Generalized positivity constraint on magnetic equivalent layers

Geophysics ◽  
2020 ◽  
Vol 85 (6) ◽  
pp. J99-J110
Author(s):  
André L. A. Reis ◽  
Vanderlei C. Oliveira Jr. ◽  
Valéria C. F. Barbosa
Keyword(s):  
Inverse Problem ◽  
Magnetic Moment ◽  
A Priori ◽  
Synthetic Data ◽  
Total Field ◽  
Magnetization Direction ◽  
Moment Distribution ◽  
Equivalent Sources ◽  
Priori Information ◽  
Equivalent Layer

It is known from the potential theory that a continuous and planar layer of dipoles can exactly reproduce the total-field anomaly produced by arbitrary 3D sources. We have proven the existence of an equivalent layer having an all-positive magnetic-moment distribution for the case in which the magnetization direction of this layer is the same as that of the true sources, regardless of whether the magnetization of the true sources is purely induced or not. By using this generalized positivity constraint, we have developed a new iterative method for estimating the total magnetization direction of 3D magnetic sources based on the equivalent-layer technique. Our method does not impose a priori information about the shape or the depth of the sources, does not require regularly spaced data, and presumes that the sources have a uniform magnetization direction. At each iteration, our method performs two steps. The first step solves a constrained linear inverse problem to estimate a positive magnetic-moment distribution over a discrete equivalent layer of dipoles. We consider that the equivalent sources are located on a plane and have a uniform and fixed magnetization direction. In the second step, we use the estimated magnetic-moment distribution and solve a nonlinear inverse problem for estimating a new magnetization direction for the dipoles. The algorithm stops when the equivalent layer yields a total-field anomaly that fits the observed data. Tests with synthetic data simulating different geologic scenarios show that the final estimated magnetization direction is close to the true one. We apply our method to field data from the Goiás alkaline province, over the Montes Claros complex, in the center of Brazil. The results suggest the presence of intrusions with remarkable remanent magnetization, in agreement with the current literature for this region.

scholarly journals The Neuroelectromagnetic Inverse Problem and the Zero Dipole Localization Error

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Rolando Grave de Peralta ◽  
Olaf Hauk ◽  
Sara L. Gonzalez
Keyword(s):  
Inverse Problem ◽  
Unique Solution ◽  
A Priori ◽  
Inverse Solution ◽  
Localization Error ◽  
A Priori Information ◽  
Dipole Localization ◽  
Inverse Solutions ◽  
Priori Information ◽  
Additional Constraints

A tomography of neural sources could be constructed from EEG/MEG recordings once the neuroelectromagnetic inverse problem (NIP) is solved. Unfortunately the NIP lacks a unique solution and therefore additional constraints are needed to achieve uniqueness. Researchers are then confronted with the dilemma of choosing one solution on the basis of the advantages publicized by their authors. This study aims to help researchers to better guide their choices by clarifying what is hidden behind inverse solutions oversold by their apparently optimal properties to localize single sources. Here, we introduce an inverse solution (ANA) attaining perfect localization of single sources to illustrate how spurious sources emerge and destroy the reconstruction of simultaneously active sources. Although ANA is probably the simplest and robust alternative for data generated by a single dominant source plus noise, the main contribution of this manuscript is to show that zero localization error of single sources is a trivial and largely uninformative property unable to predict the performance of an inverse solution in presence of simultaneously active sources. We recommend as the most logical strategy for solving the NIP the incorporation of sound additional a priori information about neural generators that supplements the information contained in the data.

