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 À1 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 € x x þ b_ x x þ cgðxÞ ¼ f cos xt can generate deterministic signals with characteristics similar to those observed in groups 2 and 3 as the values of the parameters b; c; f ; x are drifting, causing instability of orbits in the phase space.

Maximal Lyapunov exponent variations of volcanic tremor recorded during explosive and effusive activity at Mt Semeru volcano, Indonesia

“…4. Both methods give consistently the same values of the delay time (between 7 and 11) for all tremor events, which is in accordance with similar studies performed previously (e.g., Konstantinou and Lin, 2004). In order to cross-check this result we also constructed two-dimensional phase portraits for different delay times with the aim of examining the resulting orbit geometry.…”

“…Observational evidence for the existence of such a nonlinear process can be found in the systematic relationship between the frequency and amplitude of tremor signals as documented in Karimsky and Sangay volcanoes (Lees et al, 2004;Lees and Ruiz, 2008). On the other hand, several studies have applied nonlinear time series analysis methods to tremor recorded at volcanoes worldwide in order to detect the presence of deterministic structure (Chouet and Shaw, 1991;Godano et al, 1996;Carniel, 1996;Carniel and Di Cecca, 1999;Konstantinou, 2002;Konstantinou and Lin, 2004;De Lauro et al, 2008;Cannata et al, 2010). Their results revealed the existence of a strange attractor that describes the tremor source process in phase space and is characterized by a relatively small number of degrees of freedom.…”

Model for high‐frequency Strombolian tremor inferred by wavefield decomposition and reconstruction of asymptotic dynamics

“…In a heuristic way, we construct the phase space for each recording and, fixing the trajectory embedding dimension d, we investigate the scaling properties of the distances between the different trajectories by introducing a metric. The application of these methodologies to volcanic signals has been carefully discussed in other papers [e.g., Konstantinuou , 2002; Konstantinuou and Schlindenwein , 2002; Konstantinuou and Lin , 2004]. …”

Self-sustained oscillations at Volcán de Colima (México) inferred by Independent Component Analysis

“…This fraction becomes zero for dimensions not greater than 4 for the great majority of the LP events. Thus, 4 can be considered as the upper limit for the dimension of the attractor, indicating a low-dimension dynamics underlying the LP events, as already found for other volcanic signals (Chouet and Shaw 1991;Carniel 1996;Capuano and Godano 1999;De Martino et al 2002b;Konstantinou 2002;Konstantinou and Lin 2004;De Lauro et al 2008). The same analysis has been repeated for the ICs.…”