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.     

Inverse problems of gravity prospecting as a decision-making problem under uncertainty and risk

A new class of algorithms for solving the inverse problems of gravity prospecting is considered. The best interpretation is selected from the set Q of the admissible versions by the optimality criteria that are borrowed from the solution-making theory and adapted for the geophysical problems. The concept of retrieving the information about the sources of gravity anomalies, which treats the result of the interpretation as a set of locally optimal solutions of the inverse problem but not as a single globally optimal solution is discussed. The locally optimal solutions of the inverse problem are sort of singularity points of set Q. They are preferable to the other admissible solutions by a certain criterion formulated in terms of the geologically important information about the anomalous bodies. The admissible versions of the interpretation of the gravimetry data that meet the criteria of the decision-making theory are the primary candidates for the singularity points. The results of the numerical calculations are presented. The set of the admissible solutions from which the locally optimal versions of interpretation are selected is formed by the modifications of the assembly method developed by V.N. Strakhov.     

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.     

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.     

INTERPRETATION OF REFRACTED-DIFFRACTED WAVES DUE TO DYKES*

This paper gives an interpretation of a specific type of refracted-diffracted waves. The solution is supported by the good accord of data in seismic and magnetic interpretation. Beyond solving the problem of the phenomenon, the interpretation of such events shows that in the exploration of dykes and intruded formations the refracted-diffracted waves can—under favourable conditions—also render some useful information.     

Minimax approach to inverse problems of geophysics

A new approach is suggested for solving the inverse problems that arise in the different fields of applied geophysics (gravity, magnetic, and electrical prospecting, geothermy) and require assessing the spatial region occupied by the anomaly-generating masses in the presence of different types of a priori information. The interpretation which provides the maximum guaranteed proximity of the model field sources to the real perturbing object is treated as the best interpretation. In some fields of science (game theory, economics, operations research), the decision-making principle that lies in minimizing the probable losses which cannot be prevented if the situation develops by the worst-case scenario is referred to as minimax. The minimax criterion of choice is interesting as, instead of being confined to the indirect (and sometimes doubtful) signs of the “optimal” solution, it relies on the actual properties of the information in the results of a particular interpretation. In the hierarchy of the approaches to the solution of the inverse problems of geophysics ordered by the volume and quality of the retrieved information about the sources of the field, the minimax approach should take special place.     

基于GPU并行的重力、重力梯度三维正演快速计算及反演策略

利用NVIDIA CUDA编程平台,实现了基于GPU并行的重力、重力梯度三维快速正演计算方法.采用当前在重力数据约束反演或联合反演中流行的物性模型(密度大小不同、规则排列的长方体单元)作为地下剖分单元,对任意三维复杂模型体均可用很多物性模型进行组合近似,利用解析方法计算出所有物性模型在计算点的异常值并累加求和,得到整个模型体在某一计算点引起的重力(或重力梯度)值.针对精细的复杂模型体产生的问题,采用GPU并行计算技术,主要包括线程有效索引与优化的并行归约技术进行高效计算.在显卡型号为NVIDIA Quadro 2000相对于单线程CPU程序,重力和重力梯度U_xx、U_xy正演计算可以分别达到60与50倍的加速.本文还讨论了GPU并行计算在两种反演方法中的策略,为快速三维反演技术提供了借鉴.     

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.     

Increased resolution of subsurface parameters from 1D magnetotelluric modeling

In electric and electromagnetic techniques, it is well known that the principle of equivalence poses a problem in the interpretation of subsurface layers. This means the inversion problem can provide the conductivity-thickness product more confidently than the individual parameters — conductivity and thickness — separately. The principle of equivalence corresponds to the middle layer in a three-layer earth structure. In order to resolve this problem, we have touched upon the different formulae of apparent resistivity proposed by earlier workers considering the real and imaginary parts of the impedance tensor and designed a new formula to compute apparent resistivity for different models. We observed that the application of our new formula for apparent resistivity using the combination of real and imaginary parts of the impedance has a better resolution as compared to earlier conventional formulae of apparent resistivity. These results have been demonstrated through both forward and inverse modeling schemes.     

Solution of the Inverse Problem of Gravimetry for a Spherical Planetary Body Using the Decomposition of the Interior Potential into a Series of Polyharmonic Functions

The interior potential of a spherical planetary body (and the corresponding density distribution) is expressed as the sum of two parts, the first given uniquely by the external gravity field and the surface density and the second depending on an arbitrary function. The first part of the potential (density distribution) is shown to be a 3-harmonic (biharmonic) function, while the second part can be expressed as a series whose i-th term (for i 0) is an (i + 4)-harmonic ((i + 3)-harmonic) function. From this general solution a single solution is then chosen: this is done by imposing certain natural conditions on it, among others that this particular solution is an n-harmonic function for n as small as possible. The paper explains shortly this method of solving the inverse problem of gravimetry; details are presented in Pohánka (1993).     

