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Parameterization of multichromatic tornillo signals observed at Galeras Volcano (Colombia)
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Journal of Volcanology and Geothermal Research 125 (2003) 171^189 www.elsevier.com/locate/jvolgeores
Parameterization of multichromatic tornillo signals observed at Galeras Volcano (Colombia) Dieter Seidl a; , Margaret Hellweg b a
Bundesanstalt fu«r Geowissenschaften und Rohsto¡e, Seismologisches Zentralobservatorium Gra«fenberg, Mozartstr. 57, D-91052 Erlangen, Germany b Berkeley Seismological Laboratory, University of California-Berkeley, Berkeley, CA, 94520 USA Received 6 May 2002; received in revised form 10 September 2002; accepted 3 March 2003
Abstract In the past decade several of the ash eruptions at Galeras Volcano, Colombia, have been preceded by tornillos. These unusual tremor wavelets have quasi-sinusoidal waveforms with screw-like envelope profiles and can last up to several minutes. A swarm of tornillos occurred at Galeras Volcano between 8 December 1999 and 12 February 2000. These tornillos appear to be more complex than those previously recorded with the broadband instruments or with the short-period network of the Observatorio Vulcanolo¤gico y Sismolo¤gico in Pasto. They are multichromatic with a varying number of narrow spectral peaks between 1 and 20 Hz. We describe a procedure for parameterizing the tornillo signals in the time and frequency domains to determine signal parameters. In addition to wavelets like the tornillos, the procedure can be applied to random signals such as volcanic tremor. We derive distribution and correlation functions for the signal parameters determined from the swarm. These provide, along with the signal signature, constraints for modelling variations of the source process. From these observations we derive qualitative conclusions about the characteristics of differential equations which describe the underlying processes and excitation mechanisms as forced or self-excited oscillators. 7 2003 Elsevier Science B.V. All rights reserved. Keywords: Galeras Volcano; multichromatic tornillo; parameterization; linear source; cavity resonator; nonlinear source; Van der Pol equation
1. Introduction At Galeras Volcano, Colombia, several of the ash eruptions in the past decade have been preceded by tornillos, unusual tremor wavelets with sinusoidal waveforms and screw-like envelope
* Corresponding author. Tel.: +49-9131-8104011; Fax: +49-9131-8104099. E-mail address: [email protected] (D. Seidl).
pro¢les that can last for several minutes. Using recordings from the vertical seismometers of the short-period network of the Observatorio Vulcanolo¤gico y Sismolo¤gico de Pasto (OVP), Torres et al. (1996), Narva¤ez et al. (1997) and Go¤mez and Torres (1997) derived some basic signal parameters which can be used to describe these wavelets. Since 1996, a joint project between the Bundesanstalt fu«r Geowissenschaften und Rohsto¡e (BGR, Germany) and the Instituto de Investigacio¤n e Informacio¤n Geocient|¤¢ca, Minero-Ambi-
0377-0273 / 03 / $ ^ see front matter 7 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0377-0273(03)00095-7
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Frequency (Hz)
Frequency (Hz)
6
E
5 4 3 2
1 0
E
5 4 3 2
1 0
1000
Time (s)
2000
Spectral Amplitude
1000
Time (s)
2000
Spectral Amplitude
Fig. 1. Broadband multichromatic tornillo. (Top) Vertical (Z), north (N) and east (E) component seismograms of the tornillo occurring at 21.50 UTC on 12 February 2000 as recorded at the crater rim stations ANG and ACH, respectively to the east-northeast and north of the active cone. (Bottom) Spectrograms and spectra calculated from the recordings of the E components of the two stations. The peaks which can be seen in the spectra for both stations are numbered in the spectrograms. Arrows mark spectral energy present before the tornillo which corresponds to one or more of the tornillo’s peaks. In the spectra, black traces are taken during the tornillo and gray are noise samples taken from before the tornillo.
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ental y Nuclear (INGEOMINAS, Colombia) has supplemented the short-period network of the OVP with a multiparameter monitoring station in the crater region (Seidl et al., 2003), in the hope of learning more about the physical and chemical processes which generate the various tremor waveforms, and their signi¢cance in the sequence leading to ash eruptions. In addition to four broadband, three-component seismometer stations, three around the crater rim and one reference station located 5 km from the summit, the multiparameter station has equipment for continuous fumarole gas chemistry measurements, electromagnetic sensing, an infrasound sensor and weather observations (Seidl et al., 2003). The ¢rst three-component, broadband station was installed on the crater rim of Galeras in March 1996. Applying main axis transformation and using the Hilbert transform, the analysis of the broadband seismograms from a single station (Seidl et al., 1999; Go¤mez et al., 1999) revealed new insights into the kinematic and spectral behavior of tornillos. Most remarkable are the highly linear polarization of the main axis in a nearly horizontal plane, the constancy of the instantaneous frequency to better than 0.0002 Hz and the very slow exponential decay of the Hilbert envelope. A swarm of tornillos occurred at Galeras Volcano between 8 December 1999 and 12 February 2000. The tornillos were recorded well at the crater rim broadband station Anganoy (ANG) and, after 5 January 2000, also at Achalay (ACH) (¢gs. 1 and 2 of Seidl et al., 2003). These tornillos appear to be more complex than the tornillos previously recorded on the broadband instruments or those recorded with the short-period network. They are multichromatic having narrow spectral peaks at up to 10, or even more, not necessarily harmonic frequencies. This swarm of tornillos offers an opportunity for statistical parameter analysis.
2. Mode decomposition and parameterization Fig. 1 shows the three-component, broadband seismograms of a multichromatic tornillo re-
173
corded at the crater stations ANG and ACH along with spectrograms and spectra for the east component at each station. In the seismograms of the east component at ANG and the north and east components at ACH, the typical waveform of tornillos is apparent as a long-lasting, ringing coda. For both stations, the spectrograms of the east components reveal a fundamental mode in the band between 1.5 and 2.0 Hz and six spectral lines of overtones in narrow bands between 3 and 20 Hz. Spectrograms of the other components are similar, but the signal-to-noise ratio is not as good. The spectra show the spectral amplitude (black) of the tornillo as well as the spectral amplitude of noise in an interval before the beginning of the tornillo (gray). Before the tornillo onset, at least two continuous tremor bands, marked by arrows, can be seen in the spectrogram at ACH. The frequency and polarization of the low-frequency band, also visible at ANG, are similar to those of the fundamental band, while for the high-frequency band they are similar to the frequency and polarization of the third overtone at about 12 Hz. The pattern of spectral bands preceding the tornillo’s onset becomes more evident in Fig. 2, where the seismograms and highresolution spectrograms have been ¢ltered with a passband between 1 and 40 Hz. The bands of energy before and after the tornillo may be interpreted as continuous tremor. The parameterization of a three-component broadband tornillo wavelet consists of three steps. For each spectral peak in the tornillo which is present at both stations, we (1) bandpass ¢lter the three-component broadband seismograms to extract the signal; (2) determine the main axis coordinate system X1 -X2 -X3 using the eigenvectors and eigenvalues of the covariance matrix for the Z-N-E seismograms and then transform the Z-N-E seismograms into the X1 -X2 -X3 coordinate system (Matsumura, 1981; Kanasewich, 1981; Seidl and Hellweg, 1991; Hellweg, 2000a); and (3) determine signal parameters in the time and frequency domains from the X1 -X2 -X3 seismograms. Figs. 3 and 4 show the X1 -X2 -X3 seismograms at ANG (Fig. 3) and ACH (Fig. 4) for each of the
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Fig. 2. Bandpass ¢ltered multichromatic tornillo. (Top) The same tornillo as in Fig. 1, ¢ltered between 0.5 and 40 Hz. (Bottom) Spectrograms of the E component between 0 and 40 Hz. Arrows mark spectral energy present before the tornillo which corresponds to one or more of the tornillo’s peaks. Note that there are spectral peaks even at frequencies higher than 20 Hz.
