Gravity inversion using convexity constraint

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the other hand, a spread located at its top indicates that the host medium is mechanically anisotropic, exhibiting horizontal planes of weakness where the mobile ...
GEOPHYSICS, VOL. 65, NO. 1 (JANUARY-FEBRUARY 2000); P. 102–112, 16 FIGS.

Gravity inversion using convexity constraint Joao ˜ B. C. Silva∗ , Walter E. Medeiros‡ , and Valeria ´ C. F. Barbosa∗∗

ABSTRACT

ity constraint leads to a reasonable delineation of the source shape and produces a better resolution of its base as compared with the inverse method imposing smoothness constraints on the top and bottom surfaces of the anomalous source. The potential use of the convexity constraint is the possibility of combining it with other constraints to produce geologically meaningful solutions. As an example, the global convexity was combined with the minimum moment of inertia of the anomalous masses about a vertical axis passing through its center of mass. This combination allowed introducing a priori information about a diapir shape. This is important because a maximum spread at the diapir middle portion indicates that it ascended like a viscous bubble in an isotropic medium. On the other hand, a spread located at its top indicates that the host medium is mechanically anisotropic, exhibiting horizontal planes of weakness where the mobile material was forced into. So, this combination of constraints alows introducing a priori information about the diapir emplacement. The combination of global convexity and minimum moment of inertia may also be applied to isolated intrusions, as illustrated with the real gravity anomaly produced by the granitic body of Castelsarrasin, in the Aquitaine Basin, France. The estimated source has its top located between 0.45 and 0.57 km from the surface, a base 6.4 km deep, and center of mass at a depth of 3.5 km, which agree reasonably with a priori geological and geophysical information about the body.

We present a constraint which incorporates in the geophysical inverse problem a priori information about the source convexity. Two kinds of convexity are considered: directional convexity, defined as the attribute of a body being intersected at most at two points by a straight line with a fixed spatial orientation, and global convexity, defined as the attribute of a body being intersected at most at two points by any arbitrarily oriented straight line. The interpretation model consists of several juxtaposed 2-D prisms whose positions, widths, and physical property are established by the interpreter. The depth to the top and the thickness of each prism are the parameters to be determined. The directional convexity is incorporated by minimizing a functional which expresses the number of times that several lines, parallel to a given direction and spaced at regular intervals, intersect the interpretation model. On the other hand, the global convexity is introduced in an algorithmic way by constraining the depths to the top and to the bottom of each prism to be, respectively, smaller or greater than the average of the depths to the top or to the bottom of the adjacent prisms. The use of the presented convexity contraint is demonstrated using synthetic gravity data produced by an isolated source. We found that convexity constraints (either directional along the horizontal axis or global) alone are insufficient to produce strongly stable solutions, although they reduce substantially the solution instability. Despite the “slight” instability, the directional convex-

INTRODUCTION

anomalous sources. The a priori information reduces the classes of possible solutions compatible with the gravity anomaly, eliminating nonuniqueness and reducing the solution instability. The a priori information necessary to stabilize the inverse problem may originate in physical restrictions such as

The gravity inverse problem is an ill-posed problem in the sense of Hadamard (1902) because its solution is neither unique nor stable. To transform this problem into a well-posed problem, it is necessary to introduce a priori information about the

Manuscript received by the Editor December 2, 1998; revised manuscript received April 19, 1999. ∗ Federal University of Para, ´ Dep. Geof´ısica, CG, Caixa Postal 1611, 66.017-900 Belem, ´ PA, Brazil. E-mail: [email protected]. ‡Federal University of Rio Grande do Notre, Dep. Fisica/CCE, Caixa Postal 1641, CEP 59.072-970, Natal, RN, Brazil. E-mail: [email protected]. ∗∗ Formerly Federal University of Para, ´ Dep. Geof´ısia, CG, Belem, ´ PA, Brazil; presently LNCC, Av. Getulio ´ Vargas, 333, Quitandinha, Petropolis, ´ Rio de Janeiro 25651-070, Brazil. E-mail: [email protected].  c 2000 Society of Exploration Geophysicists. All rights reserved. 102

