随机Dirichlet问题解的调和展开

针对重力学随机Dirichlet问题,通过适当地对边界检验函数的分解,并在随机边界样本空间中提取确定性部分的对偶基,本文将随机Dirichlet问题的一般解展开为一随机系数的调和级数形式。     

A numerically efficient solution of the second-order design problem

The aim of the paper is the creation of an efficient algorithm to set up the normal equation system which produces the observation weights under approximation of the inverse of a given criterion matrix. The objective of minimizing the required computer memory and the number of computer operations is reached by analytically formulating the normal equation coefficients for one and two-dimensional networks.     

A linear regression solution to the spatial autocorrelation problem

The Moran Coefficient spatial autocorrelation index can be decomposed into orthogonal map pattern components. This decomposition relates it directly to standard linear regression, in which corresponding eigenvectors can be used as predictors. This paper reports comparative results between these linear regressions and their auto-Gaussian counterparts for the following georeferenced data sets: Columbus (Ohio) crime, Ottawa-Hull median family income, Toronto population density, southwest Ohio unemployment, Syracuse pediatric lead poisoning, and Glasgow standard mortality rates, and a small remotely sensed image of the High Peak district. This methodology is extended to auto-logistic and auto-Poisson situations, with selected data analyses including percentage of urban population across Puerto Rico, and the frequency of SIDs cases across North Carolina. These data analytic results suggest that this approach to georeferenced data analysis offers considerable promise. Received: 18 February 1999/Accepted: 17 September 1999     

A new series solution of Molodensky's problem

A complete series solution of Molodensky's boundary-value problem is derived using, instead of an integral equation, analytical continuation by means of power series. This solution is shown to be equivalent, term by term, to the Molodensky-Brovar series, but is simpler and practically more convenient.

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Zusammenfassung Eine vollst?ndige Reihenl?sung des Problems von Molodensky wird hergeleitet, wobei anstatt einer Integralgleichung die analytische Fortsetzung mittels Potenzreihen zugrunde gelegt wird. Es zeigt sich, dass diese L?sung gliedweise ?quivalent zur Reihe von Molodensky-Brovar ist, aber sie ist einfacher und praktisch brauchbarer.

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Résumé On déduit une série qui donne une solution complète du problème de Molodensky, en utilisant, au lieu d'une équation intégrale, la continuation analytique par une série de puissances. Il s'ensuit que cette solution est équivalente à la série de Molodensky-Brovar, mais elle est plus simple et plus pratique.

     

大地主题解算方法综述

大地主题解算是大地测量中的重要问题,由于椭球的复杂性,随之产生的解算方法也是多种多样。本文分析了大地主题解算的一般方法及其公式的适用范围,指出了它们的特点和不足,讨论了现今常用解算方法存在的问题和进一步的研究方向。     

广义非线性动态最小二乘问题的一个直接解算方法

构建“数字地球”、“数字国家”等数字化科学工程的基础是数据[1 ] ,其数据具有多源、多维、多种类型、多种时态、多种精度并具有非线性特征等特点[2 ] ,首先要进行数据处理并应采用全新的广义非线性动态最小二乘法[3] [4] ,数据处理方法的核心是广义非线性动态最小二乘问题参数估计的函数模型及其解算方法 ,迄今国内外对这方面的研究尚不多。本文在作者前期研究、提出的广义非线性动态最小二乘函数模型参数估计迭代法求解[5] 的基础上 ,进一步研究、提出了一种广义非线性动态最小二乘模型参数估计的直接解算方法 ,将问题分离 ,把待求参数减半 ,直接求解。从而大大降低求解问题的维数 ,大大减少计算难度和计算工作量 ,这是国内外首次研究提出的一种比迭代法更快速、更有效、更科学的解算方法。为多源、多类、多时态数据处理开辟了一新途径 ,也大大扩大了广义非线性动态最小二乘法的应用面     

基于虚拟观测的病态问题解法

在大地测量数据处理领域中,处理病态问题的主要方法有:岭估计方法、奇异值分解法(SVD)、Tik-honov正则化方法等,但是这些方法大多数是强调数学上的意义,没有充分联系大地测量的实际情况,因此不利于在测绘领域病态问题本质的理解和研究。为使病态问题的求解具有实际的物理意义,提出了基于虚拟观测的岭估计方法。该方法将先验约束条件作为一类互相独立的虚拟观测值,从而把病态问题转化为测量平差问题,然后运用Helmert方差估计法来确定岭参数。该方法还可以得到的参数之间的权矩阵,用它来代替虚拟观测值的权矩阵,重新对参数进行计算,则实现了该方法向广义岭估计的推广。实际算例分析的结果表明该方法不仅计算简单而且能保证结果精确。     

