Reduction to the pole and transformations of scattered magnetic data using Newtonian equivalent sources

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. L67-L73 ◽  
Author(s):  
Fernando Guspí ◽  
Iván Novara
Keyword(s):  
Real Data ◽  
Computer Time ◽  
Magnetic Data ◽  
Newtonian Potential ◽  
Two Stage ◽  
Magnetization Direction ◽  
Magnetic Dipoles ◽  
Equivalent Source ◽  
Reduction To The Pole ◽  
Equivalent Sources

We have developed an equivalent-source method for performing reduction to the pole and related transforms from magnetic data measured on unevenly spaced stations at different elevations. The equivalent source is composed of points located vertically beneath the measurement stations, and their magnetic properties are chosen in such a way that the reduced-to-the-pole magnetic field generated by them is represented by an inverse-distance Newtonian potential. This function, which attenuates slowly with distance, provides better coverage for discrete data points. The magnetization intensity is determined iteratively until the observed field is fitted within a certain tolerance related to the level of noise; thus, advantages in computer time are gained over the resolution of large systems of equations. In the case of induced magnetization, the iteration converges well for verticalor horizontal inclinations, and results are stable if noise is taken into account properly. However, for a range of intermediate inclinations near 35°, a factor tending to zero makes it necessary to perform the reduction through a two-stage procedure, using an auxiliary magnetization direction, without significantly affecting the speed and stability of the method. The performance of the procedure was tested on a synthetic example based on a field generated on randomly scattered stations by a random set of magnetic dipoles, contaminated with noise, which is reduced to the pole for three different magnetization directions. Results provide a good approximation to the theoretical reduced-to-the-pole field using a one- or a two-stage reduction, showing minor noise artifacts when the direction is nearly horizontal. In a geophysical example with real data, the reduction to the pole was used to correct the estimated magnetization direction that originates an isolated anomaly over Sierra de San Luis, Argentina.

Using an equivalent source with positivity for low-latitude reduction to the pole without striation

Geophysics ◽  
2014 ◽  
Vol 79 (6) ◽  
pp. J81-J90 ◽  
Author(s):  
Yaoguo Li ◽  
Misac Nabighian ◽  
Douglas W. Oldenburg
Keyword(s):  
Large Scale ◽  
Computational Cost ◽  
Equatorial Region ◽  
Magnetic Data ◽  
Data Sets ◽  
Low Latitude ◽  
Equivalent Source ◽  
Large Scale Data ◽  
Reduction To The Pole ◽  
Equivalent Sources

We present a reformulation of reduction to the pole (RTP) of magnetic data at low latitudes and the equator using equivalent sources. The proposed method addresses both the theoretical difficulty of low-latitude instability and the practical issue of computational cost. We prove that a positive equivalent source exists when the magnetic data are produced by normal induced magnetization, and we show that the positivity is sufficient to overcome the low-latitude instability in the space domain. We further apply a regularization term directly to the recovered RTP field to improve the solution. The use of equivalent source also naturally enables the processing of data acquired on uneven surface. The result is a practical algorithm that is effective at the equatorial region and can process large-scale data sets with uneven observation heights.

Magnetization vector imaging for borehole magnetic data based on magnitude magnetic anomaly

Geophysics ◽  
2013 ◽  
Vol 78 (6) ◽  
pp. D429-D444 ◽  
Author(s):  
Shuang Liu ◽  
Xiangyun Hu ◽  
Tianyou Liu ◽  
Jie Feng ◽  
Wenli Gao ◽  
...  
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Magnetization Vector ◽  
Real Data ◽  
Magnetic Anomalies ◽  
Magnetic Data ◽  
Preconditioned Conjugate Gradient ◽  
Magnetization Direction ◽  
Magnetization Intensity

