Composite distribution inversion applied to crosshole tomography

Abstract

Crosshole tomography requires solution of a mixed‐determined inverse problem and addition of a priori information in the form of auxiliary constraints to achieve a stable solution. Composite distribution inversion (CDI) constraints are developed by assuming parameters are drawn from a composite distribution consisting of both normally and uniformly distributed parameters. Nonanomalous parameter estimates are assumed to be Gaussian while anomalous parameters are assumed uniform. The resulting constraints are sensitive to anomaly volume and are an alternative to the usual constraints of minimizing [Formula: see text] solution length or some measure of roughness. Damped least‐squares inversion, which minimizes solution length, distributes anomalous signal through poorly resolved areas to produce in attenuated and smoothed anomalies. Similar regularization methods, such as smoothness or flatness constraints, also degrade small spatial wavelength features and produce diffuse images of distinct anomalies. CDI constraints preserve small spatial wavelength features by encouraging small amplitude anomalies to assume the value of the reference model and by allowing truly anomalous parameter estimates to assume whatever value minimizes prediction error without incurring additional penalty. CDI tomograms are characterized by nearly ideal point‐spread functions, suggesting the possibility of better quantitative parameter estimates than are produced using most existing methods. CDI tomograms of both synthetic and field data are shown to produce less diffuse images with more accurate anomaly amplitude estimates than damped least‐squares methods. The CDI algorithm is potentially applicable to nontomographic inversion problems.