Colorado School of Mines Center for Gravity, Electrical ...cgem.mines.edu/s/Krahenbuhl Thesis 2005.pdf · Ph.D. Thesis Binary inversion of gravity data for salt imaging Richard A

Ph.

D.

The

sis

Binary inversion of gravity data for salt imaging

Richard A. Krahenbuhl

Center for Gravity, Electrical & Magnetic Studies

Colorado School of Mines

Department of GeophysicsColorado School of MinesGolden, CO 80401

http://www.geophysics.mines.edu/cgem

CGEM

Ph.

D.

The

sis

Binary inversion of gravity data for salt imaging


Center for Gravity, Electrical & Magnetic Studies

Colorado School of Mines

Department of GeophysicsColorado School of MinesGolden, CO 80401

http://www.geophysics.mines.edu/cgem

CGEM

Defended: May 12, 2005

Advisor: Prof. Yaoguo Li (GP)Committee Chair: Prof. Mike Pavelich (CH)Minor: Prof. Murray Hitzman (GE)Committee Members: Prof. Misac N. Nabighian (GP)

Prof. John Scales (GP)

BINARY INVERSION OF GRAVITY DATA

FOR SALT IMAGING

by


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A thesis submitted to the Faculty and Board of Trustees of the Colorado School of

Mines in partial fulfillment of the requirements for the degree of Doctor of Philosophy

(Geophysics).

Golden, Colorado

Date June 07, 2005

Signed: on original copy Richard A. Krahenbuhl

Signed: on original copy Dr. Yaoguo Li Thesis Advisor

Golden, Colorado

Date June 07, 2005

on original copy

Dr. Terence K. Young Professor and Head

Department of Geophysics

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ABSTRACT

I present a binary inversion algorithm for inverting gravity data in salt imaging.

The density contrast is restricted to being either zero or one, where one represents the

value of density contrast of salt at a given depth. I develop this method to overcome the

difficulties associated with interface-based inversion and density-based inversion while

attempting to draw from the strengths of both existing approaches. The interface

inversion specifies the known density contrast of salt, but its parameterization can overly

restrict the model from the outset. The density inversion, on the other hand, affords great

flexibility in its model representation, but cannot directly utilize the known density

information. Binary inversion uses a similar model representation as in continuous-

density inversion by defining a density distribution as a function of spatial position, but

restricts the model values to those corresponding to two lithologic units as does the

interface inversion.

I formulate the binary inversion using Tikhonov regularization in which the

inverse solution is obtained by minimizing a weighted sum of a data misfit and a model

objective function. The model objective function serves to stabilize the solution and to

incorporate any prior information that is independent of gravity data. Because of the

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discrete nature of the problem, commonly used minimization techniques are no longer

applicable. I therefore investigate the use of genetic algorithm, quenched simulated

annealing, and a hybrid method based on these two as potential solvers for the

minimization problem associated with the binary inversion. The use of Tikhonov

regularization is well understood in continuous-variable inversion, but its application in

binary problems has yet to be explored. I investigate this aspect and conclude that

Tikhonov regularization plays a similar role in discrete inversion, and the corresponding

Tikhonov curve behaves in a similar manner. Thus the commonly used approaches for

determining the level of regularization is equally applicable in both types of inversions.

Finally, appraisal of solution is a necessary component of inversion, in which one

attempts to understand the uncertainties in the recovered model and to identify features of

high confidence. I explore the model space of binary inversion, evaluate the modality of

the objective function for this purpose, and illustrate the improved reliability of

interpretation in the process.

I illustrate binary inversion with synthetic models in 2D and 3D generated from

the SEG/EAGE salt model. As sought in development of binary inversion, the method

incorporates density information while providing a sharp contact for the subsurface. It

also allows for flexibility in model representation while solving for density distribution as

a function of spatial position. The binary condition places a strong restriction on the

admissible models so that the non-uniqueness caused by nil zones might be resolved.

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TABLE OF CONTENTS

ABSTRACT……………………………………………………………………………...iii

LIST OF FIGURES…………………………………………………………...………….ix

ACKNOWLDEGEMENTS…………………….……………………………………..…xv

DEDICATION……………………...…………………………………………………..xvii

Chapter 1: INTRODUCTION…………………………………………………………….1

Chapter 2: BINARY INVERSION……………………………………………………….8

2.1. Inversion Methods for Imaging Salt Structure……………………………....9

2.1.1. Interface Inversion…………………...……………………………9

2.1.2. Density Inversion…………………………………..…………….10

2.2. Binary Inversion…………………………………..………………...………11

2.2.1. Background……………………….…...…………………………11

2.2.2. Theory……………………………………………………………12

2.2.3. Numerical Solution………………………………………………16

2.2.3.1. Forward Modeling……………………………………..17

2.2.3.2. Data-Misfit……………………………………………..17

2.2.3.3. Model Objective Function……………………………..18

2.2.3.4. Depth Weighting……………………………………….19

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2.3. Numerical Examples………………………………………………………..21

2.3.1. Single Boxcar (1D)………...…………………………………….21

2.3.2. Gravity Problem (2D)...………………………………………….27

2.4. Summary……………………………………………………………………29

Chapter 3: SOLUTION STRATEGY FOR BINARY INVERSION…………………...30

3.1. Solution Strategies for Binary Inversion……………………...……………30

3.2. Genetic Algorithm………………………………………………………….31

3.2.1. Design of Genetic Algorithm…………………………………….32

3.2.2. Numerical Examples……………………………………………..38

3.2.2.1. Salt Body with Single Density Contrast………....…….38

3.2.2.2. Salt Body with Density Contrast Reversal……………..47

3.3. Quenched Simulated Annealing……………………………………………51

3.3.1. Numerical Example……………………………………………...53

3.4. Summary……………………………………………………………………56

Chapter 4: HYBRID ALGORITHM……………………………………………………57

4.1. Motivation for a Hybrid Algorithm………………………………………...57

4.2. Design of the Hybrid Algorithm for Binary Inversion……………………..61

4.3. Performance of the Hybrid Algorithm……………………………………...64

4.4. Application to Full 3D Binary Inversion…………………………………...71

4.5. Summary ……………………………………………………………………81

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Chapter 5: REGULARIZATION AND WEIGHTING PARAMETERS IN BINARY INVERSION……………………………………………………………………………82

5.1. Role of Regularization in Continuous-Variable Inversion…………………82

5.2. Role of Regularization in Binary Inversion………………………………...87

5.2.1. Tikhonov Curve by Genetic Algorithm………………………….88

5.2.2. Tikhonov Curve by Quenched Simulated Annealing……………92

5.3. Choice of Regularization for Binary Inversion……………………………..93

5.3.1. L-Curve……………………………………....…………………..96

5.3.2. Discrepancy Principle……………………………..……………103

5.4. Effects of Weighting Parameters in Binary Inversion……………..……...106

5.5. Depth Weighing…………………………………………………..……….114

5.6. Summary…………………………………………………..………………115

Chapter 6: EXPLORATION OF BINARY INVERSE SOLUTION………………….117

6.1. Exploring the Model Space of Binary Inversion……………………….....118

6.1.1. Multiple Inversions……………………………………………..119

6.1.2. Simple Appraisal of Binary Solution…………………………...121

6.2. Investigation of Possible Multimodality………………………………..124

6.3. Summary………………………………………………………………..130

Chapter 7: CONCLUSIONS AND FUTURE RESEARCH…………………………...132

7.1. Conclusions………………………………………………………………..132

7.1.1. Current State of Gravity Inversions……...….………………….133

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7.1.2. Contribution of Binary Inversion…………...……………..……134

7.1.3. Problems Associated with Binary Inversion………………..…..136

7.2. Future research…………………………………………………………….137

REFERENCES CITED…………………………………………………………………139

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LIST OF FIGURES

Figure 2.1. 3D binary model τv divided into cuboidal cells with constant values of either 1 or 0. While the image is generic for conceptual purposes, later images during inversion for salt have values of 0 (black) for host sediment, and 1 (white) for salt…….16 Figure 2.2. Binary inversion results for the 1D mathematical problem. The original and predicted data are illustrated in the top panel (a), with a comparison between the true model (b) and constructed model (c) beneath……………………………………………24 Figure 2.3. Progress of the genetic algorithm for the 1D binary inverse problem. The objective value of the highest-ranking individual at each generation are represented by the points, and the average of the population of solutions is represented by the line……25 Figure 2.4. Illustration of inversion result for the 1D boxcar problem with continuous variable. The true model (a) is the same as in Figure 2.2 (b), which was also used for binary inversion. The constructed model (b) with continuous variable does not have a sharp contact as does the true model or the model constructed with binary inversion, outlined with gray dashes. Likewise, the amplitude of the model is not as accurate as the binary result. Both inversions were performed with the same data set….……...……....26 Figure 2.5. Binary inversion results for a simple 2½-D gravity problem. The top panel (a) illustrates a comparison between the observed and predicted data, and the lower panels display the true (b) and constructed (c) models………………………………..…28 Figure 3.1. Flowchart of the Genetic Algorithm for the binary inverse problem. Selection, recombination, and mutation are components unique to the genetic algorithm. Modified from Pohlheim (1997)……………………………………………………...….33 Figure 3.2. Example of two starting individuals with initialization of random zeros and ones. Each individual represents a potential solution model…………...…….….…35

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Figure 3.3. SEG/EAGE seismic velocity model. Panel (a) shows a 3D perspective view of the model. One cross-section AA’ is outlined. Panel (b) shows the cross-section along AA’. This section of the salt model has variable depth to top of salt and a steeply dipping flank extending to large depth……………………………………………..……39 Figure 3.4. 3D density model generated by converting the velocity structure in the SEG/EAGE seismic model into density variations. The salt body is assumed to have a constant density contrast of -0.2g/cc. Similar to Figure 3.3, panel (a) shows a 3D perspective view and panel (b) shows the same cross-section as Figure 3.3(b)…………39 Figure 3.5. 2½-D density contrast model from the converted SEG/EAGE salt model. Panel (a) shows the cross-section, which has a density contrast of -0.2 g/cm3. Panel (b) displays the true data (line) and noise-contaminated data (points). Noise added has zero mean and standard deviation of 0.26 mGal……………………………………………...40 Figure 3.6. Progress of the genetic algorithm for the 2½-D binary inverse problem of the salt body with a single density contrast. The objective value of the highest-ranking individual at each generation are represented by the points, and the average of the population is represented by the line. Although the curve still contains a shallow slope, the top ranked and average solutions have mostly converged, and the population has evolved to a similar solution as illustrated in Figure 3.8……………………………...…42 Figure 3.7. View of the starting population with the addition of prior information. (a) is the average of the starting population of the GA. Prior information is incorporated in the form of top portion of salt. (b) is the true model we attempt to recover…………….43 Figure 3.8. Model evolution during inversion. Each image is an average of the entire population at the specified generation. The upper left model is at generation 1 and the lower right is at generation 273. By generation 103, the steep dipping flank of the salt body has started to form and the disorder beneath top of salt has decreased significantly………………………………………………………………………………45 Figure 3.9. Comparison between the inverted and true model. (a) Final constructed model. The image is the average over the final population. (b) Original model to be recovered………………………………………………………………………………....46 Figure 3.10. 2½-D section through the SEG/EAGE Salt Model in density contrast form (a), and in binary form (b). A nil zone of zero density contrast cuts horizontally through the middle of the density contrast model. The binary model is represented by background sediment in black (zeros) and salt in white (ones)…………………………………….....48

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Figure 3.11. 2½-D density contrast model from the converted SEG/EAGE salt model with density profile and nil-zone (a). The nil-zone is centered around an approximate depth of 2,000 meters. Panel (b) displays the true data (line) and noise-contaminated data (points). Data exhibit a zero crossing due to density contrast reversal………………….49 Figure 3.12. Inversion result for the 2½-D problem using GA. The model contains 5,670 cells. There is a density profile and thick nil-zone. This result is presented as an average of GA population of 1000 individuals. Therefore, regions of solid black or white are features of sediment or salt, respectively, which all members of the final population share……………………………………………………………………………………...50 Figure 3.13. Inversion result for the 2½-D problem using QSA. The model contains 5,670 cells. There is a density profile and thick nil-zone. Panel (a) illustrates the true model in density contrast form. Panel (b) illustrates the result as an average of 50 inversions with QSA. Therefore, regions of solid black or white are features of sediment or salt, respectively, which all the predicted models share……………………………....55 Figure 4.1. 2½-D section through the SEG/EAGE Salt Model in density contrast form (a), and in binary form (b). A nil zone of zero density contrast cuts horizontally through the middle of the salt model. The binary model is represented by background sediment in black (zeros) and salt in white (ones).…....…………....……….......……..….…….……65 Figure 4.2. Comparison of inversion performance between stand-alone GA (a), and the GA/QSA Hybrid (b) for the 2½-D problem. The hybrid algorithm converges to a solution in 50 GA generations with evolutionary jumps due to QSA every 5 generations……………………………………………………………………………….67 Figure 4.3. Model evolution during binary inversion with the GA/QSA Hybrid. Each image is an average of the entire population at the specified generation. The upper left model is at generation 1 and the lower right is generation 50. By generation 19, the steep dipping flank of the salt body has started to form and the disorder beneath top of salt has decreased significantly…………………………………………………………………...69 Figure 4.4. SEG/EAGE seismic velocity model. Panel (a) shows a 3D perspective view of the model. The salt model is converted into a generic model mesh (b), and then a background density profile is incorporated for gravity studies……………………….....73 Figure 4.5. Synthetic data for the SEG/EAGE 3D Salt Model. Data set contains 441 data points. Gaussian noise is added with zero mean and standard deviation of 0.1 mGal……………………………………………………………………………………...74

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Figure 4.6. Generic example of cross-over of two binary models in 3D by the Genetic Algorithm. The top figures (a & b) represent two models, the parent models, each with 28,350 cells. All the cells from (a) are white and all the cells from (b) are black in this example. The parent models are divided into 1,575 blocks, each block containing 18 cells (c & d). The blocks of cells from each parent pair are re-combined randomly to form two children models (e & f). Each child model has opposite combination of block assemblage as the other child. For the actual inverse problem, cells within each block will not necessarily have the same vales throughout the block………………………….75 Figure 4.7. Sample starting model in the GA population initialized with random zeros and ones. There are 28,350 cells in the above model region, with 201 similar starting models incorporated into the GA/QSA for binary inversion…………………………….76 Figure 4.8. Top of Salt added as prior information. The model is an average of the population of models; therefore, the grey region beneath top of salt is an average of random zeros and ones…………………………………………………………………...77 Figure 4.9. Performance plot of the 3D binary inversion problem. The black points are the objective values of the highest fit model at each generation, and the blue points are the average objective values of the GA population at each generation……………...79 Figure 4.10. True and constructed model with binary inversion. The top figure (a) is the true model which is to be reconstructed by the binary inversion algorithm. The bottom figure (b) is the constructed model by the GA/QSA Hybrid with binary inversion……..80 Figure 5.1. Tikhonov curve with continuous variable formulation: The upper left region represents underfit solutions where slight increase in model structure greatly decreases the data misfit. The lower right region represents solutions with overfit data, where large increase in model structure results in little decrease in data misfit…………86 Figure 5.2. Idealized Tikhonov curve generated from GA. Each regularization parameter is plotted by the final data misfit and model objective values. The true model is inserted in to the GA population to help understand this Tikhonov plot. Notice the curve is smoother at the lower right portion of the curve than at the upper left. This difference is primarily due to mutation and cross-over in the GA, not due to the binary constraint of the inversion………………………………………………………………..89 Figure 5.3. Tikhonov curve generated from QSA. Each regularization parameter is plotted by the final data misfit and model objective values. The curve is much smoother than the one generated by GA, allowing for easier estimation of regularization………..95

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Figure 5.4. Tikhonov curve generated from GA without true model inserted into the population. Each regularization parameter is plotted by the final data misfit and model objective values. Notice the curve is not as smooth as that illustrated in figure 6.2, where the true model is incorporated into the population. There is no clear definite ‘elbow’ in this instance. This difference illustrates the difficulty in estimating regularization based on L-curve with GA for binary inversion applied to real problems……...……….……..98 Figure 5.5. Comparison of inverted models for different regularization parameters using GA. Each panel displays the average of models in the final population for a given value of regularization, β. For small values (e.g., the top left panel), the model over-fits the data and is structurally complex. For large values (e.g., bottom right panel), the model fits the data very poorly and is structurally too simple. At intermediate value of 2.0691E-7, the data misfit is close to the expected value of 41 and the model has a reasonable amount of structure and provides a good representation of the true model…………….100 Figure 5.6. Inverse model when regularization is chosen based on discrepancy principle (a). Due of the coarse nature of the Tikhonov curve in Figure 5.4., L-curve is precluded as a means for estimating regularization with GA. Panel (b) is the true model to be recovered. The true model was not inserted into the GA population for this inversion………………………………………………………………………………...101 Figure 5.7. L-curve generated from QSA. Each regularization parameter is plotted by the final data misfit and model objective values. The curve is smooth and monotonic compared with that generated by GA, allowing for reasonable estimate of the ‘elbow’ of the plot for L-curve criterion……………………………………………………………102 Figure 5.8. Plot generated by QSA of data-misfit versus regularization value for discrepancy principle. The desired misfit of 41, equal to the number of data, is obtained with regularization of approximately 15.5……………………………………………...105 Figure 5.9. Inversion result by QSA when regularization is chosen using discrepancy principle. The result is presented as a mean of 100 binary inverse models……………106 Figure 5.10. Inversion results with the energy term in the model objective function. Results are presented over a Tikhonov loop with varying regularization parameters. With the energy term (αs), there is a gap in the nil zone where salt should be present………109 Figure 5.11. Model result, averaged over the entire population of models, with no model objective function in the inversion. Without the m.o.f., the result is overly complex, and there is no agreement among the models within the nil zone, as indicated by the gray band across the middle……………………………………………………..111

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Figure 5.12. Evolution of the model results, averaged over a population of models, when the energy term is removed from the model objective function. Nil zone is filled in with either salt or sediment. Convergence is reached in only 50 generations with the hybrid algorithm………………………………………………………………………...113 Figure 6.1. 2½-D section through the SEG/EAGE Salt Model in density contrast form (a), and in binary form (b). A nil zone of zero density contrast cuts horizontally through the middle of the density contrast model. The binary model is represented by background sediment in black (zeros) and salt in white (ones). Panel (c) displays the true data (line) and noise-contaminated data (points). Data passes through zero mGal between positive and negative anomalies due to density contrast reversal……………………………….120 Figure 6.2. Mean and variance calculated from 500 binary inversions using QSA. The mean (a) illustrates that all inversions have adequately reconstructed the model for the larger distribution of mass. The variance throughout the 500 binary inversions (b) illuminates a halo of variance around the steep dipping structure at the left of the salt body……………………………………………………………………………………..122 Figure 6.3. Distance array for 500 binary inversion models generated with QSA. The axes of the image are the model numbers, and each point within the image shows the Euclidean distance between two models………………………………………………..127 Figure 6.4. Distribution of total cells that are different between the two furthest solutions (a). The distribution of the cells largely encompasses the halo of variance generated from the 500 binary inversions, Figure 6.2(b). The second panel (b) presents the cell differences as contrasting colors to illustrate the two classes of solution to binary inversion for the 2D salt body example………………………………………………...129

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ACKNOWLEDGMENTS

I would like to first acknowledge the primary sources of my financial support

over the past several years, during my unending presence at Colorado School of Mines.

