Gravity inversion of basement relief constrained by the knowledge of depth at isolated points

Transcription

1 GEOPHYSICS, VOL. 61, NO. 6 (NOVEMBER-DECEMBER 1996); E ,13 FIGS., 5 TABLES. Gravity inversion of basement relief constrained by the knowledge of depth at isolated points Jorge W. D. Leao*, Paulo T. L. Menezest, Jacira F. Beltrao *, and Joao B. C. Silva* ABSTRACT We present an interpretation method for the gravity anomaly of an arbitrary interface separating two homogeneous media. It consists essentially of a downward continuation of the observed anomaly and the division of the continued anomaly by a scale factor involving the density contrast between the media. The knowledge of the interface depth at isolated points is used to estimate the depth d1 of the shallowest point of the interface, the density contrast Ap between the two media, and the coefficients ci and c2 of a first-order polynomial representing a linear trend to be removed from data. The solutions are stabilized by introducing a damping parameter in the computation of the downward-continued anomaly by the equivalent layer method. Different from other interface mapping methods using gravity data, the proposed method: (1) takes into account the presence of an undesirable linear trend in data; (2) requires just intervals for both Op (rather than the knowledge of its true value) and coefficients c1 and c2; and (3) does not require the knowledge of the average interface depth z0. As a result of (3), the proposed method does not call for extensive knowledge of the interface depth to obtain a statistically significant estimate of z0 ; rather, it is able to use the knowledge of the interface depth at just a few isolated points to estimate d1, zp, ci, and C2. Tests using synthetic data confirm that the method produces good and stable estimates as far as the established premises (smooth interface separating two homogeneous media and, at most, the presence of an unremoved linear trend in data) are not violated. If the density contrast is not uniform, the method may still be applied using Litinsky's concept of effective density. The method was applied to gravity data from Reconcavo Basin, Brazil, producing good correlations of estimated lows and terraces in the basement with corresponding known geological features. INTRODUCTION Mapping the depth to an interface separating two homogeneous media is a nonlinear problem that may have a unique solution if the interface intersects any vertical line only once (Smith, 1961) and the gravity anomaly is known in a continuous way with infinite precision. In practice, these last conditions are never fulfilled so that the solutions may be unstable. Therefore, methods designed to solve this problem must introduce a priori information to stabilize the solutions. Burkhard and Jackson (1976), Richardson and MacInnes (1989), and Pedersen (1977), for example, divided the upper medium into rectangular prisms and used iterative nonlinear optimization techniques to estimate the thickness of each elementary prism. The solutions are stabilized by keeping them close to a starting reference solution. This is accomplished by limiting parameter corrections at each iteration, using, for example, a parameter covariance matrix (Burkhard and Jackson, 1976; Richardson and Maclnnes, 1989) or abandoning eigenvalues smaller than a threshold value in a generalized inverse approach (Pedersen, 1977). The explicit a priori information required by these methods is a good guess for the thicknesses of the upper medium and the knowledge of the density contrast between the media. Oldenburg (1974) developed an iterative method based on Parker's (1973) expansion of the gravity anomaly into a power series of the function representing the interface in the wavenumber domain. Guspi (1993) presented a noniterative inversion of Parker's formula by expressing it as a power series expansion in the reciprocal of the density contrast. Manuscript received by the Editor November 7, 1994; revised manuscript received February 15, *Dep. Geofisica, CG, University of Para Caixa Postal 1611, , Belem, PA, Brazil. $Dep. Geofisica, Observatorio Nacional R. General Bruce, 586, , Rio de Janeiro, RJ, Brazil Society of Exploration Geophysicists. All rights reserved. 1702

