A new method for computing the ellipsoidal correction for Stokes's formula

 This paper generalizes the Stokes formula from the spherical boundary surface to the ellipsoidal boundary surface. The resulting solution (ellipsoidal geoidal height), consisting of two parts, i.e. the spherical geoidal height N ₀ evaluated from Stokes's formula and the ellipsoidal correction N ₁, makes the relative geoidal height error decrease from O(e ²) to O(e ⁴), which can be neglected for most practical purposes. The ellipsoidal correction N ₁ is expressed as a sum of an integral about the spherical geoidal height N ₀ and a simple analytical function of N ₀ and the first three geopotential coefficients. The kernel function in the integral has the same degree of singularity at the origin as the original Stokes function. A brief comparison among this and other solutions shows that this solution is more effective than the solutions of Molodensky et al. and Moritz and, when the evaluation of the ellipsoidal correction N ₁ is done in an area where the spherical geoidal height N ₀ has already been evaluated, it is also more effective than the solution of Martinec and Grafarend. Received: 27 January 1999 / Accepted: 4 October 1999     

The integral formulas of the associated Legendre functions

A new kind of integral formulas for ${\bar{P}_{n,m} (x)}$ is derived from the addition theorem about the Legendre Functions when n ? m is an even number. Based on the newly introduced integral formulas, the fully normalized associated Legendre functions can be directly computed without using any recursion methods that currently are often used in the computations. In addition, some arithmetic examples are computed with the increasing degree recursion and the integral methods introduced in the paper respectively, in order to compare the precisions and run-times of these two methods in computing the fully normalized associated Legendre functions. The results indicate that the precisions of the integral methods are almost consistent for variant x in computing ${\bar{P}_{n,m} (x)}$ , i.e., the precisions are independent of the choice of x on the interval [0,1]. In contrast, the precisions of the increasing degree recursion change with different values on the interval [0,1], particularly, when x tends to 1, the errors of computing ${\bar{P}_{n,m} (x)}$ by the increasing degree recursion become unacceptable when the degree becomes larger and larger. On the other hand, the integral methods cost more run-time than the increasing degree recursion. Hence, it is suggested that combinations of the integral method and the increasing degree recursion can be adopted, that is, the integral methods can be used as a replacement for the recursive initials when the recursion method become divergent.     

Recursive computation of finite difference of associated Legendre functions

The existing methods to compute the definite integral of associated Legendre function (ALF) with respect to the argument suffer from a loss of significant figures independently of the latitude. This is caused by the subtraction of similar quantities in the additional term of their recurrence formulas, especially the finite difference of their values between two endpoints of the integration interval. In order to resolve the problem, we develop a recursive algorithm to compute their finite difference. Also, we modify the algorithm to evaluate their definite integrals assuming that their values at one endpoint are known. We numerically confirm a significant increase in computing precision of the integral by the new method. When the interval is one arc minute, for example, the gain amounts to 2–4 digits for the degree of harmonics in the range 2 ≤ n ≤ 2,048. This improvement in precision is achieved at a negligible increase in CPU time, say less than 5%.     

New relations among associated Legendre functions and spherical harmonics

Several new relations among associated Legendre functions (ALFs) are derived, most of which relate a product of an ALF with trigonometric functions to a weighted summation over ALFs, where the weights only depend on the degree and order of the ALF. These relations are, for example, useful in applications such as the computation of geopotential coefficients and computation of ellipsoidal corrections in geoid modelling. The main relations are presented in both their unnormalised and fully normalised (4π-normalised) form. Several approaches to compute the weights involved are discussed, and it is shown that the relations can also be applied in the case of first- and second-order derivatives of ALFs, which may be of use in analysis of satellite gradiometry data. Finally, the derived relations are combined to provide new identities among ALFs, which contain no dependency on the colatitudinal coordinate other than that in the ALFs themselves.     