The primary support for my research has been the industry consortium “Gravity and

Magnetics Research Consortium”. The sponsoring companies, either continuously or

intermittently are/were ChevronTexaco, TotalFinaElf, Marathon Oil Company,

Anadarko, ConocoPhillips, Bell Geospace, and Shell Exploration. I have also received

generous support from the Department of Geophysics at Colorado School of Mines

through Colorado Fellowships and assistantships. I was also partially supported by SEG

scholarships over the past several years. And lastly, I would like to thank Citibank, to

whom I will prove my indebtedness over the next fifteen years, for the large loans

necessary to cover out-of-state tuition and monthly income during my first year as an

unsupported graduate student.

I would like to thank my thesis committee for their continuous advice, support,

instruction, criticism, and for not making my life as difficult as they have had the power

to do. I would also like to thank the support of my fellow members of GMRC and the

Center for Gravity, Electrical & Magnetic Studies [CGEM] at Colorado School of Mines.

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Their serious (and sarcastic) comments over the years, as my work has progressed, have

been wonderful. I would especially like to thank the two people I worked with the most

over the past several years: Dr. Yaoguo Li (my thesis advisor) and Dr. Misac Nabighian.

Both have been invaluable in my degree program, and none of this material would have

been possible without their constant nagging. Yaoguo Li has also been especially

valuable as a friend, in bringing the binary formulation to life, and as a co-author of my

research projects. Lastly, I would like to thank all my family, friends, and my wonderful

girlfriend for all their emotional, and sometimes financial, support over the years as well.

They made my life a lot easier when my thesis committee made it a lot harder: yin-and-

yang. Oh yes, my dog, Skylla; she always knew when I was upset about school and

always knew the right thing to say when I needed a good laugh.

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To all the people who over the years have repeatedly asked

“When are you going to graduate?!”

1

CHAPTER 1: INTRODUCTION

Conservative estimates indicate that at least 15% of U.S. domestic oil and 17% of

its natural gas production come from fields along the continental shelf margin off the

shores of Louisiana (Gibson and Millegan, 1998). To explore for future reserves,

industry has expanded towards exploration of the deeper water regions of the continental

slope. While the industry target is obviously hydrocarbon traps, the geophysical targets

are the geologic features in the sedimentary section which are responsible for these

accumulations of oil and gas. Some of these features include reefs, faults, anticlines, and

variations in thickness of horizontal salt beds.

The salt beds, including domes, ridges and pillows, are relatively incompressible

and therefore remain fairly constant in density throughout. This incompressibility

likewise allows for abundant traps throughout the Gulf of Mexico (Gibson and Millegan,

1998). As a result, they have become major targets in oil and gas exploration. Gravity

inversion is one of the tools available to geophysicists for imaging these exploration

targets.

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The gravity inverse problem for salt body imaging is one of finding the position

and shape of a constant anomalous density embedded in a sedimentary background

whose density increases with depth due to compaction. Depending on the depth and

depth extent of the salt body, three scenarios can occur. In the first scenario, the salt is

shallow enough so that its density is greater than that of the immediate sedimentary host.

This leads to a positive density contrast in the salt, and a positive gravity anomaly in

surface data. In the second scenario, the salt is positioned at depth so that its density is

less than the density of the surrounding sediments. This leads to an entirely negative

density contrast for the salt body, and therefore a negative gravity anomaly on the

surface. In the final scenario, the salt body straddles a depth at which the sediment

density is equal to the salt density. This region of equal density between salt and

sediment is referred to as a nil-zone.

In the last scenario described above, the portion of salt within the nil-zone does

not contribute to surface gravity data. This is a natural consequence of having zero

density contrast with the surrounding sediment. Likewise, portions of the salt body

above the nil-zone will have positive density contrast, producing a positive anomaly in

surface gravity data. Salt below the nil-zone, in contrast, generates a negative gravity

anomaly because it has a negative density contrast with respect to the surrounding

medium. The net result is that the positive and negative anomalies from the top and

bottom portions of salt tend to cancel out in parts of the surface gravity data, Gibson and

3

Millegan (1998). This effect is referred to as an annihilator. Zero density contrast nil-

zones, combined with annihilators in the salt body, tend to result in gravity inversions

that have little resemblance to the true geologic problem.

Current inversion methods for imaging salt structure using gravity data fall under

two general categories. The first is interface inversions. These methods assume a simple

topology for the salt body and known density contrast and construct the base of the salt

(e.g., Cheng, 2003; Jorgensen and Kisabeth, 2000). The methods have the advantage that

they directly input the known density contrast at each depth and provide a direct image of

the base of salt. However, the drawbacks are that the problem is nonlinear and can be

more difficult computationally. In addition, the assumed simple topology of salt creates

difficulties when either regional field or small-scale residuals due to shallow sources are

not completely removed. The inconsistency between the assumed model and data can

lead to large errors, or even failure of inversion.

Methods in the second category are generalized density inversions. These

methods construct a density contrast distribution as a function of spatial position and

image the base of salt by the transition in density contrast (Li, 2001). Density inversions

have the flexibility of handling multiple anomalies, highly complex shapes, and the

solution is easier to obtain because the relationship between observations and density

contrast is linear. However, as they are currently formulated, these methods are not well

4

suited for cases where nil-zones are present. They typically produce poor (if any)

resolution near these zones of zero density contrast. Likewise, when an annihilator is

present in the salt body, density inversion methods allowing continuous density values

(e.g., Li and Oldenburg, 1998) will in general produce a model that has little resemblance

to the true structure. The data are satisfied by intermediate density values and

distributions that only image a portion of the salt body.

To overcome difficulties associated with both methods, I present a binary

formulation that enables one to incorporate the density contrast values appropriate to the

geologic problem while providing a sharp boundary for the subsurface, two strength of

the interface inversion. At the same time, the binary formulation is designed to retain the

flexibility and linearity of density (cell based) inversions. Variables in the binary

formulation can only take on discrete values, 0 or 1 for sediment or salt respectively.

My thesis is divided into seven chapters. This first chapter is written as an

introduction to the problem of gravity inversion in salt imaging, and discusses techniques

currently available for this problem. I also introduce the concept of binary inversion as

an alternative method for gravity inversion.

Details of binary inversion are presented in Chapter 2. In this chapter, I develop

the theoretical and practical aspects of binary inversion method for inverting gravity data

5

in salt imaging. I start with a description of the current methods for gravity inversion in

salt imaging, and highlight the problems associated with the methods as currently

formulated when a nil-zone exists. I then present the binary formulation as an alternative

method for inversion of gravity data, and illustrate the formulation with numerical

examples.

Chapter 3 details the application of genetic algorithm and quenched simulated

annealing to binary inversion of gravity data for the salt body problem. The chapter

begins with a description of the difficulties in selecting a technique for solution to the

binary inverse problem. I then describe details of genetic algorithm for binary inversion,

and apply the method to gravity data generated above a 2D section of the SEG/EAGE

Salt Model (Aminzadeh et al., 1997). Last, I introduce and apply a modified SA called

quenched simulated annealing as a local search method for solution to the binary inverse

problem.

In the 4th chapter, I introduce a hybrid optimization algorithm as an alternative

solution technique for binary inversion. The hybrid algorithm combines genetic

algorithm with quenched simulated annealing. The former allows for easy incorporation

of prior geologic information, large number of solutions, and rapid build-up of larger

model structure, while the later guides the genetic algorithm to faster solution by rapidly

adjusting the finer model features. In this chapter, I discuss advantages of a hybrid

6

algorithm as an additional solution strategy for binary inversion, and illustrate its

improved efficiency in comparison to stand-alone genetic algorithm. The algorithm is

then applied to binary inversion of gravity data for a 3D salt problem with complex shape

and a large number of parameters.

In Chapter 5, I explore regularization and the weighting parameters for the binary

inverse problem. There are four basic components to the chapter. First I discuss the role

of regularization for continuous variable formulations, and illustrate the similarities and

differences with that of binary inversion. Second, two methods for construction of a

Tikhonov curve are analyzed to illustrate the advantages and disadvantages of each

technique for binary inversion. Third, I present two approaches for choice of

regularization to the binary inverse problem: (1) discrepancy principle and (2) L-curve

criterion. Last, I explore the weighting parameters of binary inversion and illustrate their

effects on the final model solution.

To appraise the solution, and understand the uncertainties in the recovered model,

I explore the solution space of binary inversion and evaluate the modality of the objective

function in Chapter 6. This allows for more reliable interpretation of the binary solution

through identification of high confidence features in salt structure and regions of high

variance.

7

Finally, in Chapter 7, I conclude with a discussion of binary inversion, including

advantages and disadvantages of the technique. I also present recommendations for

potential research directions on binary inversion in the future.

8

CHAPTER 2: BINARY INVERSION

In this chapter, I present a binary inversion algorithm for inverting gravity data in

salt imaging. The density contrast is restricted to being one of two possibilities: either

zero or one, where one represents the value expected at a given depth. The algorithm is

designed to easily incorporate known density contrast information, and to overcome

difficulties in salt imaging associated with nil zones. This chapter starts with a

description of the current methods for gravity inversion in salt imaging, and highlights

the problems associated with the methods as currently formulated when a nil-zone exists.

Next I present a binary formulation for inversion of gravity data, and illustrate the

formulation with numerical examples.

The examples in this chapter are solved using Genetic Algorithm. However,

details on this aspect of the solution strategy are kept to a minimum here, and the

examples are intended for illustrative purposes of binary inversion only. Solution

strategies to binary inversion, such as use of Genetic Algorithm, are presented in greater

detail in Chapter 3, along with application to more realistic geologic problems.

9

2.1. Inversion Methods for Imaging Salt Structure

Inversion methods for imaging salt structure using gravity data fall under two

general categories. The first is interface inversions. These methods assume a simple

topology for the salt body and known density contrast, and construct the base of the salt

(e.g., Cheng, 2003; Jorgensen and Kisabeth, 2000). Similar method has also been used

extensively in other applications of gravity inversion, such as in basin depth

determination (e.g., Oldenburg, 1974; Pedersen, 1977; Chai and Hinze, 1988; Reamer

and Ferguson, 1989; Barbosa et al., 1999). Methods in the second category are

generalized density inversions. These methods construct a density contrast distribution as

a function of spatial position and image the base of salt by the transition in density

contrast (Li, 2001). Similar approaches have also been used widely in mineral exploration

problems (Green, 1975; Last and Kubik , 1983; Guillen and Menichetti, 1984; Oldenburg

et al., 1998).

2.1.1. Interface Inversion

The interface inversion has the advantage that it directly inputs the known density

contrast at each depth and provides a direct image of the base of salt. However, the

drawbacks are that the problem is nonlinear and can be more difficult computationally. In

10

addition, the assumed simple topology of salt creates difficulties when either regional

field or small-scale residuals due to shallow sources are not completely removed. The

inconsistency between the assumed model and data can lead to large errors, or even

failure of inversion.

2.1.2. Density Inversion

The density inversion has the flexibility of handling multiple anomalies, more

complex shapes, and the solution is easier to obtain because the relationship between

observations and density contrast is linear. However, as they are currently formulated,

these methods are not well suited for cases where nil-zones are present. When a nil-zone

exists, a salt body of uniform density straddles a depth where the sedimentary density is

equal to the salt density within a depth interval. Within this region, salt has zero density

contrast and therefore has no contribution to surface data. Because of this relation,

gravity inversion algorithms typically produce poor (if any) resolution near the nil-zone.

A second effect on gravity data likewise occurs in the presence of nil-zones. Density

contrast reverses sign as the depth increases, and therefore parts of the gravity anomalies

due respectively to the top and bottom portions of the salt cancel out. Consequently, a

portion of the salt body is invisible to the surface gravity data. In effect, that portion of

the salt forms an annihilator. Density inversion methods allowing continuous density

11

values (e.g., Li and Oldenburg, 1998) will in general produce a model that has little

resemblance to the true structure. The data are satisfied by intermediate density values

and distributions that only image a portion of the salt body.

2.2. Binary Inversion

To overcome difficulties associated with both methods, I present a binary

formulation that enables one to incorporate the density contrast values appropriate to the

geologic problem while providing a sharp boundary for the subsurface, two strengths of

non-linear interface inversion. At the same time, the binary formulation is designed to

retain the flexibility and linearity of density (cell based) inversions. Variables in the

binary formulation can only take on discrete values, 0 or 1 for sediment or salt

respectively. In this section, I develop the theoretical and practical aspects of the binary

formulation for inversion of gravity data.

2.2.1. Background

The difficulty of an annihilator outlined in section 2.1.2 can only be overcome by

incorporating prior information to restrict the class of admissible models. I propose to

12

impose the condition that the density contrast must be the discrete values appropriate for

the geologic problem. In the simplest form, density contrast is restricted to being either

zero or a known value at a given depth. Similar binary approach has been used in both

gravity inversion and in other fields. For example, Camacho et al. (2000) invert gravity

data for a compact body with a constant density by growing the volume from an initial

guess. Litman et al. (1998) invert for the shape of a scatterer by assuming a constant

electrical conductivity value for the background and the scatter, respectively.

2.2.2. Theory

For my problem, I adopt explicitly the Tikhonov regularization approach

(Tikhonov and Arsenin, 1977) and formulate the inversion for the general case of salt

imaging at the presence of density reversal. The problem then becomes one of

minimizing an objective function subject to restricting model parameters to attain only

one of two values at each depth. The objective function consists of the weighted sum of

the model objective function mφ and data misfit dφ :

{ }.)z(,0 subject to,)()( min. d

ρρτβφρφφ

∆∈+= m (2.1)

13

Assuming that I know the standard deviation of each datum iσ , I can define the

data misfit function as

2N

1i i

prei

obsi

ddd∑

=⎟⎟⎠

⎞⎜⎜⎝

⎛ −=σ

φ (2.2)

where obsid and pre

id are the observed and predicted data, respectively. Assuming

Gaussian statistics, where the data are contaminated with independent Gaussian noise

with zero mean, the data misfit defined by eq.(2.2) is a χ2 variable with N degrees of

freedom (Hansen, 1992). As a result, the expected level of regularization through

discrepancy principle is one which sets the data misfit equal to the number of data N.

I would like to construct a compact model that is also structurally simple.

Therefore, I use the following generic model objective function (e.g., Li and Oldenburg,

1998) for 3D problems

( ) [ ]( ) ( ) [ ]

( ) [ ] ( ) [ ] dvz

zwαdvy

zw

dvx

zwdvzw

Vz

Vy

Vx

Vm

∫∫

∫∫

⎟⎠⎞

⎜⎝⎛

∂−∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

−∂+

⎟⎠⎞

⎜⎝⎛

∂−∂+−=

20

20

202

0sα

ττττα

τταττφ

vvvv

vvvv

, (2.3)

14

where τ is the binary model, 0τ is a reference model if available, V is the subsurface

region over which the model is defined, sα , xα , yα , and zα are relative weights of

the individual components of the model objective function, and ( )zw is a depth

weighting function.

Requiring dφ to achieve an expected value ensures all models that do not

adequately fit the observed data, or over fit them, be eliminated from the set of possible

solutions. The binary formulation places a limit on the number of models that fit the

data, and there are no longer infinitely many. However, the problem is still non-unique.

Therefore, I use the model objective function, mφ , to further narrow our solution set to

only geologically reasonable models. This is done by choosing the model that has the

smallest size and structural complexity among those that fit the data. The regularization

parameter β controls the balance between the data misfit and model objective function,

protecting from over fitting the data or over smoothing the model.

The binary formulation is unique in that it incorporates a binary variable τ into

the density function of eq.(2.1) through expected density contrast at depth z :

{ }1,0)( ∈rvτ . (2.4)

)()( zr ρτρ ∆=v . (2.5)

15

At a given depth, a value of zero in the model, τ , indicates a zero density contrast

(host sediments), while a value of one corresponds to the expected salt density contrast at

that depth. The minimization problem is then expressed in τ( rv ) and I can simply work

with 0 and 1 for the minimization problem. The actual density contrast value is only

incorporated into the forward modeling of predicted data during the inversion.

The solution to this problem will be better constrained than formulations that

allow continuous values within upper and lower bounds. Although still non-unique, this

problem no longer has an infinite number of possible solutions: there are a finite number

of cells within the model mesh and only two possible values for each location. For

instance, construction of an equivalent source layer at any depth is no longer possible.

The binary value of 1, at a specified depth, represents a well-defined density contrast

value - either positive or negative with corresponding magnitude. Because of this

constraint, a combination of a positive and negative anomaly in gravity data is not always

reproducible by a source distribution at one depth alone (i.e., by an equivalent source

layer). Whenever an annihilator is present, any geologically unreasonable model that

reproduces the data by a combination of density contrasts of intermediate values is

automatically eliminated with the binary constraint.

16

2.2.3. Numerical Solution

To perform numerical solution for the binary inverse problem, the model region

of interest τv is first generated as an orthogonal 3D mesh composed of cuboidal cells.

Each cell within the mesh assumes a constant binary value of 1 or 0, for salt or sediment

respectively, as illustrated in Figure 2.1. The binary model τv is therefore an M length

vector describing distribution of salt and sediment throughout the model, where M is the

total number of cells within the model.

Figure 2.1. 3D binary model τ divided into cuboidal cells with constant values of either 1 or 0. While the image is generic for conceptual purposes, later images during inversion for salt have values of 0 (black) for host sediment, and 1 (white) for salt.

17

2.2.3.1. Forward Modeling

Given N gravity observations, the data vector TNddd ),,( 1 K

v= calculated above

the model is linearly related to the subsurface density distribution by

ρvv

Gd = , (2.6)

where ρv is an M length vector of density distribution related to the binary model τv

through eq.(2.5).