seven spectral lines in the spectrograms of Fig. 1. Some of the overtones persist throughout the entire seismogram of the multichromatic tornillo while others only contribute to the turn-on transient at the tornillo’s beginning. The exponential decay of the coda for all frequencies is clear in the seismograms of the X1 components. Figs. 3 and 4 also show that while many of the lines can be rotated so that most of the energy is on the X1 component indicating linear polarization, some of the spectral lines with high frequencies are not linearly polarized. The second overtone, n = 2 with fP2 = 10.19 Hz, is a good example of this. We characterize each of the n peaks or modes
of each tornillo using nine parameters measured from the X1 -X2 -X3 seismograms. (1) The kinematic parameters, azimuth Azn , inclination Inn and rectilinearity Ren , are determined from the eigenvalues and eigenvectors. (2) The frequency of the peak’s maximum fPn and its amplitude APn are measured in the frequency domain. (3) We measure maximum velocity amplitude Vn , the rise time tRn , damping factor Qn and the signal energy ERn in the time domain. The relative signal energy ERn is estimated as the sum of the squared velocity amplitudes of the three components, while the damping factor Qn is
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2 0 -2 1 0 -1
Velocity(µm/s)
0.2 0 -0.2 0.5 0 -0.5 0.5 0 -0.5 0.1 0 -0.1
0
0.2 0 -0.2 0
50
100
0
50
100
0
50
100
Time (s) Fig. 3. Mode decomposition of multichromatic tornillos. Tornillo shown in Fig. 1, recorded at station ANG. In each row, the recorded seismograms have been ¢ltered to extract each of the spectral peaks present in Fig. 1, then transformed to the eigenvector coordinate system, X1 -X2 -X3 , using the covariance matrix for the polarization ellipsoid for that band. The peak frequencies fn are determined from the amplitude spectrum while the factor Qn is measured from the decay of the X1 component. The passbands of the ¢lter used to extract the mode are given in the X3 window.
determined from the slope of the logarithm of the Hilbert envelope (Seidl et al., 1999). Fig. 5 shows an example of this process for the second overtone of the tornillo shown in Fig. 1. Its peak frequency is 10.19 Hz at both ANG and ACH. From Figs. 3 and 4 it is clear that each individual mode can be described as an exponentially damped wavelet of the form an exp(ig n t) with complex angular frequency g n = 2Zfn +iqn and complex amplitude an . Thus, we can apply the method of parameterization shown in Fig. 5 to all the spectral peaks. In this example, the values
of fP2 measured at stations ANG and ACH di¡er by less than the frequency resolution, while di¡erences in the Q2 values at the two stations are well within the measurement error for the method, which we estimate to be about 10%. The distribution and correlation functions of the nine parameters re£ect the range of variation in geometric and physical properties of the tornillo source. The kinematic parameters, azimuth Azn , inclination angle Inn and rectilinearity Ren , and the excitation parameters, signal energy ERn , maximum velocity Vn , are strongly in£uenced by
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Fig. 4. Mode decomposition of multichromatic tornillos. The same tornillo as Fig. 3, but the decomposition is performed on the recordings from station ACH.
scattering and dissipation along the propagation path (see, for example, Hellweg, 2003). These parameters therefore only provide information about relative variation in the source location, the radiation pattern and the excitation energy. In contrast, the observed tornillo frequencies fPn and damping coe⁄cients qn can be related directly to the source process if we assume that the tornillo is the impulse response of a linear, second-order resonator. For the fundamental mode, the basic formulae relating measured values of the parameters to their values in the source are summarized in Fig. 6. These equations are also applicable to the overtones. In this example, we take the true angular frequency of the fundamental at the source to be g0 = 2Zf0 with f0 = 2.0 Hz and the
corresponding damping parameter to be Q0 = 100. We can determine the source parameters g0 = 2Zf0 and Q0 either in the time domain by measuring the frequency fP0 and the damping coe⁄cient q0 , or in the frequency domain by measuring the peak frequency fP0 and the 3-db bandwidth BP0 of the amplitude spectrum. For the weakly damped tornillos at Galeras, we determine them by measuring q0 in the time domain and fP0 in the frequency domain, as shown in Fig. 5 for the second overtone. The observed rise time tR of the tornillo, measured after ¢ltering with a bandpass of 3-db bandwidth BF , can be estimated as tR = 1/BF . For the bandpass ¢lters applied in the analysis of the the tornillo swarm, the observed rise times tR are in the range 1 s 6 tR 6 2 s. The
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VOLGEO 2612 19-6-03 Fig. 5. Parameterization of tornillo signals. Example of the parameterization procedure for the tornillo shown in Fig. 1, the left group for station ANG and the right group for station ACH. The data are bandpass ¢ltered to extract one of the spectral peaks in Fig. 1, in this case for the peak numbered 2, the low and high ¢lter corners are at 9.5 and 10.5 Hz, respectively. In each group, individual plots and signal parameters taken clockwise from the upper left show: the direction of the major axis X1 of the polarization ellipsoid (kinematic parameters: azimuth Azn measured clockwise from N, inclination Inn measured from the vertical, rectilinearity Ren = 13MV2 /V1 M with the largest and middle eigenvalues, V1 and V2 , respectively); X1 component velocity seismogram (parameters: maximum amplitude Vn , relative energy ERn , taken as the integral of the velocity amplitude squared, rise time tRn from onset to maximum); amplitude envelope of X1 component (parameter: quality factor Qn for the exponential decay segment); amplitude spectrum of X1 component (spectral parameters: peak frequency fPn , peak amplitude APn ).
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Fig. 6. Determination of source parameters from signal parameters. (Top row) Synthetic tornillo, showing the relationship of the observed time-domain signal parameters (right), damping factor Q0 and frequency f0 , and the frequency-domain signal parameters (left), 3-db bandwidth B0 and peak frequency fP , to the source parameters g0 , q0 and Q0 assuming that the tornillo is the impulse response of a linear second-order resonator. (Lower left) Relationship between the rise time tr of the recorded broadband signal and the transient rise time tR of the bandpass ¢ltered signal (tR Etr for all modes of the multichromatic tornillos investigated). (Lower right) Determination of parameters for the synthetic tornillo. For the tornillos observed at Galeras, the source parameters f0 and Q0 can be determined with the best accuracy and resolution by measuring the damping coe⁄cient q0 in time domain (Fig. 3) and peak frequency fP in frequency domain.
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20.00
Frequency (ACH)
15.00
10.00
5.00 1.5-2.0 2.0-5.0 5.0-11 11.0-15.0 15.0-20
0.00 0.00
5.00
10.00
15.00
20.00
Frequency (ANG) Fig. 7. Scatterplot comparing the frequency for each peak observed in the tornillo swarm as measured at stations ANG (horizontal axis) and ACH (vertical axis). Frequencies are grouped by symbol. The cluster of diamonds for the fundamental frequency in the band from 1.7 to 1.9 Hz represents measurments from 14 tornillos. The high correlation implies that the frequency is a characteristic of the source rather than the propagation path.
true rise time tr of the tornillo source signal can be estimated using the formula given in Fig. 6. For the tornillos at Galeras tr ItR . For example, we calculate that the true rise time tr for the second overtone in Fig. 5 is about 0.02 s. That the true rise time is so small is evidence supporting the assumption that tornillos are the response of some oscillator to an impulsive excitation.
