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We advance a principle directed to assigning numerical values to free parameters usually present in inversion methods. It may be formulated as: ‘Optimum estimates of free parameters in an inversion procedure must lead, in tests using synthetic data, to solutions whose geometrical expression reflects the main qualitative or semiquantitative geological characteristic of the study area.’ To this end, the interpreter should (i) specify a typical anomalous source geometry which incorporates the most relevant geological information for the study area, (ii) compute the corresponding gravity anomaly and (iii) invert the anomaly for the source geometry finding the numerical values of the free parameters that lead to a solution closest to the typical source. Application of the above methodology to synthetic and real data from the basement relief of a rift basin has asserted its efficacy.

We have developed a novel approach to compute, in an efficient and versatile way, the gravity anomaly produced by an arbitrary, discrete 3D distribution of density contrast. The method allows adjustable precision and is particularly suited for the interpretation of sedimentary basins. Because the gravity field decays with the square of the distance, we use a discrete Green’s operator that may be made much smaller than the whole study area. For irregularly positioned observations, this discrete Green’s operator must be computed just at the first iteration, and because each of its horizontal layers presents a center of symmetry, only one-eighth of its total elements need to be calculated. Furthermore, for gridded data on a plane, this operator develops translation symmetry for the whole region of interest and has to be computed just once for a single arbitrary observation position. The gravity anomaly is obtained as the product of this small operator by any arbitrary density contrast distribution in a convolution-like operation. We use the proposed modeling to estimate the basement relief of a [Formula: see text] basin with density contrast varying along [Formula: see text] only using a smaller Green’s operator at all iterations. The median of the absolute differences between relief estimates, using the small and a large operator (the latter covering the whole basin) has been approximately 9 m for a 3.6 km deep basin. We also successfully inverted the anomaly of a simulated basin with density contrast varying laterally and vertically, and a real anomaly produced by a steeply dipping basement. The proposed modeling is very fast. For 10,000 observations gridded on a plane, the inversion using the proposed approach for irregularly spaced data is two orders of magnitude faster than using an analytically derived fitting, and this ratio increases enormously with the number of observations.

We have developed a gravity inversion method to estimate a discontinuous basement relief based on an extended version of Bott’s method that allows variable density contrasts between sediments and basement, optimizes the modulus of the solution correction at each iteration, and provides for solution stabilization. Initially, we obtain a linear approximation stabilized by the total variation functional that correctly maps the horizontal positions of the existing high-angle faults but produced poor estimates of the basin depths. Subsequent iterations update the depth estimates toward the correct values, at the same time preserving the correct fault horizontal positions. Additionally, we stabilize each solution correction by the smoothness constraint without inverting any matrix. The method was substantially more efficient than the nonlinear method, which solves a system of linear equations by the conjugate gradient method at each iteration. For 3000 parameters, it is almost four times faster than the conjugate gradient, and this ratio increased with the number of parameters.

We have developed a new method that drastically reduces the number of the source location estimates in Euler deconvolution to only one per anomaly. Our method employs the analytical estimators of the base level and of the horizontal and vertical source positions in Euler deconvolution as a function of the [Formula: see text]- and [Formula: see text]-coordinates of the observations. By assuming any tentative structural index (defining the geometry of the sources), our method automatically locates plateaus, on the maps of the horizontal coordinate estimates, indicating consistent estimates that are very close to the true corresponding coordinates. These plateaus are located in the neighborhood of the highest values of the anomaly and show a contrasting behavior with those estimates that form inclined planes at the anomaly borders. The plateaus are automatically located on the maps of the horizontal coordinate estimates by fitting a first-degree polynomial to these estimates in a moving-window scheme spanning all estimates. The positions where the angular coefficient estimates are closest to zero identify the plateaus of the horizontal coordinate estimates. The sample means of these horizontal coordinate estimates are the best horizontal location estimates. After mapping each plateau, our method takes as the best structural index the one that yields the minimum correlation between the total-field anomaly and the estimated base level over each plateau. By using the estimated structural index for each plateau, our approach extracts the vertical coordinate estimates over the corresponding plateau. The sample means of these estimates are the best depth location estimates in our method. When applied to synthetic data, our method yielded good results if the bodies produce weak- and mid-interfering anomalies. A test on real data over intrusions in the Goiás Alkaline Province, Brazil, retrieved sphere-like sources suggesting 3D bodies.

