Reduction of variance in spectral estimates for correction of ultrasonic aberration.

A variance reduction factor is defined to describe the rate of convergence and accuracy of spectra estimated from overlapping ultrasonic scattering volumes when the scattering is from a spatially uncorrelated medium. Assuming that the individual volumes are localized by a spherically symmetric Gaussian window and that centers of the volumes are located on orbits of an icosahedral rotation group, the factor is minimized by adjusting the weight and radius of each orbit. Conditions necessary for the application of the variance reduction method, particularly for statistical estimation of aberration, are examined. The smallest possible value of the factor is found by allowing an unlimited number of centers constrained only to be within a ball rather than on icosahedral orbits. Computations using orbits formed by icosahedral vertices, face centers, and edge midpoints with a constraint radius limited to a small multiple of the Gaussian width show that a significant reduction of variance can be achieved from a small number of centers in the confined volume and that this reduction is nearly the maximum obtainable from an unlimited number of centers in the same volume.