Applying compactness constraints to differential traveltime tomography

Abstract

Tomographic imaging problems are typically ill-posed and often require the use of regularization techniques to guarantee a stable solution. Minimization of a weighted norm of model length is one commonly used secondary constraint. Tikhonov methods exploit low-order differential operators to select for solutions that are small, flat, or smooth in one or more dimensions. This class of regularizing functionals may not always be appropriate, particularly in cases where the anomaly being imaged is generated by a nonsmooth spatial process. Time-lapse imaging of flow-induced velocity anomalies is one such case; flow features are often characterized by spatial compactness or connectivity. By performing inversions on differenced arrival time data, the properties of the time-lapse feature can be directly constrained. We develop a differential traveltime tomography algorithm whichselects for compact solutions, i.e., models with a minimum area of support, through application of model-space iteratively reweighted least squares. Our technique is an adaptation of minimum support regularization methods previously explored within the potential theory community. We compare our inversion algorithm to the results obtained by traditional Tikhonov regularization for two simple synthetic models: one including several sharp localized anomalies and a second with smoother features. We use a more complicated synthetic test case based on multiphase flow results to illustrate the efficacy of compactness constraints for contaminant infiltration imaging. We apply the algorithm to a [Formula: see text]-sequestration-monitoring data set acquired at the Frio pilot site. We observe that in cases where the assumption of a localized anomaly is correct, the addition of compactness constraints improves image quality by reducing tomographic artifacts and spatial smearing of target features.