○E The Hilbert–Huang Transform: A High Resolution Spectral Method for Nonlinear and Nonstationary Time Series

The Fourier transform remains one of the most popular spectral methods in time-series analysis, so much so that the word “spectrum” is virtually equivalent to “Fourier spectrum” (Huang et al., 2001). This method assumes that a time series extends from positive to negative infinity (stationarity) and consists of a linear superposition of sinusoids (linearity). However, geophysical signals are never stationary and are not necessarily linear. This results in a trade-off between time and frequency resolution for nonstationary signals and the creation of spurious harmonics for nonlinear signals. We present an open-source implementation of the Hilbert–Huang transform (HHT), an alternative spectral method designed to avoid the linearity and stationarity constraints of Fourier analysis. The HHT defines instantaneous frequency as the time derivative of phase, illuminating previously inaccessible spectral details in transient signals. Nonlinear signals become frequency modulations rather than a series of fitted sinusoids, eliminating artificial harmonics in the resulting spectrogram. In this paper, we describe the HHT algorithm and present our recently-developed hht package for the R programming language. This package includes routines for empirical mode decomposition (EMD), ensemble empirical mode decomposition (EEMD) and Hilbert spectral analysis. It also comes with high-level plotting functions for easy and accurate visualization of the resulting waveforms and spectra. We demonstrate this code by applying it to three signals: a synthetic nonlinear waveform, a transient signal recorded at Deception Island volcano, Antarctica, and quasi-harmonic tremor from Reventador volcano, Ecuador. The synthetic signal shows how the EMD method breaks complex time series into simpler modes. It also illustrates how the Hilbert transforms of nonlinear signals produce frequency oscillations rather than harmonics. The transient signal demonstrates the high-time/frequency resolution of the HHT method. The volcanic-tremor signal has highfrequency harmonics in the Fourier spectrogram, which are not present in the Hilbert spectrogram. The EMD of the tremor signal also reveals unexpected transient events. Data and code for each analysis are included in the E electronic supplement available with this paper.