Simulation of the Rayleigh waves in the proximity of the scattering velocity heterogeneities. Exploring the capabilities of the microseismic sounding method

The interaction between the fundamental mode surface Rayleigh waves and the buried heterogeneities with various sizes and different velocity contrasts was studied on base numerical simulation. The field of surface oscillations in the proximity of the scattering heterogeneities was computed as a function of frequency. The synthetic seismograms were used for numerical simulation of the microseismic sounding technology proposed earlier, implying that the solution of the inverse problem for the structure of the medium containing inclusions can be derived from the information contained in the ambient microseismic field. It is assumed that the depth of the layer to be reconstructed is linked with the frequency of the microseisms by a simple relation with the help of a numerical coefficient equal to 0.4–0.5. The combined results of the simulation of a direct problem together with the simple inverse problem solution show that the microseismic sounding technique ensures adequate estimation of the medium structure. Previously, the technology was based on the experimental data only and was phenomenological in character. Some relations between the velocity parameters of the original model heterogeneities and their reconstructed images were also studied.     

Double-Sided Estimates of Inclusion Type for Localizing and Detailing the Location of the Gravitational Field Sources

Izvestiya, Physics of the Solid Earth - Abstract—The effective solution to the problem of the most complete extraction of reliable information about a geological object from gravimetric...     

Comparison between gradient based UCODE_2005 and the ensemble Kalman Filter for transient groundwater flow inverse modeling

Gradient based UCODE_2005 and data assimilation based on the Ensemble Kalman Filter(EnKF) are two different inverse methods. A synthetic two-dimensional flow case with four no-flow boundaries is used to compare the UCODE_2005 with the Ensemble Kalman Filter(EnKF) for their efficiency to inversely calculate and calibrate a hydraulic conductivity field based on hydraulic head data. A zonal, random heterogeneous conductivity field is calibrated by assimilating the time series of heads observed in monitoring wells. The study results indicate that the two inverse methods, UCODE_2005 and EnKF, could be used to calibrate the hydraulic conductivity field to a certain degree. More available observations and information about the conductivity field, more accurate inverse results will be obtained for the UCODE_2005. On the other hand, for a realistic zonal heterogeneous hydraulic conductivity field, EnKF can only efficiently determine the hydraulic conductivity field at the first several assimilated time steps. The results obtained by the UCODE_2005 look better than those by the EnKF. This is possibly due to the fact that the UCODE_2005 uses observed head data at every time step, while EnKF can only use observed heads at first several steps due to the filter divergence problem.     

Pitfalls in Nonlinear Inversion

— We discuss and illustrate graphically with simple 2-D problems, four common pitfalls in geophysical nonlinear inversion. The first one establishes that the Lagrange multiplier, used to incorporate a priori information in the geophysical inverse problem, should be the largest value still compatible with a reasonable data fitting. This procedure should be used only when the interpreter is sure about the importance of the a priori information used to stabilize the inverse problem relative to the geophysical observations. Because this is rarely the case, the user should use the smallest Lagrange multiplier still producing stable solutions. The second pitfall is an attempt to automatically estimate the Lagrange multiplier by decreasing it along the iterative process used to solve the nonlinear optimization problem. Consequently, at the last iteration, the Lagrange multiplier may be so small that the problem may become ill-posed and any computed solution in this case is meaningless. The third pitfall is related to the incorporation of a priori information by a technique known as “Jumping.” This formulation, from the viewpoint of the class of Acceptable Gradient Methods, is incomplete and may lead to a premature halt in the iteration, and, consequently, to solutions far from the true one. Finally, the fourth pitfall is an inadequate convergence criterion which stops the iteration when the data misfit drops just below the noise level, irrespective of the fact that the functional to be minimized may not have attained its minimum. This means that the a priori information has not been completely incorporated, so that this stopping criterion partially neutralizes the effect of the stabilizing functional, and opens the possibility of obtaining unstable, meaningless estimates.     

Combined Method of <Emphasis Type="Italic">F</Emphasis>-, <Emphasis Type="Italic">S</Emphasis>-, and <Emphasis Type="Italic">R</Emphasis>-Approximations of Increased Dimensionality in Solving the Problems of Geophysics and Geomorphology

The interrelation between different variants of the method of linear integral representations in the spaces of an arbitrary dimension is considered. The combined approximations of the topography and geopotential fields allows the selection of the optimal parameters of the method in solving a wide range of inverse problems in geophysics and geomorphology, as well as a most thorough use of the a priori information about the elevations and elements of anomalous fields. A method for numerically solving an inverse problem on finding the equivalent, in terms of the external field, mass distributions in the ordinary three-dimensional (3D) space and in the four-dimensional (4D) space is described.     

