Nonlinear Time Series Analysis of Volcanic Tremor Events Recorded at Sangay Volcano, Ecuador

Abstract

The assumption that volcanic tremor may be generated by deterministic nonlinear source processes is now supported by a number of studies at different volcanoes worldwide that clearly demonstrate the low-dimensional nature of the phenomenon. We applied methods based on the theory of nonlinear dynamics to volcanic tremor events recorded at Sangay volcano, Ecuador in order to obtain more information regarding the physics of their source mechanism. The data were acquired during 21–26 April 1998 and were recorded using a sampling interval of 125 samples s−1 by two broadband seismometers installed near the active vent of the volcano. In a previous study Johnson and Lees (2000) classified the signals into three groups: (1) short duration (<1 min) impulses generated by degassing explosions at the vent; (2) extended degassing ‘chugging’ events with a duration 2–5 min containing well-defined integer overtones (1–5 Hz) and variable higher frequency content; (3) extended degassing events that contain significant energy above 5 Hz. We selected 12 events from groups 2 and 3 for our analysis that had a duration of at least 90 s and high signal-to-noise ratios. The phase space, which describes the evolution of the behavior of a nonlinear system, was reconstructed using the delay embedding theorem suggested by Takens. The delay time used for the reconstruction was chosen after examining the first zero crossing of the autocorrelation function and the first minimum of the Average Mutual Information (AMI) of the data. In most cases it was found that both methods yielded a delay time of 14–18 samples (0.112–0.144 s) for group 2 and 5 samples (0.04 s) for group 3 events. The sufficient embedding dimension was estimated using the false nearest neighbors method which had a value of 4 for events in group 2 and was in the range 5–7 for events in group 3. Based on these embedding parameters it was possible to calculate the correlation dimension of the resulting attractor, as well as the average divergence rate of nearby orbits given by the largest Lyapunov exponent. Events in group 2 exhibited lower values of both the correlation dimension (1.8–2.6) and largest Lyapunov exponent (0.013–0.022) in comparison with the events in group 3 where the values of these quantities were in the range 2.4–3.5 and 0.029–0.043, respectively. Theoretically, a nonlinear oscillation described by the equation +β+γg(x)=fcosωt can generate deterministic signals with characteristics similar to those observed in groups 2 and 3 as the values of the parameters β,γ,f,ω are drifting, causing instability of orbits in the phase space.