2 Gravity Inversion of Basement Relief 1703 Granser (1987) presented a method in which the kernel of the nonlinear integral equation relating the gravity anomaly with the interface relief is expanded in Taylor's series and the coefficients of the inverse series are obtained from the coefficients of the original series. These methods stabilize the solutions by applying a low-pass filter to the observed anomaly, with a cutoff frequency designed to guarantee convergence of the series. The lower the cut-off frequency, the more stable the solutions and the smoother the estimated interface. The only explicit a priori information necessary is the density contrast Ap between the two media and the interface average depth z 0. Courtillot et al. (1974) and Pilkington and Crossley (1986) obtained successive approximations of the interface by computing at each iteration a relief associated to the residual between the observed and the computed anomaly using the current approximation for the interface. The "residual" topography is added to the previous approximation. In these methods, at each iteration either an integral equation (Courtillot et al., 1974) or a matrix equation (Pilkington and Crossley, 1986) is solved, which is equivalent to continuing the observed anomaly to some level below the surface. The stabilizing procedure is, therefore, reduced to stabilizing the downward continuation procedure. In the matrix formulation of Pilkington and Crossley (1986), for example, this stabilization is achieved by means of a damping parameter, which is analogous to the above-mentioned cut-off frequency. In this group of methods, the required a priori information is also the knowledge of Op and z 0. The information about Op and z, required by the methods described above, must come from geological or other geophysical data (Granser, 1987; Guspi, 1993) because when using only the gravity anomaly, one can at most estimate an upper bound for z o given Ap, or a lower bound for Ap given z o (Courtillot et al., 1974). The information about z 0, is more critical than the information about Ap and is usually obtained from seismic data interpretation, if available (Courtillot et al., 1974; Oldenburg, 1974; Guspi, 1993) or from borehole information (Chenot and Debeglia, 1990). We present a gravity interpretation method for an interface separating two homogeneous media using the linear approximation of the relief. It is based on the downward continuation of the observed anomaly to a reference level dl and its division by a scale factor involving the density contrast Ap. Parameters d, and Ap are estimated from a priori information in a different way as compared with other methods also requiring the knowledge of these parameters. In these methods, the density contrast is inferred directly from geological information, and the reference level dl is the average interface depth z 0, which is estimated by the sample mean of known depth values. In the present method, the reference level is the depth to the shallowest point of the interface, and parameters Ap, dl, and the coefficients of a first-order polynomial, representing an unknown base level in data, are estimated by the method in the following way. For fixed approximations of the above parameters, the observed anomaly is corrected for the base level, downward continued to depth dt, and transformed into a map of basement relief by dividing the continued anomaly by a constant factor involving the density contrast. Optimum values for Ap, z 0, and the polynomial coefficients are obtained by minimizing the rms of the differences between known and computed depths. The solutions are stabilized by introducing a damping parameter in the matrix formulation of the downward-continuation operator. The main advantages of this approach are: (1) only an interval, rather than a specific value for the density contrast is required; (2) any unknown, base level in data may be taken into account; and (3) the very depths at isolated points are used, instead of their average. This last point has important consequences. Methods estimating the average interface depth by the sample average of known depths may require more known depths than the present method because using just a few points may lead to a poor estimate of the average depth. In the present method, on the other hand, just a few points that may even be far from the average depth and located consistently above (or below) it, will successfully constrain the computed interface to be as close to them as possible. METHODOLOGY In this section, we first deduce the theoretical expression for the interface relief assuming that the density contrast (Op) between the two media and the depth (dl) to the shallowest point of the interface are known. We also show that the presence of an unknown constant base level in data does not affect the results. Then, we present a method for estimating the interface relief together with parameters Ap, d1 and the angular coefficients ct and c2 of a first-order