Wavelet-based group and phase velocity measurements: Method

Measurements of group and phase velocities of surface waves are often carried out by applying a series of narrow bandpass or stationary Gaussian filters localized at specific frequencies to wave packets and estimating the corresponding arrival times at the peak envelopes and phases of the Fourier spectra. However, it's known that seismic waves are inherently nonstationary and not well represented by a sum of sinusoids. Alternatively, a continuous wavelet transform (CWT) which decomposes a time series into a family of wavelets, translated and scaled copies of a generally fast oscillating and decaying function known as the mother wavelet, is capable of retaining localization in both the time and frequency domain and well-suited for the time-frequency analysis of nonstationary signals. Here we develop a wavelet-based method to measure frequency-dependent group and phase velocities, an essential dataset used in crust and mantle tomography. For a given time series, we employ the complex morlet wavelet to obtain the scalogram of amplitude modulus |Wg| and phase φ on the time-frequency plane. The instantaneous frequency (IF) is then calculated by taking the derivative of phase with respect to time, i.e., (1/2π)dφ(f, t)/dt. Time windows comprising strong energy arrivals to be measured can be identified by those IFs close to the frequencies with the maximum modulus and varying smoothly and monotonically with time. The respective IFs in each selected time window are further interpolated to yield a smooth branch of ridge points or representative IFs at which the arrival time, tridge(f), and phase, φridge(f), after unwrapping and correcting cycle skipping based on a priori knowledge of the possible velocity range, are determined for group and phase velocity estimation. We will demonstrate our measurement method using both ambient noise cross correlation functions and multi-mode surface waves from earthquakes. The obtained dispersion curves will be compared with those by a conventional narrow bandpass method.