Bayesian Evidential Learning 1D Imaging (BEL1D) has been recently introduced as a new computationally efficient tool for the interpretation of 1D geophysical datasets in a Bayesian framework. Applications have already been demonstrated for Surface Nuclear Magnetic Resonance (SNMR) data and surface waves dispersion curves. The case of SNMR is particularly relevant in hydrogeophysics, as it directly sounds the water content of the subsurface. BEL1D relies on the constitution of statistical relationships in a reduced dimension space between model parameters and simulated data using prior model samples that replicate the field experiment. In BEL1D, this relationship is deduced through Canonical Correlation Analysis (CCA). When using large prior distributions, CCA may lead to numerous poorly correlated distributions for higher dimensions. Those poorly correlated distributions are resulting in a low reduction of uncertainty on some parameters, even if the experiment is supposed to be sensitive to them. This phenomenon is related to the aggregation of multiple parameters in the same dimension, hence the possible aggregation of sensitive and insensitive parameters. However, arbitrarily reducing the extent of the prior will lead to biased estimations. To overcome this impediment, we introduce an iterative procedure, using the posterior model space of the previous iteration as prior model of the current iteration. This approach frequently reveals higher correlations between the datasets and the model parameters, while still using large unbiased priors. It enables BEL1D to produce better estimations of the posterior probability density functions of the model parameters. Nonetheless, iterating on BEL1D presents several challenges related to the presence of insensitive parameters, that will always mitigate the capacity to reduce at once the uncertainty on the whole set of parameters describing the models. On noise-free synthetic datasets, this method leads to near-exact estimation of the sensitive parameters after few (two to three) iterations. On noisy datasets, the resulting distributions bear some uncertainty, arising directly from the presence of noise, but to a lesser extent than the non-iterative approach. The procedure remains more computationally efficient than McMC.

The use of geophysical methods to characterize subsurface properties has significantly grown in the last decade. Although geophysics can bring relevant spatial and temporal information on subsurface processes, the quantitative interpretation and integration in models remain difficult. Indeed, standard deterministic solutions suffer from (excessive) smoothing and spatially variable resolution, whereas joint or coupled inversions remain difficult to apply in complex cases. Hermans et al. (2016) proved using cross-borehole ERT that physical properties distribution could be directly retrieved from data using Bayesian Evidential Learning (BEL). BEL uses a series of prior models to derive a direct relationship between data and forecast in a reduced dimension space. This can be challenging when the prediction becomes more complex with higher dimensions. In this contribution, we extend the work of Hermans et al. (2016) to a full 4D experiment (3D + time). We demonstrate that the shape and amplitude of the temperature plume can be retrieved, with uncertainty quantification, during a push-pull experiment using surface ERT. We analyze the robustness of the solution using a synthetic benchmark. The results indicate that the median of the posterior is very close to the true temperature distribution. The relative error increases at the edge of the temperature plume where the change of temperature is limited. This is related to the limited resolution of geophysics and the process of dimension reduction. We also investigate how discrete cosine transform can improve the dimension reduction process without altering the final prediction. Finally, we show that BEL is able to retrieve the spatio-temporal variability of the plume, while the smoothness constraint inversion fails to accurately image the corresponding contrast, largely underestimating the amplitude of the temperature change. BEL is therefore a well-suited framework for the interpretation of 4D geophysical data avoiding the drawbacks of standard deterministic solutions. Hermans, T., Oware, E., & Caers, J. (2016). Direct prediction of spatially and temporally varying physical properties from time-lapse electrical resistance data. Water Resources Research, 52, 7262-7283.

Prior information regarding subsurface patterns may be used in geophysical inversion to obtain realistic subsurface models. Field experiments require prior information with sufficiently diverse patterns to accurately estimate the spatial distribution of geophysical properties in the sensed subsurface domain. A variational autoencoder (VAE) provides a way to assemble all patterns deemed possible in a single prior distribution. Such patterns may include those defined by different base training images and also their perturbed versions, e.g. those resulting from geologically consistent operations such as erosion/dilation, local deformation and intrafacies variability. Once the VAE is trained, inversion may be done in the latent space which ensures that inverted models have the patterns defined by the assembled prior. Inversion with both a synthetic and a field case of cross-borehole GPR traveltime data shows that using the VAE assembled prior performs as good as using the VAE trained on the pattern with the best fit, but it has the advantage of lower computation cost and more realistic prior uncertainty. Moreover, the synthetic case shows an adequate estimation of most small scale structures. Estimation of absolute values of wave velocity is also possible by assuming a linear mixing model and including two additional parameters in the inversion.