Geoid Determination through Ellipsoidal Stokes Boundary-Value Problem

Martinec and Grafarend (1997) have shown how the construction of Green's function in the Stokes boundary-value problem with gravity data distributed on an ellipsoid of revolution is approached in the O(e ₀ ² )-approximation. They have also expressed the ellipsoidal Stokes function describing the effect of ellipticity of the boundary as a finite sum of elementary functions. We present an effective method of avoiding the singularity of spherical and the ellipsoidal Stokes functions, and also an analytical expression for the ellipsoidal Stokes integral around the computational point suitable for numerical solution. We give the numerical results of solving the ellipsoidal Stokes boundary-value problem and their difference with respect to the spherical Stoke boundary-value problem.     

Ellipsoidal Stokes Boundary-Value Problem with Ellipsoidal Corrections in the Boundary Condition

We would like to solve the Stokes boundary-value problem taking into consideration the ellipsoidal corrections in the boundary condition in ellipsoidal coordinates The original problem, i.e., the ellipsoidal Stokes boundary-value problem has been solved by Martinec and Grafarend (1997) We use the same philosophy expressed by Martinec (1998) to solve the spherical Stokes boundary-value problem with ellipsoidal corrections in the boundary condition We wish to show the magnitude of the integration kernel describing the effect of the ellipsoidal corrections in the boundary condition in a cap around the computational point.     

Geoid Determination Through Ellipsoidal Stokes Boundary-Value Problem by Splitting Its Solution to the Low-Degree and the High-Degree Parts

The ellipsoidal Stokes boundary-value problem is used to compute the geoidal heights. The low degree part of the geoidal heights can be represented more accurately by Global Geopotential Models (GGM). So the disturbing potential is splitted into a low-degree reference potential and a higher-degree potential. To compute the low-degree part, the global geopotential model is used, and for the high-degree part, the solution of the ellipsoidal Stokes boundary-value problem in the form of the surface integral is used. We present an effective method to remove the singularity of the high-degree of the spherical and ellipsoidal Stokes functions around the computational point. Finally, the numerical results of solving the ellipsoidal Stokes boundary-value problem and the difference between the high-degree part of the solution of the ellipsoidal Stokes boundary-value problem and that of the spherical Stokes boundary-value problem is presented.     

Solution to the Stokes Boundary-Value Problem on an Ellipsoid of Revolution

We have constructed Green's function to Stokes's boundary-value problem with the gravity data distributed over an ellipsoid of revolution. We show that the problem has a unique solution provided that the first eccentricity e₀ of the ellipsoid of revolution is less than 0·65041. The ellipsoidal Stokes function describing the effect of ellipticity of the boundary is expressed in the E-approximation as a finite sum of elementary functions which describe analytically the behaviour of the ellipsoidal Stokes function at the singular point = 0. We prove that the degree of singularity of the ellipsoidal Stokes function in the vicinity of its singular point is the same as that of the spherical Stokes function.     

Far-Zone Contributions to Topographical Effects in the Stokes-Helmert Method of the Geoid Determination

In the evaluation of the geoid done according to the Stokes-Helmert method, the following topographical effects have to be computed: the direct topographical effect, the primary indirect topographical effect and the secondary indirect topographical effect. These effects have to be computed through integration over the surface of the earth. The integration is usually split into integration over an area immediately adjacent to the point of interest, called the near zone, and the integration over the rest of the world, called the far zone. It has been shown in the papers by Martinec and Vaníek (1994), and by Novák et al. (1999) that the far-zone contributions to the topographical effects are, even for quite extensive near zones, not negligible.Various numerical approaches can be applied to compute the far-zone contributions to topographical effects. A spectral form of solution was employed in the paper by Novák et al. (2001). In the paper by Smith (2002), the one-dimensional Fast Fourier Transform was introduced to solve the problem in the spatial domain. In this paper we use two-dimensional numerical integration. The expressions for the far-zone contributions to topographical effects on potential and on gravitational attraction are described, and numerical values encountered over the territory of Canada are shown in this paper.     

