Three-Dimensional Gravity Modeling In All Space

We review available analytical algorithms for the gravity effect and gravity gradients especially the vertical gravity gradient due to a right rectangular prism, a right polygonal prism, and a polyhedron. The emphasis is placed on an investigation of validity, consistency, and especially singularities of different algorithms, which have been traditionally proposed for calculation of the gravity effect on ground (or outside anomalous bodies), when they are applied to all points in space. The rounding error due to the computer floating point precision is estimated. The gravity effect and vertical gradient of gravity in three dimensions caused by a cubic model are calculated by different types of algorithms. The reliability of algorithms for the calculation of gravity of a right polygonal prism and a polyhedron is further verified by using a regular polygonal prism approximating a vertical cylinder and a regular polyhedron approximating a sphere, respectively. By highlighting Haáz-Jung-Plouff and Okabe-Steiner-Zilahi-Sebess' formulae for a right rectangular prism, Plouff's algorithm for a right polygonal prism, and Gouml;tze and Lahmeyer's algorithm for a polyhedron and removing their singularities, we demonstrate that these formulae and algorithms can be used to model the gravity anomaly and its vertical gradient at all possible computation positions.     

Effective calculation of gravity effects of uniform triangle polyhedra

Uniform tetrahedra are commonly used elementary bodies for gravity calculations from which arbitrary polyhedra can be composed. A simple derivation of the gravity effect is presented for the apex P of the tetrahedron expanded from P to an arbitrarily oriented plane triangle. Integration of its potential effect in a rotated coordinate system applies vector algebra and renders the anomalous potential depending on the distance of P over the triangle plain and a function of the triangle coordinates. Partial differentiation by moving P infinitesimally in z-direction leads to two terms, a simple and a complex one; they can be understood as describing the same difference from two points of view: leaving P at the apex of the changed polyhedron or moving P off the unchanged polyhedron. Both views imply the same shape change and the sum over the polyhedron is thus numerically equal. Hence we need to calculate only the one of the terms of the differential which is simpler. The calculation of the gravity effect is numerically simplified and more stable. This has been tested for many models and is demonstrated by two examples.     

基于MPICH环境下的复杂重力异常体并行正演模拟

任意多面体重力异常正演公式常用于解决复杂几何形态地质体的正演问题。 本文以均匀物性多面体重力异常正演公式为基础, 应用有限元技术中的网格离散化思想, 以任意四面体为基本单元, 通过并行计算技术在MPICH环境下实现了任意连续空间物性分布复杂异常体网格模型的重力异常正演模拟, 通过并行处理可以有效加速正演计算速度。 本文研究结果对于联合重力异常场正演建模和开展复杂模型网格的重力场计算有一定参考意义。     

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.     

Far-zone contributions to the gravity field quantities by means of Molodensky’s truncation coefficients

To reduce the numerical complexity of inverse solutions to large systems of discretised integral equations in gravimetric geoid/quasigeoid modelling, the surface domain of Green’s integrals is subdivided into the near-zone and far-zone integration sub-domains. The inversion is performed for the near zone using regional detailed gravity data. The farzone contributions to the gravity field quantities are estimated from an available global geopotential model using techniques for a spherical harmonic analysis of the gravity field. For computing the far-zone contributions by means of Green’s integrals, truncation coefficients are applied. Different forms of truncation coefficients have been derived depending on a type of integrals in solving various geodetic boundary-value problems. In this study, we utilise Molodensky’s truncation coefficients to Green’s integrals for computing the far-zone contributions to the disturbing potential, the gravity disturbance, and the gravity anomaly. We also demonstrate that Molodensky’s truncation coefficients can be uniformly applied to all types of Green’s integrals used in solving the boundaryvalue problems. The numerical example of the far-zone contributions to the gravity field quantities is given over the area of study which comprises the Canadian Rocky Mountains. The coefficients of a global geopotential model and a detailed digital terrain model are used as input data.     

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.     

