Inversion of Gravity Data Using a Binary Formulation

Geophys. J. Int. (2006) 167, 543–556 doi: 10.1111/j.1365-246X.2006.03179.x

Inversion of gravity data using a binary formulation

Richard A. Krahenbuhl and Yaoguo Li Center for Gravity, Electrical & Magnetic Studies; Department of Geophysics, Colorado School of Mines, USA. E-mail: [email protected]

Accepted 2006 August 16. Received 2006 August 16; in original form 2005 February 24

SUMMARY We 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. The problem of salt imaging may be formulated as a general inverse problem in which a piecewise constant density contrast is constructed as an indirect means of identifying the salt boundary. Difﬁculty arises when the salt body crosses the nil zone in depth. As a result, part of

the salt structure is invisible to the surface data and many inversion algorithms have difficulties GJI Geodesy, potential field and applied geophysics in recovering the salt structure correctly. The binary condition places a strong restriction on the admissible models so that the non-uniqueness caused by nil zones might be resolved. In this paper, we will present the binary formulation for inversion of gravity data, develop the solution strategy, illustrate it with numerical examples, and discuss limitations of the technique. Key words: genetic algorithm, gravity, inversion, optimization, potential field.

a negative gravity anomaly on the surface. In the final scenario, the 1 INTRODUCTION salt body straddles a depth at which the sediment density is equal Conservative estimates indicate that at least 15 per cent of US to the salt density. This region of equal density between salt and domestic oil and 17 per cent of its natural gas production come from sediment is referred to as a nil zone. fields along the continental shelf margin off the shores of Louisiana In the last scenario described above, the portion of salt within (Gibson & Millegan 1998). To explore for future reserves, industry the nil zone does not contribute to surface gravity anomaly. This has expanded towards exploration of the deeper water regions of the is a natural consequence of having zero density contrast with the continental slope. While the industry target is obviously hydrocar- surrounding sediment. Likewise, portions of the salt body above the bon traps, the geophysical targets are the geologic features in the sed- nil zone will have positive density contrast, producing a positive imentary section that are responsible for these accumulations of oil anomaly in surface gravity data. Salt below the nil zone, in con- and gas. Some of these features include reefs, faults, anticlines and trast, generates a negative gravity anomaly because it has a negative variations in thickness of horizontal salt beds. In particular, the salt, density contrast with respect to the surrounding medium. The net including domes, ridges and pillows, are relatively incompressible result is that the positive and negative anomalies from the top and and, therefore, remain fairly constant in density throughout. This bottom portions of salt cancel out in parts of the surface gravity incompressibility likewise allows for abundant traps beneath salt data (Gibson & Millegan 1998). This effect is referred to as an an- bodies throughout the Gulf of Mexico (Gibson & Millegan 1998). nihilator. Depending on how an inversion algorithm is formulated As a result, they have become major targets in oil and gas explo- to reconstruct such a salt body, the combination of nil zones with ration. Gravity inversion is one of the tools available to geophysicists annihilators in the salt body can cause major difficulties. for imaging the base of salt as a means for exploring these targets. Inversion methods for imaging salt structure using gravity data The gravity inverse problem for salt body imaging is one of find- fall under two general categories. The first is interface inversions. ing the position and shape of an anomalous constant density em- These methods assume a simple topology for the salt body and bedded in a sedimentary background whose density increases with known density contrast and construct the base of the salt (e.g. depth due to compaction. Depending on the depth and depth extent Jorgensen & Kisabeth 2000; Cheng 2003). Similar method has also of the salt body, three scenarios can occur. In the first scenario, the been used extensively in other applications of gravity inversion, salt is shallow enough so that its density is greater than that of the such as in basin depth determination (e.g. Oldenburg 1974; Pedersen immediate sedimentary host. This leads to a positive density con- 1977; Chai & Hinze 1988; Reamer & Ferguson 1989; Barbosa et al. trast in the salt, and a positive gravity anomaly in surface data. In 1999). Methods in the second category are generalized density in- the second scenario, the salt is positioned at depth so that its density versions. These methods construct a density contrast distribution as is less than the density of the surrounding sediments. This leads to a function of spatial position and image the base of salt by the transi- an entirely negative density contrast for the salt body and, therefore, tion in density contrast (Li 2001). Similar approaches have also been

C 2006 The Author 543 Journal compilation C 2006 RAS 544 R. A. Krahenbuhl and Y. Li used widely in mineral exploration problems (Green 1975; Last & objective function φ m and data misfit φ d: Kubik 1983; Guillen & Menichetti 1984; Oldenburg et al. 1998). min.φ= φ (ρ) + βφ (τ), The interface inversion has the advantage that it directly inputs d m the known density contrast at each location in the subsurface and subject to ρ ∈{0,ρ(z)}. (1) provides a direct image of the base of salt. However, the drawbacks Assuming that we know the standard deviation of each datum σ i , are that the problem is non-linear and can be more difficult compu- we can define the data misfit function as tationally. In addition, the assumed simple topology of salt creates N obs − pre 2 difficulties when either regional field or small-scale residuals due φ = di di , d σ (2) to shallow sources are not completely removed. The inconsistency i=1 i between the assumed model and data can lead to large errors, or where dobs and dpreare the observed and predicted data, respectively. even failure of inversion. i i Assuming a Gaussian statistics, the expected value of the data misfit The density inversion has the flexibility of handling multiple is equal to the number of data, N. We would like to construct a anomalies, more complex shapes and the solution is easier to obtain compact model that is also structurally simple. Therefore, we use because the relationship between observations and density contrast the following generic model objective function that measures the is linear. However, as they are currently formulated, these methods length of the model and its flatness in different spatial directions are not well suited for cases where nil zones are present. Because (e.g. Li & Oldenburg 1998). For 2-D problems, the model objective of the presence of a nil zone and annihilators as discussed above, a function is given by, portion of the salt body is invisible to the surface gravity data. Den- sity inversion methods allowing continuous density values (e.g. Li ∂τ 2 ∂τ 2 φ = α τ 2 v + α v + α v, & Oldenburg 1998) will in general produce a model that has little m s d x ∂ d z ∂ d (3) V V x V z resemblance to the true structure. The data are satisfied by interme- where τ is the binary model, V is the subsurface region over which diate density values and distributions that only image a portion of the model is defined, and α , α and α are relative weights of the the salt body. s x z individual components of the model objective function. We note that To overcome difficulties associated with both methods, we use model values are discontinuous and, therefore, the derivative terms a binary formulation that enables one to incorporate the density in the model objective function in eq. (3) are only defined in the contrast values, a strength of non-linear interface inversion, while sense of generalized functions. We present in such a form to provide retaining the flexibility and linearity of density inversion. The diffi- conceptual understanding. However, the numerical algorithm works culty of the binary formulation, however, lies in the discrete nature of directly with the discretized form of eq. (3), and the derivative terms the density contrast. Because the variable can only take on discrete therefore represent differences between adjacent elements. values, 0 or 1 for sediment or salt, respectively, derivative-based Requiring φ to achieve an expected value ensures all models that minimization techniques are no longer applicable. In the follow- d do not adequately fit the observed data, or overfit them, be eliminated ing, we will first present the methodology of binary inversion and from the set of possible solutions. The binary formulation places a discuss the solution by genetic algorithm (GA). We then present limit on the types of models that fit the data. However, the problem results from synthetic tests of a 1-D mathematical problem and two is still non-unique. Therefore, we use the model objective function, 2-D gravity problems with finite strike length (often referred to as φ , to further narrow our solution set to only geologically reasonable 2.5-D problems). For the 2.5-D problems, we use a cross-section m models. This is done by choosing the model that has the smallest through the SEG/EAGE 3-D salt model (Aminzadeh et al. 1997) size and structural complexity among those that fit the data. The with density contrast reversal. Lastly, we discuss the role of regu- regularization parameter β controls the balance between the data larization in binary inversion and compare with that in continuous misfit and model objective function, protecting from overfitting the variable inversions. data or over smoothing the model. Our binary formulation incorporates a binary variable τ into the density function of eq. (1) through expected density contrast at depth 2 METHOD z:

The difficulty of an annihilator outlined in Section 1 can only be τ(r ) ∈{0, 1}. (4) overcome by incorporating prior information to restrict the class of admissible models. We propose to impose the condition that the ρ(r ) = τρ(z). (5) density contrast must be the discrete values appropriate for the geologic problem. In the simplest form, density contrast is restricted At a given depth, a value of zero in the model, τ, indicates a zero to being either zero or a known value at a given location. Simi- density contrast (host sediments), while a value of one corresponds to the expected salt density contrast at that depth. The minimization lar binary approach has been used in both gravity inversion and in other fields. For example, Camacho et al. (2000) invert gravity data problem is then expressed in τ(r ), where r is the location (x, z)of for a compact body with a constant density by growing the volume each cell, and we can simply work with 0 and 1 for the minimization from an initial guess. Litman et al. (1998) invert for the shape of problem. The actual density contrast value is only incorporated into a scatterer by assuming a constant electrical conductivity value for the forward modelling of predicted data during the inversion. the background and the scatter, respectively. For our problem, we The solution to this problem will be better constrained than for- adopt explicitly the Tikhonov regularization approach (Tikhonov & mulations that allow continuous values within upper and lower Arsenin 1977) and formulate the inversion for the general case of bounds. Although still non-unique, this problem no longer has salt imaging at the presence of density reversal. The problem then an infinite number of possible solutions once the problem is dis- becomes one of minimizing an objective function subject to restrict- cretized for numerical solution: there are a finite number of cells ing model parameters to attain only one of two values at each depth. within the model mesh and only two possible values for each lo- The objective function consists of the weighted sum of the model cation. For instance, construction of an equivalent source layer

C 2006 The Author, GJI, 167, 543–556 Journal compilation C 2006 RAS Inversion of gravity data using a binary formulation 545

at an arbitrary depth is no longer possible. The binary value of Mosegaard (2002) and Roy et al. (2005). To gain basic understand- 1, at a specified depth, represents a well-defined density contrast ing about the behaviour of the binary formulation, we have opted value—either positive or negative with corresponding magnitude. to start with the GA as the basic solver. The GA, designed for com- Because of this constraint, a combination of a positive and negative binatorial minimization, is well suited for minimizing an objective anomaly in gravity data is not reproducible by a source distribu- function with binary variables. We briefly describe aspects of GA tion at one depth alone (i.e. an equivalent source layer). When- that are directly pertinent to our problem in the following section. ever an annihilator is present, any geologically unreasonable model that reproduces the data by a combination of density contrasts of 3 SOLUTION USING GENETIC intermediate values is automatically eliminated with the binary ALGORITHM constraint. The minimization problem defined by eq. (1) has a deceptively 3.1 The Genetic Algorithm simple appearance, but its solution is not trivial. The difficulty lies in the discrete nature of the density contrast. Because the vari- This is a programming tool designed for solving a variety of opti- able can only take on two values, 0 or 1, derivative-based min- mization problems. It is a stochastic technique that mimics natural imization techniques are no longer applicable. There are several biological evolution by imposing the principle of ‘survival of the alternative methods for carrying out the minimization. The obvi- fittest’ on a population of individuals. For the inverse problem, fit- ous technique is mixed integer programming (e.g. Floudas 1995; ness is inversely related to a model’s objective-function value in Pardalos & Resende 2002) since our variable to be recovered can eq. (1). The main strategy of the GA is to recombine the individ- only assume a value of either 0 or 1. However, solving an integer- uals, with the better-fit individuals having higher probabilities of programming problem on a realistically sized gravity inversion is reproduction, to evolve to better solutions. The basic design of the computationally prohibitive and the cost for obtaining such an ex- GA is displayed as a flow chart in Fig. 1. Below, we briefly describe act solution far out weights the gain. Thus, practical application the components unique to the GA, including the individual, initial- requires the use of alternative methods that seek near-optimal solu- ization, rank, fitness, selection, recombination and the formation of tion to the minimization problem and demand far less computational the next generation of solutions. However, readers are also referred cost. to Goldberg (1989), Pal & Wang (1996) and Chambers (1995a,b) The second technique involves the use of a controlled random for more details. search technique such as genetic algorithm (GA) or simulated annealing (SA). Both methods are ideal for derivative-free minimiza- 3.2 Individuals tion, which is the problem we have. In addition, both GA and SA can be implemented with relative ease compared to an integer program- The basic unit of the GA is the individual. Each individual rep- ming solution and can yield solutions that are sufficiently accurate resents a potential solution to the problem, that is, a geophysical with much less computation. For information on GA, readers are model in our problem. By analogy to genetics, an individual within referred to Goldberg (1989), Pal & Wang (1996) and Chambers a GA population is commonly referred to as a chromosome. For (1995a,b). Additional information on GA, with application to geo- the binary inverse problem, models are discretized into cells with physical inversion, can be found in Sen & Stoffa (1995), Smith et al. constant values equal to zero or one. In the genetic equivalent, a (1992), Sambridge & Mosegaard (2002) and Scales et al. (1992). chromosome consists of a series of alleles, where each allele rep- For information on SA, the reader is referred to Metropolis et al. resents a cell within the model mesh. Therefore, each individual (1953), Kirkpatrick et al. (1983) and Nulton & Salamon (1988). consists of a string of allele with values of either zero or one. In this Applications of SA specific to geophysical inversion, as well as paper, we generally ignore the correspondence between natural ge- great sources of information on SA, are available in Sen & Stoffa netic terminology and that of GA, and merely refer to the individuals (1995), Nagihara & Hall (2001), Scales et al. (1992), Sambridge & as models, and the allele as cells within the model.