大地问题中截面椭圆弧长的改进算法

为了避免大椭圆弧长算法中需要对球面方位角和极距角进行繁琐的象限判断问题,该文通过空间向量分析和椭球几何关系推导,给出了一种计算简洁、具有通用性的截面椭圆弧长算法。算例分析表明,该算法可以满足椭球面上两点间大地距离计算的应用需要,当大地距离小于2000km时,求得的截面椭圆弧长与较严密公式求得的大地线长的误差仅为厘米级。     

On the geodetic boundary value problem for a fixed boundary surface—A satellite approach

Employing satellite-geometrical methods, the physical surface of the earth may be assumed to be known, while gravity measurements yield thelength of the gravity vector (including contributions from rotation). The problem then is to determine gravitational potential from such gravity observations. The corresponding linearized problem is an oblique derivative problem. The problem was discussed by Almqvist (1959), Koch (1970, 1971) and Koch and Pope (1972). Our presentation gives proofs for the existence (and uniqueness) of the solution in the non-linear case. The general implicit function theorem (in Banach spaces) is used to provewellposedness at least when data are close to given standard values (closeness is defined either in terms of Hölder or Sobolev norms). Iterative methods for solution by linear operators are given. The linearized problem is solved by harmonic reduction to an internal sphere in a generalization of the method by the first author for the Stokes problem. Also deflections of the vertical are treated.     

Computer algebra solution of the GPS N-points problem

A computer algebra solution is applied here to develop and evaluate algorithms for solving the basic GPS navigation problem: finding a point position using four or more pseudoranges at one epoch (the GPS N-points problem). Using Mathematica 5.2 software, the GPS N-points problem is solved numerically, symbolically, semi-symbolically, and with Gauss–Jacobi, on a work station. For the case of N > 4, two minimization approaches based on residuals and distance norms are evaluated for the direct numerical solution and their computational duration is compared. For N = 4, it is demonstrated that the symbolic computation is twice as fast as the iterative direct numerical method. For N = 6, the direct numerical solution is twice as fast as the semi-symbolic, with the residual minimization requiring less computation time compared to the minimization of the distance norm. Gauss–Jacobi requires eight times more computation time than the direct numerical solution. It does, however, have the advantage of diagnosing poor satellite geometry and outliers. Besides offering a complete evaluation of these algorithms, we have developed Mathematica 5.2 code (a notebook file) for these algorithms (i.e., Sturmfel’s resultant, Dixon’s resultants, Groebner basis, reduced Groebner basis and Gauss–Jacobi). These are accessible to any geodesist, geophysicist, or geoinformation scientist via the GPS Toolbox () website or the Wolfram Information Center ().

Fast numerical solution of the linearized Molodensky problem

 When standard boundary element methods (BEM) are used in order to solve the linearized vector Molodensky problem we are confronted with two problems: (1) the absence of O(|x|⁻²) terms in the decay condition is not taken into account, since the single-layer ansatz, which is commonly used as representation of the disturbing potential, is of the order O(|x|⁻¹) as x→∞. This implies that the standard theory of Galerkin BEM is not applicable since the injectivity of the integral operator fails; (2) the N×N stiffness matrix is dense, with N typically of the order 10⁵. Without fast algorithms, which provide suitable approximations to the stiffness matrix by a sparse one with O(N(logN) ^s ), s≥0, non-zero elements, high-resolution global gravity field recovery is not feasible. Solutions to both problems are proposed. (1) A proper variational formulation taking the decay condition into account is based on some closed subspace of co-dimension 3 of the space of square integrable functions on the boundary surface. Instead of imposing the constraints directly on the boundary element trial space, they are incorporated into a variational formulation by penalization with a Lagrange multiplier. The conforming discretization yields an augmented linear system of equations of dimension N+3×N+3. The penalty term guarantees the well-posedness of the problem, and gives precise information about the incompatibility of the data. (2) Since the upper left submatrix of dimension N×N of the augmented system is the stiffness matrix of the standard BEM, the approach allows all techniques to be used to generate sparse approximations to the stiffness matrix, such as wavelets, fast multipole methods, panel clustering etc., without any modification. A combination of panel clustering and fast multipole method is used in order to solve the augmented linear system of equations in O(N) operations. The method is based on an approximation of the kernel function of the integral operator by a degenerate kernel in the far field, which is provided by a multipole expansion of the kernel function. Numerical experiments show that the fast algorithm is superior to the standard BEM algorithm in terms of CPU time by about three orders of magnitude for N=65 538 unknowns. Similar holds for the storage requirements. About 30 iterations are necessary in order to solve the linear system of equations using the generalized minimum residual method (GMRES). The number of iterations is almost independent of the number of unknowns, which indicates good conditioning of the system matrix. Received: 16 October 1999 / Accepted: 28 February 2001     

