### This project is titled Uncertainty Analysis Framework in the Water Balance Equation Using Bayesian Statistical Modeling Approach

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 terms in the WBE is generally difficult even in well-instrumented watersheds.

Satellite-based data can 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. 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 accurate estimation of the parameters. However, finding parameters in Eq. (2) is ill-posed problem because of missing other hydrologic constraints in Eq. (2) (e.g., other constraints than just P and SM) and simplified assumption in f(SM). In addition, sub-optimal parameters obtained from classical maximum likelihood estimation (MLE) approaches can create significantly different values of parameters when different regulation practices made during the optimization processes (e.g., arbitrarily set starting points and range limits for each parameter), 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. Even though prediction of P with different parameter sets in WEB does not vary significantly according to optimization practice, the inference (or hydrologic interpretation of parameters) made based on the classical approach can be greatly misleading.

The Bayesian modeling approach, on the other hand, allows us to investigate the uncertainties and full 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 space and the parameter space. In the current project, we 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 physics-based WBE approach using Bayesian inference with Markov chain Monte Carlo sampling and Variational Inferences methods.