Discrete cosine transform for parameter space reduction in linear and non-linear AVA inversions

Geophysical inversions estimate subsurface physical parameters from the acquired data and because of the large number of model unknowns, it is common practice reparametrizing the parameter space to reduce the dimension of the problem. This strategy could be particularly useful to decrease the computational complexity of non-linear inverse problems solved through an iterative sampling procedure. However, part of the information in the original parameter space is lost in the reduced space and for this reason the model parameterization must always constitute a compromise between model resolution and model uncertainty. In this work, we use the Discrete Cosine Transform (DCT) to reparametrize linear and non-linear elastic amplitude versus angle (AVA) inversions cast into a Bayesian setting. In this framework the unknown parameters become the series of coefficients associated to the DCT base functions. We first run linear AVA inversions to exactly quantify the trade-off between model resolution and posterior uncertainties with and without the model reduction. Then, we employ the DCT to reparametrize non-linear AVA inversions numerically solved through the Differential Evolution Markov Chain and the Hamiltonian Monte Carlo algorithm. To draw general conclusions about the benefits provided by the DCT reparameterization of AVA inversion, we focus the attention on synthetic data examples in which the true models have been derived from actual well log data. The linear inversions demonstrate that the same level of model accuracy, model resolution, and data fitting can be achieved by employing a number of DCT coefficients much lower than the number of model parameters spanning the unreduced space. The non-linear inversions illustrate that an optimal model compression (a compression that guarantees optimal resolution and accurate uncertainty estimations) guarantees faster convergences toward a stable posterior distribution and reduces the burn-in period and the computational cost of the sampling procedure.