Initialize Evaluate objective Are optimization yes Best population function, eq.(1) criteria met? individuals

Generate Selection

Start new Result population Recombination

Mutation

Figure 1. Flowchart of the GA for the binary inverse problem. Selection, recombination and mutation are components unique to the GA. Modiﬁed from Pohlheim (1997).

C 2006 The Author, GJI, 167, 543–556 Journal compilation C 2006 RAS 546 R. A. Krahenbuhl and Y. Li

Random Starting Model #1

1 Binary Value 0 Cell Number 0 5 10 15 20 25 30 35 40 45 50

Random Starting Model #2

1 Binary Value 0 Cell Number 0 5 10 15 20 25 30 35 40 45 50

Figure 2. Example of two starting individuals with initialization of random zeros and ones. Each individual represents a potential solution model.

3.3 Initialization 3.5 Recombination The first step in applying the GA to gravity inversion is setting up an Once individuals have been chosen for recombination based on initial population, which is a community of individuals. Each indi- objective-function values and selection probabilities, offspring are vidual represents a model. For initialization, values are assigned to generated to join the population in the next generation. Selected in- the cells in each model. When prior information is not available, the dividuals are paired into parents, and a combination of their genes, starting population is initialized by assigning random zeros and ones or blocks of cells, are merged to generate offspring. Every new off- to each cell. Fig. 2 displays two examples of random initialization spring represents a new candidate solution to our problem and it is for the GA in a 1-D problem. placed into the next generation after undergoing mutation. The ability of the GA 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 3.6 Mutation models obtained from previous work. It can be an initial guess pro- duced from other geophysical data such as pre-stack depth migrated After formation of a new set of models (recombination), mutation is seismic image. This is useful in imposing features such as the known applied to the models in the newly generated population to protect top of salt. it from an irrecoverable loss of potentially useful genetic information during reproduction. Mutation in our binary problem consists of flipping randomly chosen cells from 0 to 1, or vice versa. In 3.4 Rank, Fitness and Selection addition, mutation prevents premature convergence by introducing new features into the population. It essentially expands the gene The first step in the evolutionary cycle of the GA is to rank the pool and allows different regions of the solution space to be eval- population. Objective-function values are computed for individuals uated. Mutation rates, defined as the probability of each individ- in our problem based on eq. (1). Lower objective-function values ual cell being flipped, are problem dependent and may be varied correspond to higher fitness levels and, therefore, better models. according to the performance of the GA. For every child created Rank is established by ordering the models from highest fitness val- during recombination, each cell has a low probability of being ues to lowest, that is, best model to worst. Individuals with higher mutated. fitness values will have higher probabilities of surviving the evolutionary process, as well as passing on their features to the next generation. 3.7 New Generation Next we assign selection probabilities defined as the fitness value of an individual divided by the sum of fitness of the entire population. The last step in the evolutionary cycle of the GA is to evaluate the The highest ranking model has the highest probability of surviving, children and assign objective-function values. Once these values while the lowest ranking model has a zero selection probability. The have been assigned, the children are placed into the population to final step in the selection process is choosing individuals as parents replace the least fit half of the previous generation. The new genera- for reproduction. We use Roulette Wheel Selection (Goldberg 1989) tion of potential solutions formed in this manner, therefore, consists in this algorithm. of higher-ranking individuals from the previous generation and their

C 2006 The Author, GJI, 167, 543–556 Journal compilation C 2006 RAS Inversion of gravity data using a binary formulation 547

offspring. The GA proceeds to the next evolutionary cycle and re- a single well record. The region spans from zero to one in generic peats the process until convergence. units. The interval is uniformly discretized into a 1-D mesh of 50 cells. Therefore, each cell has a length of 0.02. The simulated data from eq. (6), contaminated by uncorrelated 4 NUMERICAL EXAMPLES Gaussian noise with zero mean and standard deviation of 0.03 MGal, are shown as the dots in Fig. 3(a). These noisy data, the discretized 4.1 1-D Example model mesh of 50 cells, and our objective function eq. (1) are entered into the GA. For this example, we use the 1-D form of the model We ﬁrst illustrate the binary inversion technique using a simple objective function in eq. (3): mathematical example, and demonstrate the improved solution over traditional continuous variable inversion for certain problems. The ∂τ 2 φ = α τ 2ds + α ds, (7) forward calculation is given by the following integral, m s x ∂x S S 1 An initial population of 400 individuals is initialized. Regulariza- d = τ(z) cos( jaz) exp(− jbz) dz j tion is chosen based on discrepancy principle, where the expected 0 1 value of the data misﬁt is equal to the number of data for Gaus- ≡ τ , = , ···, (6) (z)g j (z) dz j 1 20 sian statistics. The mutation operator allows each cell within the 0 individuals to mutate with a one in 50 probability. Crossover oc- = = τ with a b 0.25. In eq. (6), (z) is the model and gj(z) are the curs in blocks of seven cells, and convergence is reached by the kernels that decay with depth. These kernels are chosen to mimic 20th generation (Fig. 4). CPU time for this example is on the or- the decaying kernels seen in many geophysical experiments. der of a few seconds with a 1.5 GHz processor. The recovered For the numerical test, we use a single boxcar model that is zero model from the binary inversion, Fig. 3(c), is a good representa- everywhere except within a central interval. The 1-D model shown tion of the true model with only one cell different from the true in Fig. 3(b) is analogous to density as a function of depth, similar to model. The predicted data from this model are shown in Fig. 3(a)

Data Results: Predicted vs. Original 0.3 Predicted data 0.2 Synthetic data with noise

0.1

0 Data Value

-0.1 a) -0.2 0 2 4 6 8 10 12 14 16 18 20 Data Point

True Model 1 e

Binary Valu b) 0 Cell Number 5 10 15 20 25 30 35 40 45 50

Constructed Model 1

Binary Value c) 0 Cell Number 5 10 15 20 25 30 35 40 45 50

Figure 3. Binary inversion results for the 1-D 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.