Improving the crosscorrelation method to estimate the total magnetization direction vector of isolated sources: A space-domain approach for unstable inclination values

Geophysics ◽  
2020 ◽  
Vol 85 (4) ◽  
pp. J59-J70 ◽  
Author(s):  
Nelson Ribeiro-Filho ◽  
Rodrigo Bijani ◽  
Cosme Ponte-Neto
Keyword(s):  
Magnetic Field ◽  
Real Data ◽  
Magnetic Anomalies ◽  
Direction Vector ◽  
Total Field ◽  
Magnetization Direction ◽  
Ambient Magnetic Field ◽  
Total Magnetization ◽  
Synthetic Test ◽  
Equivalent Sources

Knowledge of the total magnetization direction of geologic sources is valuable for interpretation of magnetic anomalies. Although the magnetization direction of causative sources is assumed to be induced by the ambient magnetic field, the presence of remanence should not be neglected. An existing method of correlating total and vertical gradients of the reduced-to-the-pole (RTP) anomaly estimates the total magnetization direction well. However, due to the numerical instability of RTP transformation in the Fourier domain, an assumption should be considered for dealing with inclination values at approximately 0°. We have adopted an extension to the standard crosscorrelation method for estimating the total magnetization direction vector, computing the RTP anomaly by means of the classic equivalent layer technique for low inclination values. Additionally, an ideal number of equivalent sources within the layer is considered for reducing the computational demands. To investigate the relevant aspects of the adopted method, two simple synthetic scenarios are presented. First, a magnetic anomaly produced by a homogeneous and isolated vertical dike is considered. This test illustrates the good performance of the adopted approach, finding the true magnetization direction, even for low inclination values. In the second synthetic test, a long-wavelength component is added to the previous magnetic total-field anomaly. In this case, the method adopted here fails to estimate a reliable magnetization direction vector, showing weak performance for strong interfering magnetic anomalies. On the real data example, the application tests an isolated total-field anomaly of the Carajás Mineral Province, in northern Brazil, where the inclination of the ambient magnetic field is close to zero. The obtained results indicate weak remanence in the estimated total magnetization direction vector, which would never be reached in the standard formulation of the crosscorrelation technique.

1D single-site and laterally constrained inversion of multifrequency and multicomponent ground-based electromagnetic induction data — Application to the investigation of a near-surface clayey overburden

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. WB19-WB35 ◽  
Author(s):  
Cyril Schamper ◽  
Fayçal Rejiba ◽  
Roger Guérin
Keyword(s):  
Magnetic Field ◽  
Electromagnetic Induction ◽  
Search Algorithm ◽  
Random Search ◽  
A Priori ◽  
Synthetic Data ◽  
Conductive Layer ◽  
Near Surface ◽  
Priori Information ◽  
Definition Of

Electromagnetic induction (EMI) methods are widely used to determine the distribution of the electrical conductivity and are well adapted to the delimitation of aquifers and clayey layers because the electromagnetic field is strongly perturbed by conductive media. The multicomponent EMI device that was used allowed the three components of the secondary magnetic field (the radial [Formula: see text], the tangential [Formula: see text], and the vertical [Formula: see text]) to be measured at 10 frequencies ranging from 110 to 56 kHz in one single sounding with offsets ranging from 20 to 400 m. In a continuing endeavor to improve the reliability with which the thickness and conductivity are inverted, we focused our research on the use of components other than the vertical magnetic field Hz. Because a separate sensitivity analysis of [Formula: see text] and [Formula: see text] suggests that [Formula: see text] is more sensitive to variations in the thickness of a near-surface conductive layer, we developed an inversion tool able to make single-sounding and laterally constrained 1D interpretation of both components jointly, associated with an adapted random search algorithm for single-sounding processing for which almost no a priori information is available. Considering the complementarity of [Formula: see text] and [Formula: see text] components, inversion tests of clean and noisy synthetic data showed an improvement in the definition of the thickness of a near-surface conductive layer. This inversion code was applied to the karst site of the basin of Fontaine-Sous-Préaux, near Rouen (northwest of France). Comparison with an electrical resistivity tomography tends to confirm the reliability of the interpretation from the EMI data with the developed inversion tool.

