Magnetotelluric Inverse Problem Research Articles

Multimodal variational autoencoder for inverse problems in geophysics: application to a 1-D magnetotelluric problem

SUMMARY Estimating subsurface properties from geophysical measurements is a common inverse problem. Several Bayesian methods currently aim to find the solution to a geophysical inverse problem and quantify its uncertainty. However, most geophysical applications exhibit more than one plausible solution. Here, we propose a multimodal variational autoencoder model that employs a mixture of truncated Gaussian densities to provide multiple solutions, along with their probability of occurrence and a quantification of their uncertainty. This autoencoder is assembled with an encoder and a decoder, where the first one provides a mixture of truncated Gaussian densities from a neural network, and the second is the numerical solution of the forward problem given by the geophysical approach. The proposed method is illustrated with a 1-D magnetotelluric inverse problem and recovers multiple plausible solutions with different uncertainty quantification maps and probabilities that are in agreement with known physical observations.

Physics-Driven Deep Learning Inversion with Application to Magnetotelluric

Due to the strong capability of building complex nonlinear mapping without involving linearization theory and high prediction efficiency; the deep learning (DL) technique applied to solve geophysical inverse problems has been a subject of growing interest. Currently, most DL-based inversion approaches are fully data-driven (namely standard deep learning), the performance of which largely depends on the training sample sets. However, due to the heavy burden of time and computational resources, it can be challenging to supply such a massive and exhaustive training dataset for generic realistic exploration scenarios and to perform network training. In this work, based on the recent advances in physics-based networks, the physical laws of magnetotelluric (MT) wave propagation is incorporated into a purely data-driven DL approach (PlainDNN) and thus builds a physics-driven DL MT inversion scheme (PhyDNN). In this scheme, the forward operator modeling MT wave propagation is integrated into the network training loop, in the form of minimizing a hybrid loss objective function composed of the data-driven model misfit and physics-based data misfit, to guide the network training. Consequently, the proposed PhyDNN method will take the advantage of the fully data-driven DL and conventional physics-based deterministic methods, allowing it to deal with complex realistic exploration scenarios. Quantitative and qualitative analysis results demonstrate that the PhyDNN can honor the physical laws of the MT inverse problem, and with other conditions unchanged, the PhyDNN outperforms the PlainDNN and the classical deterministic Occam inversion method. When processing field data, the PhyDNN method yields considerably impressive inversion results compared to the Occam method, and the corresponding simulated MT responses agree well with the real measurements, which confirms the effectiveness and applicability of the PhyDNN method.

Two-dimensional anisotropic magnetotelluric inversion using a limited-memory quasi-Newton method

We have developed a 2D anisotropic magnetotelluric (MT) inversion algorithm that uses a limited-memory quasi-Newton (QN) method for bounds-constrained optimization. This algorithm solves the inverse problem, which is nonlinear, by iterative minimization of linearized approximations of the classic Tikhonov regularized objective function. The QN approximation for the Hessian matrix is only implemented for the data-misfit term of the objective function; the part of the Hessian matrix for the regularization is explicitly computed. This adjustment results in a better approximation for the data-misfit term in particular. The inversion algorithm considers arbitrary anisotropy, and it is extended for special cases including azimuthal and vertical anisotropy. The algorithm is shown to be stable and converges rapidly for several simple anisotropic models. These synthetic tests also confirm that the anisotropic inversion produces a correct anomaly with different but equivalent anisotropic parameters. We also consider a complex 2D anisotropic model; the successful results for this model further confirm that the inversion algorithm presented here, which uses the novel modified limited-memory QN approach, is capable of solving the 2D anisotropic MT inverse problem. Finally, we have evaluated a practical application on MT data collected in northern Tibet to demonstrate the effectiveness and stability of our algorithm.