C 2006 The Author, GJI, 167, 543–556 Journal compilation C 2006 RAS 548 R. A. Krahenbuhl and Y. Li

250 Highest Fit Individual Average of Population 200

150

100 Objective Value

0 0 5 10 15 20 25 Generation Number

Figure 4. Progress of the GA for the 1-D 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. The two approach each other by the 20th generation, indicating theGAhas converged.

True Model 1 a) Binary Value 0 Cell Number 5 10 15 20 25 30 35 40 45 50

Constructed Model Continuous variable 1 b) Binary variable

Model Value 0 Cell Number 5 10 15 20 25 30 35 40 45 50

Figure 5. Illustration of inversion result for the 1-D boxcar problem with continuous variable. The true model (a) is the same as in Fig. 3(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 grey dashes. Likewise, the amplitude of the model is not as accurate as the binary result. Both inversions were performed with the same data set. as the solid line. The binary inversion has performed well in this outlined in Fig. 5(b) as a grey dashed line over the continuous vari- case. able solution to highlight the final model differences. The binary To illustrate the advantage of binary over traditional continuous formulation’s incorporation of amplitude information allows the al- variable-methods, Fig. 5(b) shows the equivalent final solution from gorithm to remain true to model’s size and position while providing continuous variable inversion. We have inverted the same 20 noisy a sharp contact in the subsurface. data. Final data misfit from the two methods are identical. How- We note that, in this example, crossover occurs with a gene length ever, Fig. 5(b) shows that the continuous formulation has spread the of seven cells, which does not evenly divide into the 50 cells of solution over a larger interval and adjusted amplitude through in- the individual. The remaining boundary cell is still exchanged dur- termediate values to satisfy the data. The binary inverse solution is ing crossover and achieves the desired value through mutation and

C 2006 The Author, GJI, 167, 543–556 Journal compilation C 2006 RAS Inversion of gravity data using a binary formulation 549

re-ranking of the population at each generation. Tests with other mutated to an undesirable value. Mutation is commonly assigned gene lengths indicate that while it may be preferred to choose with low probability values, and we assign a mutation probability of gene lengths that divide evenly into the length of the model mesh 1/50 for the 1-D problem. Therefore, 1 cell will mutate per individ- (Goldberg 1989), it is not necessary for this or later examples. Ad- ual, on average, at each generation. Further reading on optimal fixed ditional information on crossover parameters is also available in Bui mutation rates and adaptive mutation may be found in Hinterding & Moon (1995), Deb & Agrawal (1999) and Herrera et al. (1998). et al. (1995) and Bäck (1993). More significant to the progress of GA and final result than gene- length for our problem, are size of the population and rate of muta- 4.2 2-D Gravity Problem with a Constant Density tion. When we initialize a population that is too small, the GA relies Contrast heavily on mutation to compensate for the lack of ‘genetic diversity’ if convergence is to be achieved. The result is a large number of gen- The next example is a 2.5-D problem designed to illustrate binary erations over longer times to achieve the equivalent solution. The inversion of gravity data over a salt body with single density con- problem becomes inefficient and proves impractical for real world trast. For this we utilize a section of the SEG/EAGE salt model problems with tens or hundreds of thousands of parameters. In con- (Aminzadeh et al. 1997). The model is designed as a velocity model trast, when the population size is initialized too large, the method for development of imaging technology in the seismic community. becomes impractical for large problems as CPU time increases due Fig. 6 displays the perspective view (a) of the velocity model and to the increase in forward calculations at each generation. Similar one velocity section (b). We have captured the structure of the salt observation is discussed by Herrera & Lozano (1996) and Reeves body and constructed a salt model with a single density contrast to (1993). We note that optimal population size is an entire field of assist in the development of the binary inversion. A similar perspec- research and debate in the GA community (e.g. Reeves 1993; Deb tive view of the 3-D model and 2-D section are presented in Fig. 7, & Agrawal 1999), and that population size is many times chosen with density contrast in place of velocity. For the 2.5-D problem, the based on intuition, experience, and ones definition of acceptable true density contrast model consists of the simple salt body cross- CPU time. For the 1-D problem with 50 parameters, we find the section, Fig. 7(b), buried in a uniform half-space. The salt model increase in CPU time negligible for our population of 400. For later has a finite strike length of 20 km in and out of the page, and a binary inverse problems, we have observed that a population size up single density contrast of −0.2g cm−3 between salt and sediment, to twice the number of unknown parameters is generally adequate which is assumed known in the following inversion. Gravity data are for small problems with fewer than 2000 parameters. calculated along a traverse perpendicular to the strike and located Rate of mutation is another important component that has a strong central along the strike. The data are contaminated by uncorrelated influence on GA performance for binary inversion. Mutation intro- Gaussian noise with zero mean and standard deviation of 0.26 MGal duces genetic diversity and prevents the irrecoverable loss of poten- to simulate observed gravity data. Fig. 8(a) shows the true model we tially valuable genetic information if the population is too small or attempt to recover and (b) the true and noise contaminated gravity specific model features are lost during re-ordering of the population. data. There are a total of 41 data points. If mutation is too small, the GA takes a longer time to converge, and To perform binary inversion, the model region is divided into becomes inefficient for larger real-world problems. In the extreme 1407 rectangular cells (21 × 67). The model spans vertically from case when the mutation rate is zero, a small population often fails the surface to 4200 m depth, and horizontally from zero to 13 400 m. to converge. Finally, when we assign a large mutation rate, the pop- We use the model objective function shown in eq. (3) with the 5 5 ulation typically evolves quickly at the start, but the final solution weighting parameters being (αs , α x , α z ) = (1, 6.4 × 10 , 6.4 × 10 ), does not stabilize as convergence slows at the end. Resulting models which imposes the smoothness across four adjacent cells in both the contain large number of cells throughout the model region that have vertical and horizontal. We start with a population consisting of 600

Figure 6. SEG/EAGE seismic velocity model. Panel (a) shows a 3-D 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.

C 2006 The Author, GJI, 167, 543–556 Journal compilation C 2006 RAS 550 R. A. Krahenbuhl and Y. Li

Figure 7. 3-D 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 −0.2gcc−1. Similar to Fig. 6, panel (a) shows a 3-D perspective view and panel (b) shows the same cross-section as Fig. 6(b).

