Constrained inversion of gravity fields for complex 3-D structures

GEOPHYSICS, VOL. 66, NO. 2 (MARCH-APRIL 2001); P. 501–510, 6 FIGS., 1 TABLE.

Constrained inversion of gravity fields for complex 3-D structures Roberto A. V. Moraes∗ and R. O. Hansen‡ A full 3-D structural inversion algorithm is presented which is suitable for the interpreting 3-D bodies using surface gravity data. It involves the inversion for the geometry of the causative bodies (a nonlinear problem) using homogeneous polyhedral models. This modeling engine was chosen for its compatibility with detailed seismic models, and with the eventual goal of allowing subdivision of the polyhedra into tetrahedral regions for physical property modeling (Moraes, 1997). It differs from other approaches that use parametric inversion (a linear problem) (Bear et al., 1995, among others) that aim to find a physical property distribution for a parallelepiped (or cubic) cell model of the subsurface (as in Li and Oldenburg, 1996). To overcome nonuniqueness, the inversion scheme is applied to highly constrained structural models using Tikhonov regularization. Therefore, we assume that data are available to constrain the model in such a way that some parts are completely fixed, while others will be varied to fit the observed gravity data. If used with care, this procedure can lead to an inversion that is unique if not completely well posed. The motivation for developing this inversion scheme is that even simple models of comparatively modest sizes have many degrees of freedom. The forward modeling algorithm can handle models whose size is such that they are extremely tedious to adjust manually. Furthermore, when adjusting a 3-D model, one may not be capable of visualizing the correct geometry to fit the data. Inversion of the kind described in this paper can be valuable in making adjustments to limited areas of a model, since even the most user-friendly adjustment schemes designed to modify model parameters tend to be laborious to work with.

ABSTRACT

As part of a research program to develop gravity interpretation tools that can be merged with seismic techniques, a full 3-D complex structural inversion scheme for (possibly multibody) polyhedral models has been developed. The forward modeling algorithm was adopted from previous work. Because the inverse problem is generally very ill posed, several methods of regularizing the inversion were investigated and a combination of the most useful was adopted. The combination includes (i) a structured matrix formulation for the system equations, (ii) an analytical expression for the Jacobian calculation, (iii) first-derivative damping, (iv) a choice of damping parameter based on a variation of the trust region method, (v) a weighted scheme for parameter correction, and (vi) complete freezing of degrees of freedom found not to influence the gravity field significantly. This combination yields a robust inversion which was successfully demonstrated on data over the Galveston Island salt dome, offshore Texas. Variations of the technique should be applicable to magnetic data, which would make the method useful for mining problems and petroleum exploration settings involving volcanic structures.

INTRODUCTION

Possible applications of potential field methods in hydrocarbon exploration include the use of gravity data in prospect development and reservoir management. Given the tremendous advances in 3-D seismic imaging, gravity methods are most likely to be useful in areas where seismic imaging is difficult or very expensive, such as in the case of overthrusts, below salt sheets, and at salt dome flanks. Such applications require interpretation tools which can be applied to complex 3-D geometries.

INVERSE MODELING

The minimization scheme devised is a function of the nature of the problem to be solved and the metrics chosen to define the closeness of the solution. The first factor follows from the nature of the problem: the shape of the causative body, which is nonlinear in nature. The second comes from the choice of the Euclidean norm in Hilbert space to define the closeness of

Manuscript received by the Editor January 11, 1999; revised manuscript received August 29, 2000. ∗ Universidade de Bras´ılia, Instituto de Geociencias, ˆ Laboratorio ´ de Geof´ısica Aplicada, CEP 70910-900 Brasilia D.F., Brazil. E-mail: rmoraes@ unb.br. ‡Pearson, deRidder and Johnson, Inc., 12640 W. Cedar Dr., Ste. 100, Lakewood, Colorado 80228. E-mail: [email protected].  c 2001 Society of Exploration Geophysicists. All rights reserved. 501

