The use of multiwavelets in seismic array data processing

This research involves the theoretical development of new processing methods for seismic array data which exploit the unique characteristics of the multiwavelets and the associated multiwavelet transforms. The multiwavelets are a set of functions designed to be finite in time and as frequency band-limited as possible. They are determined as the eigenvectors for an eigenvalue equation where the eigenvalues represent the fraction of total signal energy contained in the frequency band of interest. These multiwavelets/eigenvectors are always a set of mutually orthogonal functions occurring in even and odd pairs where each pair emphasizes a different portion of the time and frequency bands. The pairs of multiwavelets can be combined into complex integration kernels to form the multiwavelet transforms--each analagous to a Fourier transform. These multiwavelet transforms are uniquely suited for seismic processing because the time and frequency windows can be explicitly scaled to each other and the existence of multiple transforms allows for redundancy in a given time and frequency band and better coverage of both bands than with conventional Fourier transforms. The first new processing method, multiwavelet beamforming, exploits the redundancy of the multiwavelet transforms. Each transform is used to determine a slowness vector sample and these samples are used to form a stable estimate of the slowness vector and to calculate its statistical confidence ellipse. The second method addresses the problem of removing time residuals in the station arrival times due to three dimensional variation in the earth's velocity structure. It exploits the explicit time-frequency relationship of the multiwavelets by starting with low frequency wavelets where time differences are a small percentage of the total cycle and successively working through higher frequency bands until the time residuals are known to the data sampling level. The third new method uses three component recordings to determine simultaneously the slowness vector and the particle motions at each station for a given arrival. Variations in the expected propagation direction and particle motions can be used to infer non-isotropic behaviour of local earth materials.