polynomial (representing an unremoved linear trend in data) using available knowledge about the interface depth at isolated points. Relief inversion knowing Ap and dl Let h(x, y) be a smooth surface separating two homogeneous regions with densities pl and p2 referred to an arbitrary depth d (Figure la). The gravity anomaly on plane z = dl produced by the density contrast Ap = pz p l within the slab limited by dt and d2 (Figure lb) is gz(x, y, dl) = 2 ryap(d2 d1) ga(x, y, d1), ( 1) where y is the gravitational constant and ga(x, y, dt) is the vertical field at z = dl produced by region A (Figure lc) with density Ap. To obtain a relationship between ga(x, y, d1) and h(x, y), we consider the potential U at z because of region A with density Ap: U(x,y,z) T+oc p+^ fdl+h(x,y) = YAp J dx,dy, 00 oc1 dz' X [(x x')2 + (y y')2 -t- (z z') 2] 1/2 ' z - 2) dl ( Assuming a smooth interface with I hi << dl, so that the denominator in equation (2) is nearly constant, we get the approximation /' +co +00 =YAP J f h(x', y') dx'dy' X [(x x') 2 + (y y') 2 + (z d1) 2 ]l/2 ' z d1 (3) U(x,y,z)

3 1704 Leao et al. On the other hand, solving Laplace's equation with Substituting equation (5) in equation (1) and rearranging, Neumann's boundary condition yields we get Comparing equations (3) and (4) we get as a first-order ap- g (x, y, d1) = gz (x, y, d1) + g, (7) proximation or 9A(x, y, dl) = 2rry1 p h(x, y). (5) 8z(x, y, dl) = 8^(x, y, d1) gu (8) a) 1 f +oo h(x y) 27tytp (d2 dl) gz(x, y, dl) f+o (6) U(x, y, z) 1 1 2nyop 27r J_ J_ The anomaly gz (x, y, dl), in practice, may be obtained within XgA(x', y', d1)dx'dy' Z < d (4 the range of an additive constant gz, that is, the downward- [(x x') 2 + (y y')2 + (z d1) 2] 1/21 continued(downtoleveldl)observationsg (x, y, dj)aregiven by d, d2 + + d p ^ X ^ yl `z C) FIG. 1. (a) Interface separating two homogeneous media with densities pi and p2 described by a function h(x, y), referred to level d. (b) The shallowest and deepest points of the interface are at depths dl and d2, respectively. (c) Region A, located between level d1 and the interface.

4 Gravity Inversion of Basement Relief 1705 The constant gz may be interpreted as the anomaly produced by a Bouguer slab with thickness d3 d2 (Figure 1b): gz = 2nyAp - (d3 d2). (9) The anomaly g (x, y, di) may, therefore, be seen as the anomaly produced by the interface, referred to level d3. Substituting in equation (6) the expression for g Z given in equation (8) and taking equation (9) into account, we obtain h(x, y) _ 21ryAp (d3 d1) g (x,y,dl) (10) 2nyAp Note that by hypothesis d1 is known and coincides with the shallowest point of the interface, and the distances D(x, y) = g,0(x, y, d1) (11) 27ryAp are referred to level d3. The slab thickness d3 dt is given by d3 d1 = supd(x, y). (12) Any unknown, constant base level in the observed anomaly is, therefore, canceled in obtaining h(x, y), that is, the method is not sensitive to a constant base level produced by a Bouguer slab with arbitrary thickness d3 d2. The knowledge of level d2 is, therefore, not necessary. However, in practice, the anomalous field is measured in the presence of a variable base level (from now on simply called base level) representing an unremoved regional effect. Then, to compute h(x, y) from equation (10), Ap, dl, and the base level must be known. The observed gravity anomaly alone is insufficient to estimate these quantities, and to overcome this difficulty we use a priori information about the basement depth from boreholes. Relief inversion and estimation of 0p and dl We assume that the depth to the basement h (xk, yk) is known at L points (k = 1,..., L). Then, we use this information to estimate Ap, d1, and the coefficients c - [c1 c2]t of a polynomial P(x, y) - c1x + c2y, presumably representing a local base level, by minimizing L 1/2 f(p) ([h o (Xk,Yk) h`(xk, yk, p)] 2^, (13) where hc (xk, Yk, P) P = [Ap d1 ] T, (14) 2 ryap(d3 d1) g?(xk, Yk, dl, c) 27ryAp k = 1,..., L, (15) and gz (xk, yk, di, c) is the anomaly at (xk, yk, di) corrected for the base level, it is obtained by downward continuing the anomaly gz (x, y, 0, c) given by z(x, Y, 0, c) = gz,(x, y, 0) P(x, y) (16) The above derived procedure is implemented as follows: 1) Feasible intervals for Ap and c are established a priori. 2) Level d1 is fixed at an initial approximation d1 1), necessarily smaller than the true value of dt. 3) At the ith iteration, an estimate' ) = [Ap c(1)t IT minimizing f (fir), given in equation (13) is obtained with the controlled random search algorithm in Price (1983), using the current fixed approximation d the feasible intervals defined in step 1. At this step, the anomaly, continued downward to level at points (xk, yk), k = 1,..., L, is obtained by the method is Ledo and Silva (1989) and is used in equation (15). 