Inverse Gravimetry Problem Research Articles

On the construction of an optimal network of observation points when solving inverse lin-ear problems of gravimetry and magnetometry

Unique solvability of systems of linear algebraic equations is studied to which many in-verse problems of geophysics are reduced as a result of discretization after applying the method of integral equations or integral representations. Examples of singular and nonsingular systems of vari-ous dimensions that arise when processing magnetometric and gravimetric data from experimental observations are discussed. Conclusions are drawn about methods for constructing an optimal net-work of experimental observation points.

Строение центральной части палеопротерозойского Лосевского террейна по данным плотностного моделирования

The results of detailed three-dimensional density modeling of the upper crust of the central part of the Losevsky terrane located in the northern part of the Voronezh crystal massif are considered. The distribution of absolute crustal density values of to a depth of 16 km was obtained as a result of the inversion of local anomalies of the gravitational field in the Bouguer reduction. The solution of the inverse problem of gravimetry was implemented within the framework of the initial model, which was formed using digital geological and geophysical data from the study area: the regional density model of the East European Platform lithosphere and its corresponding regional gravity field; the field of local gravity anomalies, representing the difference between the observed and regional fields of the lithosphere model; generalized information on the density of sedimentary cover rocks and the crystal basement; the thickness scheme of the «gravity-active» layer, obtained on the basis of statistical analysis of the anomalous field; geological and topographic information. Quantitative interpretation of the gravimetry data allowed us to obtain fundamentally new information about the geological structure of the upper crust of the study area. The reliability of the obtained results is confirmed by the consistency of a priori data and the correspondence of the total model (local plus regional) and observed gravitational fields.

Learning on the correctness class for domain inverse problems of gravimetry

Learning on the correctness class for domain inverse problems of gravimetry

Глубинное строение Среднего Урала по гравитационным и магнитным данным

For the Middle Urals and adjacent territories of the East European and West Siberian platforms within geographic coordinates from 56º to 60º northern latitude and from 54º to 66º east longitude, the structural features of the anomalous magnetic and gravitational fields were studied and maps of separated anomalies were constructed for three layers of the earth's crust. Using original parallel algorithms for solving forward and inverse problems of gravimetry and magnetometry, a density model of the lithosphere and sources of magnetic anomalies in the earth's crust were constructed. The results of modeling the sources of magnetic anomalies in the Earth's crust are compared with the density model in order to study the differences in the deep structure of the western paleocontinental and eastern paleo-island arc sectors of the Ural fold system, as well as the zones of junction of the orogen with the East European and West Siberian platforms.

On the Construction of an Optimal Network of Observation Points when Solving Inverse Linear Problems of Gravimetry and Magnetometry

On the Construction of an Optimal Network of Observation Points when Solving Inverse Linear Problems of Gravimetry and Magnetometry

Structural Tectonic Scheme Creation Based on Seismic-Gravity Models and Isostasy Usage: Ural Case

Process of Earth’s density models creation leads to the solution of direct and inverse gravimetry problems. The inverse problem of gravimetry is a classic example of an ill-posed problem: in the common statement, its solution is not unique and unstably depends on input data. Therefore, it is necessary to determine solutions belonging substantial sets of correctness, choosing reasonable models of an initial approximation. In this paper the application of complex interpretation methods of seismic and gravitational data for the creation of three-dimensional models of crust and the upper mantle are presented. Original algorithms and programs were developed for implementation of these methods. They contain solution of non-linear (structural) inverse problem and the solution of the linear three-dimensional inverse problem taking note of the side sources. Coefficients of the “density-velocity” correlation formulas for a number of geo-traverses were defined. Also, we suggest a technique of tectonic maps construction, which is based on the lithostatic pressure calculation. Its idea can be applied to both two- and three-dimensional cases. In the 2D case we show the way to split the mantle to blocks with vertical boundaries. If lithostatic compensation hypothesis is adopted, the method also allows one to calculate density value for each block. Such separation of the mantle helps to diminish discrepancy between model and observed fields. In 3D case we suggest a method, which can be used to construct tectonic structure maps with information about approximate depth and height of each tectonic block.

