Wavelet representation of geodetic operators.

摘要

The main objective of this research is to introduce an alternative to the FFT computational scheme using the wavelet transform for the numerical evaluation of different geodetic operators. The new wavelet representation is built on orthogonal wavelet base functions. Eight geodetic operators are evaluated in this thesis: they are classified into direct geodetic integrals, inverse geodetic integrals, and the inversion of integrals. The direct geodetic integrals are the Stokes, the Vening Meinesz, the Poisson (upward continuation), and the terrain correction integrals. The inverse geodetic integrals are the inverse Vening Meinesz integral and the deflection-geoid formula. The Stokes and Poisson (downward continuation) integrals are inverted in the wavelet domain by a conjugate gradient method.;In each case, the role of the kernel's singularity in the wavelet multi-resolution analysis is studied. The integrals are approximated in finite multi-resolution analysis subspaces. A new implementation is introduced to decrease the computational effort. The full solution with all equations requires a large computer memory. Multi-resolution properties of the wavelet transform are used to divide the full solution into parts. Each part represents a level of wavelet detail coefficients or the approximation coefficients. Hard thresholding is used for the compression of the kernels' wavelet detail coefficients. Global fixed thresholding and level/direction-wise thresholding is tested for different kernels. High compression levels are achieved with an acceptable accuracy, which leads to large savings in computer memory and storage space required for allocating the matrices, and also the ability to work with sparse matrices. In the case of the inversion of the integrals, a set of equations is formed and solved using an iterative gradient method. Soft thresholding is used for de-noising stationary and non-stationary noise because of its smoothing properties. Conclusions and recommendations are given with respect to the suitability, accuracy, and efficiency of these methods.