Knowledge Bank

Spectral Analysis in Geodetic Science and a method for the solution of least-squares collocation in the frequency domain as well as a fast Toeplitz inversion algorithm in the space domain are outlined. The mathematical background in spectral analysis as applied to geodetic applications is summarized. In geodetic problems, we usually have discrete and non-periodic data. The resolution (cut-off frequency) of the GEOS-3 altimeter data has been examined by determining the shortest wavelength (corresponding to the cut-off frequency) recoverable. The data from some 18 profiles were used in this study. The total power (variance) in the sea surface topography with respect to the reference ellipsoid as well as with respect to the GEM-9 surface was computed. A fast inversion algorithm for matrices of simple and block Toeplitz matrices and its application to least-squares collocation is explained. This algorithm yields a considerable gain in computer time and storage in comparison with conventional least-squares collocation. Frequency domain least-squares collocation techniques are also introduced and applied to estimating gravity anomalies from GEOS-3 altimeter data. These techniques substantially reduce the computer time and requirements in storage associated with the conventional least-squares collocation. Numerical examples given in this paper demonstrate the efficiency and speed of these techniques. The number of numerical operations required to calculate the signals is proportional to N log2 N (where N is the number of observations) rather than N2 with the fast Toeplitz inversion algorithm or N3 with classical collocation. [Some mathematical expressions are not fully represented in the metadata. Full text of abstract available in document.]