Multiscale Analysis of Hydrologic Time Series Data using the Hilbert–Huang Transform
For the analysis of time series data from hydrology, we used a recently developed technique that 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 using 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, an additive decomposition, called empirical mode decomposition (EMD), of the data into certain multiscale components is computed. Second, to each of these components, the Hilbert transform is applied. The resulting Hilbert spectrum of the modes provides a localized time–frequency spectrum and instantaneous (time-dependent) frequencies. In this study, we applied the HHT to hydrological time series data from the Upper Rur Catchment Area, mostly German territory, taken during a period of 20 yr. Our first observation was 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 revealed seasonal events in a particular apparent way together with their energy. We compared the Hilbert spectra with Fourier spectrograms and wavelet spectra to demonstrate a better localization of the energy components, which also exhibit strong seasonal components. The Hilbert energy spectrum of the three measurement stations appear to be very similar, indicating little local variability in drainage.