The Field of Attraction of a Polyhedron and Polygonal Plate with Linear Density

New representations of the elements of the fields of attraction (potential and its first derivatives) are presented for the important approximating models such as polyhedron and polygonal plates with the density varying by the linear law. It is shown that these elements are determined through the elements of the fields from the models with the known analytical representations (polyhedron, polygonal plate, and material segment with constant densities) and the additional integrals for which the explicit analytical expressions exist.     

GRAVITY GRADIENT TENSORS DUE TO A POLYHEDRON WITH POLYGONAL FACETS1

Using the conjugate complex variables formulation, closed-form formulae for the gravity gradient tensors of the gravitational potential due to a homogeneous polyhedral body composed of polygonal facets are derived. The treatise considers the cases of the observation point being inside the polyhedron, on the surface of a facet, or outside the polyhedron.     

Analysis of gravity source moment inversion. Part I: Using noa priori information about the source

A generic gravity source moment is an integral, over the source volume, of the product of the density distribution by a polynomia in the Cartesian coordinates of a point belonging to this volume. We obtained a formal expression for a generic moment in terms of integrals involving the gravity anomaly and the gravity potential. By analyzing the conditions under which this expression is valid, we conclude that, without usinga priori information regarding the sources, it is possible to determine, from the gravity anomaly, any moment or linear combination of moments whose associated polynomial has null Laplacian and depends only on the coordinates defining the measurement plane. Additionally, no moment whose associated polynomial has a nonnull laplacian can be determined without usinga priori information of the source.     

ON THE UPWARD CONTINUATION OF POTENTIAL FIELD DATA BETWEEN IRREGULAR SURFACES*

The potential field and its derivatives at points above an irregular surface can be approximately obtained from the sampled potential field data acquired on that surface. A method of minimizing the truncation effect, which appears when gravity and magnetic maps are processed with the aid of surface integrals, is derived. The results are compared with those of the most relevant similar methods by using a theoretical, but realistic, model.     

Extension of Vedernikov's graph for seepage from canals

In this investigation, using previously derived equations by Vedernikov and Morel-Seytoux, closed-form solutions have been obtained to compute the seepage from a slit and a strip. Also, a graphical solution as an extension of Vedernikov's graph has been presented for computing quantity of seepage from triangular, rectangular, and trapezoidal canals. The solution replaces approximately the cumbersome evaluation of improper integrals with unknown implicit transformation variables.     

Diffraction of a compressional wave by a rigid barrier in a liquid half-space

Summary The problem of diffraction of compressional waves by a rigid barrier of finite height fixed in a liquid half space has been studied. Wiener-Hopf technique forms the basis of the methods used to solve the problem. Exact solutions have been obtained in terms of Fourier integrals whose evaluation along an appropriate contour gives the transmitted, reflected and diffracted waves. The diffracted waves decay rapidly away from the barrier.     

ASH-source in an elastic half-space with a non-homogeneous surface layer

Summary The response of an elastic half-space with a non-homogeneous surface layer due to aSH-source operating in the non-homogeneous layer of an elastic half-space have been studied. The elastic and physical constants vary exponentially in the surface layer. The surface displacement consists of several integrals. On approximate evaluation of these integrals, they are identified as direct or reflected pulses.     

Global ellipsoid-referenced topographic,bathymetric and stripping corrections to gravity disturbance

This paper revisits several aspects of defining and computing the anomalous gravity data for purposes of gravimetric inversion/interpretation. Attention is paid to evaluation of a refined global topographic correction to the gravity disturbance based on the reference ellipsoid (RE) and constant reference density for solid topography onshore and sea water density for liquid topography offshore. The global bathymetric correction is discussed. Two issues associated with compilation and inversion of bathymetrically and topographically corrected gravity disturbances in regions of negative ellipsoidal (geodetic) heights are pointed out: the evaluation of normal gravity and the harmonic continuation of the gravity data. Stripping, the removal of an effect of a known density contrast, is considered also for additional geological elements such as lakes, glaciers, sedimentary basins, isostatic mountain roots, etc. The stripping corrections are discussed in the context of the gravimetric inverse problem.     