3. Parameter statistics For the tornillo shown in Fig. 1, the peak frequencies for all seven modes are the same to within the frequency resolution of the spectra, 0.01 Hz at stations ANG and ACH (see Figs. 3 and 4). Fig. 7 shows a scatter plot of fPn for the 15 events of the tornillo swarm which were recorded at both
Table 1 Mean values and standard deviations of fPn and Qn ANG
ACH
Number
W(fP0 )
c(fP0 )
W(Qn )
c(Qn )
W(fP0 )
c(fP0 )
W(Qn )
c(Qn )
0 1 2 3 4 5 6a
1.82 3.70 10.03 11.91 14.49 15.65 19.46
0.06 0.06 0.14 0.10 0.18 0.13 ^
99 47 569 369 174 263 207
41 14 152 87 88 81 ^
1.83 3.70 10.06 11.91 14.52 15.66 19.46
0.06 0.10 0.15 0.10 0.35 0.13 ^
100 59 546 366 161 221 250
46 32 210 119 43 115 ^
a
Model 6 has only one sample.
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Q (ACH)
1000
100
1.5-2.0 2.0-5.0 5.0-11 11.0-15.0 15.0-20
10 10
100
1000
Q (ANG) Fig. 8. Scatterplot comparing the Q value measured for each peak observed in the tornillo swarm as measured at stations ANG (horizontal axis) and ACH (vertical axis). The same frequency symbols are used as in Fig. 7. The high correlation between the values determined at the two stations implies that Q characterizes the source rather than the propagation path.
stations. The fundamental mode is observed in all tornillos. Higher modes are also observed for all tornillos but with decreasing probability of excitation for increasing frequency. Fig. 7 demonstrates that there is a signi¢cant correlation between the values for fPn at the two stations in all frequency bands. For individual tornillos, the frequencies fP0 (ANG) and fP0 (ACH) are the same within the frequency resolution of the spectrum, Nf = 0.01 Hz. At both stations, the mean values of fP0 for all 15 tornillos are W(fP0 ) = 1.82 Hz with standard deviations of c(fP0 ) = 0.06 Hz. Since c(fP0 ) = 6 Nf, the £uctuation of the fundamental frequency fP0 , within the band (1.82 T 0.06) Hz from one tornillo to another is signi¢cant and must be incorporated into a model for the source. The mean values and standard deviations of fPn for the spectral peaks are summarized in Table 1. In each band the exponential decay measured from the envelope of X1 is also the same at both stations but the scatter is higher for Qn than for the frequency, due to the larger error in the mea-
surement of this parameter. The determination of Qn depends strongly on the time window selected for the least-squares ¢t of the slope to the logarithm of the Hilbert envelope. We estimate that the measurement error NQn is about 10%. Values of Qn for the tornillo shown in Fig. 1 are given in Figs. 3 and 4. In the scatterplot of Fig. 8 the correlation between the values of Qn measured at the two stations is clear. For the swarm of tornillos, the mean values of the damping factor for the fundamental Q0 at the two stations are W(Q0 ) = 99 at ANG and 100 at ACH, with standard deviations of c(Q0 ) of 41 and 46, respectively. As with the frequencies, c(Qn ) s NQ and the £uctuation of Qn must be related to changes at the source and cannot be caused by noise or errors in the analysis method. Table 1 also gives the mean values and standard deviations of Qn . From these observations of the frequencies of the tornillos’ peaks and their respective damping factors, we can draw several interesting conclusions. First, as the two parameters are the same
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VOLGEO 2612 19-6-03 Fig. 9. Basic characteristics of peak frequencies and polarization. (Left) The histograms of tornillo fundamental mode and overtone frequencies observed with the short-period network of OVP from 1992 to 1996 (top) as well as with the broadband station ANG from 1997 to 2000 (middle and bottom) reveal the persistence of various frequencies over the course of time and changes in the volcano’s states of activity. They imply that there is some relatively constant factor in the source process. (Right) Polarization of a tornillo recorded at station ACH. The polarization of tornillos recorded at the crater stations ANG and ACH is mostly linear in a nearly horizontal plane with small variations of azimuth Az, inclination In and rectilinearity Re in each frequency band.
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at the two stations for any given tornillo, we consider the peak frequencies fPn and the damping factors Qn to be source parameters and not characterizations of the propagation path. Second, their £uctuations from one tornillo to the next must be related to some small changes in the physical parameters at the source. Third, with many frequencies present in the 15 multichromatic tornillos recorded at the two crater rim stations, we can divide the frequency band between 0 Hz and 16 Hz into two bands and determine the distribution of frequencies in these two bands. Are they more or less evenly distributed ? Between 0 Hz and 8 Hz the tornillos from this swarm have peaks grouped around two mean frequencies, while between 8 Hz and 16 Hz there are four.
4. Constraints and implications for tornillo source modelling In investigations of individual tornillos and tremor, proposed sources have included such models as £uid-¢lled cracks (Chouet, 1996; GilCruz and Chouet, 1997; Kumagai and Chouet, 1999; Hagerty and Benites, 2003), lumped-parameter models (Julian, 1994) or self-excited eddy shedding and turbulent slug £ow oscillations (Hellweg, 2000b). None of these models explains all available observations, especially when quantitative aspects of the models are compared with amplitude measurements from the tornillos or tremor. Often, the models are based on the interpretation of data only from vertical component recordings. For example, the coupling of seismic energy between the £uids in the conduit system and the solid edi¢ce of the volcano and the propagating medium presents particular problems for such models. Although tremor is often considered to be a direct seismic window into the conduit system, the various tremor signals actually only illuminate those small elementary volumes within the complete conduit system containing fast, seismically active turbulent or oscillating £uid movement. Geometrically, such elementary tremor sources are likely to be found at some type of constriction in a branched conduit system such as a constriction between two tubes, a constriction
leading to a branch or a constriction at the opening of an otherwise closed volume which acts as a cavity resonator (Lighthill, 1996). Thus, tremor sources must be considered to be sensors monitoring the amplitude of £uid pressure variations rather than data for an accurate mapping of the total geometry and physics of the £uid conduit system. What role do the tornillos play in the inversion of the puzzle of tremor sources, especially the multichromatic tornillos observed recently at Galeras Volcano? We review the parameters characterizing the tornillos to determine some constraints for source models. 4.1. Frequencies Tornillos have been observed at Galeras Volcano since 1992. From 1992 to 1996, the activity at Galeras was high. Six explosion eruptions occurred between July 1992 and June 1993 (Narva¤ez et al., 1997). During this time, tornillos were recorded with the network of the OVP using shortperiod, vertical component seismometers. Since 1996, the tornillos have also been recorded by one or more broadband, three-component seismometers at the crater rim. The histograms in Fig. 9 summarize the peak frequencies of the fundamental modes and overtones observed in tornillos from 1992 to 2000. The fundamental mode and the ¢rst overtone are two separate, stable bands between 1 and 4 Hz. Their mean values and standard deviations are summarized in Table 2. The persistence of the bands is an indication that there is some relatively constant factor in the process generating the tornillos.
