Using geostatistics to describe complex a priori information for inverse problems

Inverse problem theory deals with the problem of inferring properties of the subsurface based on indirect physical measurements. Although inverse problem theory can deal with complex a priori information in cases when samples from the a priori pdf can be generated, it is rarely done in practice. Instead rather simple prior constraints are often assumed, that may have little geological and petro-physical justification. Hence the solutions to inverse problems are rarely consistent with geological information. Geostatistics is a discipline where methods have been developed for simulating realistic geological subsurface structures. The first breakthrough was the application of classical covariance based geostatistics and sequential simulation. In the recent years the application of multiple point statistics have led to impressive simulation algorithms which can be used to simulate realistic geological structures. We will make use of geostatistical simulation algorithms to describe prior information for Bayesian formulated inverse problems. Using a classical georadar cross borehole tomography inverse problem, we will show how the a posteriori probability density function can be sampled, to provide solutions to inverse problems that not only match observed geophysical data within their uncertainties, but also the subsurface geology as described by the geostatistical model of the prior information. From such a sample complex questions can be answered probabilistically. The combination of geostatistics and inverse problem theory can also significantly reduce the complexity of the inverse problem. We demonstrate that when selecting an appropriate prior model based on geostatistics the computational needs for solving the inverse problem can be drastically reduced (orders of magnitude in computational gain).