Impulse noise treatment in magnetotelluric inversion

Abstract The problem of model recovering in the presence of impulse noise on the data is considered for the magnetotelluric (MT) inverse problem. The application of total variation regularization along with L1-norm penalized data fitting (TVL1) is the usual approach for the impulse noise treatment in image recovery. This combination works poorly when a high level of impulse noise is present on the data. A nonconvex operator named smoothly clipped absolute deviation (TVSCAD) was recently applied to the image recovery problem. This operator is solved using a sequence of TVL1 equivalent problems, providing a significant improvement over TVL1. In practice, TVSCAD requires the selection of several parameters, a task that can be very difficult to attain. A more simple approach to the presence of impulse noise in data is presented here. A nonconvex function is also considered in the data fitness operator, along with the total variation regularization operator. The nonconvex operator is solved by following a half-quadratic procedure of minimization. Results are presented for synthetic and also for field data, assessing the proposed algorithm’s capacity in model recovering under the influence of impulse noise on data for the MT problem.

Three-dimensional magnetotelluric inversion using L-BFGS

The gradient-based optimization methods are preferable for the large-scale three-dimensional (3D) magnetotelluric (MT) inverse problem. Compared with the popular nonlinear conjugate gradient (NLCG) method, however, the limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) method is less adopted. This paper aims to implement a L-BFGS-based inversion algorithm for the 3D MT problem. And we develop our code on top of the ModEM package, which is highly extensible and popular among the MT community. To accelerate the convergence speed, the preconditioning technique by the affine linear transformation of the original model parameters is used. Two modifications of the conventional L-BFGS algorithm are also made to get a comparable convergence rate with the NLCG method. The impacts of the preconditioner parameters, the regularization parameters, the starting model, etc., on the inversion are evaluated by synthetic examples for both L-BFGS and NLCG methods. And the real MT Kayabe dataset is also inverted by the inversion algorithms. The synthetic tests show that through our L-BFGS inversion algorithm the similar resistivity models can be obtained with that from the NLCG method. For the real data inversion, the L-BFGS method performs more efficiently and reasonable results could be obtained by less iterations of the inversion process than the NLCG method. Thus, we suggest the common usage of the L-BFGS method for the 3D MT inverse problem.

Particle swarm optimization of 2D magnetotelluric data

We implement the particle swarm optimization (PSO) algorithm for the two-dimensional (2D) magnetotelluric (MT) inverse problem. We first validate PSO on two synthetic models of different complexity and then apply it to an MT benchmark for real-field data, the COPROD2 data set (Canada). We pay particular attention to the selection of the PSO input parameters to properly address the complexity of the 2D MT inverse problem. We enhance the stability and convergence of the solution of the geophysical problem by applying the hierarchical PSO with time-varying acceleration coefficients (HPSO-TVAC). Moreover, we parallelize the code to reduce the computation time because PSO is a computationally demanding global search algorithm. The inverse problem was solved for the synthetic data both by giving a priori information at the beginning and by using a random initialization. The a priori information was given to a small number of particles as the initial position within the search space of solutions, so that the swarming behavior was only slightly influenced. We have demonstrated that there is no need for the a priori initialization to obtain robust 2D models because the results are largely comparable with the results from randomly initialized PSO. The optimization of the COPROD2 data set provides a resistivity model of the earth in line with results from previous interpretations. Our results suggest that the 2D MT inverse problem can be successfully addressed by means of computational swarm intelligence.

A reversible-jump Markov chain Monte Carlo algorithm for 1D inversion of magnetotelluric data

This paper presents a new computer code developed to solve the 1D magnetotelluric (MT) inverse problem using a Bayesian trans-dimensional Markov chain Monte Carlo algorithm. MT data are sensitive to the depth-distribution of rock electric conductivity (or its reciprocal, resistivity). The solution provided is a probability distribution - the so-called posterior probability distribution (PPD) for the conductivity at depth, together with the PPD of the interface depths. The PPD is sampled via a reversible-jump Markov Chain Monte Carlo (rjMcMC) algorithm, using a modified Metropolis-Hastings (MH) rule to accept or discard candidate models along the chains. As the optimal parameterization for the inversion process is generally unknown a trans-dimensional approach is used to allow the dataset itself to indicate the most probable number of parameters needed to sample the PPD. The algorithm is tested against two simulated datasets and a set of MT data acquired in the Clare Basin (County Clare, Ireland). For the simulated datasets the correct number of conductive layers at depth and the associated electrical conductivity values is retrieved, together with reasonable estimates of the uncertainties on the investigated parameters. Results from the inversion of field measurements are compared with results obtained using a deterministic method and with well-log data from a nearby borehole. The PPD is in good agreement with the well-log data, showing as a main structure a high conductive layer associated with the Clare Shale formation.In this study, we demonstrate that our new code go beyond algorithms developend using a linear inversion scheme, as it can be used: (1) to by-pass the subjective choices in the 1D parameterizations, i.e. the number of horizontal layers in the 1D parameterization, and (2) to estimate realistic uncertainties on the retrieved parameters. The algorithm is implemented using a simple MPI approach, where independent chains run on isolated CPU, to take full advantage of parallel computer architectures. In case of a large number of data, a master/slave appoach can be used, where the master CPU samples the parameter space and the slave CPUs compute forward solutions.

