Inference of geostatistical hyperparameters with the correlated pseudo-marginal method

We consider non-linear Bayesian inversion problems targeting the geostatistical hyperparameters of a random field describing hydrogeological or geophysical properties given data. This problem is particular importance in non-ergodic setting as there are no analytical upscaling relationships linking data to hyperparameters, such as, mean, standard deviation, and integral scales. Full local (typically involving many thousands unknowns) brings substantial computational challenges, that simplifying model assumptions (e.g., homogeneity ergodicity) typically made. To prevent errors resulting from simplified while also circumventing burden high-dimensional full inversions, we use pseudo-marginal Metropolis–Hastings algorithm treats latent variables. In this effects model, intractable likelihood observing estimated by Monte Carlo averaging over realizations field. increase efficiency method, low-variance approximations ratio obtained using sampling correlating samples used proposed current steps Markov chain. assess performance correlated method considering two representative diffusion-based wave-based physics, respectively, which infer (1) hydraulic conductivity fields apparent data-poor (2) fracture aperture borehole ground-penetrating radar (GPR) reflection more data-rich setting. For first test case, find generates similar estimates mean classical rejection sampling, an assuming ergodicity provides biased estimates. second well, computationally unfeasible leads