Generalized discrete spherical harmonic transforms

Two generalizations of the spherical harmonic transforms are provided. Firs t, they are generalized to an arbitrary distribution of latitudinal points theta(i). This unifies transforms for Gaussian and equally spaced distribut ions and provides transforms for other distributions commonly used to model geophysical phenomena. The discrete associated Legendre functions P-n(m) ( theta(i)) are shown to be orthogonal, to within roundoff error, with respec t to a weighted inner product, thus providing the forward transform to spec tral space. Second, the representation of the transforms is also generalize d to rotations of the discrete basis set P-n(m) (theta(i)). A discrete func tion basis is defined that provides an alternative to P-n(m) (theta(i)). On a grid with N latitudes, the new basis requires O (N-2) memory compared to the usual O (N-3). The resulting transforms differ in spectral space but p rovide identical results for certain applications. For example, a forward t ransform followed immediately by a backward transform projects the original discrete function in a manner identical to the existing transforms. Namely , they both project the original function onto the same smooth least square s approximation without the high frequencies induced by the closeness of th e points in the neighborhood of the poles. Finally, a faster projection is developed based on the new transforms. (C) 2000 Academic Press.