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fit, which leads to a least-squares approach. Hence, the minimization problem can be stated as

given f : n → m , m ≥ n

(1)

find ϕ∗ ∈ n , for which f (ϕ∗ ) ≤ f (ϕ) for every ϕ ∈ n , (2) where f is an appropriate functional, ϕ is a parameter vector, and ϕ∗ is its optimized version. If E i (x, ϕ), x being the position coordinate vector, is the error in the model prediction for the ith observation, given by

E i (x, ϕ) = Yi − Yic (x, ϕ);

i = 1, 2, . . . , n,

(3)

where E i (x, ϕ) is a nonlinear functional defined over n , Yi is the observed gravity value at station i, and Yic (x, ϕ) is the value of y predicted by the forward modeling algorithm at the same point, the problem is to compute estimates of ϕ such that the squared error is as small as possible, or

f (x, ϕ) =

m 1


2

E i2 (x, ϕ), m ≥ n.

(4)

i=1

Equivalently,

=

1 1 E(x, ϕ)2 = Y − Yc (x, ϕ)2 . 2 2

(5)

Although the problem under consideration leads to a constrained minimization such that

min

ϕ∈
⊂n

f : n → m , m ≥ n

(6)

where
is a closed connected region, since the solution will lie in the interior of
, it can still be viewed as an unconstrained nonlinear least-squares minimization problem. Here it is supposed that the solution must be close to the starting model. This means that the residual function is both well behaved and with terms of small magnitude compared to the signal being modeled (small residuals). Therefore, the solution given by the model expressed in equation (5) can be thought as arising from an overdetermined system of nonlinear equations E(x, ϕ) = 0 (More, 1983; Bjorck, 1996). It is then possible to approximate E(x, ϕ) by an affine model in the neighborhood of a given point ϕ + δ by

E(xi , p + δ) = E(xi , p) + J(xi , p)δ

(7)

E = Ek + Jδ,

(8)

or

which is the so-called Gauss-Newton method. The solution vector δ now appears with a linear dependence in equations (7) and (8) and can therefore be found by the standard least-squares method of setting ∂ f /∂δ j = 0 for j = 1, 2, . . . , m, or

min E(xi , p) + J(xi , p)δ2 , δ

(9)

which is an approximate solution to equation (5). Note that in equations (7) and (9), ϕ is replaced notationally by p (parameter), the final values of p being the least-squares estimate of ϕ. The vector δ is a small correction to p. The brackets  are used to distinguish expectations based upon the

linearized model from those based upon the actual nonlinear model (Marquardt, 1963). Forward model Calculating the gravity attraction using the polyhedral modeling approach at a station r is based on the evaluation of

∂U (r) = −Gρ gz (r) = − ∂z



  1 ∂ dv, ∂z r − r

(10)

where ρ is the density contrast between the model and its surroundings and G is the gravitational constant (Goetze and Bernd, 1988; Pohanka, 1988). The use of the Gauss divergence theorem gives



gz (r) = −Gρ

  1 cos(n, k) d S, r − r

(11)

surface

leading to a surface integral that has to be evaluated over the whole polyhedron surface. The cosine term determines the direction of the surface element d S with respect to the Cartesian coordinate system (Goetze and Bernd, 1988), where n is the outward normal to the surface element d S and k is the unit vector in the z-direction, taken in the present case as positive downward. Since the cosine term is constant for any polyhedral face S j for j = 1, 2, . . . , K , K being the number of faces, the attraction effect of a polyhedron can be obtained by the superposition of gravity effects of its individual surfaces S j (Goetze and Bernd, 1988), or  

gz (r) = −Gρ

K


cos(n, k)

j=1



1 r − r



ds j .

