As all kinds of physics-based and data-driven models are emerging in hydrologic and hydraulic engineering, Bayesian model averaging (BMA) is one of the popular multi-model methods used to account for various uncertainty sources in the flood modeling process and generate robust ensemble predictions. The reliability of BMA parameters (weights and variances) determines the accuracy of BMA predictions. However, the uncertainty in BMA parameters with fixed values, which are usually obtained from Expectation-Maximization (EM) algorithm, has not been adequately investigated in BMA-related applications over the past few decades. Given the limitations of the commonly used EM algorithm, Metropolis-Hastings (M-H) algorithm, which is one of the most widely used algorithms in Markov Chain Monte Carlo (MCMC) method, is proposed to estimate BMA parameters. Both numerical experiments and one-dimensional HEC-RAS models are employed to examine the applicability of M-H algorithm with multiple independent Markov chains. The performances of EM and M-H algorithms are compared based on the daily water stage predictions from 10 model members. Results show that BMA weights estimated from both algorithms are comparable, while BMA variances obtained from M-H algorithm are closer to the given variances in the numerical experiment. Moreover, the normal proposal used in M-H algorithm can yield narrower distributions for BMA weights than those from the uniform proposal. Overall, MCMC approach with multiple chains can provide more information associated with the uncertainty of BMA parameters and its performance is better than the default EM algorithm in terms of multiple evaluation metrics as well as algorithm flexibility.

Runoff in natural rivers, commonly termed as streamflow, is a major process in the water cycle and a crucial variable in water resources engineering. While the increase in extreme rainfall events over the Conterminous United States (CONUS) has been well-documented, understanding the variability of streamflow remains challenging due to the nonlinear relationship between rainfall and runoff. In this study, daily streamflow data from 18 USGS gauge stations with the largest drainage area in its respective Hydrologic Unit Code 2-digit (HUC-2) region throughout the CONUS with contiguous records spanning from 2003 to 2022 water years is used to gain insights into streamflow variability over the past two decades. The original Mann-Kendall (MK) Test is employed to assess the potential temporal trends in the basic statistics (maximum, mean, minimum, and standard deviation) of annual streamflow data over the past 20 water years. Additionally, the seasonal MK Test is performed to explore the trends in the same basic statistics of the daily streamflow on a monthly basis. Furthermore, the statistical distributions of the normalized daily streamflow within each decade (2003-2012 and 2013-2022) are compared for each HUC-2 region. The results of the original MK Test indicate that no discernible trend in the annual streamflow and its standard deviation for most of the HUC-2 regions. However, the results of the seasonal MK Test suggest either an increasing or decreasing trend in around 30% of the HUC-2 regions. Moreover, low flows demonstrate a more significant change in frequency compared to high flows between the past two decades. Overall, this study highlights the complexity of the streamflow variability and the potential implications for changes in flood or drought risk under a changing climate.  

As all kinds of physics-based and data-driven models are emerging in the fields of hydrologic and hydraulic engineering, Bayesian model averaging (BMA) is one of the popular multi-model methods used to account for the various uncertainty sources in the flood modeling process and generate robust ensemble predictions based on multiple competitive candidate models. The reliability of BMA parameters (weights and variances) determines the accuracy of BMA predictions. However, the uncertainty in the BMA parameters with fixed values, which are usually obtained from the Expectation-Maximization (EM) algorithm, has not been adequately investigated in BMA-related applications over the past few decades. Given the limitations of the commonly used EM algorithm, the Metropolis-Hastings (M-H) algorithm, which is one of the most widely used algorithms in the Markov Chain Monte Carlo (MCMC) method, is proposed to estimate the BMA parameters and quantify their associated uncertainty. Both numerical experiments and the one-dimensional HEC-RAS models are employed to examine the applicability of the M-H algorithm with multiple independent Markov chains. The performances of the EM and M-H algorithms in the BMA analysis are compared based on the daily water stage predictions from 10 model configurations. The results show that the BMA weights estimated from both algorithms are comparable, while the BMA variances obtained from the M-H MCMC algorithm are closer to the given variances in the numerical experiment. Moreover, the normal proposal distribution used in the M-H algorithm can yield narrower distributions for the BMA weights than those from the uniform prior. Overall, the MCMC approach with multiple chains can provide more information associated with the uncertainty of BMA parameters and its prediction performance is better than the default EM algorithm in terms of multiple evaluation metrics as well as algorithm flexibility.