Some space gravity formulas and the dimensions and the mass of the Earth

Summary Adopting thePizzetti-Somigliana method and using elliptic integrals we have obtained closed formulas for the space gravity field in which one of the equipotential surfaces is a triaxial ellipsoid. The same formulas are also obtained in first approximation of the equatorial flattening avoiding the use of the elliptic integrals. Using data from satellites and Earth gravity data the gravitational and geometric bulge of the Earth's equator are computed. On the basis of these results and on the basis of recent gravity data taken around the equator between the longitudes 50° to 100° E, 155° to 180° E, and 145° to 180° W, we question the advantage of using a triaxial gravity formula and a triaxial ellipsoid in geodesy. Closed formulas for the space field in which a biaxial ellipsoid is an equipotential surface are also derived in polar coordinates and its parameters are specialized to give the international gravity formula values on the international ellipsoid. The possibility to compute the Earth's dimensions from the present Earth gravity data is the discussed and the value ofMG=(3.98603×10²⁰ cm³ sec^–2) (M mass of the Earth,G gravitational constant) is computed. The agreement of this value with others computed from the mean distance Earth-Moon is discussed. The Legendre polinomials series expansion of the gravitational potential is also added. In this series the coefficients of the polinomials are closed formulas in terms of the flattening andMG.Publication Number 327, and Istituto di Geodesia e Geofisica of Università di Trieste.     

三度体（物性随深度变化模型）位场波谱的正演计算

本文将均质的任意二维、三维物体位场的波谱解析表达式的研究成果推广到变密度、变磁化强度的更一般的情形。对密度差随深度呈指数函数衰减或线性变化的模型,获得了任意倾斜多边形质量面、斜平行六面体以及一般的多面体等形体的重力谱的解析表达式。它们的结构与均质体相应表达式一样简单,易于计算。以上结果表明,在很一般的条件下,位场波谱具有指数函数和的形式。     

High speed gravity and magnetic calculations of uniform cylindrical bodies of arbitrary cross-section and finite length

Summary A simple method is designed for programming the gravity and magnetic calculations of a right circular cylinder (vertical or horizontal) by treating it as a combination of thin rectangular slabs. It takes only a few seconds to compute a profile of each kind and the accuracy is comparable to that obtained by using exact expressions (involving complete elliptic integrals) instead. The method is also applicable to cylindrical bodies of arbitrary cross-section and could as well be used for rapid computation of derivatives of gravity and magnetic anomalies.     

Solar Activity Effects on Gravity Wave Activity Inferred from Radio Wave Absorption in the Lower Ionosphere

The low frequency (LF) nighttime radio-wave absorption in the lower ionosphere has been measured at Prhonice (50°N, 15°E) in central Europe for over 35 years. Digital measurements, performed since summer 1988, allow absorption oscillations in the period range 10 – 180 mins, which are believed to reflect gravity wave activity, to be derived. Unfortunately, problems with the transmitter in recent years terminated the evaluation of gravity wave activity. The analysis of the available information (6 years of data) allows two conclusions to be drawn as to the effects of the solar activity on gravity wave activity: (1) there is no detectable effect of the solar 27-day variation on gravity wave activity; (2) there is an indication that the positive effect of the 11-year solar cycle on gravity wave activity in the winter half of the year is remarkable (lack of data in summer). The result concerning the solar cycle effect is, to a certain extent, preliminary, because the available data do not cover a complete solar cycle. A comparison with results from other stations and an interpretation of results are presented.     

三度体（均质模型）位场波谱的正演计算

本文给出任意指向的均质直线段、多边形面和多面体的重磁异常谱的解析表达式。利用它们可以进行不规则三度体的重磁异常谱的数值计算。此外,尚导出任意指向的斜平行六面体重磁异常谱的解析表达式。这些表达式结构简单,易于计算,用作位场的正演反演计算都十分方便。