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.

Core Ideas Coupling SWAP with PEST, soil moisture and ET are included in inverse modeling of soil water flow. Multi‐objective optimization could improve model predictions and reduce parameter uncertainty. Observation errors and frequency and spatial arrangement impact predictions and uncertainty. The accurate estimation of soil hydraulic properties is an important part of the application of hydrological models for quantifying water transport in the vadose zone. Various inverse models have been developed to solve hydraulic properties optimization problems, and the accuracy of the results obtained by different algorithms is still debated because of the inherent ill‐posedness of such problems. In this study, we coupled an agrohydrological Soil–Water–Atmosphere–Plant (SWAP) model with an independent parameter estimation program (PEST) to calibrate the soil hydraulic parameters and investigate the uncertainty in the parameter estimations. The objectives of this study were to assess to what extent the SWAP model can be calibrated from a single observation (only soil moisture, θ) and further to investigate whether involving additional evapotranspiration fluxes including actual evapotranspiration (ETa), actual evaporation (Ea), or actual transpiration (Ta) can lead to an improvement in model predictions and a reduction in parameter uncertainty. Extensive synthetic experiments were conducted to achieve the objectives. Both double (winter wheat [Triticum aestivum L.]–summer maize [Zea mays L.]) and single (only winter wheat or only summer maize) cropping systems were considered in the multi‐objective optimizations. The results indicate that although the addition of evapotranspiration fluxes does not necessarily improve the accuracy of the soil moisture prediction, it can reduce the parameter uncertainty for both single‐ and double‐cropping systems. Moreover, the parameter estimation of the topsoil layer could greatly benefit from the addition of Ea, whereas the addition of Ta could help reduce the parameter uncertainty of the lower layers.