ALBERT

Description: We have developed a new cost-effective method for processing large-potential-field data sets via the equivalent-layer technique. In this approach, the equivalent layer is divided into a regular grid of equivalent-source windows. Inside each window, the physical-property distribution is described by a bivariate polynomial. Hence, the physical-property distribution within the equivalent layer is assumed to be a piecewise polynomial function defined on a set of equivalent-source windows. We perform any linear transformation of a large set of data as follows. First, we estimate the polynomial coefficients of all equivalent-source windows by using a linear regularized inversion. Second, we transform the estimated polynomial coefficients of all windows into the physical-property distribution within the whole equivalent layer. Finally, we premultiply this distribution by the matrix of Green’s functions associated with the desired transformation to obtain the transformed data. The regularized inversion deals with a linear system of equations with dimensions based on the total number of polynomial coefficients within all equivalent-source windows. This contrasts with the classical approach of directly estimating the physical-property distribution within the equivalent layer, which leads to a system based on the number of data. Because the number of data is much larger than the number of polynomial coefficients, the proposed polynomial representation of the physical-property distribution within an equivalent layer drastically reduces the number of parameters to be estimated. By comparing the total number of floating-point operations required to estimate an equivalent layer via our method with the classical approach, both formulated with Cholesky’s decomposition, we can verify that the computation time required for building the linear system and for solving the linear inverse problem can be reduced by as many as three and four orders of magnitude, respectively. Applications to synthetic and real data show that our method performs the standard linear transformations of potential-field data accurately.

Description: We have presented a joint inversion of all gravity-gradient tensor components to estimate the shape of an isolated 3-D geological body located in subsurface. The method assumes the knowledge about the depth to the top and density contrast of the source. The geological body is approximated by an interpretation model formed by an ensemble of vertically juxtaposed 3-D right prisms, each one with known thickness and density contrast. All prisms forming the interpretation model have a polygonal horizontal cross-section that approximates a depth slice of the body. Each polygon defining a horizontal cross-section has the same fixed number of vertices, which are equally spaced from 0° to 360° and have their horizontal locations described in polar coordinates referred to an arbitrary origin inside the polygon. Although the number of vertices forming each polygon is known, the horizontal coordinates of these vertices are unknown. To retrieve a set of juxtaposed depth slices of the body, and consequently, its shape, our method estimates the radii of all vertices and the horizontal Cartesian coordinates of all arbitrary origins defining the geometry of all polygons describing the horizontal cross-sections of the prisms forming the interpretation model. To obtain a stable estimate that fits the observed data, we impose constraints on the shape of the estimated body. These constraints are imposed through the well-known zeroth- and first-order Tikhonov regularizations allowing, for example, the estimate of vertical or dipping bodies. If the data do not have enough in-depth resolution, the proposed inverse method can obtain a set of stable estimates fitting the observed data with different maximum depths. To analyse the data resolution and deal with this possible ambiguity, we plot the 2 -norm of the residuals ( s ) against the estimated volume ( v p ) produced by a set of estimated sources having different maximum depths. If this s x v p curve ( s as a function of v p ) shows a well-defined minimum of s , the data have enough resolution to recover the shape of the body entirely. Conversely, if the observed data do not have enough resolution, some estimates with different maximum depths produce practically the same minimum value of s on the s x v p curve. In this case, the best estimate among a suite of estimates producing equally data fits is the one fitting the gravity-gradient data and producing the minima of both the source's bottom depth and volume. The histograms of the residuals can be used to quantify and remove systematic errors in the data. After removing these errors, we confirmed the ability of our method to recover the source geometry entirely (or its upper part only), if the data have sufficient (or insufficient) in-depth resolution. By inverting the gravity-gradient data from a survey over the Vinton salt dome (Louisiana, USA) with a density contrast of 0.55 g cm –3 , we estimated a massive cap rock whose maximum depth attains 460 ± 10 m and its shallowest portion is elongated in the northeast–southwest direction.

Description: In this tutorial, we will talk about a widely used method of interpretation for potential-field data called Euler de-convolution. Our goal is to demonstrate its usefulness and, most important, to call attention to some pitfalls encountered in interpretation of the results. The code and synthetic data required to reproduce our results and figures can be found in the accompanying IPython notebooks ( ipython.org/notebook ) at dx.doi.org/10.6084/m9.figshare.923450 or github.com/pinga-lab/paper-tle-euler-tutorial . The note-books also expand the analysis presented here. We encourage you to download the data and try them on your software of choice. For this tutorial, we will use the implementation in the open-source Python package Fatiando a Terra ( fatiando.org ).

Description: We have developed codes to calculate the local amplitude, the local phase, and the local orientation of the nonscale and the Poisson’s scale-space monogenic signals of potential-field data in version 1.0 of the open-source program Monogenic. The monogenic vector of a generic function is calculated in the wavenumber domain and then transformed back into the space domain to find the monogenic signal attributes. We compare the use of the nonscale monogenic signal with the Poisson’s scale-space monogenic signal in magnetic data. This comparison shows that the latter can produce better results as an edge detection filter. The implementation of the monogenic signal can be used to enhance other geophysical data, such as seismic, ground-penetrating radar, gravity, multiple-component gravity gradiometry, and magnetic gradient data.

