Gravimetry Problem Research Articles

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.

Selection of processing parameters for solving the 3D inverse problem of gravity exploration by the discrete Fourier transform method in GIS INTEGRO

In GIS INTEGRO, a program using the discrete Fourier transform is created to solve the 3D inverse problem of gravimetry. The result of its work is a 3D distribution of the effective density. The program allows to use an extended parametrization of the form: fα,β(r,x) = f(r,zα)|z|–β. By changing the parameters α — "depth multiplier" — and β — "exponent at z"— one can obtain various equivalent distributions of effective densities and choose the most appropriate one based on a priori information. The experience of solving the 3D inverse problem by this method with different set values of these parameters allowed us to recommend the optimal values. As a criterion for choosing a solution from a variety of possible solutions, it is proposed to use a comparison of them with the data of seismic studies of the DDS and the CCP.

Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods

<p style="text-indent:20px;">We develop new efficient algorithms for a class of inverse problems of gravimetry to recover an anomalous volume mass distribution (measure) in the sense that we design fast local level-set methods to simultaneously reconstruct both unknown domain and varying density of the anomalous measure from modulus of gravity force rather than from gravity force itself. The equivalent-source principle of gravitational potential forces us to consider only measures of the form <inline-formula><tex-math id="M1">\begin{document}$ \mu = f\,\chi_{D} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> is a density function and <inline-formula><tex-math id="M3">\begin{document}$ D $\end{document}</tex-math></inline-formula> is a domain inside a closed set in <inline-formula><tex-math id="M4">\begin{document}$ \bf{R}^n $\end{document}</tex-math></inline-formula>. Accordingly, various constraints are imposed upon both the density function and the domain so that well-posedness theories can be developed for the corresponding inverse problems, such as the domain inverse problem, the density inverse problem, and the domain-density inverse problem. Starting from uniqueness theorems for the domain-density inverse problem, we derive a new gradient from the misfit functional to enforce the directional-independence constraint of the density function and we further introduce a new labeling function into the level-set method to enforce the geometrical constraint of the corresponding domain; consequently, we are able to recover simultaneously both unknown domain and varying density from given modulus of gravity force. Our fast level-set method is built upon localizing the level-set evolution around a narrow band near the zero level-set and upon accelerating numerical modeling by novel low-rank matrix multiplication. Numerical results demonstrate that uniqueness theorems are crucial for solving the inverse problem of gravimetry and will be impactful on gravity prospecting. To the best of our knowledge, our inversion algorithm is the first of such for the domain-density inverse problem since it is based upon the conditional well-posedness theory of the inverse problem.

Regularized gradient algorithms for solving the nonlinear gravimetry problem for the multilayered medium

We present new numerical algorithms for solving the structural inverse gravimetry problem for the case of multiple surfaces. The inverse problem of finding the multiple surfaces that divide the constant density layers is an ill‐posed one described by a nonlinear integral equation of the first kind. To solve it, it is necessary to apply the regularization ideas. The new regularized variants of the gradient type methods with the weighting factors are constructed, namely, the steepest descent and conjugate gradient method. We suggest the empirical rule for choosing the regularization parameters. On the basis of the constructed methods, we elaborate the parallel algorithms and implement them in the multicore CPU using the OpenMP technology. A set of experiments with the disturbed data is performed to test the gradient algorithms and study performance of the developed code. For the test problems with quasi‐real data, these new regularized algorithms increase the accuracy and speed up computation in comparison with the unregularized ones. By using the 8‐core CPU, we achieve the speedup of 8 times.

Stability and 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 boundary measurements of gravitational force outside Ω. We found that about five parameters of the unknown 𝐷 can be stably determined given data noise in practical situations. An ellipse is uniquely determined by five parameters. We prove uniqueness and stability of recovering an ellipse for the inverse problem from minimal amount of data which are the gravitational force at three boundary points. In the proofs, we derive and use simple systems of linear and nonlinear algebraic equations for natural parameters of an ellipse. To illustrate the technique, we use these equations in numerical examples with various location of measurements points on ∂ ⁡ Ω \partial\Omega . Similarly, a rectangular 𝐷 is considered. We consider the problem in the plane as a model for the three-dimensional problem due to simplicity.