Convexity Constraints

nonnegative densities or depths. The vast majority of the a priori information, however, comes from geological rather than physical restrictions. Despite the existence of huge records of geological information, their use in the gravity interpretation (or in any geophysical interpretation) is still rather limited. This imbalance is due to the considerable diversity of the geological information and to the difficulty in expressing it in mathematical terms. At present, mathematical methods allow incorporating only a few specific types of a priori information, which may be classified into two groups. The constraints in the first group are sufficient to stabilize the solutions, whereas the constraints in the second group may be not. The first group consists of two classes of a priori information: (1) proximity of all parameters to respective reference values (e.g., Braile et al., 1974; Pedersen, 1977; Vigneresse, 1978), and (2) concentration of the anomalous sources about a point or an axis (Last and Kubik, 1983, and Guillen and Menichetti, 1984, respectively). These constraints strongly stabilize the inverse geophysical problem but require concrete and precise a priori geological knowledge, such as the spatial position of an axis or a priori numerical values for all parameters defining a reference solution (which presumably displays the fundamental attributes of the true solution). As a result, if the geological a priori information is not precise, the estimated solution will be stable but biased toward a wrong solution. We will refer to these constraints as strong constraints. The second group consists also of two classes of a priori information: (1) inequality between a parameter and a numerical value specified a priori (e.g., Safon et al., 1977; Vigneresse, 1978; Silva and Hohmann, 1983), and (2) proximity between estimates of spatially adjacent parameters (e.g., deGroot-Hedlin and Constable, 1990; Ditmar, 1993). This proximity may be weighted to introduce information about areas where the spatial variation of the parameters are smooth or abrupt (Li and Oldenburg, 1996; Barbosa et al., 1999). These constraints require less restrictive geological information, so they are less probable to bias the true solution toward a wrong solution. We will refer to these constraints as weak constraints. When used alone, these constraints may be insufficient to stabilize an inverse problem. Despite the above restrictions, a judicious combination of weak constraints or weak and strong constraints may result in a stable inversion method incorporating relevant and factual geological information. By combining only weak constraints, stability may be attained if the constraints confine the possible solutions to subsets of the parameter space which, despite being individually large, have a small intersection. As pointed out before, weak constraints may represent more accurately some physical or geological characteristics, whereas strong constraints guarantee solution stability. As a result, inversion methods using a suitable weighted combinations of weak and strong constraints may produce stable and geologically meaningful solutions. As a general rule, the weight assigned to the strong constraint should be the smallest one still producing a stable solution. We explore the convexity constraint (e.g., Novikov, 1938; Leonov, 1976; Brodsky, 1983, 1986; Bakushinskii et al., 1986, 1988) as a new weak constraint in the geophysical inverse problem. The convexity of the source geometry is the attribute of a body being intersected at most at two points by any arbitrarily oriented straight line. The solution of the in-

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verse gravimetric problem for homogeneous convex sources despite being unique (Stretenski 1954), is unstable (Leonov, 1976; Bakushinskii et al., 1986). We define directional convexity as the attribute of a body being intersected at most at two points by a straight line with a fixed spatial orientation. To avoid confusion, we will refer to the usual convexity attribute as global convexity. The use of both directional and global constraints are illustrated with synthetic and real gravity data. Examples using synthetic gravity data confirm that both global and directional convexity contribute in reducing ambiguity but produce strongly stable solutions only when combined with other constraints. The introduction of directional and global convexity constraint increases the possibilities of constraint combinations, increasing in this way the chances of obtaining stable and geologically meaningful solutions to the geophysical inverse problem. As an example, we combine convexity with the minimum moment of inertia with respect to a vertical axis. This combination allows introducing information about the part of a diapir which is associated with its largest horizontal spread of mass. A spread located at its top is associated with a mechanically anisotropic host medium exhibiting horizontal planes of weakness where the mobile material is driven into, whereas a spread at its middle portion indicates mostly isotropic host rocks. This information may be obtained from the geological knowledge about the diapir emplacement and is sufficient to produce a stable and meaningful solution because the diapir geometry will, in this case, satisfy a prescribed geological attribute. An example using the anomaly produced by the Castelsarrasin intrusive body located in the Aquitaine Basin, France, produced results compatible with the available geological information. METHODOLOGY

