The gravity effect of a homogeneous polyhedron for three-dimensional interpretation

Summary As an aid to the interpretation of gravity data for three-dimensional causative bodies, a method of evaluation of the gravity effect of a homogeneous polyhedron at an external point has been worked out. It has been shown that the gravity effect is expressible in terms of some prinitive integrals over the basic triangular faces of the polyhedron. The formula for evaluation of these integrals has also been derived.Contribution from the Earth Physics Branch No. 501.     

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.     

Layer-Based Modelling of the Earth’s Gravitational Potential up to 10-km Scale in Spherical Harmonics in Spherical and Ellipsoidal Approximation

Global forward modelling of the Earth’s gravitational potential, a classical problem in geophysics and geodesy, is relevant for a range of applications such as gravity interpretation, isostatic hypothesis testing or combined gravity field modelling with high and ultra-high resolution. This study presents spectral forward modelling with volumetric mass layers to degree 2190 for the first time based on two different levels of approximation. In spherical approximation, the mass layers are referred to a sphere, yielding the spherical topographic potential. In ellipsoidal approximation where an ellipsoid of revolution provides the reference, the ellipsoidal topographic potential (ETP) is obtained. For both types of approximation, we derive a mass layer concept and study it with layered data from the Earth2014 topography model at 5-arc-min resolution. We show that the layer concept can be applied with either actual layer density or density contrasts w.r.t. a reference density, without discernible differences in the computed gravity functionals. To avoid aliasing and truncation errors, we carefully account for increased sampling requirements due to the exponentiation of the boundary functions and consider all numerically relevant terms of the involved binominal series expansions. The main outcome of our work is a set of new spectral models of the Earth’s topographic potential relying on mass layer modelling in spherical and in ellipsoidal approximation. We compare both levels of approximations geometrically, spectrally and numerically and quantify the benefits over the frequently used rock-equivalent topography (RET) method. We show that by using the ETP it is possible to avoid any displacement of masses and quantify also the benefit of mapping-free modelling. The layer-based forward modelling is corroborated by GOCE satellite gradiometry, by in-situ gravity observations from recently released Antarctic gravity anomaly grids and degree correlations with spectral models of the Earth’s observed geopotential. As the main conclusion of this work, the mass layer approach allows more accurate modelling of the topographic potential because it avoids 10–20-mGal approximation errors associated with RET techniques. The spherical approximation is suited for a range of geophysical applications, while the ellipsoidal approximation is preferable for applications requiring high accuracy or high resolution.     

High Resolution Constituents of the Earth’s Gravitational Field

During the General Assembly of the European Geosciences Union in April 2008, the new Earth Gravitational Model 2008 (EGM08) was released with fully-normalized coefficients in the spherical harmonic expansion of the Earth’s gravitational potential complete to degree and order 2159 (for selected degrees up to 2190). EGM08 was derived through combination of a satellite-based geopotential model and 5 arcmin mean ground gravity data. Spherical harmonic coefficients of the global height function, that describes the surface of the solid Earth with the same angular resolution as EGM08, became available at the same time. This global topographical model can be used for estimation of selected constituents of EGM08, namely the gravitational potentials of the Earth’s atmosphere, ocean water (fluid masses below the geoid) and topographical masses (solid masses above the geoid), which can be evaluated numerically through spherical harmonic expansions. The spectral properties of the respective potential coefficients are studied in terms of power spectra and their relation to the EGM08 potential coefficients is analyzed by using correlation coefficients. The power spectra of the topographical and sea water potentials exceed the power of the EGM08 potential over substantial parts of the considered spectrum indicating large effects of global isostasy. The correlation analysis reveals significant correlations of all three potentials with the EGM08 potential. The potential constituents (namely their functionals such as directional derivatives) can be used for a step known in geodesy and geophysics as the gravity field reduction or stripping. Removing from EGM08 known constituents will help to analyze the internal structure of the Earth (geophysics) as well as to derive the Earth’s gravitational field harmonic outside the geoid (geodesy).     

Mean Free-Air Gravity Anomalies in the Mountains

Mean free-air gravity anomalies are often needed in geodesy for gravity field modelling. Two possible ways of compiling the mean free-air gravity anomalies are discussed. One way is via simple Bouguer gravity anomalies and the second and more time consuming way is via refined Bouguer gravity anomalies. In flat areas the differences between using any of the two ways should not be significant. In the mountains however, every effect introducing a high dependency, such as e.g. terrain effect, can negatively affect the interpolation process. In fact, a numerical experiment conducted in one part of Rocky Mountains revealed large and systematic differences. The effect of these differences on the geoid model is more then two meters in the test area. Our investigation shows that this bias is caused by the location of gravity measurement points, chosen mostly on hill-tops. At such points, the terrain correction to gravity is systematically larger than the mean value of the correction. Therefore, it is not possible to prevent the mean free-air gravity anomalies obtained from simple Bouguer gravity anomalies from having a systematic bias. One can see this bias as a result of the aliasing effect because the simple Bouguer gravity anomalies in the mountains contain a higher frequency signal (terrain effect) that is, according to sampling theorem, impossible to reconstruct by sparse measured gravity data, see e.g. (Goos et al., 2003). Therefore, the more rigorous way of computing the mean free-air gravity anomalies is via refined Bouguer gravity anomalies.     

