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

• Published in: Water resources research 2021-01, Vol.57 (1), p.n/a
• Main Authors: Li, N., Yue, X. Y., Wang, W. K., Ma, Z. T.
• Format: Article
• Language: English
• Subjects: Boundaries; Boundary conditions; Computation; field case study; Field tests; Fluctuations; Hydrology; In situ measurement; Inverse estimation; Mathematical analysis; Mathematical models; Meteorological information; Modelling; Moisture content; Moisture effects; Numerical schemes; Regularization; Richards equation; Soil; Soil moisture; Soil profiles; Soil properties; Soil water; surface flux boundary conditions; Tikhonov regularization; Unsaturated flow; unsaturated soil water flow; Uptake; Water flow; Water uptake
• ISSN: 0043-1397; 1944-7973
• DOI: 10.1029/2020WR028030

Accurate 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. Key Points Spatiotemporal top flux is inversely estimated in unsaturated water flow modeling from soil moisture dynamics using Tikhonov regularization When top fluxes vary with time only (special case), the inversion reduces to be well‐posed and is explicitly solved without regularization The method's performance is evaluated by synthetic data and its applicability in the special case is further verified in field examples