The sensitivity matrix G is comprised of the elements gij, which quantify the

contribution of a unit density in the jth cell of the model to the ith datum,

∫∫∫∆ −

−=jV i

iij dv

rrzzg 3γ , (2.7)

2.2.3.2. Data-Misfit

For numerical solution, the measure of data misfit defined by eq.(2.2) becomes

18

( ) 2

2

preobsdd ddW

vv−=φ , (2.8)

where predv

are predicted data during inversion, obsdv

the observed data, and dW the N-

by-N diagonal data-weighting matrix comprised of the inverse of the estimated standard

deviations, iσ . Here I use a 2-norm measure of data misfit.

2.2.3.3. Model Objective Function

As with the data misfit, the model objective function of eq.(2.3) is written for

numerical solution to the binary inverse problem. The finite-difference approximation of

eq.(2.3) is written as:

( ) ( ) ( ) ( ) 20

2

02

02

0 ττατταττατταφ vvvvvvvv −+−+−+−= ZWZWZWZW zzyyxxssm , (2.9)

or

( ) ,20ττφ −= mm W (2.10)

where the model weighting matrix, mW , is the combined matrix of the individual

weighting terms in the model objective function. mW acts to measure the model size,

19

which is defined by the first term in eq.(2.3), and structural complexity (or change) in the

three orthogonal directions, which is defined by the derivative terms in eq.(2.3). The

former is referred to as a model energy term, and the later are often referred to as

‘smoothing’ terms in the objective function. mW also incorporates depth weighting to

counteract the decay of the kernels with depth in the sensitivity matrix. Therefore, the

model weighting matrix is defined by:

( ) ZWWWWWWWWZWW zT

zzyTyyx

Txxs

Tss

Tm

Tm αααα +++= (2.11)

where Z is diagonal matrix representing the discretized form of a depth weighting

function.

2.2.3.4. Depth Weighting

Gravity and magnetic data have no inherent depth resolution due to the rapid

decay of the kernels with depth in the sensitivity matrix. For gravity method, the kernels

decay with 21 r , eq.(2.7), where r is the distance between model and data location. As

a result, cells at depth inherently have much less influence on surface data and tend to be

small (zero for binary inversion) in the model obtained through a minimum norm

20

solution. Consequently, even with my binary constraint, there is still a tendency to

concentrate material as close to the surface as possible during inversion. The resulting

solution is not geologically meaningful.

To provide cells at depth with equal probability of obtaining non-zero values

during inversion, a generalized depth weighting function is developed to incorporate into

the model objective function (Li & Oldenburg, 2000). The depth weighting function is

designed to match the overall sensitivity of the data set to a particular cell,

2

1

2

1

2)(

λ

λ

⎟⎠

⎞⎜⎝

⎛=

=

∑

∑

=

=

N

iij

N

iij

G

Grw v

, (2.12)

where )(2 rw v is the root-mean-square sensitivity of the model, and λ is chosen to match

the 21 r decay of gravity signal away from the source. Normally, λ = ½ , but numerical

experiment indicates that λ = 0.4 works well for the experiments used in this thesis.

21

2.3. Numerical Examples

In the previous section of this chapter, I outlined the theoretical and numerical

aspects of the binary inverse formulation. In following chapters, I present tools available

for binary inversion such as Genetic Algorithm, Quenched Simulated Annealing, and a

Hybrid Algorithm. I will also apply the technique to gravity inverse problems with a

large number of parameters, complex model shape, and density reversal through nil-

zones, as well as discuss aspects of the inverse formulation such as regularization and

weighting parameters. However, it is appropriate at this point to briefly illustrate the

binary inversion by using simple numerical examples without the details on how the

solution is obtained. It suffices to state that Genetic Algorithm (GA) is used as the basic

solver for these examples. In short, the GA works with a large population of models

simultaneously, and attempts to evolve these individuals towards a final solution. More

details on the GA can be found in the following chapter.

2.3.1. Single Boxcar (1D)

I first illustrate the binary technique using a simple mathematical example. The

forward calculation is given by the following integral,

22

20,,1,)()(

)exp()2cos()(

1

0

1

0

L=≡

−=

∫

∫

jdzzgzm

dzjbzjazzmd

j

j π (2.13)

with 250ba .== . In eq.(2.13), m(z) is the model and gj(z) are the kernels that decay with

depth. These kernels are chosen to mimic the decaying kernels seen in many geophysical

experiments.

For the numerical test, I use a single boxcar model that is zero everywhere except

within a central interval. The 1D model, Figure 2.2(b), is analogous to density as a

function of depth, similar to a single well record. The region spans from zero to one in

generic units. The interval is uniformly discretized into a 1D mesh of 50 cells.

Therefore, each cell has a length of 0.02.

The 20 simulated data from eq.(2.13), with additive noise are shown as the dots in

Figure 2.2(a). These noisy data are inverted to recover a binary model defined over the

model mesh of 50 cells. The objective function is the 1D equivalent of eq.(2.1) and

includes the first two terms. For the GA, an initial population of 400 individuals is

initialized. The mutation operator allows each parameter within the individuals to mutate

with a one in fifty probability. Cross-over occurs in segments of 7 cells. Convergence of

the population of 400 solutions is reached by the 20th generation, as illustrated in Figure

23

2.3. The recovered model from binary inversion, Figure 2.2(c), is a good representation

of the true model with only one cell different from the true model. The predicted data

from this model are shown in Figure 2.2(a) as the solid line. The binary inversion has

performed well in this case.

To illustrate the differences between continuous variable and binary inversion,

Figure 2.4(b) shows the equivalent final solution from continuous variable inversion.

The data that are inverted are the same 20 noisy data generated for the binary case. Final

data fit from the two methods are equivalent. However, Figure 2.4(b) illustrates that the

continuous formulation has skewed the solution over a larger interval and adjusted

amplitude through intermediate values to satisfy the data. The binary inverse solution is

outlined in Figure 2.4(b) as gray dashes over the continuous variable solution, in order to

highlight the final model differences. The binary formulation’s incorporation of

amplitude information allows the algorithm to remain true to model’s size and position

while providing a sharp contact in the subsurface.

24

5 10 15 20 25 30 35 40 45 50

0

1

5 10 15 20 25 30 35 40 45 50

0

1 Constructed Model

Cell Number

Cell Number

Bin

ary

Val

ueB

inar

y V

alue

True Model

0 2 4 6 8 10 12 14 16 18 20-0.2

-0.1

0

0.1

0.2

0.3Predicted dataSynthetic data with noise

Data Results: Predicted vs. Original

Data Point

Dat

a V

alue

5 10 15 20 25 30 35 40 45 50

0

1

5 10 15 20 25 30 35 40 45 50

0

1 Constructed Model

Cell Number

Cell Number

Bin

ary

Val

ueB

inar

y V

alue

True Model

0 2 4 6 8 10 12 14 16 18 20-0.2

-0.1

0

0.1

0.2

0.3Predicted dataSynthetic data with noise


Data Point

Dat

a V

alue

Figure 2.2. Binary inversion results for the 1D mathematical problem. The original and predicted data are illustrated in the top panel (a), with a comparison between the true model (b) and constructed model (c) beneath.

a)

b)

c)

25

0 5 10 15 20 250

50

100

150

200

250

Highest Fit Individual

Average of Population

Generation Number

Obj

ectiv

e V

alue

0 5 10 15 20 250

50

100

150

200

250

Highest Fit Individual

Average of Population

Generation Number

Obj

ectiv

e V

alue

Figure 2.3. Progress of the genetic algorithm for the 1D binary inverse problem. The objective value of the highest-ranking individual at each generation are represented by the points, and the average of the population of solutions is represented by the line.

26

5 10 15 20 25 30 35 40 45 50

0

1

5 10 15 20 25 30 35 40 45 50

0

1

Constructed Model

Cell Number

Cell Number

Bin

ary

Val

ueM

odel

Val

ue

True Model

Continuous variable

Binary variable

5 10 15 20 25 30 35 40 45 50

0

1

5 10 15 20 25 30 35 40 45 50

0

1

Constructed Model

Cell Number

Cell Number

Bin

ary

Val

ueM

odel

Val

ue

True Model

Continuous variable

Binary variable

Figure 2.4. Illustration of inversion result for the 1D boxcar problem with continuous variable. The true model (a) is the same as in Figure 2.2 (b), which was also used for binary inversion. The constructed model (b) with continuous variable does not have a sharp contact as does the true model or the model constructed with binary inversion, outlined with gray dashes. Likewise, the amplitude of the model is not as accurate as the binary result. Both inversions were performed with the same data set.

a)

b)

27

2.3.2. Gravity Problem (2D)

This last example is a transition model to the gravity problem from the 1D

mathematical problem. The true density model, Figure 2.5(b), consists of a simple block

buried in a uniform half-space. The noise-contaminated gravity data taken along a

traverse perpendicular to the strike are shown by the dots in Figure 2.5(a). There are a

total of 60 data points. To perform binary inversion, the model region is divided into 400

rectangular cells (20x20). The model region spans vertically from the surface to 500

meters depth, and horizontally from zero to 1000 meters. Each generation has 400

individuals and the algorithm achieves convergence by 150 generations.

The predicted data from the final model are shown in Figure 2.5(a) as the solid

line, which is a smoothed version of the noisy data as expected. The recovered model,

Figure 2.5(c), compares well with the true model, with a difference of two cells at the

base edges of the block. As with the previous example, the ability to incorporate density

information appropriate for the problem has allowed the binary inversion to accurately

determine the model’s shape and position while providing a sharp contact with the

surrounding sediment.

28

-1000 -600 -200 200 600 1000 1400 1800 0

0.5

1Synthetic data with noisePredicted data

0 100 200 300 400 500 600 700 800 900 1000

100

200

300

400

500

0 100 200 300 400 500 600 700 800 900 1000

100

200

300

400

500

Constructed Model

x (meters)

x (meters)

z (m

eter

s)z

(met

ers)

True Model


x (meters)

g z(m

Gal

)

-1000 -600 -200 200 600 1000 1400 1800 0

0.5

1Synthetic data with noisePredicted data

0 100 200 300 400 500 600 700 800 900 1000

100

200

300

400

500

0 100 200 300 400 500 600 700 800 900 1000

100

200

300

400

500

Constructed Model

x (meters)

x (meters)

z (m

eter

s)z

(met

ers)

True Model


x (meters)

g z(m

Gal

)

Figure 2.5. Binary inversion results for a simple 2½-D gravity problem. The top panel (a) illustrates a comparison between the observed and predicted data, and the lower panels display the true (b) and constructed (c) models.

a)

b)

c)

29

2.4. Summary

In this chapter, I discuss problems associated with inversion of gravity data in salt

imaging when a nil-zone is present. Next, I present an alternative approach to tackle

these problems, binary inversion, which is formulated to capture the better features of

density and interface inversions. The binary condition is designed to place a strong

restriction on the admissible models so that the non-uniqueness caused by nil zones might

be resolved. The theoretical and practical aspects of binary inversion are outlined. The

final sections of the chapter illustrate the application of binary inversion to simple 1D and

2½-D numerical examples. Results demonstrate the efficacy of the binary formulation in

the 2½-D gravity problem, by allowing sharp contacts within the subsurface, and

correctly identifying size and location of the anomalous body.

In the next two chapters, I present details on the primary tools developed as

solvers for binary inversion. Those are Genetic Algorithm, Quenched Simulated

Annealing, and a hybrid optimization algorithm. Likewise, I expand upon the application

of binary inversion by introducing more realistic gravity inverse problems with a large

number of parameters (in 2D and 3D), as well as a complex background density profile.

30

CHAPTER 3: SOLUTION STRATEGY FOR BINARY INVERSION

This chapter details the application of genetic algorithm and quenched simulated

annealing to binary inversion of gravity data for the salt body problem. The chapter

begins with a description of the difficulties in selecting a technique for solution to the

binary inverse problem. Next I describe details of genetic algorithm for binary inversion,

and apply the method to gravity data generated above a 2D section of the SEG/EAGE

Salt Model (Aminzadeh et al., 1997). Last, I introduce and apply a modified SA called

quenched simulated annealing as a local search method for solution to the binary inverse

problem.

3.1. Solution Strategies for Binary Inversion

The minimization problem defined by eq.(2.1) has a deceptively simple

appearance, but its solution is not trivial. The difficulty lies in the discrete nature of the

density contrast. Because the variable can only take on two values, 0 or 1, derivative-

based minimization techniques are no longer applicable. There are several alternative

methods for carrying out the minimization. The obvious technique is mixed integer

31

programming (e.g., Floudas, 1995; Pardalos and Resende, 2002) since our variable to be

recovered can only assume a value of either 0 or 1. However, solution of the integer-

programming problem is both theoretically and numerically complicated. It is difficult to

implement and computationally costly. I have decided not to pursue this route.

The second technique involves the use of a controlled random search technique

such as genetic algorithms (GA), simulated annealing (SA), and quenched simulated

annealing (QSA). Each method is ideal for derivative free minimization, which is the

problem I have. In addition, the methods can be implemented with relative ease

compared to an integer programming solution. Therefore, in the following sections, I

present GA and QSA as solution strategies to binary inversion, illustrate them with

numerical examples, and discuss limitations of the techniques.

3.2. Genetic Algorithm

To gain basic understanding about the behavior of the binary formulation, I start

with the genetic algorithm (GA) as the basic solver. The GA is a derivative-free

minimization technique which is well suited for the binary problem. It generates updated

solutions to the inverse problem by combining and modifying the property values from

multiple models to create new inverse solutions at each iteration. Due to the binary

32

nature of my problem, the GA does not have to deal with the magnitude of these property

values, as with continuous-variable applications. If a model parameter is selected for

change, the GA merely changes it to the only other possible value, 0 or 1. This is the

ideal scenario for a GA. Components of the GA are described next, and then the GA is

applied to binary inverse problems immediately following.

3.2.1. Design of Genetic Algorithm

The GA is a programming tool designed for solving a variety of optimization

problems. It is a stochastic technique that mimics natural biological evolution by

imposing the principle of ‘survival of the fittest’ on a population of individuals. For the

inverse problem, fitness is derived from a model’s total objective value, eq.(2.1). Lower

objective values translate to higher fit solutions. I note that the fitness of a model in GA

should not be confused with the data misfit. The main objective of the GA is to

recombine the individuals, with the better-fit individuals having higher probabilities of

reproduction, to evolve to better solutions. The basic design of the GA is displayed as a

flow chart in Figure 3.1. Below, I briefly describe the components unique to the GA,

including the individual, initialization, rank, fitness, selection, recombination and the

formation of the next generation of solution. However, readers are also referred to

Goldberg (1989), Pal and Wang (1996), and Chambers (1995) for more details on basic

33

Genetic Algorithm. Additional information on GA, with application specific to

geophysical inversion, is also available in Sen and Stoffa (1995), Smith et al. (1992),

Sambridge and Mosegaard (2002), Scales et al. (1992).

Individuals:

The basic unit of the genetic algorithm is the individual. Each individual

represents a potential solution to the problem, i.e. a geophysical model in our problem.

Initialize population

Start

Mutation

Recombination

Selection

Evaluate objective function, eq.(1)

Are optimization criteria met?

Best individuals

Result

no

yes

Generate

new

population


Start

Mutation

Recombination

Selection



Best individuals

Result

no

yes

Generate

new

population


Start

Mutation

Recombination

Selection



Best individuals

Result

no

yes

Generate

new

population

Figure 3.1. Flowchart of the Genetic Algorithm for the binary inverse problem. Selection, recombination, and mutation are components unique to the genetic algorithm. Modified from Pohlheim (1997).

34

For the binary inverse problem, models are discretized into cells with constant values

equal to zero or one. The corresponding individual consists of a series of chromosomes,

where each chromosome represents a cell within the model mesh. Therefore, each

individual consists of a string of chromosomes with values of either zero or one.

Initialization:

The first step in applying the genetic algorithm to gravity inversion is setting up

an initial population, which is a community of individuals. Each individual represents a

model. For initialization, values are assigned to the cells in each model. When prior

information is not available, the starting population is initialized by assigning random

zeros and ones to each cell. Figure 3.2 displays two examples of random initialization for

the GA in 1D.

The ability of the genetic algorithm to work with multiple models at one time,

through the creation of a population, also allows the user to incorporate prior information.

One form of such prior information is the models obtained from previous work. It can be

an initial guess produced from other geophysical data such as pre-stack depth migrated

seismic image. This is useful in imposing features such as the known top of salt.

35

Rank, Fitness, and Selection:

The first step in the evolutionary cycle of the GA is to rank the population.

Individuals for my problem are assigned objective values based on eq.(2.1). Lower

objective values correspond to higher fitness levels and, therefore, better models. Rank is

established by ordering the models from highest fitness values to lowest, i.e. best model

0 5 10 15 20 25 30 35 40 45 50

0

1

0 5 10 15 20 25 30 35 40 45 50

0

1

Random Starting Model #1


Cell Number

Cell Number

Bin

ary

Val

ueB

inar

y V

alue

0 5 10 15 20 25 30 35 40 45 50

0

1

0 5 10 15 20 25 30 35 40 45 50

0

1



Cell Number

Cell Number

Bin

ary

Val

ueB

inar

y V

alue

Figure 3.2. Example of two starting individuals with initialization of random zeros and ones. Each individual represents a potential solution model.

36

to worst. Individuals with higher fitness values will have higher probabilities of

surviving the evolutionary process, as well as passing on their genes to the next

generation.

Next I assign selection probabilities defined as the fitness value of an individual

divided by the sum of fitness of the entire population. The highest-ranking model has the

highest probability of surviving, while the lowest ranking model has a zero selection

probability. The final step in the selection process is choosing individuals as parents for

reproduction. I use Roulette Wheel Selection (Goldberg, 1989) in this algorithm.

Recombination:

Once individuals have been chosen for recombination based on objective values

and selection probabilities, offspring are generated to join the population in the next

generation. Selected individuals are paired into parents, and a combination of their

chromosomes, i.e. model features, are merged to generate offspring. Every new offspring

represents a new candidate solution to the problem. For my problem, I have formulated

selection and recombination such that the population of models is paired into parent

solutions at each generation. Each pair is crossed-over to generate two new solutions

from their combined model features. The new generation of solutions, i.e. children, will

replace the least fit half of the previous population of solutions once mutation has been

applied to them.

37

Mutation:

After formation of a new set of models (recombination), mutation is applied to the

newly generated solutions, i.e. children, to protect the population from an irrecoverable

loss of potentially useful genetic information during reproduction. Mutation in the binary

problem consists of flipping randomly chosen cells from 0 to 1, or vise versa. In

addition, mutation prevents premature convergence by introducing new genes into the

population. It essentially expands the gene pool and allows different regions of the

solution space to be explored. Mutation rates, i.e. the probability of each individual cell

being flipped, are problem-dependent and may be varied according to the performance of

the GA. For every child created during recombination, each chromosome has a low

probability of being mutated.

