• nonlinear;
  • nonstationary;
  • time series transform

Climatic and hydrologic time series often display periodicities, and thus Fourier spectral analysis sometimes is appropriate. However, time series that are nonstationary, and also perhaps nonlinear, are not well handled by standard Fourier spectral analysis.

Methods to handle nonstationarity, such as moving-time-window Fourier spectral analysis, assume linearity and have known limitations regarding the combined frequency and time resolution. For example, if the time series is stationary, then it is well known that better frequency resolution can be achieved by observing a longer time series (more time points). However, if the time series is nonstationary, then shorter time windows are required to estimate the “local in time” spectrum, analogous to using short-memory moving averages that use only the recent past few values to forecast the next value (P. Bloomfield, Fourier Analysis of Time Series: An Introduction, 2nd ed., John Wiley, 2000) because the mean value is changing over time. Therefore, in nonstationary time series analysis, there is a tension between the competing goals of time and frequency resolution. This tension is the reason that N. Huang et al. (Proc. R. Soc. A, 454, 903–995, 1998) introduced the Hilbert-Huang Transform (HHT) as an alternative to moving-time-window Fourier spectral analysis (Bloomfield, 2000).