计算Legendre函数导数的非奇异方法

Non-Singular Formulae for Computing Derivatives of Legendre Functions

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。

Since the singularities occur near the poles in computing gradients of the gravitational field, this can cause a lot of difficulties in dealing with the gravitational field＇s data such as GOCE. In order to overcome the so-called singularities, the reasons for the singularities are analyzed firstly, i.e. , it is caused by adopting the spherical coordinates that are singular at the poles in mathematical meaning. Then non-singular computational formulae for derivatives of the Legendre＇s functions are derived with the help of their fundamental properties. At last, comparing computational results of obtained formulae with ones from general methods, it is illustrated that the arithmetic of the derivatives given in the paper is much more accurate than general ones.