Inverse Gravimetry Problem Research Articles

The method of functional representations in the solution of inverse problems of gravimetry

The paper describes the method for solving the inverse problems of gravimetry based on the functional representations, which follow from the variational principles in the uniform metrics with respect to density models of the geological medium. The functional representations are obtained for both the linear problem (which study local density distributions) and nonlinear problem (which study a system of structural models). The explicit formulas for calculating density models are derived for the particular cases based on the introduced functional representations of the obtained solutions. In the general case, the converging iterative processes providing the solution for both the density distribution models and structural models are constructed. The relationship is established between the functional representations implementing the variational principle in the uniform metric and linear integral representations corresponding to the optimization in the quadratic norm, on one hand, and the other known density models, on the other hand.

On the solution of inverse problems of gravimetry by the modified method of S-approximations

Ways to solve the inverse linear problems of gravimetry and constructing linear transformations of the anomalous physical fields based on the modified method of S-approximations of the initial field element are discussed. The results of the mathematical experiment are presented.

A quadratic convergence yielding iterative method for the implementation of Lavrentiev regularization method for ill-posed equations

George and Elmahdy (2012), considered an iterative method which converges quadratically to the unique solution xαδ of the method of Lavrentiev regularization, i.e., F(x)+α(x-x0)=yδ, approximating the solution xˆ of the ill-posed problem F(x)=y where F:D(F)⊆X⟶X is a nonlinear monotone operator defined on a real Hilbert space X. The convergence analysis of the method was based on a majorizing sequence. In this paper we are concerned with the problem of expanding the applicability of the method considered by George and Elmahdy (2012) by weakening the restrictive conditions imposed on the radius of the convergence ball and also by weakening the popular Lipschitz-type hypotheses considered in earlier studies such as George and Elmahdy (2012), Mahale and Nair (2009), Mathe and Perverzev (2003), Nair and Ravishankar (2008), Semenova (2010) and Tautanhahn (2002). We show that the adaptive scheme considered by Perverzev and Schock (2005) for choosing the regularization parameter can be effectively used here for obtaining order optimal error estimate. In the concluding section the method is applied to numerical solution of the inverse gravimetry problem.

An improved fast local level set method for three-dimensional inverse gravimetry

We propose an improved fast local level set method for the inverse problem ofgravimetry by developing two novel algorithms: one is of linear complexitydesigned for computing the Frechet derivative of the nonlinear domain inverseproblem, and the other is designed for carrying out numerical continuationrapidly so as to obtain fictitious full measurement data from partialmeasurement. Since the Laplacian kernel is symmetric and translationally invariant, wedesign certain affine transformations to speed up the computational process inevaluating the Frechet derivative; since it decays rapidly away from diagonal,we carry out low-rank matrix approximation to reduce storage requirements. Theseproperties are eventually translated into an algorithm of linear complexity andlinear storage requirement for computing the derivative. Since the single layerdensity function, used in representing the potential, is smooth and periodic onan artificial hypersurface enclosing the target domain, the spectral expansionis allowed to approximate this density function, which eventually leads to rapidalgorithms for carrying out the numerical continuation in both 2-D and 3-Dcases. 2-D and 3-D numerical examples illustrate that this improved level-setmethod is capable of computing high-resolution inversions and handling 3-Dlarge-scale inverse gravimetry problems.

A derivative free iterative method for the implementation of Lavrentiev regularization method for ill-posed equations

In this work, we develop a derivative free iterative method for the implementation of Lavrentiev regularization for approximately solving the nonlinear ill-posed operator equation F(x) = y. Convergence analysis shows that the method converges quadratically. Apart from being totally free of derivatives, the method, under a general source condition provide an optimal order error estimate. We use the adaptive method introduced in Pereverzyev and Schock (SIAM J. Numer. Anal. 43, 2060---2076, 2005) for choosing the regularization parameter. In the concluding section the method is applied to numerical solution of the inverse gravimetry problem.

Hidden equivalence and efficiency of gravity data interpretation

Analytical properties of the solutions of the inverse problem of gravimetry are studied in the context of the approximative approach and method of linear integral representations. A new effect of the inheritance of specific analytical properties in the solutions, which is named the effect of hidden equivalence, is revealed and investigated. This effect significantly influences the interpretation informativity in the complex models of a medium, and it should be taken into account in the geological interpretation of gravity data. Hidden equivalence is studied for both linear and nonlinear inverse problems. As an example of a nonlinear problem, the inverse problem of structural gravimetry is analyzed. The correlation between the boundary equations and boundary values of the harmonic functions is demonstrated. Methods are suggested to allow for the effects that occur by expanding the approximative approach for complex conditions (the models of the media with spatially distributed parameters) by the dynamical and criterion principles.

