In this article, the use of adaptive convolutional neural network architectures is examined for solving three-dimensional inverse problems in structural geophysics. An object-oriented approach to modeling geophysical data has been developed, which includes the creation of a database for typical hierarchical structural models (grabens, horsts, normal faults, and reverse faults) and the computation of forward problems. The application of convolutional neural networks—incorporating architectures that extract local features, as well as transfer learning methods—has enabled an improvement in the models’ generalization capabilities and facilitated their adaptation to the specifics of different geophysical classes, compared to multi-layer perceptron-based approaches. The numerical results presented demonstrate the fundamental feasibility of applying the proposed approach to the solution of structural inverse problems in geophysics. #CSOC1120.

Neural networks (NNs) are successfully used to solve inverse and other problems in geophysics. The aim of this work, which is a continuation of a series of works by a group of authors, is to improve the efficiency of the NN method for solving nonlinear inverse 3D problems of geoelectrics, based on the construction of the author's convolutional neural network. The network includes a number of additional special transformations (data compression, suppression of the influence of an unknown background environment, etc.) preceding the training of a classical MLP neural network and adapted to the inverse problem that is being solved. This allows us to formally, excluding the human factor, solve inverse problems of geoelectrics of large dimensions without specifying a first approximation based on data measured in areas whose dimensions exceed the dimensions of the network training area. The inversion speed is a few tens of seconds and does not depend on the physical dimensionality (2D or 3D) of the data. The solution to the inverse problem found using a trained neural network can, if necessary, be refined using a random search method. Numerical results of solving 3D geoelectric problems on model and field data are presented, confirming the stated development parameters.

Previously, it was shown that integration (joint use of data) of several geophysical methods allows one to obtain a higher quality of the solution of the inverse problem of exploration geophysics in comparison with the individual use of each of these methods. However, there may be a situation when for some measurement points there is no data from one of the geophysical methods used. At the same time, the data spaces of different integrated geophysical methods are interconnected. Therefore, the missing data of one method can be recovered from the known data of another one by constructing a preliminary adaptive mapping of one of the spaces to another. In this study, we investigate the solution of the inverse problem with integration of geophysical methods on the recovered data obtained based on noise addition during training of the neural networks performing the mapping from the data space of the method(s) with all data present to the data space of the method with missing data.

This study is devoted to solving inverse problems of exploration geophysics, which consist in reconstructing the spatial distribution of the properties of the medium in the thickness of the earth from the geophysical fields measured on its surface. We consider the methods of gravimetry, magnetometry, and magnetotelluric sounding, as well as their integration, i.e. simultaneous use of data from several geophysical methods to solve the inverse problem. To implement such integration, in our previous studies we have proposed a parameterization scheme that describes a layered geophysical model with fixed layer properties, in which the determined parameters were the positions of the boundaries between the layers. In the present study, this parameterization scheme is complicated so that the properties of the layers vary from pattern to pattern in the data set. To improve the quality of neural network solution of the described inverse problem, we consider an approach based on the use of a priori information about the physical properties of the layers, in which this information is used directly as additional input features for the neural network.

Exploration geophysics requires solving specific inverse problems — reconstructing the spatial distribution of the medium properties in the thickness of the earth from the geophysical fields measured on its surface. We consider inverse problems of gravimetry, magnetometry, magnetotelluric sounding, and their integration, which means simultaneous use of various geophysical fields to reconstruct the desired distribution. Integration requires the determined parameters for all the methods to be the same. This may be achieved by the spatial statement of the problem, in which the task is to determine the boundaries of geophysical objects. In our previous studies, we considered the parameterization scheme where the inverse problem was to determine the lower boundary of several geological layers. Each layer was characterized by variable values of the depth of the lower boundary along the section, and by fixed values of density, magnetization, and resistivity, both for the layer and over the entire dataset. It was demonstrated that the integration of geophysical methods provides significantly better results than the use of each of the methods separately. The present study considers an extended and more realistic model of data—a parameterization scheme with variable properties of the medium, both along each layer and over the dataset.

