  In hydrology, modeling streamflow remains a challenging task due to the
limited availability of basin characteristics information such as soil geology
and geomorphology. These characteristics may be noisy due to measurement errors
or may be missing altogether. To overcome this challenge, we propose a
knowledge-guided, probabilistic inverse modeling method for recovering physical
characteristics from streamflow and weather data, which are more readily
available. We compare our framework with state-of-the-art inverse models for
estimating river basin characteristics. We also show that these estimates offer
improvement in streamflow modeling as opposed to using the original basin
characteristic values. Our inverse model offers 3\% improvement in R$^2$ for
the inverse model (basin characteristic estimation) and 6\% for the forward
model (streamflow prediction). Our framework also offers improved
explainability since it can quantify uncertainty in both the inverse and the
forward model. Uncertainty quantification plays a pivotal role in improving the
explainability of machine learning models by providing additional insights into
the reliability and limitations of model predictions. In our analysis, we
assess the quality of the uncertainty estimates. Compared to baseline
uncertainty quantification methods, our framework offers 10\% improvement in
the dispersion of epistemic uncertainty and 13\% improvement in coverage rate.
This information can help stakeholders understand the level of uncertainty
associated with the predictions and provide a more comprehensive view of the
potential outcomes.
