应用Bjerhammar方法确定GPS重力似大地水准面

GPS技术的发展提出了新的大地边值问题——GPS重力边值问题.本文将Bjerhammar方法应用于GPS重力问题的求解,并在给出理论公式的基础上,针对实际计算中虚拟场元的分布和求解、虚拟球半径的确定及奇异积分等问题提出了具体的解决方案.文中通过比例因子k在虚拟球半径和GPS重力数据密度间建立起联系,并推导出其近似值.在...     

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.     

GPS/重力边值问题的求解及应用

从分析GPS技术在确定地球形状中的作用入手，论述了建立一类新的大地边值问题——GPS/重力边值问题的意义，给出了GPS/重力边值问题的定义及数学描述，推导出GPS/重力边值问题的逼近解式，并给出了应用GPS/重力边值问题确定（似）大地水准面、地面垂线偏差及外部重力场的基本公式. 对GPS技术用于物理大地测量的优势及有待解决的问题进行了简要归纳.     

Joint solution of leveling and gravity data in crustal deformation monitoring

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电导率分段线性变化的水平层的点电源电场的数值解

首先给出柱坐标系中电导率分段线性变化的水平层的点源电场的二维边值问题,然后用变分法将边值问题转变为变分问题。用有限单元法解变分问题,将区域剖分成矩形单元,在单元中进行双线性函数插值,将变分方程化为线性代数方程组。解方程组,得各节点的电位值,由此可计算地表的视电阻率。 算例表明,本方法计算结果与精确解十分符合。本文还举了一个定量分析视电阻率年变化的例子。 本方法占用计算机内存约100K数量级。在MV/6000超小型计算机上计算一条电测深曲线的时间为几十秒钟。     

A Brovar-type solution of the fixed geodetic boundary-value problem

For more than 150 years gravity anomalies have been used for the determination of geoidal heights, height anomalies and the external gravity field. Due to the fact that precise ellipsoidal heights could not be observed directly, traditionally a free geodetic boundary-value problem (GBVP) had to be formulated which after linearisation is related to gravity anomalies. Since nowadays the three-dimensional positions of gravity points can be determined by global navigation satellite systems very precisely, the modern formulation of the GBVP can be based on gravity disturbances which are related to a fixed GBVP using the known topographical surface of the Earth as boundary surface. The paper discusses various approaches into the solution of the fixed GBVP which after linearization corresponds to an oblique-derivative boundary-value problem for the Laplace equation. Among the analytical solution approaches a Brovar-type solution is worked out in detail, showing many similarities with respect to the classical solution of the scalar free GBVP.     

Ellipsoidal Neumann geodetic boundary-value problem based on surface gravity disturbances: case study of Iran

An ellipsoidal Neumann type geodetic boundary-value problem (GBVP) for the computation of disturbing potential on the surface of the Earth based on the surface gravity disturbance as the boundary data is formulated. The solution methodology of the GBVP can be algorithmically summarized as follows: (i) using global navigation satellite systems (GNSS) coordinates of the gravity stations, the surface gravity disturbances are generated as the boundary data. (ii) Applying the deflection correction to the gravity disturbances to arrive at the derivative of the surface disturbing potential along the ellipsoidal normal. (iii) Removing the low frequencies part of the gravity field using harmonic expansion to degree and order 110. (iv) Using the short wavelength part of the corrected gravity disturbances derived in the previous section as the boundary data within the constructed GBVP to derive the short wavelength disturbing potential over the Earth surface. (v) The computed shortwave length signals of disturbing potentials are converted to disturbing potential values by restoring the removed effects.     

重力－重力梯度边值问题的准解

依据超定边值问题的准解理论，推导出关于重力－重力梯度边值问题的准解，其简洁易算的解析形式给联合重力和卫星重力梯度数据精化地球重力场提供了一种新方法．     

环域大地逆边值问题

从确定大地水准面实际出发,提出环域大地逆边值问题．文中首先建立环域大地道边值问题的数理模型．由于环域内边界待定,属自由边界,本质上是非线性问题．循传统给出环域逆边值问题的线性化形式．重点讨论并构造了线性化问题的解式,包括谱域内的解．     

THE USE OF VARIATIONAL METHODS AND OF ERROR DISTRIBUTION PRINCIPLES IN GROUNDWATER HYDRAULICS

Abstract

The aim of the present paper is to present some mathematical techniques for the solution of problems connected with three-dimensional steady-state groundwater flow with a free surface. The validity of Darcy's law is assumed. As no use is made of the Dupuit-Forschheimer approximation, the shape of the free surface and the velocity potential must be determined simultaneously from a non-linear boundary value problem. In order to demonstrate the use of a variational method and of error distribution principles for the solution of those problems by an example as simple as possible, we investigate the gravity flow of incompressible, homogeneous groundwater towards a circular well completely penetrating an isotropic, homogeneous, inelastic aquifer resting on a horizontal, impermeable substratum.     

