Error bars for sonic slowness measurements

Abstract

Abstract Semblance provides a quality indicator for beam-former array processing methods such as slowness/time coherence (STC) and dispersive STC (DSTC). Semblance does not indicate the slowness variance because the array design, the processing time-bandwidth parameters, and the signal spectral content are not taken into account. For the classical case of a single propagating wave with added Gaussian noise, semblance is distributed according to the noncentral beta distribution. This distribution is parameterized by the number of receivers, M, the processing time-bandwidth product, BT, and signal-to-noise ratio. Given M and BT, the measured semblance can be inverted to give a high-quality estimate of the signal-to-noise ratio. The Cramer-Rao lower bounds on slowness variance for the classical case of signals in added Gaussian noise depend on the array dimensions, the signal power spectrum, and the signal-to-noise ratio. Estimates of the Cramer-Rao bounds can be calculated from the estimated signal-to-noise ratio and the measured signal power spectrum. Monte-Carlo simulations show that these bounds approximate the actual slowness variance for high, but practical, signal-to-noise ratios. Preliminary results on field data show that error bars based on Cramer-Rao bounds are credible, and that error bars yield slowness standard deviations comparable to those from repeat runs.