Figure 8. 2.5-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 cm−3. Panel (b) displays the true data (line) and noise-contaminated data (points). Noise added has zero mean and standard deviation of 0.26 MGal. individuals. To simulate a realistic application of the algorithm, we of grey, between zero (black), and one (white). When the models generated the individual models in the initial generation by incor- are displayed in this manner, the ﬂuctuation of grey may be viewed porating known top of salt into each model. Such prior information as an expression of entropy of the population, such as described is routinely available from seismic imaging or other independent by Rubinstein & Kroese (2004), with higher entropy in the lower sources. The unknown portion of these models is initialized with portion representing increased disorder. random zeros and ones. Fig. 9(a) shows the initial population, which While the individuals of the GA population are typically stored can be compared with the true model in Fig. 9(b). Although top of as algebraic vectors for ease of numerical computation, crossover salt is added as prior information, this feature is not enforced as a segments (or gene lengths) are actually small blocks in the cross- constant. In other words, all regions of the model may be altered section of models. One should not simply cut the model vector into during inversion to allow for uncertainty in the top of salt. segments. Instead, we cut the model in both dimensions and divide The model displayed in Fig. 9(a) presents the values for each cell the model cross-section into small blocks. This is a more physical of the model mesh, averaged over the entire starting population. approach to crossover in 2-D (and 3-D). A detailed implementation Since each model is initialized with random zeros and ones for all and analysis of this form of multidimensional crossover for GA is cells in the lower portion of salt, the average values appear as shades available in B¨ack (1993). The same boundary issue occurs as in 1-D

C 2006 The Author, GJI, 167, 543–556 Journal compilation C 2006 RAS Inversion of gravity data using a binary formulation 551

Figure 9. 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 and (b) is the true model we attempt to recover.

case when the dimensions of the model are not evenly divisible by Associated with each stage of convergence, there is a population the chosen gene lengths in the respective dimensions. However, it of models that is a snapshot of the evolutionary process. Mapping does not cause difficulties for the same reason discussed in the 1-D the evolution of the models at difference stages of the inversion is case. a valuable tool for understanding the evolutionary progress of the To carry out the inversion, we chose to use 3 × 3 cell blocks models. Fig. 11 displays the average of the population at selected as the basic genes for crossover, a mutation rate of 1/1100, and a generations (referred to as average models) as the GA progresses. regularization parameter of 2.0 × 10−7. Details of the choice of There are two dominant trends apparent in the model evolution. regularization parameter will be discussed in the next section, but First, visible by the 35th generation, entropy within the lower re- it suffices to say that this chosen value yields final data misfit of 41, gion of the models has decreased as inversion attempts to minimize which is the same as the number of data. Inversion achieves conver- structural complexity of the model and construct base of salt. Sec- gence by 300 generations. The CPU time required for this inversion ond, the steeply dipping flank on the left portion of the model starts with fixed regularization parameter is approximately 30 min on the to fill in by the 103rd generation. By generation 273, all individu- equivalent 1.5 GHz processor. Fig. 10 shows the convergence curve. als in the population have mostly converged to a similar solution,

Figure 10. Progress of the GA for the 2.5-D binary inverse problem of the salt body with a constant 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. The two merge by 300 generations, indicating the GA has converged.

C 2006 The Author, GJI, 167, 543–556 Journal compilation C 2006 RAS 552 R. A. Krahenbuhl and Y. Li

Figure 11. 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.

Constructed Model among the final population of solution. The final result illustrates 0 that binary inversion has successfully imaged the lower portion of 1000 salt in this problem. The steeply dipping structure at the left of the 2000 model has been successfully filled in with salt, and the grey region

z (meters) of disorder in the starting models beneath top of salt has been filled 3000 a) in with sediment (represented by 0s). Isolated cells of white and grey 4000 throughout the model region represent attempts to either fit noise in 0 2000 4000 6000 8000 10000 12000 x (meters) the data, or to compensate at depth for increased shallow salt while maintaining an appropriate data fit. True Model 0 We remark that the manner in which this inversion was set up

1000 and carried out also illustrates one of the greatest advantages of the formulation, that is, the ability to incorporate easily any prior 2000 information about the salt body into the inversion. Known top of

z (meters) 3000 the salt from seismic imaging or other independent sources can b) 4000 be easily input into the inversion through the initialization of the 0 2000 4000 6000 8000 10000 12000 starting population. Such features can be fixed during the inversion x (meters) if our confidence in them is high, or they can be altered during the inversion if their uncertainty is large. Such flexibility enables us to Figure 12. Comparison between the inverted and true model: (a) final con- make use of a wide range of prior information to enhance the final structed model. The image is the average over the final population and (b) original model to be recovered. inversion results.

4.3 2-D Gravity Problem with a Variable Density Contrast with only slight differences visible as grey cells throughout the model. The preceding example deals with single density contrast and rep- The final results are presented in Fig. 12. The top panel (a) illus- resents the simplest binary gravity inversion. We now proceed to trates the average solution over the entire final population. While the illustrate our algorithm using a more realistic example that incorpo- average model may not be the best solution, and in fact, the average rates density contrast reversal into the model to simulate the effect does not necessarily fit the data, this method of display illustrates of an annihilator. In such cases, data from the upper and lower por- common features present in the population of solutions. Regions of tions of the source body reverse sign. The effect on surface gravity solid black or solid white illustrate agreement for location of sed- data is that a portion of the measured signal is zero, which cre- iment or salt, respectively, while regions of grey indicate variance ates difficulties for continuous variable gravity inverse formulations.