Table 2 Mean values and standard deviations of the fundamental and ¢rst overtone since 1992
OVP ANG Swarm
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Interval
f0 (Hz)
f1 (Hz)
1992^1996 1997^1998 1999^2000
1.40 T 0.10 1.90 T 0.10 1.82 T 0.06
3.10 T 0.60 2.80 T 0.70 3.70 T 0.06
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Fig. 10. Helmholtz cavity resonator. (Top left) Patterns of tremor signals observed at Galeras Volcano which can be explained as responses of a cavity source described by a linear di¡erential equation to excitation by several di¡erent types of impulsive or turbulent variations in pressure. (Top right) Helmholtz resonator, a £uid-¢lled cavity with a small opening which can be described approximately by a linear, second-order di¡erential equation with coe⁄cients related to the cavity’s geometrical parameters (volume V0 , Rayleigh conductivity, U = 2R for a circular opening and U = F/LP for a cylindrical neck with e¡ective length LP and area F) and physical parameters (velocity of sound in the £uid, c). (Bottom left) Relationships between the observed tornillo parameters, f0 and Q0 , and cavity parameters for the fundamental mode, under the assumption of acoustic emission. (Bottom right) Estimation of characteristic cavity dimensions for a Helmholtz resonator ¢lled with a gas of 80% H2 O+20% CO2 at a temperature of 723 K and for typical observed parameters f0 = 1.82 Hz and Q0 = 100.
4.2. Polarization The orientation and rectilinearity of the polarization ellipsoids change apparently randomly from one frequency band to another (see ¢g. 12 of Hellweg, 2003). This may be the result of a volume source with a complex shape and a nonuniform wall response. It is exacerbated by the fact that for low frequencies, the stations are in
the near-¢eld of the source. For high frequencies, scattering in the complex medium that is the volcanic edi¢ce is also important (Hellweg, 2003). Nevertheless, at least two qualitative conclusions may be drawn from the observed polarization parameters : (1) The mostly horizontal polarization (example in Fig. 9; Hellweg, 2003) for the fundamental modes indicates that the source is near the surface
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Fig. 11. Observed and synthetic tornillos as an onset transient for tremor oscillations. (Top left) Tornillo followed by a sequence of harmonic tremor, recorded at station ANG on 3 March 2000 at 21.20 UTC. The tornillo and the tremor have the same peak frequency f0 = 2.0 Hz, the tremor lasted about 1.5 h. (Top right) Details of tremor before (precursor) and after (coda) the tornillo, respectively, showing the increase in the amplitude of the tremor after the tornillo. The precursor has the same peak frequency f0 = 2.0 and polarization as the tornillo and the tremor. (Bottom left) Synthetic tornillo with tremor coda generated by numerical integration of the di¡erential equation for a self-excited, nonlinear Van der Pol oscillator for the parameters W = 0.25 and g = 2Zf with f = 1.5 Hz (Fig. 12). (Bottom right) Details of the synthetic harmonic tremor before (zero) and after the synthetic tornillo.
within the active crater, and that it radiates longitudinal waves. (2) The nearly linear polarization and the £uctuations of the polarization parameters suggest the radiation comes from small elementary sources within a larger volume. The polarization depends on the complex three-dimensional eigenfunction of the source as well as the location
of the excitation pulse within the larger volume. 4.3. Tornillo source as a linear oscillator and the Helmholtz cavity resonator The multichromatic tornillos observed in the tornillo swarm from late 1999 and early 2000
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Fig. 12. Synthetic tremor generated by numerical integration of the di¡erential equation for a self-excited, nonlinear Van der Pol oscillator with g = 2Zf and f = 1.5 Hz, for di¡erent damping parameters W. (Top) Tornillo with tremor coda for two damping coe⁄cients W = 0.25 and W = 0.5, respectively. (Bottom left) Harmonic tremor as the result of the random £uctuations of the damping coe⁄cient around a mean value of W = 0.05 (parametric oscillator). (Bottom right) Cyclic tremor for a high damping coe⁄cient W = 20.
can be explained, at least qualitatively, as free vibrations of a £uid-¢lled cavity, excited by a pressure pulse either at the wall or within the cavity. The impulse response of a cavity of volume V, ¢lled with a £uid having sound velocity c,
is characterized by two basic theoretical results (Morse and Ingard, 1968): b The resulting pressure wave ¢eld is a superposition of the fundamental and higher-mode eigenvibrations pn = an (x)exp(ig n t). an (x) are the
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eigenfunctions (position vector x) and g n = 2Zfn +iqn are the complex eigenfrequencies of the cavity. The frequencies fn and the damping coef¢cients qn are related to the damping factor Qn and the bandwidth Bn of the mode’s amplitude spectrum by the formula : Qn = Zfn /qn = fn /Bn . Such behavior is observed in the example of mode decomposition given for the multichromatic tornillos in Figs. 3 and 4. b The number vN of eigenfrequencies fn in an interval vf increases with frequency and can be approximated for a cavity with a small opening by the formula vN = 4 V/(c3 f2 vf). Although the number of tornillos of the swarm which occurred during January and February 2000 may be too small for signi¢cant statistics, the values listed in Table 1 are in good agreement with this formula. We would like to mention an interesting implication of this formula. The mean separation Df between the frequencies fn is given by: Df = vf/ vN = c3 /(4 V f2 ). If Df 6 Bn = fn /Qn , the bands Bn of the modes will overlap and the discrete mode spectrum will converge to a quasi-continuum. This corresponds to a transition from successive tornillos or tremor packets to continuous tremor in the time domain or a transition from multichromatic peaked spectra to broadband spectra in the frequency domain. Such a transition can be achieved in several ways: by shifting the eigenfrequencies to lower values for ¢xed Qn , by decreasing the Qn (increasing the damping coe⁄cients qn ) for ¢xed fn , or by a combination of both e¡ects. One special case of a cavity resonator is the Helmholtz resonator. A wine bottle is a wellknown example of a Helmholtz resonator, a cavity with a small opening such as a circular hole or a cylindrical neck of length L (Fig. 10). When excited by an external source, the cavity responds like a precisely tuned system with a very small resonance impedance (Skudrzyk, 1971). Its impulse response is a sinusoidal signal with a sharp onset followed by exponentially decaying amplitude similar to the typical shape of a tornillo (Fig. 6). In his systematic, theoretical investigation of the Helmholtz resonator, Howe (1976) includes the transfer function of the resonator as well as excitation by a pressure pulse, a Karman vortex
street and a turbulent pressure ¢eld. For the fundamental mode, the transfer function can be approximated by a second-order, linear di¡erential equation with two parameters, a damping coef¢cient q0 and a natural angular frequency g0 = 2Zf0 with frequency f0 (Fig. 10). The frequency f0 of the fundamental mode depends mainly on the cavity volume V0 , the geometry and dimension of the opening, described by the Rayleigh conductivity U and the acoustic velocity c of the £uid. However, to ¢rst approximation f0 is independent of the shape of the cavity. This explains the stability of f0 during a single tornillo as well as its variation from one tornillo to the next over the course of hours or days. These observations are both related to changes in the acoustic velocity of the £uid in the cavity due to £uctuations of the pffiffiffiffitemperature, as given by the relationship cV T . The eigenfunctions and eigenfrequencies of the higher modes depend very strongly on the shape of the cavity and the local geometry of the cavity wall. The e¡ects are even more pronounced for the damping factors which are very sensitive to the local acoustic impedance of the cavity wall and the frequency-dependent thermodynamic and viscous coupling of the £uid pressure £uctuations to the seismic motion in the solid medium. These problems cannot be treated analytically but only numerically for particular realizations of a conduit system. For acoustic radiation, the damping coe⁄cient q0 is related to the parameters c, V0 and U (Howe, 1976). Using this assumption, the Rayleigh conductivity U and the volume V0 can be estimated from the observed values of f0 and Q0 (Fig. 10). The example given in Fig. 10 assumes the cavity is ¢lled with a gas consisting of 80% H2 O and 20% CO2 at a temperature of 723 K. This gives a typical linear dimension of D = 15^20 m for a cavity which generates a tornillo with a frequency f0 = 1.82 Hz and a damping factor Q0 = 100. Although no cavity of such a size is known to exist at Galeras, the size is not unreasonable. 4.4. Tornillo source as a nonlinear oscillator and the Van der Pol equation For some strong tornillos at Galeras Volcano,
