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: 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 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.

Description: 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.

Description: 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.

Description: An aeromagnetic data filter was tested as a tool in enhancing the magnetic responses of deep sources associated with marginal basins. Our approach consists in amplifying the responses of magnetic sources which extend through the continental slope and continental rise. This amplification compensates for the magnetic response attenuation over the continental slope as a result of (1) the seaward increase of the distance between the source and the magnetometer and (2) the seaward decrease in the volume of sources within the continental crust. To amplify the source response, we multiply the total-field anomaly at a given location by the exponential of the bathymetric depth at the same point. Tests on synthetic and real data showed that the filter was able to recover the locations of linear magnetic responses not clearly seen in the total-field-anomaly map. When compared with the analytic signal amplitude and the total horizontal derivative filters, the proposed filter proved to be complementary in enhancing lineaments with different orientations. When applied in conjunction with these filters, the proposed filter produced a synergistic effect, revealing lineaments that were not present in individual applications of the filters. The application of the filter to an offshore Brazilian margin comprising the Camamu-Almada and Jequitinhonha marginal basins highlighted northwest–southeast and northeast–southwest lineaments not clearly seen in the total-field data. The northeast–southwest lineaments were interpreted as related to fracture zones that may cut across the continental margin, controlling basement structures and fluid migration paths.

Description: We have developed a new gravity gradient inversion method for estimating a 3D density-contrast distribution defined on a grid of rectangular prisms. Our method consists of an iterative algorithm that does not require the solution of an equation system. Instead, the solution grows systematically around user-specified prismatic elements, called “seeds,” with given density contrasts. Each seed can be assigned a different density-contrast value, allowing the interpretation of multiple sources with different density contrasts and that produce interfering signals. In real world scenarios, some sources might not be targeted for the interpretation. Thus, we developed a robust procedure that neither requires the isolation of the signal of the targeted sources prior to the inversion nor requires substantial prior information about the nontargeted sources. In our iterative algorithm, the estimated sources grow by the accretion of prisms in the periphery of the current estimate. In addition, only the columns of the sensitivity matrix corresponding to the prisms in the periphery of the current estimate are needed for the computations. Therefore, the individual columns of the sensitivity matrix can be calculated on demand and deleted after an accretion takes place, greatly reducing the demand for computer memory and processing time. Tests on synthetic data show the ability of our method to correctly recover the geometry of the targeted sources, even when interfering signals produced by nontargeted sources are present. Inverting the data from an airborne gravity gradiometry survey flown over the iron ore province of Quadrilátero Ferrífero, southeastern Brazil, we estimated a compact iron ore body that is in agreement with geologic information and previous interpretations.

Description: 〈span〉〈div〉ABSTRACT〈/div〉In most applications, the Euler deconvolution aims to define the nature (type) of the geologic source (i.e., the structural index [SI]) and its depth position. However, Euler deconvolution also estimates the horizontal positions of the sources and the base level of the magnetic anomaly. To determine the correct SI, most authors take advantage of the clustering of depth estimates. We have analyzed Euler’s equation to indicate that random variables contaminating the magnetic observations and its gradients affect the base-level estimates if, and only if, the SI is not assumed correctly. Grounded on this theoretical analysis and assuming a set of tentative SIs, we have developed a new criterion for determining the correct SI by means of the minimum standard deviation of base-level estimates. We performed synthetic tests simulating multiple magnetic sources with different SIs. To produce mid and strongly interfering synthetic magnetic anomalies, we added constant and nonlinear backgrounds to the anomalies and approximated the simulated sources laterally. If the magnetic anomalies are weakly interfering, the minima standard deviations either of the depth or base-level estimates can be used to determine the correct SI. However, if the magnetic anomalies are strongly interfering, only the minimum standard deviation of the base-level estimates can determine the SI correctly. These tests also show that Euler deconvolution does not require that the magnetic data be corrected for the regional fields (e.g., International Geomagnetic Reference Field [IGRF]). Tests on real data from part of the Goiás Alkaline Province, Brazil, confirm the potential of the minimum standard deviation of base-level estimates in determining the SIs of the sources by applying Euler deconvolution either to total-field measurements or to total-field anomaly (corrected for IGRF). Our result suggests three plug intrusions giving rise to the Diorama anomaly and dipole-like sources yielding Arenópolis and Montes Claros de Goiás anomalies.〈/span〉

Description: 〈span〉〈div〉ABSTRACT〈/div〉The presence of fractures in a reservoir can have a significant impact on its effective mechanical and hydraulic properties. Many researchers have explored the seismic response of fluid-saturated porous rocks containing aligned planar fractures through the use of analytical models. However, these approaches are limited to the extreme cases of regular and uniform random distributions of fractures. The purpose of this work is to consider more realistic distributions of fractures and to analyze whether and how the frequency-dependent anisotropic seismic properties of the medium can provide information on the characteristics of the fracture network. Particular focus is given to fracture clustering effects resulting from commonly observed fracture distributions. To do so, we have developed a novel hybrid methodology combining the advantages of 1D numerical oscillatory tests, which allows us to consider arbitrary distributions of fractures, and an analytical solution that permits extending these results to account for the effective anisotropy of the medium. A corresponding numerical analysis indicates that the presence of clusters of fractures produces an additional attenuation and velocity dispersion regime compared with that predicted by analytical models. The reason for this is that a fracture cluster behaves as an effective layer and the contrast with respect to the unfractured background produces an additional fluid pressure diffusion length scale. The characteristic frequency of these effects depends on the size and spacing between clusters, the latter being much larger than the typical spacing between individual fractures. Moreover, we find that the effects of fracture clustering are more pronounced in attenuation anisotropy than velocity anisotropy data. Our results indicate that fracture clustering effects on fluid pressure diffusion can be described by two-layer models. This, in turn, provides the basis for extending current analytical models to account for these effects in inversion schemes designed to characterize fractured reservoirs from seismic data.〈/span〉

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.