Remanent magnetization and self-demagnetization change the magnitude and direction of the magnetization vector, which complicates the interpretation of magnetic data. To deal with this problem, we evaluated a method for inverting the distributions of 2D magnetization vector or effective susceptibility using 3C borehole magnetic data. The basis for this method is the fact that 2D magnitude magnetic anomalies are not sensitive to the magnetization direction. We calculated magnitude anomalies from the measured borehole magnetic data in a spatial domain. The vector distributions of magnetization were inverted methodically in two steps. The distributions of magnetization magnitude were initially solved based on magnitude magnetic anomalies using the preconditioned conjugate gradient method. The preconditioner determined by the distances between the cells and the borehole observation points greatly improved the quality of the magnetization magnitude imaging. With the calculated magnetization magnitude, the distributions of magnetization direction were computed by fitting the component anomalies secondly using the conjugate gradient method. The two-step approach made full use of the amplitude and phase anomalies of the borehole magnetic data. We studied the influence of remanence and demagnetization based on the recovered magnetization intensity and direction distributions. Finally, we tested our method using synthetic and real data from scenarios that involved high susceptibility and complicated remanence, and all tests returned favorable results.

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.

Evaluating the accuracy of equivalent-source predictions using cross-validation

2020 ◽  
Author(s):  
Leonardo Uieda ◽  
Santiago Soler
Keyword(s):  
Prediction Accuracy ◽  
Cross Validation ◽  
Point Sources ◽  
Magnetic Data ◽  
Random Permutations ◽  
Training Set ◽  
Equivalent Source ◽  
Upward Continuation ◽  
Reduction To The Pole ◽  
Validation Set

<p>We investigate the use of cross-validation (CV) techniques to estimate the accuracy of equivalent-source (also known as equivalent-layer) models for interpolation and processing of potential-field data. Our preliminary results indicate that some common CV algorithms (e.g., random permutations and k-folds) tend to overestimate the accuracy. We have found that blocked CV methods, where the data are split along spatial blocks instead of randomly, provide more conservative and realistic accuracy estimates. Beyond evaluating an equivalent-source model's performance, cross-validation can be used to automatically determine configuration parameters, like source depth and amount of regularization, that maximize prediction accuracy and avoid over-fitting.</p><p>Widely used in gravity and magnetic data processing, the equivalent-source technique consists of a linear model (usually point sources) used to predict the observed field at arbitrary locations. Upward-continuation, interpolation, gradient calculations, leveling, and reduction-to-the-pole can be performed simultaneously by using the model to make predictions (i.e., forward modelling). Likewise, the use of linear models to make predictions is the backbone of many machine learning (ML) applications. The predictive performance of ML models is usually evaluated through cross-validation, in which the data are split (usually randomly) into a training set and a validation set. Models are fit on the training set and their predictions are evaluated using the validation set using a goodness-of-fit metric, like the mean square error or the R² coefficient of determination. Many cross-validation methods exist in the literature, varying in how the data are split and how this process is repeated. Prior research from the statistical modelling of ecological data suggests that prediction accuracy is usually overestimated by traditional CV methods when the data are spatially auto-correlated. This issue can be mitigated by splitting the data along spatial blocks rather than randomly. We conducted experiments on synthetic gravity data to investigate the use of traditional and blocked CV methods in equivalent-source interpolation. We found that the overestimation problem also occurs and that more conservative accuracy estimates are obtained when applying blocked versions of random permutations and k-fold. Further studies need to be conducted to generalize these findings to upward-continuation, reduction-to-the-pole, and derivative calculation.</p><p>Open-source software implementations of the equivalent-source and blocked cross-validation (in progress) methods are available in the Python libraries Harmonica and Verde, which are part of the Fatiando a Terra project (www.fatiando.org).</p>

Stepped inversion of magnetic data

Geophysics ◽  
2007 ◽  
Vol 72 (3) ◽  
pp. L21-L30 ◽  
Author(s):  
Soraya Lozada Tuma ◽  
Carlos Alberto Mendonça
Keyword(s):  
Magnetic Anomaly ◽  
Shape Function ◽  
Source Parameters ◽  
Real Data ◽  
Magnetic Data ◽  
Total Field ◽  
Magnetization Direction ◽  
Magnetization Intensity ◽  
Magnetic Inversion ◽  
Total Gradient