4) If f (pl` 1 ) < f (131` t) ), increment dl' 1 by a prespecified step Ad, increment the iteration number by one and go to step 3; otherwise, obtain dl as the minimum of f( li _^1)) a parabola that interpolates points (d1 2) (di' t), f(p(1-1) )), and (di` ), f(p (` ) )), and obtain the solution p* _ [Ap* dl c* T ] T by repeating step (3) with dl` = dj, and proceed to step (5). 5) Compute the anomaly gz (x1, yf, d, c*), j = 1,..., N at every observation point and use it in equation (15), replacing k by j and L by N with Ap = Ap* and d3 d1 given by equation (12) to obtain estimates h`'(x1, yj, p*), j = 1,..., N at every observation point. It should be stressed that the algorithm of Ledo and Silva (1989), when applied to the above described method, presents two advantages over downward continuation using frequency domain filters. First, it allows us to obtain the downward continued field at only the L points required to compute the objective function. This substantially reduces unnecessary computations because the values of the downward-continued anomaly at every observation point are not necessary at intermediary iterations. This represents a substantial savings in computer time because L is usually much smaller than N. The second advantage is the simplicity in stabilizing the downward-continuation operation by means of a single damping parameter that can be determined by objective criteria (Ledo and Silva, 1989) as compared with more complex criteria for low-pass filter designs (e.g., Oldenburg, 1974). APPLICATION TO SYNTHETIC DATA We simulated an 80 km x 80 km sedimentary basin having steep borders and a smooth, undulating bottom using several adjacent vertical prisms. Figure 2 shows the isopachs of the sedimentary cover, and Figure 3 displays the associated gravity anomaly obtained by assigning a density contrast of 0.4 g/cm3 between the basement and the sediments. The dashed rectangle in both figures represents the study area (magnified in Figure 4) where a 30 x 40 grid of observations was produced with a grid spacing of 1 km in both x (north-south) and y (east-west) directions. A polynomial P(x, y) = 0.1y was added to the theoretical observations to simulate the presence of an unremoved component of a regional field. The depth to the basement inside the study area ranges from 1900 m to 3370 m. We suppose that the true depths to the basement are known at seven points (A-G) shown in Figure 4. The depth and horizontal coordinates of each point are given in Table 1. In applying Ledo and Silva's (1989) algorithm, we used a discrete, 4-km deep equivalent layer consisting of a 25 x 25 grid of point masses located directly below the observations. The grid spacing is

5 1706 Leao et al. 1 km in both x- and y-directions, the data window is 7 x 7, and a value of 0.01 was assigned to the damping parameter. The feasible intervals for parameters d1, Ap, and the base-level coefficients are shown in Table 2. Parameter dl is incremented at steps of 25 m starting at the surface. Figure 4 displays, in solid Iines, the isopachs of the true sedimentary cover and in dashed lines the isopachs produced by the proposed method. Three rows and three columns at the borders of the original gridded data were eliminated because the downward continuation at these points are subject to edge effects (Lego and Silva,1989). The true and computed depths agree reasonably, except at the regions labeled T, U, and V in Figure 4, where an inflexion in the 3000 m and 3100 m contours reflects a basement feature that cannot be resolved by the data. The curve of 3300 m is shifted slightly from the correct position. Note that even though we know the depths at D and E, we should not force the computed relief to pass through these points because at these localities the hypothesis of a smooth surface is violated. The estimates for d1, Ap, cl, and c2 are, respectively, 2081 m, 0.45 g/cm 3, mgal/km, and Table 1. Coordinates and depths of points of known depths to the basement for the test with synthetic data. Point A B C D E F G x-coordinate (km) y-coordinate (km) depth (km) Table 2. Feasible intervals for the continuation level, the density contrast, and the polynomial coefficients for the test with synthetic data. Parameters Interval dl (km) Ap (g/cm3 ) Cl (mgal/km) c2 (mgal/km) km FIG. 2. Isopachs of sediments in a simulated sedimentary basin. The dashed rectangle indicates the study area shown in detail in Figure 4. Contour interval 100 m.

