A spectral-domain approach for the calibration of shot noise-based models of daily streamflows

Abstract

Since 50's the scientific community has been strongly interested in the modeling of several classes of stochastic processes, among which a particular attention has been attracted by hydro-meteorological phenomena. Indeed, both their synthetic reproduction and forecasting is a central point in the resolution of a wide class of problems, as the design and management of water resources systems and flood risk analysis. Concerning the modeling of generic stochastic variables, there are two crucial aspects to be addressed: the identification of the most appropriate model able to correctly reproduce the statistical features of the real process (i.e., model selection), and the estimation of the parameters of the selected model (i.e., model calibration) from available data, concerning both input and output processes of the system to be identified. During the last decades, considerable efforts have been undertaken by researchers to provide the scientific community with suitable calibration techniques of several class of models, resulting in the current availability of numerous procedures for the estimation of the selected model parameters. A general distinction can be made between time-domain and frequency-domain (or spectral-domain) calibration approaches, whose main difference consists in the typology of information adopted in the parameters estimation. Indeed, while the former are usually based on a numerical comparison between historical and synthetic series of the output process, spectral-domain procedures adopt, in several ways, the frequency information content of recorded input and output series. Such a substantial difference results in some considerable advantages for frequency-domain techniques, especially in the case of unavailability of sufficiently long and simultaneous records of both input and output variables. This latter condition, in particular, is not rare in the case of hydrologic model calibration problems, since the model input sequences (e.g., rainfall and air temperature series) and the output sequence (e.g., the streamflow series) can be usually both available but not simultaneously or even unavailable (i.e., poorly gauged or ungauged basins). These considerations make recommendable the adoption of frequency-domain calibration techniques in hydrologic applications. Starting from this proposed framework, in this thesis the author focuses on the spectral-domain calibration problem of a widely developed class of models for the modeling of daily streamflow processes, the so-called shot noise models. These models consider the river flow process as the result of a convolution of Poisson-distributed occurrences, representing the rainfall process, and a linear response function, depending on the parameters to be estimated, representing the natural basin transformations. The technical literature provides several techniques for the calibration of this class of models, both in the time and in the frequency domain. Nevertheless, none of the existing procedures is found to take advantage of a remarkable property of shot noise models, i.e. the impulsive nature of the autocorrelation function of the input process. On the contrary, starting from this relevant feature, the proposed calibration technique allow the estimation of the basin response function parameters only through the knowledge of the power spectral density of the recorded streamflow series. Hence, on the one hand, the main drawbacks of classical time-domain calibration approaches are solved and, on the other hand, the dependence of existing frequency-domain techniques on the availability of both input and output data is overcome. The effectiveness of the proposed procedure is widely proved through its application to three daily streamflow series, associated to three watersheds located in the Italian territory. In particular, performances of the approach in the reproduction of the recorded flow series statistical properties are ascertained through a simulation analysis.