Inverse Problem Of Potential Theory Research Articles

Алгоритм визначення параметрів включення при розв’язуванні обернених задач геоелектророзвідки методом про

The paper aims to develop an algorithm for recognizing the physical and geometric parameters of inclusion, using indirect methods of boundary, near-boundary, and partially-boundary elements based on the data of the potential field. Methodology. The direct and inverse two-dimensional problems of the potential theory concerning geophysics were solved when modeling the earth's crust with a piecewise-homogeneous half-plane composed of a containing medium and inclusion that are an ideal contact. To construct the integral representation of the solution of the direct problem, a special fundamental solution for the half-plane (Green's function) of Laplace's equation, which automatically satisfies the zero-boundary condition of the second kind on the day surface, and a fundamental solution for inclusion were used. To find the intensities of unknown sources introduced in boundary, near-boundary, or partially-boundary elements, the collocation technique was used, i.e. the conditions of ideal contact are satisfied in the middle of each boundary element. After solving the obtained SLAE, the unknown potential in the medium and inclusion and the flow through their boundaries are found, considering that the medium and inclusion are considered as completely independent domains. Results. The computational experiment for the task of electric prospecting with a constant artificial field using the resistance method, in particular, electrical profiling, was carried out, while focusing on the physical and geometric interpretation of the data. Initial approximations for the electrical conductivity of the inclusion, its center of mass, orientation and dimensions are determined by the nature of the change in apparent resistivity. To solve the inverse problem two cascades of iterations are organized: the first one is to specify the location of the local heterogeneity and its approximate dimensions, the second one is to specify its shape and orientation in space. At the same time, the minimization of the functional considered on the section of the boundary, where an excess of boundary conditions is set, is carried out. Originality. The method of boundary integral equations is shown to be effective for constructing numerical solutions of direct and inverse problems of potential theory in a piecewise homogeneous half-plane, using indirect methods of boundary, near-boundary, and partial-boundary elements as variants. Practical significance. The proposed approach for solving the inverse problem of electrical exploration with direct current is effective because it allows fora step-by-step, "cascade" recognition of the shape, size, orientation, and electrical conductivity of the inclusion. We follow the principle of not using all the details of the model and not attempting to recognize parameters with little effect on the result, especially with imprecise initial approximations.

Algorithm for constructing the subsoil distribution function considering its value on the surface

The conventional approach to constructing a three-dimensional distribution of the Earth's masses involves using Stokes constants incrementally up to a certain order. However, this study proposes an algorithm that simultaneously considers all of these constants, which could potentially provide a more efficient method. The basis for this is a system of equations obtained by differentiating the Lagrange function, which takes into account the minimum deviation of the three-dimensional mass distribution of the planet's subsoil from one-dimensional referential one. An additional condition, apart from taking into account the Stokes constants, for an unambiguous solution to the problem is to specify the value of the function on the surface of the ellipsoidal planet. It is possible to simplify the calculation process by connecting the indices of summation values in a series of expansions to their one-dimensional analogues in the system of linear equations. The study presents a control example illustrating the application of the given algorithm. In its implementation, a simplified variant of setting the density on the surface of the ocean is taken. The preliminary results of calculations confirm the expediency of this approach and the need to expand such a technique with other conditions for unambiguously solving the inverse problem of potential theory. Objectives. To create and implement the algorithm that takes into account the density of the planet’s subsoil on its surface. Method. The mass distribution function of the planet's subsoil is represented by a decomposition into biorthogonal series, the coefficients of decomposition which are determined from a system of linear equations. The system of equations is obtained from the condition of minimizing the deviation function of the desired mass distribution from the initially determined two-dimensional density distribution (PREM reference model). Results. On the basis of the described algorithm, a three-dimensional model of the density distribution of subsoil masses in the middle of the Earth is obtained, which takes into account Stokes constants up to the eighth order inclusively and corresponds to the surface distribution of masses of the oceanic model of the Earth. Its concise interpretation is also presented.