SUMMARY We present a gravity-inversion method for estimating the geometry of an isolated 3-D source, assuming prior knowledge about its top and density contrast. The subsurface region containing the geological sources is discretized into an ensemble of 3-D vertical prisms juxtaposed in the vertical direction of a right-handed coordinate system. The prisms’ thicknesses and density contrasts are known, but their horizontal cross-sections are described by unknown polygons. The horizontal coordinates of the polygon vertices approximately represent the edges of horizontal depth slices of the 3-D geological source. The polygon vertices of each prism are describedbypolarcoordinateswithanunknownoriginwithintheprism.Ourmethodestimates the horizontal Cartesian coordinates of the unknown origin and the radii associated with the vertices of each polygon for a fixed number of equally spaced central angles from 0 o to 360 o . By estimating these parameters from gravity data, we retrieve a set of vertically stacked prisms with polygonal horizontal sections that represents a set of juxtaposed horizontal depth slices of the estimated source. This set, therefore, approximates the 3-D source’s geometry. To obtain stable estimates we impose constraints on the source shape. The judicious use of first-order Tikhonov regularization on either all or a few parameters allows estimating both vertical and inclined sources whoseshapes can beisometricoranisometric. Theestimated solution, despite being stable and fitting the data, will depend on the maximum depth assumed for the set of juxtaposed 3-D prisms. To reduce the class of possible solutions compatible with the gravity anomaly and the constraints, we use a criterion based on the relationship between the datamisfit measure and the estimated total-anomalous mass computed along successive inversions, using different tentative maximum depths for the set of assumed juxtaposed 3-D prisms. In applying this criterion, we plotted the curve of the estimated total-anomalous mass mt versus data-misfit measure s for the range of different tentative maximum depths. The tentative value forthe maximum depth producing thesmallest value of data-misfit measure inthemt ×s curve is the best estimate of the true (or minimum) depth to the bottom of the source, depending on whether the true source produces a gravity anomaly that is able (or not) to resolve the depth to the source bottom. This criterion was theoretically deduced from Gauss’ theorem. Tests with synthetic data shows that the correct depth-to-bottom estimate of the source is obtained if the minimum of s on the mt × s curve is well defined; otherwise this criterion provides just a lower bound estimate of the source’s depth to the bottom. These synthetic results show that the method efficiently recovers source geometries dipping at different angles. Test on real data from the Matsitama intrusive complex (Botswana) retrieved a dipping intrusion with variable dips and strikes and with bottom depth of 8.0 ± 0.5 km.

We have developed an iterative scheme for inverting gravity data produced by salt bodies with density contrasts relative to the sediments varying from positive to negative, crossing, in this way, the nil zone. Our inversion method estimates a 3D density-contrast distribution, through a piecewise constant function defined on a user-specified grid of cells. It consists of two nested iterative loops. The outer loop uses an adaptive learning strategy that starts with a coarse grid of cells, a set of first-guess geometric elements (axes and points) and the corresponding assigned density contrasts. From the second iteration on, this strategy refines the grid and automatically creates a new set of geometric elements (points only) and associated density contrasts. Each geometric element operates as the first-guess skeletal outline of a section of the salt body to be imaged. The inner loop estimates the 3D density-contrast distribution for the grid of cells and for the set of geometric elements defined in the outer loop. The outer loop allows for easy incorporation of prior geologic information about the lithologic units and automatic evolution of the prior information. The inner loop forces the estimated density contrast of each cell to be close either to a null or to a non-null prespecified value. The iteration stops when the geometries of the estimated salt bodies are invariant along successive iterations. We apply our method to synthetic gravity data produced by a homogeneous salt body embedded in heterogeneous sediments. We tested two geologic hypotheses about the real gravity data from Galveston Island salt dome, USA. In the first, the estimated salt body attains a maximum bottom depth of 5 km, whereas in the second hypothesis, it is shallower and discloses an overhang. Both solutions fit the data and are feasible geologically, so both hypotheses are acceptable.

We have developed an inversion approach that estimates the basement relief of a fault-bounded sedimentary basin. The sedimentary pack is approximated by a grid of 3D or 2D vertical prisms juxtaposed in the horizontal directions of a right-handed coordinate system. The prisms’ thicknesses represent the depths to the basement and are the parameters to be estimated from the gravity data. To obtain depth-to-basement estimates, we introduce the total variation (TV) regularization as a stabilizing function. This approach lets us estimate a nonsmooth basement relief because it does not penalize sharp features of the solution. We have deduced a compact matrix form of the gradient vector and the Hessian matrix of the approximation to the TV function that allows a regularized Gauss-Newton minimization approach. Because the Hessian matrix of the approximation to the TV function is ill conditioned, we have modified this Hessian matrix to improve its condition and to accelerate the convergence of the Gauss-Newton algorithm. Tests conducted with synthetic data show that the inversion method can delineate discontinuous basements presenting large slips or sequences of small-slip step faults. Tests on field data from the Almada Basin, Brazil, and from the San Jacinto Graben, California, U.S.A., confirm the potential of the method in detecting and locating in-depth normal faults in the basement relief of a sedimentary basin.