Let a homogeneous, 2-D gravity source be approximated by the interpretation model consisting of a set of juxtaposed 2-D vertical prisms (Figure 1) defined within a discretized

FIG. 1. Discretized area containing the true source. The width w, the position xo j , and the density contrast ρ of each prism are fixed by the interpreter. The depth to the top h j and the thickness t j ( j = 1, 2 · · · M) are the parameters to be determined.

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area which presumably contains the anomalous source. The horizontal position, the width, and the density contrast of each prism are fixed by the interpreter. The jth prism is defined by the position xo j of its center, the width w, the depth to the top h j , the thickness t j , and the density contrast ρ (Figure 1). The depth to the top and the thickness of each prism are the parameters to be estimated, and are related to the computed gravity anomaly by a nonlinear relationship given by

gi =

M 

F( p j , ri ),

i = 1, 2, . . . , N ,

(1)

j=1

where N is the number of observations, M is the number of parameters, F( p j , ri ) is the nonlinear function (Telford et al., 1976) associating the ith gravity observation with the jth parameter p j (depth to the top or thickness) of the interpretation model, and ri is the position vector of the ith observation in the x-z space. The term gi defines the ith element of the vector g ≡ g(p) ≡ {g1 , g2 , . . . , g N }T containing the computed gravity anomaly, where p ≡ { p1 , p2 , . . . , p M }T is the vector of parameters of the interpretation model, and the superscript T stands for transposition. The nonlinear inverse problem of estimating p ∈ R M from g may be formulated as the minimization of the function

φ g (g, go ) =

1 g − go 2 , M

φ(p) = φ g (g, go ) + µ(δ)K (p),

(3)

where µ(δ) is a nonnegative real number. This problem is nonlinear and is, therefore, solved iteratively. Moreover, the function K (p) is not differentiable with respect to the components of p. In this way, to minimize the function (3), we use Nelder and Mead’s (1965) method because it does not require differentiability of the objective function with respect to the parameters. Global convexity The directional convexity constraint is a weaker constraint compared with the global convexity constraint because it leads to a smaller reduction of the class of possible solutions, that is, the stability of the solutions obtained with the directional convexity constraint is smaller compared with the global convexity constraint. These constraints are compared in Figure 3. The inverse problem of estimating p from the gravity anomaly go and incorporating global convexity constraint is

(2)

where go ≡ {g1o , g2o , . . . , g oN }T is the vector containing the N gravity observations, and  ·  is the Euclidean norm. This is an ill-posed problem because its solution is unstable. To reduce the instability, we introduce two kinds of convexity constraints: directional and global.

Directional convexity We define directional convexity as the attribute of a body’s surface to be intersected at most at two points by any straight line parallel to a fixed direction. The assumed interpretation model shown in Figure 1 incorporates implicitly the property of convexity in the z-direction. We introduce explicitly the convexity in the x-direction in the following way. Let u 1 , u 2 · · · u L be L horizontal lines, separated vertically by a distance d which intersect a tentative estimate of the anomalous body at points 1, 2 · · · K (Figure 2 shows an example for L = 3). If the estimated geometry is convex in the x-direction, K is equal to 2 × L; otherwise it is greater than 2 × L. So, by minimizing K we are favoring estimates displaying convexity along x. However, the estimated source will not be strictly convex in the x-direction unless the distance d could be made infinitely small. In other words, by minimizing K , oscillations with amplitude smaller than d may show up on the top or on the base of the estimated source. These oscillations may, nevertheless, be accidentaly minimized if they are intersected by a line u i , as in the case of the oscillation located between points 2 and 3 in Figure 2. Because K is a positive integer, the inverse problem incorporating the x-convexity constraint may be formulated as: minimize K (p) subject to φ g (g, go ) = δ, where δ is the expected root-mean-square of the noise realizations in data. This constrained problem may be solved by minimizing the function

FIG. 2. Tentative estimate of the anomalous body intersected by a set of horizontal lines u 1 , u 2 , and u 3 , at points 1 through 10.