We present a three-step magnetic inversion procedure in which invariant quantities with respect to source parameters are inverted sequentially to give (1) shape cross section, (2) magnetization intensity, and (3) magnetization direction for a 2D (elongated) magnetic source. The quantity first inverted (called here the shape function) is obtained from the ratio of the gradient intensity of the total-field anomaly to the intensity of the anomalous vector field. For homogenous sources, the shape function is invariant with source magnetization and allows reconstruction of the source geometry by attributing an arbitrary magnetization to trial solutions. Once determined, the source shape is fixed and magnetization intensity is estimated by fitting the total gradient of the total-field anomaly (equivalent to the amplitude of the analytic signal of magnetic anomaly). Finally, the source shape and magnetization intensity are fixed and the magnetization direction is determined by fitting the magnetic anomaly. As suggested by numerical modeling and real data application, stepped inversion allows checking whether causative sources are homogeneous. This is possible because the shape function from inhomogeneous sources can be fitted by homogeneous models, but a model obtained in this way fits neither the total gradient of the magnetic anomaly nor the magnetic anomaly itself. Such a criterion seems effective in recognizing strongly inhomogeneous sources. Stepped inversion is tested with numerical experiments, and is used to model a magnetic anomaly from intrusive basic rocks from the Paraná Basin, Brazil.

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.

scholarly journals Interpolation of magnetic anomalies over an oceanic ridge region using an equivalent source technique and crust age model constraint

2021 ◽  
Author(s):  
Duan Li ◽  
Jinsong Du ◽  
Chao Chen ◽  
Qing Liang ◽  
Shida Sun
Keyword(s):  
Plate Tectonics ◽  
Magnetic Anomalies ◽  
Background Field ◽  
Source Model ◽  
Magnetic Data ◽  
Magnetization Distribution ◽  
Equivalent Source ◽  
Oceanic Ridge ◽  
Age Model ◽  
Equivalent Sources

Abstract. Marine magnetic surveys over oceanic ridge regions are of great interest for investigations of structure and evolution of oceanic crust, and have played a key role in developing the theory of plate tectonics (Dyment, 1993; Maus et al, 2007; Vine and Matthews, 1963). In this study, we propose an interpolation approach based on the dual-layer equivalent source model for the generation of a magnetic anomaly map based on sparse survey line data over oceanic ridge areas. In this approach, information from an ocean crust age model is utilized as constraint for the inversion procedure. The constraints can affect the magnetization distribution of equivalent sources following crust age. The results of synthetic tests show that the obtained magnetic anomalies have higher accuracy than those obtained by other interpolation methods. Meanwhile, considering the unclear on the true magnetization directions of sources and the background field in the synthetic model, well interpolation result can still be obtained. We applied the approach to magnetic data obtained from five survey lines east of the Southeast Indian Ridge. This prediction result is useful to improve the lithospheric magnetic field models WDMAMv2 and EMAG2v3, in the terms of spatial resolution and the consistency with observed data.

Rapid construction of equivalent sources using wavelets

Geophysics ◽  
2010 ◽  
Vol 75 (3) ◽  
pp. L51-L59 ◽  
Author(s):  
Yaoguo Li ◽  
Douglas W. Oldenburg
Keyword(s):  
Wavelet Transform ◽  
Wavelet Transforms ◽  
Sparse Matrix ◽  
Coefficient Matrix ◽  
Magnetic Data ◽  
Dense Matrix ◽  
Wavelet Domain ◽  
Space Domain ◽  
Equivalent Source ◽  
Equivalent Sources

We have developed a fast algorithm for generating an equivalent source by using fast wavelet transforms based on orthonormal, compactly supported wavelets. We apply a 2D wavelet transform to each row and column of the coefficient matrix and subsequently threshold the transformed matrix to generate a sparse representation in the wavelet domain. The algorithm then uses this sparse matrix to construct the the equivalent source directly in the wavelet domain. Performing an inverse wavelet transform then yields the equivalent source in the space domain. Using upward continuation of total-field magnetic data between uneven surfaces as examples, we have compared this approach with the direct solution using the dense matrix in the space domain. We have shown that the wavelet approach can reduce the CPU time by as many as two orders of magnitude.

