SHAVEL: A program for the spherical harmonic analysis of a horizontal vector field sampled in an equiangular grid on a sphere

Abstract

A method for performing a spherical harmonic analysis, using observed horizontal components of a tangent vector on a sphere, is presented. The vector data samples are assumed to be provided in an equiangular grid, which essentially simplifies the least-squares analysis by making use of (1) the block diagonal structure of the normal equations of least squares, (2) the even–odd symmetry of the associated Legendre functions, and (3) the fast Fourier transform of mix-radix. The correct function of the program and its numerical precision is verified by applying it to a data set, derived by evaluating a given set of vector spherical harmonic coefficients. That the program works correctly is demonstrated by the excellent agreement between the input and output spherical harmonic coefficients. Program summaryProgram Title: SHAVELProgram Files doi:http://dx.doi.org/10.17632/nppz4y7wg7.1Licensing provisions: GPLv3Programming language: Fortran 2003, LinuxExternal routines: FFTPACK5.1, CPC program library SPHANClassification: 4.9, 4.10, 4.11Nature of problem: The least-squares analysis of horizontal vector field sampled in an equiangular grid on a sphere in terms of horizontal vector spherical harmonics.Solution method: The vector spherical harmonic coefficients of a horizontal vector field are estimated by the method of least-squares adjustment of data samples distributed in an equiangular grid on a sphere. For such a regular grid the normal matrix is sparse and allows the system of the normal equations to be decomposed into a series of subsystems according to azimuthal order m. The solution of each subsystem is sought by the Gauss elimination. The fast Fourier transform of mix-radix is implemented in (i) setting up the right-hand sides of the normal equations, and (ii) performing the spherical harmonic synthesis where the series of spherical harmonics are summed.