Combined method of <Emphasis Type="Italic">F</Emphasis>-, <Emphasis Type="Italic">S</Emphasis>-, and <Emphasis Type="Italic">R</Emphasis>-approximations in solving the problems of geophysics and geomorphology

The interrelation between different modifications of the method of linear integral representation is studied. Combined approximations of the topography and geopotential fields enable more refined tuning of the method in solving inverse problems of geophysics and geomorphology and provide a more complete allowance for the a priori information about the surface elevation data and elements of anomalous fields. A technique for finding the numerical solution for the inverse problem for determining the mass distributions equivalent in terms of the external field is presented. The results of the mathematical experiment are discussed.     

用于速度图象重建的层析成象法

本文提出了一种用于速度图象重建的层析成象法。特别注意到地震学上的ST与医学上的CT的不尽相同之处,并处理了由此引起的困难。与速度图象重建有关的正问题的分析表明,不同频率范围的资料其分辨能力是不同的,因此,区域地震与远震资料的解释应予注意。本文有关速度图象重建的讨论不仅涉及到与成象方法有关的反演理论和方法,还给出了进行数值计算的内存量和运算量的分析。关于重建图象的质量评价,还给出了反演解的可靠性分析,为分辨特性的描述提供了一种既简便又形象直观的方案。 本文的方法与现行主要方法的比较表明:1.本文的方法允许存在速度间断面,有利于揭示不同构造区域的地壳厚度的差异;2.在常用的方法中,作为正问题假定的常速度块同反演结果解释时非常速度块之间不协调。由于我们把给定网格点的速度值的插值函数作为速度的空间函数,这一困难已被解决;3.相对于Aki等的一般方法,本文的内存量节省约一个量级、运算量减少约一半;4.相对于ART类方法本身不能给出成象结果的可靠性分析,本文的方法则具有显著的优点。 作者在分析了天然地震资料的状况之后相信,对充分利用我国地震台网的现有观测资料,本文的方法是有效的。     

Trefftz method for the solution of three-dimensional forward and inverse problems of geoelectrics

The Trefftz method is applied to the numerical solution of the three-dimensional (3D) forward problem for an electromagnetic field harmonically oscillating with time in a 3D environment, which ensures the possibility to solve a 3D inverse problem. Here, the known idea of simultaneous joint interpretation of the tangential components of electric and magnetic fields measured on the Earth's surface, which was suggested by A.N. Tikhonov for the development of the electromagnetic sounding method, is employed in the context of the numerical-analytical representation of the forward problem solution in accordance with the Trefftz method. Application of this method allows enables one to generalize the basic plane-stratified vertically 1D model of the medium by allowing for lateral variations in electric conductivity.     

Applied Mathematics in EM Studies with Special Emphasis on an Uncertainty Quantification and 3-D Integral Equation Modelling

Despite impressive progress in the development and application of electromagnetic (EM) deterministic inverse schemes to map the 3-D distribution of electrical conductivity within the Earth, there is one question which remains poorly addressed—uncertainty quantification of the recovered conductivity models. Apparently, only an inversion based on a statistical approach provides a systematic framework to quantify such uncertainties. The Metropolis–Hastings (M–H) algorithm is the most popular technique for sampling the posterior probability distribution that describes the solution of the statistical inverse problem. However, all statistical inverse schemes require an enormous amount of forward simulations and thus appear to be extremely demanding computationally, if not prohibitive, if a 3-D set up is invoked. This urges development of fast and scalable 3-D modelling codes which can run large-scale 3-D models of practical interest for fractions of a second on high-performance multi-core platforms. But, even with these codes, the challenge for M–H methods is to construct proposal functions that simultaneously provide a good approximation of the target density function while being inexpensive to be sampled. In this paper we address both of these issues. First we introduce a variant of the M–H method which uses information about the local gradient and Hessian of the penalty function. This, in particular, allows us to exploit adjoint-based machinery that has been instrumental for the fast solution of deterministic inverse problems. We explain why this modification of M–H significantly accelerates sampling of the posterior probability distribution. In addition we show how Hessian handling (inverse, square root) can be made practicable by a low-rank approximation using the Lanczos algorithm. Ultimately we discuss uncertainty analysis based on stochastic inversion results. In addition, we demonstrate how this analysis can be performed within a deterministic approach. In the second part, we summarize modern trends in the development of efficient 3-D EM forward modelling schemes with special emphasis on recent advances in the integral equation approach.