Inversion of magnetic anomaly on rugged observation surface in the presence of strong remanent magnetization

Geophysics ◽  
2014 ◽  
Vol 79 (2) ◽  
pp. J11-J19 ◽  
Author(s):  
Shu-Ling Li ◽  
Yaoguo Li
Keyword(s):  
Remanent Magnetization ◽  
Country Rock ◽  
Magnetization Vector ◽  
Magnetic Data ◽  
Magnetization Direction ◽  
Equivalent Source ◽  
Total Magnetization ◽  
Amplitude Inversion ◽  
Amplitude Data ◽  
And Inversion

We study the inversion of magnetic data acquired over a rugged observation surface and where the buried source bodies have strong remanent magnetization that leads to unknown total magnetization directions. These factors pose significant challenges for processing and inversion of such data. To tackle the challenges from both a rugged observation surface and an unknown magnetization direction, we propose a strategy through the joint use of the equivalent source technique and 3D amplitude inversion to obtain 3D magnetization strength. We use equivalent source processing to calculate the amplitude data in the space domain because the use of the wavenumber-domain method is invalid due to large variations in the data elevation. We then carried out an amplitude inversion to generate a 3D subsurface distribution of the magnitude of the total magnetization vector. The results from a synthetic example and aeromagnetic data in Daye Mine in China showed that this approach is effective and images the magnetic units whose contact zones with the limestone country rock host the mineralization. The method is general and can be applied to a variety of cases with similar challenges.

Gradient-boosted equivalent sources for gridding large gravity and magnetic datasets

2021 ◽  
Author(s):  
Santiago Rubén Soler ◽  
Leonardo Uieda
Keyword(s):  
Jacobian Matrix ◽  
Least Squares Method ◽  
Computational Cost ◽  
Magnetic Data ◽  
Gradient Boosting ◽  
Equivalent Source ◽  
Harmonic Field ◽  
Gravity And Magnetic ◽  
Data Points ◽  
Equivalent Sources

<p>The equivalent source technique is a well known method for interpolating gravity and magnetic data. It consists in defining a set of finite sources that generate the same observed field and using them to predict the values of the field at unobserved locations. The equivalent source technique has some advantages over general-purpose interpolators: the variation of the field due to the height of the observation points is taken into account and the predicted values belong to an harmonic field. These make equivalent sources a more suited interpolator for any data deriving from a harmonic field (like gravity disturbances and magnetic anomalies). Nevertheless, it has one drawback: the computational cost. The process of estimating the coefficients of the sources that best fit the observed values is very computationally demanding: a Jacobian matrix with number of observation points times number of sources elements must be built and then used to fit the source coefficients though a least-squares method. Increasing the number of data points can make the Jacobian matrix to grow so large that it cannot fit in computer memory.</p><p>We present a gradient-boosting equivalent source method for interpolating large datasets. In it, we define small subsets of equivalent sources that are fitted against neighbouring data points. The process is iteratively carried out, fitting one subset of sources on each iteration to the residual field from previous iterations. This new method is inspired by the gradient-boosting technique, mainly used in machine learning solutions.</p><p>We show that the gradient-boosted equivalent sources are capable of producing accurate predictions by testing against synthetic surveys. Moreover, we were able to grid a gravity dataset from Australia with more than 1.7 million points on a modest personal computer in less than half an hour.</p>

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