Efficient probabilistic inversion using the rejection sampler—exemplified on airborne EM data

Abstract

Summary Probabilistic inversion methods, typically based on Markov chain Monte Carlo, exist that allow exploring the full uncertainty of geophysical inverse problems. The use of such methods is though limited by significant computational demands, and non-trivial analysis of the obtained set of dependent models. Here, a novel approach, for sampling the posterior distribution is suggested based on using pre-calculated lookup tables with the extended rejection sampler. The method is 1) fast, 2) generates independent realizations of the posterior, and 3) does not get stuck in local minima. It can be applied to any inverse problem (and sample an approximate posterior distribution) but is most promising applied to problems with informed prior information and/or localized inverse problems. The method is tested on the inversion of airborne electromagnetic data and shows an increase in the computational efficiency of many orders of magnitude as compared to using the extended Metropolis algorithm.

Use of Machine Learning to Estimate Statistics of the Posterior Distribution in Probabilistic Inverse Problems—An Application to Airborne EM Data

“…The extended variations of the Metropolis algorithm (Mosegaard & Tarantola, 1995) and the rejection sampler (Hansen, 2021;Hansen et al, 2016) do not require that an analytical description of the prior exists, as evaluation of the prior is not needed. It is sufficient that an algorithm exists that can generate a realization from the prior.…”

“…For a single sounding this may take at least 10 min per sounding, requiring access to supercomputers for the application of real‐world data sets (Foks & Minsley, 2020). Hansen (2021) proposed 1D probabilistic inversion based on the extended rejection sampler (using lookup tables, similar to [ N *, M *, D *]) that relies on the construction of a large sample for the prior along with the forward responses (generated once). This is then used to generate independent realizations of the posterior distribution numerically more efficiently than is possible using Markov Chain‐based algorithms, and at the same time avoids issues related to model equivalences.…”

“…A sufficient sampling of the 1D posterior distribution of resistivity values, to obtain a limited set of independent realizations, may require hundreds of thousands of McMC iterations, and hence forward model evaluations. For a single sounding this may take at least 10 min per sounding, requiring access to supercomputers for the application of real-world data sets(Foks & Minsley, 2020) Hansen (2021). proposed 1D probabilistic inversion based on the extended rejection sampler (using lookup tables, similar to [N*, M*, D*])…”

Use of machine learning to estimate statistics of the posterior distribution in probabilistic inverse problems - an application to airborne EM data.

“…Once trained, the prediction of the posterior mean and standard deviation for the 451 soundings takes around 4 ms. Around 100000 1D soundings can be analyzed per second. A similar analysis will take around 451 seconds using the extended rejection sampler (T. M. Hansen, 2021), and around 6 hours using the extended Metropolis algorithm (T. M. Hansen & Minsley, 2019).…”

Instantaneous Inversion of Airborne Electromagnetic Data Based on Deep Learning

“…Naturally, data inversion has become the essential part of AEM survey, the result of which contributes significantly to our understanding of the subsurface structure. AEM inversion techniques can basically be categorized into either deterministic or stochastic methods (Bai et al., 2021; Blatter et al., 2018; Christiansen et al., 2016; Cox et al., 2012; Hansen, 2021; Hauser et al., 2015; Liu & Yin, 2016; McMillan et al., 2015; Minsley et al., 2021). Deterministic methods suffer from the inherent non‐uniqueness of the solution and are prone to instability, particularly under severe noise conditions.…”