Complexity reduction in inverse problems: Wavelet transforms, PCA and geological bases

Most geoscience inversion algorithms require a complicated forward model involving a large number of parameters needed for accuracy of the data prediction. However the model parameterization used in the forward problem may not be the best choice for inversion since the data does not inform about all the components of the model. In the linear case this is of course related to the dimension of the null space. Model parameterization is a key concept in order to make the inverse problem less ill conditioned. Adopting the right parameterization (i.e. basis set) also reduces the number of dimensions in which the inverse problem is going to be solved, allowing performing posterior uncertainty analysis more easily. In this poster we demonstrate this approach to complexity reduction by various examples related to geophysical and reservoir inversion. Particularly we compare several mathematical and geological bases for model parameterization. In the first example we compare the performance of two different basis sets -pixel and wavelets - for a prototype 1-D linear inverse problem. The use of wavelet basis gives better reconstruction for the same number of model parameters. The second example concerns seismic history matching where we invert for reservoir facies by matching production and seismic data. In this problem the combined use of geological bases build up from multiple conditional geostatistical simulations and spatial Principal Component Analysis allows the model parameter reduction as well as a geological consistent updating of the model to obtain the optimal solution. In addition we also obtain a sample of equivalent solutions allowing us to perform posterior uncertainty analysis. These multi-resolution approaches to inverse problems can help to regularize the problem by matching the appropriate scales in the model parameter space that are informed by the data. The telescopic nature also allows the inversion of one scale at a time with different prior information at different scales.