Ellipsoids (v1.0): 3D Magnetic modelling of ellipsoidal bodies

A considerable amount of literature has been published on the magnetic modelling of uniformly magnetized ellipsoids since the second half of the nineteenth century. Ellipsoids have flexibility to represent a wide range of geometrical forms, are the only known bodies which can be uniformly magnetized in the presence of a uniform inducing field and are the only finite bodies for which the self-demagnetization can be treated analytically. This property makes ellipsoids particularly useful for modelling compact orebodies having high susceptibility. In this case, neglecting the self-demagnetization may strongly mislead the interpretation of these bodies by using magnetic methods. A number of previous studies consider that the self-demagnetization can be neglected for the case in which the geological body has an isotropic susceptibility lower than or equal to 0.1 SI. This limiting value, however, seems to be determined empirically and there has been no discussion about how this value was determined. Besides, the geoscientific community lacks an easy-to-use tool to simulate the magnetic field produced by uniformly magnetized ellipsoids. Here, we present an integrated review of the magnetic modelling of arbitrarily oriented triaxial, prolate and oblate ellipsoids. Our review includes ellipsoids with both induced and remanent magnetization, as well as with isotropic or anisotropic susceptibility. We also discuss the ambiguity between confocal ellipsoids with the same magnetic moment and propose a way of determining the isotropic susceptibility above which the self-demagnetization must be taken into consideration. Tests with synthetic data validate our approach. Finally, we provide a set of routines to model the magnetic field produced by ellipsoids. The routines are written in Python language as part of the Fatiando a Terra, which is an open-source library for modelling and inversion in geophysics

Tópicos de inversão em geofísica

<p>Apostila para o curso "Tópicos de inversão em geofísica".</p> <p>Código Latex para gerar a apostila e apresentações do curso podem ser encontradas no endereço abaixo.</p

Presentation: Source geometry estimation using the mass excess criterion to constrain 3-D radial inversion of gravity data

<p><strong>Slides for the oral presentation "Source geometry estimation using the mass excess criterion to constrain 3-D radial inversion of gravity data" presented at the SEG International Exposition and Eighty-First Annual Meeting in San Antonio, Texas.<br></strong></p> <p><strong><br></strong></p> <p><strong>ABSTRACT</strong></p> <p>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 0 to 360 degrees. 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-misfitmeasure 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<br>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.</p

Iterative fast equivalent-layer technique

<p><b>Poster: In 79th EAGE Conference and Exhibition 2017, Paris</b></p><p><br></p><p><b>Abstract</b></p> <p>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 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 modelling 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. The algorithm requires neither matrix multiplications nor the solution of linear systems. The desired processed data is obtained by multiplying the matrix of Green's functions associated with the desired processing by the estimated 2D mass distribution. Tests on synthetic and field gravity data from the Vinton salt dome, USA, confirms the potential of our approach in processing large gravity data set over on undulating surface.</p> <p><b> </b></p

Synthetic total-field magnetic anomaly data and code to perform Euler deconvolution on it

<p>Synthetic data, source code, and supplementary text for the article "Euler deconvolution of potential field data" by Leonardo Uieda, Vanderlei C. Oliveira Jr., and Valéria C. F. Barbosa.</p> <p>This is part of a tutorial submitted to The Leading Edge (http://library.seg.org/journal/tle).</p> <p>Results were generated using the open-source Python package Fatiando a Terra version 0.2 (http://www.fatiando.org).</p> <p>This material along with the manuscript can also be found at https://github.com/pinga-lab/paper-tle-euler-tutorial</p> <p><strong>Synthetic data and model</strong></p> <p>Examples in the tutorial use synthetic data generated with the IPython notebook create_synthetic_data.ipynb. File synthetic_data.txt has 4 columns: x (north), y (east), z (down) and the total field magnetic anomaly. x, y, and z are in meters. The total field anomaly is in nanoTesla (nT). File metadata.json contains extra information about the data, such as inclination and declination of the inducing field (in degrees), shape of the data grid (number of points in y and x, respectively), the area containing the data (W, E, S, N, in meters), and the model boundaries (W, E, S, N, top, bottom, in meters).</p> <p>File model.pickle is a serialized version of the model used to generate the data. It contains a list of instances of the PolygonalPrism class of Fatiando a Terra. The serialization was done using the cPickle Python module.</p> <p><strong>Reproducing the results in the tutorial</strong></p> <p>The notebook euler-deconvolution-examples.ipynb runs the Euler deconvolution on the synthetic data and generates the figures for the manuscript. It also presents a more detailed explanation of the method and more tests than went into the finished manuscript.</p

Code samples in "Modeling the Earth with Fatiando a Terra"

<p>This file set includes an IPython notebook file with the code samples in the proceedings of the 2013 Scipy Conference "Modeling the Earth with Fatiando a Terra".</p> <p>Also included are a plain text Python file with the code and a pickle file with the polygonal prism model used in the proceedings.</p> <p>The links bellow are:</p> <p>* The git repository of the proceedings</p> <p>* A rendered version of the code_samples.ipynb notebook on http://nbviewer.ipython.org/</p

Supplement to "Estimation of the total magnetization direction of approximately spherical bodies"

<p>Supplementary data and source code to "Estimation of the total magnetization direction of approximately spherical bodies".</p> <p>Article submitted for publication to Nonlinear Processes in Geophysics.</p> <p>The original submission and open peer-review can be viewed at http://dx.doi.org/10.5194/npgd-1-1465-2014</p> <p>This is an archive of the git version controlled repository containing: the manuscript LaTeX source files; IPython notebooks (www.ipython.org) containing source code to perform the applications to synthetic and real data; the total field magnetic anomaly data used in the real data application.</p> <p>See the original repository at https://github.com/pinga-lab/Total-magnetization-of-spherical-bodies</p> <p>The source code that implements the proposed methodology is included in version 0.3 of the open-source software Fatiando a Terra (www.fatiando.org). See module fatiando.gravmag.magdir.</p

A single Euler Solution Per Anomaly

<p>Poster (Part 2) presented at EAGE 2014 - 76th EAGE Conference & Exhibition 2014 (Amsterdam).</p> <p> </p> <p>We have presented a new method for selecting the best source location estimates in 3D Euler deconvolution. Our approach has drastically reduced the number of the selected 3D Euler solutions to only one per anomaly. This is possible because our method does not select Euler solutions based on their statistical consistency after a cluster analysis. Rather, in our method this selection is grounded on the theoretical analysis of the estimators for the horizontal and vertical source positions in 3D Euler deconvolution as a function of the x- and y-coordinates of the observations. Our approach consists in detecting automatically the regions of the anomaly producing consistent estimates of the source horizontal coordinates.</p