Abstract
This paper demonstrates, by means of a systematic uncertainty analysis, that the use of outputs from more than one model can significantly improve conditional forecasts of discharges or water stages, provided the models are structurally different. Discharge forecasts from two models and the actual forecasted discharge are assumed to form a three-dimensional joint probability density distribution (jpdf), calibrated on long time series of data. The jpdf is decomposed into conditional probability density distributions (cpdf) by means of Bayes formula, as suggested and explored by Krzysztofowicz in a series of papers. In this paper his approach is simplified to optimize conditional forecasts for any set of two forecast models. Its application is demonstrated by means of models developed in a study of flood forecasting for station Stung Treng on the middle reach of the Mekong River in South-East Asia. Four different forecast models were used and pairwise combined: forecast with no model, with persistence model, with a regression model, and with a rainfall-runoff model. Working with cpdfs requires determination of dependency among variables, for which linear regressions are required, as was done by Krzysztofowicz. His Bayesian approach based on transforming observed probability distributions of discharges and forecasts into normal distributions is also explored. Results obtained with his method for normal prior and likelihood distributions are identical to results from direct multiple regressions. Furthermore, it is shown that in the present case forecast accuracy is only marginally improved, if Weibull distributed basic data were converted into normally distributed variables.
Highlights
Our objective is to provide a systematic error analysis based on Bayes formula in order to obtain the optimum forecast from up to two models, and to determine the probability density function of forecast errors eF(i+m)
The second step is model calibration because hydrological models generally depend on empirical parameters which are specific for the basin considered
We start with given forecast models, which are supposed to have been developed in a design phase
Summary
The best forecast is defined as that linear combination of model outputs from the set which minimizes the forecast error. The result is a simplified version of the Bayesian framework developed by Krzysztofowicz [1], and Todini [2] This approach is generally applicable to any two forecast models. For each of the flood forecast models that set of model parameters is determined which minimizes the error between forecast and observation of long historical time series. Due to model-, parameter- , and other uncertainties only estimates for the true record are obtained, and the purpose of the analysis in this paper is to minimize total uncertainty, which shall be expressed through the error of observed record minus forecast.
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