FIG. 3. (a) Source with convexity in the x- and z-directions, but exhibiting no global convexity: there is at least one straight line u intersecting the body’s surface at more than two points. (b) Source with global convexity.

Convexity Constraints

nonlinear and is solved iteratively by Nelder and Mead’s (1965) method. We minimize the objective function φ g (g, go ) given in equation (2) and incorporate the global convexity constraint in the algorithmic way described below. At the kth iteration, we check if the depth to the top h i and the depth to the bottom Hi of each prism satisfy the inequalities (see Appendix A for details)

hi ≤

h i−1 + h i+1 , 2

(4)

Hi ≥

Hi−1 + Hi+1 . 2

(5)

and

If the inequalities hold, parameters h i and Hi are not changed; otherwise, h i or Hi are redefined as

h i−1 + h i+1 , hi = 2

(6)

and

Hi =

Hi−1 + Hi+1 , 2

(7)

respectively. In both cases (directional and global convexities), to guarantee convergence of the algorithm and to exclude physically meaningless solutions, we impose positivity constraint to all parameters by means of a logarithmic transformation. Comparison between directional and global convexity The global convexity constraint may be interpreted in the following way. Let a homogeneous source with arbitrary shape be limited by a surface S belonging to A∞ , the space of continuous functions with first- and second-order continuous derivatives with respect to the spatial coordinates. The surface S may be approximated by the infinite series

S≈

∞ 

cn f n ,

(8)

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ci be closest to zero, that is, by imposing that the body surface be smooth. TESTS WITH GRAVITY SYNTHETIC DATA

In all tests presented in this section, we employed 30 equispaced noise-corrupted Bouguer anomaly observations defined in the interval x ∈ [−7 km, 23 km] and an interpretation model consisting of 15 juxtaposed, 1 km wide prisms in the interval x ∈ [0 km, 15 km]. The pseudorandom noise added to data is Gaussian with zero mean and standard deviation of 1.0 mGal. The positivity constraint is imposed in all tests. Convexity along x Figure 5a shows the noise-corrupted Bouguer anomaly produced by a simulated batholith exhibiting convexity along the x-direction (shaded area in Figure 5b). The density contrast assigned to this body is 0.3 g/cm3 , and the initial guess is displayed in Figure 6. The reason for selecting this initial guess was to define a body presenting limited a priori information about the actual top and bottom reliefs and whose mass distribution would follow from inspection of the anomaly: most of the mass is concentrated close to the anomaly peak. The inversion result using parameters µ = 0.5 and d = 0.4 km (Figure 5b) shows that the estimated source is virtually convex in the x-direction and delineates the main features of the source top and bottom while minor artifacts occur at the source borders. The intersecting horizontal lines are shown on both sides of Figure 5b. The choice of parameter d is critical to obtain meaningful results. For very small values of d (0.01 km), the estimated source presents a flat top and a flat bottom; larger values (0.1 km) lead to a flat bottom and a curved top. Still larger values (0.25 km) produce oscillations on top and bottom, but the latter presents a smaller amplitude as compared with the true oscillation on the bottom. The selected value of 0.4 km is the maximum value still producing convexity in the x-direction, allowing a

n=0

where cn are coefficients and f n are basis functions spanning the infinite-dimensional space A∞ . An example of basis functions f n similar to the Legendre’s polynomials in spherical coordinates is shown in Figure 4. By inspecting this figure, we observe that imposing global convexity constraint to the solution of the inverse problem is equivalent to imposing that

co ci ,

i = 2, 3 · · · ∞

(9)

c1 ci ,

i = 2, 3 · · · ∞,

(10)

and

limiting, in this way, the class of possible solutions. Global convexity may be seen as a severe low-pass filter applied on the harmonics defining the source geometry. On the other hand, directional convexity is analogous to a less severe filter, allowing the presence of oscillations on the surface of the estimated source which are controlled by parameter d. For comparison, we note that Bakushinkii et al. (1986) proposed a stable inversion of gravity anomalies produced by homogeneous star-shaped bodies with known center and density using a procedure equivalent to requiring that the coefficients

FIG. 4. Basis functions which may be used to expand the geometry of an arbitrary source.