Recurrence relations for integrals of Associated Legendre functions

Recurrence relations for the evaluation of the integrals of associated Legendre functions over an arbitrary interval within (0°, 90°) have been derived which yield sufficiently accurate results throughout the entire range of their possible applications. These recurrence relations have been used to compute integrals up to degree 100 and similar computations can be carried out without any difficulty up to a degree as high as the memory in a computer permits. The computed values have been tested with independent check formulae, also derived in this work; the corresponding relative errors never exceed 10⁻²³ in magnitude. Contribution from the Earth Physics Branch No. 719     

A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions

 Spherical harmonic expansions form partial sums of fully normalised associated Legendre functions (ALFs). However, when evaluated increasingly close to the poles, the ultra-high degree and order (e.g. 2700) ALFs range over thousands of orders of magnitude. This causes existing recursion techniques for computing values of individual ALFs and their derivatives to fail. A common solution in geodesy is to evaluate these expansions using Clenshaw's method, which does not compute individual ALFs or their derivatives. Straightforward numerical principles govern the stability of this technique. Elementary algebra is employed to illustrate how these principles are implemented in Clenshaw's method. It is also demonstrated how existing recursion algorithms for computing ALFs and their first derivatives are easily modified to incorporate these same numerical principles. These modified recursions yield scaled ALFs and first derivatives, which can then be combined using Horner's scheme to compute partial sums, complete to degree and order 2700, for all latitudes (except at the poles for first derivatives). This exceeds any previously published result. Numerical tests suggest that this new approach is at least as precise and efficient as Clenshaw's method. However, the principal strength of the new techniques lies in their simplicity of formulation and implementation, since this quality should simplify the task of extending the approach to other uses, such as spherical harmonic analysis. Received: 30 June 2000 / Accepted: 12 June 2001     

A formula for computing the gravity disturbance from the second radial derivative of the disturbing potential

 A formula for computing the gravity disturbance and gravity anomaly from the second radial derivative of the disturbing potential is derived in detail using the basic differential equation with spherical approximation in physical geodesy and the modified Poisson integral formula. The derived integral in the space domain, expressed by a spherical geometric quantity, is then converted to a convolution form in the local planar rectangular coordinate system tangent to the geoid at the computing point, and the corresponding spectral formulae of 1-D FFT and 2-D FFT are presented for numerical computation. Received: 27 December 2000 / Accepted: 3 September 2001     

完全规格化缔合勒让德函数及其导数的递推算法与适用范围比较

完全规格化缔合勒让德函数及其导数常用标准向前列递推算法和标准向前行递推算法进行计算。基于第一、第二相对数值精度标准对两种算法的适用范围进行分析比较,计算结果表明,标准向前列递推算法的适用范围大于标准向前行递推算法,说明前者优于后者;结果同时还表明,完全规格化缔合勒让德函数与其导数同一种算法的适用范围也相同,并指出了二者适用范围相同的原因。     

A method of evaluating the truncation error coefficients for geoidal height

Neglecting distant zones in the computation of geoidal height using Stokes' formula gives rise to some truncation error. This truncation error is expressible as a weighted summation of the zonal harmonic components of the gravity anomaly. Making use of the well-known properties of Legendre polynomials, a compact method of computing these theoretical coefficients has been developed in this paper.     

伴随勒让德函数的一个注记

证明了伴随勒让德函数Ｐｎｋ （ｘ） 的导函数Ｐ′ｎｋ （ｘ） 相应于次ｋ 在〔－ １ ， １〕上关于权函数ρ（ｘ） ＝ １ － ｘ２ 具有部分正交性。     

基于扩展空间句法的城区人口密度估算方法

城市人口密度是人口空间分布的重要指标,对城市交通、安全、规划和管理都有着重要意义。针对传统的人口密度估算方法在数据要求、可操作性以及精度方面的不足,本文提出一种基于扩展空间句法的城市人口密度估算方法。该方法通过城市空间可达性与城市行人流量的相关关系,估算城市人口密度的空间分布情况。经过实验验证表明,与传统估算方法相比,该方法需要的人口统计数据较少,可操作性强,与G IS软件无缝融合,通过反距离权重法(IDW)可使模拟估算结果连续,试验验证表明该方法适用于城镇地区人口密度估算。     

