Multiscale Analysis of Hydrologic Time Series Data using the Hilbert-Huang-Transform (HHT)

For the analysis of time series data from hydrology, we employ a technique recently developed in Huang et al., 1998, which is by now widely known as the Hilbert-Huang-Transform (HHT). Specifically, it is designed for nonlinear and nonstationary data. In contrast to data analysis techniques employing the short-time/windowed Fourier transform or the continuous wavelet transform, the new technique is empirically adapted to the data in the following sense. First, one computes an additive decomposition called empirical mode decomposition (EMD) of the data into certain multiscale components. Second, to each of these components, the Hilbert transform is applied. The resulting Hilbert spectrum of the modes provides a localized timefrequency spectrum and instantaneous (time-dependent) frequencies. In this paper, we recall the necessary components of the HHT and apply it to hydrological time series data from the Upper Rur Catchment Area, mostly German territory, taken over a period of twenty years. Our first observation is that a coarse approximation of the data can be derived by truncating the EMD representation. This can be used to better model patterns like seasonal structures. Moreover, the corresponding time/frequency energy spectrum applied to the complete EMD reveals in a particular apparent way seasonal events together with their energy. We provide a comparison of the Hilbert spectra with Fourier spectrograms and wavelet spectra in order to demonstrate a better localization of the energy components which also exhibit strong seasonal components. Finally, the Hilbert energy spectrum of the three measurement stations appear to be very similar, indicating little local variability in drainage.