An analysis of Lavrentiev regularization method and Newton type process for nonlinear ill-posed problems

In this paper we consider the Lavrentiev regularization method and a modified Newton method for obtaining stable approximate solution to nonlinear ill-posed operator equations F(x)=y where F:D(F)⊆X⟶X is a nonlinear monotone operator or F′(x0) is nonnegative selfadjoint operator defined on a real Hilbert space X. We assume that only a noisy data yδ∈X with ‖y-yδ‖⩽δ are available. Further we assume that Fréchet derivative F′ of F satisfies center-type Lipschitz condition. A priori choice of regularization parameter is presented. We proved that under a general source condition on x0-xˆ, the error ‖xˆ-xn,αδ‖ between the regularized approximation xn,αδ(x0,αδ≔x0) and the solution xˆ is of optimal order. In the concluding section the algorithm is applied to numerical solution of the inverse gravimetry problem.

Making use of a partial order in solving inverse problems

In many applications, the concepts of inequality and comparison play an essential role, and the nature of the objects under consideration is better described by means of partial order relations. To reflect this nature, the conventional problem statements in normed spaces have to be modified. There is a need to enrich the structure of the functional spaces employed. In this paper, we consider inverse problems in Banach lattices—functional spaces endowed with both norm and partial order. In this new problem statement, we are able to construct a linearly constrained set that contains all admissible approximate solutions given the approximate data, the approximation errors and prior restrictions that are available. We do not suppose the problem to be either well posed or ill posed since we believe that the concepts and techniques we describe here can be useful in both cases. The range of applications includes medical imaging (CT, PET, SPECT, photo- and thermoacoustics), geophysics (e.g., inverse gravimetry problems), engineering design and other inverse problems.

The Levenberg-Marquardt method and its modified versions for solving nonlinear equations with application to the inverse gravimetry problem

The Levenberg-Marquardt method and its modified versions are studied. Under some local conditions on the operator (in a neighborhood of a solution), strong and weak convergence of iterations is established with the solution error monotonically decreasing. The conditions are shown to be true for one class of nonlinear integral equations, in particular, for the structural gravimetry problem. Results of model numerical experiments for the inverse nonlinear gravimetry problem are presented.

Algorithms for solving inverse geophysical problems on parallel computing systems

For solving inverse gravimetry problems, efficient stable parallel algorithms based on iterative gradient methods are proposed. For solving systems of linear algebraic equations with block-tridiagonal matrices arising in geoelectrics problems, a parallel matrix sweep algorithm, a square root method, and a conjugate gradient method with preconditioner are proposed. The algorithms are implemented numerically on a parallel computing system of the Institute of Mathematics and Mechanics (PCS-IMM), NVIDIA graphics processors, and an Intel multi-core CPU with some new computing technologies. The parallel algorithms are incorporated into a system of remote computations entitled “Specialized Web-Portal for Solving Geophysical Problems on Multiprocessor Computers.” Some problems with “quasi-model” and real data are solved.

A three-dimensional inverse gravimetry problem for ice with snow caps

We propose a model for the gravitational field of a floating iceberg $D$ with snow on its top.The inverse problem of interest in geophysics is to find $D$ and snow thickness $g$ on itsknown (visible) top from remote measurements of derivatives of the gravitational potential.By modifying the Novikov's orthogonality method we prove uniqueness of recovering $D$ and $g$ forthe inverse problem. We design and test two algorithms for finding $D$ and $g$. One is based on a standardregularized minimization of a misfit functional. The second one applies the level set method to our problem.Numerical examples validate the theory and demonstrate effectiveness of the proposed algorithms.