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the amplitude level of continuous tremor increases immediately following the tornillo, as though it had an extended coda. One example of a recording of such a combined tornillo^tremor is shown in Fig. 11. In a case like this, we may consider the tornillo to be the initial transient of a self-excited, nonlinear oscillator activated by an impulsive input signal. After the transient the process continues, feeding on heat or pressure energy from the volcano. Such a transient signal has both of the characteristic properties of a tornillo, a sinusoidal waveform with constant frequency and exponential amplitude decay. A synthetic signal with similar characteristics to the tornillo^tremor combination can be generated by integrating a nonlinear di¡erential equation, such as the Van der Pol equation (given in Fig. 12), for an impulsive input (Kirbani, 1983; Berge¤ et al., 1984). The Van der Pol equation, the most simple second-order, nonlinear di¡erential equation is analogous to the linear equation for the cavity shown in Fig. 10. Rather than the linear damping term there, the Van der Pol equation has a damping term which changes as a function of x. When x is greater than some critical or threshold value, the system is damped and it radiates energy. The signal amplitude decreases and disappears. If x is less than the critical value, the system absorbs energy from a heat or pressure reservoir by some internal feedback mechanism and the signal amplitude increases or remains constant. Thus, the threshold parameter controls the transition of the damping term between negative and positive values. In the Van der Pol equation shown in Fig. 12, we have normalized the variable x so that it is equal to 1 when the threshold parameter changes between negative and positive. There are many phenomena in volcanoes which could provide the mechanism needed for such a feedback system. A pressureactivated system which functions like a valve (Hellweg, 2000b) or bubble mixtures almost certainly present in magma or hydrothermal £uid systems (Martinelli, 1991) are two likely possibilities. Fig. 12 gives several examples of waveforms that can be generated by changing the value of the damping parameter, W, in the Van der Pol equation. The top two graphs are clearly tornillos
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of various lengths with associated extended tremor codas. The bottom two graphs are similar to two other types of volcanic tremor, harmonic tremor (more or less sinusoidal, but varying amplitude) and cyclic tremor (typically has a spectrum with numerous peaks at harmonic frequencies). A model such as this one presents one major problem to interpretation. Unlike the case for the linear di¡erential equation describing the cavity resonator, it is di⁄cult to relate characteristics of the volcanic system, such as conduit geometry or the process providing the feedback mechanism, to the parameters determining the behavior of the nonlinear di¡erential equation, such as W. We are investigating this aspect of the model further to relate the parameters of the nonlinear di¡erential equation to physical reality.
5. Conclusions Recordings of a swarm of tornillos occurring in January and February 2000, at two three-component, broadband stations installed on the crater rim of Galeras Volcano, Colombia, a¡ord a new opportunity. With the insights from a detailed investigation of the characteristics of these tornillos at the two stations, we have developed new constraints for modelling their source. The tornillos of this swarm were multichromatic, having between two and 10 narrow spectral peaks between 1 and 20 Hz. Some tornillos at Galeras were followed by a coda consisting of tremor with amplitudes higher than those before the tornillo. We used mode decomposition to investigate the signal characteristics of the individual spectral peaks and parameterize them individually. We treated these tornillos of the swarm as members of an ensemble, and determined distribution and correlation functions of the parameters. They were then used as additional constraints for describing and classifying the basic source and excitation processes. Tremor is a collective term for a large variety of impulsive or continuous seismic signals with random, harmonic or cyclic signature generated in the £uid conduit system of the volcano. The variety of signal forms may re£ect a corresponding
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variety of possibilities either for the source process, such as forced or self-excited oscillating and resonating vibrations, or for the excitation process, such as shock waves, turbulent £ow or convective transport phenomena. Data from the myriad forms that tremor takes in the seismic record provide necessary, but far from su⁄cient information for constraining unique physical and morphological models of the £uid dynamics describing these source processes. Individual models for the source of tremor are still very much a topic of discussion, especially with regard to their quantitative aspects and their uniqueness. There seems to be general agreement, however, that nonlinear and two-phase phenomena in the £uid ¢lling the conduit system of the volcano play a fundamental role in the processes generating tremor (Martinelli, 1991; Julian, 2000; Ferrick et al., 1982). Nonlinearities can be caused by many e¡ects. These include phenomena such as £uid^£uid or £uid^solid coupling and feedbacktype boundary conditions in the conduit geometry. Other possible sources of nonlinearity are £ow transients and instabilities, transitions to turbulence or critical thresholds of material parameters such as the strong dependence of the sound velocity on the content of free gases in two-phase £uids (Martinelli, 1991). The similarity of polarization and characteristic frequencies for tremor and tornillo signals at Galeras Volcano suggests that they have a common source with di¡erent states of oscillation and excitation. The observations of multichromatic tornillos as well as of tornillos followed by an increased level of tremor suggest two possible models for the source: a linear model in which the tornillo is the free vibration response of a £uid-¢lled cavity to a pressure pulse; or a nonlinear model in which the tornillo is an initial transient leading into a tremor sequence generated by a nonlinear, self-excited oscillator. A Helmholtz resonator is a simple realization of the linear model (Fig. 10). It is supported by the waveforms and the distribution of frequencies observed in the multichromatic tornillos of the swarm. For such a model, the parameters of the di¡erential equation are related to the geometry of the resonator or conduit and the velocity of sound in the medium
¢lling the resonator. The simplest description of a nonlinear oscillator is the Van der Pol equation. We have used this equation to generate synthetic waveforms which, under some conditions, mimic the tornillos as initial transients to sequences of harmonic tremor (Fig. 12). It is, however, di⁄cult to relate the parameters of the nonlinear di¡erential equation to parameters describing physical characteristics of some elements of the volcanic system. However, Fig. 12 demonstrates that an oscillator which can be described by the Van der Pol equation can generate tornillos as initial transients, continuous tremor as parametric oscillator signals when the damping £uctuates and cyclic tremor, such as that observed at Lascar Volcano, Chile (Hellweg, 2000b). The tornillos are only one member of the wide range of impulsive, harmonic or random tremor signals observed at active volcanoes. While such signals are clearly di¡erent from one volcano to the next and even from one occurrence to the next, they nevertheless also exhibit some similarities on which to base classes of models. The variation among these di¡erent forms of signals must be related to di¡erences in the geometry of channels, conduits and cavities, as well as to the variety of £ow phenomena possible when the moving £uid is a mixture of liquids and gases. Many different classes of mechanisms are possible as sources of tremor in such branched conduit systems. They include free, forced, resonant, parametric and self-excited motion in systems with linear or nonlinear responses. Using tremor data with low seismic amplitudes from an interval at Galeras Volcano with relatively low activity, we have only been able to derive a few possibilities for basic models for the source of tornillos and tremor. The mechanisms which may be described by these models suggest that seismic signals only offer us a small window into the seismically active portion of the volcano’s conduit system. In order to improve our understanding of the physics of volcanoes, we hope to combine the tremor observations and analysis with other data, such as gaschemical or electromagnetic signals, in a multiparameter approach (Seidl et al., 2003; Faber et al., 2003).
