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.     

A method for computing the coefficients in the product-sum formula of associated Legendre functions

The product of two associated Legendre functions can be represented by a finite series in associated Legendre functions with unique coefficients. In this study a method is proposed to compute the coefficients in this product-sum formula. The method is of recursive nature and is based on the straightforward polynomial form of the associated Legendre function's factor. The method is verified through the computation of integrals of products of two associated Legendre functions over a given interval and the computation of integrals of products of two Legendre polynomials over [0,1]. These coefficients are basically constant and can be used in any future related applications. A table containing the coefficients up to degree 5 is given for ready reference.     

On the integral formulas of Stokes and Vening Meinesz

Recently much work has been done concerning the behavior of the truncation errors of the integral formulas of Stokes and Vening Meinesz. In our paper we examine the theoretical foundations of truncation error behavior.     

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     

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 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     

Spherical integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients

New integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients are derived in this article. They provide more options for continuation of gravitational gradient combinations and extend available mathematical apparatus formulated for this purpose up to now. The starting point represents the analytical solution of the spherical gradiometric boundary value problem in the spatial domain. Applying corresponding differential operators on the analytical solution of the spherical gradiometric boundary value problem, a total of 18 integral formulas are provided. Spatial and spectral forms of isotropic kernels are given and their behaviour for parameters of a GOCE-like satellite is investigated. Correctness of the new integral formulas and the isotropic kernels is tested in a closed-loop simulation. The derived integral formulas and the isotropic kernels form a theoretical basis for validation purposes and geophysical applications of satellite gradiometric data as provided currently by the GOCE mission. They also extend the well-known Meissl scheme.     

球冠谐分析中非整阶Legendre函数的性质及其计算

局部重力场的谱方法是当前重力学的研究方向,该方法的核心问题是如何构造合适的谱函数以及如何对谱函数实施快速、有效的计算。当所研究的区域近似一个球冠时,迂冠谐函数是该区域对应的谱函数,它由非整阶勒让德（Ｌｅｇｅｎｄｒｅ）函数和三角函数组成,显然非整阶勒让德函数的构造和计算是研究球冠谐函数的关键。本文研究了非高小阶勒让德函数的性质和实用计算方法,包括如何对非整阶勒让德函数实施规格化处理。     

计算Legendre函数导数的非奇异方法

引力场关于经度和纬度方向的梯度在两极附近会产生奇异性现象,这将会给诸如重力场和静态洋流探索(GOCE,Gravityfield and stesdy-state Oceam Circulation Explorer)数据处理等引力场的研究工作带来诸多不便和困难.这里首先分析了该奇异性产生的原因,即目前采用的球坐标系自身在两极处是奇异的;然后利用Legendre函数的性质推导了一组不含任何奇异性的计算引力场梯度的计算公式;最后与常用的迭代方法进行了实例计算比较,结果表明所导出的公式不仅计算精度大大提高,而且计算用时也不会增加.     

计算Legendre函数导数的非奇异方法

引力场关于经度和纬度方向的梯度在两极附近会产生奇异性现象,这将会给诸如重力场和静态洋流探索(GOCE,Gravity field and stesdy-state Oceam Circulation Explorer)数据处理等引力场的研究工作带来诸多不便和困难。这里首先分析了该奇异性产生的原因,即目前采用的球坐标系自身在两极处是奇异的;然后利用Legendre函数的性质推导了一组不含任何奇异性的计算引力场梯度的计算公式;最后与常用的迭代方法进行了实例计算比较,结果表明所导出的公式不仅计算精度大大提高,而且计算用时也不会增加。     

Fourier transform summation of Legendre series and D-functions

The relation between D- and d-functions, spherical harmonic functions and Legendre functions is reviewed. Dmatrices and irreducible representations of the rotation group O(3) and SU(2) group are briefly reviewed. Two new recursive methods for calculations of D-matrices are presented. Legendre functions are evaluated as part of this scheme. Vector spherical harmonics in the form af generalized spherical harmonics are also included as well as derivatives of the spherical harmonics. The special dmatrices evaluated for argument equal to/2 offer a simple method of calculating the Fourier coefficients of Legendre functions, derivatives of Legendre functions and vector spherical harmonics. Summation of a Legendre series or a full synthesis on the unit sphere of a field can then be performed by transforming the spherical harmonic coefficients to Fourier coefficients and making the summation by an inverse FFT (Fast Fourier Transform). The procedure is general and can also be applied to evaluate derivatives of a field and components of vector and tensor fields.     

