João B. C. Silva

Reduction to the pole as an inverse problem and its application to low‐latitude anomalies

Traditionally, reduction to the pole has been accomplished either by space- or wavenumber-domain filtering. In the two-dimensional case, this procedure is stable regardless of the latitude, as long as the source strike is not parallel to the horizontal projection of the geomagnetic field. In the three-dimensional case, however, reduction-to-the-pole filtering is stable only at high magnetic latitudes. At latitudes lower than 15 degrees, it is of no practical use due to a sharply increasing instability toward the magnetic equator.The three-dimensional instability of this filtering technique is demonstrated, and the reduction-to-the-pole problem is formulated in the context of a general linear inverse problem. As a result, stable solutions are found by using well-known stabilizing procedures developed for the inverse linear problem. The distribution of magnetization of an equivalent layer of doublets that reproduces the observed data is computed. The magnetic doublets are parallel to the magnetization direction which is assumed constant throughout the sources. The magnetic field reduced to the pole is then obtained by changing the inclinations of the geomagnetic field and the doublets to 90 degrees and recalculating the total field.The usefulness and limitations of the method at low magnetic latitudes are assessed using theoretical data. The effects of noise and anomaly truncation are also investigated for both high and low latitudes. In all cases, application of the proposed method produced meaningful results regardless of the latitude. The method is applied to field data from two different low-latitude anomalies. The first anomaly is due to a seamount in the Gulf of Guinea with reversed magnetization. The geomagnetic field at this location is about -23 degrees. The second anomaly is an intrabasement anomaly from Parnaiba Basin, Brazil, where the magnetization is assumed to be induced by a geomagnetic field with -1.4 degree inclination. The results obtained confirm that the proposed method produces stable, meaningful, reduced-to-the-pole maps.

Robust polynomial fitting method for regional gravity estimation

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 Ceara state, Brazil, and produced information on zones of possible crustal thickening and the occurrence of lower‐crustal granulitic rocks thrust into the shallow subsurface.

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.

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 ...

Nonlinear magnetic inversion using a random search method

A robust magnetic interpretation technique is described. It is based on the minimization of an objective function by a random search algorithm. Instead of just one estimate, this algorithm yields several points in parameter space, all producing objective function values below an assumed noise level. The centroid of the search points is in general a better estimate than each individual search point, especially when data are corrupted by noise. Since no derivatives are employed, even nondifferentiable objective functions, such as the l1 norm of residuals, can be used. Computation of parameter covariance matrices and confidence ellipses is possible using simple formulas. The usefulness of R‐ and Q‐mode factor analysis applied to the correlation matrix is demonstrated. The R‐mode factor analysis can be used in parameter variance control, and the Q‐mode factor analysis provides means to find models which are extreme in some sense. The technique was applied to theoretical data and to field data from three diffe...

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

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 Δρ between the two media, and the coefficients c1 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 Δρ (rather than the knowledge of its true value) and coefficients c1 and c2...

Discrete linear transformations of potential field data

We present a new approach to perform any linear transformation of gridded potential field data using the equivalent‐layer principle. It is particularly efficient for processing areas with a large amount of data. An N × N data window is inverted using an M × M equivalent layer, with M greater than N so that the equivalent sources extend beyond the data window. Only the transformed field at the center of the data window is computed by premultiplying the equivalent source matrix by the row of the Green’s matrix (associated with the desired transformation) corresponding to the center of the data window. Since the inversion and the multiplication by the Green’s matrix are independent of the data, they are performed beforehand and just once for given values of N, M, and the depth of the equivalent layer. As a result, a grid operator for the desired transformation is obtained which is applied to the data by a procedure similar to discrete convolution. The application of this procedure in reducing synthetic anoma...

The equivalent data concept applied to the interpolation of potential field data

The equivalent layer calculation becomes more efficient by first converting the observed potential data set to a much smaller equivalent data set, thus saving considerable CPU time. This makes the equivalent‐source method of data interpolation very competitive with other traditional gridding techniques that ignore the fact that potential anomalies are harmonic functions. The equivalent data set is obtained by using a least‐squares iterative algorithm at each iteration that solves an underdetermined system fitting all observations selected from previous iterations and the observation with the greatest residual in the preceding iteration. The residuals are obtained by computing a set of “predicted observations” using the estimated parameters at the current iteration and subtracting them from the observations. The use of Cholesky’s decomposition to implement the algorithm leads to an efficient solution update everytime a new datum is processed. In addition, when applied to interpolation problems using equiva...

Transformation of nonlinear problems into linear ones applied to the magnetic field of a two-dimensional prism

I present a magnetic interpretation method which transforms into a linear problem the nonlinear problem of obtaining the geometric and position parameters of a two‐dimensional vertical, infinite prism. The magnetization, the only linear parameter, becomes nonlinear after the transformation. By assuming a few discrete values over a prescribed interval for the magnetization, I obtain several solutions for the geometric and position parameters. By storing only the extreme solutions, bounds for each parameter are produced. The method was applied to synthetic anomalies due to isolated and interfering sources for which robust alternatives performed better than the least‐squares method. The correlation between the magnetization and the prism width is the most important factor controlling ambiguity of parameters. The horizontal position is the least affected parameter, followed by the depth to the top of the prism. Application to a real anomaly confirmed the results from synthetic data, except for a greater uncer...

Apparent-density mapping using entropic regularization

We present a new apparent-density mapping method on the horizontal plane that combines the minimization of the first-orderentropywiththemaximizationofthezeroth-order entropyoftheestimateddensitycontrasts.Theinterpretation modelconsistsofagridofvertical,juxtaposedprismsinboth horizontal directions. We assume that the top and the bottom of the gravity sources areflat and horizontal and estimate the prisms’densitycontrasts.Theminimizationofthefirst-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, ajudiciouscombinationofbothconstraintsmayleadtosolutions 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 presentflat 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, thatthetopsofthesourcesareneitherflatnorhorizontal.