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good definition of top and bottom at the expense of a slight instability; solutions obtained with the same value of d = 0.4 km but using different sequences of pseudorandom noise differ from each other, but the oscillations on the top and on the base

of the true source are always detected and correctly positioned along the x-direction. In addition, all solutions obtained with µ = 0.5 and d = 0.4 km and different pseudorandom sequences are virtually convex. These results support the previously mentioned notion that the convexity constraint works as a low-pass filter on the harmonics defining an arbitrary geometry. Values of d larger than 0.4 km produce nonconvex and increasingly unstable solutions. The inversion results will be somewhat dependent on the initial guess because, as pointed out before, directional convexity alone is not sufficient to completely stabilize the solution. Inverting the same gravity anomaly using only the positivity constraint, we obtain an estimate still fitting the data (Figure 7a) but exhibiting spurious oscillations on top and bottom (Figure 7b).

Comparison with the smoothness estimator.—The spatial smoothness constraint operating on the top and base of the anomalous body is incorporated via minimization of the

FIG. 5. Simulated batholith. Convexity constraint along x. (a) Observed Bouguer anomaly (crosses) and fitted anomaly (solid line). (b) True source (shaded area) and inversion result using µ = 0.5 and d = 0.4 km. The horizontal lines separated by 0.4 km are shown on both sides of the figure.

FIG. 6. Initial guess for the solution shown in Figure 5b.

FIG. 7. Simulated batholith. Positivity constraint only. (a) Observed Bouguer anomaly (crosses) and fitted anomaly (solid line). (b) True source (shaded area) and inversion result.

Convexity Constraints

function



φ(p) = φ g (g, go ) + λ(δ)

M−1 



(h i+1 − h i )2

i=1



+ (h i+1 + ti+1 − h i − ti )

2



,

(11)

where φ g and δ have the same meaning presented before, and λ is the Lagrange multiplier. Qualitatively, imposing smoothness on the top and bottom surfaces of the interpretation model is comparable to imposing convexity in the x-direction. For a high degree of smoothness (large λ), the tops and bases of the prisms will be virtually invariant, leading to an estimated body similar to a single prism. On the other hand, for a low degree of smoothness (small λ), the depths to the top and to the bottom of each prism become virtually independent of each other, alowing spurious oscillations to occur on top and base of the estimated model. To compare the smoothness and convexity contraints, we minimize φ(p) in equation (11) using the observations shown in Figure 5a (the same used in the previous tests) and different values for λ in the interval [0.1, 1.0]. Values smaller than 0.5 lead to nonconvex solutions. For values between 0.1 and 0.2, the top and bottom of the solutions are very irregular, whereas for values between 0.2 and 0.5, the irregularities decrease, but the solutions present a pronounced concavity between x = 6 km and x = 10 km. For values of λ larger than 0.5, this concavity becomes increasingly smaller, but the relief of the base becomes too flat. Figure 8a shows the fitted anomaly corresponding to the solution exhibiting the best trade-off between resolution and stability, obtained with λ = 0.5 (Figure 8b). By comparing Figures 5b and 8b, we note that the bottom surface of the anomalous body is better delineated with the convexity constraint. Similar results are obtained using other pseudorandom noise sequences in the data. Note that the convexity constraint produced better results in this example because the actual model is convex and this attribute was imposed to the solutions. It does not mean that the convexity constraint is always superior to smoothness constraint. In fact, when applied to gravity inversion of anomalies produced by a single interface separating two homogeneous media (such as the basement relief of a sedimentary basin), both convexity and smoothness constraints produce virtually the same results.