Quantitative interpretations of potential fields: from parametric to geostructural recalculations

The relevance. Need to development of an applied algorithm for combined interpretation of heterogeneous geophysical data based on unified physical analogy, focused on solving the inverse problem of potential theory. The wave approximation of the cross-section of mining massif fits into the paradigm of dissipative structuring of nonequilibrium media acts as a special case of the above analogy. The task of the study is to adapt this algorithm to recalculations of both potential and non-potential geofields. In the list of nonpotential geofields the remote sensing data and digital terrain model are the most interesting as the data with non-regulated access. Practical significance of the research is associated with the approbation of a family of algorithms in production projects of both geological and engineering nature. The aim. To minimize the family of equivalent solutions to the inverse problem by mutual verification of heterogeneous models of the anomaly-forming source. The objects. Development of the geofields combined interpretation algorithm and recalculation of geofields in depth for tracing elements of stratification and geological fractures. The methods. Formed by software and algorithmic development, which is based on deterministic analytical representations: computation of Cauchy integral, iterative application of direct solution, spectrum estimation. The results. Reduced to the set of parametric cross-sections derived with recalculations with minimum a priori geological information with final geological consideration. The combination of visualizations of analytical continuation, singular points and results of automated fitting admits the reconstruction of geostructural model of parametric cross-section with significant reliability. Extension of standard methods of quantitative interpretation by means of wave approximations includes quasi-seismic halftone visualizations verifying the mentioned reconstructions.

ПРИМЕНЕНИЕ МЕТОДА НАИБОЛЬШИХ ВКЛАДОВ В ТЕХНИКЕ ИНВЕРСИИ МАГНИТОГРАММ

Fundamentals of the spherical harmonic analysis (SHA) of the geomagnetic field were created by Gauss. They acquired the classical Chapman — Schmidt form in the first half of the XXth century. The SHA method was actively developed for domestic geomagnetology by IZMIRAN, and then, since the start of the space age, by ISTP SB RAS, where SHA became the basis for a comprehensive method of MIT (magnetogram inversion technique). SHA solves the inverse problem of potential theory and calculates sources of geomagnetic field variations (GFV) - internal and external electric currents. The SHA algorithm forms a system of linear equations (SLE), which consists of 3K equations (three components of the geomagnetic field, K is the number of ground magnetic stations). Small changes in the left and (or) right side of such SLE can lead to a significant change in unknown variables. As a result, two consecutive instants of time with almost identical GFV are approximated by significantly different SHA coefficients. This contradicts both logic and real observations of the geomagnetic field. The inherent error of magnetometers, as well as the method for determining GFV, also entails the instability of SLE solution. To solve such SLEs optimally, the method of maximum contribution (MMC) was developed at ISTP SB RAS half a century ago. This paper presents basics of the original method and proposes a number of its modifications that increase the accuracy and (or) speed of solving the SLEs. The advantage of MMC over other popular methods is shown, especially for the Southern Hemisphere of Earth.

APPLYING THE METHOD OF MAXIMUM CONTRIBUTIONS TO THE MAGNETOGRAM INVERSION TECHNIQUE

Fundamentals of the spherical harmonic analysis (SHA) of the geomagnetic field were created by Gauss. They acquired the classical Chapman — Schmidt form in the first half of the XXth century. The SHA method was actively developed for domestic geomagnetology by IZMIRAN, and then, since the start of the space age, by ISTP SB RAS, where SHA became the basis for a comprehensive method of MIT (magnetogram inversion technique). SHA solves the inverse problem of potential theory and calculates sources of geomagnetic field variations (GFV) - internal and external electric currents. The SHA algorithm forms a system of linear equations (SLE), which consists of 3K equations (three components of the geomagnetic field, K is the number of ground magnetic stations). Small changes in the left and (or) right side of such SLE can lead to a significant change in unknown variables. As a result, two consecutive instants of time with almost identical GFV are approximated by significantly different SHA coefficients. This contradicts both logic and real observations of the geomagnetic field. The inherent error of magnetometers, as well as the method for determining GFV, also entails the instability of SLE solution. To solve such SLEs optimally, the method of maximum contribution (MMC) was developed at ISTP SB RAS half a century ago. This paper presents basics of the original method and proposes a number of its modifications that increase the accuracy and (or) speed of solving the SLEs. The advantage of MMC over other popular methods is shown, especially for the Southern Hemisphere of Earth.

