Gravimetry Problem Research Articles

РЕШЕНИЕ ОБРАТНЫХ ЗАДАЧ ГРАВИМЕТРИИ С ПОМОЩЬЮ МЕТОДОВ МАТЕМАТИЧЕСКОЙ СТАТИСТИКИ

The purpose of this article is to study the possibilities of mathematical statistics methods under the quantitative interpretation of gravity anomalies. The study uses dispersion and regression methods of mathematical analysis for gravitation field transformation and inversion on example of the Bogdoulin granitoid massif (Mongolia) with the abnormally high content of rare and non-ferrous metals (Sn, Zn, W). As a result, quantitative characteristics of the massif thickness have been obtained as well as the data on the morphological features of its shape. The maximum thickness of granitoids is 12 km. The obtained estimates are confirmed by the results of solving the inverse problem of gravimetry by the method of model selection. Therefore, the application of methods of mathematical statistics for determining intrusive body morphology allows to obtain petrophysical models of geological objects coinciding with the results of quantitative interpretation performed with the use of the program of semi-automatic 3D-Markquardt-inversion of the gravitational field.

A greedy algorithm for nonlinear inverse problems with an application to nonlinear inverse gravimetry

Based on the Regularized Functional Matching Pursuit (RFMP) algorithm for linear inverse problems, we present an analogous iterative greedy algorithm for nonlinear inverse problems, called RFMP_NL. In comparison to established methods for nonlinear inverse problems, the algorithm is able to combine very diverse types of basis functions, for example, localized and global functions. This is important, in particular, in geoscientific applications, where global structures have to be distinguished from local anomalies. Furthermore, in contrast to other methods, the algorithm does not require the solution of large linear systems. We apply the RFMP_NL to the nonlinear inverse problem of gravimetry, where gravitational data are inverted for the shape of the surface or inner layer boundaries of planetary bodies. This inverse problem is described by a nonlinear integral operator, for which we additionally provide the Frechet derivative. Finally, we present two synthetic numerical examples to show that it is beneficial to apply the presented method to inverse gravimetric problems.

Two-Stage Method of Construction of Regularizing Algorithms for Nonlinear Ill-Posed Problems

For an equation with a nonlinear differentiable operator acting in a Hilbert space, we study a two-stage method of construction of a regularizing algorithm. First, we use the Lavrentiev regularization scheme. Then we apply to the regularized equation either Newton’s method or nonlinear analogs of α-processes: the minimum error method, the minimum residual method, and the steepest descent method. For these processes, we establish the linear convergence rate and the Fejer property of iterations. Two cases are considered: when the operator of the problem is monotone and when the operator is finite-dimensional and its derivative has nonnegative spectrum. For the two-stage method with a monotone operator, we give an error bound, which has optimal order on the class of sourcewise representable solutions. In the second case, the error of the method is estimated by means of the residual. The proposed methods and their modified analogs are implemented numerically for three-dimensional inverse problems of gravimetry and magnetometry. The results of the numerical experiment are discussed.

On Solving the Forward Problem of Gravimetry in Curvilinear and Cartesian Coordinates: Krasovskii’s Ellipsoid and Plane Modeling

Correcting the effects of the sphericity of the Earth in the results of the interpretation of gravimetric data is a topical issue in modern gravimetry. Estimating the error of the gravity field calculations due to the replacement of the spherical Earth model by the plane model is an important part of this problem. In this paper, a method is proposed for transforming the plane density models into spherical ones and vice versa. Algorithms for calculating the vertical component of gravity field for both model types are presented. For two extensive plane models of the Earth’s density, their transformation into spherical models is carried out and the resulting gravity fields are compared. The relative root mean square residuals between the fields calculated with this replacement are at most 5%.

Inverse gravimetry: background material and multiscale mollifier approaches

This paper represents an extended version of the publications Freeden (in: Freeden, Nashed, Sonar (eds) Handbook of Geomathematics, 2nd edn, vol 1, Springer, New York, pp 3–78, 2015) and Freeden and Nashed (in: Freeden, Nashed (eds) Handbook of Mathematical Geodesy, Geosystems Mathematics, Birkhauser, Basel, pp 641–685, 2018c). It deals with the ill-posed problem of transferring input gravitational potential information in the form of Newtonian volume integral values to geological output characteristics of the density contrast function. Some essential properties of the Newton volume integral are recapitulated. Different methodologies of the resolution of the inverse gravimetry problem and their numerical implementations are examined including their dependence on the data source. Three types of input information may be distinguished, namely internal (borehole), terrestrial (surface), and/or external (spaceborne) gravitational data sets. Singular integral theory based inversion of the Newtonian integral equation such as a Haar-type solution is handled in a multiscale framework to decorrelate specific geological signal signatures with respect to inherently given features. Reproducing kernel Hilbert space regularization techniques are studied (together with their transition to certain mollified variants) to provide geological contrast density distributions by “downward continuation” from terrestrial and/or spaceborne data. Numerically, reproducing kernel Hilbert space spline solutions are formulated in terms of Gaussian approximating sums for use of gravimeter data systems.