6 Gravity Inversion of Basement Relief mgal/km. The minimum objective function at level d* is 10 m. Note also that the basin depocenter (represented by the isovalue curve of 3300 m) is well defined despite the absence of control points in its vicinity. Figure 5 Shows the behavior of the minimum of f (p) against the corresponding level of continuation d, in the interval [0 m, 3500 m]. The interval for dl is extended beyond d, defined in previous section, to illustrate that the function f (p) presents just one well-defined minimum. The a priori information about the source depth at just a few points is, therefore, sufficient to estimate an optimum value for the reference level dl. In addition, the estimated interface is stable because of the a priori information incorporated by the method that the interface is smooth and closest to the points of known depth, which, in this case, are representative of the interface. The degree of smoothness of the computed interface is controlled by the damping parameter. APPLICATION TO REAL DATA We apply the proposed method to the gravity anomaly of Reconcavo Basin, Brazil with the purpose of obtaining a map of the basement. This basin, together with Tucano and Jatoba basins form a 100-km wide rift extending for more than 400 km in the north-south direction. The Reconcavo Basin lies at the southern most portion of this rift and Jatoba Basin at the northern most portion. The rift developed from a Late Jurassic to Early Cretaceous rupture feature implanted over terrains of different natures and ages. The origin of this rift is related to the breakup of Gondwana and the opening of the South Atlantic Ocean. The sedimentary cover in the Reconcavo Basin comprises rocks from Upper Jurassic to Tertiary and locally may be 6000 m thick (Milani and Davison, 1988). The basement is segmented by several northeast- and northwest-trending normal faults that may have been reactivated and produced vertical 2 iiii I _24 22O' 2 _ O km FIG. 3. Gravity anomaly produced by the sedimentary basin simulated in Figure 2 with a density contrast of 0.4 g/cm3 between the basement and the sediments. The dashed rectangle indicates the study area. Contour interval 2 mgal.

7 ' 1708 Ledo et al. offsets across the sedimentary layers. Beneath the basin, an eastward crustal thinning is inferred from geophysical observations (Karner et al., 1992). The geological setting described above suggests that, for interpreting the basement relief, the gravity anomaly over Reconcavo Basin should be corrected for the effects of crustal thinning and lateral density contrasts in the sediments produced by the vertical offsets. We chose to make these corrections by successive regional-residual separations. The terms "residual" and "regional" are used to differentiate between anomalies from local, near surface masses and those arising from larger and generally deeper features, respectively. Regional residual separation is usually performed either by spectral methods or by polynomial fitting. Spectral methods characterize the smoothness of a regional field by its predominantly low-frequency spectral content so that regionals are obtained by low-pass filtering the observed anomaly. One problem with spectral methods is that they usually assign null frequency content to the zero frequency in the residual, producing, in this way, residuals contaminated by pseudoanomalies (Ulrich, 1968). These pseudoanomalies have the opposite sign from the real residual anomaly. They arise because assigning a null spectral content to the residual zero frequency is equivalent to obtaining a residual with a null spatial mean. In other words, there must always be residual anomalies of both signs even though the real residual anomaly may be of just one sign. Polynomial fitting methods assume that a polynomial surface adequately models the regional field whose assumed smoothness is controlled by the polynomial order (Agocs, 1951; Simpson, 1954). Because the polynomial is fitted to the total field, any attempt to model a complex regional by a high-order polynomial will force the polynomial to represent part of the residual field, distorting, in this way, the estimated regional relative to the true one. The fitting polynomial usually includes a constant term, and if the least-squares method is employed, the sum of residuals will be zero. As in spectral methods, this implies the presence of positive and negative residual anomalies even though the real sources may be of one sign. These undesirable effects are reduced by using robust polynomial fitting (Beltrao et al., 1991) and for this reason this method was employed. 0 E E E oo 5nn 7nnn lsnn 9nnn 9sn0 anon 201 FIG. 5. Minimum objective function f (p), relative to parameters Ap, c1, and c2, against fixed values of d1 in the interval [0 m, 3500 m] for the test using synthetic data. x 1 / 1 / /' /' 25 00, J j Ii Ii I l N j Ii II ji l II O N N^ 11 N / N I N / / NN O C / N o I Nb 1 S 1 O 1 q I N I I I I 1' / T' t I / / t C o O O ^ O.) O Eli O O y (km) FIG. 4. True (solid line) and estimated (dashed lines) depths to the basement for part of the basin simulated in Figure 2. A through G are the points where the depths to the basement are known. Contour interval 100 m.

