不同Legendre函数递推公式对计算球谐函数定积分的影响

Effects on the Definite Integrals of Spherical Harmonic Functions Using Different Recursive Formulas for Computing Legendre Functions

针对球谐函数定积分计算中Legendre函数递推问题展开研究,分析了标准向前列推法、Belikov法、跨阶次法、X数法以及顾及麦克劳林级数展开式对球谐函数定积分计算的影响。利用Eigen6c-4地球重力场模型计算扰动引力梯度径向分量,分析不同方法之间的差异。实验表明,不考虑麦克劳林级数展开式时4种方法的相对精度在高纬度地区较差,但计算模型扰动引力径向分量的精度一致,结合麦克劳林级数式可提高高纬度地区定积分计算的相对精度,但会降低中低纬度地区定积分计算的精度,并且对高纬度地区扰动引力径向分量的影响极小,但会严重降低低纬度地区扰动引力梯度计算的精度。

Aimed at different recursive formulas for computing Legendre functions in the computation of definite integrals of spherical harmonic functions,this paper analyzes the effects of standard forward column method,Belikov method,Swarztrauber method and X-Number method on computing Legendre functions.The radial components of mean disturbing gravity gradient are computed using Eigen6c-4 earth gravity field model in the application of standard forward column method,Belikov method,Swarztrauber method and X-Number method.From the tests,it can be claimed that these four methods have poor accuracies in the high-latitude areas but have the same accuracies in the computation of radial component of mean disturbing gravity gradient.If the Maclaurin formula is applied,the precision of definite integrals of spherical harmonic functions can be improved significantly in the high-latitude areas but can deteriorate the accuracies in mid-and low-latitude areas,and Maclaurin formula has little effect on the radial components of mean disturbing gravity gradient in high-latitude areas but can seriously reduce the accuracies of mean disturbing gravity gradient in low-latitude areas.