abstract From a probabilistic point-of-view, the solution to an inverse problem can be seen as a combination ofindependent states of information quantiﬁed by proba bility density functions. Typically, these states ofinformation 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 functiontypically referred to as the a posteriori probability density function. We present a generic toolbox for Matlaband Gnu Octave called SIPPI that implements a number of methods for solving such probabilisticallyformulated inverse problems by samp ling the a posteriori probability density function. In order to describethe a priori probability density function, we conside r both simple Gaussian models and more complex (andrealistic) a priori models based on higher order stati stics. These a priori models can be used with both linearand non-linear inverse problems. For linear inverse G aussian problems we make use of least-squares andkriging-based methods to describe the a posteriori pro bability density function directly. For general non-linear (i.e. non-Gaussian) inverse problems, we make u se of the extended Metropolis algorithm to sample thea posteriori probability density function. Together wit h the extended Metropolis algorithm, we use sequentialGibbs sampling that allow computationally efﬁcient sa mpling of complex a priori models. The toolbox can beapplied to any inverse problem as long as a way of solving the forward problem is provided. Here wedemonstrate the methods and algorithms available in S IPPI. An application of SIPPI, to a tomographic crossborehole inverse problems, is presented in a second part of this paper.& 2012 Elsevier Ltd. All rights reserved.
['Thomas Mejer Hansen', 'Knud Skou Cordua', 'Majken C. Looms', 'Klaus Mosegaard']
SIPPI: A Matlab toolbox for sampling the solution to inverse problems with complex prior information Part 1—Methodology