Imaging the subsurface of the Earth is of prime concern in geosciences. In this scope, geophysics offers a wide range of methods that are able to produce models of the subsurface, classically through inversion processes. Deterministic inversions lack the ability to produce satisfactory quantifications of uncertainty, whereas stochastic inversions are often computationally demanding. In this paper, a new method to interpret geophysical data is proposed in order to produce 1D imaging of the subsurface along with the uncertainty on the associated parameters. This new approach called Bayesian Evidential Learning 1D imaging (BEL1D) relies on the constitution of statistics-based relationships between simulated data and associated model parameters. The method is applied to surface nuclear magnetic resonance for both a numerical example and field data. The obtained results are compared to the solutions provided by other approaches for the interpretation of these datasets, to demonstrate the robustness of BEL1D. Although this contribution demonstrates the framework for surface nuclear magnetic resonance geophysical data, it is not restricted to this type of data but can be applied to any 1D inverse problem.

Summary The non-uniqueness of the solution of inverse geophysical problem has been recognized for a long-time. Although stochastic inversion methods have been developed, deterministic inversion using subsequent regularization are still more widely applied. This is likely due to their efficiency and robustness, compared to the computationally expensive and sometimes difficult to tune to convergence stochastic methods. Recently, Bayesian Evidential Learning 1D imaging has been presented to the community as a viable tool for the efficient stochastic 1D imaging of the subsurface based on geophysical data. The method has been proven to be as fast, or sometimes even faster, than deterministic solution. However, the method has a significant drawback when dealing with large prior uncertainty as often encountered in geophysical surveys: it tends to overestimate the uncertainty range. In this paper, we provide an efficient way to overcome this limitation through Iterative Prior Resampling (IPR) followed by rejection sampling. IPR adds the posterior distribution calculated at a former iteration to the prior distribution in a subsequent iteration. This allows to sharpen the learning phase of the algorithm and improve the estimation of the final posterior distribution while rejection sampling eliminates models not fitting the data. In this contribution, we demonstrate that this new approach allows BEL1D to converge towards the true posterior distribution. We also analyze the convergence behavior of the algorithm and derive guidelines for its application. We apply the approach for the interpretation of surface waves dispersion curves but the approach can be generalized to other geophysical methods.