In this paper, we consider a neural network solution of the inverse problem (IP) of magnetotelluric sounding (MTS) which consists in constructing the electrical conductivity distribution in the Earth's interior from the values of the electromagnetic field components measured on its surface. It has a high input dimension (thousands of features), so it is necessary to reduce the input data dimension to achieve a more accurate and stable solution while reducing computational complexity. Neighboring measurement points and neighboring frequencies carry similar information dictating the need to use a selection method that considers this feature. The present work is devoted to the study of a method based on the iterative selection of features with the highest correlation with respect to the target variable and the exclusion of features with high cross-correlation. This method was compared with the traditional selection method, the cross-correlation filter.

Abstract—The approximation neural network (ANN) method for solving the inverse geoelectric problem in the piecewise constant classes of media with the subsequent refinement of the model by random search method is presented. A conditionally well-posed magnetotelluric inverse problem is considered on a regularized parameterization grid. The total number of the sought-for parameters in this problem can be $$\sim n \times {\text{1}}{{0}^{{\text{3}}}}$$ . The work of the algorithm is illustrated by the example of 3D inversion of the field survey data. The posterior ambiguity characteristics of the obtained solution were calculated. The proposed methods are implemented using supercomputing cluster resources and parallel computing technology.

The inverse problem (IP) of exploration geophysics consists in reconstructing the spatial distributionf of the properties of the medium in the Earth’s interior from measurements on its surface. This IP is a non-linear ill-posed ill-conditioned problem with high dimensionality both by input and by output. One of the approaches free of many shortcomings inherent for traditional methods of IP solving, is the use of artificial neural networks (NN). In this study, it has been suggested to use an integration of geophysical methods to improve the quality of the solution obtained by NN. The considered model combines three geophysical methods: gravimetry, magnetometry, and magnetotellurics. The problem considered is that of determining the structural boundaries separating the geological layers with constant values of the parameters: density in gravimetry, magnetization in magnetometry, electrical resistivity in magnetotellurics. In this study, a four-layer 2D model was considered. It is demonstrated that integration of geophysical methods provides significantly better results that use of each of the methods separately. It is also shown that in some cases it is also possible to improve the quality of the IP solution using multitask learning—simultaneous determination of the positions of two or all three layer boundaries.

This study is devoted to the inverse problems of exploration geophysics, which consist in reconstructing the spatial distribution of the properties of the medium in the Earth’s thickness from the geophysical fields measured on its surface. We consider the methods of gravimetry, magnetometry, and magnetotelluric sounding, as well as their integration, i.e. simultaneous use of data from several geophysical methods to solve the inverse problem. In their previous studies, the authors have shown that the integration of geophysical methods allows improving the quality of the solution of the inverse problem in comparison with the individual use of each of them. One of the obstacles to using the integration of geophysical methods can be the situation when for some measurement points there is no data from one of the geophysical methods used. At the same time, the data spaces of different integrated geophysical methods are interconnected, and the values of the observed quantities (fields) for one of the methods can be possibly recovered from the known values of the observed quantities of another geophysical method by constructing a preliminary adaptive mapping of one of the spaces to another. In this study, we investigate the neural network recovery of missing data of one geophysical method from the known data of another one and compare the quality of the solution of the inverse problem on full and on recovered data.

In their previous studies, the authors have shown that the integration of geophysical methods allows improving the quality of the solution of an inverse problem of exploration geophysics in comparison with the individual use of each of them. However, in practice, it is possible that for some measurement points, data from one of the geophysical methods used is missing. In this study, we investigate an approach associated with neural network recovery of the missing data of one geophysical method from the known data of another, and their further joint application to solve the inverse problem. In addition, we explore the effectiveness of applying multitask learning approach at the data recovery stage for the subsequent solution of the inverse problem.