Contribution of subducted Pacific slab to Late Cretaceous mafic magmatism in Qingdao region, China: A petrological record

Abstract The occurrence of the Pishikou mafic dike in the Qingdao region, China provides important constraints on the origin of Late Cretaceous (86–78 Ma) mafic magmatism on the eastern North China craton. The Pishikou mafic dike is distributed in the Cretaceous Laoshan granitoid body, Qingdao region and contains peridotitic and granulitic xenoliths, xenocrysts, and megacrysts. Rocks from the Pishikou mafic dike are basanites and have low SiO₂ (< 42 wt%) and Al₂O₃ (12.5 wt%) contents, and high MgO (> 8 wt%), total alkalis (Na₂O + K₂O > 4.8 wt%, Na₂O/K₂O > 1), TiO₂ (> 2.5 wt%), CaO (> 9 wt%) and P₂O₅ (> 1 wt%). In trace element abundances, they are highly enriched in large ion lithophile elements (LILEs) and light rare‐earth elements (LREEs) (ΣREE = 339–403 ppm, (La/Yb)_N = 39–42) without high field strength element (HFSE) depletion. These rocks have radiogenic Sr and Pb, and less radiogenic Nd isotopic compositions [(⁸⁷Sr/⁸⁶Sr)_i > 0.7059, εNd ≈ 2.7–3.8 (²⁰⁶Pb/²⁰⁴Pb)_i ≈ 18.0 ± 0.1]. The diagnostic elemental ratios, such as Nb/La, Nb/U, and Nb/Th, are compatible with those of mid‐oceanic ridge basalts (MORBs) and oceanic island basalts (OIBs). Therefore, the Pishikou mafic dike has a geochemical feature completely different from those of the Early Cretaceous mafic dikes from the Qingdao region, but similar to those of back‐arc basalts from the Japan Sea. This geochemical feature suggests that the Pishikou mafic dike was derived from an asthenosphere source, but contaminated by materials from the subducted Pacific slab. The discovery of this mafic dike thus provides a petrological evidence for the contribution of subducted Pacific slab to the Late Cretaceous magmatism in the Qingdao region of the eastern North China craton.     

The discrete wave number formulation of boundary integral equations and boundary element methods: A review with applications to the simulation of seismic wave propagation in complex geological structures

We review the application of the discrete wave number method to problems of scattering of seismic waves formulated in terms of boundary integral equation and boundary element methods. The approach is based on the representation of the diffracting surfaces and interfaces of the medium by surface distributions of sources or by boundary source elements, the radiation from which is equivalent to the scattered wave field produced by the diffracting boundaries. The Green's functions are evaluated by the discrete wave number method, and the boundary conditions yield a linear system of equations. The inversion of this system allows the calculation of the full wave field in the medium. We investigate the accuracy of the method and we present applications to the simulation of surface seismic surveys, to the diffraction of elastic waves by fractures, to regional crustal wave propagation and to topographic scattering.     

重力异常场空间-波数混合域三维数值模拟

重力勘探中复杂条件下的三维正演计算量大存储要求高,使得这种条件下重力勘探高效、精细正反演变得困难.针对这一问题,提出一种空间-波数混合域数值模拟方法,该方法将空间域引力位积分进行水平方向二维傅里叶变换,将三维空间域卷积问题转换为多个不同波数之间相互独立的空间垂向一维积分问题,一维积分垂向可离散为多个单元积分之和,每个单元采用二次形函数表征密度变化,可得出单元积分的解析表达式.该方法计算量和存储需求少,算法高度并行;保留垂向为空间域,优势之一在于可根据实际情况合理调整单元疏密程度,准确模拟任意复杂地形和密度异常体的重力异常,兼顾计算精度与计算效率;优势之二在于用形函数拟合求得积分的解析解,计算精度和效率高;充分利用一维形函数积分的高效和高精度,不同波数之间一维积分高度并行性及快速傅里叶变换的高效性,实现重力异常场三维数值模拟.设计棱柱体模型,通过数值解和解析解对比验证了该方法的正确性、适用性和高效性.针对任意复杂地形条件下的重力场及其张量的模拟问题,提出一种快速算法,对其有效性进行了验证.探究标准FFT法的截断效应对计算精度的影响,对比分析Gauss-FFT法和标准FFT扩边法两种方法的计算精度和效率,总结了二者的选取策略,结果表明选用标准FFT扩边法计算效率更高.实际地形的数值模拟表明本文算法适用于任意复杂地形的高效计算.