Evaluation of the performance of hydrologic and hydraulic models is a crucial step in the modeling process. Considering the limitations of single statistical metrics, such as the Nash Sutcliffe efficiency (NSE), the Kling Gupta efficiency (KGE), and the coefficient of determination (R2), which are widely used in the evaluation of model performance, an evaluation framework that incorporates multiple criteria and based on the generalized likelihood uncertainty estimation (GLUE) is proposed to demonstrate the uncertainty in the evaluation criteria and hence to quantify the overall uncertainty of flood models in a comprehensive way. This framework is applied to the one-dimensional HEC-RAS models of six reaches located in States of Indiana and Texas of the United States to quantify the uncertainty associated with the channel roughness and upstream flow input. Specifically, the effects of different prior distributions of the uncertainty sources, multiple high-flow scenarios, and various types of measurement errors (white noise, positive bias, and negative bias) in observations on the evaluation metrics are investigated by using the bootstrapping method and Monte Carlo simulations. The results show that the model performances based on the uniform and normal priors are comparable. The distributions of all the evaluation metrics in the framework are significantly different for the flood model under different high-flow scenarios, and it further indicates that the metrics are essentially random statistical variables. Additionally, the white-noise error in observations has the least impact on the metrics, while the positive and the negative biases would have opposite impacts, which depends on whether the model overestimated or underestimated the hydrologic variable.

Flood Insurance Rate Maps (FIRMs) managed by the Federal Emergency Management Agency (FEMA) have been providing ongoing flood information to most of the communities in the United States over the past half century. However, the uncertainty associated with the modeling of the FIRMs, some of which are created by using a single HEC-RAS one-dimensional (1D) steady flow model, may have adverse effects on the reliability of flood stage and inundation extent. Therefore, a systematic understating of the uncertainty in the modeling process of FIRMs is important and necessary. The Bayesian model averaging (BMA), which is a statistical approach that can combine estimations from multiple models and produce reliable probabilistic predictions, is applied to evaluating the uncertainty associated with the FIRMs. In this study, both the BMA and HBMA approaches are used to quantify the uncertainty within the detailed FEMA models of the Deep River and the Saint Marys River in the state of Indiana based on water stage predictions from 150 HEC-RAS 1D unsteady flow model configurations that incorporate four uncertainty sources including the bridges, channel roughness, floodplain roughness, and upstream flow input. The BMA weight and the variance for each model member are obtained given the ensemble predictions and the observed water stage data in the training period, and then the BMA prediction ability is validated for the observed data from the later period. The results indicate that BMA prediction is more robust than the original FEMA model as well as the ensemble mean. Different types of uncertainty coefficients based on the BMA prediction distribution are also proposed to evaluate the FEMA models. Furthermore, the HBMA framework shows that both the channel roughness and the upstream flow input have a larger impact on prediction variance than bridges, and hence provides some insights for modelers into the relative impact of individual uncertainty sources in the flood modeling process.

The Natural Resources Conservation Service (NRCS, formerly the Soil Conservation Service, SCS) unit hydrograph (UH) is one of the most commonly used synthetic UH methods for hydrologic modeling and engineering design all over the world. However, previous studies have shown that the application of the NRCS UH method for some ungauged watersheds in the state of Indiana produced unrealistic flood predictions for both the peak discharge and the time to peak. The objective of this work is to customize the NRCS UH by analyzing the role of its two key parameters, namely, the peak rate factor (PRF) and the lag time, in creating the runoff hydrograph. Based on 120 rainfall-runoff events collected from 30 small watersheds in Indiana over the past two decades, the observed UHs are derived and the corresponding PRF and lag time are extracted. The observed UHs in Indiana show that the mean value of PRF is 371, which is lower than the standard PRF of 484, and the NRCS lag time equation tends to underestimate the “true” lag time. Moreover, a multiple linear regression method, especially the stepwise selection technique, is employed to relate the NRCS UH parameters to the most appropriate geomorphic attributes extracted from the study watersheds. Both the statewide and regional regression models show that the main channel slope is a major factor in determining the PRF and lag time. A customized Indiana unit hydrograph (INUH) is derived with updated parameters and the Gamma function. Validation results show that the INUH provides more reliable and accurate predictions in terms of the peak discharge and the time to peak than the original NRCS UH for the watersheds in Indiana.