Finite-dimensional iteratively regularized processes with an a posteriori stopping for solving irregular nonlinear operator equations

Abstract We construct and study a class of numerically implementable iteratively regularized Gauss–Newton type methods for approximate solution of irregular nonlinear operator equations in Hilbert space. The methods include a general finite-dimensional approximation for equations under consideration and cover the projection, collocation and quadrature discretization schemes. Using an a posteriori stopping rule for the iterative processes and the standard source condition on the solution, we establish accuracy estimates for the approximations generated by the methods. We also investigate projected versions of the processes which take into account a priori information about a convex compact containing the solution. An iteratively regularized quadrature process is applied to an inverse 2D problem of gravimetry.

Determination of Global Density Inhomogeneities and Stresses Inside the Moon

The possible distribution of density and stress anomalies in the interior of the Moon is determined. For this purpose, we have developed a new solution of the incorrect inverse gravimetry problem developed by the authors, tested for the Earth and Mars. The idea of this solution is that the isostatic compensation depths are uniquely determined for the harmonics of the expansion of the relief heights in spherical functions based only on cosmic data on the gravitational field and the relief of the planet. We have shown that possible isostatic relief compensation is carried out in the depth range of 0–530 km. The most probable compensation depths in this range were determined and the distribution of compensation masses at these depths was found. For the remaining harmonics, two other compensation options were chosen: (1) compensation is carried out at two levels, the first of which is the mean depth of the crust of 2.7 km, corresponding to a layer of 0–10 km; the possible depths for the second level were determined from an analysis of the results obtained for isostatically compensated harmonics; (2) uncompensated harmonics of the relief lead to vertical stresses in the Moon’s lithosphere reaching the compressive strength (16 MPa) and tensile strength (15 MPa).

Application of Neural Networks in Nonlinear Inverse Problems of Geophysics

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.

METHODS FOR SOLVING FREDHOLM INTEGRAL EQUATIONS

The article deals with integral equations that are widely used in various sections of physics (theory of waves on the surface of liquids, quantum mechanics, problems of spectroscopy, crystallography, acoustics, analysis and diagnostics of plasma, etc.), Geophysics (problems of gravimetry, kinematic problems of seismics), mechanics (vibrations of structures), etc. When the physics introduced aftereffect, it is not enough ordinary differential equations or partial differential equations, otherwise the initial data would determine the future state. To take into account the continuous sequence of previous States, we need to use integral and integro-differential equations, where the sign of the integral appears functions of parameters that characterize the system, which depend on time for some period preceding the moment under consideration. In this article we have considered the solution of Fredholm integral equations of the second kind by the method of successive approximations and the method of iterated nuclei.

Аналитическое моделирование матрично-векторного произведения на многоядерных процессорах

Many areas of study and their applications such as machine learning, data mining, quantum chemistry, mathematical physics, and high-performance computing require effective implementation of matrix-vector multiplication. In this work, we present an overview of the algorithm for automatic optimization of matrix vector multiplication. The algorithm models computations on the hypothetical multicore processor, which is introduced by the authors, and applies polyhedral modeling. In comparison with methods, which rely on manual tuning and auto-tuning, the algorithm can be utilized when the execution time is a critical factor and the target platform is not accessible. We apply the approach to optimize an implementation of the solution of the inverse gravimetry problem of finding an interface between the layers, which uses the iterative Levenberg-Marquardt method. The performance of the obtained implementation is compared with the performances produced by implementations, which are based on MKL, BLIS, and OpenBLAS. Results of the experimental evaluation show that the considered approach is comparable with the approaches, which are created on target architectures using manual tuning.