(12)

surface

The evaluation of this surface integral can be further simplified by its transformation into a line integral around the perimeter of each of the polygonal facets by using either the Gauss theorem in two dimensions (Pohanka, 1988) or Stokes’ theorem (Holstein and Ketteridge, 1996; Ivan, 1996). It is important to note that, according to equation (12), the contribution of a given facet to the anomalous gravity value at any given station is governed by the angle that its outward normal makes with the vertical direction (z-axis). This angle will be maximum when the facet is perpendicular to the z-axis and nil when parallel to it. Among the derivations worked out to make this approach operational (Bott, 1963; Paul, 1974; Barnett, 1976; Okabe, 1979; Goetze and Bernd, 1988; Furness, 1994; Ivan, 1994, 1996; Holstein and Ketteridge, 1996; among others), the expressions developed by Pohanka (1988) for the gravity field seem to be most concise and best adapted to numerical implementation. His paper summarizes the theoretical concepts involved and gives a full and precise account of the important steps he took in this direction. Pohanka’s formula for the gravity field can be given as

gz (r) = −Gρ

K
k=1

nk

L(k)


φ[u k,l (r), vk,l (r), wk,l (r), z k (r)],

l=1

(13) c

where gz is the gravity field vector [Y (r) in equation (4)], r is the field position, G is the universal gravity constant, ρ is

Structural Gravity Inversion

the density of the polyhedron, K is the number of faces of the polyhedron, nk is the outward unit normal to the kth face, L(k) is the number of edges on the kth face, u k,l (r) is the projection on the unit vector along the {k, l}th edge of the vector from the beginning point of the edge to the field position, vk,l (r) is the projection on the unit vector along the {k, l}th edge of the vector from the end point of the edge to the field position, wk,l (r) is the projection on the unit vector perpendicular to the {k, l}th edge and to the unit normal to the face, directed outward from the edge, of a vector from a point along the edge to the field position, and z k (r) is the projection along the unit normal vector to the face of the vector from a point on the face to the field position. Jacobian The computation of the Jacobian used in the present approach relies on an analytical solution of

J(x, p) =

 N  M

∂(gz )i (x, p) i=1 j=1

,

∂pj

(14)

where (gz )i (x, p) is given in equation (13), M is the total number of stations or sampling points, and N is the total number of parameters of the model. Expressing the gravity gradient in arbitrary components of the gravity field and using equation (13),

∇ j (gz )i (x, p) = −Gρ

K


nki

k=1 3


∇ j φ[u k,l (x, p)vk,l (x, p)wk,l (x, p)z k (x, p)],

(15)

l=1

where K is the number of facets defining the model. This result shows that it is only necessary to consider the derivative of the function φ (Ivan, 1996). Hence, the gravity gradient can be expressed as

(gz )i j (x, p) = −Gρ

K


nki

ψ j [u k,l (x, p)vk,l (x, p)wk,l (x, p)z k (x, p)].

(16)

l=1

This expression is used here to compute the Jacobian and is similar to that obtainable from equation (16). Regularization Ill-posed problems can be dealt with satisfactorily by using some kind of regularization (Dmitriev and Karus, 1987; Glasko et al., 1987; Goncharsky, 1987; Zhdanov, 1993), which means restricting the solution to a subset of the parameter space, or p ∈ Pc , with Pc ⊂ P, by using prior information. If this also minˆ imizes a positive functional R(p), a conditional extremum is present (Dmitriev and Karus, 1987), or

 ˆ min{R(p)} p∈Pc

E(p ) + J(p )δ ≤  k k 2

where  is the noise level in the data or the tolerance. Instead, it may be easier to find an unconditional extremum of the parametric functional (Dmitriev and Karus, 1987), expressed by the following regularized problem (Bjorck, 1996):



ˆ · δ22 , min E(pk ) + J(pk )δ22 + R δ

,

(17)

(18)

ˆ is the roughening or regularizing matrix, expressed as where R

ˆ = R

   λk Dk · ∂ i ,

i = 0, 1, 2

(19)