New Generation:

The last step in the evolutionary cycle of the genetic algorithm is to evaluate the

children and assign objective values. Once these values have been assigned, the children

are placed into the population to replace the least fit half of the previous generation. The

new generation of potential solutions formed in this manner therefore consists of higher-

ranking individuals from the previous generation and their offspring. The GA proceeds

to the next evolutionary cycle and repeats the process until completion. For the binary

inverse problem, I have formulated the GA to run until the models within the population

38

have converged to similar solution. At this stage, there are little to no changes which can

occur within the population of solutions by GA.

3.2.2. Numerical Examples

Numerical examples of binary inversion, in 1D and 2D, were presented at the end

of Chapter 2 using GA. However, details of the GA were kept to a minimum because the

examples were presented for illustrative purposes of binary inversion only. The models

were simple and meant for illustration only. In this section, I present two examples of

binary inversion using GA for more realistic gravity problems.

3.2.2.1. Salt Body with Single Density Contrast

The SEG/EAGE salt model (Aminzadeh et al., 1997) is designed as a velocity

model for development of imaging technology in the seismic community. Figure 3.3

displays the perspective view (a) of the velocity model and one velocity section (b). I

have converted the velocity model to a density model to assist in the development of the

binary inversion. A similar perspective view of the 3D model and 2D section are

39

A

A’

A A’

A

A’

A A’

Figure 3.3. SEG/EAGE seismic velocity model. Panel (a) shows a 3D perspective view of the model. One cross-section AA’ is outlined. Panel (b) shows the cross-section along AA’. This section of the salt model has variable depth to top of salt and a steeply dipping flank extending to large depth.

a)

b)

Figure 3.4. 3D density model generated by converting the velocity structure in the SEG/EAGE seismic model into density variations. The salt body is assumed to have a constant density contrast of -0.2g/cc. Similar to Figure 3.3, panel (a) shows a 3D perspective view and panel (b) shows the same cross-section as Figure 3.3(b).

a)

b)

40

presented in Figure 3.4, with density in place of velocity. The section has been simplified

to a single density contrast of –0.2g/cm3 between salt and sediment.

Gravity data are calculated above the 2D section along a traverse perpendicular to

the strike, Figure 3.5(b). There are a total of 41 data points. Noise has been added to the

data with zero mean and a standard deviation of 0.26 mGal. Figure 3.5(a) shows the

original 2½-D model I attempt to recover.

-13000 7000 27000 -6

-4

-2

0

True data before noise

Data with noise added

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

z (m

eter

s)

2½-D Salt Model: Single Density Contrast

Forward Data With and Without Noise

x (meters)

g z(m

Gal

)

x (meters)-13000 7000 27000

-6

-4

-2

0

True data before noise

Data with noise added

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

z (m

eter

s)

2½-D Salt Model: Single Density Contrast


x (meters)

g z(m

Gal

)

x (meters)

Figure 3.5. 2½-D density contrast model from the converted SEG/EAGE salt model. Panel (a) shows the cross-section, which has a density contrast of -0.2 g/cm3. Panel (b) displays the true data (line) and noise-contaminated data (points). Noise added has zero mean and standard deviation of 0.26 mGal.

a)

b)

41

To perform binary inversion, I utilize the 2D form of the model objective function

in eq.(2.3). The model region is divided into 1407 rectangular cells (21x67), each with a

200m length in the x- and z-directions. Regularization is chosen such that the final data-

misfit, eq.(2.2), equals the number of data. Weighting parameters in eq.(2.3), measuring

the size and change of the solution, are set to 5104.6,1 xa zxs === αα . An evaluation

of the choice of regularization and weighting parameters is presented in Chapter 5:

Regularization and weighting parameters in binary inversion. Each generation of the

genetic algorithm has 600 individuals and the population evolves to similar solution by

300 generations, as illustrated in Figures 3.6 and 3.8.

The ease with which prior information can be incorporated from seismic imaging

or other inversions is one of the greatest advantages of the formulation. To demonstrate

this, I have incorporated the top part of salt into the initial population as shown in Figure

3.7(a), with (b) representing the true model. The inversion will attempt to recover base of

salt, including the steeply dipping slope on the left side of the model (north-west in the

3D model).

Although top of salt is added as prior information, this feature is not enforced as a

constant. In other words, all regions of the model may be altered during inversion to

allow for uncertainty in the initial estimate of top of salt. The model displayed in Figure

3.7(a) presents the values for each cell of the model, averaged over the entire starting

42

population. Since each model is initialized with random zeros and ones for all cells in the

lower portion of salt, the average values appear as shades of gray, between zero (black),

and one (white). When the models are displayed in this manner, the fluctuation of gray

may be viewed as an expression of entropy of the population, such as described by

Rubinstein and Kroese (2004), with higher entropy in the lower portion representing

increased disorder.

Generation Number

Obj

ectiv

e V

alue

0 50 100 150 200 250 300 350101

102

103

104

Best IndividualAverage of Population

Generation Number

Obj

ectiv

e V

alue

0 50 100 150 200 250 300 350101

102

103

104


Figure 3.6. Progress of the genetic algorithm for the 2½-D binary inverse problem of the salt body with a single density contrast. The objective value of the highest-ranking individual at each generation are represented by the points, and the average of the population is represented by the line. Although the curve still contains a shallow slope, the top ranked and average solutions have mostly converged, and the population has evolved to a similar solution as illustrated in Figure 3.8.

43

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

z (m

eter

s)

True Model

x (meters)

Average of Models from Starting Population: Top-of Salt Added

0

0

1000

2000

3000

40000 2000 4000 6000 8000 10000 12000

x (meters)

z (m

eter

s)

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

z (m

eter

s)

True Model

x (meters)

Average of Models from Starting Population: Top-of Salt Added

0

0

1000

2000

3000

40000 2000 4000 6000 8000 10000 12000

x (meters)

z (m

eter

s)

Figure 3.7. View of the starting population with the addition of prior information. (a) is the average of the starting population of the GA. Prior information is incorporated in the form of top portion of salt. (b) is the true model I attempt to recover.

a)

b)

44

Mapping the evolution of the models over time is a valuable tool for

understanding the evolutionary progress of the models during inversion. This feature

also allows one to stop the process if one feels inversion is moving in the wrong

direction, as is often the practice in evolutionary computing. Figure 3.8 displays the

average of the population, or average of the models, as the GA progresses. There are two

dominant trends apparent in the model evolution. First, visible by the 35th generation,

entropy within the lower region of the models has decreased as inversion attempts to

minimize structural complexity of the model and constructs base of salt. Second, the

steeply dipping slope on the left portion of the model starts to fill in by the 103rd

generation. By generation 273, all individuals in the population have mostly converged

to a similar solution, with only slight differences visible as gray cells throughout the

model.

The final results are presented in Figure 3.9. The top panel (a) illustrates the

average solution over the entire final population. While the average model may not be

the best solution and, in fact, the average does not correspond to any possible solution in

my binary inversion, this method of display illustrates common features present in the

population of solutions. Regions of solid black or solid white illustrate agreement for

location of sediment or salt respectively, while regions of gray indicate variance among

the final population of solution. The final result illustrates that binary inversion has

successfully solved for the lower portion of salt in this problem. The steep dipping

45

structure at the left of the model has been successfully filled in, and the gray region of

disorder in the starting models beneath top of salt has been filled in with sediment

(represented by 0’s). Isolated cells of white and gray throughout the model region

represent attempts to either fit noise in the data, or to compensate at depth for increased

shallow salt while maintaining an appropriate data fit.

Generation 1 Generation 35 Generation 69

Generation 171Generation 137Generation 103


Figure 3.8. Model evolution during inversion. Each image is an average of the entire population at the specified generation. The upper left model is at generation 1 and the lower right is at generation 273. By generation 103, the steep dipping flank of the salt body has started to form and the disorder beneath top of salt has decreased significantly.

46

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

x (meters)

z (m

eter

s)

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

True Model

x (meters)

z (m

eter

s)

Constructed Model

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

x (meters)

z (m

eter

s)

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

True Model

x (meters)

z (m

eter

s)

Constructed Model

Figure 3.9. Comparison between the inverted and true model. (a) Final constructed model. The image is the average over the final population. (b) Original model to be recovered.

a)

b)

47

3.2.2.2. Salt Body with Density Contrast Reversal

The last example of binary inversion with straight GA utilizes the same cross-

section through the SEG/EAGE salt model as with the previous example. The model

however is more complex for this example: there are a larger number of parameters since

I use a finer model mesh, and a continuous density profile is incorporated to generate a

thick nil-zone. As density of host sediment increases with depth and salt density remains

constant, the top portion of the salt body attains a positive density contrast, the bottom

has a negative contrast, and a nil zone is present around 2000 m depth, Figure 3.10(a).

This type of problem is the motivation for development of binary inversion. The same

2D section is also presented in binary form in the lower panel, Figure 3.10(b).

The model contains 5,670 cells, each 100m by 100m in dimension. 41 gravity

data for the binary inversion are generated over the model perpendicular to strike. Noise

with zero mean and a standard deviation of 0.025 mGal have been added to the data,

Figure 3.11. As with the previous example, I incorporate top of salt as prior information,

along with the expected density contrast function. This is the similar information

incorporated into non-linear, interface inversion algorithms for gravity method (e.g.

Cheng, 2003). The same model, density profile, and data are utilized in the following

section on quenched simulated annealing and in Chapter 4 for evaluation of a hybrid

algorithm for binary inversion.

48

-0.1 -0.05 0 0.05 0.1 0.15

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

x (meters)

z (m

eter

s)

True Model: Binary Form

x (meters)

z (m

eter

s)

True Model: Density Contrast Form

∆ρ (g/cm3)

-0.1 -0.05 0 0.05 0.1 0.15

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

x (meters)

z (m

eter

s)


x (meters)

z (m

eter

s)


∆ρ (g/cm3)

Figure 3.10. 2½-D section through the SEG/EAGE Salt Model in density contrast form (a), and in binary form (b). A nil zone of zero density contrast cuts horizontally through the middle of the density contrast model. The binary model is represented by background sediment in black (zeros) and salt in white (ones).

a)

b)

49

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

-13000 7000 27000

0

0.5

1

-0.1 -0.05 0 0.05 0.1 0.15

OriginalNoisy

z (m

eter

s)

2½-D Salt Model with Density Profile & Nil-zone


x (meters)

g z(m

Gal

)

x (meters)

∆ρ (g/cm3)

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

-13000 7000 27000

0

0.5

1

-0.1 -0.05 0 0.05 0.1 0.15

OriginalNoisy

z (m

eter

s)

2½-D Salt Model with Density Profile & Nil-zone


x (meters)

g z(m

Gal

)

x (meters)

∆ρ (g/cm3)

Figure 3.11. 2½-D density contrast model from the converted SEG/EAGE salt model with density profile and nil-zone (a). The nil-zone is centered around an approximate depth of 2,000 meters. Panel (b) displays the true data (line) and noise-contaminated data (points). Data exhibit a zero crossing due to density contrast reversal.

a)

b)

50

The GA contains a constant population of 1000 individuals for this problem.

Convergence of the highest ranking and average solutions is reached by 500 generations.

The inversion result, Figure 3.12, is presented as an average of the final population of

solutions. As with the previous example, the average of a population of binary models

does not translate to a real solution; however, the image illustrates features of

commonality in each model of the GA population, and parameters that vary from solution

to solution. The deep structure to the left of the model has been successfully filled in

with salt, and the remainder of the model region has been reduced to zero, representing

sediment, while achieving an expected data fit. Results, therefore, illustrates that binary

inversion can adequately solve for base of salt in the presence of density contrast

reversal.

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

z (m

eter

s)

x (meters)0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

z (m

eter

s)

x (meters) Figure 3.12. Inversion result for the 2½-D problem using GA. The model contains 5,670 cells. There is a density profile and thick nil-zone. This result is presented as an average of GA population of 1000 individuals. Therefore, regions of solid black or white are features of sediment or salt, respectively, which all members of the final population share.

51

3. 3. Quenched Simulated Annealing

Simulated Annealing (SA) is another global search technique well suited for

gravity inversion with binary variable. SA formulation is designed to mimic the process

of chemical annealing, where the final energy state of a crystal lattice is determined by its

rate of cooling through the melting point. To achieve a lower energy state with highly

ordered crystals, the material is usually cooled slowly. When the material is cooled too

rapidly, the lattice may not reach the lowest possible energy state. The analogy in an

inversion for this latter case is pre-mature convergence where the final solution has

missed the desired global minimum. For geophysical inversion, the SA typically starts

with a model at random, and calculates the model’s energy based on its objective value,

eq.(2.1). Perturbations are then applied to the model and the new objective values are

calculated at each iteration. If the new objective value decreases or remains the same, the

model is accepted as a replacement. If the objective value increases, the model is

accepted by a thermally controlled probability function often referred to as the Metropolis

criterion:

⎟⎠⎞

⎜⎝⎛ ∆−=

TP φexp , (3.1)

where φ∆ is the difference between objective values of the old and new models, and T is

52

a temperature parameter designed to decay (or cool) over time. For more information on

SA, the reader is referred to Metropolis et al. (1953), Kirkpatrick et al. (1983), and

Nulton and Salamon (1988). Applications of SA specific to geophysical inversion, as

well as great sources of information on SA, are available in Sen and Stoffa (1995),

Nagihara and Hall (2001), Scales et al. (1992), Sambridge and Mosegaard (2002), and

Roy et al. (2005).

Similar to GA, simulated annealing is well suited for the binary inverse problem.

This is because perturbations to the geophysical model require a mere binary flip from 1

to 0, or vice versa. If a parameter is selected by the SA to perturb the model, there is only

one change that can occur. Magnitude of the change is not relevant with the binary

inverse formulation. Forward calculation of the gravity response is therefore rapid, with

the contribution of the flipped cell either added or subtracted from predicted data based

on the new value of 0 or 1. With a temperature controlled cooling function, SA works as

a global search technique. As a result, this places SA in a similar category with GA as a

solver for binary inversion. Results and processing time have likewise proven to be

similar for both GA and SA.

I desire an alternative method capable of working with the binary formulation. I

have opted to use a modified SA, called Quenched Simulated Annealing (QSA), as a

local search tool for this. QSA in its simplest form is Simulated Annealing described

53

previously, without the Metropolis or any other cooling criteria such as eq.(3.1). The

algorithm works as follows. After a change is performed to a model by QSA, i.e. a cell is

flipped from 1 to 0 or vice versa for my binary problem, the objective value of the new

solution is calculated through eq.(2.1). If the change decreases the total objective value

of the model in eq.(2.1), or makes no change to the objective value, the perturbation is

accepted. Therefore the algorithm only accepts downhill and lateral moves.

3.3.1. Numerical Example

To illustrate the application of quenched simulated annealing for binary inversion,

I test it on the same model and data from the previous example under GA. The model,

illustrated in density contrast form, is presented in Figure 3.13(a). Density of host

sediment increases with depth while the salt density remains constant. As a result, the

top portion of the salt body attains a positive density contrast, the bottom has a negative

contrast, and a nil zone is present around 2000 m depth.

The model region is divided into 5,670 cells. The dimensions of each cell are

100m by 100m. The data used here for binary inversion with QSA are the same 41 noisy

data presented in Figure 3.11. The noise added has a zero mean and standard deviation of

54

0.025 mGal. As with the examples for GA, I incorporate top of salt as prior information,

along with the expected density contrast function.

To apply QSA for binary inversion here, the algorithm was run for 75,000

iterations, rejecting moves which would increase the model’s objective value. The choice

of regularization for this problem is selected such that the data-misfit is equal to the

number of data. Details on choice of regularization are presented in Chapter 5:

Regularization and weighting parameters in binary inversion.

The final result from QSA is presented in Figure 3.13(b). The image is generated,

not from a single inversion, but from 50 separate inversions. The image therefore

represents an average over the 50 solutions. As with presentation of the GA solution

averaged over the entire population, the average of a set of inversions with my binary

formulation with QSA does not translate to any real solution in density contrast form.

Each binary solution allows it’s parameters to take on either salt or sediment, with no

values in-between. However, presenting the average from multiple QSA inversions

allows one to gain a basic understanding of which model features are present in each

inversion. The advantages of running multiple inversions with QSA are likewise

discussed in more detail in Chapter 6. The solution presented in Figure 3.13(b) is a good

representation of the true model. QSA has performed well with binary inversion for this

55

problem. In addition, QSA has illustrated, as with GA that it can solve for complex

shapes, incorporate density and other prior information such as top of salt, as well as

provide sharp contacts in the sub-surface.

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

z (m

eter

s)

x (meters)

-0.1 -0.05 0 0.05 0.1 0.15

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

x (meters)

z (m

eter

s)

Constructed Model


∆ρ (g/cm3)

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

z (m

eter

s)

x (meters)0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

z (m

eter

s)

x (meters)

-0.1 -0.05 0 0.05 0.1 0.15

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

x (meters)

z (m

eter

s)

Constructed Model


∆ρ (g/cm3)

Figure 3.13. Inversion result for the 2½-D problem using QSA. The model contains 5,670 cells. There is a density profile and thick nil-zone. Panel (a) illustrates the true model in density contrast form. Panel (b) illustrates the result as an average of 50 inversions with QSA. Therefore, regions of solid black or white are features of sediment or salt, respectively, which all the predicted models share.

a)

b)

56

3.4. Summary

In this chapter, I illustrate two solution strategies for inversion of gravity data

using the binary formulation introduced in Chapter 2. They are quenched simulated

annealing (QSA) and genetic algorithm (GA). Each proves effective as a solver for

binary inversion, generating consistent results between them. In addition, the methods

illustrate the effectiveness of binary inversion for the salt imaging problem in the

presence of a nil-zone and density contrast reversal. Likewise, density profile, top of salt,

and other prior geologic information can be incorporated into both GA and QSA with

relative ease. This is necessary for any solver of binary inversion, as with most interface

inversions.

The limitations of working with GA and QSA for binary inversion is in the

efficiency of GA, and the limited number of solutions and prior information available in

QSA. GA requires many forward calculations at each iteration, and therefore requires

large processing times as the scale of the problem increases. QSA is much faster in

generating solution than GA. However, because the method only works with individual

models at a time, smaller amounts of prior geologic information can be incorporated into

the binary inversion. In addition, the level of information from a single QSA inversion is

limited. Several inversions with QSA are therefore desired, which raises the processing

time to similar levels as GA.