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ure 11a), the solution (Figure 11b) is substantially more stable compared with the solution shown in Figure 9b. The base and the right side of the source are well delineated, but its left side is not. The estimated source is too wide at the base, demonstrating that the global convexity constraint together with positivity constraints may be insufficient to produce stable solutions. Estimated solutions will, therefore, depend on the initial guess; this dependency, however is less pronounced relative to the inversions using directional convexity. In next section, we combine global convexity and minimum moment of inertia constraints to obtain stable and meaningful solutions in the case of a specific geological setting. Global convexity and minimum moment of inertia Depending on the geological process controlling the diapir emplacement, two different shapes for this source are expected. If the host rocks behaves mainly in a plastic way, the diapir will rise and assume the shape of an inverted drop (Figure 12a). In this case, the maximum lateral spread will be in the median portion of the diapir. On the other hand, if the host

Global convexity Figure 9a shows the noise-corrupted Bouguer gravity anomaly produced by a simulated diapir (shaded area in Figure 9b) presenting global convexity, except for the irregularities related to the finite width of each elementary prism defining the interpretation model. The density contrast assigned to this body is −0.3 g/cm3 , and the initial guess is displayed in Figure 10. The reason for selecting this initial guess is to introduce minimum information about the actual source besides the position of its center of mass. This anomaly was inverted using only positivity constraints, producing, as expected, unstable results (Figure 9b). We also inverted the anomaly using the global convexity constraint. The result is shown in Figure 11. Besides fitting the data (Fig-

FIG. 8. Simulated batholith. Smoothness constraint. (a) Observed Bouguer anomaly (crosses) and fitted anomaly (solid line). (b) True source (shaded area) and inversion result using λ = 0.5.

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rocks behaves in an elastic way, the diapir will ascend by forcing its way along horizontal weakness planes located close to the surface. In this case, the maximum lateral spread will be in the uppermost portion of the diapir, producing a shape similar to a lacolith (Figure 12b). This kind of geological information may be easily introduced in the gravity inversion by combining the convexity and positivity constraints with the minimization of the moment of inertia of the estimated source with respect to an axis. So, we minimize the function

φ(p) = φ g (g, go ) + β(δ)ψ(p),

The parameter β allows the incorporation of the geological information about the diapir emplacement in the following way. The larger the value of β, the smaller the moment of inertia of the estimated source. This minimization implies that prisms which are distant from the axis [associated with large value of i in function ψ(p) defined by equation (13)] must present small thicknesses ti . To fit the observations, this decrease in thickness

(12)

subject to conditions (4) and (5), with

ψ(p) = ρ

M 

ti · wi · i2 ,

(13)

i=1

where i is the distance from the center of the ith prism to a vertical axis passing through the center of mass of the anomalous source. The estimation of the horizontal coordinate of the center of mass of an isolated source from the gravity anomaly only is a well-posed problem, as demonstrated by Medeiros and Silva (1995). As a result, the distances i may always be determined with reasonable precision. FIG. 10. Initial guess for the tests using the simulated diapir model.

FIG. 9. Simulated diapir. Positivity constraint only. (a) Observed Bouguer anomaly (crosses) and fitted anomaly (solid line). (b) True source (shaded area) and inversion result.

FIG. 11. Simulated diapir. Global convexity constraint. (a) Observed Bouguer anomaly (crosses) and fitted anomaly (solid line). (b) True source (shaded area) and inversion result.

Convexity Constraints

must be compensated with a reduction in the depth to the top h i (Fraiha and Silva, 1994). In this way, the whole estimated source becomes wider at top. This process is illustrated in Figures 13 and 14 for β equal to 0.15 and 0.55, respectively, and a vertical axis located at x = 7.5 km. Note that for β = 0.15, the estimated source maximum spread is at its median portion, forcing the solution to assume a lenticular shape (Figure 13b), whereas for β = 0.55, the spread is at the uppermost portion (Figure 14b). In both cases, the observed anomaly is fitted within the experimental error precision (Figures 13a and 14a). Therefore, any choice between the solutions shown in Figures 11b, 13b, and 14b must be grounded on the a priori geological knowl-