Summary The paper presents an approximation neural network algorithm for solving conditionally correct coefficient inverse problems of geoelectrics in the class of media with piecewise constant electrical conductivity given on a parametrization grid. It is shown that the degree of ambiguity (error) of solutions monotonically increases with an increase in the dimension of the parametrization grid. A method is proposed for constructing an optimal parametrization grid, which has the maximum dimension provided that the a priori estimates of the ambiguity of the solutions do not exceed a given value. It is shown that the inverse problem in the considered class of media is reduced to the classical approximation-interpolation problem using neural network polynomials, the solution of which is the essence of the approximation neural network (ANN) method. The intrinsic error of the ANS method is determined, a posteriori estimates of the ambiguity (error) of the obtained approximate solutions are calculated with the achieved synthesis discrepancy. The method makes it possible to formalize and uniformly obtain solutions to the inverse problem of geoelectrics with the total number of the required parameters of the medium ∼ n 10 ^ 3.

The inverse problems of exploration geophysics are to reconstruct the spatial distribution of the properties of the medium in the Earth's thickness from the geophysical fields measured on its surface. In particular, this paper deals with the problems of gravimetry, magnetometry, and magnetotelluric sounding, as well as their integration, i.e., the simultaneous use of several geophysical fields to restore the desired distribution. To implement the integration, a 4-layer 2D model was used, where the inverse problem was to determine the lower boundary of the layers, and each layer was characterized by variable values of the depth of the lower boundary along the section and fixed values of density, magnetization, and resistivity, both for the layer and for the entire data set. To implement the neural network solution of the inverse problem, a data set was generated by solving the direct problem, where for each pattern, the distribution of layer depth values was set randomly in a given range and with a given step, i.e. it took discrete values from a certain set. In this paper, we consider an approach involving the use of neural networks to solve the problem of multiclass classification, where class labels correspond to discrete values of the determined layer depths. The results of the solution are compared with the results of the solution of the same inverse problem in the formulation of the regression problem, in terms of the error in determining the depth of the layers.

Summary The paper presents an example of the application of approximation neural network structures to the problem of reconstructing the resistivity distributions of 2D and 3D piecewise linear media from geoelectric data. This problem is reduced to solving a nonlinear operator equation of the first kind. An algorithm was proposed [ Shimelevich et al, 2018 , Obornev et al, 2020 ] for finding an approximate solution of this equation with a total number of parameters of the order of ∼ n 10 ^ 3, based on the use of neural (Kolmogorov) networks of the multilayer perceptron type. This approach, which allows real-time data inversion, is illustrated both on model examples and on profile and areal field survey data.

Рассматриваются априорные оценки неоднозначности (погрешности) приближенных решений условно-корректных нелинейных обратных задач, основанные на модуле непрерывности обратного оператора и его модификациях. Установлена связь модуля непрерывности обратного оператора с разрешающей способностью геофизического метода. Показано, что в классе кусочно-постоянных решений, определенных на заданной сетке параметризации, модуль непрерывности обратного оператора и его модификации монотонно возрастают с увеличением размерности сетки. Предложен метод построения оптимальной сетки параметризации, которая имеет максимальную размерность при условии, что модуль непрерывности обратного оператора не превышает заданной величины. Представлен численный алгоритм расчета модуля непрерывности обратного оператора и его модификаций с использованием алгоритмов Монте-Карло, исследуются вопросы сходимости алгоритма. Предлагаемый метод применим также для расчета классических апостериорных оценок погрешности. Приводятся численные примеры для нелинейных обратных задач геоэлектрики. The article considers a priori estimates of the ambiguity (error) of approximate solutions of conditionally correct nonlinear inverse problems based on the modulus of continuity of the inverse operator and its modifications. It is shown that in the class of piecewise constant solutions defined on a given parametrization grid, the modulus of continuity of the inverse operator and its modifications monotonously increase with increasing mesh dimension. A method is proposed for constructing an optimal parameterization grid that has a maximum dimension provided that the modulus of continuity of the inverse operator does not exceed a given value. A numerical algorithm for calculating the modulus of continuity of the inverse operator and its modifications using Monte Carlo algorithms is presented; questions of convergence of the algorithm are investigated. The proposed method is also applicable for calculating classical posterior error estimates. Numerical examples are given for nonlinear inverse problems of geoelectrics.