We have developed a gravity interpretation method for estimating the discontinuous basement relief of a sedimentary basin. The density contrast between the basement and the sediments is assumed to be known, and it could be either constant or vary monotonically with depth. The interpretation model consists of a set of vertical, juxtaposed prisms, whose thicknesses are the parameters to be estimated. We used the entropic regularization that combines the minimization of the first-order entropy measure with the maximization of the zeroth-order entropy measure of the solution vector. We validated the method by applying it to synthetic data produced by a simulated basin bordered by high-angle step faults; we obtained a good definition of the relief, particularly of the discontinuities. We also applied the method to a profile across the Büyük Menderes Valley in West Turkey and obtained a solution exhibiting a gravity fault with large slip on the northern border of the valley. When applied to the interpretation of a discontinuous basement relief, the method has a better performance than the global smoothness method. It is comparable to the weighted smoothness method, but it does not require the a priori knowledge about the maximum basin depth.

We have developed a gravity-inversion method for simultaneously estimating the 3D basement relief of a sedimentary basin and the parameters defining a presumed parabolic decay of the density contrast with depth in a sedimentary pack, assuming prior knowledge about the basement depth at a few points. The sedimentary pack is approximated by a grid of 3D vertical prisms juxtaposed in both horizontal directions of a right-handed coordinate system. The prisms’ thicknesses represent the depths to the basement and are the parameters to be estimated from the gravity data. To estimate the parameters defining the parabolic decay of the density contrast with depth and to produce stable depth-to-basement estimates, we imposed smoothness on the basement depths and proximity between estimated and known depths at boreholes. We applied our method to synthetic data from a simulated complex 3D basement relief with two sedimentary sections having distinct parabolic laws describing the density-contrast variation with depth. The results provide good estimates of the true parameters of the parabolic law of density-contrast decay with depth and of the basement relief. Inverting the gravity data from the onshore and part of the shallow offshore Almada Basin on Brazil’s northeastern coast shows good correlation with known structural features.

A new apparent-magnetization mapping method on the horizontal plane combines minimization of first-order entropy with maximization of zeroth-order entropy of the estimated magnetization. The interpretation model is a grid of vertical, juxtaposed prisms in both horizontal directions. To estimate the magnetization of the prisms, assume that the top and bottom of the magnetic sources are horizontal. Minimization of the first-order entropy favors solutions with sharp borders, and the maximization of zeroth-order entropy prevents the tendency of the estimated source to become a single prism with large magnetization. Thus, a judicious combination of both constraints can lead to solutions characterized by regions with virtually constant magnetizations separated by sharp discontinuities. This is applied to synthetic data from simulated intrusive bodies in sediments that have horizontal tops. By comparing the results with those obtained with the common Tikhonov regularization (smoothness constraint) method, it is shown that both methods produce good and equivalent locations of the central positions of the sources. However, entropic regularization delineates the boundaries of the bodies with greater detail. Both the proposed and the smoothness constraints are applied to real anomaly data over a magnetic skarn in Butte Valley, Nevada, U.S.A. Entropic regularization produced an estimated magnetization distribution with sharper boundaries, smaller volume, and higher apparent magnetization as compared with results produced by incorporating the smoothness constraint.

We have developed a gravity inversion method to estimate a 3D density-contrast distribution producing strongly interfering gravity anomalies. The interpretation model consists of a grid of 3D vertical, juxtaposed prisms in the horizontal and vertical directions. Iteratively, our approach estimates the 3D density-contrast distribution that fits the observed anomaly within the measurement errors and favors compact gravity sources closest to prespecified geometric elements such as axes and points. This method retrieves the geometry of multiple gravity sources whose density contrasts (positive and negative values) are prescribed by the interpreter through the geometric element. At the first iteration, we set an initial interpretation model and specify the first-guess geometric elements and their target density contrasts. Each geometric element operates as the first-guess skeletal outline of the entire homogeneous gravity source or any of its homogeneous parts to be reconstructed. From the second iteration on, our method automatically redefines a new set of geometric elements, the associated target density contrasts, and a new interpretation model whose number of prisms increases with the iteration. The iteration stops when the geometries of the estimated sources are invariant along successive iterations. Tests on synthetic data from geometrically complex bodies and on field data collected over a mafic-ultramafic body and a volcanogenic sedimentary sequence located in the Tocantins Province, Brazil, confirmed the potential of our method in producing a sharp image of multiple and adjacent bodies.

