Time-varying autoregressive model for spectral analysis of microseismic experiments and long-period volcanic events

S U M M A R Y Recent studies show that the frequency content of continuous passive recordings contains useful information for the study of hydraulic fracturing experiments as well as longstanding applications in volcano and global seismology. The short-time Fourier transform (STFT) is usually used to obtain the time–frequency representation of a seismic trace. Yet, this transform has two main disadvantages, namely its fixed time–frequency resolution and spectral leakage. Here, we describe two methods based on autoregressive (AR) models: the short-time autoregressive method (ST-AR) and the Kalman smoother (KS). These two methods allow for the AR coefficients to vary over time in order to follow time-varying frequency contents. The outcome of AR methods depends mainly on the number of AR coefficients. We use a robust approach to estimate the optimum order of the AR methods that best matches the spectral comparison between Fourier and AR spectra. Comparing the outcomes of the three methods on a synthetic signal, a long-period volcanic event, and microseismic data, we show that the STFT and both AR methods are able to track fast changes in frequency content. The STFT provides reasonable results even for noisy data using a simple and effective algorithm. The coefficients of the AR filter are defined at all time in the case of the KS. However, its better time resolution is slightly offset by a lower frequency resolution and its computational complexity. The ST-AR has a high spectral resolution and the lowest sensitivity to background noises, facilitating the identification of the various frequency components. The estimated AR coefficients can also be used to extract parts of the signal. The study of long-term phenomena, such as resonance frequencies, or transient events, such as long-period events, could help to gain further insight on reservoir deformation during hydraulic fracturing experiments as well as global or volcano seismological signals.