贝赛尔大地主题解算分析

贝赛尔大地主题解算是少数适合长距离大地主题计算的方法之一。文章通过对贝赛尔大地主题解算进行计算分析,发现贝赛尔大地主题反算中的大地线长计算精度受起点方位角的影响很大,误差可达8m。为了消去这一巨大误差,本文提出在大地主题反算时互换大地线起点和终点的方法,计算结果表明该方法可以有效消除方位角对大地线长误差的影响。     

Numerical solution of the linearized fixed gravimetric boundary-value problem

The fixed gravimetric boundary-value problem (FGBVP) represents an exterior oblique derivative problem for the Laplace equation. Terrestrial gravimetric measurements located by precise satellite positioning yield oblique derivative boundary conditions in the form of surface gravity disturbances. In this paper, we discuss the boundary element method (BEM) applied to the linearized FGBVP. In spite of previous BEM approaches in geodesy, we use the so-called direct BEM formulation, where a weak formulation is derived through the method of weighted residuals. The collocation technique with linear basis functions is applied for deriving the linear system of equations from the arising boundary integral equations. The nonstationary iterative biconjugate gradient stabilized method is used to solve the large-scale linear system of equations. The standard MPI (message passing interface) subroutines are implemented in order to perform parallel computations. The proposed approach gives a numerical solution at collocation points directly on the Earth’s surface (on a fixed boundary). Numerical experiments deal with (i) global gravity field modelling using synthetic data (surface gravity disturbances generated from a global geopotential model (GGM)) (ii) local gravity field modelling in Slovakia using observed gravity data. In order to extend computations, the memory requirements are reduced using elimination of the far-zone effects by incorporating GGM or a coarse global numerical solution obtained by BEM. Statistical characteristics of residuals between numerical solutions and GGM confirm the reliability of the approach and indicate accuracy of numerical solutions for the global models. A local refinement in Slovakia results in a local (national) quasigeoid model, which when compared with GPS-levelling data, does not make a large improvement on existing remove-restore-based models.     

Advances in the numerical solution of the linear molodensky problem

The solution of the linear Molodensky problem by analytical continuation to point level is numerically the most convenient of all the theoretically equivalent solutions. It is obtained by successively applying the same integral operator and it does not depend explicitly on the terrain inclination. However, its dependence on the computation point restricts somehow the computational efficiency. The use of the Fourier transform for the evaluation of the integral operator in planar approximation lessens significantly the burden of computations. Using this spectral approach, the problem has been reformulated and solved in the frequency domain. Moreover, it is shown that the solution can be easily split into two steps: (a) “downward” continuation to sea level, which is independent of the computation point, and (b) “upward” continuation from sea to point level, using the values computed at sea level. Such a treatment not only simplifies the formulas and increases the numerical efficiency but also clarifies the physical interpretation and the theoretical equivalence of the continuation solution with respect to the other solution types. Numerical tests have been performed to investigate which terms in the Molodensky series are of significance for geoid and deflection computations. The practical difficulty of differences in the grid spacings of gravity and height data has been overcome by frequency domain interpolation. Presented at theXIX IUGG General Assembly, Vancouver, B.C., August 9–22, 1987.     