Inverse problem of potential theory

P. Novikov in 1938 has proved that if u 1 ( x ) = u 2 ( x ) for | x | > R , where R > 0 is a large number, u j ( x ) ≔ ∫ D j g 0 ( x , y ) d y , g 0 ( x , y ) ≔ 1 4 π | x − y | , and D j ⊂ R 3 , j = 1 , 2 , D j ⊂ B R , are bounded, connected, smooth domains, star-shaped with respect to a common point, then D 1 = D 2 . Here B R ≔ { x : | x | ≤ R } . Our basic results are: (a) the removal of the assumption about star-shapeness of D j , (b) a new approach to the problem, (c) the construction of counter-examples for a similar problem in which g 0 is replaced by g = e i k | x − y | 4 π | x − y | , where k > 0 is a fixed constant.

On mother body measures with algebraic Cauchy transform

Below we discuss the existence of a mother body measure for the exterior 1 inverse problem in potential theory in the complex plane. More exactly, we study the 2 question of representability almost everywhere (a.e.) in C of (a branch of) an irreducible 3

On an inverse problem in potential theory

The Newtonian potential of a Euclidean ball $B$ of $\mathbb R^n$ centered at $x\_0$ is proportional, outside $B$, to the Newtonian potential of a mass concentrated at $x\_0$. Vice-versa, as proved by Aharonov, Schiffer and Zalcman, if $D$ is a bounded open set in $\mathbb R^n$, containing $x\_0$, whose Newtonian potential is proportional, outside $D$, to the one of a mass concentrated at $x\_0$, then $D$ is a Euclidean ball with center $x\_0$. In this paper we generalize this last result to more general measures and domains.

The method of linear integral representations for solving 3D structural inverse problems of potential theory

The algorithm for the solution of a three-dimensional (3D) structural inverse problem in potential theory is described. The algorithm is based on the method of linear integral representations and on the method of extending compacts. The proposed approach is verified on model examples for an anomalous gravity field.

Convexity and the Exterior Inverse Problem of Potential Theory

Let Ω 1 and Ω 2 be bounded solid domains such that their associated volume potentials agree outside Ω 1 U Ω 2 . Under the assumption that one of the domains is convex, it is deduced that Ω 1 =Ω 2 .

Angular Distribution of Zeros of the Partial Sums of ez via the Solution of Inverse Logarithmic Potential Problem

We continue the work of Szegő [18] on describing the angular distribution of the zeros of the normalized partial sum sn(nz) of ez, where \(s _{n}(z):={\sum _{k=0} ^{n}}z ^k/k!\). We imbed this problem in some inverse problem of potential theory and prove a so-called Erdős-Turan-type theorem, which is of interest in itself.

Self-similar relaxation dynamics of a fluid wedge in a Hele-Shaw cell

Let the interface between two immiscible fluids in a Hele-Shaw cell have, at t = 0, a wedge shape. As a wedge is scale-free, the fluid relaxation dynamics are self-similar. We find the dynamic exponent of this self-similar flow and show that the interface shape is given by the solution of an unusual inverse problem of potential theory. We solve this problem analytically for an almost flat wedge, and numerically otherwise. The wedge solution is useful for analysis of pinch-off singularities.

Computer simulation of the inverse problem of electrocardiography: use of properties of harmonic functions

The problem of reconstructing the pattern of heart excitation from body surface potentials is simulated. The problem is well known as the inverse problem of electrocardiography and in a general case this problem has a non-unique solution. The relationship of the problem with the inverse problem of potential theory is shown. From this relationship a new excitation propagation model for the heart ventricles is developed. The model is based on a classical multidipole cardiogenerator, but is stable, flexible, and provides a unique solution for the inverse problem of electrocardiography.