C 2006 The Author, GJI, 167, 543–556 Journal compilation C 2006 RAS Inversion of gravity data using a binary formulation 553

−

− Figure 13. 2.5-D density contrast model from the converted SEG/EAGE salt model with density contrast reversal (a). Panel (b) displays the true data (line) and noise-contaminated data (points). Data passes through 0 MGal Figure 14. Inversion result for the 2.5-D problem with density contrast between positive and negative anomalies due to density contrast reversal. reversal. The top panel (a) is the constructed model in binary form and panel (b) is the inverted model converted back to density contrast. Developing an inversion method that can handle this type of problem is the motivation for our research on binary inversion. serve as an effective solver for binary inversion. As for any inversion The same 2-D section through the SEG salt model is modified to algorithms, however, the success of our algorithm depends to a large include density contrast reversal as shown in Fig. 13(a). As with the extent on the choice of pertinent parameters. Some are related to the previous example, the model region is divided into 1407 rectangular parameters that define the GA, but others are more fundamental cells (21 × 67). The model spans vertically from the surface to 4200 to the formulation itself. We have addressed some of these details m depth, and horizontally from zero to 13 400 m. Above 1800 m the in this section. The remaining crucial aspect is the choice of the salt body has a density contrast value of +0.2gcc−1, whereas below regularization parameter. We address it in the next section. that depth salt density contrast is −0.2gcc−1. The 2.5-D model has a finite strike length of 20 km in and out of the page and gravity 5 ROLE OF REGULARIZATION IN data are calculated along a traverse perpendicular to the strike and BINARY INVERSIONS located central along the strike. True data are calculated with 41 stations along a 40-km survey line. Observed data are generated by Regularization plays a crucial role in the inverse formulation. In the addition to the true data of uncorrelated Gaussian noise with the traditional inversion formulated with continuous variables, the zero mean and a standard deviation of 0.1 MGal, Fig. 13(b). The data misfit increases with the regularization parameter while the effect of density contrast reversal is apparent in the data with an model complexity decreases with it. The Tikhonov curve formed by anomaly varying between −1 and +2 MGal. plotting the data misfit as a function of model norm is a smooth, For the variable density contrast problem, we use the model objec- monotonically decreasing curve (Parker 1994; Hansen 1998). It is tive function shown in eq. (3) with the weighting parameters being important to understand, first, whether the same property holds true 6 5 (αs , α x , α z ) = (1, 1.4 × 10 , 6.4 × 10 ), which imposes the smooth- for the binary problem, and second, whether a Tikhonov curve can ness across four adjacent cells in the vertical and 6 in the horizontal. be stably calculated using GA, which is based on random search We chose to use 3 × 3 cell blocks as the basic genes for crossover, and combinatorial minimization. The importance of answering these a mutation rate of 1/900, and a regularization parameter of 5.0 × questions is twofold. First, they are fundamental to understanding 10−5. binary inversion using Tikhonov regularization. Secondly, the an- By 500 generations, a population of 2000 individuals has evolved swers directly affect how we can reliably and efficiently select a to a solution that defines a similar salt geometry while achieving regularization parameter. In the following, we first investigate these an expected data fit as in the preceding example. The total CPU two issues and then discuss the methods for choosing an optimal time is approximately 50 min on the same single processor PC. regularization parameter.

Fig. 14 presents the inverted model in binary form, that is, τ(r ), panel (a), and in density contrast form (b). The deep structure to the 5.1 Tikhonov curve left of the model has been successfully filled in with salt, and the remainder of the model region has been reduced to zero, representing The Tikhonov curve is the plot of the model objective function versus sediment. These results, therefore, illustrates that binary inversion the data misfit of the final models obtained by using different reg- can adequately solve for base of salt in the presence of density ularization parameters. The curve has a well-understood behaviour contrast reversal. in the case of continuous variable inversion (e.g. Parker 1994). To To summarize, the algorithm has worked well on a number of test investigate the role of Tikhonov regularization in binary inversion, a cases ranging from a simple 1-D mathematical example to synthetic Tikhonov curve is generated for the inversion of the data set shown 2.5-D gravity problems with different density contrast structures. in Fig. 8(b), which is associated with the 2.5-D constant density These results demonstrate the efficacy of the binary formulation in contrast model. We have performed 20 inversions with regulariza- the 2.5-D gravity problem. They have also shown that the GA can tion parameter β varying over 3 orders of magnitude (10−9 –10−6).

C 2006 The Author, GJI, 167, 543–556 Journal compilation C 2006 RAS 554 R. A. Krahenbuhl and Y. Li

β β

Figure 15. Tikhonov curve generated with regularization parameters vary- Figure 16. Tikhonov curve generated for the 2.5-D constant density salt −9 −6 ing from 10 to 10 . The plot illustrates that although the curve is coarse problem with model inlays to demonstrate regularization for binary inver- and non-monotonic on a fine scale, Tikhonov regularization plays a similar sion. Each regularization parameter is plotted by the final data misfit and role with binary inversion as with continuous variable inversion, and the model objective values. Based on discrepancy principle, the expected data Tikhonov curve behaves in a similar manner. The result also demonstrates misfit value is 41. This corresponds to a regularization value of approxi- that the GA can successfully construct a Tikhonov curve for binary inversion. mately 2.07E-7. The corresponding inverse model is inlayed within the plot. The true model is also inlayed for comparison, along with the two end models which overfit the data at small regularization, and underfit the data at The resulting Tikhonov curve is shown in Fig. 15. Numerous other larger regularization. examples have shown the same behaviour as in this example. These results have shown empirically that the two questions posed earlier 5.2 Choice of regularization have clear answers. First, Tikhonov regularization does play a similar role with binary inversion as with continuous variable inversion, Two techniques are typically utilized for determining appropriate and the Tikhonov curve behaves also in a similar manner. Secondly, regularization in continuous variable inversions. The first assumes the GA can successfully construct a Tikhonov curve. However, the known standard deviations of the observed data, and therefore, curve is monotonic at large scales only and has noticeable local chooses a regularization value from the Tikhonov curve so that the features that are non-monotonic. Furthermore, it is not precisely data misfit is equal to the number of data, which is the expected repeatable due to the random nature and combinatorial features of value of data misfit. This is called the discrepancy principle (Parker GA. Details are presented in the following. 1994; Hansen 1998). The second technique is to choose regulariza- Fig. 16 displays the same Tikhonov curve with overlays of differ- tion corresponding to the ‘elbow’ of the Tikhonov curve, which is ent models obtained from different regularization parameters. The the regularization value corresponding to the optimal trade-off be- plot shows that resulting models increase in structural complexity tween data misfit and model complexity. This is called the L-curve with overfit data for low regularization values, corresponding to so- criterion (Regiska 1996; Hansen 1998, Johnston & Gulrajani 2002). lutions at the lower right end of the Tikhonov curve. Likewise, as β The elbow point is given by the point of maximum curvature on the increases, the solutions tend to under-fit the data while the resulting Tikhonov curve. Our understanding about the role and behaviour of models are overly smooth, corresponding to the upper left portion Tikhonov regularization in binary inversions means that we can use of the curve. This trend is similar to that of the Tikhonov curve the same methods to determine the value of regularization param- generated for a continuous variable inversion. eters in binary inversions. We illustrate this point in the following There are also differences in the Tikhonov curve generated by using the first 2.5-D example. GA that are restricted to details. The Tikhonov curve is rougher and Let us return to the Tikhonov curve in Fig. 16. To assist in choos- non-monotonic on finer scales. This difference does not appear to ing between discrepancy principle and L-curve criterion, we exam- be a result of the binary nature of our inverse problem. It is more ine model results as a function of regularization, along with the related to the two primary components of GA, namely, crossover associated Tikhonov curve. Fig. 17 displays the constructed mod- and mutation, and to the fact that our algorithm seeks a near-optimal els from the binary inversion as a function of regularization value. solution. Termination of GA with finite accuracy in search of a near- As predicted by the Tikhonov curve in Fig. 16, the models have optimal solution means that the final curve will reflect the difference a trend from highly complex for low regularization values at the in evolution histories. The non-repeatability is also related to the top of Fig. 17, to overly smooth models for higher values near the termination of evolutions that involve random components before bottom. The best representation of the true model for the varying exact solutions are obtained. regularization values appears in Row 4, Column 3 of Fig. 17. The To produce idealized results for theoretic understanding, we can result from this chosen regularization value is presented separately obtain increasingly smoother and repeatable curves by using more in Fig. 18(a), along with the true model (b). Based on the Tikhonov stringent termination conditions, but at a huge computational cost. curve in Fig. 16, this corresponds to the point with data-misfit equal Seeking a near-optimal solution with a realistic accuracy means that to the number of data, that is, 41. To better illustrate the similar such roughness on the finer scales is acceptable. It does not seem to behaviour of regularization for binary inversion to continuous vari- impact the practical application of the algorithm. able, the true model is inlayed within the Tikhonov curve in Fig. 16,