8 Gravity Inversion of Basement Relief 1709 Figure 6 shows the Bouguer anomaly map of Reconcavo Basin, Brazil, corrected for the crustal thinning effect (Menezes, 1990). The correction consisted in removing from the observations a fourth-order polynomial fitted to the gravity data of Bahia state, which comprises the study area (shown as thick line in Figure 7). The fourth-order was selected because regionals of orders five and up display a well-defined anomaly coinciding with the Reconcavo-Tucano-Jatoba rift (R, T, and J in Figure 7d). On the other hand, fitted polynomials of orders two through four are very close to each other (Figures 7a-7c). To isolate the effect of the basement topography, the data of Figure 6 were separated into their regional and residual components by the method of Beltran et al. (1991) using a thirteenthorder polynomial. These data were gridded at intervals of 1 km in both x (north-south) and y (east-west) directions. Two criteria were used to select this polynomial order. The first one was based on the qualitative stability, that is, the persistence of the same features in the regional field for polynomials above certain order (Beltran et al., 1991). This stability can be monitored by plotting the rms of the absolute residuals (observed field minus fitted regional) for different polynomial orders (Figure 8). In this example, qualitative stability was attained around order 12. The second criterion was the ability of the fitted polynomial surface to define a low related to the basin depocenter (Camacari Low, labeled U in Figure 6). The thirteenth-order regional is shown in Figure 9 and the absolute difference between this and the twelfth-order regional is displayed in Figure 10. Note that the main difference between them is exactly at the Camacari Low which the thirteenth-order regional is able to resolve. Borehole information at seven points were used (A G in Figure 9) whose locations are given in Table 3. Table 4 displays the feasible intervals for dl, Ap, and the polynomial coefficients. Parameter dl is incremented at intervals of 25 m starting at the surface. The optimum value for dl is 1027 m, as shown by the plot of the minimum of f (p) against d1 in the interval [0 m, 1600 m] (Figure 11). When applying Leao and Silva's (1989) algorithm, we used the same damping parameter, the same spatial distribution of the equivalent layer sources, and the same data window dimensions employed in the synthetic test. The estimates for the density contrast and the polynomial coefficients cl and c2 are 0.46 g/cm 3, mgal/km, and mgal/km, respectively. The negative value for the estimate of cl suggests that part of the regional field removed by Menezes (1990), FIG. 6. Bouguer anomaly map from Reconcavo Basin, Brazil, corrected for the effect of an eastward crustal thinning. Contour interval 2 mgal.

