عنوان مقاله [English]
Spherical harmonic transform (SHT) provides a standard tool for spectral analysis of data measured on a spherical manifold particularly in geodesy and geophysics, such as Earth’s gravity/magnetic field, topography, atmosphere, etc. Thanks to recent satellite missions such as SRTM, the spatial resolution of geoscience data decreases, leading to an increase in the degree/order of spherical harmonic. The accurate and fast computation of fully normalized associated Legendre functions (fnALF) is the main part of spherical harmonic computations.In polar region, for high degrees (n>2000), the arithmetic underflow may occur and Extended Range Arithmetic (ERA) can be applied to fix underflow problem. In Belikov method the four terms row-wise reccurrence formula is used for fnALF computation without any numerical issues.
Previous studies have only addressed the efficiency of the ERA and Belikov method without mentioning to their performnace in spherical harmonic synthesis (SHS) and analysis (SHA). This study aims at investigating the effect of row-wise and colunm-wise fnALF on SHS/SHA for high harmonic degree/order. For this purpose, we developed parallel FORTRAN95 software that utilizes Belikov and ERA to perform fnALF in the row-wise and column-wise SHT, respectively. For SHA, exact Gauss-Legendre quadrature was used that allows one to determine the level error of SHT.
In the first part of numerical results, the computation time and error of Belikov and ERA methods for calculating fnALF are compared in serial and parallel processing. Based on the results of the developed software, Belikov method is approximately 2 times faster than ERA for degrees 2160 to 64800 in both serial and parallel mode. Also for mid-latitudes, the errors for both methods are nearly the same. In polar region the error in ERA grows, while Belikov method has not latitudinal dependencies. To evaluate the performance of fnALF method on error in SHS/SHA, calculations were performed using random spherical harmonic coefficients (Cnm , Snm) between [-1, +1]. Using SHS and SHA in a closed cycle, the computed coefficients (C׳nm , S׳nm) were compared with original coefficients (Cnm , Snm). The RMS and maximum error for different degrees from N=2160 to 21600 were computed. The level error of two methods in terms of RMS is in the same level. Numerical results show that in serial mode, the row-wise SHS/SHA is approximately 1.8 times faster than column-wise method. However, in parallel computing with 10 threads, the speed up factor decreases to about 1.2.