On the Distribution of Velocity Measurements From Pulse-to-Pulse Coherent Doppler Sonar

Abstract

Knowledge of the velocity measurement distribution is useful for designing new coherent Doppler systems, for resolving velocity ambiguity, and for optimally combining measurements from multiple receivers and multiple carrier frequencies. In this paper, the effects of pulse-pair averaging and pulse-to-pulse autocorrelation are analyzed for ensembles consisting of up to ten pulse pairs. A formula is presented for the probability distribution of velocity measurements from a single pulse pair. This distribution is nonnormal for all values of the autocorrelation coefficient. In the limit of perfect pulse-to-pulse correlation, single pulse-pair measurements obey a Pearson Type VII distribution which has a higher peak and broader tails than a normal distribution. Second and fourth moments of the multiple pulse-pair measurement distributions are evaluated using a Monte Carlo method. The applicability of perturbation analysis for predicting velocity standard deviation is investigated. In the analysis of Zrnić (1977), it is assumed that perturbations in autocorrelation phase are small compared to the mean phase. Three prediction failure mechanisms are demonstrated through changes in the distribution of the complex autocorrelation coefficient; these occur for both high and low autocorrelation magnitude, and for short ensemble lengths. Measurement distributions from the Monte Carlo method are also used to determine the autocorrelation magnitude necessary to resolve velocity ambiguity using a dual-frequency coherent Doppler sonar. Simulations with a high-fidelity coherent Doppler sonar model show how the distribution of velocity measurements is affected by finite particle concentrations. While velocity standard deviation is insensitive to concentration above a threshold of approximately 1 particle per sample volume, kurtosis exhibits a strong dependence on concentration in the range of 0.1-5 particles per sample volume.