In the present study, using the inverse problem (IP) of magnetotelluric sounding (MTS) as an example, we consider the use of neural networks to solve high-dimensional coefficient inverse problems. To reduce the incorrectness, a complex approach is considered related to the use of narrow classes of geological models, with prior selection of the model class by solving the classification problem by MTS data. Within the framework of this approach, the actual direction of work is to reduce the volume of calculations when re-building the system for another set of geological models. This goal can be achieved by selecting the essential features. The present paper is devoted to the study of the applicability of various selection methods to the MTS IP. Also, in this paper we consider taking into account domain knowledge about the studied object in the process of selection of essential features using methods such as wrapper.

Neural networks (NN) are widely used for solving various problems of geophysical data interpretation and processing. The application of the neural network approximation (NNA) method for solving inverse problems, including inverse multicriteria problems of geophysics that are reduced to a nonlinear operator equation of first kind (respectively, to a system of operator equations) is considered. The NNA method assumes the construction of an approximate inverse operator of the problem using neural network approximation designs (MLP networks) on the basis of a preliminary constructed set of reference solutions to direct and inverse problems. A review of the application of the NNA method for solving nonlinear inverse problems of geophysics is given. Techniques for estimating the practical ambiguity (error) of approximate solutions to inverse multicriteria problems are considered. Results of solving the inverse two-criteria 2D gravimetry problem in combination with magnetometry are presented.

Summary Neural networks (NN) are widely used for solving various problems of geophysical data interpretation and processing. The application of the neural network approximation (NNA) method for solving inverse problems, including inverse multi-criteria problems of geophysics that are reduced to a nonlinear operator equation of the first kind (respectively, to a system of operator equations) is considered. The NNA method assumes the construction of an approximate inverse operator of the problem using neural network approximation designs (MLP networks) on the basis of a preliminary constructed set of reference solutions to direct and inverse problems. Techniques for estimating the practical ambiguity (error) of approximate solutions to inverse multicriteria problems are considered. Results of solving the inverse two-criteria 3D problem in combination with magnetometry are presented. It is shown that the NNA method allows one to stably solve nonlinear multicriteria inverse 3D problems with many desired parameters in real-time, with accuracy acceptable for practice. The experience of calculations shows that the error in solving the two-criteria problem is less than the one-criterion.

The approximating neural network algorithm for solving the inverse problems of geoelectrics in the class of grid (block) models of the medium is presented. The algorithm is based on constructing an approximate inverse operator using neural networks and makes it possible to formally obtain the solutions of the geoelectrics inverse problem with a total number of the sought parameters of the medium \( \sim n\, \times \, 10^{ 3} \). The questions concerning the correctness of the problem of constructing the inverse neural network operators are considered. The a posteriori estimates of the degree of ambiguity in the inverse problem solutions are calculated. The work of the algorithm is illustrated by the examples of 2D and 3D inversions of the synthesized data and the real magnetotelluric sounding data.

Summary The task of building a geoelectric profile based on geophysical fields measured on the surface can be compared with the task of recognizing an object based on its quantitative characteristics. For example, a face can be recognized by the numerical values characterizing the geometry of its constituent parts. In our case, the role of these quantitative characteristics is played by the measured values of the electromagnetic field, and the restored image is the geoelectric section in the color legend, where the color corresponds to a certain value of the specific resistance of the medium. Approximate neural network methods of geophysics are based on the construction of an approximate inverse problem operator in a given class of media using a multilayered neural network - a neural network approximator (NNA). For the construction of the NNA, the problem of its training is solved, which lies in the fact that the coefficients of the NNA are adjusted by the training sample of known solutions of direct problems obtained using the direct operator of the problem being solved. The task of adjusting the coefficients of the approximator is reduced to an optimization problem, which is solved using the methods of the Monte Carlo group.