完全正常化缔合Legendre函数及其导数计算的综合分析

地球重力场模型的建立和应用,其数值精度和稳定性主要取决于完全正常化的缔合Legendre函数及其导数的计算精度和稳定性.总结了各种常见的计算缔合Legendre函数及其导数的方法,对各种方法进行了数值测试,分析了其精度和效率情况,提出了改进缔合Legendre函数及其导数计算精度和效率的几点意见.     

勒让德函数数值积分方法的改进

在分析已有的关于缔合勒让德函数数值积分的基础上,在数值积分上下限不确定(不是通常的±1)的情况下提出一种新的递推方法,计算结果表明新的方法既可以减少数值计算的递推过程,快速地得到计算结果,又可以节省计算机的内存。该方法可以广泛应用于需要进行勒让德函数数值积分的各个方面。     

Optimized formulas for the gravitational field of a tesseroid

Various tasks in geodesy, geophysics, and related geosciences require precise information on the impact of mass distributions on gravity field-related quantities, such as the gravitational potential and its partial derivatives. Using forward modeling based on Newton’s integral, mass distributions are generally decomposed into regular elementary bodies. In classical approaches, prisms or point mass approximations are mostly utilized. Considering the effect of the sphericity of the Earth, alternative mass modeling methods based on tesseroid bodies (spherical prisms) should be taken into account, particularly in regional and global applications. Expressions for the gravitational field of a point mass are relatively simple when formulated in Cartesian coordinates. In the case of integrating over a tesseroid volume bounded by geocentric spherical coordinates, it will be shown that it is also beneficial to represent the integral kernel in terms of Cartesian coordinates. This considerably simplifies the determination of the tesseroid’s potential derivatives in comparison with previously published methodologies that make use of integral kernels expressed in spherical coordinates. Based on this idea, optimized formulas for the gravitational potential of a homogeneous tesseroid and its derivatives up to second-order are elaborated in this paper. These new formulas do not suffer from the polar singularity of the spherical coordinate system and can, therefore, be evaluated for any position on the globe. Since integrals over tesseroid volumes cannot be solved analytically, the numerical evaluation is achieved by means of expanding the integral kernel in a Taylor series with fourth-order error in the spatial coordinates of the integration point. As the structure of the Cartesian integral kernel is substantially simplified, Taylor coefficients can be represented in a compact and computationally attractive form. Thus, the use of the optimized tesseroid formulas particularly benefits from a significant decrease in computation time by about 45 % compared to previously used algorithms. In order to show the computational efficiency and to validate the mathematical derivations, the new tesseroid formulas are applied to two realistic numerical experiments and are compared to previously published tesseroid methods and the conventional prism approach.     

Closed formulas for the direct and reverse geodetic problems

A new solution of the direct and reverse geodetic problems has been deduced without series expansion or coordinate transformation. The unknown parameters are directly expressed as explicit functions of the given parameters; the forms of the functions are closed formulas deduced by elementary mathematics using the chord of normal section. Numerical examples prove that the formulas are valid for distances from 40 km to 15 000 km on the surface of the ellipsoid.     

勒让德级数计算大地坐标主题反解的迭代算法

在地球椭球面上如果已知两点的大地经、纬度,求两点间的大地线长度及其正、反大地方位角的过程称为大地主题反解.大地主题计算用于空间技术、航空、航海、国防等现代科学技术领域.勒让德级数是解决短程大地主题计算的一种经典的方法.文献[1]中给出勒让德级数正解公式,现在给出该级数反解的算法,即迭代算法.这种迭代算法形式简单,便于理解与编程,避免了枯燥的反解公式的推导.     

The Abel-Poisson kernel and the Abel-Poisson integral in a moving tangent space

The upward-downward continuation of a harmonic function like the gravitational potential is conventionally based on the direct-inverse Abel-Poisson integral with respect to a sphere of reference. Here we aim at an error estimation of the “planar approximation” of the Abel-Poisson kernel, which is often used due to its convolution form. Such a convolution form is a prerequisite to applying fast Fourier transformation techniques. By means of an oblique azimuthal map projection / projection onto the local tangent plane at an evaluation point of the reference sphere of type “equiareal” we arrive at a rigorous transformation of the Abel-Poisson kernel/Abel-Poisson integral in a convolution form. As soon as we expand the “equiareal” Abel-Poisson kernel/Abel-Poisson integral we gain the “planar approximation”. The differences between the exact Abel-Poisson kernel of type “equiareal” and the “planar approximation” are plotted and tabulated. Six configurations are studied in detail in order to document the error budget, which varies from 0.1% for points at a spherical height H=10km above the terrestrial reference sphere up to 98% for points at a spherical height H = 6.3×10⁶km. Received: 18 March 1997 / Accepted: 19 January 1998