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edge about the diapir emplacement. Note that the shape of true source is closer to an inverted drop than to a lacolith’s shape. The inversion performed under this assumption (Figure 13b) produced a reasonable delineation of the source. We stress that the above results are obtained by combining convexity and minimum moment of inertia contraints. Using just the minimum moment of inertia constraint, we obtain a solution which is stable but too narrow and displaying a geologically unrealistic thicknesses of about 50 km. Because the estimated solutions are stable for a fixed value of β, they will be virtually independent of the initial guess (except, possibly for extremely unreasonable guesses). REAL DATA EXAMPLE

Figure 15a shows the Bouguer anomaly produced by the granitic body of Castelsarrasin in the Aquitaine Basin, France (Guillen and Menichetti, 1984). The 37 observations were digitized directly from the profile presented by Guillen and Menichetti (1984) (previously enlarged photographycally) with a spacing of 1 km. We estimate that the error introduced in the discretizing operation may attain 1.5 mGal.

FIG. 12. Schematic evolution of a diapir (D). (a) The host rocks H are isotropic and have a plastic behavior. (b) The host rocks H present fractures (F) close to the surface.

FIG. 13. Simulated diapir. Global convexity and minimum moment of inertia relative to a vertical axis. (a) Observed Bouguer anomaly (crosses) and fitted anomaly (solid line). (b) True source (shaded area), vertical axis AA passing through its center of mass, and inversion result using β = 0.15.

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The granitic body was identified by a drillhole that hit its uppermost part at a depth of 500 m (Guillen and Menichetti, 1984). Density contrasts smaller than −0.15 g/cm3 were considered acceptable by Guillen and Menichetti (1984), who used an algorithm that minimizes the moment of inertia of the estimated source with respect to an axis with known spatial position and orientation. Figure 15b shows the initial guess and the inversion result using the global convexity constraint associated with the minimization of the moment of inertia with respect to a vertical axis located at x = 10 km and setting β = 0.1. The initial guess was influenced by the a priori information that the shallowest part of the body is at 500 m and by the anomaly asymmetry, indicating that the mass deficiency dips to the northeast. The value of β was selected to produce a geometry where the mass is concentrated about the middle of the estimated source, based on the interpretation of Guillen and Menichetti (1984) with a density contrast of −0.20 g/cm3 . The interpretation model consists of ten prisms 2 km wide with density contrast of −0.15 g/cm3 . This value was selected [and not the value −0.20 g/cm3 assumed by Guillen and Menichetti (1984)] because −0.15 g/cm3 was considered an extreme value by Guillen and Menichetti (1984); that is, we want to find an extreme geometry for the

FIG. 14. Simulated diapir. Global convexity and minimum moment of inertia relative to a vertical axis. (a) Observed Bouguer anomaly (crosses) and fitted anomaly (solid line). (b) True source (shaded area), vertical axis AA passing through its center of mass, and inversion result using β = 0.55.

estimated source. The estimated depth to the top of the intrusion varies between 450 m and 570 m, which agrees reasonably well with the figure of 500 m obtained from the drillhole. The estimated source base is at 6.4 km, and its center of mass, calculated from the estimated solution, is 3.5 km deep [Guillen and Menichetti (1984) found, respectively, 6.5 km and 4.0 km on the basis of their gravity inversion]. Finally, we note that the estimated source presented a dip to the northeast without the necessity of minimizing the moment of inertia with respect to an inclined axis. We stress that because the inversion method is stable, the estimated source geometry in Figure 15b is not substantially affected by the observational errors, either in the original data or as a result of the discretizing process. CONCLUSIONS

We present a mathematical constraint which incorporates information about the source convexity. Two kinds of convexity were considered: global and directional. Different “degrees of convexity” may be imposed to the solution, increasing, in this way, the inversion flexibility in incorporating a priori information about different source geometry. The potential use of the convexity contraint is illustrated with synthetic and real

FIG. 15. (a) Anomaly produced by the intrusive body of Castelsarrasin in the Aquitaine Basin, France. (b) Initial guess and inversion result obtained by combining global convexity and minimum moment of inertia relative to axis AA with β = 0.1.