Estimation of density, magnetization, and depth to source: A nonlinear and noniterative 3-D potential‐field method

Geophysics ◽  
1997 ◽  
Vol 62 (3) ◽  
pp. 814-830 ◽  
Author(s):  
Maurizio Fedi
Keyword(s):  
A Priori ◽  
Real Data ◽  
Field Method ◽  
Magnetic Data ◽  
System Of Equations ◽  
Source Depth ◽  
A Priori Information ◽  
Magnetization Direction ◽  
Priori Information ◽  
Magnetic Case

The depth to the top, or bottom, and the density of a 3-D homogeneous source can be estimated from its gravity or magnetic anomalies by using a priori information on the maximum and minimum source depths. For the magnetic case, the magnetization direction is assumed to be constant and known. The source is assumed to be within a layer of known depth to the top h and thickness t. A depth model, satisfying both the data and the a priori information is found, together with its associated density/magnetization contrast. The methodology first derives, from the measured data, a set of apparent densities [Formula: see text] (or magnetizations), which do not depend on the layer parameters h and t, but only on source thickness. A nonlinear system of equations based on [Formula: see text], with source thicknesses as unknowns, is constructed. To simplify the solution, a more practical system of equations is formed. Each equation depends on only one value of thickness. Solving for the thicknesses, taking into account the above a priori information, the source depth to the top (or to the bottom) is determined uniquely. Finally, the depth solutions allow a unit‐density gravity model to be computed, which is compared to the observed gravity to determine the density contrast. A similar procedure can be used for magnetic data. Tests on synthetic anomalies and on real data demonstrate the good performance of this method.

Isostatic constraint for 2D nonlinear gravity inversion on rifted margins

Geophysics ◽  
2019 ◽  
Vol 85 (1) ◽  
pp. G17-G34
Author(s):  
B. Marcela S. Bastos ◽  
Vanderlei C. Oliveira Jr.
Keyword(s):  
Gravity Data ◽  
A Priori ◽  
Petroleum Industry ◽  
Synthetic Data ◽  
Gravity Inversion ◽  
Rifted Margins ◽  
Isostatic Equilibrium ◽  
Priori Information ◽  
Pelotas Basin ◽  
Nonlinear Gravity

We have developed a nonlinear gravity inversion for simultaneously estimating the basement and Moho geometries, as well as the depth of the reference Moho along a profile crossing a passive rifted margin. To obtain stable solutions, we impose smoothness on basement and Moho, force them to be close to previously estimated depths along the profile and also impose local isostatic equilibrium. Different from previous methods, we evaluate the information of local isostatic equilibrium by imposing smoothness on the lithostatic stress exerted at depth. Our method delimits regions that deviate and those that can be considered in local isostatic equilibrium by varying the weight of the isostatic constraint along the profile. It also allows controlling the degree of equilibrium along the profile, so that the interpreter can obtain a set of candidate models that fit the observed data and exhibit different degrees of isostatic equilibrium. Our method also differs from earlier studies because it attempts to use isostasy for exploring (but not necessarily reducing) the inherent ambiguity of gravity methods. Tests with synthetic data illustrate the effect of our isostatic constraint on the estimated basement and Moho reliefs, especially at regions with pronounced crustal thinning, which are typical of passive volcanic margins. Results obtained by inverting satellite data over the Pelotas Basin, a passive volcanic margin in southern Brazil, agree with previous interpretations obtained independently by combining gravity, magnetic, and seismic data available to the petroleum industry. These results indicate that combined with a priori information, simple isostatic assumptions can be very useful for interpreting gravity data on passive rifted margins.

scholarly journals Estimation of the total magnetization direction of approximately spherical bodies

2015 ◽  
Vol 22 (2) ◽  
pp. 215-232 ◽  
Author(s):  
V. C. Oliveira ◽  
D. P. Sales ◽  
V. C. F. Barbosa ◽  
L. Uieda
Keyword(s):  
Inverse Problem ◽  
Synthetic Data ◽  
Magnetization Vector ◽  
Time Interval ◽  
Alkaline Complex ◽  
Multiple Sources ◽  
Geological Time ◽  
Magnetization Direction ◽  
Reduction To The Pole ◽  
Horizontal Grid

