On one method of solving the three-dimensional direct magnetic problem

Summary The solution of the direct magnetic problem is derived for three-dimensional bodies under the assumption that the components of the magnetization vector are analytical functions of the co-ordinates within the body being considered. The solution to the problem is given in the form of Green-type integrals and, much like with gravitational fields of inhomogeneous bodies, it may serve the purpose of solving the problem of the analytical continuation of the external field into the body.     

Numerical Studies on the Harmonic Downward Continuation of Band-Limited Airborne Gravity

In this paper, the numerical stability and efficiency of methods of harmonic downward continuation from flying altitudes are treated for sampled gravity field data. The problem is first formulated in its continuous form, i.e. as the inverse solution of the spherical Dirichlet problem, and is then approximated by Gaussian quadrature to yield a finite system of linear equations. The numerical stability of this system is investigated for both error-free gravity data and for the noisy and band-limited gravity measurements usually obtained from airborne gravity surveys. It can be shown that the system becomes ill-conditioned, once the ratio between flying altitude and data sampling rate exceeds a certain limit. It can also be shown that noisy measurements tend to generate a solution that is practically useless, long before the system becomes ill-conditioned. Therefore, instead of treating the general solution of the discrete downward continuation problem, the more modest question is studied, for which range of flying altitudes and sampling rates, the numerical solution of the discrete linear system can be considered as practically useful. Practically useful will be defined heuristically as of sufficient accuracy and stability to satisfy the requirements of the user. The question will be investigated for the specific application of geoid computation from gravity data sampled at flying altitudes. In this case, a stable solution with a standard deviation of a few centimeters is required. Typical flight parameters are heights of 2–6 km, a minimum half-wavelength resolution of 2 km, and data noise between 0.5 and 1.5 mGal. Different methods of geoid determination, different solution techniques for the resulting systems of linear equations, and different minimization principles will be compared. As a result operational parameters will be defined which, for a given noise level, will result in a geoid accuracy of a few centimeters for the estimated band-limited gravity field spectrum.     

On one method of solving the direct gravimetric problem

Summary The general solution of the direct gravimetric problem for homogeneous and inhomogeneous bodies was derived in[1] in the form of Green-type integrals. In the present paper it is proved that this solution simplifies considerably, if the solution of the external Dirichlet problem, defined by conditions(2), is known for a closed surface bounding the attracting body.Dedicated to RNDr Jan Pícha, CSc., on his 60th Birthday     

Die Schwerewirkungen eines elliptischen Zylindersegmentes

Summary A method has been derived for computing the gravity effect of a segment of an infinite homogeneous elliptical cylinder. The initial data was represented by formulae expressing the components of the field of gravity of a homogeneous two-dimensional body by means of line integrals. The method is based on the integration of theln R function over the boundary of the cross-section of the attracting body, R being the distance from a fixed point in which the gravity effect is determined. The problem was solved in confocal co-ordinates.     

On one method of solving the two-dimensional direct magnetic problem

Summary In this paper the solution of the direct magnetic problem for two-dimensional bodies, founded on the application of Green's theorem is derived. This solution is derived under the assumption that the components of the magnetization vector have continuous derivatives with respect to the coordinates and that they are continuous within the body. The problem is solved in terms of Green-type integrals for the scalar and vector potential of the magnetostatic field and it may serve the purpose of solving the problem of the analytical continuation of the external field into the body.     

One Solution of the Forward Problem of DC Resistivity Well Logging by the Method of Volume Integral Equations with Allowance for Induced Polarization

For theoretically studying the intensity of the influence exerted by the polarization of the rocks on the results of direct current (DC) well logging, a solution is suggested for the direct inner problem of the DC electric logging in the polarizable model of plane-layered medium containing a heterogeneity by the example of the three-layer model of the hosting medium. Initially, the solution is presented in the form of a traditional vector volume-integral equation of the second kind (IE2) for the electric current density vector. The vector IE2 is solved by the modified iteration–dissipation method. By the transformations, the initial IE2 is reduced to the equation with the contraction integral operator for an axisymmetric model of electrical well-logging of the three-layer polarizable medium intersected by an infinitely long circular cylinder. The latter simulates the borehole with a zone of penetration where the sought vector consists of the radial J_r and J_z axial (relative to the cylinder’s axis) components. The decomposition of the obtained vector IE2 into scalar components and the discretization in the coordinates r and z lead to a heterogeneous system of linear algebraic equations with a block matrix of the coefficients representing 2x2 matrices whose elements are the triple integrals of the mixed derivatives of the second-order Green’s function with respect to the parameters r, z, r', and z'. With the use of the analytical transformations and standard integrals, the integrals over the areas of the partition cells and azimuthal coordinate are reduced to single integrals (with respect to the variable t = cos ? on the interval [?1, 1]) calculated by the Gauss method for numerical integration. For estimating the effective coefficient of polarization of the complex medium, it is suggested to use the Siegel–Komarov formula.     

