Simulation models of watershed hydrology (also referred to as “rainfall-runoff models”) are calibrated to the best available streamflow data, which are typically published discharge time series at the outlet of the watershed. Even after calibration, the model generally cannot replicate the published discharges because of simplifications of the physical system embedded in the model structure and uncertainties of the input data and of the estimated model parameters, which, although optimized for the given calibration data, remain uncertain. The input data errors are caused by uncertainties in the forcing data, such as precipitation and other climatological data, and in the published discharges used for calibration. In the numerical algorithms used for calibration, the published discharges are often assumed to be without error, but they are themselves uncertain, typically having been computed using ratings, which are models fitted to uncertain discharge measurements.

In this study, uncertainty of published daily discharge data and how the discharge uncertainty is transmitted to the parameter values of the Hydrological Simulation Program–FORTRAN (HSPF) rainfall-runoff model and to the simulated discharge at both calibration and prediction locations were investigated for the Lake Michigan diversion in northeastern Illinois and northwestern Indiana. The HSPF model used in this study is used by the U.S. Army Corps of Engineers as part of quantifying the diversion of water from Lake Michigan by the State of Illinois. In this study, the model is calibrated jointly at two watersheds in the study area; the resulting model is considered the base model in this study. Seven other gaged watersheds in the study area are used for testing predictive simulations. A Bayesian rating curve estimation (BaRatin) approach, the BaRatin stage-period-discharge (SPD) method, was used to estimate the uncertainty of the published discharge from the calibration watersheds. To characterize the effect of the discharge uncertainty on parameter values, the HSPF model parameters were recalibrated to 17 nonrandomly selected pairs of discharge series from the BaRatin SPD analysis. To provide an indicator of the effect of parameter uncertainty to compare to the effect of discharge uncertainty, 1,000 parameter sets also were randomly generated from the estimated parameter covariance matrix of the base model. The recalibrated and random parameter sets were then used in HSPF simulations of discharge at the two calibration watersheds and at the seven prediction watersheds. Selected discharge summary statistics—the period-of-study (POS, water years 1997 to 2015) mean discharge, selected flow-duration curve (FDC) quantiles, and water year mean discharges—are used to characterize the variability between simulated and published discharge.

A normalized variability index (V_N) is used as a measure of the uncertainty of flow statistics arising from the uncertainty of the sources considered in this study. When this index is at least 1, the variability of the simulations is large enough to explain the median error between simulated and published values, although offsetting errors from other sources are also likely. When the index is appreciably less than 1, the variability of the simulations is clearly insufficient to explain the median error between simulated and published values. At the two calibration watersheds and for results of the two simulation sets considered together, the V_N values ranged from 0.2 to 0.8 for POS mean discharge, from 0.3 to 0.6 in the median for a set of FDC quantiles, and from 0.1 to 0.2 in the median for water year mean discharges. These values indicate that substantial uncertainty remains unexplained. Even though two watersheds were used in calibration, that calibration was highly constrained because it was applied to the watersheds simultaneously and was subject to parameter regularization that constrained the adjustment of the parameters from their initial values. These constraints were applied to avoid overfitting to the calibration watersheds and thus to increase the likelihood that the resulting parameters would give accurate results at watersheds not used in the calibration, but they created a parameter transfer error in the calibration watershed results shown by the balancing of errors between the two watersheds. Additional remaining error sources include model structural error and meteorological forcing error to the degree that the calibration was unable to adjust the parameters to account for these errors. At the prediction watersheds, the corresponding V_N values were almost always substantially lower than those values at the calibration watersheds. This result is expected because the prediction watersheds have additional uncertainty, including parameter transfer error.

The work described in this report provides preliminary estimates of a limited range of sources of error in predicted discharge uncertainty. Future work would be beneficial to obtain a better statistical characterization of the effect of the uncertainty of calibration discharge series and to address additional sources of uncertainty, such as from precipitation input data used in calibration and prediction and from structural (model) errors.