Pareto-optimal solutions of the inverse problem of gravimetry with indeterminate a priori information

The paper discusses theoretical aspects of solving the nonlinear inverse problem of gravimetry with uncertainty of a priori information. The a priori information is described by fuzzy sets. Special-purpose geophysical problem with uncertain a priori information is transformed into a multi-objective optimization problem. One of the criteria is the membership function of a fuzzy set of possible solutions. Solution of the problem is a set of Pareto-optimal solutions, which is constructed in the parametric space applying a three-step search algorithm. The advantage of the proposed method is that it provides a possibility of including the wide range of non-probabilistic a priori information in the inversion procedure and can be applied to the solution of highly nonlinear problems. This reduces the number of direct computing problems by selective modeling of sample points in the parametric space.A test example has been given of the algorithm applied to the inverse problem of gravimetry for a single contact surface.

Convergence of Tikhonov regularization for solving ill--posed operator equations with solutions defined on surfaces

We study Tikhonov regularization for solving ill-posed operator equations where the solutions are functions defined on surfaces. One contribution of this paper is an error analysis of Tikhonov regularization which takes into account perturbations of the surfaces, in particular when the surfaces are approximated by spline surfaces. Another contribution is that we highlight the analysis of regularization for functions with range in vector bundles over surfaces. We also present some practical applications, such as an inverse problem of gravimetry and an imaging problem for denoising vector fields on surfaces, and show the numerical verification.

On the new approach to solving the inverse problem of gravimetry

The results of the studies within the new approach to solving the inverse problem of gravimetry are considered. This approach consists in direct (analytical) continuation of the anomalous gravitational field specified on the Earth’s surface into the lower half-space with the use of the method of discrete approximations. The solution of the problem of analytical continuation is demonstrated by the model example. In the solution of the problem of analytical continuation, the developed algorithms and computer programs were implemented in two program packages which are used both in the model computations and in practice.

A level-set method for imaging salt structures using gravity data

We have developed a level-set method for the inverse gravimetry problem of imaging salt structures with density contrast reversal. Under such a circumstance, a part of the salt structure contributes two completely opposite anomalies that counteract with each other, making it unobservable to the gravity data. As a consequence, this amplifies the inherent nonuniqueness of the inverse gravimetry problem so that it is much more challenging to recover the whole salt structure from the gravity data. To alleviate the severe nonuniqueness, it is reasonable to assume that the density contrast between the salt structure and the surrounding sedimentary host depends upon the depth only and is known a priori. Consequently, the original inverse gravity problem reduces to a domain inverse problem, where the supporting domain of the salt body becomes the only unknown. We have used a level-set function to parametrize the boundary of the salt body so that we reformulated the domain inverse problem into a nonlinear optimization problem for the level-set function, which was further solved for by a gradient descent method. Both 2D and 3D experiments on the SEG/EAGE salt model were carried out to demonstrate the effectiveness and efficiency of the new method. The algorithm was able to recover dipping flanks of the salt model, and it only took 40 min in a 2.5 GHz CPU to invert for a 3D model of 97,000 unknowns.

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.

Обратные линейные задачи гравиметрии и магнитометрии с итерационными формулами аналогов фильтров Винера-Калмана

Усовершенствованы итерационные методы решения обратных линейных задач гравиметрии и магнитометрии на основе аналогов фильтров Винера–Калмана для двух интерпретационных моделей, использующих разделение карты поля на два массива четных и нечетных профилей. Приведены новые итерационные формулы, обеспечивающие, в отличие от известных методов, устойчивое и геологически содержательное решение с любых начальных условий без предварительного решения другими методами. Это дает два близких решения, независимых от решений обратной задачи другими методами. Приведены примеры практической реализации метода для магнитометрии в рудном районе на Украинском щите.

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.

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.

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.

How to solve systems of linear algebraic equations with symmetrical positive semidefinite matrices in the problems of gravimetry and magnetometry

Highly efficient new methods for solving the systems of linear algebraic equations with symmetrical positive semidefinite matrices are considered.