[G02-1] Theory and methods

[G02-1-01] Rectangular rotation of spherical harmonic expansion of arbitrary high degree and order

^○Toshio Fukushima (National Astronomical Observatory, Tokyo, Japan)

In order to move the polar singularity of arbitrary spherical harmonic expansion to a point on the equator, we rotate the expansion around the y-axis by 90 degrees such that the original x-axis becomes a new pole. The spherical harmonic expansion coefficients are transformed by multiplying a special value of Wigner's D-matrix and a normalization factor. Thanks to the choice of not the x- but the y-axis as the rotation axis, the special value becomes a real-valued function: Wigner's d-function of the angle as 90 degrees. Its realness results the transformation of Cnm and Snm completely separated. The transformation matrix is unchanged whether the coefficients are fully-normalized or Schmidt quasi-normalized. This is because the rotation is restricted within the coefficients of the same degree, and therefore the difference in the normalization factors do not alter the transformation formula. The d-function is stably computed by an increasing-degree fixed-orders three-term recurrence formula. The two seed values can be also computed recursively. The underflow problem during the recursions is effectively resolved by the so-called X-number formulation (Fukushima, 2012, J. Geodesy, 86, 271--285). As an example, we obtained 2190x2190 coefficients of the rectangular rotated spherical harmonic expansion of EGM2008. The obtained coefficients are of the 15 digits consistency with the original one because the transformation is conducted by the quadruple precision X-number formulation. A proper combination of the original and the rotated expansions is realized by switching them at the parallels of 45 degrees latitude. The combined spherical harmonic expansions will be useful in (i) integrating the polar orbits of artificial satellites precisely, and (ii) synthesizing/analyzing the gravitational/geomagnetic potentials and their derivatives accurately in the high latitude regions including the arctic and antarctic area.