The mathematical challenges in agrogeophysics: examples and ways ahead

Electrical resistivity tomography (ERT) is increasingly used in the context of agriculture since the measured resistivity distribution can be linked to soil moisture, soil structure or pore water salinity. Due to its minimally invasive character, its spatial coverage and its monitoring abilities, ERT can be used to study field heterogeneity and competition between plants, quantify water fluxes throughout a growing season or distinguish preferential flow pathways in soils. Nevertheless, a lot of challenges still remain. From a mathematical point of view, the inverse problem linked to ERT is ill-posed. To solve it, the inverse problem is often regularized with a Tikhonov-type approach. The latter is typically done using a gradient operator, resulting in smoothed resistivity distribution. However, strong contrasts can exist due to e.g. compacted soil layers due to ploughing, water infiltration fronts, etc. In such a case, other operators such as the total variation or the minimum gradient support may be used. In such approaches, the selection of the regularization parameter with respect to the data quality and the definition of image appraisal indicators still remains a challenge. Uncertainty quantification of ERT-derived results often relies on data-error propagation around the inverse solution. Given the inherent non-uniqueness of the problem, both mathematically but also from a pedological point of view, challenges for stochastic approaches lie in providing realistic uncertainty estimation, encompassing all uncertainties (e.g. prior, pedophysics or data error). Monitoring data allows further elements to constrain the inverse problems, data can be replaced by data difference and regularization may incorporate the temporal dimension for instance. However, such constraints require their compatibility with the studied temporal process. Whereas the above challenges stay true for monitoring data, several alternative strategies are being developed more specifically, such as coupled hydrogeophysical inversion, with the challenge of addressing the non-stationarity of pedophysical relationships and the accuracy of the conceptual flow and transport model using deterministic approaches. Stochastic approaches allow to a certain extent to tackle those challenges in particular using a prior falsification/validation approach following a Popper-Bayes philosophy. In this presentation, we will illustrate the challenges and some of the recent developments with numerical and field examples.eRoo