On the Uniqueness of Solution to Systems of Linear Algebraic Equations to Which the Inverse Problems of Gravimetry and Magnetometry Are Reduced: A Regional Variant

On the Uniqueness of Solution to Systems of Linear Algebraic Equations to Which the Inverse Problems of Gravimetry and Magnetometry Are Reduced: A Regional Variant

On the Uniqueness of Solution to Systems of Linear Algebraic Equations to Which the Inverse Problems of Gravimetry and Magnetometry Are Reduced: A Regional Variant

Conditions for the unique solvability of systems of linear algebraic equations to which many inverse problems of gravitational and magnetic exploration are reduced are considered. The mathematical statements of inverse problems take into account the sphericity of the Earth.

Piecewise Acoustic Source Imaging with Unknown Speed of Sound Using a Level-Set Method.

We investigate the following inverse problem: starting from the acoustic wave equation, reconstruct a piecewise constant passive acoustic source from a single boundary temporal measurement without knowing the speed of sound. When the amplitudes of the source are known a priori, we prove a unique determination result of the shape and propose a level set algorithm to reconstruct the singularities. When the singularities of the source are known a priori, we show unique determination of the source amplitudes and propose a least-squares fitting algorithm to recover the source amplitudes. The analysis bridges the low-frequency source inversion problem and the inverse problem of gravimetry. The proposed algorithms are validated and quantitatively evaluated with numerical experiments in 2D and 3D.

On the Uniqueness of Solutions to Systems of Linear Algebraic Equations Resulting from the Reduction of Linear Inverse Problems of Gravimetry and Magnetometry: a Local Case

The paper considers issues of unique solvability of systems of linear algebraic equations to which many inverse problems of geophysics are reduced as a result of discretization. Examples of degenerate and nondegenerate systems of different dimensions arising from the interpretation of gravity and magnetometric data are given.

On the Uniqueness of Solutions to Systems of Linear Algebraic Equations Resulting from the Reduction of Linear Inverse Problems of Gravimetry and Magnetometry: a Local Case

On the Uniqueness of Solutions to Systems of Linear Algebraic Equations Resulting from the Reduction of Linear Inverse Problems of Gravimetry and Magnetometry: a Local Case

On the Unique Solvability of Inverse Problems of Magnetometry and Gravimetry

This article deals with the question of the unique solvability of systems of linear algebraic equations, to the solution of which many inverse problems of geophysics are reduced as a result of discretization when applying the methods of integral equations or integral representations. Examples are given of degenerate and nondegenerate systems of different dimensions that arise in the processing of magnetometric and gravimetric data from experimental observations. Conclusions are drawn about the methods for constructing the optimal grid of experimental observation points.

Lithosphere Density Model of the Middle Urals Segment

For the Middle and North Urals segments and adjacent territories of the East European Platform and West Siberian Plate (within the graticular trapezoid with geographic coordinates 53–65° N, 48–72° E), ten deep seismic sounding (DSS) profiles were reprocessed using two-dimensional (2D) seismic tomography method, and gradient sections of crustal velocities were constructed in the form of grid functions. In the same format, density sections were constructed. The coefficients of density–velocity empirical correlation were calculated using the solution algorithm of 2D inverse problem of gravimetry. The method and calculation technology of a three-dimensional (3D) density distribution with tying to 2D data from reference seismic sections are embedded in the procedure of quantitative interpretation of potential fields with construction of 3D geophysical models. A stable solution of the 3D inverse problem of gravimetry is sought on the correctness set of the family of horizontal layers with 2D density distribution.

On the inverse gravimetry problem with minimal data

Abstract The inverse problem in gravimetry is to find a domain 𝐷 inside the reference domain Ω from measurements of gravitational force outside Ω. We consider the problem in three dimensions where we found that a few parameters of the unknown 𝐷 can be stably determined given data noise in practical situations. An ellipsoid is the best approximation of 𝐷. We prove uniqueness of recovering an ellipsoid in a particular case for the inverse problem from minimal amount of data which are the approximated gravitational force at nine boundary points. In the proofs, we derive and use simple systems of linear and nonlinear algebraic equations for the parameters of an ellipsoid. Similarly, a rectangular parallelepiped 𝐷 is considered. To support our theory, we use numerical examples with different location of measurements points on ∂ ⁡ Ω \partial\Omega .

