Data assimilation (DA) makes the best combination of model simulation results and observed, or measured, values. Ensemble methods are a form of DA that generates multiple equally good, or equally calibrated, models using a description of model and observation uncertainty. Uncertainty is lack of knowledge. The collection of equally good, in the presence of uncertainty, models is an ensemble of models. An observation error model provides the means to describe the amount of uncertainty in model simulation results and in observed values as part of assimilation. Model-related uncertainty comes from model representation limitations created by differences between what the model represents, or simulates, and by what is measured to make an observation. Observation uncertainty comes from observation error. When an observed value is calculated or estimated, rather than measured, additional uncertainty is generated by the estimation procedure.

An observation error model is developed and presented for river discharge observations made at a stream gauging station using a measured water depth value with a derived rating curve to calculate discharge from observed water depth. A rating curve is a poor hydrodynamics model. Consequently, large estimation errors are expected for river discharge calculated using a rating curve, which generates correspondingly large amounts of observation uncertainty for assimilation. Uncertainty is propagated through DA to the spread, or variability, of model outcomes provided by the ensemble of models. When assimilating simulation results and data with significant uncertainty, the goal of assimilation is to optimize the bias-variance tradeoff and thus the spread of ensemble outcomes. Optimizing this tradeoff involves limiting the amount of uncertainty as much as possible to make informed decisions while including sufficient uncertainty to avoid overfitting. The risk from overfitting is production of biased model outcomes and spurious decision support.

Abstract: A discharge uncertainty envelope is presented that provides an observation error model for data assimilation (DA) using discharge observations derived from measurement of stage using a rating curve. It uniquely represents the rating curve representation error, which is due to scale and process incompatibility between the rating curve hydrodynamic model and “true” discharge, within the observation error model. Ensemble methods, specifically, the iterative ensemble smoother (IES) algorithms in PEST++, provide the DA framework for this observation error model. The purpose of the uncertainty envelope is to describe prior observation uncertainty for ensemble methods of DA. Envelope implementation goals are (1) limiting the spread of the envelope to avoid conditioning to extreme parameter values and producing posterior parameter distributions with increased variance, and (2) incorporating a representative degree of observation uncertainty to avoid overfitting, which will introduce bias into posterior parameter estimates and predicted model outcomes. The expected uncertainty envelope is flow regime dependent and is delineated using stochastic, statistical methods before undertaking history matching with IES. Analysis of the goodness-of-fit between stochastically estimated “true” discharge and observed discharge provides criteria for the selection of best-fit parameter ensembles from IES results.