We present a new apparent-density mapping method on the horizontal plane that combines the minimization of the first-order entropy with the maximization of the zeroth-order entropy of the estimated density contrasts. The interpretation model consists of a grid of vertical, juxtaposed prisms in both horizontal directions. We assume that the top and the bottom of the gravity sources are flat and horizontal and estimate the prisms’ density contrasts. The minimization of the first-order entropy favors solutions presenting sharp borders, and the maximization of the zeroth-order entropy prevents the tendency of the source estimate to become a single prism. Thus, a judicious combination of both constraints may lead to solutions characterized by regions with virtually constant estimated density contrasts separated by sharp discontinuities. We apply our method to synthetic data from simulated intrusive bodies in sediments that present flat and horizontal tops. By comparing our results with those obtained with the smoothness constraint, we show that both methods produce good and equivalent locations of the sources’ central positions. However, the entropic regularization delineates the boundaries of the bodies with greater resolution, even in the case of 100-m-wide bodies separated by a distance as small as [Formula: see text]. Both the proposed and the global smoothness constraints are applied to real anomalies from the eastern Alps and from the Matsitama intrusive complex, northeastern Botswana. In the first case, the entropic regularization delineates two sources, with a horizontal and nearly flat top being consistent with the known geologic information. In the second case, both constraints produce virtually the same estimate, indicating, in agreement with results of synthetic tests, that the tops of the sources are neither flat nor horizontal.

We have developed a new approach for estimating the location and geometry of several density anomalies that give rise to a complex, interfering gravity field. The user interactively defines the assumed outline of the true gravity sources in terms of points and line segments, and the method estimates sources closest to the specified outline to achieve a match between the predicted and observed gravity fields. Each gravity source is assumed to be a homogeneous body with a known density contrast; different density contrasts may be assigned to each source. Tests with synthetic data show that the method can be of use in estimating (1) multiple laterally adjacent and closely situated gravity sources, (2) single gravity sources consisting of several homogeneous compartments with different density contrasts, and (3) two gravity sources with different density contrasts of the same sign, one totally enclosed by the other. The method is also applied to three different sets of field data where the gravity sources belong to the same categories established in the tests with synthetic data. The method produces solutions consistent with the known geologic attributes of the gravity sources, illustrating its potential practicality.

We present a new stable gravity inversion method applied to the mapping of an interface separating two homogeneous media. In contrast with previous similar methods, it does not impose an overall smoothness on the estimated interface to stabilize the solution. The density contrast between the media is assumed to be known. The interpretation model for the upper medium consists of rectangular juxtaposed prisms whose thicknesses represent the depths to the interface and are the parameters to be estimated. The true interface is assumed to be flat everywhere except at faults. To incorporate this attribute into the estimated relief, we developed an iterative process in which three kinds of constraints are imposed on parameters: (1) proximity between values of adjacent parameters, (2) lower and upper bounds to parameters, and (3) proximity between the values of parameters and fixed numerical values. Starting with an initial solution which presents an overall smooth relief, the method enhances initially estimated geometric features of the interface; that is, flat areas will tend to become flatter and steep areas will tend to become steeper. This is accomplished by weighting the constraints, which requires proximity between adjacent parameters. The weights are initialized with values equal to unity and are updated automatically to enhance any discrepancy between adjacent depths that have been detected at the initial solution. Constraints 2 and 3 are used both to compensate for the decrease in solution stability caused by the introduction of small weights and to reinforce flatness at the basin bottom. Constraint 2 imposes that any depth be nonnegative and smaller than an a priori known maximum depth value, whereas constraint 3 imposes that all depths be closest to a value greater than the maximum depth. The trade‐off between these conflicting constraints is attained with a final relief presenting flat bottom and steep borders. The method was tested with a synthetic gravity anomaly produced by a simulated sedimentary cratonic extensional basin whose basement consists of steep edges and a flat bottom. The results showed an improvement in the resolution of the relief, leading to a reliable mapping both of the sharp discontinuities at the borders and of the lateral extent of the base of the basin. Additionally, the method produced excellent estimates for the average dip angles of the basin edges (presumably controlled by normal faults), indicating, in this way, its potential in interpreting data produced by this kind of basin. The method was applied to the Bouguer anomaly from the northern portion of Steptoe Valley, Nevada, delineating an isolated basin with a wider, flat base and relatively straight borders as compared with the estimate imposing overall smoothness on the relief.