CROSS-CORRELATION AS AN AID IN SIMULTANEOUS GRAVITY AND MAGNETIC ANALYSIS *

The properties of cross-correlation functions for the case of gravity and magnetic total field anomalies produced by three geometric models of geological bodies (point-mass, horizontal line-mass, and vertical semi-infinite rectangular prism mass distributions) are studied. This study is carried out on four cases of cross-correlation: the whole curve of anomalies, a single branch of anomalies, the branch between the two inflexion points and the branch between the apex point and the inflexion point. In all cases, the crosscorrelation function can serve as a good indicator to discriminate the so-called genetically related anomalies from those produced by different geological bodies situated on the same vertical line; but the best results are obtained using the cross-correlation of the branches between the apex point and the inflexion point of the two geophysical anomalies. A practical procedure is developed in order to undertake such analysis. The tests in some cases of real gravity and magnetic anomalies mapped in Romania show the validity of this practical procedure.     

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.     

On correct application of one-step inversion of gravity data

One of the main problems on the numerical solution of integral equations is the resolution of input data. Among the integral equations used in geodesy we have the “onestep inversion” based on the first derivative of the Poisson integral, which transforms gravity values on the Earth’s surface to the gravity potential on the reference ellipsoid. In this study, it is shown that the required spatial resolution of the input gravity data on the Earth’s surface for correct one-step inversion depends on the height of the computational region, the fact that if overlooked can cause totally wrong results. Consequently the following two major questions are posed: (i) How could one know whether the spatial resolution of the input gravity data for correct one-step inversion is sufficient? (ii) What should be done if the spatial resolution is not sufficient? By studying the behaviour of the integral kernel, an algorithm is presented which enables an appropriate answer to the former question. In order to address the latter question, a method is proposed to modify the integral kernel which overcomes the adverse effect of insufficient spatial resolution of the input gravity data. Our answers, which possess the novelty of the study, are numerically verified by means of real and simulated gravity data. The numerical results approve the efficiency of the proposed method in solving the problem of insufficient spatial resolution of the input gravity data for correct one-step inversion.     

Large‐scale 3D inversion of potential field data

Inversion of gravity and/or magnetic data attempts to recover the density and/or magnetic susceptibility distribution in a 3D earth model for subsequent geological interpretation. This is a challenging problem for a number of reasons. First, airborne gravity and magnetic surveys are characterized by very large data volumes. Second, the 3D modelling of data from large‐scale surveys is a computationally challenging problem. Third, gravity and magnetic data are finite and noisy and their inversion is ill posed so regularization must be introduced for the recovery of the most geologically plausible solutions from an infinite number of mathematically equivalent solutions. These difficulties and how they can be addressed in terms of large‐scale 3D potential field inversion are discussed in this paper. Since potential fields are linear, they lend themselves to full parallelization with near‐linear scaling on modern parallel computers. Moreover, we exploit the fact that an instrument’s sensitivity (or footprint) is considerably smaller than the survey area. As multiple footprints superimpose themselves over the same 3D earth model, the sensitivity matrix for the entire earth model is constructed. We use the re‐weighted regularized conjugate gradient method for minimizing the objective functional and incorporate a wide variety of regularization options. We demonstrate our approach with the 3D inversion of 1743 line km of FALCON gravity gradiometry and magnetic data acquired over the Timmins district in Ontario, Canada. Our results are shown to be in good agreement with independent interpretations of the same data.     

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。     

THE GRAVIMETRIC DETECTION OF MINING SUBSIDENCE*

A series of gravity measurements were taken over a period of time (c. 150 days each) above two adjacent working coal faces. Precise levelling and gravity measurements were taken along the same profile before, during, and after seam extraction. The observed change in gravity agrees well with the levelling data. The combined data sets illustrate the validity of a simple Bouguer relationship for the gravity gradient. The results of two-dimensional modelling are used to estimate the gravitational effect of the extracted material. A local feature detected only on the gravity signal may be due to an outcropping limestone layer. This controlled experiment demonstrates the possibility of using high-precision gravity measurements as a substitute for levelling, particularly in surveys of large areal extent where the cost of obtaining comprehensive coverage by levelling may be prohibitive.     