(Parker, 1994; Scales and Smith, 1996), where λk ≥ 0 is a parameter (Lagrange multiplier) that limits the size of δ k (Marquardt, 1963; Bjorck, 1996); Dk is the scaling diagonal matrix (More et al., 1980; More, 1983), both at iteration k; and ∂ i ∈ n×n is a partial derivative operator of the ith order. The ∂ i operator is a desirable feature when a smooth solution for the inverted model is sought. In the present formulation, both the first and second partial derivative can be used and are approximated by difference operators. The first difference (i = 1) yields the flattest and the second (i = 2) the smoothest models in the inversion process (Vandecar and Snieder, 1994). Formulated that way, this problem is also called damped least squares (Bjorck, 1996). The regularization scheme proves to be the heart of the inversion algorithm. It is an extremely important factor in driving this process to a minimum. Therefore, as defined here, the regularizing matrix is composed of three terms: (1) the Lagrange multiplier λk , (2) the scaling matrix Dk , and (3) the derivative operator ∂ i . The scheme chosen to control and determine λk belongs to a class known as trust region methods. Its development can be traced back to the works of Levenberg (1944) and Marquardt (1963) on nonlinear least-squares problems (More et al., 1980; Sorensen, 1982; More, 1983; More and Sorensen, 1983; Bjorck, 1996). The guiding principle in this class of methods is that λk is given as a solution to a subproblem with a bound on the step (More, 1983). It is chosen so the solution δ k+1 is likely to improve the current approximation to the optimization problem. Equation (18) is related to the least-squares problem with quadratic constraint

min{E(pk ) + J(pk )δ2 : Dk · δ2 ≤ k }.

k=1 3


503

(20)

The constraint given in this equation is binding, meaning that λk > 0. Hence, the set of feasible vectors δ, such that Dk · δ2 ≤ k in equation (20), can be thought of as a region of trust for the affine model (Bjorck, 1996), or

E(p) ≈ E(pk ) + J(pk )δ k ;

with δ k = p − pk .

(21)

An important consideration in solving many practical problems is that independent (and/or dependent) variables may vary greatly in magnitude. The ranges in which these variables are defined are referred to as the scales of these respective variables (Dennis and Schnabel, 1996). Scaling the independent variables of a given problem is important in calculating terms such as the Euclidean norm of two vectors, or p+ − pc 2 . In the present formulation, the scaling matrix was set as the column norm of initial Jacobian, or (Diag )i 1 = max1≤i≤n { ji 1 }. Derivative operators, approximated by difference operators, ˆ to smooth the can be set to act upon the regularizing matrix R model obtained from the inversion process. They were found

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necessary to prevent some ill-conditioned parameters from behaving unpredictably (damping action) and to drive the convergence of the inversion process. Parameter calculation The parameter vector can be updated at each iteration with the results of the last one using

pk+1 = pk + δ k ,

(22)

where pk is the parameter vector at iteration k (current), δ k is the correction computed by solving equation (18), and pk+1 is the new parameter vector for the next iteration. Tests performed with the algorithm showed that if the parameter vector is perturbed by a fraction of δ k or υ kT · δ k , there is an enhancement in the convergence of the inversion process. The elements of the scaling vector υ k are computed separately for each element of the δ k vector as uniform deviates in the interval [0, 1] of a minimal random number generator of Parker and Miller type, with Bays-Durham shuffle and added safeguards (Press et al., 1992). The seed used as input to the random number generator routine is kept constant within each iteration. These corrections are further reduced by multiplying υ kT · δ k by √ λk , or

pk+1 = pk +

  T  λk υ k · δ k .

