第二类勒让德函数的重规格化及其优化  

Renormalization and its optimization of the Legendre function of the second kind

作　　者：张捍卫 杨永勤[2] 李晓玲 张华 ZHANG Hanwei;YANG Yongqin;LI Xiaoling;ZHANG Hua(School of Civil Engineering and Geomatics,Shandong University of Technology,Zibo 255000,China;School of Surveying and Land Information Engineering,Henan Polytechnic University,Jiaozuo 454003,China)

机构地区：[1]山东理工大学建筑工程与信息工程学院,山东淄博255000 [2]河南理工大学测绘与国土信息工程学院,河南焦作454003

出　　处：《测绘学报》2024年第7期1298-1307,共10页Acta Geodaetica et Cartographica Sinica

基　　金：国家自然科学基金(42074002;41931075)。

摘　　要：椭球谐函数级数展开是地球重力场椭球谐建模的基础。然而,处理椭球谐函数级数的主要困难在于计算第二类勒让德函数。而Jekeli的重规格化方法简化了这一计算过程。本文在Jekeli重规格化的基础上,详细推导了两种基于高斯超几何函数变换的优化递归方法,同时利用这两种优化递归的方法计算了第二类勒让德函数,并将其展开到二阶导数。通过数值计算,证明了优化递归方法可以有效加快收敛,缩短计算时间,并且适用阶数更高,这使得椭球谐函数级数在实际应用中更加方便、可行。The series expansion of ellipsoidal harmonic functions is the basis for ellipsoid harmonic modeling of the Earth s gravity field.However,the main difficulty in dealing with ellipsoidal harmonics series lies in the calculation of Legendre functions of the second kind.Jekeli s renormalization method simplifies this calculation process.Based on Jekeli s renormalization,this paper deduces two optimization recursive methods based on transformations of Gaussian hypergeometric functions are derived in details.At the same time,these two optimization recursive methods are used to calculate the second type of Legendre function,and expand it to the second derivative.Numerical calculations have proven that the optimization recursive method can effectively accelerate convergence,shorten calculation time,and is applicable to higher orders,which makes the ellipsoid harmonic function series more convenient and feasible in practical applications.

关 键 词：第二类勒让德函数 缔合勒让德微分方程 重规格化 高斯超几何函数 

分 类 号：P223[天文地球—大地测量学与测量工程]