Orthogonal functions over the oceans and applications to the determination of orbit error, geoid and sea surface topography from satellite altimetry

The goal of this study is to look for a set of orthogonal functions over the oceans and then to apply the functions to the expansions of oceanic signals. Ultimately these functions are incorporated in the parameter estimation problem using a model that simultaneously reduces satellite radial orbit errors, improves the geoid and estimates the sea surface topography.;To construct a possible set of orthogonal functions (over the oceans), three methods have been studied. In one method, an attempt was made to solve the spherical Helmholtz equation over the oceans with either the Dirichlet boundary condition or the Newmann Boundary condition or the mixed boundary condition. This method leads to successful developments of the spherical cap harmonics and generalized Fourier-Bessel series for some regularized oceanic domains. The other study employs the Shwarz-Christoffel conformal mapping, as well as an auxiliary mapping, to transform the irregular domain (the oceans) onto the interior of a unit disk where a set of orthogonal functions are easy to find. The set of orthogonal functions over the oceans are then found through the relationship between the two domains implied by the mappings. The third method involves the Gram-Schmidt process for which a new technique for computing the integrals of the products of two associated Legendre functions is developed and a FFT method is used to compute the inner products of spherical harmonics over the oceans. Also, for the entire unit sphere, the generalized 2-D Fourier series and the generalized Fourier-Tschebycheff series are proposed as alternatives for the spherical harmonics.;The expansions of the Levitus SST into the orthonormal functions constructed by the Gram-Schmidt process show that 98.5% of the energy of that signal is contained within degree 10 of the orthonormal functions. Such expansions also render regular spectral behavior of oceanic signal as compared to that from spherical harmonic expansions. A method of detecting band limited oceanic signal is also developed using these orthonormal functions. The applications of these functions to the simultaneous model yield lower correlations between expansion coefficients and show that the cut-off frequency (the highest degree determinable) of the SST from the Geosat data is in the vicinity of degree 15 of the functions. Some other improved simultaneous models have also been tested in an attempt to better separate the geoid and the SST signal. This study concludes that the orthonormal functions are suitable for representing oceanic signal with excellent spectral behavior and can be applied to future altimetric mission such as TOPEX/POSEIDON.