(23)

The random multiplier vector acting upon the perturbation vector also plays an important role in breaking some local minima and allowing the process to seek other directions toward the global one (see Figure 2). This comes from the fact that when √ the process is stuck in some minimum, different values of λk (υ kT · δ k ) are tried in combination with pk to produce a new pk+1 ; pk+1 is only accepted if there is an improvement in the convergence of the process. Final discussion Equation (18) could also be written as



min g(pk ) + A(pk )δ22 + λk Dk · δ22 ,

(24)

with

A[m×m] = JT J 

and

g

[m×1]

=



m


∂ fi (Yi − f i ) ∂ pj j=1

(25)

 = JT (Y − fk ) = JT Ek . (26)

The system of linearized equations to be solved takes the form



 A + λk DkT Dk δ k = gk ,

(27)

which is the regularized normal equations system. Thus, the simplest way to obtain the correction δ k is to use Cholesky decomposition on the linearized system of equation (27). Another way is to recognize that equation (34) are the normal equations for the problem



   J(pk ) E(pk ) δk = √ λk Dk 0

(28)

and to solve this structured least-squares problem using QR decomposition with column pivoting (More, 1978). The main advantage of the normal equations formulation is speed. It is possible to solve the system in (27) twice as fast than that of equation (28). The problem is that normal equations are particularly unreliable when λk = 0 and J is nearly rank deficient. Moreover, the formation of JT J or DT D can lead to unnecessary underflow and overflow, which is not the case with the system in equation (28). Hence, in this case it is more advantageous to sacrifice speed and gain reliability (More, 1978). Another important point to consider is that the condition number of the system in equation (28) is the square root of that expressed by equation (27). This also improves the resolution of the system of equations, which has a system matrix that is almost singular in many of its terms to the numerical evaluation. Having decided to use the structured matrix formulation, we focused attention on which regularizing scheme to use. Tests on synthetic data using the possible combinations of the ele√ ˆ = λk (Dk · ∂ i ), with ments forming the regularizing matrix R i = 0, 1, and 2, showed that the best results were obtained √ ˆ = λk (Dk · ∂ 1 ), where ∂ 1 is the first difference when using R operator. With this approach the system of equations to be solved in the least-squares sense can be presented as



   J(pk ) r(pk )  δk = , √  0 λk Dk · ∂ 1

(29)

which is the adopted formulation. This expression is recommended by Parker (1994) as one of the better formulations of the inversion process (Moraes, 1997). COMPUTATIONAL ALGORITHM

The computation relies on the forward modeling concept adopted: homogeneous polyhedra. To fit into this concept, the 3-D geological problem to be modeled must first be broken into closed units or blocks where the density is assumed to be constant. The final surfaces these blocks assume can be any that the subsurface geologic prior information supports. The best representation of the polyhedron surface is in terms of triangles because the vertices of a triangle always define a planar figure in 3-D space. No matter how the vertices change position during the inversion process, each triangle always remains a planar figure. This is an important and desirable characteristic; it validates all the assumptions used in the derivation of the forward model formulation. With any other polygon the distortion from a plane in the surface of the facet, caused by movement of the vertices during the inversion, would make the problem handling awkward and the formulation more complicated. Since its three vertices lie in a plane in space, there will always be a single outward normal to any given facet of the polyhedron. As such, the elements that define the vectorial reference system in the facet (nk , ui,i+1 , vi,i+1 , wi,i+1 ) will hold, with K being the number of facets and i = 1, 2, 3, labeling the edges of a triangular facet. The decision to use the triangular facet as the building block for the polygonal surfaces defining the bodies also renders the data structure very simple. The model can be specified by (i) the number of vertices forming it, (ii) the number of facets that define it, (iii) a list of all model vertex coordinates, (iv) the

Structural Gravity Inversion

physical property distribution on both sides of each facet and (v) a list of the triplets of vertices defining each facet. With this arrangement, calculating the gravity model response is simple. VALIDATION TESTS