The hunt property and unique solvability of the inverse problem of potential theory for one class of generalized Ukawa potentials

The hunt property and unique solvability of the inverse problem of potential theory for one class of generalized Ukawa potentials

Variational approach to three-dimensional inverse problems of potential theory

A method of regularization and solving problems of flow around bodies according to a desired distribution of the characteristics of the fields on the surface of the bodies themselves is described. Examples of solutions of such problems for potential flows are given, and a method of reducing inverse problems of the fluid mechanics of an effectively inviscid liquid to potential problems is presented.

Mapping EEG-potentials on the surface of the brain: a strategy for uncovering cortical sources.

This paper describes a uniform method for calculating the interpolation of scalp EEG potential distribution, the current source density (CSD), the cortical potential distribution (cortical mapping) and the CSD of the cortical potential distribution. It will be shown that interpolation and deblurring methods such as CSD or cortical mapping are not independent of the inverse problem in potential theory. Not only the resolution but also the accuracy of these techniques, especially those of deblurring, depend greatly on the spatial sampling rate (i.e., the number of electrodes). Using examples from simulated and real (64 channels) data it can be shown that the application of more than 100 EEG channels is not only favourable but necessary to guarantee a reasonable accuracy in the calculations of CSD or cortical mapping. Likewise, it can be shown that using more than 250 electrodes does not improve the resolution.

Reconstruction of the velocity distribution in conducting melts from induced magnetic field measurements

A possibility to reconstruct the velocity field in a conducting melt without using tracer particles or any mechanical contact to the melt is presented. To recover the flow distribution the liquid under study is placed in an external magnetic field and the induced magnetic field due to the currents appearing in such a liquid is measured. The problem of velocity field reconstruction from these measurement data is closely related to inverse problems of potential theory. It is shown that in contrast to the latter problem the reconstruction of the velocity field can be performed uniquely if there exists a possibility to measure the induced field for different applied field configurations. A stable solution of the corresponding integral equation (arising from a Poisson equation for the induced magnetic field), which is known as an ill-conditioned problem, is obtained by means of the method of singular value analysis. Demonstration results for the case of a two-dimensional flow pattern in an infinitely long pipe are presented.

Convexity and Uniqueness in an Inverse Problem of Potential Theory

Let ${\Omega _1}$ and ${\Omega _2}$ be two bounded domains in ${\mathbb {R}^n}$ whose intersection is convex. Suppose moreover that their volume potentials coincide in the complement of their union. Then ${\Omega _1} = {\Omega _2}$.

Stability for an Inverse Problem in Potential Theory

Stability for an Inverse Problem in Potential Theory

Stability for an inverse problem in potential theory

Let D D be a subdomain of a bounded domain Ω \Omega in R n {\mathbb {R}^n} . The conductivity coefficient of D D is a positive constant k ≠ 1 k \ne 1 and the conductivity of Ω ∖ D \Omega \backslash D is equal to 1 1 . For a given current density g g on ∂ Ω \partial \Omega , we compute the resulting potential u u and denote by f f the value of u u on ∂ Ω \partial \Omega . The general inverse problem is to estimate the location of D D from the known measurements of the voltage f f . If D h {D_h} is a family of domains for which the Hausdorff distance d ( D , D h ) d(D,{D_h}) equal to O ( h ) O(h) ( h h small), then the corresponding measurements f h {f_h} are O ( h ) O(h) close to f f . This paper is concerned with proving the inverse, that is, d ( D , D h ) ≤ 1 c ‖ f h − f ‖ d(D,{D_h}) \leq \frac {1}{c}\left \| {f_h} - f\right \| , c > 0 c > 0 ; the domains D D and D h {D_h} are assumed to be piecewise smooth. If n ≥ 3 n \geq 3 , we assume in proving the above result, that D h ⊃ D {D_h} \supset D (or D h ⊂ D {D_h} \subset D ) for all small h h . For n = 2 n = 2 this monotonicity condition is dropped, provided g g is appropriately chosen. The above stability estimate provides quantitative information on the location of D h {D_h} by means of f h {f_h} .