Numerical Methods for Harmonic Analysis on the Sphere

TLDR: In this article, the authors present some numerical methods for estimating spherical harmonic coefficients from data sampled on the sphere, where the data may be given in the form of area means or of point values, and it may be free from errors or affected by measurement noise.

Abstract: : This report presents some numerical methods for estimating spherical harmonic coefficients from data sampled on the sphere. The data may be given in the form of area means or of point values, and it may be free from errors or affected by measurement 'noise'. The case discussed to greatest length is that of complete, global data sets on regular grids (i.e., lines of latitude and longitude, the latter, at least, separated by constant interval); the case where data are sparsely and irregularly distributed is also considered in some detail. The first section presents some basic properties of spherical harmonics, stressing their relationship to two-dimensional Fourier series. Algorithms for the evaluation of the harmonic coefficients by numerical quadratures are given here, and it is shown that the number of operations is the order of N cubed for equal angular grids, where N is the number of lines of latitude, or 'Nyquist frequency', of the grid. The second section introduces a quadratic measure for the error in the estimation of the coefficients by linear techniques. This is the error measure of least squares collocation, which is a method that can be used for harmonic analysis. Efficient algorithms for implementing collocation on the whole sphere are described. a formal relationship between collocation and least squares adjustment is used to obtain an alternative form of the collocation algorithm that is likely to be stable with dense data sets and, with a minor modification, can be used to implement least squares adjustment as well. The basic principle is that for regular grids the variance-convariance matrix of the data consists of Toeplitz-circulant blocks, so it can be both set up and inverted very efficiently.