Multiscale Analysis of Hydrologic Time Series Data using the Hilbert–Huang Transform

Abstract

For the analysis of me series data from hydrology, we used a recently developed technique that is by now widely known as the Hilbert-Huang transform (HHT). Specifi cally, it is designed for nonlinear and nonsta onary data. In contrast to data analysis techniques using the short-me, windowed Fourier transform or the con nuous wavelet transform, the new technique is empirically adapted to the data in the following sense. First, an addive decomposi on, called empirical mode decomposi on (EMD), of the data into certain mul scale components is computed. Second, to each of these components, the Hilbert transform is applied. The resul ng Hilbert spectrum of the modes provides a localized me-frequency spectrum and instantaneous ( me-dependent) frequencies. In this study, we applied the HHT to hydrological me series data from the Upper Rur Catchment Area, mostly German territory, taken during a period of 20 yr. Our fi rst observa on was that a coarse approxima on of the data can be derived by trunca ng the EMD representaon. This can be used to be er model pa erns like seasonal structures. Moreover, the corresponding me-frequency energy spectrum applied to the complete EMD revealed seasonal events in a par cular apparent way together with their energy. We compared the Hilbert spectra with Fourier spectrograms and wavelet spectra to demonstrate a be er localiza on of the energy components, which also exhibit strong seasonal components. The Hilbert energy spectrum of the three measurement sta ons appear to be very similar, indica ng li le local variability in drainage.Abbrevia ons: EMD, empirical mode decomposi on; HHT, Hilbert-Huang transform; IMF, intrinsic mode func on.Given empirical data, the detection and parameterization of multiscale patterns and shapes in the measurements is an important task. Specifi cally, to study the eff ect of patterns on water and solute fl uxes, temporal and spatial data have to be analyzed at various stages so that their parameterization can eventually be used in simulation fl ux models.Th is study was part of a SFB/TR32 project (Transregional Collaborative Research Centre 32, www.tr32.de; verifi ed 3 June 2010) for which the overall objective is to improve our knowledge about the mechanisms leading to spatial and temporal patterns in energy and matter fl uxes of the soil-vegetation-atmosphere system. Part of the objectives is the determination, description, and analysis of patterns derived from diff erent sources. For instance, the large spatial and temporal variability of soil moisture patterns is determined by factors like atmospheric forcing, topography, soil properties, and vegetation, which interact in a complex, nonlinear way (see, e.g., Grayson and Blöschl, 2001;Western et al., 2004). Th us, a very large number of continuous soil moisture measurements are necessary to adequately capture this variability. In the framework of the TR32, a dense soil moisture sensor network for monitoring soil water content changes at high spatial and temporal scales has been set up (see Bogena et al., ...

Patterns in Soil–Vegetation–Atmosphere Systems: Monitoring, Modeling, and Data Assimilation

“…Application to fi eld data corroborated these fi ndings and showed that the proposed approach is useful in estimating shallow soil properties using areal infrared measurements. Rudi et al (2010) applied the Hilbert-Huang transform (HHT) to long-term hydrologic time series of river discharge to extract information on local patterns and structures. Th e advantage of HHT over commonly applied Fourier and wavelet analyses is its applicability to nonuniform time grids.…”

“…Rudi et al (2010) applied the Hilbert–Huang transform (HHT) to long‐term hydrologic time series of river discharge to extract information on local patterns and structures. The advantage of HHT over commonly applied Fourier and wavelet analyses is its applicability to nonuniform time grids.…”

Time–frequency characterization of sub-divisional scale seasonal rainfall in India using the Hilbert–Huang transform

“…The time-frequency distribution of the amplitude is designated as the Hilbert amplitude spectrum (or Hilbert spectrum) H(x,t), which can be defined as (Rudi et al 2010) Hðx; tÞ ¼ H½xðtÞ; t ¼ A k ðtÞ on the curve f½t; xðtÞ : t 2 Rg…”

“…To rectify such issues a normalization of Hilbert transform was proposed by Huang et al (2009b). In recent past, the HHT method gaining popularity to analyse the hydro-climatic time series signals worldwide (Huang et al 2009a;Rudi et al 2010;Massei and Fournier 2012;Antico et al 2014;Adarsh and Janga Reddy 2015b). Few studies also applied the EMD procedure for the decomposition of hydrologic time series from India Raghukanth 2005, 2006;Karthikeyan and Nagesh Kumar 2013), but none of the studies used advanced variants of HHT for performing spectral analysis of hydro-climatic data.…”

Demodulation of time series highlights impacts of hydrologic drivers on the Everglades ecosystem

“…In hydrological modelling, in fact, such techniques have been primarily adopted for streamflow modelling and reconstruction (Rodriguez-Iturbe et al, 1971;Humphries et al, 1992) and have not performed well enough compared with traditional modelling frameworks to justify the increase in complexity. On the other hand, other advanced applications of spectral analysis, such as Short-Term Fourier Transform, Wavelets Transform or Hilbert-Huang Transform, were successfully applied to the analysis of non-stationary hydrological series (Kumar and Foufoula-Georgiou, 1997;Kang and Lin, 2007;Rudi et al, 2010) . However, while the latter techniques are designed to localize instantaneous or time-dependent frequencies in the time-frequency domain, demodulation enables the investigation of the time dependence of the amplitude and phase of a given frequency, which makes it perfectly suited for the analysis of trends in the energy associated to the harmonics of interest.…”