We present a new, stable method for interpreting the basement relief of a sedimentary basin which delineates sharp discontinuities in the basement relief and incorporates any law known a priori for the spatial variation of the density contrast. The subsurface region containing the basin is discretized into a grid of juxtaposed elementary prisms whose density contrasts are the parameters to be estimated. Any vertical line must intersect the basement relief only once, and the mass deficiency must be concentrated near the earth’s surface, subject to the observed gravity anomaly being fitted within the experimental errors. In addition, upper and lower bounds on the density contrast of each prism are introduced a priori (one of the bounds being zero), and the method assigns to each elementary prism a density contrast which is close to either bound. The basement relief is therefore delineated by the contact between the prisms with null and nonnull estimated density contrasts, the latter occupying the upper part of the discretized region. The method is stabilized by introducing constraints favoring solutions having the attributes (shared by most sedimentary basins) of being an isolated compact source with lateral borders dipping either vertically or toward the basin center and having horizontal dimensions much greater than its largest vertical dimension. Arbitrary laws of spatial variations of the density contrast, if known a priori, may be incorporated into the problem by assigning suitable values to the nonnull bound of each prism. The proposed method differs from previous stable methods by using no smoothness constraint on the interface to be estimated. As a result, it may be applied not only to intracratonic sag basins where the basement relief is essentially smooth but also to rift basins whose basements present discontinuities caused by faults. The method’s utility in mapping such basements was demonstrated in tests using synthetic data produced by simulated rift basins. The method mapped with good precision a sequence of step faults which are close to each other and present small vertical slips, a feature particularly difficult to detect from gravity data only. The method was also able to map isolated discontinuities with large vertical throw. The method was applied to the gravity data from Reco⁁ncavo basin, Brazil. The results showed close agreement with known geological structures of the basin. It also demonstrated the method’s ability to map a sequence of alternating terraces and structural lows that could not be detected just by inspecting the gravity anomaly. To demostrate the method’s flexibility in incorporating any a priori knowledge about the density contrast variation, it was applied to the Bouguer anomaly over the San Jacinto Graben, California. Two different exponential laws for the decrease of density contrast with depth were used, leading to estimated maximum depths between 2.2 and 2.4 km.

We present a gravity interpretation method for estimating the relief of an arbitrary interface separating two homogeneous media. The upper medium is discretized into rectangular, juxtaposed prisms whose thicknesses represent the depths to the interface and are the parameters to be estimated from the gravity anomaly. The density contrast of each prism is assumed to be constant and known. To stabilize the inversion, we introduce two kinds of constraints on the depths. The first one requires proximity between the observed and computed depths at isolated points such as those obtained from boreholes (absolute equality constraint). The second one requires that groups of depths approximately follow an established linear relationship among the depths (relative equality constraint). Both kinds of constraints are imposed in the least‐squares sense. We illustrate the method performance by applying it to a synthetic anomaly produced by a simulated basement relief consisting of four narrow and adjacent structural lows. Only two structural lows produced isolated gravity lows. Nonetheless, the whole basement topography was successfully reconstructed with an average error of 4% of the maximum relief amplitude. In this example, the relative constraints established that the thicknesses of adjacent prisms should be as close as possible to each other (overall relief smoothness). As absolute constraints we used point information about the basement depth at five points. In addition to the overall relief smoothness, other relevant geologic information, such as localized relief smoothness, occurring at structural terraces, for example, can be incorporated by assigning different weights to the relative equality constraints. The method was applied to the gravity anomaly of Recôncavo Basin, Brazil, leading to a sharper definition of the basement features relative to previous gravity interpretations of this area.

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 [Formula: see text] of the shallowest point of the interface, the density contrast Δρ between the two media, and the coefficients [Formula: see text] and [Formula: see text] 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 Δρ (rather than the knowledge of its true value) and coefficients [Formula: see text] and [Formula: see text]; and (3) does not require the knowledge of the average interface depth [Formula: see text]. 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 [Formula: see text]; rather, it is able to use the knowledge of the interface depth at just a few isolated points to estimate [Formula: see text], Δρ, [Formula: see text], and [Formula: see text]. 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 Recôncavo Basin, Brazil, producing good correlations of estimated lows and terraces in the basement with corresponding known geological features.

