Regularization methods improve the stability of ill-posed inverse problems by introducing some a priori characteristics for the solution such as smoothness or sharpness. In this contribution, we propose a multidimensional, scale-dependent wavelet-based ℓ₁-regularization term to cure the ill-posedness of the airborne (time-domain) electromagnetic induction inverse problem. The regularization term is flexible, as it can recover blocky, smooth and tunable in-between inversion models, based on a suitable wavelet basis function. For each orientation, a different wavelet basis function can be used, introducing an additional relative regularization parameter. We propose a calibration method to determine (an educated initial guess for) this relative regularization parameter, which reduces the need to optimize for this parameter, and, consequently, the overall computation time is under control. We apply our novel scheme to a time-domain airborne electromagnetic data set in Belgian saltwater intrusion context, but the scheme could equally apply to any other 2D or 3D geophysical inverse problem.

Human activities, such as managed aquifer recharge (MAR) and groundwater pumping, are influencing the natural groundwater flow in coastal areas. This might induce saltwater intrusion and impact fresh submarine groundwater discharge (FSGD). The use of resistivity methods (electrical resistivity tomography and continuous resistivity profiling) in coastal studies is very effective to investigate the salt–freshwater distribution, but it can be difficult to interpret quantitatively. In this study, the Western Belgian coast is investigated with resistivity methods, and image appraisal tools are systematically used to quantitatively interpret inversion models. Synthetic resistivity models, which reflect the existing situation at the Western Belgian coast, are first created and assessed quantitatively by means of the model resolution matrix, cumulative sensitivity matrix, and depth of investigation index. They reveal that no quantitative interpretation is possible of the FSGD, although lateral qualitative changes can be deduced from the inversion models. The field data show freshwater outflow from the lower beach to below the low water line, and they indicate that MAR has a positive impact on FSGD, while groundwater extraction reduces the outflow of freshwater to the North Sea. A comparison with existing groundwater models indicates their inability to reproduce the actual FSGD footprints.

SUMMARY: The inversion of electromagnetic induction data to a conductivity profile is an ill-posed problem. Regularization improves the stability of the inversion and a smoothing constraint is typically used. However, the conductivity profiles are not always expected to be smooth. Here, we develop a new inversion scheme in which we transform the model to the wavelet space and impose a sparsity constraint. This sparsity constrained inversion scheme will minimize an objective function with a least-squares data misfit and a sparsity measure of the model in the wavelet domain. A model transform to the wavelet domain allows to investigate the temporal resolution (periodicities at different frequencies) and spatial resolution (location of the peaks) characteristics of the model, and penalizing small-scale coefficients effectively reduces the complexity of the model. The novel scale-dependent regularization term can be used to favour either blocky or smooth structures, as well as high-amplitude models in globally smooth structures in the inversion. Depending on the expected conductivity profile, a suitable wavelet basis function can be chosen. The scheme supports multiple types of regularization with the same algorithm and is thus flexible. Finally, we apply this new scheme on a frequency domain electromagnetic sounding data set, but the scheme could equally apply to any other 1-D geophysical method.

Full 3D inversion of time-domain Airborne ElectroMagnetic (AEM) data requires specialists’ expertise and a tremendous amount of computational resources, not readily available to everyone. Consequently, quasi-2D/3D inversion methods are prevailing, using a much faster but approximate (1D) forward model. We propose an appraisal tool that indicates zones in the inversion model that are not in agreement with the multidimensional data and therefore, should not be interpreted quantitatively. The image appraisal relies on multidimensional forward modeling to compute a so-called normalized gradient. Large values in that gradient indicate model parameters that do not fit the true multidimensionality of the observed data well and should not be interpreted quantitatively. An alternative approach is proposed to account for imperfect forward modeling, such that the appraisal tool is computationally inexpensive. The method is demonstrated on an AEM survey in a salinization context, revealing possible problematic zones in the estimated fresh–saltwater interface.

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.