Uncertainty Estimation of Transient Storage Model Based on Bayesian Inference Framework Using the Formal Likelihood

Though the parameter in transient storage model (TSM) should be well estimated for interpretation and prediction of river mixing processes, it has been widely known that the parameters in TSM are highly uncertain to estimate. However, several studies in hydrology field have shown that the high uncertainty of parameters could be attributed to the use of informal likelihood based on goodness-of-fit. Therefore, there is a need to investigate whether the use of informal likelihood has led to high uncertainty of parameters in TSM and the parameter uncertainty can decrease by using the formal likelihood based on the sturdy probabilistic grounds.

In this study, the parameter uncertainty of TSM was estimated with informal and formal likelihood in order to suggest the better way of uncertainty estimation in TSM. For uncertainty estimation, the Generalized Likelihood Uncertainty Estimation (GLUE), which is one of the most widespread uncertainty estimation methods, was used. Two types of informal likelihood were formulated based on root-mean-squared-error (RMSE) and Nash-Sutcliffe efficiency (NSE). Meanwhile, the formal likelihood was formulated using the probabilistic distribution under the assumption of heteroscedastic and autocorrelated residual. This whole uncertainty estimation process was performed using the tracer test data achieved in Uvas Creek and Cheongmi Creek.

The reliability measure and the mean width of the prediction interval were computed to check the appropriateness of the estimated prediction intervals. The reliability measure was calculated as the percentage of the observations that are included within the prediction interval and mean width was obtained by averaging the width of prediction interval over the time. As a result, the parameter uncertainty of TSM was overestimated by the informal likelihood by showing the reliability measures as 100% under the 95% confidence level. On the other hand, the parameter uncertainty of TSM was well estimated by showing the coverage of 94.3% and 93.9%, that were almost equivalent to 95%, for Uvas Creek and Cheongmi Creek, respectively. Although the parameter uncertainty still existed when using the formal likelihood, it showed the smaller uncertainty level compared to the use of informal likelihood. This is because the use of informal likelihood incorrectly classified non-behavioral parameters into behavioral parameters.