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

Abstract

AbstractAccurate estimation of boundary conditions is essential to the success of numerical schemes for modeling soil water flow processes. The flux‐type boundary condition, which is integration of many boundary fluxes and subsurface conditions, is traditionally estimated from hydrologic and meteorological information. However, such information could be rapidly variable in space and time and difficult to measure and quantify accurately. This study inversely estimates upper flux boundary conditions by calibrating interior observations of soil moisture. Unknown flux boundaries with both spatiotemporal and temporal only variabilities are studied. In the case of spatiotemporal variability, we formulate the inverse model by extending a Tikhonov regularization framework proposed in our previous study, which calibrated a macroscopic root water uptake model in one‐dimensional soil water flow. The developed method is efficient because it takes advantage of the strategy in the previous study that the inversion is transformed to a variational problem, and also because it incorporates locally one‐dimensional operator splitting technique to overcome the computation complexity when dealing with multidimensional flow processes. When flux boundaries vary with time only (special case), we have found that the inversion degenerates into a well‐posed problem and formulated the inversion explicitly without use of regularization. Numerical tests are conducted through both one‐dimensional (well‐posed) and two‐dimensional (ill‐posed) heterogeneous soil profiles to illustrate the effectiveness of the proposed inverse formulation. Particularly, the applicability of the inverse formulation in the special case is further validated in field experiments using in situ measurements in arid and semiarid northern China.