Valéria C. F. Barbosa

Stability analysis and improvement of structural index estimation in Euler deconvolution

Euler deconvolution has been widely used in automatic aeromagnetic interpretations because it requires no prior knowledge of the source magnetization direction and assumes no particular interpretation model, provided the structural index defining the anomaly falloff rate related to the nature of the magnetic source, is determined in advance. Estimating the correct structural index and electing optimum criteria for selecting candidate solutions are two fundamental requirements for a successful application of this method. We present a new criterion for determining the structural index. This criterion is based on the correlation between the total‐field anomaly and the estimates of an unknown base level. These estimates are obtained for each position of a moving data window along the observed profile and for several tentative values for the structural index. The tentative value for the structural index producing the smallest correlation is the best estimate of the correct structural index. We also propose a n...

Gravity inversion of basement relief using approximate equality constraints on depths

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...

Generalized compact gravity inversion

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.

Potential‐field inversion: Choosing the appropriate technique to solve a geologic problem

To produce a unique and stable solution in potential‐field interpretation, an inversion method must introduce particular constraints. These constraints will inevitably restrict the type of geological setting where the method may be applied. We present a nonmathematical overview of most stabilizing constraints used in inversion methods. Our purpose is to demonstrate that the inversion results are valuable only if the mathematical stabilizing constraints are translated from the geological setting. We identify five basic types of constraints: (1) lower and upper bounds of parameter estimates; (2) proximity of a parameter estimate to a specified value; (3) proximity between pairs of parameter estimates; (4) concentration of the anomalous source about a geometrical element such as an axis; and (5) source compactness. In practice, if used in isolation, constraints (1), (2), (4), and (5) will not produce geologically meaningful results, regardless of the geological setting of the interpretation area. Constraint ...

3D Euler deconvolution: Theoretical basis for automatically selecting good solutions

We derive the analytical estimators for the horizontal and vertical source positions in 3D Euler deconvolution as a function of the x-, y-, and z-derivatives of the magnetic anomaly within a data window. From these expressions we show that, in the case of noise-corrupted data, the x-, y-, and z-coordinate estimates computed at the anomaly borders are biased toward the respective horizontal coordinate of the data window center regardless of the true or presumed structural indices and regardless of the magnetization inclination and declination. On the other hand, in the central part of the anomaly, the x- and y-coordinate estimates are very close to the respective source horizontal coordinates regardless of the true or presumed structural indices and regardless of the magnetization inclination and declination. This contrasting behavior of the horizontal coordinate estimates may be used to automatically delineate the region associated with the best solutions. Applying the Euler deconvolution operator inside this region would decrease the dispersion of all position estimates, improving source location precision.

Gravity inversion of a discontinuous relief stabilized by weighted smoothness constraints on depth

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 ...

Interactive gravity inversion

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.

Gravity inversion of basement relief and estimation of density contrast variation with depth

We present a method to estimate the basement relief as well as the density contrast at the surface and the hyperbolic decaying factor of the density contrast with depth, assuming that the gravity anomaly and the depth to the basement at a few points are known. In both cases, the interpretation model is a set of vertical rectangular 2D prisms whose thicknesses are parameters to be estimated and that represent the depth to the interface separating sediments and basement. The solutions to both problems are stable because of the incorporation of additional prior information about the smoothness of the estimated relief and the depth to the basement at a few locations, presumably provided by boreholes. The method was tested with synthetic gravity anomalies produced by simulated sedimentary basins with smooth relief, providing not only well-resolved estimated relief, but also good estimates for the density contrasts at the surface and for the decaying factors of the density contrast with depth. The method was ap...

Stable inversion of gravity anomalies of sedimentary basins with nonsmooth basement reliefs and arbitrary density contrast variations

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...

3D gravity inversion through an adaptive-learning procedure

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.