NATURAL SMOOTHNESS CONSTRAINTS IN CROSS‐HOLE SEISMIC TOMOGRAPHY1

Abstract

AbstractTomographic inversion problems are ill‐posed and therefore solutions must be either damped or regularized to produce results that are geologically reasonable. We introduce a priori information in terms of parameter covariances to constrain the solution. We use slowness logs to determine the appropriate parameter covariances for the inversion of traveltime data collected for a cross‐hole geometry. We find that the logs exhibit power spectra proportional to a power α of the frequency. The value of α controls the smoothness of the inversion solution. For α < 0, the solution smoothness is maximized. Thus, knowing the correct value of α, which can be found from well logs, we can specify the appropriate amount of smoothing, rather than using some arbitrary level.Inversions were carried out on synthetic data for the case of α= 0 and α=−1. The use of α= 0 implies uncorrelated model parameters and is equivalent to standard damped least‐squares methods. We find that solutions for α= 0 show greater complexity than for α=−1 but this level of resolution can be illusory. An example from the Midale field, Saskatchewan, Canada, is inverted using both α= 0 and α=−2, given by the observed parameter covariances. The solution for α= 0 exhibits spurious detail which must be smoothed away. For α=−2, the solution smoothness is maximized and we recover only that structure which is required to fit the data.