陕西重力测网起算基准稳定性分析

采用经典平差方法处理重力网数据时,判断起算基准是否稳定是一个关键性问题。本文在精细化处理陕西重力测网观测资料基础上,采用线性回归法,对1992—2011年陕西重力测网起算基准进行了稳定性分析。并以泾阳M_S 4.8地震为例,分析了重力起算基准稳定性对重力网计算的影响及线性回归方法的有效性,结果表明:(1)起算基准重力值随时间变化明显;(2)起算基准扰动影响改正后,地震前后重力场及跨断裂重力段差变化特征更加符合实际情况;(3)处理重力数据及分析重力场变化特征时,必须考虑起算基准稳定性的影响。     

Gravity field determination using boundary element methods

The Boundary Element Method (BEM), a numerical technique for solving boundary integral equations, is introduced to determine the earth's gravity field. After a short survey on its main principles, we apply this method to the fixed gravimetric boundary value problem (BVP), i.e. the determination of the earth's gravitational potential from measurements of the intensity of the gravity field in points on the earth's surface. We show how to linearize this nonlinear BVP using an implicit function theorem and how to transform the linearized BVP into a boundary integral equation using the single layer representation. A Galerkin method is used to transform the boundary integral equation using the single layer representation. A Galerkin method is used to transform the boundary integral equation into a linear system of equations. We discuss the major problems of this approach for setting up and solving the linear system. The BVP is numerically solved for a bounded part of the earth's surface using a high resolution reference gravity model, measured gravity values of high density, and a 50 50 m² digital terrain model to describe the earth's surface. We obtain a gravity field resolution of 1 1 km² with an accuracy of the order 10^–3 to 10^–4 in about 1 CPU-hour on a Siemens/Fujitsu SIMD vector pipeline machine using highly sophisticated numerical integration techniques and fast equation solvers. We conclude that BEM is a powerful numerical tool for solving boundary value problems and may be an alternative to classical geodetic techniques.     

基于二次场电导率分块连续变化的三维可控源电磁有限元数值模拟

从电偶源三维地电断面可控源电磁法的二次电场边值问题及其变分问题出发,采用任意六面体单元对研究区域进行剖分,并且在单元分析中同时对电导率及二次电场进行三线性插值,实现电导率分块连续变化情况下,基于二次场的可控源电磁三维有限元数值模拟.这个新的可控源电磁三维正演方法可以模拟实际勘探中地下任意形状及电性参数连续变化的复杂模型.理论模型的计算结果表明,均匀大地计算的视电阻率误差和相位误差分别为0.002%和0.0005°.分层连续变化模型的有限元计算结果表明,其与对应的分层均匀模型解析结果有明显差异.三维异常体组合模型以及倾斜异常体等复杂模型的有限元计算结果也有效地反映了异常形态.     

A new ellipsoidal gravimetric, satellite altimetry and astronomic boundary value problem, a case study: The geoid of Iran

A new gravimetric, satellite altimetry, astronomical ellipsoidal boundary value problem for geoid computations has been developed and successfully tested. This boundary value problem has been constructed for gravity observables of the type (i) gravity potential, (ii) gravity intensity (i.e. modulus of gravity acceleration), (iii) astronomical longitude, (iv) astronomical latitude and (v) satellite altimetry observations. The ellipsoidal coordinates of the observation points have been considered as known quantities in the set-up of the problem in the light of availability of GPS coordinates. The developed boundary value problem is ellipsoidal by nature and as such takes advantage of high precision GPS observations in the set-up. The algorithmic steps of the solution of the boundary value problem are as follows:

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and of the ellipsoidal centrifugal field for the removal of the effect of global gravity and the isostasy field from the gravity intensity and the astronomical observations at the surface of the Earth.

- Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the gravity intensity and the astronomical observations at the surface of the Earth the effect of the residual masses at the radius of up to 55 km from the computational point.

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and ellipsoidal centrifugal field for the removal from the geoidal undulations derived from satellite altimetry the effect of the global gravity and isostasy on the geoidal undulations.

- Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the geoidal undulations derived from satellite altimetry the effect of the water masses outside the reference ellipsoid within a radius of 55 km around the computational point.

- Least squares solution of the observation equations of the incremental quantities derived from aforementioned steps in order to obtain the incremental gravity potential at the surface of the reference ellipsoid.

- The removed effects at the application points are restored on the surface of reference ellipsoid.

- Application of the ellipsoidal Bruns’ formula for converting the potential values on the surface of the reference ellipsoid into the geoidal heights with respect to the reference ellipsoid.

- Computation of the geoid of Iran has successfully tested this new methodology.