Accuracy of deflections of vertical determined from discrete solution of the geodetic boundary value problem

Summary The general problem of determining the figure of the earth leads to the solution of the geodetic boundary value problem. By its discrete approximation we obtain the discrete disturbing potential that maintains all properties of the original problem. Thus, the discrete approximation of the disturbing potential can be used in studying the behaviour of the earth's gravity field outside the disturbing masses. The deflections of the vertical are one of the quantities describing the behaviour of the earth's gravity field. A method for their computation from the discrete solution of the geodetic boundary value problem is put forth and estimates for its accuracy are given.     

To Pizzetti's Theory of the Level Ellipsoid

The topic of the Earth's reference body, which has now been established as Pizzetti's level rotational ellipsoid, is analysed. Such a body is fully determined by four parameters: a, GM, J ₂ and . At present, the largest discrepancy in determining these parameters occurs in the value of a, which may in future be replaced by the gravity potential of the mean sea level W _o, with respect to Brovar's condition.Pizzetti's four parameters of the reference body are determined by solving the Dirichlet boundary value problem. The Dirichlet problem has only a unique solution, which, however, can be expressed in infinitely many ways. It turns out that the most important part in the form of the solution is played by Lamé's conditions, which determine the type of ellipsoidal coordinates.The solutions given by Pizzetti, Molodensky and another variant are considered. The last variant leads to a simple formula for the potential of the reference ellipsoid, but the formulae for Lamé's coefficients are inconvenient. Of course, all the methods lead to identical solutions, but some of them are more convenient for the historical use of logarithms, whereas others are more appropriate for use in computers.     

Gravity Anomalies of Arbitrary 3D Polyhedral Bodies with Horizontal and Vertical Mass Contrasts

During the last 15 years, more attention has been paid to derive analytic formulae for the gravitational potential and field of polyhedral mass bodies with complicated polynomial density contrasts, because such formulae can be more suitable to approximate the true mass density variations of the earth (e.g., sedimentary basins and bedrock topography) than methods that use finer volume discretization and constant density contrasts. In this study, we derive analytic formulae for gravity anomalies of arbitrary polyhedral bodies with complicated polynomial density contrasts in 3D space. The anomalous mass density is allowed to vary in both horizontal and vertical directions in a polynomial form of \(\lambda =ax^m+by^n+cz^t\), where m, n, t are nonnegative integers and a, b, c are coefficients of mass density. First, the singular volume integrals of the gravity anomalies are transformed to regular or weakly singular surface integrals over each polygon of the polyhedral body. Then, in terms of the derived singularity-free analytic formulae of these surface integrals, singularity-free analytic formulae for gravity anomalies of arbitrary polyhedral bodies with horizontal and vertical polynomial density contrasts are obtained. For an arbitrary polyhedron, we successfully derived analytic formulae of the gravity potential and the gravity field in the case of \(m\le 1\), \(n\le 1\), \(t\le 1\), and an analytic formula of the gravity potential in the case of \(m=n=t=2\). For a rectangular prism, we derive an analytic formula of the gravity potential for \(m\le 3\), \(n\le 3\) and \(t\le 3\) and closed forms of the gravity field are presented for \(m\le 1\), \(n\le 1\) and \(t\le 4\). Besides generalizing previously published closed-form solutions for cases of constant and linear mass density contrasts to higher polynomial order, to our best knowledge, this is the first time that closed-form solutions are presented for the gravitational potential of a general polyhedral body with quadratic density contrast in all spatial directions and for the vertical gravitational field of a prismatic body with quartic density contrast along the vertical direction. To verify our new analytic formulae, a prismatic model with depth-dependent polynomial density contrast and a polyhedral body in the form of a triangular prism with constant contrast are tested. Excellent agreements between results of published analytic formulae and our results are achieved. Our new analytic formulae are useful tools to compute gravity anomalies of complicated mass density contrasts in the earth, when the observation sites are close to the surface or within mass bodies.     

