On the use of the Whittle likelihood for calibrating hydrological models in scarcely gauged catchments

When extended hydrological information is lacking, it becomes extremely important to be able to calibrate hydrological models by taking advantage of sparse data observed at different time scales. The present paper proposes a methodology for estimating the parameters of any hydrological model, which is based on the use of the likelihood function proposed by Whittle in 1953. This latter is based on the approximate independence of the discrete Fourier transforms of the data at certain frequencies. Roughly speaking, the estimation is carried out by comparing the spectral density of the model with the discrete periodogram of the data. Whittle's likelihood has been widely used in hydrology, but exclusively for estimating the parameters of stochastic processes. It provides asymptotically consistent estimates for Gaussian and non-Gaussian data, even in the presence of long-range dependence, which is deemed to be a possible explanation for the well-known Hurst effect. This approach presents two main advantages: first, the model can be calibrated by using only certain Fourier frequencies, depending on data availability; second, the periodogram of the data can be estimated by using sparse observations. The spectral density of the model is approximated with the discrete periodogram of a long simulation provided by the model, with the addition of a term that allows to account for the presence of correlation in the model residuals. This latter behavior of the Whittle's estimator constitutes a potential advantage with respect to the traditionally used objective functions, like the sum of squared errors or the Nash coefficient of efficiency. The proposed procedure is applied to the case study of a Italian river basin, for which a lumped rainfall-runoff model has been calibrated. It is shown that the Whittle's estimator can be successfully applied within the GLUE methodology, therefore being a potentially useful tool for the application of hydrological models in the presence of equifinality