During the minimization of the residuals, only the gravity contributions of the facets defined by those vertices free to relax are relevant. These contributions are maximized when the facets formed by these free vertices are perpendicular to the z-axis. Therefore, the inversion process will best resolve vertices forming facets lying nearly parallel to the x–y-plane. Conversely, vertices forming facets parallel to the z-direction will behave as singularities in the inversion process. It is also interesting to consider that vertices forming facets that project themselves on, or are contained within a larger boundary of the polyhedral model (like the vertices in the center of the faces of the reference cube model), are singular in the inverse process. They are almost certainly poorly resolved or not resolved at all. We used a simple geometrical model to test the validity of the method. It is represented by a cube with one extra vertex placed in the center of each of its faces. This gives a model with a total of 14 vertices, allowing for a symmetrical triangulation at each face. Hence, each face is discretized into four triangles, given a total of 24 triangular facets and 42 parameters to be inverted for. The reference system that ties the cube to the measurement grid is right handed, with the y-axis pointing north, the x-axis pointing east, and the vertical z-axis positive downward. The cube has 2000 m edges, with its top face 500 m from the zerolevel observation grid. The center of the cube has coordinates (7000, 7000, 1500). The density contrast used was −400 kg/m3 (Moraes, 1997). The anomalous gravity response was sampled on a regular grid located at z = 0. Its origin has coordinates (0, 0, 0) and extends 15 000 m in each of the horizontal positive directions, with 15 sampling points in each direction. Tests were run on models obtained by distorting vertices of the reference cube by an amount approximately ±15% of the length of its edge, successively, at its bottom (model A), upper (model B), and lateral sides (model C, face perpendicular to the positive x-axis), forming prismatic bodies with a pyramidal termination on the deformed side (Moraes, 1997). The reference (noise-free) data used were obtained by forward modeling the reference model. Tests were also carried out on noise-corrupted data, supposing that the noise is randomly distributed over the measured data and has a Gaussian distribution with zero mean and standard deviation, which is the desired fraction of contamination (used: 2%, 5%, and 10%) of the standard deviation of the data (Moraes, 1997). When performing these tests, three setups√ for the regu√ ˆ = λ(D · ∂ 1 ), and ˆ = λD, R larization matrix were tried: R √ √ ˆ = λ(D · ∂ 1 ), yielded the ˆ = λ(D · ∂ 2 ). The second setup, R R best performance. The parameter vector, in all cases, was corrected using pk+1 = pk + λk (υ kT · δ k ) whenever the perturbed parameter vector produced convergence in the inversion process. It could be thought at first that using random multipliers to modify the correction made on the parameters would give a random behavior in the final result of an inversion process. This is not

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the case. In many runs of the same test, the repeatability was excellent. We must remember that just a small part of the parameter is subject to this random process: the correction that is added to it. This procedure helps break the encasing effect that some local minima have on the process, allowing convergence to other levels in the minimization of the residuals (as can be seen in Figure 1a). It represents an original contribution in enhancing the convergence of the optimization process beyond that mentioned in the published literature.

Limitations The resolution of inverse modeling has limitations inherent in the forward-modeling algorithm. These limitations can be traced back to the general principles of gravity modeling and the concept of contact surfaces. Therefore, it is not surprising that the worst scenario in the tests is represented by model C. Since the contribution of a given facet to the vertical anomalous gravity field response is proportional to its horizontal projection, it is easy to understand why model C is a difficult case. What makes the inversion of such a model possible is that the perturbed triangular facets have horizontal projection at the beginning of the inversion. The inversion process tries to bring the perturbed facets to the position they have in the reference cube model. In doing so, it ends up bringing them nearly to a plane surface, slightly angled with respect to the vertical direction (see Figures 1 and 2; b, c, and d). Furthermore, the noise level presented in the data works against resolution (compare the results in Figures 1a and 2a). Another desirable feature of the proposed algorithm is that when convergence is achieved, the volume of the perturbed model is reduced to one closest to that of the reference model. Therefore, the final model may have a distorted shape when compared to the reference, but the volume is the smallest possible to justify the anomalous gravity field being modeled—in this case, the volume of the reference cube model. The simulation with the same model and different random noise populations in the contamination of the reference data, when analyzed in a reverse sense, showed how critical the initial model is to convergence of the inversion process. It must be as close as possible to the most geologically plausible model. If, in this situation, convergence is not attained or is attained but not as expected, it would point out that it would be advisable to revise the working hypothesis. The case of model A is interesting in this respect. Another issue illustrated by these tests concerns vertices that prove to be in a singular position during inversion. These vertices can possibly be eliminated from the model and facets formed by them redefined, to give a more stable solution to all vertices in the model. As such, to some extent this condition spots vertices that might be eliminated from the model. GRAVITY INVERSION FOR COMPLEX 3-D STRUCTURES