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Acknowledgements The Multiparameter Station at Galeras Volcano is a cooperative project of the Federal Institute of Geosciences and Natural Resources (BGR) in Hannover (Germany) and the Instituto de Investigacio¤n e Informacio¤n Geocient|¤¢ca, MineroAmbiental y Nuclear (INGEOMINAS) in Bogota¤ (Colombia). We would like to express our gratitude to the Deutsches Zentrum fu«r Luft- und Raumfahrt (DLR) for providing ¢nancial assistance for travel and transport expenses. This work would not have been possible without the data collected through the hard work and system maintenance of our Colombian colleagues at the Observatorio Vulcanolo¤gico y Sismolo¤gico de Pasto. We would like to thank Guiseppe Lombardo and an anonymous reviewer for their comments and suggestions for improving this presentation of our work. References Berge¤, P., Pomeau, Y., Vidal, Ch., 1984. Order within Chaos: Towards a Deterministic Approach to Turbulence. John Wiley and Sons, New York, 329 pp. Chouet, B.A., 1996. New methods and future trends in seismological volcano monitoring. In: Scarpa, R., Tilling, R.I. (Eds.), Monitoring and Mitigation of Volcano Hazards. Springer-Verlag, Berlin, pp. 23^97. Faber, E., Moran, C., Garzon, G., Poggenburg, J., Teschner, M., 2003. Continuous gas monitoring at Galeras Volcano, Colombia: ¢rst evidence J. Volcanol. Geotherm. Res. 125 (this issue). doi: 10.1016/S0377-0273(03)00086-3 Ferrick, M.G., Qamar, A., Lawrence, W.F.St., 1982. Source mechanism of volcanic tremor. J. Geophys. Res. 87 (B10), 8675^8683. Gil-Cruz, F., Chouet, B., 1997. Long-period events, the most characteristic seismicity accompanying the emplacement and extrusion of a lava dome in Galeras Volcano, Colombia, in 1991. J. Volcanol. Geotherm. Res. 77, 121^158. Go¤mez, D.M., Torres, R.A., 1997. Unusual low-frequency volcanic seismic events with slowly decaying coda waves observed at Galeras and other volcanoes. J. Volcanol. Geotherm. Res. 77, 173^193. Go¤mez, D.M., Torres, R.A., Seidl, D., Hellweg, M., Rademacher, H., 1999. Tornillo seismic events at Galeras Volcano, Colombia: a summary and new information from broadband three-component measurements. Ann. Geo¢s. 42, 365^378. Hagerty, M., Benites, R., 2003. Tornillos beneath Tongariro Volcano, New Zealand, J. Volcanol. Geotherm. Res. 125 (this issue) doi: 10.1016/S0377-0273(03)00094-5.
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Hellweg, M., 2000a. Volcanic Tremor and Physical Source Models: Lascar Volcano, Chile [Doctoral Thesis]. Ber. Inst. Geophys. Univ. Stuttgart 14, 132 pp. Hellweg, M., 2000b. Physical models for the source of Lascar’s harmonic tremor. J. Volcanol. Geotherm. Res. 101, 183^ 198. Hellweg, M., 2003. The polarization of volcanic seismic signals: Medium or source. J. Volcanol. Geotherm. Res. in press. Howe, M.S., 1976. On the Helmholtz resonator. J. Sound Vib. 45, 427^440. Julian, B.R., 1994. Volcanic tremor: nonlinear excitation by £uid £ow. J. Geophys. Res. 99 (B6), 11859^11877. Julian, B.R., 2000. Period doubling and other nonlinear phenomena in volcanic earthquakes and tremor. J. Volcanol. Geotherm. Res. 101, 19^26. Kanasewich, E.R., 1981. Time Sequence Analysis in Geophysics. The University of Alberta Press, Edmonton, AB, 480 pp. Kirbani, S.B., 1983. Analysis and Interpretation of Volcanic Tremor at Etna. Institute of Geophysics, University of Stuttgart, Publication No. 210. Kumagai, H., Chouet, B., 1999. The complex frequencies of long-period seismic events as probes of £uid composition beneath volcanoes. Geophys. J. Int. 138, F7^F12. Lighthill, J., 1996. Waves in Fluids. Cambridge University Press, Cambridge, 503 pp. Martinelli, B., 1991. Fluidinduzierte Mechanisman fu«r die Entstehung von vulkanischen Tremor-Signalen. ETH Zu«rich, Dissertation No. 9376. Matsumura, S., 1981. Three-dimensional expression of seismic particle motion by the trajectory ellipsoid and its application to the seismic data observed in the Kanto district, Japan. J. Phys. Earth 29, 221^239. Morse, P.M., Ingard, K.U., 1968. Theoretical Acoustics. McGraw-Hill, New York, 927 pp. Narva¤ez, M.L., Torres, R.A., Go¤mez, D.M., Corte¤s, G.P.J., Cepeda, H.V., Stix, J., 1997. Tornillo-type seismic signals az Galeras Volcano, Colombia, 1992^1993. J. Volcanol. Geotherm. Res. 77, 159^171. Seidl, D., Hellweg, M., 1991. Volcanic tremor recordings: Polarization analysis. In: Schick, R., Mugiono, R. (Eds.), Volcanic Tremor and Magma Flow. Forschungszentrum Ju«lich, Ju«lich, pp. 31^46. Seidl, D., Hellweg, M., Rademacher, H., Go¤mez, D.M., Torres, R.A., 1999. The anatomy of a tornillo: puzzles from three-component measurements at Galeras Volcano (Colombia). Ann. Geo¢s. 42, 355^364. Seidl, D., Hellweg, M., Calvache, M., Gomez, D., Ortega, A., Torres, R., Bo«ker, F., Buttkus, B., Faber, E., Greinwald, S., 2003. The multiparameter station at Galeras Volcano, Colombia: Concept and realization J. Volcanol. Geotherm. Res. 125 (this issue). doi: 10.1016/S0377-0273(03)00075-1 Skudrzyk, E., 1971. The Foundations of Acoustics. SpringerVerlag, Vienna, 790 pp. Torres, R.A., Go¤mez, D.M., Narva¤ez, M.L., 1996. Unusual seismic signals associated with the activity at Galeras Volcano, Colombia, from July 1992 to September 1994. Ann. Geo¢s. 39, 299^310.
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Parameterization of multichromatic tornillo signals observed at Galeras Volcano (Colombia) Dieter Seidl a; , Margaret Hellweg b a
Bundesanstalt fu«r Geowissenschaften und Rohsto¡e, Seismologisches Zentralobservatorium Gra«fenberg, Mozartstr. 57, D-91052 Erlangen, Germany b Berkeley Seismological Laboratory, University of California-Berkeley, Berkeley, CA, 94520 USA Received 6 May 2002; received in revised form 10 September 2002; accepted 3 March 2003
Abstract In the past decade several of the ash eruptions at Galeras Volcano, Colombia, have been preceded by tornillos. These unusual tremor wavelets have quasi-sinusoidal waveforms with screw-like envelope profiles and can last up to several minutes. A swarm of tornillos occurred at Galeras Volcano between 8 December 1999 and 12 February 2000. These tornillos appear to be more complex than those previously recorded with the broadband instruments or with the short-period network of the Observatorio Vulcanolo¤gico y Sismolo¤gico in Pasto. They are multichromatic with a varying number of narrow spectral peaks between 1 and 20 Hz. We describe a procedure for parameterizing the tornillo signals in the time and frequency domains to determine signal parameters. In addition to wavelets like the tornillos, the procedure can be applied to random signals such as volcanic tremor. We derive distribution and correlation functions for the signal parameters determined from the swarm. These provide, along with the signal signature, constraints for modelling variations of the source process. From these observations we derive qualitative conclusions about the characteristics of differential equations which describe the underlying processes and excitation mechanisms as forced or self-excited oscillators. 7 2003 Elsevier Science B.V. All rights reserved. Keywords: Galeras Volcano; multichromatic tornillo; parameterization; linear source; cavity resonator; nonlinear source; Van der Pol equation
1. Introduction At Galeras Volcano, Colombia, several of the ash eruptions in the past decade have been preceded by tornillos, unusual tremor wavelets with sinusoidal waveforms and screw-like envelope
* Corresponding author. Tel.: +49-9131-8104011; Fax: +49-9131-8104099. E-mail address: [email protected] (D. Seidl).