Efficient numerical algorithm for solving the gravimetry problem of finding a lateral density in a layer: Parallel implementation

The paper is devoted to developing the new time‐ and memory‐efficient algorithm BiCGSTABmem for solving the inverse gravimetry problem of determination of a variable density in a layer using the gravitational data. The problem is in solving the linear Fredholm integral equation of the first kind. After discretization of the domain and approximation of the integral operator, this problem is reduced to solving a large system of linear algebraic equations.It is shown that the matrix of coefficients is the Toeplitz‐block‐Toeplitz one in the case of the horizontal layer. For calculating and storing the elements of this matrix, we construct an efficient method, which significantly reduces the required memory and time. For the case of the curvilinear layer, we construct a method for approximating the parts of the matrix by a Toeplitz‐block‐Toeplitz one. This allows us to exploit the same efficient method for storing and processing the coefficient matrix in the case of a curvilinear layer. To solve the system of linear equations, we constructed the parallel algorithm on the basis of the stabilized biconjugated gradient method with using the Toeplitz‐block‐Toeplitz structure of the matrix.We implemented the BiCGSTAB and BiCGSTABmem algorithms for the Uran cluster supercomputer using the hybrid MPI + OpenMP technology. A model problem with synthetic data was solved for a large grid. It was shown that the new BiCGSTABmem algorithm reduces the computation time in comparison with the BiCGSTAB. Scalability of the parallel algorithm was studied.

On the simultaneous restoration of the density and the surface equation in the inverse gravimetry problem for a contact surface

On the simultaneous restoration of the density and the surface equation in the inverse gravimetry problem for a contact surface

Морфология Горихинского гранитного массива с помощью методов математической статистики

The Gorikhinsky granite massif is located in the marginal part of the Khentei highlands (Mongolian-Transbaikalian folded system, Central Mongolia). The massif is composed of coarse-grained weak-porphyry leucocratic granites, granite porphyry, and pegmatites. In the rocks of the massif, rock crystal deposits have been discovered. In certain pegmatite bodies, in addition to smoky quartz and morion crystals, the study has found the presence of fluorite, topaz, biotite, tourmaline, beryl, garnet, and allanite. The accessory minerals are cyrtolite, xenotime, zircon, sphene, rutile, apatite, cassiterite, etc. Thus, the study of the geological structure of the above object is of interest not only from a scientific but also from a practical point of view. The research data have become the basis for setting areal gravimetric work to determine the features of the massif’s geological structure. The work has been performed by the standard procedure using Scintrex Autograv CG-5 gravimeters. The height of the observation points has been determined by Trimble 5700 GPS devices. The RMS accuracy of the gravitational anomalies calculation as well as the density of the observation network meet the technical requirements for 1:100000 scale gravimetric surveys. Using the results of the gravity survey, a map of gravity anomalies in Bouguer has been compiled. Based on the interpretation of the map, with the use of the methods of mathematical statistics, quantitative characteristics of the shape and size of the body have been defined and a three-dimensional model of the massif has been built. For the purpose of the transformation and inversion of the gravitational field, the variance and regression methods of mathematical analysis have been used. The maximum thickness of the granites is 3.5 km; the area of the earth's surface projection is twice that of the granite outcrop. The obtained estimates are confirmed by the results of solving the inverse problem of gravimetry using the method of model selection. The article concludes that the use of the regression and variance analysis methods of mathematical statistics in the studies of the morphology of the intrusive bodies has proved most effective. The information on the shape and size of the massif will be used in concluding on the prospects of the massif’s metallogenic trend.

Crustal structure of the Ulaanbaatar region, Mongolia according to gravimetric data