The algorithm was tested over the gravity anomaly of the Galveston Island salt dome in the Gulf of Mexico. The study area is located approximately 100 km south of Galveston, Texas, on the Continental Shelf in the Gulf of Mexico. The salt dome is centered in the A64 block of the Galveston lease area (Figure 3). The compiled Bouguer gravity map encompasses an area delimited by the coordinates 307 and 336 km east, 3129 and

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3156 km north. The anomalous signature is centered around the coordinates 321 km east and 3143 km north (Figure 4). The gravity relief is represented by a somewhat concave surface closing to the southeast. It is roughly oriented northeast– southwest and dips to the southeast corner of the area, exhibiting a gradient on the order of 7 mGal/km. The anomaly associated with the salt dome itself is defined by a subcircular feature represented by a gravity high with an amplitude of 2.5 mGal. Seismic reflection data and well-log density data were used to build geological models for the site. The model extended down to 4600 m (15 000 ft) and included a horizontal layered model and the salt structure that pinched through it (Moraes, 1997). This salt dome was once thought to be deeply rooted, attached to a source bed at depth. The forward modeling of its gravity signature, however, suggested that it might be detached or only thinly rooted, with an associated overhang of some 15 km2 (Fueg, 1995). The forward approach was able to model quite well the gravity signature observed over the salt struc-

ture (Fueg, 1995). However, some small residual differences are not accounted for by this model (Figure 5). Hence, we decided to use it as a starting guess and see if improvements could be reached using the inversion procedure in question. The complete model used, obtained from the assemblage of its corresponding cut layer and edge extension models, had a total of 3186 vertices and is defined by 3425 triangular facets. The model was generated using GOCADTM (Moraes, 1997). The detached salt model was generated with its upper topography controlled by seismic data. The bottom surface was modeled by 2-D forward gravity modeling of the original gravity profiles. Its final discretization produced an anomalous body composed of 1285 vertices delimiting 1736 facets. The inversion test used the original Bouguer gravity data, comprised of 389 stations. It started with 89 free vertices (or 267 parameters) representing the delimiters of all faces whose outward normal pointed downward (base salt surface), while the 3336 remaining vertices were kept fixed (top salt surface

FIG. 1. Performance of the inversion using model C as the starting model and noiseless data. The right side (east) x-coordinates were distorted with respect to the reference model; vertices 1 and 4 are the uppermost ones, and vertex 6 is in the center of the facet. (a) Decrease of the residual rms error during the inversion process (50 iterations). (b)–(d) Behavior of the relative displacement (in meters) of the free vertices coordinates during the inversion process (Moraes, 1997).

Structural Gravity Inversion

and layered model). During the inversion process vertices behaving as singularities were frozen at their original positions after a cautious study of the inversion reports. The reason for this action was to achieve the smallest combination of free vertices whose modifications give the best fit. The process was carried out in 11 different trials of 25 iterations each. During these assays, the variance decreased 17 orders of magnitude. The condition number increased three orders of magnitude at the end of the process, as compared with the first iteration (Moraes, 1997). The last iteration adjusted only 12 vertices (or 36 parameters) among the 89 original ones. As can be seen (Figure 6), the adjustment performed in the starting model enhanced the fit not only in central region of the area where the shallow gravity signature of the dome occurs but also in its surroundings as well (compare with Figure 4). This can be observed especially in the southeast corner of the map, where a previous gradient was almost entirely compensated. The differences in the other corners lowered as well. The essential complement to these observations is the analysis of how the anomalous salt body was affected by the

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FIG. 3. Location map of the case study area (after Fueg, 1995).