航空重力测量数据向下延拓方法比较

首先介绍了空中重力异常向下延拓的直接代表法、正则化法、点质量法和基于球内Dirichlet问题的调和解法,分析了它们各自的特点.然后利用这些方法对中国大同航空重力测量数据进行向下延拓计算.通过对延拓结果与地面实测数据的比较分析,比较客观地评价了不同延拓方法的计算精度、可靠性及适应性.实际计算结果表明,直接代表法和正则化法的延拓结果具有较高的精度和可靠性,而球内Dirichlet问题调和解法的精度较差.     

基于向上延拓的航空重力向下解析延拓解

位场向下与向上延拓之间存在固有的内在联系,向上延拓解算具有稳定可靠的优良特性,本文据此提出了借助向上延拓信息实现航空重力向下延拓稳定解算的两种方法,分别建立了点对点向下解析延拓模型和最小二乘向下解析延拓模型.其核心思想是,依据泰勒级数展开模型,将位场向下延拓解算过程转换为向上延拓计算和垂向偏导数解算两个步骤,通过第一步的处理有效抑制数据观测噪声对解算结果的干扰,通过第二步的处理成功实现向下延拓反问题的稳定解算,较好地解决了向下延拓解算固有的不适定性问题.分析研究了两种解析延拓模型的计算精度及适用条件,利用超高阶位模型EGM2008建立的模拟标准场数据对两种模型解算结果的合理性和有效性进行了数值验证,证明本文新方法实用易行,具有较高的应用价值.     

Evaluating BOD and the coefficient of oxidation rate: Monitoring,direct and inverse problems,formulas, calculations and tables

It is proposed to evaluate two theoretical characteristics, i.e., BOD (biochemical oxygen demand) and k ₀ (the coefficient of oxidation rate by new formulas based on two experimental variables: BOD _T and BOD_2T (biochemical oxygen consumption in two periods T and 2T day, respectively). The formulation and an analytical solution are given for a direct problem describing the process of biochemical oxidation of organic matter (OM) in a water volume in the absence of aeration (e.g., in a water body under ice or in a sealed flask used to measure biochemical oxygen consumption). The problem is solved based on the closed (modified) Streeter–Phelps system. Unlike the classical Streeter–Phelps system, the closed system excludes physically incorrect solutions (e.g., negative concentrations of dissolved oxygen (DO)) [4]. The solution of the direct problem is used to formulate an inverse problem, whose solution is given in the form of formulas for evaluating BOD and k ₀. These formulas are used to compile tables to illustrate the essence of the proposed method.     

Semi-analytical evaluation of integral forms associated with Lamb's problem

The present work gives solutions of integrals resulting from Lamb's problem solution by an accurate and efficient approximation procedure. The formulation of the solutions requires the study of Green's function of Lamb problem solution. Unfortunately, this problem is associated to great mathematical difficulties and needs the evaluation of some complex integrals. We handled these integrals by an appropriate strategy based on decoupling their effects mathematically and solved by using the least squares method. The present solutions can be applied to study the dynamical interaction between soil and structures, to be used to determine the transient wave fields caused by a seismic source, and also to assess numerical computations with a different numerical methods program.     

用压缩质面法反演重力资料以估算地壳构造

本文用压缩质面来近似二维地质体。先由地面上测得的重力数据用矩阵方法反演此压缩质面各单元的面密度,然后从面密度与体密度差的关系求得各二维质体单元的厚度,进而得到各单元的近似地壳厚度。经正演校验和反复调整结果,使计算的重力异常值与实际测量值之残差小到满足要求,从而得到较准确的地壳底部界面。 由上地幔,玄武岩层和花岗岩层的密度差异及已算出的地壳厚度,从重力异常中分解出莫霍界面和康腊界面起伏所分别引起的重力异常。将后者同样用反演地壳厚度的压缩质面法进行计算,得到康腊界面。 文中以三种假想的地壳模型和一个实测剖面为例来检验本方法,并用其他确定地壳界面方法所得到的结果相比较,表明本文提出的压缩质面法结果较好。     

Conditionality of inverse solutions to discretised integral equations in geoid modelling from local gravity data

The eigenvalue decomposition technique is used for analysis of conditionality of two alternative solutions for a determination of the geoid from local gravity data. The first solution is based on the standard two-step approach utilising the inverse of the Abel-Poisson integral equation (downward continuation) and consequently the Stokes/Hotine integration (gravity inversion). The second solution is based on a single integral that combines the downward continuation and the gravity inversion in one integral equation. Extreme eigenvalues and corresponding condition numbers of matrix operators are investigated to compare the stability of inverse problems of the above-mentioned computational models. To preserve a dominantly diagonal structure of the matrices for inverse solutions, the horizontal positions of the parameterised solution on the geoid and of data points are identical. The numerical experiments using real data reveal that the direct gravity inversion is numerically more stable than the downward continuation procedure in the two-step approach.     