基于电场矢量波动方程的三维可控源电磁法有限单元法数值模拟

从可控源电磁法的基本原理出发,推导了基于电场矢量波动方程的三维边值问题,利用广义变分原理,把边值问题转换为变分问题,并引入散度条件,避免了伪解的出现,使有限元计算在理论上更加完备.在准静态近似条件下,把水平电偶极子在空中和大地的远区电场闭合表达式作为有限元计算中的区域外边界条件,解决了边界条件加载的困难;把应用于地震模拟中的伪delta函数引入到可控源电磁法中的三维有限元模拟中,消除了源点的奇异性,提高了方程组的稳定性.通过对均匀大地和层状介质模型的模拟,检验了程序的正确性,并对典型的地质体模型进行了数值模拟,分析了其变化规律.     

The GPS-gravimetry boundary value problem

Boundary value problem (BVP) plays a funda-mental role in physical geodesy that aims at determin-ing the earth’s shape and its external gravity field. TheMolodensky BVP and the Stokes BVP are typical inphysical geodesy, and the gravity anomaly is a kind ofbasic data. With the wide use of GPS, measurementaccuracy of the earth’s surface can reach one centime-ter, while that of the gravity measurement can reachμgals. Hence, it is necessary to establish a new kind ofBVP which can satisfy…     

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.     

A refined method of recovering potential coefficients from surface gravity data

Summary A new method for computing the potential coefficients of the Earth's external gravity field is presented. The gravimetric boundary-value problem with a free boundary is reduced to the problem with a fixed known telluroid. The main idea of the derivation consists in a continuation of the quantities from the physical surface to the telluroid by means of Taylor's series expansion in such a way that the terms whose magnitudes are comparable with the accuracy of today's gravity measurements are retained. Thus not only linear, but also non-linear terms are taken into account. Explicitly, the terms up to the order of the third power of the Earth's flattening are retained. The non-linear boundary-value problem on the telluroid is solved by an iteration procedure with successive approximations. In each iteration step the solution of the non-linear problem is estimated by the solutions of two linear problems utilizing the fact that the non-linear boundary condition may be split into two parts; the linear spherical approximation of the gravity anomaly whose magnitude is significantly greater than the others and the non-linear ellipsoidal corrections. Finally, in order to solve the problem in terms of spherical harmonics, the transform method composed of the fast Fourier transform and Gauss Legendre quadrature is theoretically outlined. Immediate data processing of gravity data measured on the physical Earth's surface without any continuation of gravity measurements to a reference level surface belongs to the main advantage of the presented method. This implies that no preliminary data handling is needed and that the error data propagation is, consequently, maximally suppressed.     

On the discrete outer gravity field

Summary In the present paper the gravity field of the earth in the neighbourhood of the local disturbing masses is studied. The object of the method presented consists of the approximation of the disturbing potentialT _h , which fulfils Laplace's equation outside disturbing masses, on the earth's surface the fundamental boundary value condition of gravity and in infinity it is to be regular by the approximation of the disturbing potential (or by the discrete disturbing potential)T _h , which fulfils the respective finite difference approximation of Laplace's equation and the boundary value conditions in infinity and on the earth's surface. It is also shown that the approximation of the disturbing potentialT _h has the same properties as the disturbing potentialT. The method under consideration will be derived quite generally without any hypothesis about the distribution of the mass between the earth's surface and the geoid. It commences from the gravity data related to the earth's surface only-from the given geodetic measurements.     

Direct BEM for high-resolution global gravity field modelling

The paper presents a high-resolution global gravity field modelling by the boundary element method (BEM). A direct BEM formulation for the Laplace equation is applied to get a numerical solution of the linearized fixed gravimetric boundary-value problem. The numerical scheme uses the collocation method with linear basis functions. It involves a discretization of the complicated Earth’s surface, which is considered as a fixed boundary. Here 3D positions of collocation points are simulated from the DNSC08 mean sea surface at oceans and from the SRTM30PLUS_V5.0 global topography model added to EGM96 on lands. High-performance computations together with an elimination of the far zones’ interactions allow a very refined integration over the all Earth’s surface with a resolution up to 0.1 deg. Inaccuracy of the approximate coarse solutions used for the elimination of the far zones’ interactions leads to a long-wavelength error surface included in the obtained numerical solution. This paper introduces an iterative procedure how to reduce such long-wavelength error surface. Surface gravity disturbances as oblique derivative boundary conditions are generated from the EGM2008 geopotential model. Numerical experiments demonstrate how the iterative procedure tends to the final numerical solutions that are converging to EGM2008. Finally the input surface gravity disturbances at oceans are replaced by real data obtained from the DNSC08 altimetryderived gravity data. The ITG-GRACE03S satellite geopotential model up to degree 180 is used to eliminate far zones’ interactions. The final high-resolution global gravity field model with the resolution 0.1 deg is compared with EGM2008.