Uncertainty Analysis Framework in the Water Balance Equation Using Bayesian Statistical Modeling Approach
Abstract
The water balance equation (WBE) describes the relationship between net inflows into a water system and concurrent changes in storage during a time interval (d). The general WBE for the land surface is P (precipitation) = dS (change in storage) + ET (evapotranspiration) + D (drainage) + R (runoff) + etc. (e.g., vegetation-intercepted water) (Eq. 1). The WBE can be a critically important tool for estimating water supply, understanding the terrestrial water cycle, determining where there may be water shortages under climate change, and more. However, estimating the terms in the WBE is generally difficult, even in well-instrumented watersheds.
Satellite-based data can play a pivotal role in large-scale WBE analysis over ungauged areas by providing spatially continuous estimates of, for example, P and surface soil moisture (SM). By utilizing time-series estimates of satellite-based P and SM in order to partition ET versus D versus R, a simplified form of Eq. (1), such as P = ZdSM + f(SM) (Eq. 2), can be utilized, where Z is the representative depth (or hydrological storage length) of SM and f is a function which takes satellite-based SM information as an independent variable to bridge to other hydrologic variables in WBE (e.g., ET+D = aSM^b). The key to success in making rigorous hydrologic insights from estimated parameters (i.e., understanding partitioning of P into other variables) in Eq. (2) is the accurate estimation of these parameters. However, finding the parameters in Eq. (2) is an ill-posed problem because of the other missing hydrologic constraints in Eq. (2) (e.g., constraints other than just P and SM) and simplifying assumptions imbedded in f(SM). In addition, sub-optimal parameters obtained from classical maximum likelihood estimation (MLE) approaches can lead to significantly different parameter sets when different regulation practices made during the optimization processes (e.g., arbitrarily set starting points and range limits for each parameter), and different realizations (i.e., different sets of data sets), different optimization algorithms (e.g., Broyden-Fletcher-Goldfarb-Shanno algorithm, Newton-Conjugate-Gradient algorithm, etc.) and different objective and negative paneity functions are used. Even though estimates of P with different parameter sets in WEB do not vary significantly according to different optimization practices, the inference (or hydrologic interpretation of parameters) made based on the classical approach can be misleading. The Bayesian modeling approach, on the other hand, allows us to fully investigate the uncertainties and range of each parameter, as well as their dependency of our prior knowledge on varying parameters. In other words, we can use the Bayesian statistical modeling approach to understand the uncertainties when we determine the partitioning of P into different hydrologic variables. Thus, we can incorporate robust scientific knowledge into the water balance model. As an example, we can use prior predictive checks to determine the credibility of the link function (i.e., WBE), which, in turn, can help us understand the outcome and parameter space. In this current study, we will show the importance of more (intelligent) regularization of the problem and the need to exercise caution regarding the hydrologic interpretation of parameters obtained in a physically based WBE approach using Bayesian inference with the Markov chain Monte Carlo sampling and Variational Inferences methods.- Publication:
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AGU Fall Meeting Abstracts
- Pub Date:
- December 2022
- Bibcode:
- 2022AGUFM.H25J1231K