Effect of Heteroscedasticity Treatment in Residual Error Models on Model Calibration and Prediction Uncertainty Estimation

Formal Bayesian approaches are widely adopted for parameter calibration and prediction uncertainty estimation, but can also cause unrealistic distributions due to inappropriate statistical assumptions of the model residual errors, which are usually nonnormal, autocorrelated and heteroscedastic. The heteroscedasticity has a direct impact on the model calibration and prediction uncertainty estimation. This study compares three methods to deal with the heteroscedasticity, including the explicit linear modeling (LM) method and nonlinear modeling (NL) method using the hyperbolic tangent function, as well as the implicit Box-Cox transformation (BC). Then a combined approach (CA) which combines the advantages of both LM and BC methods has been proposed. The autocorrelation is represented using a first order autoregressive (AR) model (AR(1)) and the nonnormality is modeled using the skew exponential power (SEP) distribution. Altogether four residual error models, namely LM-SEP, NL-SEP, BC-SEP and CA-SEP, and their corresponding likelihood functions are applied for the Variable Infiltration Capacity (VIC) hydrologic model over the Huaihe River basin, China, where streamflow indicates strong heteroscedasticity and seasonality. Results show that the LM-SEP generates quite different parameter posterior distributions compared to three other residual error models and yields the poorest streamflow predictions with the widest uncertainty band and unrealistic negative flows because the LM method can’t represent the complicated heteroscedasticity. The NL and BC methods can better deal with the heteroscedasticity and hence their corresponding predictive performance is improved, yet the negative flows cannot be avoided. The combination of LM and BC is more suitable for addressing the complicated heteroscedasticity over the Huaihe River basin. The CA-SEP produces the optimal predictions with the highest precision and reliability, and effectively avoids the negative flows.