9 1710 Table 3. Geographical coordinates and depths of points of known depths to the basement in Reconcavo Basin. Point Latitude Longitude Depth (km) A 12 01'26.4"S 37 59'16.8"W 2.29 B 12 19'54.9"S 38 14'42.6"W 2.88 C 12 47' 34.8" S 38 26' 45.6" W 3.80 D 12 08' 09.6" S 38 11' 08.5" W 2.95 E 12 15' 09.6" S 38 04' 15.2" W 1.62 F 12 10'08.4"S 38 12'27.7"W 2.97 G 12 11'20.4"S 38 13'19.2"W 3.14 Table 4. Reconcavo Basin. Feasible intervals for the continuation level, the density contrast, and the polynomial coefficients. Parameters a) Interval d1 (km) Ap (g/cm3 ) ct (mgal/km) c2 (mgal/km) Lebo et al. b) along the north-south direction, was in fact produced by the basement of the basin and not by a northward Moho uprise. The present method permitted a fine tunning of the base level and a restoration of the removed part of the signal. We stress that the flattening of the curve shown in Figure 11 about its minimum does not reflect an instability of the method because the objective function is not the rms of the residuals between the observed and fitted gravity anomaly data. The solutions (interface depths) obtained with depths dt located close to the minimum are, in fact, very close to each other. The computed basement topography corresponding to the thirteenth-order polynomial and the difference between this relief and the one computed with the twelfth-order regional are shown in Figures 12 and 13, respectively. Again, the main difference lies at the Camacari Depocenter. The major features in the map of Figure 12 correlate with known geologic structures. The Alagoinhas and the Camacari structural lows (T and U, respectively) coincide with lows in this figure; the Cassarongongo terrace coincides with a low-gradient area (V). In addition, the absolute depths are in reasonable agreement with depths C) d) FiG. 7. Polynomials or orders 2 (a), 3 (b), 4 (c), and 5 (d) fitted to the gravity data of Bahia state, which comprises the study area (shown in thick line). Areas labeled R, T, and J stand for Reconcavo, Tucano, and Jatoba Basins, respectively. Contour interval 5 mgal.

10 Gravity Inversion of Basement Relief 1711 reported by Milani (1985) for specific profiles across the basin. Table 5 shows the observed and computed depths at points A G. The "mis-ties" at wells A, C, and E, located at the borders, are related mainly to the fact that the premise of an interface relief presenting no abrupt, high-amplitude discontinuities is violated at the borders but not at the center of the basin. The minimum objective function (rms of the difference bem I... S polynomial order FIG. 8. Root mean squares of the residual (observed anomaly minus fitted regional) for successive polynomial orders. tween the observed and computed depths) at level dl = 1027 m is 236 m. CONCLUSIONS We presented a constrained method to determine, from the gravity anomaly, the depth to an irregular interface separating two homogeneous media. The constraints consist of borehole information about the depth to the interface. If the density contrast is not uniform, the method may still be applied if an effective density contrast (Litinsky, 1989) is available. Tests using synthetic data showed that the depths to a smooth, low-amplitude interface are well determined by the method. However, interfaces presenting abrupt, highamplitude discontinuities are not expected to be well determined by the method, not only because the depth estimates are linear transformations of the gravity field (which is continuous and presents first- and second-order continuous derivatives), but also because of the damping parameter employed to reduce the instability of the downward-continuation operator. '4 FIG. 9. Regional Bouguer anomaly from Reconcavo Basin, Brazil, obtained with the method of Beltrao et al. (1991) and using a thirteenth-order polynomial. Contour interval 5 mgal.

11 1712 Leao et al. The method may be extended to incorporate depth information from seismic or other geophysical data interpretation. In this case, differential weights should be given to all this information, according to their estimated accuracy. In computing the objective function, the 2 norm of the differences between the true and computed depths was used. However, if one suspects that a few data, although accurate, are not Table 5. Reconcavo Basin. Observed and computed depths at points of known depths to the basement, and residual (observed minus computed) depths. Point Observed depth (km) Computed depth (km) Residual (km) A B C D E F G FIG. 11. Minimum objective function f (p), relative to parameters Op, cl, and C2, against fixed values of d1 in the interval [0 m,1600 m] for the RecBncavo Basin, Brazil. FIG. 10. Absolute difference between thirteenth- and twelfth-order regional Bouguer anomalies from Reconcavo Basin, Brazil. Contour interval 2 mgal.

