SIPPI: A Matlab toolbox for sampling the solution to inverse problems with complex prior information: Part 1 - Methodology

From a probabilistic point-of-view, the solution to an inverse problem can be seen as a combination of independent states of information quantified by probability density functions. Typically, these states of information are provided by a set of observed data and some a priori information on the solution. The combined states of information (i.e. the solution to the inverse problem) is a probability density function typically referred to as the a posteriori probability density function. We present a generic toolbox for Matlab and Gnu Octave called SIPPI that implements a number of methods for solving such probabilistically formulated inverse problems by sampling the a posteriori probability density function. In order to describe the a priori probability density function, we consider both simple Gaussian models and more complex (and realistic) a priori models based on higher order statistics. These a priori models can be used with both linear and non-linear inverse problems. For linear inverse Gaussian problems we make use of least-squares and kriging-based methods to describe the a posteriori probability density function directly. For general non-linear (i.e. non-Gaussian) inverse problems we make use of the extended Metropolis algorithm to sample the a posteriori ∗Corresponding author. Tel.:+45 45253086, Fax.: +45 45882673 Email addresses: [email protected] (Thomas Mejer Hansen), [email protected] (Knud Skou Cordua), [email protected] (Majken Caroline Looms), [email protected] (Klaus Mosegaard) Preprint submitted to Computers & Geosciences September 27, 2012 probability density function. Together with the extended Metropolis algorithm we use sequential Gibbs sampling that allow computationally efficient sampling of complex a priori models. The toolbox can be applied to any inverse problem as long as a way of solving the forward problem is provided. Here we demonstrate the methods and algorithms available in SIPPI. An application of SIPPI, to a tomographic cross borehole inverse problems, is presented in a second part of this paper.