Analytical Solution of Direct and Inverse Problems in the Internal Gravity Waves Studies by the Doppler Frequency Shift Method

An analytical solution of direct and inverse problems arising in the study of the internal gravity waves (IGWs) dynamic via recording of the Doppler frequency shift, is presented. The direct problem is to determine the response of the Doppler shift to IGWs in the region of the radio wave reflection point; the inverse problem is the determination of IGW parameters from data on the Doppler frequency shift. Solutions were obtained in an approximation of the isothermal ionosphere for the heights of the F-region. They are presented in a form convenient for their practical use and can have a wide range of applications, including the detection of soliton-like wave structures in the F-region of the ionosphere.     

Dimension of the phase space of a dynamic system from geophysical data

The possibility of determining parameters of complex geophysical processes is considered in terms of nonlinear dynamics. In accordance with modern approaches in the theory of nonlinear dynamic systems, the number of independent parameters controlling the behavior of a nonlinear system can be estimated from the available time realization of only one of these parameters. Model calculations showed that the dimension of the phase space of a dynamic system can be estimated from a sample of one variable. Experimental data on variations in the apparent electric resistivity (AER) and the relative vertical movement of the surface (RVMS) in a seismically active region are analyzed and the dimension of the dynamic system determining its behavior in the phase space is estimated. The resulting estimates of the embedding dimension m = 7?8 for AER variations and m = 6 for RVMS variations possibly characterize the complexity of the dynamic system describing the given fields. The method presented in the paper is also applied to the analysis of the degree of connectivity of different dynamic systems and their parameters. By the connectivity we mean the number of independent parameters simultaneously involved in the formation of the dynamic behavior of various physical fields. The model estimates demonstrate the possibility of such an approach. It is shown that variations in the AER in perpendicular directions are described by a general system of dynamic equations, whereas dynamic systems controlling the AER field and the RVMSs are interconnected only partially. The resulting dimension m = 12 estimated for the AER-RVMS system provides an estimate for the number of common controlling parameters: n = (8 + 6)?12 = 2. The methods and results presented in the paper are applicable to the construction of models of complex geophysical processes and the development and the development of new approaches and methods of identification of prognostic characteristics for the behavior of physical fields of various origins.     

重力垂直梯度的测定及其应用与潜力

近年来,随着重力垂直梯度测量精度的提高及间接推求方法的改善,它在地球科学的研究中的作用日益扩大,本文叙述了利用高精度重力仪直接测量垂直梯度的方法与注意事项,阐明了垂直梯度在地学研究中的意义与作用,其中包括(1)可以更精确测定大地水准面与正高;(2)可以更好探测地下的物质的分布及界面起伏;(3)可根据垂直梯度及形变资料反演场源及解释引起上述变化的动力机制.关于如何进一步发掘其潜力,这里也提出了建议.     

LINEAR TRANSFORMATIONS OF GRAVITY AND MAGNETIC FIELDS*

The spectral representation of gravity and magnetic fields shows that the mathematical expressions describing these fields are the result of convolution of factors which depend on the geometry of the causative body, the physical properties of the body and the type of field being observed. If a field is known, it is possible to remove or alter these factors to map other fields or physical parameters which are linearly related to the observed field. The transformations possible are: continuation, reduction to the pole, converting between gravity and magnetic fields, converting between components of measurement, calculation of derivatives, and mapping magnetization and density distribution, relief on interfaces, and vertical thicknesses of layers.     

位场向下延拓组合滤波器的设计和应用

一 重、磁异常向下延拓能将水平方向迭加的异常分开,并能突出异常细节,从而丰富异常的解释内容,提高异常解释的可靠性.本文从对引起向下延拓不稳定的各种因素的讨论入手,结合对一些向下延拓方法在频率域的分析,提出了一种与已知资料频谱特征相“匹配”的组合滤波器。经理论模型和实例考核,说明该滤波器既能保证向下延拓有较高的精度,又有较强的抗干扰能力。