Inverse problems of gravimetry as retrieval of reliable information under uncertainty

The guaranteed approach to the solution of inverse problems of gravimetry, which is fundamentally different from the common solution, is presented. Instead of providing single (optimum) estimates of the model parameters, whose quality is, generally, random, the interpretation in the suggested approach yields that volume of reliable information on disturbing objects, which is contained in the field measurements, together with the a priori constraints. The method of solving the inverse problem of gravimetry is developed within this approach, where, in contrast to the conventional approach, both the geometrical parameters of the geological bodies and the densities of their composing rocks are treated as unknowns. The generalized assembly algorithm suggested by V.N. Strakhov is proposed as a basic working tool to implement this approach. The results of testing this algorithm and the guaranteed approach itself on the model and practical examples are discussed.

A Fast Local Level Set Method for Inverse Gravimetry

AbstractWe propose a fast local level set method for the inverse problem of gravimetry. The theoretical foundation for our approach is based on the following uniqueness result: if an open set D is star-shaped or x3-convex with respect to its center of gravity, then its exterior potential uniquely determines the open set D. To achieve this purpose constructively, the first challenge is how to parametrize this open set D as its boundary may have a variety of possible shapes. To describe those different shapes we propose to use a level-set function to parametrize the unknown boundary of this open set. The second challenge is how to deal with the issue of partial data as gravimetric measurements are only made on a part of a given reference domain Ω. To overcome this difficulty we propose a linear numerical continuation approach based on the single layer representation to find potentials on the boundary of some artificial domain containing the unknown set D. The third challenge is how to speed up the level set inversion process. Based on some features of the underlying inverse gravimetry problem such as the potential density being constant inside the unknown domain, we propose a novel numerical approach which is able to take advantage of these features so that the computational speed is accelerated by an order of magnitude. We carry out numerical experiments for both two- and three-dimensional cases to demonstrate the effectiveness of the new algorithm.

Iterative processes of gradient type with applications to gravimetry and magnetometry inverse problems

Abstract The iterative processes of gradient type for nonlinear equations with differentiable operator satisfying a local condition in the neighborhood of solution are investigated. The theorems on weak and strong convergence of iterations constructed by these methods and their modified analogs are established. The inverse gravimetry problem is considered as the application of the developed methods: retrieval of the interface between the media with different constant densities. For stable solution of the nonlinear inverse magnetometry problem the additional regularization by the Tikhonov method is used and for approximation of the regularized solution one variant of the conjugate gradient method is applied. The numerical results for model and real gravitational and magnetic data are considered.

Three-dimensional deep density model of the Pautovaya Horst and its geological interpretation (Northeast Russia)

The quantitative interpretation of gravity anomalies reveals a concealed granitoid pluton within the core of the Pautovaya Horst (southeastern extremity of the Iniyali-Debin Synclinorium, northeast Russia). Using a new interpretative gravimetry method, a 3D density model is developed for the study area with its geological-petrological explanation. The model satisfies the necessary condition that the solution of inverse gravimetry problems provides the true solution.

On the use of the fitting method in solving inverse problems of gravimetry and magnetometry

An analytical approach to the gravity and magnetic data inversion by the fitting method is described. The problems of the uniqueness of the solution and the convergence and stability of the iterative process are considered.

Iteratively Regularized Gradient Method with a Posteriori Stopping Rule for 2D Inverse Gravimetry Problem

Iteratively Regularized Gradient Method with a Posteriori Stopping Rule for 2D Inverse Gravimetry Problem

Numerical solution of an inverse problem of gravimetry for a contact surface

An inverse problem of gravimetry for a contact surface is considered. Using the critical component method, shapes of a surface are determined numerically at a known depth.

CONTINUOUS ANALOG OF THE GAUSS–NEWTON METHOD

A Continuous Analog of discrete Gauss–Newton Method (CAGNM) for numerical solution of nonlinear problems is suggested. In order to avoid the ill-posed inversion of the Fréchet derivative operator, some regularization function is introduced. For the CAGNM, a convergence theorem is proved. The proposed method is illustrated by a numerical example in which a nonlinear inverse problem of gravimetry is considered. Based on the results of the numerical experiments, practical recommendations for the choice of the regularization function are given.

A numerical method for solving nonlinear ill-posed problems

A two-step iterative process for the numerical solution of nonlinear problems is suggested. In order to avoid the ill-posed inversion of the Fréchet derivative operator, some regularization parameter is introduced. A convergence theorem is proved. The proposed method is illustrated by a numerical example in which a nonlinear inverse problem of gravimetry is considered. Based on the results of the numerical experiments practical recommendations for the choice of the regularization parameter are given. Some other iterative schemes are considered.