C 2006 The Author, GJI, 167, 543–556 Journal compilation C 2006 RAS Inversion of gravity data using a binary formulation 555

β β β

Figure 18. Inverse model when regularization is chosen based on discrepancy principle (a). Panel (b) is the true model to be recovered. β β β section of the SEG/EAGE salt model with density contrast reversal. We also illustrate that regularization behaves similarly for binary inversion as continuous variable inversions. As a result, one has the ability to successfully construct a Tikhonov curve for estimating β β β regularization given ones understanding of the errors in their data. This includes estimation by discrepancy principle when errors in the data are know, and more significantly, with L-curve for more practical applications where little to no information may be available on the noise in ones data. Future research on binary inversion will β β β focus on an increase in the number of parameters, a more realistic background density profile with sizeable nil zone, and application Figure 17. Comparison of inverted models for 18 different regularization to full 3-D inverse problems. Practical applications may require parameters. Each panel displays the average of models in the final population more efficient solution methods. Currently, we are exploring other β for a given value of regularization, . For small values (e.g. the top left derivative-free methods as well as more efficient implementations panel), the model overfits the data and is structurally complex. For large of GA, such as parallel processing of sub-populations, by utilizing 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 the special structure of the problem. 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. ACKNOWLEDGMENTS This work was supported by the ‘Center for Gravity, Electrical, & along with the solutions for the high, low, and optimal regularization Magnetic Studies’ (CGEM) and the industry consortium ‘Grav- parameters. Thus, choosing regularization based on data-misfit, that ity and Magnetics Research Consortium’ at the Colorado School is discrepancy principle, is the appropriate choice for our problem. of Mines. The sponsoring companies are ChevronTexaco, Cono- We note that this choice falls in the close vicinity of the elbow on coPhilips, Total and Anadarko. the Tikhonov curve. Therefore, choosing regularization parameter by L-curve criterion would have worked well in this case also.

REFERENCES 6 CONCLUSIONS Aminzadeh, F., Brac, J. & Kunz, T., 1997. 3-D salt and overthrust mod- We have developed a binary inversion method for inverting gravity els, SEG/EAGE 3-D modeling series, No. 1: distribution CD of salt and data for subsurface structure that has well-defined density contrasts, overthrust models available through SEG. and analysed its behaviour. The method is designed to overcome the Bäck, T., 1993. Optimal mutation rates in genetic algorithm. In Proceedings difficulties introduced by the presence of nil zones in salt imaging. of the 5th International Conference on Genetic Algorithms, pp. 2–8, ed. Tests show that the binary formulation provides an effective means Forrest, S., Morgan Kaufmann Publishers, Inc., San Mateo, CA, USA. to incorporate known density into the inversion. In addition, it can Barbosa, V.C.F., Silva, J.B.C. & Medeiros, W.E., 1999. Gravity inversion of provide a sharp boundary for the subsurface while maintaining the a discontinuous relief stabilized by weighted smoothness constraints on depth, Geophysics, 64, 1429. flexibility of density inversions. The trade-off is the increased com- Bui, T.N. & Moon, B.R., 1995. On multi-dimensional encoding/crossover, in putational complexity of minimization with binary constraints. Proceedings of the 6th International Conference on Genetic Algorithms, We have explored the utility of the GA and shown that it can pp. 49–56, ed. Eshelman, L.J., Morgan Kaufmann Publishers, Inc., San serve as an effective solver for this problem, generating geologically Francisco, CA, USA. reasonable models that fit the data. To date, the method has solved Camacho, A., Montesinos, F.& Vieira, R., 2000. Gravity inversion by means several 2.5-D gravity inverse problems successfully, including a 2-D of growing bodies, Geophysics, 65, 95–101.

C 2006 The Author, GJI, 167, 543–556 Journal compilation C 2006 RAS 556 R. A. Krahenbuhl and Y. Li