57

CHAPTER 4: HYBRID ALGORITHM

This chapter introduces a hybrid optimization algorithm as an alternative solution

technique for binary inversion. The hybrid algorithm combines Genetic Algorithm with

Quenched Simulated Annealing. The former allows for easy incorporation of prior

geologic information, large number of solutions, and rapid build-up of larger model

structure, while the later guides the Genetic Algorithm to faster solution by rapidly

adjusting the finer model features. In this chapter, I discuss advantages of a hybrid

algorithm as an additional solution strategy for binary inversion, and illustrate its

improved efficiency in comparison to stand-alone Genetic Algorithm. The algorithm is

then applied to binary inversion of gravity data for a 3D salt problem with complex shape

and a large number of parameters.

4.1. Motivation for a Hybrid Algorithm

As described in Chapter 2, genetic algorithm (GA) and quenched simulated

annealing (QSA) generate solutions to the inverse problem by utilizing information about

the objective function directly rather than using derivative information. Because of this,

58

they are among a small list of techniques available for the binary inverse problem. There

are features of GA and QSA, however, which one might consider undesirable, or might

want to improve upon in choosing a minimization tool for binary inversion. In this

section, I therefore discuss some of these features in order to identify which aspects of

GA and QSA one might want to capture in a hybrid algorithm, and which aspects one

might want to eliminate.

Advantages and disadvantages of GA

There are two primary advantages of the GA for binary inversion. First, the GA

is ideally suited for minimization with the binary variable/model τ, composed entirely of

zeros and ones. Cells within a geophysical model translate directly to a string of

chromosomes for combination and mutation, and are more manageable as an array of

zeros and ones. For example, initialization and mutation by GA are straightforward,

where only values of 0 and 1 are assigned to the models during the former, and binary

flips from 1 to 0 or vice versa are performed by the later. Lack of magnitude in model

perturbation during these stages of GA make it more efficient than with continuous

variable problems. The second advantage of GA for binary inversion is its ability to

work with multiple models at one time through the formation of a population, which

allows the user to easily incorporate prior geologic information. This information may be

in the form of models generated from previous inversions, or top of salt from pre-stack

depth migrated seismic data.

59

The difficulty of working with GA for large inverse problems is in the forward

calculation of data at each generation. Because the GA works with a large population of

solutions simultaneously, it may require hundreds to thousands of forward calculations at

each iteration. In addition, the efficiency problem is compounded for larger inverse

problems because increased number of parameters typically requires a larger population

of solutions and an increased number of generations. For this reason, I find GA may not

be a reasonable solution strategy for binary inversion beyond simple 2D geophysical

problems with more than a few thousand parameters. It should be mentioned, however,

that literature on application of GA, over time, tend to increase this number as computing

technology progresses. It is therefore possible that GA will one day be capable of

efficiently working with such large real world geophysical inverse problems.

Advantages and disadvantages of QSA

Similar to GA, QSA is well suited for the binary inverse problem. Perturbation to

the geophysical model requires a mere binary flip from 1 to 0, or vice versa. Forward

calculation of the gravity response is therefore rapid, with the contribution of the flipped

cell either added or subtracted based on the new value of 0 or 1. Without a temperature

controlled cooling function as with SA, QSA also works as a local search technique,

providing an alternative solution strategy to binary inversion over GA.

60

There are two principle disadvantages of working with QSA over GA which one

may want to improve upon. The first is quality of information in the final inverse

solution, and the second is limitation on quantity of prior geologic information during

inversion. Both stem from the methods inability to work with multiple models during

inversion.

The final solution presented by QSA is a single model, with no statistical

information or visual understanding of features present throughout the model region.

Multiple inversions may be performed to generate such information. However, as the

number of inversions increases for this purpose, the total processing time approaches that

of GA solution, especially for large geophysical inverse problems. In parallel to this

issue, QSA has limited ability to incorporate prior geologic information into the

inversion. It can, in practice, only work with a single model from prior inversions, unlike

GA, which has the ability to incorporate unlimited number of prior inverse solutions and

geologic models into the population. For these reasons, one may wish to generate an

alternative solver for binary inversion, which combines the speed of QSA with the

advantages of a GA population. I next discuss development of such a hybrid technique.

61

4.2. Design of the Hybrid Algorithm for Binary Inversion

To improve computational efficiency over stand-along GA, and to generate better

solution information over QSA for large inverse problems, I present an additional

solution method to the binary inverse problem. I employ a hybrid algorithm which

combines the GA with QSA. Components and application of stand-alone GA and QSA

are detailed in Chapter 3, and as such are not repeated here. This section outlines the

design of the hybrid algorithm and it’s affect on the inverse solution over time.

Background

Hybrid optimization algorithms are typically designed as a means of capturing

desired characteristics from different minimization techniques. The motivation may be to

generate a more efficient algorithm, to capture broader frequency information, or to

provide a balance of one method’s speed with another’s ability to incorporate prior

geologic information. Often times, this is performed by designing an algorithm which

blends together both global and local search techniques.

There are a number of excellent publications on the application and cost

advantages of hybrid optimization for geophysical inversion. For example, Cary and

Chapman (1988) use a Monte Carlo method combined with a gradient algorithm to obtain

low and high frequency information, respectively, in seismic waveform inversion. Stork

62

and Kusuma (1992) apply wave form steepest descent to the initial population of a

genetic algorithm, as well as intermittently throughout the GA process for the residual

statics problem. A year later, Porsani et al. (1993) increased the activation interval of

local search over Stork and Kusuma to every generation of the GA, applied to the top-fit

individual only, for seismic waveform inversion. Chunduru et al. (1997), as well as the

Ph.D. dissertation of Chunduru (1996), evaluate performance of several hybrid

algorithms, and successfully illustrate application and cost advantage of the algorithms

for inversion of electrical and seismic data. For my binary problem, I develop a hybrid

algorithm that combines genetic algorithm with quenched simulated annealing.

Hybrid Formulation

The motivation behind my hybrid algorithm is to develop an efficient, although

not necessarily optimal (Chunduru, 1996), technique for working with my binary inverse

formulation. In addition, the method must be able to work with binary variable, and have

the ability to incorporate density and other prior geologic information into the inversion.

As described in Chapter 3, both GA and QSA are capable of working with binary

variable and can manage prior geologic information; however, the methods by

themselves, as described in the previous section, contain undesirable features which one

may wish to improve upon.

I therefore apply a hybrid algorithm in the fashion of Porsani et al. (1993) by

63

implementing local search to the top-fit individual of a GA (global) population of

solutions. However, in my implementation, I do not activate QSA for local search at

every generation. Rather, the GA is provided several generations to evolve a population

of solutions, and QSA is implemented only intermittently. The frequency of QSA

activation throughout the larger genetic algorithm varies based on the problem, i.e.

number of parameters. My goal in incorporating QSA, however, is to merely speed up

the evolution of a large GA population of solutions. The equivalent of implementing

such a hybrid is application of a pure GA for binary inversion, with a rapid mutation

operator applied to one individual every couple generations by QSA.

The effects of the hybrid algorithm on model structure during binary inversion, as

a result, are twofold. First, the GA efficiently develops large subsurface structure.

Second, complementary to GA, QSA is observed to improve the top fit individual after a

prescribed number of generations by adjusting the finer details of the solution. This is

similar observation to that presented by Cary and Chapman (1988). Since the highest fit

individual in the GA population has a strong influence on the evolution of the entire

population, the “evolutionary jumps” provided by QSA result in faster convergence of a

large number of solutions. I next illustrate implementation and results of binary inversion

with the hybrid algorithm, and compare performance to that of GA.

64

4.3. Performance of the Hybrid Algorithm

To illustrate the performance of the hybrid algorithm, I apply it to an example that

has the characteristics of a difficult salt imaging problem: i.e. complex density profile,

irregular shape of anomaly source, and a large number of parameters. A 2½-D model is

again extracted from a section through the SEG/EAGE salt model of Aminzadeh et al.

(1997). Density of the surrounding sediment increases with depth while that of the salt

body remains constant. As a result, the top portion of the salt body attains a positive

density contrast, the bottom has a negative contrast, and a nil zone is present around

2000-m depth, Figure 4.1(a). The same 2½-D section is also presented in binary form,

Figure 4.1(b).

There are 5670 cells in the model and 41 noisy data with zero mean and 0.025

mGal standard deviation are simulated in profile above the model. To carry out inversion,

I incorporate the top of salt as prior information, along with the expected density contrast

function. The data are then inverted with the binary formulation by both GA and the

hybrid algorithm to compare performance. The two methods will solve for the shape of

the lower portion of the salt body.

Before applying the hybrid to this problem, a stand-alone GA is first implemented

with a population size of 1000 individuals. The performance from this GA inversion will

65

be compared to that of the hybrid. Half the GA population are advanced at each

generation, while the other half is created through selection, cross-over, and mutation as

described in Chapter 3. Regularization is chosen by discrepancy principle such that the

final data-misfit in eq.(2.2) must equal the number of data. Additional details on the level

of regularization chosen for binary inversion are discussed in greater detail in Chapter 5:

Regularization and weighting parameters in binary inversion.

-0.1 -0.05 0 0.05 0.1 0.15

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

x (meters)

z (m

eter

s)


x (meters)

z (m

eter

s)


∆ρ (g/cm3)

-0.1 -0.05 0 0.05 0.1 0.15

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

x (meters)

z (m

eter

s)


x (meters)

z (m

eter

s)


∆ρ (g/cm3)

Figure 4.1. 2½-D section through the SEG/EAGE Salt Model in density contrast form (a), and in binary form (b). A nil zone of zero density contrast cuts horizontally through the middle of the salt model. The binary model is represented by background sediment in black (zeros) and salt in white (ones).

66

The GA reaches its final solution at approximately 500 generations. At this stage,

the highest fit and average objective values of the population have converged, as

illustrated in Figure 4.2(a). The feature of interest in the GA convergence plot, which is

representative of the GA’s performance, is the close spacing between the top fit

individual and average of the population. There are little differences between the

solutions at each generation, and therefore evolution of the population takes 500

generations and a total CPU time of 5 hours on a 2.4-GHz PC.

I next implement the hybrid to illustrate its performance in comparison to the GA

solution described above. For this problem, the GA/QSA hybrid is initialized with a

population size of 100 individuals. All other GA parameters, such as selection, cross-

over, and mutation are the same as with the previous example. QSA is incorporated into

the inversion every 5 generations of the GA, acting on the top-fit solution for 5,000

iterations. The goal in choosing frequency and number of QSA iterations is to allow the

GA time to evolve a population of solutions while rapidly mutating a single individual. I

desire to have neither long processing time due to GA evolution, or rapid solution of a

single model by QSA. The hybrid is implemented to balance these two.

67

Generation Number

Obj

ectiv

e V

alue

0 100 200 300 400 500 600102

103

104

105


Performance Plot: Genetic Algorithm

Generation Number

Obj

ectiv

e V

alue

0 100 200 300 400 500 600102

103

104

105


Generation Number

Obj

ectiv

e V

alue

0 100 200 300 400 500 600102

103

104

105


Performance Plot: Genetic Algorithm

Generation Number

Obj

ectiv

e V

alue

0 10 20 30 40 50 60101

102

103

104

105


Performance Plot: GA/QSA Hybrid

Generation Number

Obj

ectiv

e V

alue

0 10 20 30 40 50 60101

102

103

104

105


Generation Number

Obj

ectiv

e V

alue

0 10 20 30 40 50 60101

102

103

104

105


Performance Plot: GA/QSA Hybrid

Figure 4.2. Comparison of inversion performance between stand-alone GA (a), and the GA/QSA Hybrid (b) for the 2½-D problem. The hybrid algorithm converges to a solution in 50 GA generations with evolutionary jumps due to QSA every 5 generations.

a)

b)

68

Convergence of the population of solutions by the hybrid algorithm is illustrated

in Figure 4.2(b), below that of stand-alone GA (a). The population converge to similar

solution by 50 GA generations, versus 500 for pure GA. The total CPU time is

approximately 4 minutes on the same PC.

The observation is that the hybrid algorithm reduces the need for a large GA

population with large genetic diversity, as well as reduces the number of generations

required for convergence. GA tends to dominate in the build-up of the larger model

features throughout the population, whereas QSA modifies the top-ranking individual by

rapidly developing the finer details in the model. The improvement achieved with the

top-ranking individual leads to faster evolution of the entire population through

evolutionary jumps in the hybrid algorithm. This is apparent in the performance plot in

Figure 4.2(b) with evolutionary jumps occurring every 5 generations due to QSA.

Figure 4.3 shows inversion results using the GA/QSA hybrid by model evolution,

i.e. solution as a function of generation. The images are presented as an average of the

population of 100 models at specified generations, similar to results presented in Chapter

3 for GA inversions. By the 50th generation of the hybrid, the solutions are good

representations of the true model. In contrast to continuous variable formulations,

inversion results with the binary constraint illustrate the technique’s ability to properly

image the salt body by filling in the nil-zone.

69







Figure 4.3. Model evolution during binary inversion with the GA/QSA Hybrid. Each image is an average of the entire population at the specified generation. The upper left model is at generation 1 and the lower right is generation 50. By generation 19, the steep dipping flank of the salt body has started to form and the disorder beneath top of salt has decreased significantly.

70

The desired aspects of the GA and QSA algorithms are successfully captured by

the hybrid algorithm, which is the motivation for developing this solution strategy. First,

the population of solutions allows one to incorporate larger amounts of prior geologic

information, such as previous inversions, top of salt, and density information. I note,

however, that no previous inversions were placed in the population for the above

simulation. Second, the final solution may be represented as geologic information shared

by multiple inverse models. Figure 4.3 represents an average of 100 final solutions

generated by the hybrid algorithm. The average, in itself, does not represent an actual

inverse model, but rather illustrates those parameters which are common or vary within

the 100 binary models. This is a feature not available for single QSA inversions. Last,

the algorithm has significantly decreased processing time over that of GA. For the

current salt example, the difference in total CPU time is minutes for the hybrid versus

hours for GA.

The hybrid algorithm has successfully solved the problem of salt imaging for

binary inversion of gravity data. The method demonstrates its ability to adequately

balance speed with quantity of information, providing an alternative solution strategy for

binary inversion over GA or QSA. I next demonstrate binary inversion on the full 3D

SEG/EAGE salt model (Aminzadeh et al, 1997) using the hybrid algorithm.

71

4.4. Application to Full 3D Binary Inversion

So far, I have focused on 2D examples for illustration in my efforts to develop a

robust and efficient binary inversion algorithm. However, the ultimate goal is to invert

gravity data in 3D. I now turn my attention to this goal. In principle, any numerical

algorithm for inversion developed in 2D can be applied in 3D. The challenge is the

computational cost. With pure GA, 3D binary inversion of gravity data would be

prohibitively expensive, even for a moderate-sized problem. The development of the

hybrid algorithm has overcome this limitation to a large extent. In the following, I apply

the binary inversion to a 3D problem derived by simplifying the SEG/EAGE salt model.

The model, Figure 4.4(a), was developed as a velocity model for seismic studies.

For purposes of gravity studies, I have converted the velocity model to a generic cell

based model and incorporated density information in place of velocity, Figure 4.4(b).

The density model contains 28,350 cells with dimensions of 300 X 300 X 300 m each.

As with the 2D problem in the previous section, I incorporate a background density

profile with sediment density increasing continuously with depth. Given such a profile,

the density contrast reverses sign from positive at the top of salt to negative at the bottom.

The nil-zone occurs at a depth of approximately 2000m. In response to the model,

surface gravity data contain components of both positive and negative anomalies. To

simulate the response of the density model, 441 gravity data are generated above the

72

model. Gaussian noise with zero mean and standard deviation of 0.1 mGal is added to

the data, Figure 4.5.

During binary inversion for the 3D salt model, evolution is performed by the GA

on a population of solutions, with local acceleration by QSA. At each generation, GA

cross-over, the step of generating new models is performed by cutting two parent models

into blocks of cells. These blocks each contain 18 cells, and there are 1,575 blocks

within each model (Figure 4.6(a-d)). Next, the blocks of cells are combined randomly to

generate the next generation of solution, Figure 4.6(e-f), from each parent pair. Mutation

is incorporated into the GA by allowing each cell within the offspring models to undergo

a binary flip with a probability of 1/3000. This corresponds to an approximate ten cell

mutation for each offspring model. QSA is applied to the hybrid algorithm every 25

generations of the GA on the top ranking model. At this stage, cells within the model are

flipped randomly for 20,000 iterations. Any binary flips that do not increase the

objective value of the model in eq.(2.1) are accepted. The effect of QSA on the GA

population, as described in the previous section, is to incorporate evolutionary jumps

periodically, thereby speeding up the inversion over stand-alone GA.

73

Figure 4.4. SEG/EAGE seismic velocity model. Panel (a) shows a 3D perspective view of the model. The salt model is converted into a generic model mesh (b), and then a background density profile is incorporated for gravity studies.

a)

b)

74

-10 -5 0 5 10 15 20 25

-10

-5

0

5

10

15

20

25

0

2

4

6

8

10

Nor

thin

g (k

m) g

z (mG

al)

Easting (km)-10 -5 0 5 10 15 20 25

-10

-5

0

5

10

15

20

25

0

2

4

6

8

10

Nor

thin

g (k

m) g

z (mG

al)

Easting (km) Figure 4.5. Synthetic data for the SEG/EAGE 3D Salt Model. Data set contains 441 data points. Gaussian noise is added with zero mean and standard deviation of 0.1 mGal.

75

Figure 4.6. Generic example of cross-over of two binary models in 3D by the Genetic Algorithm. The top figures (a & b) represent two models, the parent models, each with 28,350 cells. All the cells from (a) are white and all the cells from (b) are black in this example. The parent models are divided into 1,575 blocks, each block containing 18 cells (c & d). The blocks of cells from each parent pair are re-combined randomly to form two children models (e & f). Each child model has opposite combination of block assemblage as the other child. For the actual inverse problem, cells within each block will not necessarily have the same vales throughout the block.

76

Initialization of the hybrid algorithm is performed by generating a starting

population of 201 models, with each model composed of uncorrelated random zeros and

ones, Figure 4.7. Next, top of salt is incorporated into the population as prior information

to guide the inversion, Figure 4.8. Top of salt may be forced to remain constant

throughout inversion, or may be allowed to change as preferred. For this problem, top of

salt is permitted to change by both GA mutation and QSA according to the conditions

described above.

13500 m 4200

m

1350

0 m13500 m 4200

m

1350

0 m

Figure 4.7. Sample starting model in the GA population initialized with random zeros and ones. There are 28,350 cells in the above model region, with 201 similar starting models incorporated into the GA/QSA for binary inversion.