FIG. 2. Performance of the inversion using model C as the starting model and 5% Gaussian distributed noise contaminated data. (a) Decrease of the residual rms error during the inversion process (50 iterations). Stabilization occurred around 5% noise floor. (b)–(d) Behavior of the relative displacement (in meters) of the free vertices coordinates during the inversion process (Moraes, 1997).

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adjustments produced by the inversion process (Table 1). The most important information from the tabulation was that the body increased its volume downward by a maximum value of 0.554 km while gaining 0.44 km west and 0.123 km south in its horizontal dimensions. One can use this information to derive a better forward model to fit the data more appropriately. This approach was not tried because of the poor data quality and the poor sampling of the local gravity field. CONCLUSIONS

The interpretation of the boundaries of structures by salt intrusions is an important matter in the correct use of both prestack and poststack depth migration techniques. This is particularly true if the goal is a good image of the subsalt features, a subject of overwhelming importance in many regions of the world (as can be seen in Abriel and Wright, 1994; Hodgkins and O’Brien, 1994; House and Pritchett, 1994; Lee and HouseFinch, 1994; Lewis et al., 1994; MacKay and Dragoset, 1994; and Wyatt et al., 1994, among others). Gravity data are often acquired along with seismic reflection data. The gravity data can enhance the interpretation of a better velocity model. In this connection it should be interesting to try some kind of simultaneous inversion of gravity and seismic reflection data. If even one of the iterations in the migration process could be eliminated using gravity data, its usefulness would be justified.

Another area where this formulation could be valuable is interpreting 2-D seismic data for clues about 3-D subsurface structures, as was done in the case study. Many hydrocarbon prospects fit into this situation. Some authors have already used 2-D seismic data iteratively along with 3-D gravity and seismic modeling to achieve this goal (Starich et al., 1994). This is one example where the 3-D gravity inversion scheme can be of some help, taking care of the fine adjustments in the interpreted 3-D density model. These are only a few examples of the many applications an algorithm like this can have. The only requirement is that the model to be adjusted be constrained by some kind of prior information. Thus, mining problems can be handled adequately as well. The algorithm can also be modified easily to handle magnetic inverse modeling, based on the same guidelines, broadening its usefulness. It can even be worked out to handle the inversion of gravity and magnetic data simultaneously, as long as density and magnetization contrasts can be assigned to the same regions in space. ACKNOWLEDGMENTS

The authors thank Dr. Yaoguo Li, Dr. John Mariano, and an anonymous reviewer for their critical reading and for their suggestions to improve the text. The authors also thank the following companies for their data: Exxon Exploration Company

FIG. 4. Bouguer gravity signature (mGal) of the Galveston salt dome (after Fueg, 1995).

Structural Gravity Inversion

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FIG. 5. Contour map (mGal) of the residual difference between the measured Bouguer gravity data and the anomalous field of the initial model (according to Fueg, 1995).

FIG. 6. Contour map (mGal) of the residual difference between the measured Bouguer gravity data and that produced by the detached model after the inversion (Moraes, 1997).

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Table 1.

Moraes and Hansen

Comparison between the data sets with the vertex coordinates of the final detached anomalous body (Fueg, 1995) and that adjusted by the proposed inversion procedure.

Model Coordinates (km) Minimum Maximum Average Variance Std. deviation

Inverted x 316.905 325.318 321.244 6.049 2.459

y 3138.760 3147.130 3142.800 4.677 2.163

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