Trans-dimensional Monte Carlo sampling applied to the magnetotelluric inverse problem

The data required to build geological models of the subsurface are often unavailable from direct measurements or well logs. In order to image the subsurface geological structures several geophysical methods have been developed. The magnetotelluric (MT) method uses natural, time-varying electromagnetic (EM) fields as its source to measure the EM impedance of the subsurface. The interpretation of these data is routinely undertaken by solving inverse problems to produce 1D, 2D or 3D electrical conductivity models of the subsurface. In classical MT inverse problems the investigated models are parametrized using a fixed number of unknowns (i.e. fixed number of layers in a 1D model, or a fixed number of cells in a 2D model), and the non-uniqueness of the solution is handled by a regularization term added to the objective function. This study presents a different approach to the 1D MT inverse problem, by using a trans-dimensional Monte Carlo sampling algorithm, where trans-dimensionality implies that the number of unknown parameters is a parameter itself. This construction has been shown to have a built-in Occam razor, so that the regularization term is not required to produce a simple model. The influences of subjective choices in the interpretation process can therefore be sensibly reduced. The inverse problem is solved within a Bayesian framework, where posterior probability distribution of the investigated parameters are sought, rather than a single best-fit model, and uncertainties on the model parameters, and their correlation, can be easily measured.

Magnetotelluric inversion for azimuthally anisotropic resistivities employing artificial neural networks

An extension of an artificial neural network (ANN) approach to solve the magnetotelluric (MT) inverse problem for azimuthally anisotropic resistivities is presented and applied for a real dataset. Three different model classes, containing general 1-D and 2-D azimuthally anisotropic features, have been considered. For each model class, characteristics of three-layer feed forward ANNs trained through an error back propagation algorithm have been adjusted to approximate the inverse modeling function. It appears that, at least for synthetic models, reasonable results would be obtained by applying the amplitudes of the complex impedance tensor elements as inputs. Furthermore, the Levenberg-Marquart algorithm possesses optimal performance as a learning paradigm for this problem.

A regularized three-dimensional magnetotelluric inversion with a minimum gradient support constraint

SUMMARY Most 3-D magnetotelluric (MT) inversions are classified as a regularized inversion with a smoothness constraint. These inverse algorithms provide smooth solutions but cannot clearly image sharp geo-electrical interfaces. In this paper, we introduce the minimum gradient support (MGS) functional to regularize the 3-D MT inverse problem. This functional has a property whereby the functional seeks a structure with minimum volume containing large conductivity gradients. Therefore, the MGS functional can be used to search for a model with a sharp boundary. We apply the MGS functional to 3-D MT inversion to obtain a clear and accurate image of geo-electrical interfaces. In addition, the modified scattering equation approach introduced in the modified iterative dissipative method (MIDM) is applied to forward calculation, which is based on integral equation (IE) formulation and allows us to efficiently reduce the time required for forward calculation with high accuracy. The quasi-Newton iterative method is used to optimize the objective functional. It is a kind of iterative method with simplified calculation of the inverse Hessian matrix using Broyden–Fletcher–Goldfarb–Shanno (BFGS) update. The convergence of this iterative method is guaranteed with inexact line searches. We also modify the adaptive approach for optimum selection of the regularization parameter so as to fit the inverse algorithm of this study. Three synthetic models are investigated, and the obtained results are compared with those obtained by a smoothing inversion. Based on the comparison, we confirm that the MGS inversion can provide higher resolution when geo-electrical interfaces are sharp. This property will help us to determine reliable electrical structures by the MT exploration method.