We present a 3-D gravity interpretation method based on the inversion of source moments employing the series derived from the gravity anomaly expansion in multipoles and retaining moments up to second order only. It presents the advantages of being essentially linear and allowing a straightforward computer implementation. The method requires neither an explicit assumption about the source geometry nor a priori knowledge about the density contrast distribution, which may even be nonuniform. The method assumes implicitly: (1) that the source fits in the interior of an imaginary sphere whose center coincides with the source center of mass and whose radius is smaller than the depth to the source center of mass relative to the measuring plane, and (2) that the spatial density distribution presents three ortogonal planes of symmetry intersecting at the source center of mass. The first assumption can be met by upward continuing the observed anomaly. When both assumptions are met, the method produces reliable and stable estimates of the total anomalous mass, the coordinates of the center of mass, the three principal axes directions of the anomalous body, and the relative importance among the axes. The method is particularly suited for interpreting compact, isolated or disjoint, but spatially correlated sources. The method is sensitive to an incomplete removal of the regional field (presumably superimposed on the residual anomaly of interest) because the unremoved part of the regional field may strongly affect the estimates of the residual source moments.

A generic gravity source moment is an integral, over the source volume, of the product of the density distribution by a polynomia in the Cartesian coordinates of a point belonging to this volume. We obtained a formal expression for a generic moment in terms of integrals involving the gravity anomaly and the gravity potential. By analyzing the conditions under which this expression is valid, we conclude that, without usinga priori information regarding the sources, it is possible to determine, from the gravity anomaly, any moment or linear combination of moments whose associated polynomial has null Laplacian and depends only on the coordinates defining the measurement plane. Additionally, no moment whose associated polynomial has a nonnull laplacian can be determined without usinga priori information of the source.

Interpolation using only the observations at discrete points is an ill‐posed problem because it admits infinite solutions. Usually, to reduce ambiguity, a priori information about the sample function is introduced. Current interpolation methods in mineral exploration introduce only the constraints of continuity and smoothness of the interpolating function. In interpolating potential‐field anomalies, the constraint that the sampled function is harmonic may be introduced by the equivalent‐layer method (ELM). We compare the performance of the ELM and the minimum curvature method (MCM) in interpolating potential‐field anomalies by applying these methods to synthetic magnetic data simulating an aeromagnetic survey. In the case the anomaly flanks and peak are undersampled, the ELM performs better than the MCM in recovering the anomaly gradients and peak. In the case of elongated linear anomalies, the ELM recovers the exact linear pattern, but the MCM introduces spurious oscillations in the linear pattern. Also, the ELM is able to reduce the data from a survey flown at different heights to a common level. In contrast, the MCM, being a 2-D interpolation method, cannot account for variations in the vertical coordinates of the observation points.

Extending the compact gravity inversion technique by incorporating a priori information about the maximum compactness of the anomalous sources along several axes provides versatility. Thus, the method may also incorporate information about limits in the axes lengths or greater concentration of mass along one or more directions. The judicious combination of different constraints on the anomalous mass distribution allows the introduction of several kinds of a priori information about the (arbitrary) shape of the sources. This method is particularly applicable to constant, linear density sources such as mineralizations along faults and intruded sills, dikes, and laccoliths in a sedimentary basin. The correct source density must be known with a maximum uncertainty of 40 percent; otherwise, the inversion produces thicker bodies for densities smaller than the true value and vice‐versa. Because of the limitations of the inverse gravity problem, the proposed technique requires an empirical technique to analyze the sensitivity of solutions to uncertainties in the a priori information. The proposed technique is based on a finite number of acceptable solutions, presumably representative of the ambiguity region. By using standard statistical techniques, each parameter is assigned a coefficient measuring its uncertainty. The known hematite and magnetite ore body shape, in the vicinity of Iron Mountain, MO, was reproduced quite well using this inversion technique.

