ESTIMATION OF SLOWNESS VECTORS AND THEIR UNCERTAINTIES USING MULTI-WAVELET SEISMIC ARRAY-PROCESSING

We have developed a new seismic array data processing method to produc e slowness vector estimates and an objective measure of their uncertai nties in the form of statistical confidence intervals. The slowness ve ctor, which is typically transformed into bearing and velocity, is a k ey parameter used for identifying seismic phases and for event source location. Our method, multi-wavelet beamforming, is closely related to both time-domain and frequency-domain beamforming. The major advantag e of multi-wavelet beamforming is that it produces multiple estimates of the slowness vector that are approximately statistically independen t. First, a set of wavelet transforms is applied to the data in a mann er analogous to the use of the windowed Fourier transform. Next, for e ach wavelet transform, we calculate semblance, a measure of signal coh erence, for a range of possible slowness vectors. Then, the slowness v ector estimate associated with that transform is the vector that produ ces the largest semblance value. The multiple slowness vector estimate s can be treated as samples from a probability distribution, whose ''c enter'' we estimate using the mean, the median, and an M-estimator. Un certainty intervals are calculated for these estimators by applying th e jackknife statistical method. The intervals for the mean estimator a ppear to be true statistical confidence intervals, but the estimates c an be biased by a directional noise field in low signal-to-noise circu mstances. The median estimates are less biased by a directional noise field but sometimes underestimate the uncertainty. The M-estimator pro duces less-biased estimates while appearing to estimate correctly thei r uncertainty.