Abstract. 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.

scholarly journals Linear inverse Gaussian theory and geostatistics

Geophysics ◽  
2006 ◽  
Vol 71 (6) ◽  
pp. R101-R111 ◽  
Author(s):  
Thomas Mejer Hansen ◽  
Andre G. Journel ◽  
Albert Tarantola ◽  
Klaus Mosegaard
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Covariance Function ◽  
A Priori ◽  
Model Space ◽  
Sequential Gaussian Simulation ◽  
A Priori Information ◽  
Sequential Approach ◽  
Linear Inverse Problem ◽  
Priori Information

Inverse problems in geophysics require the introduction of complex a priori information and are solved using computationally expensive Monte Carlo techniques (where large portions of the model space are explored). The geostatistical method allows for fast integration of complex a priori information in the form of covariance functions and training images. We combine geostatistical methods and inverse problem theory to generate realizations of the posterior probability density function of any Gaussian linear inverse problem, honoring a priori information in the form of a covariance function describing the spatial connectivity of the model space parameters. This is achieved using sequential Gaussian simulation, a well-known, noniterative geostatisticalmethod for generating samples of a Gaussian random field with a given covariance function. This work is a contribution to both linear inverse problem theory and geostatistics. Our main result is an efficient method to generate realizations, actual solutions rather than the conventional least-squares-based approach, to any Gaussian linear inverse problem using a noniterative method. The sequential approach to solving linear and weakly nonlinear problems is computationally efficient compared with traditional least-squares-based inversion. The sequential approach also allows one to solve the inverse problem in only a small part of the model space while conditioned to all available data. From a geostatistical point of view, the method can be used to condition realizations of Gaussian random fields to the possibly noisy linear average observations of the model space.

On an inverse spectral problem for one integro-differential operator of fractional order

2019 ◽  
Vol 27 (1) ◽  
pp. 17-23 ◽  
Author(s):  
Mikhail Ignatiev
Keyword(s):  
Inverse Problem ◽  
Differential Operator ◽  
Spectral Problem ◽  
Fractional Order ◽  
Inverse Spectral Problem ◽  
A Priori ◽  
A Priori Information ◽  
Inverse Spectral ◽  
Priori Information

Abstract An inverse spectral problem for some integro-differential operator of fractional order {\alpha\in(1,2)} is studied. We show that the specification of the spectrum together with a certain a priori information about the structure of the operator determines such operator uniquely. The proof is constructive and provides a procedure for solving the inverse problem.

Selection of processing parameters for solving the 3D inverse problem of gravity exploration by the discrete Fourier transform method in GIS INTEGRO

2021 ◽  
pp. 47-52
Author(s):  
Nadezhda Pimanova ◽  
Viktor Spiridonov
Keyword(s):  
Inverse Problem ◽  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
A Priori ◽  
Processing Parameters ◽  
Effective Density ◽  
Transform Method ◽  
Optimal Values ◽  
Priori Information ◽  
Selection Of

In GIS INTEGRO, a program using the discrete Fourier transform is created to solve the 3D inverse problem of gravimetry. The result of its work is a 3D distribution of the effective density. The program allows to use an extended parametrization of the form: fα,β(r,x) = f(r,zα)|z|–β. By changing the parameters α — "depth multiplier" — and β — "exponent at z"— one can obtain various equivalent distributions of effective densities and choose the most appropriate one based on a priori information. The experience of solving the 3D inverse problem by this method with different set values of these parameters allowed us to recommend the optimal values. As a criterion for choosing a solution from a variety of possible solutions, it is proposed to use a comparison of them with the data of seismic studies of the DDS and the CCP.

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