Apparent density maps are derived from observed residual Bouguer maps under the assumptions that the sources are restricted to a depth interval and that the density distribution is not a function of the vertical coordinate. If the depth interval containing the sources is known, the computed apparent density maps are very close to the true density distributions. Even when the depth interval of the sources is not known, but the sources have constant density, their horizontal extent may be delineated by the half‐maximum contours of the apparent densities. We developed an interpretation method using a family of curves based on source density, depth to top of sources, and source thickness. The main advantage in the use of these families of curves is the broad range of possible interpretations involving the above‐mentioned parameters. Rough semiquantitative interpretations are possible in the absence of any further a priori information; more refined semiquantitative interpretations require additional quantitative knowledge of one parameter and just a lower or upper bound for another parameter. Finally, quantitative interpretations are possible only with additional quantitative knowledge of two of the three parameters involved.

Standard polynomial fitting methods are inconsistent in their formulation. The regional field is approximated by a polynomial fitted to the observed field. As a result, in addition to the nonuniqueness in the definition of the regional field, the fitted polynomial is strongly influenced by the residual field (observed field minus regional field). We present a regional‐residual separation method for gravity data which uses a robust procedure to determine the coefficients of a polynomial fitted to the observations. Under the hypothesis that the regional can be modeled correctly by the polynomial surface, the proposed method minimizes the influence of the residual field in the fitted surface. The proposed method was applied to real gravity data from Ceará state, Brazil, and produced information on zones of possible crustal thickening and the occurrence of lower‐crustal granulitic rocks thrust into the shallow subsurface.

We present a 3D gravity inversion for estimating the location and geometry of 3D salt bodies. Iteratively, our approach estimates a 3D density-contrast distribution by assuming a piecewise constant function defined on a userspecified grid of cells. To estimate a unique and stable density-contrast distribution, we look for the solution that fits the observed anomaly within the measurement errors and favors compact salt bodies closest to pre-specified geometric elements, such as axes and points. Our inversion method consists of two nested iterative loops. The outer loop uses an adaptive learning strategy that starts with a coarse grid of cells, a set of first-guess geometric elements (axes and points) and the corresponding assigned density contrast. From the second iteration on, this strategy refines the grid and creates a new set of geometric elements (points only) and associated density contrasts. The inner loop estimates the density-contrast distribution for the grid of cells and for the set of geometric elements defined in the outer loop. Each geometric element operates as the firstguess skeletal outline of a particular, homogeneous section of the salt body to be imaged. The algorithm easily incorporates known density contrast information by assigning these values to each geometric element. The iteration stops when the geometries of the estimated salt bodies are invariant along successive iterations. We apply our method to synthetic gravity data obtained from a salt body, whose density contrast with the host rocks varies from positive to negative. We tested two different geologic hypotheses about the geometric elements and the assigned density contrast on the real gravity anomaly from Galveston Island salt dome, offshore Texas, USA. In the first one, we estimated a vertically elongated salt dome with depth to the bottom at 5 km. In the second hypothesis, we estimated a cylindrical salt dome with an associated overhang and depth to the bottom at 4 km.

We develop a 3D gravity anomaly inversion to estimate a 3D density-contrast distribution that gives rise to a complex, interfering gravity field. Our approach estimates the 3D density-contrast distribution that fits the observed anomaly within the measurement errors and favors compact gravity sources closest to pre-specified geometric elements such as lines and points. This method retrieves the geometry of multiple gravity sources with prescribed density contrasts (positive and negative values) assigned to each geometric element. In this way, each geometric element operates as the first-guess skeletal outline of a particular, homogeneous section of the gravity source to be reconstructed. Based on an iterative learning scheme, the number of parameters, the number of geometric elements, their spatial positions and the associated target density contrasts may increase automatically. Tests on synthetic data from geometrically complex bodies and on field data collected over a granitic pluton, located in the Carajas province of the eastern Amazonian Craton, Brazil, illustrate that our method makes its possible to reconstruct a sharp image of multiple and closely located gravity sources.

We present a new gravity inversion method, which produces an apparent density contrast mapping on the horizontal plane, based on the combination of the minimization of the first-order entropy with the maximization of the zeroth-order entropy. The interpretation model consists of a horizontal grid of 3D vertical prisms, and the parameters to be estimated are the prisms density contrasts. The maximization of the zeroth-order entropy is equivalent to the global smoothness constraint whereas the minimization of the first-order entropy favors solutions presenting abrupt borders, so a judicious combination of both constraints may lead to solutions characterized by regions where the estimated density contrasts are virtually constant, separated by abrupt discontinuities. The method has been applied to synthetic data simulating the presence of intrusive bodies in sediments. The comparison of the results with those obtained with the global smoothness constraint applied to the same data, showed that that both methods locate the sources with good precision, but the entropic regularization delineates the contour of the bodies with greater resolution, even in the case of 100 m wide bodies separated by a distance as small as 50 m