12 Gravity Inversion of Basement Relief ' I F ' 2.4 /i 1) Le s. e &0 a.l V L8 s0 u C ]s 0 20,0 kw FIG. 12. Map of estimated depths to the basement in the Reconcavo Basin, Brazil, computed from the thirteenthorder regional. Contour interval 0.2 km. representative of their neighborhood, robust norms such as the 21, could be employed because the minimization of such norms allows for the presence of large discrepancies between true and computed depths. In other words, the 21 norm will not force the estimated interface to pass close to nonrepresentative points. Higher-order polynomials might also be used to represent an unwanted trend in data. However, the computed depth to the interface will tend to be more unstable. ACKNOWLEDGMENTS We thank the Associate Editor Peter Graebner for suggestions to improve the text, and the reviewers for their comments. We thank Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq) for supporting this research. REFERENCES Agocs, W. B., 1951, Least-squares residual anomaly determination: Geophysics, 16, Beltrao, J. F., Silva, J. B. C., and Costa, J. C., 1991, Robust polynomial fitting method for regional gravity estimation: Geophysics, 56, Burkhard, N., and Jackson, D. D., 1976, Application of stabilized linear inverse theory to gravity data: J. Geophys. Res., 81, Courtillot, V., Ducruix, J., and Le Mouel, J. L., 1974, A solution of some inverse problems in geomagnetism and gravimetry: J. Geophys. Res., 79, Chenot, D., and Debeglia, N., 1990, Three-dimensional gravity or magnetic constrained depth inversion with lateral and vertical variation of contrast: Geophysics, 55, Granser, H., 1987, Nonlinear inversion of gravity data using the Schmidt-Lichtenstein approach: Geophysics, 52, Guspi, F., 1993, Noniterative, nonlinear gravity inversion: Geophysics, 58, Karner, G. D., Egan, S. S., and Weissel, J. K., 1992, Modeling the tectonic development of the Tucano and Sergipe-Alagoas rift basins, Brazil: Tectonophysics, 215, Ledo, J. W. D., and Silva, J. B. C.,1989, Discrete linear transformations of potential-field data: Geophysics, 54, Litinsky, V. A., 1989, Concept of effective density: Key to gravity depth determinations for sedimentary basins: Geophysics, 54, Menezes, P. T. L., 1990, Uma nova abordagem na interpretacao de anomalias gravimetricas em bacias sedimentares Exemplo da Bacia do Reconcavo, Bahia, Brazil: M.Sc. thesis, Federal Univ. of Para, Brazil. Milani, E. J., 1985, Aspectos da evolucao tectonica das bacias do Rec6ncavo e Tucano Sul, Bahia, Brazil: M.Sc. thesis, Federal Univ. of Ouro Preto, Brazil. and Davison I., 1988, Basement control and transfer tectonics in the Reconcavo Tucano Jatoba rift, Northeast Brazil: Tectonophysics, 154,

13 1714 Leao et al. FIG. 13. Map of absolute differences between the estimated depths to the basement in the Reconcavo Basin, Brazil, computed from the thirteenth- and twelfth-order regionals. Contour interval 0.1 km. Oldenburg, D. W., 1974, The inversion and interpretation of gravity anomalies: Geophysics, 39, Parker, R. L., 1973, The rapid calculation of potential anomalies: Geophys. J. Roy. Astr. Soc., 31, Pedersen, L. B., 1977, Interpretation of potential field data a generalized inverse approach: Geophys. Prosp., 25, Pilkington, M., and Crossley, D. J., 1986, Determination of crustal interface topography from potential fields: Geophysics, 51, Price, W. L., 1983, Global optimization by controlled random search: J. Optimization Theory and Appl., 40, Richardson, R. M., and MacInnes, S. C., 1989, The inversion of gravity data into three-dimensional polyhedral models: J. Geophys. Res., 94, Simpson, S. M. Jr., 1954, Least-squares polynomial fitting to gravitational data and density plotting by digital computers: Geophysics, 19, Smith, R. A., 1961, A uniqueness theorem concerning gravity fields: Cambridge Phil. Soc. Proc., 57, Ulrich, T. J., 1968, Effect of wavelength filtering on the shape of the residual anomaly: Geophysics, 33,