Chai, Y. & Hinze, W.J., 1988. Gravity inversion of an interface above which Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller,A. & Teller,E., 1953. the density contrast varies exponentially with depth, Geophysics, 53, 837– Equation of state calculation by fast computing machines, J. Chem. Phys., 845. 21, 1087–1092. Chambers, L., 1995a. Practical Handbook of Genetic Algorithms, Applica- Mickus, K.L. & Peeples, W.J., 1992. Inversion of gravity and magnetic data tions, Vol. I., CRC Press, Boca Raton, Florida. for the lower surface of a 2.5 dimensional sedimentary basin, Geophys. Chambers, L., 1995b. Practical Handbook of Genetic Algorithms, New Fron- Prospect., 40, 171–193. tiers, Vol. II, CRC Press, Boca Raton, Florida. Nagihara, S. & Hall, S.A., 2001. Three-dimensional gravity inversion using Cheng, D., 2003. Inversion of gravity data for base salt, PhD dissertation, simulated annealing: Constraints on the diapiric roots of allochthonous Colorado School of Mines. salt structures, Geophysics, 66, 1438–1449. Deb, K. & Agrawal, S., 1999. Understanding interactions among genetic Nulton, J.D. & Salamon, P., 1988. Statistical mechanics of combinatorial algorithm parameters, in Foundations of Genetic Algorithms, pp. 265– optimization, Phys. Rev., A37, 1351–1356. 286, eds, Banzhaf, W. & Reeves, C., Morgan Kaufmann Publishing, San Oldenburg, D.W., Li, Y., Farquharson, C.G., Kowalczyk, P., King, A., Ara- Francisco, CA. vanis, T.C.G., Zhang, P. & Watts, A., 1998. Applications of geophysical Floudas, C.A., 1995. Nonlinear and Mixed-integer Optimization: Funda- inversions in mineral exploration, Leading Edge, 17,(4), 461–465. mentals and Applications, Oxford University Press, New York. Oldenburg, D.W., 1974. The inversion and interpretation of gravity anoma- Gibson, R.I. & Millegan, P.S., 1998. Geologic applications of gravity and lies, Geophysics, 4, 526–536. magnetics: case histories, SEG Geophysical Reference Series, No. 8, Pal, S.K. & Wang, P.P., 1996. Genetic Algorithms for Pattern Recognition, AAPG Studies in Geology, 43. CRC Press, Boca Raton, Florida. Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization and Ma- Pardalos, P.M. & Resende, M.G.C., 2002. Handbook of Applied Optimiza- chine Learning, Addison-Wesley, New York. tion, Oxford University Press, New York. Green, W.R., 1975. Inversion of gravity profiles by use of a Backus-Gilbert Parker, R.L., 1994. Geophysical Inverse Theory, Princeton University Press, approach, Geophysics, 40, 763–772. Princeton, New Jersey. Guillen, A. & Menichetti, V., 1984. Gravity and magnetic inversion with Pedersen, L.B., 1977. Interpretation of potential field data: a generalized minimization of a specific functional, Geophysics, 49, 1354–1260. inverse approach, Geophys. Prospect., 25, 199–230. Hansen, P.C., 1998. Rank-deficient and Discrete Ill-posed Problems: Nu- Pohlheim, H., October 13, 1997. (last modified), January 17, 2006 (last merical Aspects of Linear Inversion, Society for Industrial and Applied viewed). Evolutionary algorithms: principles, methods and algorithms. Mathematics (SIAM), Philadelphia, PA. http://www.systemtechnik.tu-ilmenau.de/∼pohlheim/GA Toolbox/ Herrera, F. & Lozano, M., 1996. Adaptation of genetic algorithm pa- algoverv.html. rameters based on fuzzy logic controllers, in Genetic Algorithms Reamer, S.K. & Ferguson, J.F., 1989. Regularized two-dimensional Fourier and Soft Computing, pp. 95–125, eds Herrera, F., Verdegay, J.L., gravity inversion method with application to the Silent Canyon Caldera, Physica-Verlag, Heidelberg, Germany. Nevada, Geophysics, 54, 486–496. Herrera, F., Lozano, M. & Verdegay, J.L., 1998. Tackling real-coded genetic Reeves, C.R., 1993. Using genetic algorithms with small populations, in algorithms: operators and tools for the behaviour analysis, Art. Intell. Rev., Proceedings of the 5th International Conference on Genetic Algorithms, 12, 265–319. pp. 92–99, ed. Forrest, S., Morgan Kaufmann Publishers, Inc., San Mateo, Hinterding, R., Gielewski, H. & Peachey, T.C., 1995. The nature of mutation CA, USA. in genetic algorithms, in Proceedings of the 6th International Conference Regiska, T., 1996. A regularization parameter in discrete ill-posed problems, on Genetic Algorithms, pp. 65–72, ed. Eshelman, L.J., Morgan Kaufmann SIAM J. Sci. Comput., 17(3), 740–749. Publishers, Inc., San Francisco, CA, USA. Roy, L., Sen, M.K., Blankenship, D.D., Stoffa, P.L. & Richter, T.G., 2005. Johnston, P.R. & Gulrajani, R.M., 2002. An analysis of the zero-crossing Inversion and uncertainty estimation of gravity data using simulated an- method for choosing regularization parameters, SIAM J. Sci. Comput., nealing: an application over Lake Vostok, East Antarctica, Geophysics, 24(2), 428–442. 70, J1–J12. Jorgensen, G. & Kisabeth, J., 2000. Joint 3-D Inversion of gravity, magnetic Rubinstein, R.Y.& Kroese, D.P., 2004. The Cross-entropy Method: A Unified and tensor gravity fields for imaging salt formations in the deepwater Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Gulf of Mexico, 70th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Machine Learning, Springer Science+Business Media Inc., New York. Abstracts. Sambridge, M. & Mosegaard, K., 2002. Monte Carlo methods in geopohys- Kirkpatrick, S.C., Gelatt, D. & Vecchi, M.P., 1983. Optimization by simu- ical inverse problems, Rev. Geophys., 40, 3-1–3-29. lated annealing, Science, 220, 671–680. Scales, J.A., Smith, M.L. & Fischer, T.L., 1992. Global optimization meth- Last, B.J. & Kubik, K., 1983. Compact gravity inversion, Geophysics, 48, ods for multimodal inverse problems, J. Comp. Phys., 103(2), 258– 713–721. 268. Li, Y. & Oldenburg, D.W., 1998. 3-D inversion of gravity data, Geophysics, Sen, M.K. & Stoffa, P.L., 1995. Global Optimization Methods in Geophys- 63, 109–119. ical Inversion, Advances in Exploration Geophysics, V4, Elsevier, Ams- Li, Y., 2001. 3-D inversion of gravity gradiometer data, in 71st Ann. Internat. terdam, the Netherlands. Mtg., Soc. Expl. Geophys., Expanded Abstracts. Smith, M.L., Scales, J.A. & Fischer, T.L., 1992. Global search and genetic Litman, A., Lesselier, D. & Santosa, F., 1998. Reconstruction of a two- algorithms, Geophys.: Leading Edge Explor., 11(1), 22–26. dimensional binary obstacle by controlled evolution of a level-set, Inverse Tikhonov, A. & Arsenin, V., 1977. Solutions of Ill-posed Problems, V.H. Problems, 14, 685. Winston & Sons, Washington DC.

C 2006 The Author, GJI, 167, 543–556 Journal compilation C 2006 RAS