E stablish ing relationship s between the regional seismicity and the depth geometry and composition s of large -size density inhomogeneities of the upper crust is one of the ways to determine the p arameters of mean long-term seismicity . The origin of geological bodies and mechanism s of penetration of intrusive bodies in to the upper crust are detectable from the shape s and size s of such bodies. K nowledge o n the shape s and size s of large intrusive bodies is important for prospecting and exploration of mineral resources. Based on t he medium - scale areal gravimetric survey data, a regional map of Bouguer gravity anomalies was constructed for the Ulaanbaatar region , Central Mongolia. I nterpret ed gravimetric data from the sol utions of the inverse problem of gravimetry based on the body - shape selection were used to develop the 3D models showing the largest crustal density inhomogeneities of the study area – the Tolskaya ( Ula a n ba at a r) basin , Nalaiha depression and Bogdoulin granite massif. It is confirmed that the basement of these depression s has a block structure. The basement of the Tolskaya basin includes two depressions , the western and eastern ones . In the western depression, t he sediment thickness is 150 m . It amounts to 400 m i n the eastern depression at the intersection of the Selbin and Tolsk aya deep fault systems. The i nformation on the shape s and thicknesses of the sediment bodies is used in the seismic studies aimed at obtaining a more precise assessment of seismic hazard of the Ul a an ba at a r city and the region . In the Nalai ha depression, the sediments are 8 00 m thick . The morphology of the basement and the thickness of sediments in this depression can be used to update the mining prospects of the Nalaiha coal deposit. The maximum thickness of granitoids in the Bogdoulin massif is 9.0 km . T he area of its projection to the ground surface is twice as big as the area of the outcrops . Our research results can be useful for assessing the metallogenic specialization of granitoids, which indicators in relation to some minerals are related to the specific features o f the Bogdoulinsky massif . Such indicators have been discovered by the geological, geophysical and geochemical methods . Our study shows that t he geophysical methods (in particular, gravimetr y ) can considerably expand the knowledge of the morphology of geological objects at depth and facilitate reaching a new level in structural analysis, wh ich is critical for develop ing new ideas to clarify the geological history and tectonic conditions of the formation of the Mongolia-Siberian region.

A 2D Forward Gravimetry Problem for a Polygon with Parabolic Density

Abstract—This article describes the formula of gravity of a polygon with a density varying according to a parabolic law (2D model). Calculation of gravity according to this formula can be made for both the external and internal area.

Third-order derivative-free methods in Banach spaces for nonlinear ill-posed equations

We develop three third order derivative-free iterative methods to solve the nonlinear ill-posed oprerator equation $$F(x)=f$$ approximately. The methods involve two steps and are free of derivatives. Convergence analysis shows that these methods converge cubically. The adaptive scheme introduced in Pereverzyev and Schock (SIAM J Numer Anal 43(5):2060–2076, 2005) has been employed to choose regularization parameter. These methods are applied to the inverse gravimetry problem to validate our developed results.

Mathematical problems of gravimetry and its applications

Abstract. Gravimetry is associated with analysis of the gravitational field. The gravitational field is characterized by its potential. This is described by the Poisson equation, the right side of which includes the density of the environment. There exists direct and inverse problems of gravimetry. Direct gravimetry problems involve the determination of the potential of the gravitational field in a given region. The inverse problems of gravimetry imply the restoration of the structure of a given area from the results of measuring the characteristics of the gravitational field. Such studies are needed to assess on the basis of gravimetric geodynamic events occurring in oil and gas fields. The relevance of such research is necessary, because with prolonged development of the oil and gas fields, negative consequences may occur. This paper discusses some of the features of direct and inverse gravimetry problems. A description of the mathematical model of the processes under consideration is given. Different direct and inverse gravimetry problems are posed. Describes the methods of its solving. Based on the analysis of the results of a computer experiment, appropriate conclusions are made.

Two-dimensional Kalman filter approach to airborne vector gravimetry

Abstract The paper presents a new approach to the airborne vector gravimetry problem. The idea of the approach is to take into account spatial correlation of the gravity field to improve observability of horizontal components of the gravity disturbance vector (GDV). We consider the GDV determination problem given airborne data at a set of parallel survey lines assuming that lines are flown in the same direction at a constant height above the reference ellipsoid. We use a 2-D random field model for the gravity field at the flight height. The random field is governed by two autoregressive equations (one in the direction along the lines, the other across the lines). Then we pose the estimation problem simultaneously for the GDV horizontal components and systematic errors of an inertial navigation system at all the lines simultaneously. The developed estimation algorithm is based on 2D Kalman filtering and smoothing techniques. Numerical results obtained from simulated data processing showed improved accuracy of the gravity horizontal component determination.