Bayesian estimation of medium properties in wavefield downward extrapolation problems

Abstract

Acoustic waves are used in many areas (ultrasonic medicine, geophysical oil prospecting, etc.) for nondestructive imaging of the interior of objects such as the earth, the human body, etc. The wavefield measurements recorded on the surface of the object are extrapolated according to the wave equation to give an "image" of the object by using an imaging principle. The extrapolation propagates backward the noise present in the measurements. Thus, the quality of the final image is deteriorated, unless statistical restoration techniques are used. Another source of degradation of the image is the insufficient knowledge of some properties of the medium such as the wave propagation velocity. The aim of this work is to use estimation theory to determine good estimates of the velocity and of the backward extrapolated wavefield from the known surface wavefield measurements.

Source imaging with minimum mean-squared error

“…Rx=H Cov[ss*]H*+Cov[nn*] =HKsH*+Rn, (8) and in general there is a diagonal (uncorrelated) noise component in Rn, (7) is directly invertible, and we find w* ( u ) =Cov [s( u ) s* ] H*R• -1.…”