扰动引力梯度张量无奇异性最小二乘法计算模型的建立

Construction of Nonsingular Least Squares Computational Model of Disturbing Gravity Gradient Tensors

利用地心球坐标来表示扰动引力梯度张量，当计算点趋近于两极时，由于Legendre函数的一阶和二阶导数以及分母上所含余纬的正弦函数，将导致扰动引力梯度张量的计算出现无穷大。本文引入了Legendre函数的一阶和二阶导数以及mPnm（cosθ）／sinθ无奇异性的计算公式，进一步推导了m2Pnm（cosθ）／sin2θ无奇异性的计算公式，并且将Legendre函数的一阶和二阶导数以及mPnm（cosθ）／sinθ、m2Pnm。（cosθ）／sin2θ无奇异性的计算公式代入到了扰动引力梯度张量的最小二乘法计算模型中，建立了扰动引力梯度张量无奇异性的最小二乘法计算模型。

If the disturbing gravity gradient tensors are expressed by earth - centered spherical coordinates in the local north - oriented reference frame, the first - and second - order derivatives of the Legendre functions and the sine function of colatitude in the denominator will go to infinity when the computational point is approaching the poles. The nonsingular computational formulas of the first - and second - order derivatives of the Legendre functions of the colatitude and the mPnm （cosθ）/sinθ nonsingular formula were introduced, and the nonsingular computational formula of m2Pnm（cosθ）/sin2θ were also deduced. Then the nonsingular computational formulas of the first - and second - order derivatives of the Legendre functions and the mPnm（cosθ）/sinθ,m2Pnm （cosθ）/sin2θ were taken into the least squares computational model of the disturbing gravity gradient tensors, and the nonsingular least squares computational models of the disturbing gravity gradient tensors were constructed.