77

The hybrid solutions converge by 300 generations with a processing time of

approximately forty-five minutes. Figure 4.9 illustrates the performance of the

population of models at each generation of the GA, with the highest fit model (lowest

objective value) undergoing evolutionary jumps every twenty-five generations due to

QSA. Consistent with the convergence plot for the 2D example, Figure 4.2(b), the best

fit individual here changes little by GA in-between each QSA iteration. In contrast, the

average of the population changes drastically during these intervals by GA evolution. An

exception is noted between the 75th and 100th generations. The top model undergoes

13500 m

4200

m

13500 m13500 m

4200

m

13500 m

Figure 4.8. Top of Salt added as prior information. The model is an average of the population of models; therefore, the grey region beneath top of salt is an average of random zeros and ones.

78

minimization due to GA similar to that of the rest of the population. Regularization for

eq.(2.1) is chosen to minimize model size and structure, eq.(2.3), subject to fitting the

data to the appropriate degree, eq.(2.2). Detailed information on selection of

regularization for binary inversion is provided in Chapter 5.

Results of the 3D inversion are presented in Figure 4.10. The top image shows

the true 3D salt model, and the bottom image is the model recovered by the hybrid

algorithm. Top of salt, while incorporated as a starting guide, is not held constant, and

therefore slight changes to the upper portion of the salt body are observed. Locations of

structure beneath top of salt, as well as depths to base-of-salt, correlate closely with the

true model. Individual cells throughout the model region, which indicate isolated cells of

salt, are attempts to fit noise during inversion.

The GA/QSA hybrid algorithm has been successfully applied to binary inversion

of the full 3D SEG/EAGE Salt Model. Results indicated that the formulation has

adequately resolved structure beneath top-of-salt and maintained appropriated depths to

base-of-salt. The formulation has the ability to easily incorporate density information,

and therefore can resolve base of salt in the presence of density contrast reversal.

79

0 50 100 150 200 250 300 350103

104

105

106

107

Top IndividualAverage of Population

Figure 4.9. Performance plot of the 3D binary inversion problem. The black points are the objective values of the highest fit model at each generation, and the blue points are the average objective values of the GA population at each generation.

80

13500 m

4200

m

13500 m

True model

13500 m

4200

m

13500 m

True model

13500 m

4200

m

13500 m

Constructed model

13500 m

4200

m

13500 m

Constructed model

Figure 4.10. True and constructed model with binary inversion. The top figure (a) is the true model which is to be reconstructed by the binary inversion algorithm. The bottom figure (b) is the constructed model by the GA/QSA Hybrid with binary inversion.

a)

b)

81

4.5. Summary

I this chapter, I have developed a hybrid optimization algorithm as an alternative

solution strategy for inversion of gravity data using the binary formulation presented in

Chapter 2. The hybrid algorithm, utilizing the Genetic Algorithm and Quenched

Simulated Annealing, has significantly decreased computational cost over stand-alone

GA. Likewise, the algorithm can incorporate larger quantities of prior information, and

can present more valuable information in the final solution over QSA due to the large

population of models. The hybrid algorithm has also proven feasible for tackling binary

inversions for realistic 3D salt problems with complex background density profiles,

density contrast reversal, nil-zones, and a large number of parameters.

In the next chapter, I discuss methods for choosing regularization in binary

inversion and illustrate the effects of regularization in comparison to continuous variable

inversions. Also, I discuss the role of the weighting parameters in the model objective

function, eq.(2.3), and illustrate their effects on the final binary inverse solution when a

nil zone exists.

82

CHAPTER 5: REGULARIZATION AND WEIGHTING PARAMETTERS IN

BINARY INVERSION

In this chapter, I explore regularization and the weighting parameters for the

binary inverse problem. There are four basic components to the chapter. First I discuss

the role of regularization for continuous variable formulations, and illustrate the

similarities with and differences from that of binary inversion. Second, two methods for

constructing a Tikhonov curve are analyzed to illustrate the advantages and

disadvantages of each technique for binary inversion. Third, I apply two approaches for

choice of regularization to the binary inverse problem: (1) discrepancy principle and (2)

L-curve criterion, and discuss the associated issue. Last, I explore the weighting

parameters of binary inversion and illustrate their effects on the final model solution.

5.1. Role of Regularization in Continuous-Variable Inversions

Tikhonov regularization (Tikhonov and Arsenin, 1977) plays an important role in

continuous-variable inversion of geophysical data. We deal with a finite number of

inaccurate data, and attempt to construct a function that has many more degrees of

83

freedom. Consequently, if there is one model that fits the observed data, there must be

infinitely many that will fit the data equally well. In order to overcome this non-

uniqueness, Tikhonov regularization imposes certain conditions on the complexity of the

model to restrict the class of admissible solutions and thus allows one to select, from the

multitude of models fitting the data, the ones that are simplest and geologically

interpretable. The underlying goal is to allow one the ability to eliminate models or

features that may be mathematically acceptable, but geologically or physically

unreasonable. For example, direct inversion of gravity and magnetic data may generate

solutions which concentrate density or susceptibility near the surface, in a single thin

layer (equivalent source layer), extremely oscillatory, or with unrealistic values.

Regularization therefore incorporates into the inversion some form of a model objective

function that penalizes such undesired features. For example, it can penalize the total

energy in the model, penalize the roughness of the model in the three spatial directions,

or incorporate a reference model that may be available independently from the data. One

type of commonly used model objective function has the form, also shown in eq.(2.3):

( ) [ ]( ) ( ) [ ]

( ) [ ] ( ) [ ] dvz

mmzwαdvy

mmzw

dvx

mmzwdvmmzw

Vz

Vy

Vx

Vm

∫∫

∫∫

⎟⎠⎞

⎜⎝⎛

∂−∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

−∂+

⎟⎠⎞

⎜⎝⎛

∂−∂+−=

20

20

202

0sα

vvvv

vvvv

α

αφ , (5.1)

84

where 0m allows for incorporation of a reference model, and the alphas are relative

weights. The first term in eq.(5.1) limits the size of the final solution, referred to as an

energy term, and the remaining derivative terms penalize for roughness, often referred to

as smoothing terms.

The use of a model objective function has been extended from its original purpose

of stabilizing the inverse solution to acting as a vehicle for incorporating different user

supplied information and assumptions about the model. For example, in gravity

inversion, the parameter )(zw in eq.(5.1) incorporates a depth weighting function.

Solution to an inversion by Tikhonov regularization then seeks to minimize the

model objective function, subject to fitting the data. The total objective function to be

minimized therefore consists of the weighted sum of a model objective function mφ and

data misfit dφ :

)()( min. d mm mβφφφ += . (5.2)

The parameter, β , is referred to as a Tikhonov regularization parameter, which acts to

balance the two components of the total objective function.

85

A curve formed by plotting the data misfit as a function of model-norm is

generated for the purpose of estimating one’s desired level of regularization. The curve,

Figure 5.1, is referred to as a Tikhonov curve, and is generally smooth and monotonic for

continuous variable inversions. Each point on the curve provides a measure of the

inverse model’s data-misfit ( dφ ) and structure ( mφ ) for every regularization parameter

evaluated.

The use of regularization and the Tikhonov curve for solving linear inverse

problems has been extensively studied for continuous-variable formulations (e.g. Lawson

and Hanson, 1974; Parker, 1994; Hansen, 1998). For inversions where data are

contaminated with uncorrelated Gaussian noise and zero mean, regularization parameter

β is chosen such that the expectation of data-misfit must equal to the number of data.

This is common for synthetic problems where one generates noise in their observed data.

In practice, however, noise in the observed data provided to us does not conform to

Gaussian statistics, and/or the standard deviations of the data are not generally available.

For these situations, regularization is often estimated through use of the Tikhonov curve

to find an acceptable balance between data-misfit and model objective value.

86

Model Objective Value: ϕm

Dat

a M

isfit

Val

ue: ϕ d

small β

large β


Dat

a M

isfit

Val

ue: ϕ d

small β

large β

Figure 5.1. Tikhonov curve with continuous variable formulation: The upper left region represents underfit solutions where slight increase in model structure greatly decreases the data misfit. The lower right region represents solutions with overfit data, where large increase in model structure results in little decrease in data misfit.

87

5.2. Role of Regularization in Binary Inversion

It is important to understand, first, whether the same condition holds true for

binary inversion, and second, whether a Tikhonov curve can be stably calculated using

Genetic Algorithm (GA) and/or Quenched Simulated Annealing (QSA), both derivative-

free minimization techniques incorporating random search. In addition, the

combinatorial nature of GA raises the additional concerns of repeatability and continuity

during generation of a Tikhonov curve. In the next section, I illustrate the role of

regularization for binary inversion, through use of GA and QSA, and discuss the

similarities and differences of their Tikhonov curve(s) to those of continuous-variable

formulations.

To investigate the role of regularization in binary inversion, I generate two

Tikhonov curves using Genetic Algorithm (GA) and Quenched Simulated Annealing

(QSA), respectively. While only one method is needed to determine if regularization

plays a similar role in binary inversion as continuous-variable inversion, and whether the

Tikhonov curves behave similarly, both methods are presented to illustrate which

technique (GA or QSA) allows for best estimate of regularization in binary inversion. In

the section after, I illustrate two methods for estimation of regularization: L-curve

criterion and discrepancy principle.

88

5.2.1. Tikhonov Curve by Genetic Algorithm

To illustrate the role of regularization in binary inversion first, and to determine if

genetic algorithm is an appropriate technique for estimating regularization with the

binary constraint, I use GA to invert the data from section 3.2.2.1 (Figure 3.5(b)). The

true model used to generate the data is the 2D section through the SEG/EAGE Salt model

with single density contrast, Figure 3.5(a).

An idealized Tikhonov curve, Figure 5.2, is generated by performing 27

inversions with regularization parameter β varying over nine orders of magnitude (10-10 –

10-1). Each point on the curve corresponds to the final data-misfit and model structure

when binary inversion is performed with a single regularization parameter. The curve is

idealized because the true model is inserted into the starting population of each run, so

one can understand how regularization affects the final solution; i.e. how the solution

pulls away from the true model as regularization changes. Note: insertion of the true

model is only performed here to present the smoothest possible curve by GA, and to

illustrate how the combinatorial nature of GA affects choice of regularization, even under

idealized (unrealistic) circumstances. This approach also enables me to construct a

Tikhonov curve with moderate computational cost.

89

108

102

103


4 x 107

Dat

a M

isfit

Val

ue: ϕ d

108

102

103


4 x 107

Dat

a M

isfit

Val

ue: ϕ d

Figure 5.2. Idealized Tikhonov curve generated from GA. Each regularization parameter is plotted by the final data misfit and model objective values. The true model is inserted in to the GA population to help understand this Tikhonov plot. Notice the curve is smoother at the lower right portion of the curve than at the upper left. This difference is primarily due to mutation and cross-over in the GA, not due to the binary constraint of the inversion.

90

The curve shows that the questions posed earlier have clear answers. First,

regularization does play a similar role with binary inversion as with continuous-variable

inversion, and the Tikhonov curve behaves in a similar manner. Secondly, the GA can

successfully construct a Tikhonov curve. However, the curve is monotonic at large

scales only and has noticeable local features that are non-monotonic. Furthermore, it is

not precisely repeatable due to the random nature and combinatorial features of GA, as

well as limited numerical precision in minimization. Details are presented in the

following.

Figure 5.2 displays the Tikhonov curve constructed by GA for the 2½-D salt

problem with a single density contrast. The plot show that resulting models increase in

structural complexity with over-fit data for low regularization values, corresponding to

solutions at the lower right end of the Tikhonov curve. Likewise, as β increases, the

solutions tend to under-fit the data while the resulting models are overly smooth,

corresponding to the upper left portion of the curve. This trend is similar to that of the

Tikhonov curve generated for a continuous-variable inversion, Figure 5.1.

However, the Tikhonov curve generated by GA is rough, non-monotonic, and not

precisely repeatable. To be more precise, the upper left portion of the curve turns rough

and non-monotonic, while the lower right portion is smoother and more similar to

continuous-variable Tikhonov curves. The presence of these slightly different behaviors

91

in my Tikhonov curve does not appear to be a result of the binary nature of the inverse

problem, but rather from the two primary components of GA, namely, cross-over and

mutation. To expand upon this, I first discuss the upper left portion of the curve. As

regularization increases, one moves up the curve towards poorer data-fit and simpler

model structure. This segment of the curve becomes rough and non-monotonic as cross-

over ignores data-fit, and combines large blocks to generate over-smoothed solutions.

The linking of large blocks of parameters from separate models does not appear capable

of generating a smooth curve over varying regularization values while ignoring data-fit.

The lower portion of the curve, in contrast, is smoother and more continuous as

regularization decreases. For inversions with this level of regularization, minimization

occurs through attempts to fit the noise in the data, while increasing model complexity.

This is achieved by the GA by slowly flipping parameters (0 or 1) throughout the model

region, fitting noise in the data at each generation. To generate this behavior and

maintain a smooth curve on the Tikhonov plot, mutation, not cross-over, appears as the

dominant component of GA minimization at these levels of regularization. Mutation

slowly adds or subtracts salt from the model to decrease the data-misfit function. Cross-

over of large blocks of cells, therefore, plays a decreased role at this end of the Tikhonov

curve.

92

5.2.2. Tikhonov Curve by Quenched Simulated Annealing

The previous section illustrates the role of regularization in binary inversion.

However, estimation of regularization for binary inversion will be more complicated than

continuous-variable inversions when GA is the tool of choice. This is due to the

combinatorial nature of GA, as discussed in the preceding section. Monotonic features

within the Tikhonov curve generated by GA are due primarily to mutation, not cross-

over. As a natural extension, QSA, with similar structure to the mutation component of

GA, should be a better choice for generation of a smooth Tikhonov curve for binary

inversion. In this section, I present the Tikhonov curve generated by QSA for binary

inversion and show that it is the preferred tool for estimating regularization in the binary

inverse problem.

To generate a Tikhonov curve with QSA, 18 inversions with regularization

parameter β varying over 10 orders of magnitude (10-6 to 104) are performed. The data

which are inverted are from the 2D section of the SEG/EAGE salt model with complex

density profile, section 3.2.2.2. The true model is illustrated in Figure 3.11(a). Unlike

the preceding section that applies GA for generating the Tikhonov curve, inversions with

QSA here do not incorporate the true model for generation of the Tikhonov curve.

93

The Tikhonov curve generated with QSA, Figure 5.3, supports the primary

conclusions obtained from application of GA to regularization for binary inversion:

predicted models with binary inversion increase in structural complexity with over-fit

data for low regularization values, and under-fit the data with overly smooth features at

larger values. However, it is apparent that the curve generated with QSA is more similar

to that predicted by continuous-variable inversions (Figure 5.1). The QSA Tikhonov

curve is smoother and more monotonic than that generated by GA, Figure 5.2. As a

result, while GA can be used in selection of regularization, Quenched Simulated

Annealing is the preferred tool for generating Tikhonov curves in binary inversion,

regardless of one’s choice for estimating the level of regularization, i.e. L-curve versus

discrepancy principle.

5.3. Choice of Regularization for Binary Inversion

In this section, I present two techniques for determining appropriate regularization

for binary inversion. The first assumes known standard deviations of the observed data,

and therefore chooses regularization so that the data-misfit is equal to the expected value

of the misfit function. Because this method specifies regularization based on an assumed

level of data-misfit, it is equivalent to having an estimate of the noise in the data prior to

the inversion. The technique proves valuable for synthetic problems, where one often has

94

an estimate of the noise in their data. This method is referred to as the discrepancy

principle (Parker, 1994; Hansen, 1998).

The second method, L-curve criterion, is often more practical for estimating

regularization in applied inverse problems. This is because, in practice, noise in the

observed data provided to us do not conform to Gaussian statistics, and/or the standard

deviations of the data are not generally available. As a result, an alternative method for

determining an appropriate level of regularization is necessary. This second method

chooses regularization corresponding to the ‘elbow’ of the Tikhonov curve, which is the

regularization value corresponding to the optimal tradeoff between data-misfit and model

complexity. This is called the L-curve criterion (Regińska, 1996; Hansen, 1998;

Johnston and Gulrajani, 2002), and the ‘elbow’ is defined as the point of maximum

curvature on the Tikhonov curve. In this section, I analyze the two methods, L-curve and

discrepancy principle, for estimation of regularization in the binary inverse problem.

95

102

102

103


Dat

a M

isfit

Val

ue: ϕ d

102

102

103


Dat

a M

isfit

Val

ue: ϕ d

Figure 5.3. Tikhonov curve generated from QSA. Each regularization parameter is plotted by the final data misfit and model objective values. The curve is much smoother than the one generated by GA, allowing for easier estimation of regularization.

96

5.3.1. L-Curve

The first method I explore for estimating regularization is L-curve criterion. As

described in the preceding section, this technique is often utilized when noise in the

observed data do not conform to Gaussian statistics, and/or the standard deviations of the

data are not available. This is often the case in practice, particularly for gravity data

gathered for the salt body problem.

The result of plotting the data-misfit (eq.(2.2)) versus model objective function

(eq(2.3)) is described as an L-curve because, on a log-log plot, it contains a sharp curve

connecting two straight regions. For solutions with large regularization parameters,

corresponding to the upper left portion of the L-curve, small changes in model structure

mφ of the final inverse solution tend to have large effects on data-misfit dφ . The final

inverse models tend to be overly simple with poor data fit for large regularization

parameters. In contrast, as the regularization parameter decreases, the final inverse

solution moves to the lower right region of the L-curve. In this region, large changes in

model structure mφ produce small changes in the data-misfit dφ for the final solution.

The final inverse models tend to be structurally complex with over-fit data for small

regularization parameters. The corner, often referred to as the ‘elbow’ or ‘knee’ of the L-

curve, and defined by the point of maximum curvature, is considered the optimal tradeoff

between data-misfit and model structure. At this location, regularization generates

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inverse models that reproduce the dominant trends of the data while limiting the effects

of noise in the data on model structure.

To illustrate L-curve for binary inversion, I present two Tikhonov curves: one by

GA and one by QSA. As I will demonstrate, the combinatorial nature of GA generates

sharp breaks in the L-curve, rendering calculation of the true corner of the curve

impossible. QSA on the other hand, has the ability to successfully generate a smooth,

monotonically decreasing L-curve.

I start by generating a Tikhonov curve, Figure 5.4, using GA as described in

section 3.2. This time the true model is not inserted into the population and I simulate a

practical scenario where we don’t know very much about the final solution nor do we

know much about the data errors. The L-curve in Figure 5.4 does not have a pronounced

‘elbow’, but the transition between the two regions is still clear. The corner region is

coarse and non-monotonic due to the combinatorial nature of the GA. Therefore, to

better understand the curve generated by GA, I present inversion results as a function of

regularization, along with the associated L-curve.

