Calibrating the Spatiotemporal Root Density Distribution for Macroscopic Water Uptake Models Using Tikhonov Regularization

Abstract

Macroscopic root water uptake models proportional to a root density distribution function (RDDF) are most commonly used to model water uptake by plants. As the water uptake is difficult and labor intensive to measure, these models are often calibrated by inverse modeling. Most previous inversion studies assume RDDF to be constant with depth and time or dependent on only depth for simplification. However, under field conditions, this function varies with type of soil and root growth and thus changes with both depth and time. This study proposes an inverse method to calibrate both spatially and temporally varying RDDF in unsaturated water flow modeling. To overcome the difficulty imposed by the ill‐posedness, the calibration is formulated as an optimization problem in the framework of the Tikhonov regularization theory, adding additional constraint to the objective function. Then the formulated nonlinear optimization problem is numerically solved with an efficient algorithm on the basis of the finite element method. The advantage of our method is that the inverse problem is translated into a Tikhonov regularization functional minimization problem and then solved based on the variational construction, which circumvents the computational complexity in calculating the sensitivity matrix involved in many derivative‐based parameter estimation approaches (e.g., Levenberg‐Marquardt optimization). Moreover, the proposed method features optimization of RDDF without any prior form, which is applicable to a more general root water uptake model. Numerical examples are performed to illustrate the applicability and effectiveness of the proposed method. Finally, discussions on the stability and extension of this method are presented.

Inverse Estimation of Spatiotemporal Flux Boundary Conditions in Unsaturated Water Flow Modeling

“…The aim of this study is to propose a deterministic inverse method to parameterize and reconstruct the spatiotemporally varying flux‐type boundary conditions of unsaturated soil water flow modeling conditioned on the inexpensive measured soil moisture dynamics. We adapt and develop our previous work (Li & Yue, 2018) by employing a nonlinear Tikhonov regularization technique to construct the inverse problem. Efforts are made in deducing the derivative function of the observation on the unknown, deriving the more appropriate boundary information of the unknown, employing efficient algorithms of operator splitting to solve multidimensional flow equation.…”

“…Before the final step to solve from Equation , one needs to define the boundary condition for . In our pervious study (Li & Yue, 2018), an essential boundary condition was assumed for the unknown root density function, that is, the boundary condition was prescribed as a prior. Herein, to be general and to be realistic, we assume there is no precise information on the boundary conditions.…”

“…Following Li and Yue (2018), here we choose a stronger norm, norm of the first derivative of the unknown top flux as one of the penalty term. Such incorporation of a derivative operator has a smoothing effect and in many cases leads to a considerable improvement of the results (Franklin, 1974; Kirsch, 2011).…”

“…The first two terms in the objective functional are referred to as spatial and temporal constraints, respectively, and the third term is the data misfit between the predicted and the observed data. This objective function differs from that of the previous study (Li & Yue, 2018) in that the constraint terms and the residual error are integrated over different spaces denoted by…”

“…To cope with the ill-posedness, the concept of nonlinear Tikhonov regularization (Tikhonov & Arsenin, 1977;Tikhonov et al, 1995) is employed in the construction of the objective function. This research work is an adaption and development of our previous study (Li & Yue, 2018), which calibrated a spatiotemporal root water uptake model in one-dimensional soil water flow processes. The present study shares the same concept to formulate the inverse problem in the framework of nonlinear Tikhonov regularization, which adds smoothing factors to the original minimization problem and thus add additional constraints to reduce the solution space, and takes advantage of the strategy to translate the inverse problem into a functional minimization problem and frame it in a spatially continuous system, which circumvents the computational complexity of the calculation of the sensitivity matrix.…”