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We have combined modeling and gravity inversion to estimate an overall smooth but locally discontinuous basement relief of a sedimentary basin. Through an initial global smoothness solution, the method indicates fault positions that may be accepted or modified. Then, the interpreter specifies the fault dip and a tentative fault slip, so the fault plane is fixed on the estimated relief. The segments of the current interpreted relief that are not accepted faults are estimated through smoothness inversion, and the response of the current estimated relief is computed. The slip of the fault being incorporated in the estimated relief is then modified interactively until the solution presents no side lobes about the fault extremes. The process is repeated for all faults. The iteration stops when the tentative solution becomes compatible with the geologic knowledge of the area and produces an acceptable data fit. The method was applied to data produced by a simulated graben defined by step faults dipping 60° and by inclined and arcuate terraces. The solution virtually coincided with the true source. The method's ability in testing geologic hypotheses was demonstrated in gravity profiles across the Büyük Menderes Valley in west Turkey and the San Jacinto graben in southern California. [ABSTRACT FROM AUTHOR]

Gravity interpretation of large sedimentary basins requires a large number of observations and of parameters estimations, so the development of very efficient gravity inversion methods applicable to such environments is of utmost importance. We found that the highly efficient Bott's method can be cast in the framework of Gauss-Newton's method for minimizing nonlinear functions. Also, we extended Bott's method to (1) be applicable to sedimentary basins with density contrast between the sediments and the basement decreasing with depth according to different laws, (2) optimize the modulus of the parameter correction vector, (3) stabilize the solution by imposing that it be smooth, and (4) quit the iteration according to an objective and effective criterion. To keep up the efficiency of Bott's method, stable solutions were obtained by applying a moving average to the usually unstable solution obtained at the last iteration. Solutions as efficacious as those obtained by the usual nonlinear method were produced by the present extension at a much smaller computation time. Nonlinear methods, with different implementations for solving a linear system of equations, required between one and two orders of magnitude more time to produce similar solutions using between 2000 and 3000 parameters and observations, for example. [ABSTRACT FROM AUTHOR]

We found that minimum ℓ2-norm and smoothness-constrained continuous solutions of the linear inverse problem of potential field data are harmonic and biharmonic, respectively. In the case of a discrete distribution, the minimum ℓ2-norm and smoothness-constrained solutions become biased toward being harmonic or biharmonic, respectively. As a result, the estimated discrete distribution of density or magnetization contrast tends to be smooth and to satisfy the maximum principle, which forces the solution maxima and minima to lie on any boundary of the discretized region. The above findings were illustrated with 2D numerical examples. The harmonic or biharmonic bias is brought forth when the strengths of the minimum ℓ2-norm or the smoothness constraint are enhanced (relative to all other constraints) by approximating the continuous case (a large number of discretizing cells relative to the number of independent observations) and/or by using a regularizing parameter. We discovered that, by inspecting the rearranged normal equations, it is possible to qualify three different possibilities of designing nonharmonic estimators. Then we found that all three possibilities have in fact already been implemented in the literature, reinforcing, in this way, the validity of the theoretical demonstration. [ABSTRACT FROM AUTHOR]

ABSTRACT We present a comprehensive review of the most common gravity and magnetic interpretation methods to in depth retrieval of the geometry of geologic bodies associated with a sedimentary basin. We identify three types of bodies: 1) the sedimentary basement relief, 2) salt bodies and 3) mafic intrusions in a sedimentary section. In reconstructing basement topography through gravity and/or magnetic data we identify three groups of methods: the automatic, the spectral and the nonspectral methods. The reconstruction of salt bodies from gravity data usually uses interactive forward modelling but recently gravity inversion methods have been developed to interpret this kind of geologic environment. Finally, the problem of reconstructing intrusive bodies using magnetic and/or gravity data employs three strategies to interpret mafic or ultramafic intrusions in a sedimentary section: the automatic methods, interactive forward modelling and the inversion methods. [ABSTRACT FROM AUTHOR]

