Reduction of variance in statistical estimates of spectra for use in correction of ultrasonic aberration

Abstract

Parameters of ultrasonic aberration can be obtained from power spectra of scattering when individual scattering measurements from which the spectra are estimated have a common aberration and the same nominal geometry. However, the scattering volumes are then confined to a small spatial region and use of finitely many overlapping volumes that result in a nonzero variance is necessary for the measurements. Assuming the scattering is from a spatially uncorrelated medium, the variance of the spectral estimates is expressed as the product of the variance for a single measurement and a reduction factor that depends on the amount of overlap between each volume pair. This factor describes the rate of convergence and the accuracy of the estimates as a function of the number and the overlap of the scattering volumes. Assuming further that the individual volumes are localized by a Gaussian window and that the centers of the volumes are located on orbits of an icosahedral rotation group, the variance is minimized by adjusting the weight and radius of each orbit. Numerical evaluations using orbits formed by icosahedral vertices, face centers, and edge midpoints show that a significant reduction of variance can be achieved from volumes in a confined region.