Neste tutorial apresentamos uma revisão da deconvolução de Euler que consiste de três partes. Na primeira parte, recordamos o papel da clássica formulação da deconvolução de Euler 2D e 3D como um método para localizar automaticamente fontes de campos potenciais anômalas e apontamos as dificuldades desta formulação: a presença de uma indesejável nuvem de soluções, o critério empírico usado para determinar o índice estrutural (um parâmetro relacionado com a natureza da fonte anômala), a exeqüibilidade da aplicação da deconvolução de Euler a levantamentos magnéticos terrestres, e a determinação do mergulho e do contraste de susceptibilidade magnética de contatos geológicos (ou o produto do contraste de susceptibilidade e a espessura quando aplicado a dique fino). Na segunda parte, apresentamos as recentes melhorias objetivando minimizar algumas dificuldades apresentadas na primeira parte deste tutorial. Entre estas melhorias incluem-se: i) a seleção das soluções essencialmente associadas com observações apresentando alta razão sinal-ruído; ii) o uso da correlação entre a estimativa do nível de base da anomalia e a própria anomalia observada ou a combinação da deconvolução de Euler com o sinal analítico para determinação do índice estrutural; iii) a combinação dos resultados de (i) e (ii), permitindo estimar o índice estrutural independentemente do número de soluções; desta forma, um menor número de observações (tal como em levantamentos terrestres) pode ser usado; iv) a introdução de equações adicionais independentes da equação de Euler que permitem estimar o mergulho e o contraste de susceptibilidade das fontes magnéticas 2D. Na terceira parte apresentaremos um prognóstico sobre futuros desenvolvimentos a curto e médio prazo envolvendo a deconvolução de Euler. As principais perspectivas são: i) novos ataques aos problemas selecionados na segunda parte deste tutorial; ii) desenvolvimento de métodos que permitam considerar interferências de fontes localizadas ao lado ou acima da fonte principal, e iii) uso das estimativas de localização da fonte anômala produzidas pela deconvolução de Euler como vínculos em métodos de inversão para obter a delineação das fontes em um ambiente computacional amigável. In this tutorial we present a historical overview of the evolution of Euler deconvolution method consisting of three sections. In the first section, we recall the Euler deconvolution 2D and 3D classic formulation as a method to automatically locate sources of potential fields anomalies and point out the difficulties of this formulation: the presence of an undesirable spray of solutions, the empiric criteria used to determine the structural index (a parameter related with the source nature), the feasibility of applying Euler deconvolution to small ground data survey, and the determination of the dip and susceptibility contrast of geological contact (or the product of the susceptibility contrast and the thickness when applied to thin dike). In the second section, we present recent improvements aiming at minimizing some of the difficulties presented in the first section. They include: i) selecting the solutions essentially associated with observations presenting the largest signal-to-noise ratios; ii) using the correlation between the estimated anomaly baselevel and the observed anomaly itself or combining Euler deconvolution with the analytic signal to determine the structural index; iii) combining the results from (i) and (ii) to allow estimating the structural index independently of the number of solutions so that a small number of observations (such as in a ground survey) may be used; and iv) introducing additional equations, which are independent from Euler's equation to estimate the dip and susceptibility of 2D sources. In the third section we provide a short- to medium-term outlook into future developments. The main prospects are: i) furthering the improvements on the problems singled out in the second section; ii) developing methods to take vertical and lateral interferences into accounts; and iii) incorporating Euler deconvolution to constrain the source location estimates and use them in inversion methods to obtain a source delineation in a user-friendly computer environment.

We present a gravity interpretation method for estimating the relief of an arbitrary interface separating two homogeneous media. The upper medium is discretized into rectangular, juxtaposed prisms whose thicknesses represent the depths to the interface and are the parameters to be estimated from the gravity anomaly. To stabilize the inversion, we introduce two kinds of constraints on the depths. The first one requires proximity between the observed and computed depths at isolated points such as those obtained from boreholes (absolute equality constraint). The second one requires that groups of depths approximately follow an established linear relationship among them (relative equality constraint). Both kinds of constraints are imposed in the least-squares sense. This stabilization procedure does not bias the solution either toward an initial approximation, or toward the null solution. The method is applied to synthetic data, producing a good reconstruction of a complex basement consisting of several structural high and lows. It is also applied to real data of Reconcavo Basin, Brazil, leading to a sharper definition of the basement structures as compared with previous gravity interpretations of the area.