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10830

40

50

60

70

80

90

4 x 107

β ~ 2.07E-7


Dat

a M

isfit

Val

ue: ϕ d

10830

40

50

60

70

80

90

4 x 107

β ~ 2.07E-7


Dat

a M

isfit

Val

ue: ϕ d

Figure 5.4. Tikhonov curve generated from GA without true model inserted into the population. Each regularization parameter is plotted by the final data misfit and model objective values. Notice the curve is not as smooth as that illustrated in figure 6.2, where the true model is incorporated into the population. There is no clear definite ‘elbow’ in this instance. This difference illustrates the difficulty in estimating regularization based on L-curve with GA for binary inversion applied to real problems.

99

Figure 5.5 displays the constructed models by GA as a function of regularization

parameter. As predicted by the Tikhonov curve, the models have a trend from highly

complex for low regularization values at the top of Figure 5.5, to overly smooth models

for higher values near the bottom. The best representation of the true model for the

varying regularization parameters appears in Row 4, Column 3 of Figure 5.5. The result

for this regularization parameter is presented separately in Figure 5.6(a), along with the

true model (b). Although the elbow of the curve is not clearly defined using GA, the

solution corresponds to a point which one might approximate as the elbow of the L-

curve, Figure 5.4. Therefore, while GA may not be ideal for generating an L-curve, as

apparent by the non-monotonic nature of the curve, it does provide an adequate estimate

of regularization parameters.

Next, I demonstrate generation of an L-curve with QSA for binary inversion. For

this, I repeat the Tikhonov curve from section 5.2.2 below, Figure 5.7. Unlike the L-

curve generated with GA, the curve from QSA is relatively monotonic and has a well

defined corner. As a result, location of the elbow of the curve for estimating

regularization parameter with L-curve is straightforward using QSA.

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β = 7.8476 E -8 β = 1.2743 E -7 β = 2.0691 E -7

β = 3.3598 E -7 β = 5.4556 E -7 β = 8.8587 E -7

β = 1.4384 E -6 β = 2.3357 E -6 β = 3.7927 E -6

β = 1.0 E -9 β = 1.6238 E -9 β = 2.6367 E -9

β = 4.2813 E -9 β = 6.9519 E -9 β = 1.1288 E -8

β = 1.833 E -8 β = 2.9764 E -8 β = 4.8329 E -8

β = 7.8476 E -8 β = 1.2743 E -7 β = 2.0691 E -7

β = 3.3598 E -7 β = 5.4556 E -7 β = 8.8587 E -7

β = 1.4384 E -6 β = 2.3357 E -6 β = 3.7927 E -6

β = 1.0 E -9 β = 1.6238 E -9 β = 2.6367 E -9

β = 4.2813 E -9 β = 6.9519 E -9 β = 1.1288 E -8

β = 1.833 E -8 β = 2.9764 E -8 β = 4.8329 E -8

Figure 5.5. Comparison of inverted models for different regularization parameters using GA. Each panel displays the average of models in the final population for a given value of regularization, β. For small values (e.g., the top left panel), the model over-fits the data and is structurally complex. For large values (e.g., bottom right panel), the model fits the data very poorly and is structurally too simple. At intermediate value of 2.0691E-7, the data misfit is close to the expected value of 41 and the model has a reasonable amount of structure and provides a good representation of the true model.

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x (meters)

z (m

eter

s)

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

True Model

x (meters)

z (m

eter

s)

Inverse Model Predicted by Discrepancy Principle

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0

1000

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3000

4000

x (meters)

z (m

eter

s)

0 2000 4000 6000 8000 10000 12000

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4000

True Model

x (meters)

z (m

eter

s)

Inverse Model Predicted by Discrepancy Principle

0 2000 4000 6000 8000 10000 120000

0

1000

2000

3000

4000

Figure 5.6. Inverse model when regularization is chosen based on discrepancy principle (a). Due of the coarse nature of the Tikhonov curve in Figure 5.4., L-curve is precluded as a means for estimating regularization with GA. Panel (b) is the true model to be recovered. The true model was not inserted into the GA population for this inversion.

a)

b)

102

102

102

103


Dat

a M

isfit

Val

ue: ϕ d

102

102

103


Dat

a M

isfit

Val

ue: ϕ d

Figure 5.7. L-curve generated from QSA. Each regularization parameter is plotted by the final data misfit and model objective values. The curve is smooth and monotonic compared with that generated by GA, allowing for reasonable estimate of the ‘elbow’ of the plot for L-curve criterion.

103

Estimation of regularization parameter with L-curve criterion can work well for

binary inversion. This is important because, often, little to no information is available on

the noise in one’s data. However, use of L-curve in conjunction with Genetic Algorithm

to estimate an appropriate regularization parameter is not ideal for binary inversion. QSA

is the preferred method in this instance. I note, also, that the final solution in Figure 5.6

has a data-misfit value predicted by discrepancy principle, as well as a regularization

value near the elbow of the L-curve.

5.3.2. Discrepancy Principle

Next I discuss choice of regularization through use of the discrepancy principle.

This method works well for numerical studies, where one often generates the noise in

their data, and the statistical properties of the noise are therefore known. However, the

method does not often prove as valuable as L-curve for real problems, where little to no

information on errors in the data is available.

The diagonal matrix Wd in eq.(2.8) represents the reciprocal of the standard

deviations of the N observed data. Assuming Gaussian statistics, where the data are

contaminated with uncorrelated Gaussian noise with zero mean and known standard

deviations, the data-misfit defined by eq.(2.2) is a χ2 variable with N degrees of freedom

104

(Hansen, 1992). As a result, the expected level of regularization through discrepancy

principle is one which sets the data-misfit in the total objective function, eq.(2.1), equal to

the number of data N.

As mentioned in the preceding section, the estimate of regularization parameter,

which approximates the elbow of the L-curve in Figure 5.4, also generates a final inverse

solution by GA with data-misfit equal to the number of observations. This indicates that

regularization estimated from L-curve and discrepancy principle are similar for the 2D

salt problem with single density contrast. Therefore, I present here regularization

estimation with discrepancy principle using Quenched Simulated Annealing for binary

inversion. I start by performing 18 inversions with regularization values varying from

10-6 to 104 for the 2D salt model with complex density profile (section 3.3.1). The final

data-misfit values of the inversions are next plotted for each regularization value, Figure

5.8. Since the observed data are simulated for this problem with known standard

deviations, the desired regularization value, β ~ 15.5, is easily identified from Figure 5.8,

such that the data-misfit will equal the number of data, 41.

The final inversion result, illustrated as a mean of 100 inversions with this level of

regularization, Figure 5.9, is a good representation of the true model. Therefore,

choosing regularization based on data-misfit, i.e. discrepancy principle, is a valid method

for this numerical example. In addition, the smooth, monotonic curve generated by QSA

105

allows for simpler identification of regularization over GA based on expected data-misfit.

This does not, however, imply that discrepancy principle provides the best estimate of

regularization for real geologic problems, where noise in the data cannot be assumed to

have the same statistical properties as used here.

10-6

10-4

10-2

100

102

10410

1

102

103

Regularization Value: β

Regularization Value, β ~ 15.5

Data Misfit, Φd = 41

Dat

a M

isfit

Val

ue: ϕ d

10-6

10-4

10-2

100

102

10410

1

102

103

Regularization Value: β

Regularization Value, β ~ 15.5

Data Misfit, Φd = 41

Dat

a M

isfit

Val

ue: ϕ d

Figure 5.8. Plot generated by QSA of data-misfit versus regularization value for discrepancy principle. The desired misfit of 41, equal to the number of data, is obtained with regularization of approximately 15.5.

106

5.4. Effects of Weighting Parameters in Binary Inversion

The model objective function defined in eq.(2.3) has several weighting

parameters: sα , xα , yα , and zα . These parameters have a strong influence on the

character of the final solution, particularly when a salt body straddles a nil-zone. It is

therefore important to understand how these parameters affect the final solution of

models. This section demonstrates how one can change these parameters and how these

changes affect the final solution.

0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

z (m

eter

s)

x (meters)0 2000 4000 6000 8000 10000 12000

0

1000

2000

3000

4000

z (m

eter

s)

x (meters)

Figure 5.9. Inversion result by QSA when regularization is chosen using discrepancy principle. The result is presented as a mean of 100 binary inverse models.

107

To better understand the results, visually, I will work with a 2D section of the

SEG/EAGE salt model instead of the full 3D model. The model is that introduced in

section 3.2.2.2, containing 5,670 cells, density contrast reversal, and a thick nil-zone.

The model objective function, eq.(2.3), is therefore adjusted for the 2D problem to:

( ) [ ]( ) ( ) [ ] ( ) [ ] dvz

zwαdvx

zwdvzwV

zV

xV

m ∫∫∫ ⎟⎠⎞

⎜⎝⎛

∂−∂+⎟

⎠⎞

⎜⎝⎛

∂−∂+−=

20

202

0sατττταττφvvvv

vv (5.3)

and I now have only three weighting parameters to adjust: sα , xα , and zα . Inversions

are performed for these experiments with the GA/QSA hybrid algorithm presented in

Chapter 4. The hybrid contains a population of 100 individuals; therefore, the results

presented below are always an average of the 100 solutions. I present this section by

demonstrating varying combinations of the weighting parameters in eq.(5.3), and

illustrating their effects on the final inverse solutions when a nil-zone is present.

αs = 1, αx = αz = 2.5*105: Without further discussion of the weighting components in

eq.(5.1), a value of 2.5*105 for αx and αz is rather obscure. However, it suffices to state

that the quanity sx αα / defines a correlation length for the model in the x-direction.

The longer the correlation length, the smoother the model is. The above choice of the

coefficients corresponds to a correlation length of 5 cells in the x and z directions.

108

Model results for this set of weighting parameters are presented in Figure 5.10

with regularization values spanning several orders of magnitude. The purpose of

presenting the solutions for each regularization value is to illustrate the common features

present throughout each model, regardless of one’s choice of regularization.

There are two distinct trends present in the inversion models presented in Figure

5.10. First, the solutions vary from structurally complex models at low regularization

values to overly-smooth models at large values. This is expected. The second and more

important feature here is that the nil-zone is never filled in within the models, for any

level of regularization. This result exhibits the influence of the first term in eq.(5.3), the

energy term, which seeks to minimizing the size of the final model. Non-zero cells in the

nil-zone will not contribute to surface gravity data, and therefore will not affect data-

misfit, eq.(2.2). However, they would act to increase the size or energy of the solution,

eqs.(2.3 & 5.3). Minimization of the objective function therefore seeks a solution which

contains zero values within the nil-zone when an energy term is present in the inversion.

109

β = 2.2539 E -6 β= 1.145 E -5 β = 5.8171 E -5

β = 1.0 E -12 β = 5.0802 E -12 β = 2.5809 E -11

β = 1.3111 E -10 β = 6.6608 E -10 β = 3.3839 E -9

β = 1.7191 E -8 β = 8.7333 E -8 β = 4.4367 E -7

β = 2.2539 E -6 β= 1.145 E -5 β = 5.8171 E -5

β = 1.0 E -12 β = 5.0802 E -12 β = 2.5809 E -11

β = 1.3111 E -10 β = 6.6608 E -10 β = 3.3839 E -9

β = 1.7191 E -8 β = 8.7333 E -8 β = 4.4367 E -7

Figure 5.10. Inversion results with the energy term in the model objective function. Results are presented over a Tikhonov loop with varying regularization parameters. With the energy term (αs), there is a gap in the nil zone where salt should be present.

110

αs = αx = αz = 0: This combination of weighting parameters in the objective function is

equivalent to setting the regularization parameter to zero. Therefore, I have eliminated

the model objective function, eq.(5.3), and attempt to find the model or models that best

fit the data without placing constraints on model structure. This method is not

recommended for applied inversions, as it will produce overly complex models which

attempt to fit noise in the data. The combination is merely implemented here to add to

our overall understanding of the weighting parameters, and to illustrate how they affect

the final binary inverse solution with nil-zones.

The result for this set of weighting parameters is presented in Figure 5.11. As

with the results generated from the previous set of weighting parameters, there are two

important features present again. First, while inversion has identified the general region

of salt throughout the models, the solution is structurally complex. This is expected

without the model objective function, eq.(5.3). The second feature is the disagreement

among the resulting models within the nil-zone. Figure 5.11 presents the average of the

population of models, with white and black regions generally accepted as salt or sediment

respectively by all the solutions. However, the gray horizontal band across the middle of

the figure indicates that there is no bias toward either salt or sediment in a nil-zone when

the model objective function is absent.

111

Figure 5.11. Model result, averaged over the entire population of models, with no model objective function in the inversion. Without the m.o.f., the result is overly complex, and there is no agreement among the models within the nil zone, as indicated by the gray band across the middle.

112

Given the understanding of my inversion algorithm with respect to the nil-zone, it

is illustrative to next select a set of parameters for the inversion such that the nil-zone

might be naturally filled.

αs = 0, αx = αz = 1: With this set of weighting parameters, I remove the first component

of eq.(5.3), which seeks to minimize the size or energy of the final solution. The

remaining weighting values of 1 correspond to maximum correlation in the x and z

directions for the derivative terms. Model evolution is displayed in Figure 5.12 for this

inversion. After 50 generations, the resulting model is a good representation of the true

model. Likewise, by maximally correlating the derivative terms and eliminating the

energy term from the model objective function, inversion has filled in the gap (nil-zone)

separating the top and bottom portions of the salt body. It should be pointed out,

however, that cells within the nil-zone do not contribute to surface gravity data, and

therefore the result presented in Figure 5.12 merely illustrates a feature resulting from the

weighting parameters in the model objective function.

The explanation for this result is that the first term of eq.(5.3), the energy term, is

satisfied by keeping the nil-zone empty, as illustrated in Figure 5.10. In contrast, the

maximally correlated derivative terms of the model objective function are satisfied by

eliminating the gap between the top and bottom regions of the nil-zone.

113







Figure 5.12. Evolution of the model results, averaged over a population of models, when the energy term is removed from the model objective function. Nil zone is filled in with either salt or sediment. Convergence is reached in only 50 generations with the hybrid algorithm.

114

5.5. Depth Weighting

Gravity and magnetic data have no inherent depth resolution due to the rapid

decay of the kernels with depth in the sensitivity matrix. For gravity method, the kernels

decay with 21 r , eq.(2.7), where r is the distance between model and data location. As

a result, cells at depth inherently have much less influence on surface data and tend to be

zero in the binary model obtained through a minimum norm solution. Consequently,

even with my binary constraint, there is still a tendency to concentrate material as close to

the surface as possible during inversion. The resulting solution is not geologically

meaningful.

To provide cells at depth with equal probability of obtaining non-zero values

during inversion, a generalized depth weighting function is developed to incorporate into

the model objective function (Li & Oldenburg, 2000). The depth weighting function is

designed to match the overall sensitivity of the data set to a particular cell,

2

1

2

1

2)(

λ

λ

⎟⎠

⎞⎜⎝

⎛=

=

∑

∑

=

=

N

iij

N

iij

G

Grw v

, (2.12)

115

where )(2 rw v is the root-mean-square sensitivity of the model, and λ is chosen to match

the 21 r decay of gravity signal away from the source. Normally, λ = ½ . For my

problem, however, I have experimented with this parameter and found that λ = 0.4

provides better depth placement in the final inverse solutions. When λ = ½ , inversions

tend to place undesired structure within the deepest parameters of the model region,

particularly when there is no density contrast reversal with the model. In contrast, when

λ < 0.4, structures tend to concentrate at shallow depths: around the upper surface of the

model for reproduction of positive gravity anomalies, and just beneath the nil-zone to

reproduce negative anomalies.

5.6. Summary

In this chapter, I have explored the effects of regularization and other weighting

parameters in binary inversion. This includes the alpha weighting parameters within the

model objective function, as well as depth weighting. This is important for practical

applications since the final result for binary inversion depends on choice of these

parameters.

In the first part of the chapter, I demonstrate that regularization behaves similarly

for binary inversion as continuous variable inversions. As a result, one has the ability to

116

successfully construct a Tikhonov curve for estimating regularization given one’s

understanding of the errors in their data. This includes estimation by discrepancy

principle when errors in the data are known, and more significantly, with L-curve for

more practical applications where little to no information may be available on the noise in

one’s data.

In the later part of the chapter, I illustrate how choice of weighting parameters in

binary inversion affects the character of the final solution. This includes the nature of the

solution when one emphasizes, or de-emphasizes, the separate components of the model

objective function associated with model size and complexity. These parameters are

found to have an especially strong influence on the nature of the recovered model in salt

imaging when the anomalous body straddles a nil-zone.

117

CHAPTER 6: EXPLORATION OF BINARY INVERSE SOLUTION

In the previous chapters, I develop binary inversion as a means of constructing a

meaningful solution to the salt body problem, and illustrate performance of several

optimization tools as solvers for binary inversion. While construction is a valuable

component of geophysical inversion (e.g., Parker, 1977; Sabatier, 1971; Oldenburg,

1984; Menke, 1984), appraisal of the solution is also a necessary component in order to

assess the reliability of specific model features (Scales and Snieder, 2000).

To derive meaningful information from the binary inverse solution through

appraisal, and more importantly to make decisions when provided with such information,

one needs a method for exploring the model space. The reason for such a need is to

answer two fundamental questions: (1) are there more than one ‘class’ of models that

solve the minimization problem? (2) if there are more than one class of solutions, how

does one derive meaningful information from them? This information should ultimately

present itself visually in order to identify areas throughout the model region where

potential inverse solutions agree, and regions where they disagree.

In this chapter, I appraise the solution of binary inversion by focusing on the 2D

118

salt problem with density contrast reversal as a means of understanding the uncertainties

in the recovered model, and to identify features of high confidence. I explore the model

space of binary inversion, evaluate the modality of the objective function for this

purpose, and illustrate improved reliability of interpretation in the process.

6.1. Exploring the Model Space of Binary Inversion

The first step to understanding model uncertainty in binary inversion is

exploration of the model space. The issues to address for this step are, first, whether the

objective function is multimodal, and if it is multimodal, how far separated the minima

are. In the end, the goal of exploration is to assess the reliability and uncertainties of the

model by identifying what these separate minima, if they exist, tell us about our solution.

That is, do these minima each represent a completely separate class of model, or do they

represent a similar inverse solution overall, with only slight variations throughout the

model region?

