Stochastic parameter estimation procedures for hydrologie rainfall‐runoff models: Correlated and heteroscedastic error cases

TLDR

The authors propose a maximum‑likelihood calibration method that simultaneously handles correlated and heteroscedastic streamflow measurement errors and introduces residual tests to detect these error structures. The method models correlated errors with a first‑order autoregressive scheme, estimates the error correlation coefficient for use in the objective function, and treats heteroscedastic errors via a power‑transformation weighting approach, evaluated on synthetic data for a two‑parameter rainfall‑runoff model. Compared with simple and weighted least squares, the new procedures consistently yield more accurate parameter estimates, are straightforward to implement without convergence issues, produce identical results to least squares when errors are uncorrelated, and automatically reduce the objective function as data become homoscedastic.

Abstract

A maximum likelihood estimation procedure is presented through which two aspects of the streamflow measurement errors of the calibration phase are accounted for. First, the correlated error case is considered where a first‐order autoregressive scheme is presupposed for the additive errors. This proposed procedure first determines the anticipated correlation coefficient of the errors and then uses it in the objective function to estimate the best values of the model parameters. Second, the heteroscedastic error case (changing variance) is considered for which a weighting approach, using the concept of power transformation, is developed. The performances of the new procedures are tested with synthetic data for various error conditions on a two‐parameter model. In comparison with the simple least squares criterion and the weighted least squares scheme of the HEC‐1 of the U.S. Army Corps of Engineers for the heteroschedastic case, the new procedures constantly produced better estimates. The procedures were found to be easy to implement with no convergence problem. In the absence of correlated errors, as theoretically expected, the correlated error procedure produces the exact same estimates as the simple least squares criterion. Likewise, the self‐correcting ability of the heteroscedastic error procedure was effective in reducing the objective function to that of the simple least squares as data gradually became homoscedastic. Finally, the effective residual tests for detection of the above‐mentioned error situations are discussed.