A bias-free geodetic boundary value problem approach to height datum unification

A geodetic boundary value problem (GBVP) approach has been formulated which can be used for solving the problem of height datum unification. The developed technique is applied to a test area in Southwest Finland with approximate size of 1.5° × 3° and the bias of the corresponding local height datum (local geoid) with respect to the geoid is computed. For this purpose the bias-free potential difference and gravity difference observations of the test area are used and the offset (bias) of the height datum, i.e., Finnish Height Datum 2000 (N2000) fixed to Normaal Amsterdams Peil (NAP) as origin point, with respect to the geoid is computed. The results of this computation show that potential of the origin point of N2000, i.e., NAP, is (62636857.68 ± 0.5) (m²/s²) and as such is (0.191 ± 0.003) (m) under the geoid defined by W ₀ = 62636855.8 (m²/s²). As the validity test of our methodology, the test area is divided into two parts and the corresponding potential difference and gravity difference observations are introduced into our GBVP separately and the bias of height datums of the two parts are computed with respect to the geoid. Obtaining approximately the same bias values for the height datums of the two parts being part of one height datum with one origin point proves the validity of our approach. Besides, the latter test shows the capability of our methodology for patch-wise application.     

A geostatistical approach to the change-of-support problem and variable-support data fusion in spatial analysis

A key issue to address in synthesizing spatial data with variable-support in spatial analysis and modeling is the change-of-support problem. We present an approach for solving the change-of-support and variable-support data fusion problems. This approach is based on geostatistical inverse modeling that explicitly accounts for differences in spatial support. The inverse model is applied here to produce both the best predictions of a target support and prediction uncertainties, based on one or more measurements, while honoring measurements. Spatial data covering large geographic areas often exhibit spatial nonstationarity and can lead to computational challenge due to the large data size. We developed a local-window geostatistical inverse modeling approach to accommodate these issues of spatial nonstationarity and alleviate computational burden. We conducted experiments using synthetic and real-world raster data. Synthetic data were generated and aggregated to multiple supports and downscaled back to the original support to analyze the accuracy of spatial predictions and the correctness of prediction uncertainties. Similar experiments were conducted for real-world raster data. Real-world data with variable-support were statistically fused to produce single-support predictions and associated uncertainties. The modeling results demonstrate that geostatistical inverse modeling can produce accurate predictions and associated prediction uncertainties. It is shown that the local-window geostatistical inverse modeling approach suggested offers a practical way to solve the well-known change-of-support problem and variable-support data fusion problem in spatial analysis and modeling.     

测地主题正反解解算

由测地坐标系中大地线的微分方程式推导出其微分关系式，得出在地球椭球面上基于测地坐标进行正反解的算法和公式，它与大地主题解算公式相比，更为简捷明了。由实际计算数据表明，对于100km以下的距离解算，它亦能达到相当高的精度。因此测地坐标的点位表述不仅可用于DEM和GIS三维可视化，也可用于三维GIS建模以及空间度量和分析。     

A continuous velocity field for Norway

In Norway, as in the rest of Fennoscandia, the process of Glacial Isostatic Adjustment causes ongoing crustal deformation. The vertical and horizontal movements of the Earth can be measured to a high degree of precision using GNSS. The Norwegian GNSS network has gradually been established since the early 1990s and today contains approximately 140 stations. The stations are established both for navigation purposes and for studies of geophysical processes. Only a few of these stations have been analyzed previously. We present new velocity estimates for the Norwegian GNSS network using the processing package GAMIT. We examine the relation between time-series length and precision. With approximately 3.5 years of data, we are able to reproduce the secular vertical rate with a precision of 0.5 mm/year. To establish a continuous crustal velocity field in areas where we have no GNSS receivers or the observation period is too short to obtain reliable results, either interpolation or modeling is required. We experiment with both approaches in this analysis by using (i) a statistical interpolation method called Kriging and (ii) a GIA forward model. In addition, we examine how our vertical velocity field solution is affected by the inclusion of data from repeated leveling. Results from our geophysical model give better estimates on the edge of the network, but inside the network the statistical interpolation method performs better. In general, we find that if we have less than 3.5 years of data for a GNSS station, the interpolated value is better than the velocity estimate based on a single time-series.     

The solution of the general geodetic boundary value problem by least squares

 A general scheme is given for the solution in a least-squares sense of the geodetic boundary value problem in a spherical, constant-radius approximation, both uniquely and overdetermined, for a large class of observations. The only conditions are that the relation of the observations to the disturbing potential is such that a diagonalization in the spectrum can be found and that the error-covariance function of the observations is isotropic and homogeneous. Most types of observations used in physical geodesy can be adjusted to fit into this approach. Examples are gravity anomalies, deflections of the vertical and the second derivatives of the gravity potential. Received: 3 November 1999 / Accepted: 25 September 2000