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 described by polar coordinates with an unknown origin within the prism. Our method estimates 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 0o to 360o. 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 whose shapes can be isometric or anisometric. The estimated 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 data-misfit 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 for the maximum depth producing the smallest value of data-misfit measure in the mt× 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The gravity and magnetic data measured on the Earth's surface or above it (collected from an aircraft flying at low altitude) can be used to assist in geologic mapping by estimating the spatial density and magnetization distributions, respectively, presumably confined to the interior of a horizontal slab with known depths to the top and bottom. To estimate density or magnetization distributions we assume a piecewise constant function defined on a user-specified grid of cells and invert the gravity or magnetic data by using the entropic regularization as a stabilizing function that allows estimating abrupt changes in the physical-property distribution. The entropic regularization combines the minimization of the first-order entropy measure with the maximization of the zeroth-order entropy measure of the solution vector. The aim of this approach is to detect sharp-bounded geologic units through the discontinuities in the estimated density or magnetization distributions. Tests conducted with synthetic data show that the entropic regularization can delineate discontinuous geologic units, allowing a better mapping of sharp-bounded (but buried) geologic bodies. We demonstrate the potential of the entropic regularization to assist a geologist in obtaining a geologic map by analyzing the estimated magnetization distributions from field magnetic data over a magnetic skarn in Butte Valley, Nevada, U.S.A. We show that it is an exoskarn where the ion exchange between the intrusive and the host rock occurs along a limited portion of the southern intrusive border. [ABSTRACT FROM AUTHOR]

We applied the mathematical basis of the total variation (TV) regularization to analyze the physicogeologic meaning of the TV method and compared it with previous gravity inversion methods (weighted smoothness and entropic Regularization) to estimate discontinuous basements. In the second part, we analyze the physicogeologic meaning of the TV method and compare it with previous gravity inversion methods (weighted smoothness and entropic regularization) to estimate discontinuous basements. Presenting a mathematical review of these methods, we show that minimizing the TV stabilizing function favors discontinuous solutions because a smooth solution, to honor the data, must oscillate, and the presence of these oscillations increases the value of the TV stabilizing function. These three methods are applied to synthetic data produced by a simulated 2D graben bordered by step faults. TV regularization and weighted smoothness are also applied to the real anomaly of Steptoe Valley, Nevada, U.S.A. in all applications, the three methods perform similarly. TV regularization, however, has the advantage, compared with weighted smoothness, of requiring no a priori information about the maximum depth of the basin. As compared with entropic regularization, TV regularization is much simpler to use because it requires, in general, the tuning of just one regularization parameter. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 cohtrasts. 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 50 m. 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. [ABSTRACT FROM AUTHOR]

We demonstrate the potential of gravity data to detect and to locate in-depth subtle normal faults in the basement relief of a sedimentary basin. This demonstration is accomplished by inverting the gravity data with the constraint that the estimated basement relief presents local abrupt faults and is smooth elsewhere. We inverted the gravity data from the onshore Almada Basin in northeastern Brazil, and we mapped several normal faults whose locations and plane geometries were already known from seismic imaging. The inversion method delineated well both the discontinuities with small or large slips and a sequence of step faults. Using synthetic data, we performed a systematic search of normal fault slips versus fault displacement depths to map the fault-detectable region in this space. This mapping helps to assess the ability of gravity inversion to detect normal faults. Mapping shows that normal faults with small (<0.5 km), medium (about 1 km), and large (about 2 km) vertical slips can be detected if the maximum midpoint depths of the fault planes are smaller than 1.8, 3.8, and 6.8 km, respectively. [ABSTRACT FROM AUTHOR]

We present a new semiautomatic gravity interpretation method for estimating a complex interface between two media containing density heterogeneities (referred to as interfering sources) that give rise to a complex and interfering gravity field. The method combines a robust fitting procedure and the constraint that the interface is very smooth near the interfering sources, whose approximate horizontal coordinates are defined by the user. The proposed method differs from the regional-residual separation techniques by using no spectral content assumption about the anomaly produced by the interface to be estimated, i.e., the interface can produce a gravity response containing both low- and high-wavenumber features. As a result, it may be applied to map the relief of a complex interface in a geologic setting containing either shallow or deep-seated interfering sources. Tests conducted with synthetic data show that the method can be of utility in estimating the basement relief of a sedimentary basin in the presence of salt layers and domes or in the presence of mafic intrusions in the basement or in both basement and the sedimentary section. The method was applied to real gravity data from two geologic settings having different kinds of interfering sources and interfaces to be interpreted: (1) the interface between the upper and lower crusts over the Bavali shear zone of southern India and (2) the anorthosite-tonalite interface over the East Bull Lake gabbro-anorthosite complex outcrop in Ontario, Canada. [ABSTRACT FROM AUTHOR]