cos 휓0 )} 푙0

3 푟0 2 퐿020 = ( ) cos 휑 (1 − 2 sin 휑0) cos 훿휆 푙0 2 푟0 2 2 + 5 {−푟(푟 + 푟0 ) cos 휑0 푙0

+ 푟0 sin 휑 [−푟푟0(sin 휑 cos 휑0 − cos 휑 sin 휑0 cos 훿휆) 2 2 + sin 휑0 cos 휑0 (2푟 + 4푟0 − 3푟푟0 sin 휑 sin 휑0)] 2 2 + 푟0 cos 휑 cos 훿휆 (1 − 2 sin 휑0)

∗ [푟0 + 푟 cos 휑 cos 휑0 cos 훿휆] 2 + 푟푟0 cos 휑 sin 휑0 cos 휑0 cos 훿휆 ∗ [3 sin 휑 cos 휑 − 4 cos 휑 sin 휑 cos 훿휆]} 0 0 ( 101 3 5푟푟0 2 2 + 7 {−푟(푟 + 푟0 ) sin 휑0 ) 푙0 2 + 푟0 cos 휑 sin 휑0 cos 휑0 cos 훿휆

∗ (푟0 + 푟 cos 휑 cos 휑0 cos 훿휆) 2 2 2 + 푟0 sin 휑 [2푟 − 푟0 − 푟푟0 cos 휓0 + sin 휑0 2 2 ∗ (푟 + 2푟0 − 푟푟0 sin 휑 sin 휑0)]}

∗ (sin 휑 cos 휑0 − cos 휑 sin 휑0 cos 훿휆)

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3 푟0 2 퐿002 = ( ) cos 휑 cos 휑0 푙0

∗ {cos 훿휆 ( 102 3푟 2 − 2 [2푟0 cos 휑 cos 휑0 sin 훿휆 + (푟 − 푟0 cos 휓0) cos 훿휆] ) 푙0 2 15푟 푟0 2 + 4 cos 휑 cos 휑0 (푟 − 푟0 cos 휓0) sin 훿휆} 푙0

The compensating effect of each tesseroid is evaluated at its geometrical center, which implies:

푟1 + 푟2 ( 103 푟0 = 2 )

휑1 + 휑2 ( 104 휑0 = 2 )

휆 + 휆 ( 105 휆 = 1 2 0 2 )

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APPENDIX E: LITHOSPHERIC MODEL PARAMETERS AND VALUES

Table 17: Non-controlling model parameters

Parameter: Model Value: Units: H 60 km

TA 1600 K

TC 900 K L 120 km  3 x 10-5 [ ] 3 W 1030 kg/m W 2.5 km

3 B 2960 kg/m

Figure 55: Scenario I Model with Densities [kg/m3] 136