colorado school of mines center for gravity, electrical ...cgem.mines.edu/s/krahenbuhl thesis...
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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
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
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)
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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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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
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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
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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
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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.
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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.
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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 Results: Predicted vs. Original
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
Data Results: Predicted vs. Original
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
Data Results: Predicted vs. Original
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
Initialize population
Start
Mutation
Recombination
Selection
Evaluate objective function, eq.(1)
Are optimization criteria met?
Best individuals
Result
no
yes
Generate
new
population
Initialize population
Start
Mutation
Recombination
Selection
Evaluate objective function, eq.(1)
Are optimization criteria met?
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
Random Starting Model #2
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
Random Starting Model #1
Random Starting Model #2
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
Forward Data With and Without Noise
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
Best IndividualAverage of Population
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
Generation 205 Generation 239 Generation 273
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)
True Model: Binary Form
x (meters)
z (m
eter
s)
True Model: Density Contrast Form
∆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
Forward Data With and Without Noise
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
Forward Data With and Without Noise
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
True Model: Density Contrast Form
∆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
True Model: Density Contrast Form
∆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
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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.
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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
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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)
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)
True Model: Binary Form
x (meters)
z (m
eter
s)
True Model: Density Contrast Form
∆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
Best IndividualAverage of Population
Performance Plot: Genetic Algorithm
Generation Number
Obj
ectiv
e V
alue
0 100 200 300 400 500 600102
103
104
105
Best IndividualAverage of Population
Generation Number
Obj
ectiv
e V
alue
0 100 200 300 400 500 600102
103
104
105
Best IndividualAverage of Population
Performance Plot: Genetic Algorithm
Generation Number
Obj
ectiv
e V
alue
0 10 20 30 40 50 60101
102
103
104
105
Best IndividualAverage of Population
Performance Plot: GA/QSA Hybrid
Generation Number
Obj
ectiv
e V
alue
0 10 20 30 40 50 60101
102
103
104
105
Best IndividualAverage of Population
Generation Number
Obj
ectiv
e V
alue
0 10 20 30 40 50 60101
102
103
104
105
Best IndividualAverage of Population
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)
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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
Generation 1 Generation 7 Generation 13
Generation 31Generation 25Generation 19
Generation 50Generation 43Generation 37
Generation 1 Generation 7 Generation 13
Generation 31Generation 25Generation 19
Generation 50Generation 43Generation 37
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.
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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
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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.
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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)
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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.
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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
Model Objective Value: ϕm
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
Model Objective Value: ϕm
4 x 107
Dat
a M
isfit
Val
ue: ϕ d
108
102
103
Model Objective Value: ϕm
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.
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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.
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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.
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102
102
103
Model Objective Value: ϕm
Dat
a M
isfit
Val
ue: ϕ d
102
102
103
Model Objective Value: ϕm
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.
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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
Model Objective Value: ϕm
Dat
a M
isfit
Val
ue: ϕ d
10830
40
50
60
70
80
90
4 x 107
β ~ 2.07E-7
Model Objective Value: ϕm
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.
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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)
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x (meters)
z (m
eter
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Inverse Model Predicted by Discrepancy Principle
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z (m
eter
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True Model
x (meters)
z (m
eter
s)
Inverse Model Predicted by Discrepancy Principle
0 2000 4000 6000 8000 10000 120000
0
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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
Model Objective Value: ϕm
Dat
a M
isfit
Val
ue: ϕ d
102
102
103
Model Objective Value: ϕm
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.
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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
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(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.
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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.
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z (m
eter
s)
x (meters)0 2000 4000 6000 8000 10000 12000
0
1000
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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.
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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.
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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.
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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.
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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.
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Generation 1 Generation 7 Generation 13
Generation 31Generation 25Generation 19
Generation 50Generation 43Generation 37
Generation 1 Generation 7 Generation 13
Generation 31Generation 25Generation 19
Generation 50Generation 43Generation 37
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.
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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)
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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
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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.
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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
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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-
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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.
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-0.1 -0.05 0 0.05 0.1 0.15
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s)
True Model: Binary Form
x (meters)
z (m
eter
s)True Model: Density Contrast Form
∆ρ (g/cm3)
-13000 7000 27000
0
0.5
1 OriginalNoisy
Forward Data With and Without Noise
g z(m
Gal
)
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
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
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x (meters)
z (m
eter
s)
True Model: Binary Form
x (meters)
z (m
eter
s)True Model: Density Contrast Form
∆ρ (g/cm3)
-13000 7000 27000
0
0.5
1 OriginalNoisy
Forward Data With and Without Noise
g z(m
Gal
)
x (meters)-13000 7000 27000
0
0.5
1 OriginalNoisy
Forward Data With and Without Noise
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)
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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
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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)
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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
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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.
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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
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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.
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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.
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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 belonging to model 1
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 belonging to model 2
Cells belonging to model 1
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.
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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.
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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
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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).
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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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