Convexity Constraints

gravity data; some results, therefore, hold only for the gravity method. The convexity constraint, however, may be used in the inversion of any geophysical data. The results obtained with synthetic gravity data showed that both kinds of convexity reduce the solution instability, and that solutions obtained with the global convexity constraint are more stable than those obtained with the directional convexity constraint. The convexity constraint proved helpful in interpreting anomalies produced by diapiric structures, particularly when the global convexity constraint is combined with the minimization of the moment of inertia with respect to a vertical axis. This combination allows us to introduce in the geophysical inverse problem relevant geological information about the portion of the anomalous body where the horizontal spread of mass is largest. Applying this combination of constraints to the Castelsarrasin anomaly in the Aquitaine Basin, France, produced results consistent with a priori geological information (depth to the top at 500 m) and with previous geophysical interpretation (anomalous source dipping to northeast, center of mass 4 km deep, and depth to the bottom at 6.5 km). The convexity constraint may also be combined with other strong constraints, namely the proximity of some parameters to respective a priori reference values, such as depths to the source obtained from boreholes at isolated points. Because the convexity constraint is introduced in terms of nondifferentiable functions, any inversion algorithm incorporating this constraint must use search methods. These are less efficient than methods using derivatives. The Nelder-Mead’s method used in the present paper takes about 5–7 s to invert 30 observations using an interpretation model defined by 30 parameters using a 300-MHz Pentium processor. ACKNOWLEDGMENTS

We thank Richard O. Hansen, Robert S. Pawlowski, and an anonymous reviewer for suggestions improving the text clarity. The authors were supported in this research by fellowships from Conselho de Desenvolvimento Cient´ıfico e Tecnologico ´ (CNPq), Brazil. REFERENCES Bakushinskii, A. B., Goncharskii, A. V., and Stepanova, L. D., 1986, The use of iterative regularization algorithms for the solution of

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APPENDIX A DERIVATION OF EQUATIONS (4) AND (5)

Let C be a two-dimensional body such that any vertical line intersects it at only two points (Figure A-1). The top of C is defined by the function f t (x), which describes an arbitrary line constrained only to be continuous and present first- and second-order continuous derivatives with respect to the space variable x. The base of C is defined by the function f b (x) subject to the same restrictions imposed to f t (x). Both functions are defined in the interval of x ∈ [x1 , x2 ]. A simple manner of imposing global convexity to body C is to require that the concavities of f t and f b be directed downward and upward, respectively. This is attained by requiring simultaneously

∂ 2 f t (x) ≥0 ∂x2

and

∂ 2 f b (x) ≤ 0. ∂x2

(A-2)

In the case that C is approximated by the interpretation model consisting of vertical juxtaposed prisms, the derivatives in equations (A-1) and (A-2) may be approximated by the corresponding numerical derivatives evaluated at the jth prism, yielding

and

(A-1)

h j+1 − 2h j + h j−1 ≥0 w2

(A-3)

H j+1 − 2H j + H j−1 ≤ 0, w2

(A-4)

where h j , H j , and w are, respectively, the depths to the top and to the bottom and the width of the jth prism. By isolating variables h j and H j in equations (A-3) and (A-4), respectively, we get

hi ≤

h i−1 + h i+1 , 2

(A-5)

Hi ≥

Hi−1 + Hi+1 , 2

(A-6)

and

which are the equations (4) and (5), respectively. To make the method more flexible in incorporating a priori information about the source geometry, the strict convexity constraint may be relaxed by using the extensions of equations (6) and (7) given by

hi =

h i−1 + h i+1 + , 2

(A-7)

Hi =

Hi−1 + Hi+1 − . 2

(A-8)

and

FIG. A-1. Arbitraty body C defined by the functions f t (x) (solid line) and f b (x) (dashed line), which represent, respectively, the top and the base of C.

If is negative, the estimated source will be strictly convex; otherwise, the estimated source may present oscillations (that is, concavities) with amplitudes controlled by .