pro¢les that can last for several minutes. Using recordings from the vertical seismometers of the short-period network of the Observatorio Vulcanolo¤gico y Sismolo¤gico de Pasto (OVP), Torres et al. (1996), Narva¤ez et al. (1997) and Go¤mez and Torres (1997) derived some basic signal parameters which can be used to describe these wavelets. Since 1996, a joint project between the Bundesanstalt fu«r Geowissenschaften und Rohsto¡e (BGR, Germany) and the Instituto de Investigacio¤n e Informacio¤n Geocient|¤¢ca, Minero-Ambi-
0377-0273 / 03 / $ ^ see front matter 7 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0377-0273(03)00095-7
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Frequency (Hz)
Frequency (Hz)
6
E
5 4 3 2
1 0
E
5 4 3 2
1 0
1000
Time (s)
2000
Spectral Amplitude
1000
Time (s)
2000
Spectral Amplitude
Fig. 1. Broadband multichromatic tornillo. (Top) Vertical (Z), north (N) and east (E) component seismograms of the tornillo occurring at 21.50 UTC on 12 February 2000 as recorded at the crater rim stations ANG and ACH, respectively to the east-northeast and north of the active cone. (Bottom) Spectrograms and spectra calculated from the recordings of the E components of the two stations. The peaks which can be seen in the spectra for both stations are numbered in the spectrograms. Arrows mark spectral energy present before the tornillo which corresponds to one or more of the tornillo’s peaks. In the spectra, black traces are taken during the tornillo and gray are noise samples taken from before the tornillo.
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ental y Nuclear (INGEOMINAS, Colombia) has supplemented the short-period network of the OVP with a multiparameter monitoring station in the crater region (Seidl et al., 2003), in the hope of learning more about the physical and chemical processes which generate the various tremor waveforms, and their signi¢cance in the sequence leading to ash eruptions. In addition to four broadband, three-component seismometer stations, three around the crater rim and one reference station located 5 km from the summit, the multiparameter station has equipment for continuous fumarole gas chemistry measurements, electromagnetic sensing, an infrasound sensor and weather observations (Seidl et al., 2003). The ¢rst three-component, broadband station was installed on the crater rim of Galeras in March 1996. Applying main axis transformation and using the Hilbert transform, the analysis of the broadband seismograms from a single station (Seidl et al., 1999; Go¤mez et al., 1999) revealed new insights into the kinematic and spectral behavior of tornillos. Most remarkable are the highly linear polarization of the main axis in a nearly horizontal plane, the constancy of the instantaneous frequency to better than 0.0002 Hz and the very slow exponential decay of the Hilbert envelope. A swarm of tornillos occurred at Galeras Volcano between 8 December 1999 and 12 February 2000. The tornillos were recorded well at the crater rim broadband station Anganoy (ANG) and, after 5 January 2000, also at Achalay (ACH) (¢gs. 1 and 2 of Seidl et al., 2003). These tornillos appear to be more complex than the tornillos previously recorded on the broadband instruments or those recorded with the short-period network. They are multichromatic having narrow spectral peaks at up to 10, or even more, not necessarily harmonic frequencies. This swarm of tornillos offers an opportunity for statistical parameter analysis.
2. Mode decomposition and parameterization Fig. 1 shows the three-component, broadband seismograms of a multichromatic tornillo re-
173
corded at the crater stations ANG and ACH along with spectrograms and spectra for the east component at each station. In the seismograms of the east component at ANG and the north and east components at ACH, the typical waveform of tornillos is apparent as a long-lasting, ringing coda. For both stations, the spectrograms of the east components reveal a fundamental mode in the band between 1.5 and 2.0 Hz and six spectral lines of overtones in narrow bands between 3 and 20 Hz. Spectrograms of the other components are similar, but the signal-to-noise ratio is not as good. The spectra show the spectral amplitude (black) of the tornillo as well as the spectral amplitude of noise in an interval before the beginning of the tornillo (gray). Before the tornillo onset, at least two continuous tremor bands, marked by arrows, can be seen in the spectrogram at ACH. The frequency and polarization of the low-frequency band, also visible at ANG, are similar to those of the fundamental band, while for the high-frequency band they are similar to the frequency and polarization of the third overtone at about 12 Hz. The pattern of spectral bands preceding the tornillo’s onset becomes more evident in Fig. 2, where the seismograms and highresolution spectrograms have been ¢ltered with a passband between 1 and 40 Hz. The bands of energy before and after the tornillo may be interpreted as continuous tremor. The parameterization of a three-component broadband tornillo wavelet consists of three steps. For each spectral peak in the tornillo which is present at both stations, we (1) bandpass ¢lter the three-component broadband seismograms to extract the signal; (2) determine the main axis coordinate system X1 -X2 -X3 using the eigenvectors and eigenvalues of the covariance matrix for the Z-N-E seismograms and then transform the Z-N-E seismograms into the X1 -X2 -X3 coordinate system (Matsumura, 1981; Kanasewich, 1981; Seidl and Hellweg, 1991; Hellweg, 2000a); and (3) determine signal parameters in the time and frequency domains from the X1 -X2 -X3 seismograms. Figs. 3 and 4 show the X1 -X2 -X3 seismograms at ANG (Fig. 3) and ACH (Fig. 4) for each of the
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Fig. 2. Bandpass ¢ltered multichromatic tornillo. (Top) The same tornillo as in Fig. 1, ¢ltered between 0.5 and 40 Hz. (Bottom) Spectrograms of the E component between 0 and 40 Hz. Arrows mark spectral energy present before the tornillo which corresponds to one or more of the tornillo’s peaks. Note that there are spectral peaks even at frequencies higher than 20 Hz.
seven spectral lines in the spectrograms of Fig. 1. Some of the overtones persist throughout the entire seismogram of the multichromatic tornillo while others only contribute to the turn-on transient at the tornillo’s beginning. The exponential decay of the coda for all frequencies is clear in the seismograms of the X1 components. Figs. 3 and 4 also show that while many of the lines can be rotated so that most of the energy is on the X1 component indicating linear polarization, some of the spectral lines with high frequencies are not linearly polarized. The second overtone, n = 2 with fP2 = 10.19 Hz, is a good example of this. We characterize each of the n peaks or modes
of each tornillo using nine parameters measured from the X1 -X2 -X3 seismograms. (1) The kinematic parameters, azimuth Azn , inclination Inn and rectilinearity Ren , are determined from the eigenvalues and eigenvectors. (2) The frequency of the peak’s maximum fPn and its amplitude APn are measured in the frequency domain. (3) We measure maximum velocity amplitude Vn , the rise time tRn , damping factor Qn and the signal energy ERn in the time domain. The relative signal energy ERn is estimated as the sum of the squared velocity amplitudes of the three components, while the damping factor Qn is
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2 0 -2 1 0 -1
Velocity(µm/s)
0.2 0 -0.2 0.5 0 -0.5 0.5 0 -0.5 0.1 0 -0.1
0
0.2 0 -0.2 0
50
100
0
50
100
0
50
100
Time (s) Fig. 3. Mode decomposition of multichromatic tornillos. Tornillo shown in Fig. 1, recorded at station ANG. In each row, the recorded seismograms have been ¢ltered to extract each of the spectral peaks present in Fig. 1, then transformed to the eigenvector coordinate system, X1 -X2 -X3 , using the covariance matrix for the polarization ellipsoid for that band. The peak frequencies fn are determined from the amplitude spectrum while the factor Qn is measured from the decay of the X1 component. The passbands of the ¢lter used to extract the mode are given in the X3 window.
