A Novel Bayesian Framework for Inverse Problems in Precision Agriculture

An essential problem in precision agriculture is to accurately model and predict root-zone (top 1 m of soil) soil moisture profile given soil properties and precipitation and evapotranspiration information. This is typically achieved by solving agro-hydrological models. Nowadays, most of these models are based on the standard Richards equation (RE), a highly nonlinear, degenerate elliptic-parabolic partial differential equation that describes irrigation, precipitation, evapotranspiration, runoff, and drainage through soils. Recently, the standard RE has been generalized to time-fractional RE with any fractional order between 0 and 2. Such generalization allows the characterization of anomalous soil exhibiting non-Boltzmann behavior due to the presence of preferential flow. In this work, we focus on inverse modeling of time-fractional RE; that is, how to accurately estimate the fractional order and soil property parameters of the fractional RE given soil moisture content measurements. Specifically, we introduce a novel Bayesian variational autoencoder (BVAE) framework that synergistically integrates our in-house developed fractional RE solver and adaptive Fourier decomposition (AFD) to accurately estimate the parameters of time-fractional RE. Our proposed AFD-enhanced BVAE framework consists of a probabilistic encoder, latent-to-kernel neural networks and convolutional neural networks. The BVAE framework is theoretically explainable and enhanced by the AFD theory, a novel signal processing technique that achieves superior computationally efficiency. Through illustrative examples, we demonstrate the efficiency and reliability of our AFD-enhanced BVAE framework.