Stochastic inversion for soil hydraulic parameters in the presence of model error: An example involving ground-penetrating radar monitoring of infiltration

Proxy forward solvers are commonly used in Bayesian solutions to inverse problems in hydrology and geophysics in order to make sampling of the posterior distribution, for example using Markov-chain-Monte-Carlo (MCMC) methods, computationally tractable. However, use of these solvers introduces model error into the problem, which can lead to strongly biased and overconfident parameter estimates if left uncorrected. Focusing on the specific example of estimating unsaturated hydraulic parameters in a layered soil from time-lapse ground-penetrating radar data acquired during a synthetic infiltration experiment, we show how principal component analysis, conducted on a suite of stochastic model-error realizations, can for some problems be used to build a sparse orthogonal basis for the model error arising from known forward solver approximations and/or simplifications. Projection of the residual onto this basis during MCMC permits identification and removal of the model error before calculation of the likelihood. Our results indicate that, when combined with an informal likelihood metric based on the expected behaviour of the ℓ2-norm of the residual, this methodology can yield posterior parameter estimates exhibiting a marked reduction in bias and overconfidence when compared to those obtained with no model-error correction, at reasonable computational cost.