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 is 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 1-D imaging has been presented to the community as a viable tool for the efficient stochastic 1-D 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 analyse the convergence behaviour 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.

Understanding the subsurface is of prime importance for many geological and hydrogeological applications. Geophysical methods offer an economical alternative for investigating the subsurface compared to costly borehole investigations. However, geophysical results are commonly obtained through deterministic inversion of data whose solution is non-unique. Alternatively, stochastic inversions investigate the full uncertainty range of the obtained models, yet are computationally more expensive. In this research, we investigate the robustness of the recently introduced Bayesian evidential learning in one dimension (BEL1D) for the stochastic inversion of time-domain electromagnetic data (TDEM). First, we analyse the impact of the accuracy of the numerical forward solver on the posterior distribution, and derive a compromise between accuracy and computational time. We also introduce a threshold-rejection method based on the data misfit after the first iteration, circumventing the need for further BEL1D iterations. Moreover, we analyse the impact of the prior-model space on the results. We apply the new BEL1D with a threshold approach on field data collected in the Luy River catchment (Vietnam) to delineate saltwater intrusions. Our results show that the proper selection of time and space discretization is essential for limiting the computational cost while maintaining the accuracy of the posterior estimation. The selection of the prior distribution has a direct impact on fitting the observed data and is crucial for a realistic uncertainty quantification. The application of BEL1D for stochastic TDEM inversion is an efficient approach, as it allows us to estimate the uncertainty at a limited cost.

SUMMARY: Understanding processes in the Critical Zone requires reliable information about the vadose-zone aquifer, its geometry, and spatial variability. Commonly, such information is obtained from boreholes, yet large areas might render their application prohibitively expensive. Additionally, limited geological a-priori information might bias the interpretation due to lateral geological changes smaller than the borehole sampling scale. The transient electromagnetic method (TEM) has emerged in the last decades as a well-suited method to efficiently investigate the subsurface, as required for many hydrogeological applications. The interpretation of TEM measurements relies mainly on deterministic inversions, offering only a limited insight on the uncertainty of the subsurface model. Uncertainty quantification, however, is essential for integrating TEM results into hydrogeological models. Hence, we propose a combined approach using both deterministic and stochastic inversion of TEM soundings to investigate the uncertainty of shallow ($<$ $40 \,\mathrm{m}$) aquifers. Current stochastic approaches for TEM data rely on Markov chain Monte Carlo algorithms, which have to be run from scratch for each individual sounding. Alternatively, machine learning approaches, such as Bayesian Evidential Learning (BEL), can be much faster because they do not require retraining for every new data set. Hence, we investigate, in particular, the application of a single, common prior model space instead of multiple, individual prior model spaces to directly estimate the uncertainty of multiple TEM soundings. To this end, we combine forward modelling routines with the stochastic inversion approach BEL1D and assess our approach using both field data and numerical experiments.

Detailed information on the clay content of the subsurface and its spatial distribution plays a critical role in the interaction between surface- and groundwater. In this study, we investigate a new methodology to integrate data measured with electromagnetic and electrical geophysical methods, namely, the transient electromagnetic (TEM) and the spectral induced polarization (SIP) to quantify subsurface clay content in an imaging framework. The methodology is tested in data sets collected at a quarry close to Vienna and consists of a ca. 10 m thick clay layer below a ca. 8 m thick overburden of sandy silts. Our data set includes SIP data collected along a 315 m long profile with an electrode separation of 5 m in a frequency range from 0.1 to 225 Hz. Along this profile, we measured 26 TEM soundings using a 12.5 m loop with 24 windows recording in a time range between 4 and 140 μs. Ground truth information corresponds to grain size analysis conducted in 25 soil samples collected in a depth from 5 to 28 m. SIP inversion results at a single frequency provided structural a-priori information to improve the inversion of the TEM data. The inverted TEM conductivity model, nearest to the position of soil sample collection, was correlated to the grain size distribution and the resulting positive exponential relationship was used to obtain vertical 1D variations of clay content with depth. All sounding positions were interpolated to obtain a 2D image of subsurface clay content. This clay content variations were then compared to images of the Cole-Cole parameters, describing the frequency dependence of SIP imaging results. To evaluate the uncertainty in our clay estimations, we applied the Bayesian evidential learning 1D imaging (BEL1D). We obtained uncertainties of layer thickness, resistivity, and clay content by integrating the clay-conductivity relationship derived from TEM data into the BEL1D framework.

SUMMARY: Surface waves such as Rayleigh, Love and Scholte waves can exhibit dispersion, that is, variations in phase velocity with wavelength as a function of frequency. This property enables the inversion of 1-D models of seismic velocity and density in the subsurface. Conventional deterministic and stochastic inversion schemes are widely applied to surface wave data but face two main challenges. The first is the identification of dispersion curves for fundamental and higher modes on wavefield-transformed images, which is often done manually. The second is the quantification of uncertainty, which can be computationally expensive in stochastic approaches or limited to data-propagated uncertainty in deterministic inversions. Our objectives are to (1) eliminate the need for manual or automatic dispersion curve picking, and (2) directly infer ensembles of 1-D velocity models—and their associated uncertainties—from the full velocity spectrum, that is, the complete dispersion image containing all modes. To this end, we employ Bayesian Evidential Learning, a predictive framework that reproduces experimental data from prior information while allowing prior falsification. In our application, ensembles of prior Earth models are sampled to predict 1-D subsurface structures in terms of seismic velocity and, where applicable, attenuation from near-surface seismic wave data. This approach bypasses traditional inversion schemes and provides a computationally efficient tool for uncertainty quantification.