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Uncertainty quantification in linear inverse problems with dimension reduction

Dimension reduction is a process used to simplify high dimensional inverse problems in geophysics. In this work, we analyze the impact of the model and data reduction on the uncertainty quantification of the inversion results. In the Bayesian approach for inverse problems, the solution is given by the posterior distribution and the uncertainty is interpreted as the posterior variance. Under the Gaussian assumption for the noise and prior distribution and if the forward model and reduction operator are linear, we analytically obtain the posterior distribution. We use the analytical result to compute the solution of the acoustic inversion problem with different reduction levels of both data and model parameters. The method allows us to analyze the impact of the linear reduction on the uncertainty quantification of linear inverse problems.