Description: Estimating the relief of the Moho from gravity data is a computationally intensive nonlinear inverse problem. What is more, the modelling must take the Earths curvature into account when the study area is of regional scale or greater. We present a regularized nonlinear gravity inversion method that has a low computational footprint and employs a spherical Earth approximation. To achieve this, we combine the highly efficient Bott's method with smoothness regularization and a discretization of the anomalous Moho into tesseroids (spherical prisms). The computational efficiency of our method is attained by harnessing the fact that all matrices involved are sparse. The inversion results are controlled by three hyperparameters: the regularization parameter, the anomalous Moho density-contrast, and the reference Moho depth. We estimate the regularization parameter using the method of hold-out cross-validation. Additionally, we estimate the density-contrast and the reference depth using knowledge of the Moho depth at certain points. We apply the proposed method to estimate the Moho depth for the South American continent using satellite gravity data and seismological data. The final Moho model is in accordance with previous gravity-derived models and seismological data. The misfit to the gravity and seismological data is worse in the Andes and best in oceanic areas, central Brazil and Patagonia, and along the Atlantic coast. Similarly to previous results, the model suggests a thinner crust of 30–35 km under the Andean foreland basins. Discrepancies with the seismological data are greatest in the Guyana Shield, the central Solimões and Amazonas Basins, the Paraná Basin, and the Borborema province. These differences suggest the existence of crustal or mantle density anomalies that were unaccounted for during gravity data processing.

Description: We have developed a new iterative scheme for processing gravity data using a fast equivalent-layer technique. This scheme estimates a 2D mass distribution on a fictitious layer located below the observation surface and with finite horizontal dimensions composed by a set of point masses, one directly beneath each gravity station. Our method starts from an initial mass distribution that is proportional to the observed gravity data. Iteratively, our approach updates the mass distribution by adding mass corrections that are proportional to the gravity residuals. At each iteration, the computation of the residual is accomplished by the forward modeling of the vertical component of the gravitational attraction produced by all point masses setting up the equivalent layer. Our method is grounded on the excess of mass and on the positive correlation between the observed gravity data and the masses on the equivalent layer. Mathematically, the algorithm is formulated as an iterative least-squares method that requires neither matrix multiplications nor the solution of linear systems, leading to the processing of large data sets. The time spent on the forward modeling accounts for much of the total computation time, but this modeling demands a small computational effort. We numerically prove the stability of our method by comparing our solution with the one obtained via the classic equivalent-layer technique with the zeroth-order Tikhonov regularization. After estimating the mass distribution, we obtain a desired processed data by multiplying the matrix of the Green’s functions associated with the desired processing by the estimated mass distribution. We have applied the proposed method to interpolate, calculate the horizontal components, and continue gravity data upward (or downward). Testing on field data from the Vinton salt dome, Louisiana, USA, confirms the potential of our approach in processing large gravity data set over on undulating surface.

Description: We have developed a fast total-field anomaly inversion to estimate the magnetization direction of multiple sources with approximately spherical shapes and known centres. Our method is an overdetermined inverse problem that can be applied to interpret multiple sources with different but homogeneous magnetization directions. It requires neither the prior computation of any transformation-like reduction to the pole nor the use of regularly spaced data on a horizontal grid. The method contains flexibility to be implemented as a linear or non-linear inverse problem, which results, respectively, in a least-squares or robust estimate of the components of the magnetization vector of the sources. Applications to synthetic data show the robustness of our method against interfering anomalies and errors in the location of the sources' centre. Besides, we show the feasibility of applying the upward continuation to interpret non-spherical sources. Applications to field data over the Goiás alkaline province (GAP), Brazil, show the good performance of our method in estimating geologically meaningful magnetization directions. The results obtained for a region of the GAP, near to the alkaline complex of Diorama, suggest the presence of non-outcropping sources marked by strong remanent magnetization with inclination and declination close to −70.35 and −19.81°, respectively. This estimated magnetization direction leads to predominantly positive reduced-to-the-pole anomalies, even for other region of the GAP, in the alkaline complex of Montes Claros de Goiás. These results show that the non-outcropping sources near to the alkaline complex of Diorama have almost the same magnetization direction of those ones in the alkaline complex of Montes Claros de Goiás, strongly suggesting that these sources have been emplaced in the crust within almost the same geological time interval.

Description: We have developed a fast total-field anomaly inversion to estimate the magnetization direction of multiple sources with approximately spherical shape and known centres. Our method can be applied to interpret multiple sources with different magnetization directions. It neither requires the prior computation of any transformation like reduction to the pole nor the use of regularly spaced data on a horizontal grid. The method contains flexibility to be implemented as a linear or non-linear inverse problem, which results, respectively, in a least-squares or robust estimate of the components of the magnetization vector of the sources. Applications to synthetic data show the robustness of our method against interfering anomalies and errors in the location of the sources' centre. Besides, we show the feasibility of applying the upward continuation to interpret non-spherical sources. Applications to field data over the Goiás Alkaline Province (GAP), Brazil, show the good performance of our method in estimating geological meaningful magnetization directions. The results obtained for a region of the GAP, near from the alkaline complex of Diorama, suggest the presence of non-outcropping sources marked by strong remanent magnetization with inclination and declination close to -70.35° and -19.81°, respectively. This estimated magnetization direction leads to predominantly positive reduced-to-the-pole anomalies, even for other region of the GAP, in the alkaline complex of Montes Claros de Goiás. These results show that the non-outcropping sources near from the alkaline complex of Diorama have almost the same magnetization direction of that ones in the alkaline complex of Montes Claros de Goiás, strongly suggesting that these sources have emplaced the crust almost within the same geological time interval.