A formal likelihood function for parameter and predictive inference of hydrologic models with correlated, heteroscedastic, and non‐Gaussian errors

Abstract

Estimation of parameter and predictive uncertainty of hydrologic models has traditionally relied on several simplifying assumptions. Residual errors are often assumed to be independent and to be adequately described by a Gaussian probability distribution with a mean of zero and a constant variance. Here we investigate to what extent estimates of parameter and predictive uncertainty are affected when these assumptions are relaxed. A formal generalized likelihood function is presented, which extends the applicability of previously used likelihood functions to situations where residual errors are correlated, heteroscedastic, and non‐Gaussian with varying degrees of kurtosis and skewness. The approach focuses on a correct statistical description of the data and the total model residuals, without separating out various error sources. Application to Bayesian uncertainty analysis of a conceptual rainfall‐runoff model simultaneously identifies the hydrologic model parameters and the appropriate statistical distribution of the residual errors. When applied to daily rainfall‐runoff data from a humid basin we find that (1) residual errors are much better described by a heteroscedastic, first‐order, auto‐correlated error model with a Laplacian distribution function characterized by heavier tails than a Gaussian distribution; and (2) compared to a standard least‐squares approach, proper representation of the statistical distribution of residual errors yields tighter predictive uncertainty bands and different parameter uncertainty estimates that are less sensitive to the particular time period used for inference. Application to daily rainfall‐runoff data from a semiarid basin with more significant residual errors and systematic underprediction of peak flows shows that (1) multiplicative bias factors can be used to compensate for some of the largest errors and (2) a skewed error distribution yields improved estimates of predictive uncertainty in this semiarid basin with near‐zero flows. We conclude that the presented methodology provides improved estimates of parameter and total prediction uncertainty and should be useful for handling complex residual errors in other hydrologic regression models as well.