Solving a geophysical inverse problem means determining the parameters of an earth model given a set of measurements. In solving many practical inverse problems, accounting for the uncertainty of the solution is very important to aid in decision-making. In this work, we address the problem of determining the posterior uncertainty of the solution for models that arise from decimated wavelet bases using a simple 1-dimensional seismic travel time inversion problem.
Our inversion methodology is to pick a model decimation, prepare a prior mean and covariance matrix of the wavelet coefficients, compute a posterior mean and covariance, and then to sample from this posterior distribution. We also sample different choices of model decimation in proportion to their posterior probability. These samples span the uncertainty of the inverse problem solution, accounting for both the uncertainty in the choice of model decimation and of wavelet coefficients. We note that a re-normalization of the decimated prior covariance matrix of the wavelet coefficients is required to properly account for the amount of variance in the prior distribution. Further, we present a fast algorithm for computing this normalized decimated prior covariance matrix.