On the numerical problems of spherical harmonics: Numerical and algebraic methods avoiding instabilities of the associated legendre's functions

摘要

Spherical harmonics are widely used to describe the structure of the Earth's gravity field. When detailed features of the Earth's gravity field are required one may be confronted with numerical problems of spherical harmonics. These problems can be treated by numerical and algebraic methods. Extending the range of real numbers on computer is preferred in numerical methods. This can be achieved by choosing a proper standard floating point arithmetic format or by extendedrange arithmetic and arbitrary precision libraries. In algebraic methods the range of the magnitudes of the spherical harmonics is reduced by algebraic manipulations. Based on the algebraic methods normalized and scaled equivalents of the spherical harmonics can be derived. In the present contribution numerical and algebraic methods avoiding the numerical problems of the spherical harmonics are discussed. Limits of the numerical and algebraic methods are studied in a numerical experiment. It is shown that current computer facilities and simple algebraic manipulations allow evaluation of the spherical harmonic expansions above degree and order 20000.