HYDRUS-1D Simulation of Soil Water Dynamics and Response of Different ETo Models to Crop Evapotranspiration (ETc) Under a Rainfed Condition in Southern Manitoba

Soil water content (SWC) plays a critical role in crop yield, irrigation scheduling, and water resources management. In the Canadian Prairies, the water content in the rootzone replenished by rainfall is rarely sufficient to satisfy crop water requirements. Thus, the need for robust and effective water management. Hydrologic modelling provides the opportunity to understand the underlying processes controlling and affecting soil water movement and distribution. Evapotranspiration (ET) is an important input of hydrologic models; thus, the estimation of ET could have significant consequences on modelling outcome and inference. The FAO Penman-Monteith (PM) is the recommended model for estimating the reference crop evapotranspiration (ETo). However, it is limited by requiring too many weather variables that are not readily available. Simple empirical ETo models have been developed as an alternative. In this study, six ETo models with different inputs were used to simulate soil water dynamics in a rainfed potato farm in Winkler, Manitoba, using the HYDRUS-1D model. The results showed that when used to simulate SWC, all the models followed a similar pattern, although a significant difference was observed at shallow depth (20 cm). Specifically, a significant difference (p < 0.05) was observed between observed and simulated SWC from Hargreaves Samani, Romanenko, Penman, and FAO-PM (missing) models. When used to simulate the crop evapotranspiration (ETc), there was no significant difference (p > 0.05) between observed and simulated ETc from FAO PM, Irmak, and Priestly – Taylor models. Hence, ETo models with fewer data inputs such as Irmak and Priestly – Taylor models can provide accurate and reliable results for water management in southern Manitoba.

Large Time Convergence of the Non-homogeneous Goldstein-Taylor Equation

The Goldstein-Taylor equations can be thought of as a simplified version of a BGK system, where the velocity variable is constricted to a discrete set of values. It is intimately related to turbulent fluid motion and the telegrapher’s equation. A detailed understanding of the large time behaviour of the solutions to these equations has been mostly achieved in the case where the relaxation function, measuring the intensity of the relaxation towards equally distributed velocity densities, is constant. The goal of the presented work is to provide a general method to tackle the question of convergence to equilibrium when the relaxation function is not constant, and to do so as quantitatively as possible. In contrast to the usual modal decomposition of the equations, which is natural when the relaxation function is constant, we define a new Lyapunov functional of pseudodifferential nature, one that is motivated by the modal analysis in the constant case, that is able to deal with full spatial dependency of the relaxation function. The approach we develop is robust enough that one can apply it to multi-velocity Goldstein-Taylor models, and achieve explicit rates of convergence. The convergence rate we find, however, is not optimal, as we show by comparing our result to those found in [8].

Implementation of Taylor models in CORA 2018

Tool Presentation: Computing guaranteed bounds of function outputs when their input variables are bounded by intervals is an essential technique for many formal methods. Due to the importance of bounding function outputs, several techniques have been proposed for this problem, such as interval arithmetic, affine arithmetic, and Taylor models. While all methods provide guaranteed bounds, it is typically unknown to a formal verification tool which approach is best suitable for a given problem. For this reason, we present an implementation of the aforementioned techniques in our MATLAB tool CORA so that advantages and disadvantages of different techniques can be quickly explored without hav- ing to compile code. In this work we present the implementation of Taylor models and affine arithmetic; our interval arithmetic implementation has already been published. We evaluate the performance of our implementation using a set of benchmarks against Flow* and INTLAB. To the best of our knowledge, we have also evaluated for the first time how a combination of interval arithmetic and Taylor models performs: our results indicate that this combination is faster and more accurate than only using Taylor models.