New analytic solutions for the 2-D TE mode MT problem

SUMMARY A closed-form solution is given for a 2-D, transverse electric mode, magnetotelluric (MT) problem. The model system consists of a finite vertical thin conductor with variable integrated conductivity over a perfectly conducting base. A notable property of the solution is that the frequency response possesses a single pole in the complex plane. Systems with finitely many resonances play a central role in the 1-D MT inverse problem based on finite data sets, but until now, no 2-D system of this kind was known. The particular model is shown to be just one of a large class of thin conductors with same the property, and further examples are given. The solutions of the induction problem for members of this family can often be written in compact closed form, making them the simplest known solutions to the 2-D MT problem.

Non-linearity in Bayesian 1-D magnetotelluric inversion

This paper applies a Bayesian approach to examine non-linearity for the 1-D magnetotelluric (MT) inverse problem. In a Bayesian formulation the posterior probability density (PPD), which combines data and prior information, is interpreted in terms of parameter estimates and uncertainties, which requires optimizing and integrating the PPD. Much work on 1-D MT inversion has been based on (approximate) linearized solutions, but more recently fully non-linear (numerical) approaches have been applied. This paper directly compares results of linearized and non-linear uncertainty estimation for 1-D MT inversion; to do so, advanced methods for both approaches are applied. In the non-linear formulation used here, numerical optimization is carried out using an adaptive-hybrid algorithm. Numerical integration applies Metropolis-Hastings sampling, rotated to a principal-component parameter space for efficient sampling of correlated parameters, and employing non-unity sampling temperatures to ensure global sampling. Since appropriate model parametrizations are generally not known a priori, both under- and overparametrized approaches are considered. For underparametrization, the Bayesian information criterion is applied to determine the number of layers consistent with the resolving power of the data. For overparametrization, prior information is included which favours simple structure in a manner similar to regularized inversion. The data variance and/or trade-off parameter regulating data and prior information are treated in several ways, including applying fixed optimal estimates (an empirical Bayesian approach) or including them as hyperparameters in the sampling (hierarchical Bayesian). The latter approach has the benefit of accounting for the uncertainty in the hyperparameters in estimating model parameter uncertainties. Non-linear and linearized inversion results are compared for synthetic test cases and for the measured COPROD1 MT data by considering marginal probability distributions and marginal profiles. In some cases, important differences are indicated, including poorer sensitivity to thin and/or low-conductivity layers for linearized inversion, and multimodal PPDs which cannot be addressed within a linearized approach.

The 2-D magnetotelluric inverse problem solved with optimization

SUMMARY The practical 2-D magnetotelluric inverse problem seeks to determine the shallow-Earth conductivity structure using finite and uncertain data collected on the ground surface. We present an approach based on using PLTMG (Piecewise Linear Triangular MultiGrid), a special-purpose code for optimization with second-order partial differential equation (PDE) constraints. At each frequency, the electromagnetic field and conductivity are treated as unknowns in an optimization problem in which the data misfit is minimized subject to constraints that include Maxwell's equations and the boundary conditions. Within this framework it is straightforward to accommodate upper and lower bounds or other conditions on the conductivity. In addition, as the underlying inverse problem is ill-posed, constraints may be used to apply various kinds of regularization. We discuss some of the advantages and difficulties associated with using PDE-constrained optimization as the basis for solving large-scale nonlinear geophysical inverse problems. Combined transverse electric and transverse magnetic complex admittances from the COPROD2 data are inverted. First, we invert penalizing size and roughness giving solutions that are similar to those found previously. In a second example, conventional regularization is replaced by a technique that imposes upper and lower bounds on the model. In both examples the data misfit is better than that obtained previously, without any increase in model complexity.