一种利用磁偏角地磁图自动计算磁偏角的方法

磁偏角地磁图是较理想的确定磁偏角的工具。当需要实时跟踪磁偏角时,可以根据地磁图采集数据建立数据库进行实时查询,但是这种方法工作量较大,过程也比较繁琐。因此,文中提出了一种利用磁偏角地磁图基于二维拉格朗日插值多项式的磁偏角自动计算方法,并详细论述了其原理。实验结果表明,这种方法是可行的,精度较高并可大大提高工作效率。     

基于动态水位的涉水距离和深度计算方法

涉水距离和深度计算是登陆决策分析中的重要内容。针对传统方法的不足,本文提出了一种基于动态水位的涉水距离和深度的计算方法。根据瞬时水位,动态地求定实时水线和登陆点之间的坡度,从而可动态地得到涉水距离和深度。实验证明,此方法与传统方法相比,所计算的涉水距离和深度在准确性上有明显提高。     

A strict formula for geoid-to-quasigeoid separation

The paper presented by Flury and Rummel (J Geod 83:829–847, 2009) discusses an important topographic correction to the traditional formula for the quasigeoid-to-geoid separation. Nevertheless, as their formula is approximate, the reader may ask for its relation to the strict one (defined as the one consistent with Bruns’s formula and the boundary condition of physical geodesy), which is now derived. Although the result formally differs from that of Flury and Rummel, we show that the two formulas agree to the centimetre level all over the Earth. We also discuss the practical computation of the topographic correction.     

采用Belikov列推和跨阶次递推方法计算超高阶缔合勒让德函数

超高阶球谐重力场模型的精确构制与快速计算取决于缔合勒让德函数的计算方法。在前人研究的基础上,文中对适合超高阶缔合勒让德函数计算的Belikov列推和跨阶次递推方法进行介绍,为验证精度,通过两种途径对计算结果进行检验,并比较其计算速度。结果表明,采用两种算法得到的每个勒让德函数的绝对精度均优于10-12,在低阶,跨阶次递推方法的计算用时大约是Belikov列推法的2倍,随着阶数的升高,跨阶次递推算法表现出明显的速度优势。     

一种云计算系统的瓦片数据组织方法

针对云计算系统中文件数据存储效率较低的问题,文章以Hadoop云计算系统为研究对象,改进了Hadoop-pvfs分布式系统,以提高Hadoop底层文件系统的存储效率,同时使得数据的更新可以在文件的任何位置;并提出一种多层次、位置相关的瓦片数据组织方法。实验结果表明,上述方法相对于Hadoop瓦片服务器系统吞吐效率有较大提高,可以为建设高性能、海量存储的GIS数据库服务系统提供参考。     

一种利用扩展磁偏角数据自动计算磁偏角的方法

提出一种基于扫描磁偏角地磁图自动计算磁偏角的方法,把一维磁偏角数据扩展到二维数据,将有关图像校正的方法用在磁偏角的求取上,即在扫描磁偏角地磁图中以图上的地理坐标、扩展磁偏角等值线坐标以及扩展磁偏角等变线坐标所对应的像素坐标为媒介,通过非线性变换,最终建立起地理坐标、磁偏角等值线以及磁偏角等变线之间的相互对应关系,以解决...     

A new formula for evaluating the truncation error coefficient

In this paper, a new formula for evaluating the truncation coefficientQ _n is derived from recurrence relations of Legendre polynomials. The present formula has been conveniently processed by an electronic computer, providing the value ofQ _n up to a degreen=49 which are exactly equal to those of Paul (1973).     

边缘系数建筑物雷达点云数据简化方法

针对基于LiDAR点云数据进行建筑物自动重建中存在的数据冗余问题,该文设计了一种定量描述激光点位于地物边缘区几率大小的指标——边缘系数,并据此提出了基于边缘系数的建筑物LiDAR点云数据简化方法。该方法利用激光点与其邻域点的位置、数量及分布计算该点的边缘系数,通过试验分析确定边缘系数的阈值并对点云数据进行分割,最后保留建筑物边缘区域的点,实现点云数据的简化。实验表明,该方法在对点云数据进行高效压缩的同时有效保留了位于地物边缘处的点云,有助于提高海量点云数据处理能力和建筑物重建效率。