Разработка комплекса параллельных программ решения обратных задач гравиметрии и магнитометрии для сеток большой размерности

The most important problems in studying of the structure of the earth’s crust are the inverse problems of gravimetry and the problems of magnetometry. The problem is in finding interfaces between layers with different constant densities using known gravitational data. The methods for solving these problems are based on the ideas of iterative regularization. After discretization, these are reduced to systems of nonlinear functions of large dimensions. The need to improve the accuracy of the results of solving problems entails an increase in the computation time. The paper describes a remote computing system and an integrated program complex for graphics accelerators that implement the fastest and most memory-efficient algorithms developed earlier and based on gradient methods. The remote computing system considered in the work is a web portal that is a universal solution for launching tasks on remote clusters. The most important advantage of the portal is the simplicity of its use: when connecting to a cluster to carry out calculations, you no longer need to install additional software on the cluster itself, and you do not need a privileged account to work with the cluster. All that is required is a valid account on the remote cluster, the rest of the work on communication with the data processing center (DPC) is taken over by the portal. The portal can easily scale with the growth of the number of users who can download the necessary algorithms and perform calculations using the data center.

Применение теории потенциала при численном решении обратной задачи гравиметрии

This article discusses the results of a study in solving the inverse problem of gravimetry. It is shown that the solution of the inverse problem should be conducted in two stages. At the first step, the contour of the area occupied by gravitation masses is determined using the Laplace equation. At the second step, a generalization of the Poisson equation is used to determine the density distribution of the masses that fill the area, whose contour is determined at the first step. In this case, when setting the problem of analytic continuation, the provisions of potential theory are applied. The results of numerical calculations are presented on a model example with the determination of unknown values of density and values of the gravitational field inside a region filled with gravitation masses.

Kantorovich-Rubinstein metric based level-set methods for inverting modulus of gravity-force data

We propose to use the Kantorovich-Rubinstein (K-R) metric as a novel misfit function for the level-set based inverse gravity problems, where modulus of gravity-force data is used. By using the modulus data, we can satisfy the non-negativity requirement of distribution for the K-R metric naturally. Moreover, the K-R metric based level-set method can tolerate high level noise in the modulus data so that we can solve the domain inverse problem of gravimetry to high resolution. We develop the computational framework systematically. Numerical examples demonstrate the performance and effectiveness of the proposed algorithms.

ТРЕХМЕРНАЯ ПЛОТНОСТНАЯ МОДЕЛЬ ВЕРХНЕЙ КОРЫ В ОБЛАСТИ СОЧЛЕНЕНИЯ ЛОСЕВСКОГО И ВОРОНЦОВСКОГО ТЕРРЕЙНОВ (ВОРОНЕЖСКИЙ КРИСТАЛЛИЧЕСКИЙ МАССИВ)

The results of detailed three-dimensional density modeling of the upper part of the Earth's crust for the area located in the northeastern part of the Voronezh crystalline massif in the area of the junction of the Losevsky and Vorontsovsky terranes are considered. The model obtained by solving the inverse gravimetry problem characterizes the density distribution of the rocks of the crystalline basement of the territory to a depth of 16 km. An important role in such constructions is given to the starting model of the studied environment, which was formed taking into account the available a priori geological and geophysical information: the regional density model of the East European Platform lithosphere and the gravity field, the data on the density of rocks of the sedimentary cover and the crystalline basement, estimates of the “gravity” layer thickness, the geological map of the crystalline basement, and the digital elevation models of the region. The three-dimensional density model allowed specifying the geological structure of the upper crust of the study area and tracing the deep continuation of the surface structures of the basement. The reliability of the obtained results is determined by the consistency of the accepted a priori data on the environment and the consistency of the model and observed gravity fields of the study area.

The technique of recovery of causative sources in case of strong interference of gravity fields using the continuous wavelet transform

The technique of processing gravimetric data is offered in this study. Offered technique based on wavelet transform with so-called «native» wavelet basis functions. Distinctive feature of the technique is a close relationship with both direct and inverse problems of gravimetry. It was shown that the peculiarity allows to quite simply and quickly location of causative sources even under of strong interference of gravity fields. Keywords: gravimetry; wavelet transform; anomaly; inverse problem.

Numerical method for solving the piecewise constant source inverse problem of an elliptic equation from a partial boundary observation data

We present results of numerical investigation of the source term recovery in a boundary value problem for an elliptic equation. An additional information about the solution is considered as its normal derivative taken on a part of the boundary. Such source inverse problem is related with inverse gravimetry problem of determining an inhomogeneity from gravitational potential anomalies on the Earth’s surface. We propose an iterative method for numerical recovery of the source term on the base of minimization of the observation residual by a gradient type method. The numerical implementation is based on finite element approximation using the FEniCS scientific computing platform and the dolfin-adjoint package. The capabilities of the developed computational algorithm are illustrated by results of numerical solutions of two dimensional test problems.