Stochastic interpretation of magnetotelluric data, comparison of methods

Global optimization and stochastic approaches to the interpretation of measured data have recently gained particular attraction as tools for directed search for and/or verification of characteristic structural details and quantitative parameters of the deep structure, which is a task often arising when interpreting geoelectrical induction data in seismoactive and volcanic areas. We present a comparison of three common global optimization and stochastic approaches to the solution of a magnetotelluric inverse problem for thick layer structures, specifically the controlled random search algorithm, the stochastic sampling by the Monte Carlo method with Markov chains and its newly suggested approximate, but largely accelerated, version, the neighbourhood algorithm. We test the algorithms on a notoriously difficult synthetic 5-layer structure with two conductors situated at different depths, as well as on the experimental COPROD1 data set standardly used to benchmark 1D magnetotelluric inversion codes. The controlled random search algorithm is a fast and reliable global minimization procedure if a relatively small number of parameters is involved and a search for a single target minimum is the main objective of the inversion. By repeated runs with different starting test model pools, a sufficiently exhaustive mapping of the parameter space can be accomplished. The Markov chain Monte Carlo gives the most complete information for the parameter estimation and their uncertainty assessment by providing samples from the posterior probability distribution of the model parameters conditioned on the experimental data. Though computationally intensive, this method shows good performance provided the model parameters are sufficiently decorrelated. For layered models with mixed resistivities and layer thicknesses, where strong correlations occur and even different model classes may conform to the target function, the method often converges poorly and even very long chains do not guarantee fair distributions of the model parameters according to their probability densities. The neighbourhood resampling procedure attempts to accelerate the Monte Carlo simulation by approximating the computationally expensive true target function by a simpler, piecewise constant interpolant on a Voronoi mesh constructed over a set of pre-generated models. The method performs relatively fast but seems to suggest systematically larger uncertainties for the model parameters. The results of the stochastic simulations are compared with the standard linearized solutions both for thick layer models and for smooth Occam solutions.

Data space conjugate gradient inversion for 2-D magnetotelluric data

SUMMARY A data space approach to magnetotelluric (MT) inversion reduces the size of the system of equations that must be solved from M × M, as required for a model space approach, to only N × N, where M is the number of model parameter and N is the number of data. This reduction makes 3-D MT inversion on a personal computer possible for modest values of M and N. However, the need to store the N × M sensitivity matrix J remains a serious limitation. Here, we consider application of conjugate gradient (CG) methods to solve the system of data space Gauss‐Newton equations. With this approach J is not explicitly formed and stored, but instead the product of J with an arbitrary vector is computed by solving one forward problem. As a test of this data space conjugate gradient (DCG) algorithm, we consider the 2-D MT inverse problem. Computational efficiency is assessed and compared to the data space Occam’s (DASOCC) inversion by counting the number of forward modelling calls. Experiments with synthetic data show that although DCG requires significantly less memory, it generally requires more forward problem solutions than a scheme such as DASOCC, which is based on a full computation of J.

A limited-memory quasi-Newton inversion for 1D magnetotellurics

We apply a limited-memory quasi-Newton (QN) method to the 1D magnetotelluric (MT) inverse problem. Using this method we invert a realistic synthetic MT impedance data set calculated for a layered earth model. The calculation of gradients based on the adjoint method speeds up the inverse problem solution many times. In addition, regularization stabilizes the QN inversion result and a few correction pairs are sufficient to produce reasonable results. Comparison with the L-BFGS-B algorithm shows similar convergence rates. This study is a first step towards the solution of large-scale electromagnetic problems, with a full treatment of the 3D conductivity structure of the earth.

Determination of the regularization level of truncated singular‐value decomposition inversion: The case of 1D inversion of MT data

ABSTRACTThe total mean‐square error (MSE) of the estimated model, defined as the sum of the standard model variance and the bias variance, is used to define the truncation level of the singular‐value decomposition to give a reasonable balance between model resolution and model variance. This balance is determined largely by the data and no further assumptions are necessary except that the bias terms are estimated sufficiently well. This principle has been tested on the 1D magnetotelluric inverse problem with special emphasis on high‐frequency radio magnetotelluric (RMT) data. Simulations clearly demonstrate that the method provides a good balance between resolution and variance. Starting from a homogeneous half‐space, the best solution is sought for a fixed set of singular values. The model variance is estimated from the sum of the inverse eigenvalues squared, up to a certain threshold, and the bias variance is estimated from the model projections on the remaining eigenvectors. By varying the threshold, the minimum of the MSE is found for an increasing number of fixed singular values until the number of active singular values becomes greater than or equal to the estimated number. As a side‐effect, the depth of penetration of a given set of measurements can be estimated very efficiently by simply noting at which depth the final model deviates little from the starting homogeneous half‐space model. A suite of synthetic data is inverted and an example of inversion of one site is shown to illustrate how the truncation is carried out as the non‐linear inversion process proceeds. A field example with a profile across a plume of contaminated groundwater in the Netherlands shows good agreement with the electrical resistivity obtained in a nearby borehole.