Recent work on exploring the model space and characterizing the modality of

high-dimensional objective functions is presented by Deng (1996). In her work, Deng

(1996) utilizes an entropy-based criterion to analyze the ‘clustering’ of solutions by

performing multiple independent random local-descent searches, referred to as ‘ball-

119

rolling’ experiments. The outcome is to identify the number of local minima, the widths

or separation of the corresponding basins, and their relative depths. While her methods

are applied to continuous variable problems with a small number of parameters, relative

to the salt body problem, the foundation of her work is applied in this chapter for simple

appraisal of binary inversion.

I carry out the exploration by performing multiple independent random local-

descent searches, adapted from Deng (1996) for binary variable, by utilizing QSA for the

binary problem. A parallel approach is performed by Roy et al. (2005) for appraisal of

gravity inversion results over east Antarctica. In their application however, Roy et al.

(2005) evaluate the models sampled by standard SA. For my problem, multiple

minimizations are carried out independently from different, random starting models. I

then evaluate a measure of the Euclidean distance between the final solutions, attempting

to connect the two furthest models through additional minimization, and verify that the

model space is multi-modal on local scale.

6.1.1. Multiple Inversions

For exploration, I invert data for the 2D salt model with nil-zone, which was first

introduced in section 3.2.2.2. The data and true model are presented below in Figure 6.1.

120

-0.1 -0.05 0 0.05 0.1 0.15

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x (meters)

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eter

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∆ρ (g/cm3)

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1 OriginalNoisy


g z(m

Gal

)

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eter


∆ρ (g/cm3)

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1 OriginalNoisy


g z(m

Gal

)

x (meters)-13000 7000 27000

0

0.5

1 OriginalNoisy


g z(m

Gal

)

x (meters)

Figure 6.1. 2½-D section through the SEG/EAGE Salt Model in density contrast form (a), and in binary form (b). A nil zone of zero density contrast cuts horizontally through the middle of the density contrast model. The binary model is represented by background sediment in black (zeros) and salt in white (ones). Panel (c) displays the true data (line) and noise-contaminated data (points). Data passes through zero mGal between positive and negative anomalies due to density contrast reversal.

a)

b)

c)

121

There are 5,670 cells comprising the model. Observed data are simulated above the

model with additive noise of zero mean and standard deviation of 0.025 mGal.

Exploration is performed with 500 starting models, each initialized with top of

salt, as in the previous examples in this thesis. Beneath top of salt, the models are filled

with random zeros and ones in order to adequately span the model space. I perform 500

inversions using QSA, with 75,000 iterations each. Observations indicate that the models

are minimized well before this level, with no further downhill motions apparent well

before the last iteration.

6.1.2. Simple Appraisal of Binary Solution

Results from the 500 binary inversions are presented in Figure 6.2(a). The image

is an average of the 500 solutions defined by:

MjN

kjij ,...,1,1 N

1k== ∑

=ττ (6.1)

where N is the number of solutions, M is the number of cells in the model, and jτ are the

individual model solutions from each binary inversion. The advantage of averaging over

122

z (m

eter

s)

Mean

x (meters)

mean value from 500 binary solutions

Variance0

2000

4000

z (m

eter

s)

0

2000

40000 6000 12000

0 0.5 10.25 0.75

x (meters)

variance from 500 binary solutions

0 6000 12000

0 0.1 0.250.05 0.15 0.2

z (m

eter

s)

Mean

x (meters)

mean value from 500 binary solutions

Variance0

2000

4000

z (m

eter

s)

0

2000

40000 6000 12000

0 0.5 10.25 0.75

x (meters)

variance from 500 binary solutions

0 6000 12000

0 0.1 0.250.05 0.15 0.2

Figure 6.2. Mean and variance calculated from 500 binary inversions using QSA. The mean (a) illustrates that all inversions have adequately reconstructed the model for the larger distribution of mass. The variance throughout the 500 binary inversions (b) illuminates a halo of variance around the steep dipping structure at the left of the salt body.

a) b)

123

a population of binary solutions is direct separation of high confidence zones from

regions of uncertainty. Regions illustrated in Figure 6.2(a) as solid white or black

directly indicate features of commonality among all models, i.e. salt and sediment

respectively. These are the high confidence zones in the solution which are present in all

500 solutions, and are therefore interpreted as required features for given minimization

problem.

Mid values between 0 and 1 in the average solution, illustrated in gray in Figure

6.2(a), highlight portions of the model region that vary from solution to solution. These

are the regions, or parameters, of uncertainty in the final solutions.

To further explore binary inversion, I evaluate the regions of uncertainty within

model displayed in Figure 6.2(a). For this, I calculate the variances of the model

parameters, as defined by (Gonick & Smith, 1993):

( ) MjN j

kjj ,...,1,

11 2N

1k

2 =−−

= ∑=

ττσ (6.2)

where jτ is the average of the binary inverse solutions defined by eq.(6.1), jτ are the

individual binary solutions, M is the number of cells in the model, and N is the total

number of inversions performed by QSA. The regions of black, or zero variances, in

124

Figure 6.2(b) reiterate the high confidence zones throughout the model regions in which

all 500 solutions agree. They have each identified the steep dipping flank beneath top of

salt at the left of the model region. The gray to white parameters, in contrast, illustrate a

‘halo of high variance’ surrounding this dipping feature.

Most assessment of the inverse solution stop at this point, if this stage is ever

reached. The final solution is typically presented as an identification of features present

in several inversions, and possibly regions of uncertainty. As I will show in the next

section, further information may be gathered from this halo of high variance, allowing for

better interpretation of the final inverse solution.

6.2. Investigation of Possible Multimodality

In this section, I illustrate that the halo of high variance identified by the 500

binary inversions from the previous section is directly related to the modality of the

objective function. In return, these multiple minima may represent multiple solutions to

the binary inverse problem, providing varying widths of the dipping slab and depths to

base of salt. This is significant because depth to base of salt is one of the primary

motivations behind the salt imaging problem with gravity inversion.

125

To investigate the modality of the model space, I first seek a means of

representing some relation between model pairs from the 500 binary inversions generated

in the previous section. For this, I use a measure of Euclidean distance between the 500

models:

( )2M

121∑

=

−=j

lj

kjkld ττ , (6.4)

where M is the number of cells, and ( lj

kj ττ − ) is the difference between the jth parameter

of the two solutions. The results of the distance calculations are presented in Figure 6.3.

The axes of the image are the model numbers and each point within the image shows the

distance between two models according to eq.(6.4). The distance image is symmetric and

has zeros along the main diagonal, since each model solution has zero distance with

itself. The purpose in generating such an array is to identify any distinct patterns, or

breaks, in the distances among the 500 models. Models with large differences in

structure, will naturally have large and distinct distances between them. Such a pattern

would indicate that the two solutions lie in separate basins in the model space. However,

no such feature is present in Figure 6.3. The differences between models are relatively

continuous, indicating that the model space is most likely uni-modal on the larger scale.

This is consistent with results from the previous section (Figure 6.2), in which the large

126

distribution of salt throughout the model region is identified with large confidence, and

variances occur only around the edges of the solutions.

Next I investigate the possible modality of the model space at local levels. At this

point, I assume that the model space is uni-modal on the larger scale, representing a

global minima for the minimization problem that solves the original inverse problem; and

that the problem may be multi-modal locally, representing different interpretations of the

edge and base of salt within the halo of high variance. In another word, the total objective

function has a single global basin but there are small-scale multiple minima in the form

of micro-topography within the basin.

For this purpose, I identify the two furthest solutions as calculated by eq.(6.4) and

attempt to ascertain whether they are in the vicinity of same minimum or they truly

represent two different minima. This is tested by attempting to connect them through

further minimization. Two scenarios may occur from this process. First, the two models

may be connected through continued minimization, the natural conclusion being that one

or both of the solutions were not fully minimized, and they were not models in two

separated local minima. The second, the models may not be connected, in which then

they are separated locally and the features representing these local minima must be

incorporated into one’s final appraisal of the inverse solution.

127

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

2

4

6

8

10

12

14

16

Figure 6.3. Distance array for 500 binary inversion models generated with QSA. The axes of the image are the model numbers, and each point within the image shows the Euclidean distance between two models.

128

I start by examining the parameters that vary between the two furthest solutions,

Figure 6.4(a). The feature of interest is that the total cells different between the two

furthest models largely encompass the halo of high variance in Figure 6.2(b). Next I

focus on the parameters which vary between the two models and attempt to further

minimize each solution by applying QSA to only these cells. Thus, I attempt to connect

the two models by modifying only the parameters which vary between the two. The

result shows that the objective function cannot be further reduced from either model.

Since only a small number of cells are involved and the attempts to further reduce the

objective function have test all possibilities of model perturbation, I conclude that both

models are at their respective minimum. This indicates that the model space is very

likely multi-modal on a local level.

The value of this information is not in the modality of the function itself, but in

the character of the two solutions which the separate minima represent. To illustrate this,

the parameter differences are separated into the two models for which they belong.

Figure 6.4(b) shows the parameter differences once more, with those belonging to the

first model in white, and those from the second model in black. Gray cells are those that

are similar between the two models (zero variance), and encompass the background

sediment, top of salt, and the central region of the dipping slab to the left of the salt body.

129

z (m

eter

s)

x (meters)

0

2000

4000

z (m

eter

s)

0

2000

40000 6000 12000

Cells that are the same

x (meters)0 6000 12000

Cells that are different

Cells belonging to model 2


Cells that the two models

share

Total Cells Different Cells From Each Model

z (m

eter

s)

x (meters)

0

2000

4000

z (m

eter

s)

0

2000

40000 6000 12000

Cells that are the same

x (meters)0 6000 12000

Cells that are different



Cells that the two models

share

Total Cells Different Cells From Each Model

Figure 6.4. Distribution of total cells that are different between the two furthest solutions (a). The distribution of the cells largely encompasses the halo of variance generated from the 500 binary inversions, Figure 6.2(b). The second panel (b) presents the cell differences as contrasting colors to illustrate the two classes of solution to binary inversion for the 2D salt body example.

a) b)

130

The two models have distinct differences in density distribution. The first model

with the white cells in Figure 6.4(b) is narrower and has a deeper base of salt within the

steeply dipping flank of the model. The second solution represented by the black cells

has a shallower base of salt with a wider and less steeply dipping flank. Therefore,

results indicate there are at least two distinct classes of solution to the binary inverse

problem for this example.

In this section, I have illustrated that the halo of high variance identified by the

500 binary inversions from the previous section, is related to the modality of the

objective function. The model space is potentially uni-modal on the larger scale, with

one class of solution imaging the larger distribution of salt throughout the model region.

In contrast, the model space is estimated to contain multiple minima locally, which allow

for at least two interpretations of width and depth to base salt.

6.3. Summary

In this chapter, I have attempted a simple appraisal of the solution of binary

inversion to understand the uncertainties in the recovered model, and to identify features

of high confidence. I explore the model space of binary inversion, evaluate the modality

of the objective function for this purpose, and illustrate the improved reliability of

131

interpretation in the process. For the 2D salt problem with density contrast reversal, the

model space appears to be uni-modal on a larger scale, while containing multiple minima

on a local scale. The interpretation is that binary inversion successfully identifies the

larger distribution of salt within the model region with high confidence. However, a halo

of high variance around the edges and base of salt indicate regions of uncertainty in the

final model which one must incorporate into their appraisal of the inverse solution.

132

CHAPTER 7: CONCLUSIONS AND FUTURE RESEARCH

7.1. Conclusions

The gravity inverse problem for salt body imaging is one of finding the position

and shape of a constant anomalous density embedded in a sedimentary background

whose density increases with depth. Difficulty arises when the body crosses a region at

depth where background density is equal to that of salt. When this occurs, the salt body

straddles what is referred to as a nil-zone, and salt within the nil-zone has a zero density

contrast with its host. As a result, this portion of the salt body is invisible to surface

gravity data. A second scenario also occurs due to nil-zones which can adversely affect

gravity inversions. Salt above the nil-zone has a positive density contrast and generate a

positive anomaly in surface gravity data. In contrast, portions of the salt body below the

nil-zone attain a negative density contrast, and therefore produce a negative anomaly in

the gravity data. The net result is that the positive and negative anomalies from the top

and bottom of salt cancel out in parts of the surface gravity data. This effect is referred to

as an annihilator. Zero density contrast nil-zones, combined with annihilators in the salt

body, often result in gravity inversions that have little resemblance to the true geologic

problem.

133

7.1.1. Current State of Gravity Inversions

Current gravity inversion methods fall under two general categories, each of

which have desirable and undesirable features when applied to the salt imaging problem.

Interface inversions construct base of salt directly by incorporating density contrast at

each depth. The discrete nature of these methods proves valuable in restricting the class

of admissible models, and therefore limiting the non-uniqueness of the problem. The

disadvantage of interface inversion is that the problem is non-linear, and can therefore be

more difficult computationally. In addition, parameterization can overly restrict the

model from the outset. For example, the methods typically assume simple topology for

the salt body, and they cannot handle multiple anomaly sources that are not accounted

for at the start. This inconsistency between assumed model and data can lead to large

errors, or even failure of inversion.

The second methods are generalized density inversions, which construct density

contrast distribution as a function of spatial position and image base of salt by the

transition in density contrast. The advantages of these methods are that the problem is

linear, they can handle multiple anomaly sources, and they afford great flexibility in

model representation. The drawbacks of density inversion are that they cannot directly

utilize the known density information, they typically produce poor, if any, resolution near

134

nil-zones. The data are satisfied by intermediate density values and distributions that

only image a portion of the salt body.

7.1.2. Contribution of Binary Inversion

In this thesis, I have developed a binary inversion algorithm in which density

contrast is restricted to being either zero or the value of density contrast of salt at a given

depth. The method is developed to overcome the difficulties associated with interface-

based inversion and density-based inversion while drawing from the strengths of both

existing approaches. It uses a similar model representation as in continuous-density

inversion by defining a density distribution as a function of spatial position, but restricts

the model values to those corresponding to two lithologic units as does the interface

inversion. Therefore, binary inversion enables one to incorporate density contrast values

appropriate to the geologic problem while providing a sharp boundary for the subsurface.

I have formulated the binary inversion using Tikhonov regularization in which the

inverse solution is obtained by minimizing a weighted sum of a data misfit and a model

objective function. The model objective function serves to stabilize the solution and to

incorporate any prior information that is independent of gravity data. Because of the

discrete nature of the problem, commonly used minimization techniques are not

135

applicable for binary problem. I therefore investigated the use of genetic algorithm,

quenched simulated annealing, and a hybrid method based on these two as potential

solvers for the minimization problem associated with binary inversion. The use of

Tikhonov regularization is well understood in continuous-variable inversion, but its

application in binary problems had yet to be explored. I investigated this aspect and

conclude that Tikhonov regularization plays a similar role in discrete inversion, and the

corresponding Tikhonov curve behaves in a similar manner. Thus the commonly used

approaches for determining the level of regularization are equally applicable in both types

of inversions. In conjunction with the regularization, I examined the effect of various

parameters in the model objective function, and illustrated that they can have a strong

influence on the binary solution when a nil-zone is present. Finally, to appraise the

solution, and to understand the uncertainties in the recovered model, I explored the

solution space of binary inversion and evaluated the modality of the objective function.

This allowed for more reliable interpretation of the binary solution through identification

of high confidence features in salt structure and regions of high variance.

I illustrated binary inversion with synthetic salt models in 2D and 3D generated

from the SEG/EAGE salt model. As sought in the motivation behind binary inversion,

the method proves it can easily incorporate density information while providing a sharp

contact for the subsurface. It also allows for great flexibility in model representation

while solving for density distribution as a function of spatial position. The binary

136

condition places a strong restriction on the admissible models so that the non-uniqueness

caused by nil zones might be resolved.

7.1.3. Problems Associated with Binary Inversion

In practical applications, there are inevitably errors introduced both from prior

information assumed to be ‘known’, and from data errors resulted from both acquisition

and processing techniques. In the former, uncertainties associated with top of salt,

density contrast information, number of anomalous source bodies, and perhaps regions of

base of salt, will degrade the quality of the recovered model. Severe errors in these

assumptions may even lead to failure of the inversion. An example of the later is the

assumption that the regional field has been successfully removed from observed gravity

data. Binary inversion, as with interface inversions, can introduce large errors in the

recovered model when any one of these situations exists. Analysis of the sensitivity of a

model to errors in known prior information and data processing is of practical

importance, and is as valuable for binary inversion as with other formulations. However,

these are issues not specific to binary inversion and, therefore, are not addressed in this

work. Recent studies on assessing the sensitivity of a model to uncertainties in prior

information are available in Cheng (2003).

137

7.2. Future Research

Future work in, and advances stemming from binary inversion fall under two

categories. The first is the practical aspects of algorithmic development. The most

obvious is development of faster and more efficient implementations of binary inversion,

and study of additional minimization methods outside genetic algorithm and quenched

simulated annealing. GA, for example, has proven to be a valuable solver for binary

inversion for the simple 2D gravity inverse problem. However, as the number of

dimensions in the problem increases, GA proves inefficient and not practical for

inversions with thousands of unknown parameters. To tackle real-world problems with

tens to hundreds of thousands of dimensions, more efficient optimization techniques may

be required which can solve the binary problem. Integer programming may prove a more

valuable solver for binary inversion in these cases.

The second area of future research would be to expand the application of binary

inversion beyond the salt problem. One particular problem is application of the current

formulation to the time-lapse gravity problem. These problems attempt to invert for

changes in the gravity field associated with fluid flow, such as monitoring oil reservoirs

during enhanced oil recovery (EOR) process, and the monitoring groundwater reservoirs

in aquifer storage and recovery (ASR) processes. In such cases, the change if density can

be well-defined. Furthermore, the regions of density change over time typically do not

138

form a simply connected domain. Therefore, imaging such regions with a binary

formulation is ideal.

Another promising direction for future research is the expansion of binary

inversion to lithologic inversion. Binary inversion has illustrated an ability to

successfully invert gravity data for discrete lithologic units, namely, the salt and sediment

in the salt imaging problem. However, since the formulation is binary in nature, the

method can naturally only solve for two distinct geologic units. While this may be

adequate for the salt problem, many practical applications seek lithologic definition

beyond this binary scenario. This is supported by the growing interest from both

petroleum and mineral exploration companies in lithologic inversion (e.g., Bosch and

McGaughey, 2001). In a lithologic inversion, distribution of a number of distinct

lithologic units having well-defined physical property values is sought such that the

observed geophysical data sets are reproduced. Binary inversion, as presented in this

thesis, has laid a solid foundation for such problems by solving for distribution of

separate lithologic units as a function of spatial position, while providing a sharp contact

in the subsurface. The discrete formulation is easily extended to lithologic inversion with

multiple units, and the computational methods developed for binary inversion are also

directly applicable. This is expected to be a fruitful area of research that may contribute

greatly to a number of problems in resource exploration and production.

139

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