Solution of an Acoustic Transmission Inverse Problem by Extended Inversion: Theory

A single-trace transmission inverse problem for the wave equation seeks to determine both the wave velocity in a homogenous acoustic medium and the transient waveform of an isotropic point source. The duration (support) of the source waveform and the source-to-receiver distance are assumed known. A least squares formulation of this problem exhibits the “cycle-skipping” behaviour observed in field scale problems of this type, with many local minima differing greatly from the global minimizer. This behaviour is eliminated by dropping the hard support constraint on the source waveform, replacing it by a soft penalty implemented as a weighted mean-square of the source waveform. For properly chosen weight function, penalizing nonzero values away from t = 0, the velocity component of any stationary point differs from the global minimizer of the constrained least-squares formulation by a linear combination of the source waveform support radius and data noise-to-signal ratio. Given an estimate of data noise, the penalty weight can be dynamically adjusteded during iterative optimization to maximize predicted data accuracy and closely approximate a support-constrained source waveform.