Minimum support nonlinear parametrization in the solution of a 3D magnetotelluric inverse problem

In this paper we describe a new approach to sharp boundary geophysical inversion. We demonstrate that regularized inversion with a minimum support stabilizer can be implemented by using a specially designed nonlinear parametrization of the model parameters. This parametrization plays the same role as transformation into the space of the weighted model parameters, introduced in the original papers on focusing inversion. It allows us to transform the nonquadratic minimum support stabilizer into the traditional quadratic minimum norm stabilizer, which simplifies the solution of the inverse problem. This transformation automatically ensures that the solution belongs to the class of models with a minimum support. The method is illustrated with synthetic examples of 3D magnetotelluric inversion for an earth conductivity structure. To simplify the calculations, in the initial stage of the iterative inversion we use the quasi-analytical approximation developed by Zhdanov and Hursán (2000 Inverse Problems 16 1297–322). However, to increase the accuracy of inversion, we apply rigorous forward modelling based on the integral equation method at the final stage of the inversion. To obtain a stable solution of a 3D inverse problem, we use the Tikhonov regularization method with a new nonlinear parametrization. This technique leads to the generation of a sharp image of anomalous conductivity distribution. The inversion is based on the regularized conjugate gradient method.

Two‐dimensional magnetotelluric inversion of blocky geoelectrical structures

This paper demonstrates that there are alternative approaches to the magnetotelluric (MT) inverse problem solution based on different types of geoelectrical models. The traditional approach uses smooth models to describe the conductivity distribution in underground formations. In this paper, we present a new approach, based on approximating the geology by models with blocky conductivity structures. We can select one or another class of inverse models by choosing between different stabilizing functionals in the regularization method. The final decision, whose approach may be used for the specific MT data set, is made on the basis of available geological information. This paper describes a new way of stabilizing two‐dimensional MT inversion using a minimum support functional and shows the improvement that it provides over traditional methods for geoelectrical models with blocky structures. The new method is applied to MT data collected for crustal imaging in the Carrizo Plain in California and to MT data collected for mining exploration by INCO Exploration.

Nonlinear conjugate gradients algorithm for 2-D magnetotelluric inversion

We investigate a new algorithm for computing regularized solutions of the 2-D magnetotelluric inverse problem. The algorithm employs a nonlinear conjugate gradients (NLCG) scheme to minimize an objective function that penalizes data residuals and second spatial derivatives of resistivity. We compare this algorithm theoretically and numerically to two previous algorithms for constructing such “minimum‐structure” models: the Gauss‐Newton method, which solves a sequence of linearized inverse problems and has been the standard approach to nonlinear inversion in geophysics, and an algorithm due to Mackie and Madden, which solves a sequence of linearized inverse problems incompletely using a (linear) conjugate gradients technique. Numerical experiments involving synthetic and field data indicate that the two algorithms based on conjugate gradients (NLCG and Mackie‐Madden) are more efficient than the Gauss‐Newton algorithm in terms of both computer memory requirements and CPU time needed to find accurate solutions to problems of realistic size. This owes largely to the fact that the conjugate gradients‐based algorithms avoid two computationally intensive tasks that are performed at each step of a Gauss‐Newton iteration: calculation of the full Jacobian matrix of the forward modeling operator, and complete solution of a linear system on the model space. The numerical tests also show that the Mackie‐Madden algorithm reduces the objective function more quickly than our new NLCG algorithm in the early stages of minimization, but NLCG is more effective in the later computations. To help understand these results, we describe the Mackie‐Madden and new NLCG algorithms in detail and couch each as a special case of a more general conjugate gradients scheme for nonlinear inversion.