The 3-D Truncation Filtering Methodology Defined for Planar and Spherical Models: Interpreting Gravity Data Generated by Point Masses

This paper focuses on one particular way of linear filtering the gravity data to facilitate gravity inversion or interpretation. With the use of integral transforms the gravity anomalies are transformed into new quantities that allow an easier interpretation with the help of pattern recognition. As the integral transforms are in fact filters, and as the regions of integration are caps with a variable radius, which can be systematically changed as a free parameter, we refer to such methodology as the truncation filtering. Such filters may be understood as weighted spherical windows moving over the surface, on which the gravity anomaly is defined, the kernel of the transform being the weight function. The objective of this paper is to define and deploy the truncation filtering for a planar model, i.e. for a homogenous horizontally infinite layer with embedded anomalous masses, and for a spherical model, i.e., for a homogenous massive sphere with embedded anomalous masses. Instead of the original gravity anomaly, the quantities resulting from the truncation filtering are interpreted/inverted. As we shall see, this approach has certain benefits. The fundamental concept of the truncation filtering methodology is demonstrated here in terms of the model consisting of one point mass anomaly.The relationship between the depth of the point mass and the instant of the onset of the dimple pattern observed in sequences produced by truncation filtering the synthetic gravity data generated by point masses is, for both the planar and spherical models, compiled by computer simulations, as well as derived analytically. It is shown, that the dimple pattern is a consequence of truncating the domain of the filter and is free of the choice of the kernel of the filter. It is shown, that in terms of the mean earth and depths of point masses no greater than some 100 km the spherical model may be replaced by a planar model from the perspective of the truncation filtering methodology. It is also shown, that from the viewpoint of the truncation filtering methodology the rigorous gravity anomaly may be approximated by the vertical component of the gravity disturbance. The relationship between the instant of the dimple onset and the depth of the point mass thus becomes linear and independent of the magnitude (mass) of the point mass.     

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。     

地球横向不均匀结构对地表以及空间固定点同震重力变化的影响

本文提出一个新算法,用来高精度计算三维不均匀地球模型中地震位错引起的地表以及空间固定点同震重力变化.具体地说,我们首先把实际三维不均匀地球分解成球对称地球模型和对应的横向不均匀增量,分别进行计算,二者对应的计算结果分别称为球对称解和三维响应.由于球对称解可直接利用球对称地球模型位错理论计算得到,本文的目标是计算三维响应即地球的横向不均匀结构对同震重力变化的影响.然后,我们把三维响应再分为震源的响应和地球横向不均匀结构的响应,它们可分别借助对震源函数的扰动以及对平衡方程式的变分求解.本文推导出六个特殊点源位错引起的地表以及空间固定点同震重力变化计算公式(一个垂直走滑位错,两个相互垂直的倾滑位错,三个开裂位错),对这些公式进行适当组合就可以计算任意位置任意类型位错产生的同震重力变化,对应的计算公式同步给出.接着,依据36阶P波速度模型,我们利用岩石试验经验关系式推导出三维S波速度模型,密度模型,位场模型以及重力模型.最后,利用上述三维模型,本文计算出三种典型类型的点源位错产生的同震重力变化,结果显示三维响应与位错类型,震源深度都有关系,其最大响应占球对称解的0.5%左右,且在所有影响因素中S波速度模型影响最大.数值结果同时表明,三维响应中震源的响应与地球横向不均匀构造的响应处于同一量级.本文给出的地表和空间固定点同震重力变化计算公式可分别高精度解析地表重力和卫星重力观测数据(GRACE、GOCE等),提高大地测量数据理论解析水平.     

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。     

A sidelight on long range forecasting

Summary A gradually cumulative error due to the use of the hydrostatic approximation is examined through consideration of a simple mathematical model taken from the classical theory of gravity waves. Two cases are chosen which differ solely in the circumstance that in one the hydrostatic assumption is made while in the other it is not. A large divergence of the results present in the later stages is then to be referred to the neglect of terms contained in the exact equation of motion for the vertical direction in one of the solutions. The inference is drawn that comparable phenomena might arise in the integration of the atmospheric equations over long time periods.     

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.     

Computing three-dimensional gravitational fields with equivalent sources

Existing techniques for computing the gravitational field due to a homogeneous polyhedron all transform the required volume integral, expressing the field due to a volume distribution of mass, into a surface integral, expressing the potential due to a surface mass distribution over the boundary of the source body. An alternative representation is also possible and results in a surface integral expressing the potential due to a variable-strength double layer located on the polyhedral source boundary. Manipulation of this integral ultimately allows the gravitational field component in an arbitrary direction to be expressed as a weighted sum of the potentials due to two basic source distributions. These are a uniform-strength double layer located on all faces and a uniform-strength line source located along all edges. The derivatives of the gravitational field components can also be expressed in a similar form as can the magnetic field components due to a homogeneous magnetic polyhedron. It follows that the present approach can be used to generate a universal program capable of modelling all the commonly used potential field responses due to 3D bodies of arbitrary shape.     

离散时变重力数据的可视化、指标量定义与解释

针对流动重力观测获得的数据成果特点,提出一种基于流动重力段差变化的时变重力数据可视化方法,并定义了两个指标量G值和C值,用以评价区域性重力场变化的显著性程度.在此基础上,应用该方法处理和分析了首都圈地区的流动重力数据.结果表明:与传统采用等值线方式来刻画时变重力场特征不同,该方法更能突出发生变化的重力测点位置、测量误差及其变化的显著性程度,可为研究与地震孕育、发生过程有关的重力场变化提供更多的定量依据.