determined from the slope of the logarithm of the Hilbert envelope (Seidl et al., 1999). Fig. 5 shows an example of this process for the second overtone of the tornillo shown in Fig. 1. Its peak frequency is 10.19 Hz at both ANG and ACH. From Figs. 3 and 4 it is clear that each individual mode can be described as an exponentially damped wavelet of the form an exp(ig n t) with complex angular frequency g n = 2Zfn +iqn and complex amplitude an . Thus, we can apply the method of parameterization shown in Fig. 5 to all the spectral peaks. In this example, the values
of fP2 measured at stations ANG and ACH di¡er by less than the frequency resolution, while di¡erences in the Q2 values at the two stations are well within the measurement error for the method, which we estimate to be about 10%. The distribution and correlation functions of the nine parameters re£ect the range of variation in geometric and physical properties of the tornillo source. The kinematic parameters, azimuth Azn , inclination angle Inn and rectilinearity Ren , and the excitation parameters, signal energy ERn , maximum velocity Vn , are strongly in£uenced by
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Fig. 4. Mode decomposition of multichromatic tornillos. The same tornillo as Fig. 3, but the decomposition is performed on the recordings from station ACH.
scattering and dissipation along the propagation path (see, for example, Hellweg, 2003). These parameters therefore only provide information about relative variation in the source location, the radiation pattern and the excitation energy. In contrast, the observed tornillo frequencies fPn and damping coe⁄cients qn can be related directly to the source process if we assume that the tornillo is the impulse response of a linear, second-order resonator. For the fundamental mode, the basic formulae relating measured values of the parameters to their values in the source are summarized in Fig. 6. These equations are also applicable to the overtones. In this example, we take the true angular frequency of the fundamental at the source to be g0 = 2Zf0 with f0 = 2.0 Hz and the
corresponding damping parameter to be Q0 = 100. We can determine the source parameters g0 = 2Zf0 and Q0 either in the time domain by measuring the frequency fP0 and the damping coe⁄cient q0 , or in the frequency domain by measuring the peak frequency fP0 and the 3-db bandwidth BP0 of the amplitude spectrum. For the weakly damped tornillos at Galeras, we determine them by measuring q0 in the time domain and fP0 in the frequency domain, as shown in Fig. 5 for the second overtone. The observed rise time tR of the tornillo, measured after ¢ltering with a bandpass of 3-db bandwidth BF , can be estimated as tR = 1/BF . For the bandpass ¢lters applied in the analysis of the the tornillo swarm, the observed rise times tR are in the range 1 s 6 tR 6 2 s. The
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VOLGEO 2612 19-6-03 Fig. 5. Parameterization of tornillo signals. Example of the parameterization procedure for the tornillo shown in Fig. 1, the left group for station ANG and the right group for station ACH. The data are bandpass ¢ltered to extract one of the spectral peaks in Fig. 1, in this case for the peak numbered 2, the low and high ¢lter corners are at 9.5 and 10.5 Hz, respectively. In each group, individual plots and signal parameters taken clockwise from the upper left show: the direction of the major axis X1 of the polarization ellipsoid (kinematic parameters: azimuth Azn measured clockwise from N, inclination Inn measured from the vertical, rectilinearity Ren = 13MV2 /V1 M with the largest and middle eigenvalues, V1 and V2 , respectively); X1 component velocity seismogram (parameters: maximum amplitude Vn , relative energy ERn , taken as the integral of the velocity amplitude squared, rise time tRn from onset to maximum); amplitude envelope of X1 component (parameter: quality factor Qn for the exponential decay segment); amplitude spectrum of X1 component (spectral parameters: peak frequency fPn , peak amplitude APn ).
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Fig. 6. Determination of source parameters from signal parameters. (Top row) Synthetic tornillo, showing the relationship of the observed time-domain signal parameters (right), damping factor Q0 and frequency f0 , and the frequency-domain signal parameters (left), 3-db bandwidth B0 and peak frequency fP , to the source parameters g0 , q0 and Q0 assuming that the tornillo is the impulse response of a linear second-order resonator. (Lower left) Relationship between the rise time tr of the recorded broadband signal and the transient rise time tR of the bandpass ¢ltered signal (tR Etr for all modes of the multichromatic tornillos investigated). (Lower right) Determination of parameters for the synthetic tornillo. For the tornillos observed at Galeras, the source parameters f0 and Q0 can be determined with the best accuracy and resolution by measuring the damping coe⁄cient q0 in time domain (Fig. 3) and peak frequency fP in frequency domain.
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20.00
Frequency (ACH)
15.00
10.00
5.00 1.5-2.0 2.0-5.0 5.0-11 11.0-15.0 15.0-20
0.00 0.00
5.00
10.00
15.00
20.00
Frequency (ANG) Fig. 7. Scatterplot comparing the frequency for each peak observed in the tornillo swarm as measured at stations ANG (horizontal axis) and ACH (vertical axis). Frequencies are grouped by symbol. The cluster of diamonds for the fundamental frequency in the band from 1.7 to 1.9 Hz represents measurments from 14 tornillos. The high correlation implies that the frequency is a characteristic of the source rather than the propagation path.
true rise time tr of the tornillo source signal can be estimated using the formula given in Fig. 6. For the tornillos at Galeras tr ItR . For example, we calculate that the true rise time tr for the second overtone in Fig. 5 is about 0.02 s. That the true rise time is so small is evidence supporting the assumption that tornillos are the response of some oscillator to an impulsive excitation.
3. Parameter statistics For the tornillo shown in Fig. 1, the peak frequencies for all seven modes are the same to within the frequency resolution of the spectra, 0.01 Hz at stations ANG and ACH (see Figs. 3 and 4). Fig. 7 shows a scatter plot of fPn for the 15 events of the tornillo swarm which were recorded at both
Table 1 Mean values and standard deviations of fPn and Qn ANG
ACH
Number
W(fP0 )
c(fP0 )
W(Qn )
c(Qn )
W(fP0 )
c(fP0 )
W(Qn )
c(Qn )
0 1 2 3 4 5 6a
1.82 3.70 10.03 11.91 14.49 15.65 19.46
0.06 0.06 0.14 0.10 0.18 0.13 ^
99 47 569 369 174 263 207
41 14 152 87 88 81 ^
1.83 3.70 10.06 11.91 14.52 15.66 19.46
0.06 0.10 0.15 0.10 0.35 0.13 ^
100 59 546 366 161 221 250
46 32 210 119 43 115 ^
a
Model 6 has only one sample.
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Q (ACH)
1000
100
1.5-2.0 2.0-5.0 5.0-11 11.0-15.0 15.0-20
10 10
100
1000
Q (ANG) Fig. 8. Scatterplot comparing the Q value measured for each peak observed in the tornillo swarm as measured at stations ANG (horizontal axis) and ACH (vertical axis). The same frequency symbols are used as in Fig. 7. The high correlation between the values determined at the two stations implies that Q characterizes the source rather than the propagation path.
stations. The fundamental mode is observed in all tornillos. Higher modes are also observed for all tornillos but with decreasing probability of excitation for increasing frequency. Fig. 7 demonstrates that there is a signi¢cant correlation between the values for fPn at the two stations in all frequency bands. For individual tornillos, the frequencies fP0 (ANG) and fP0 (ACH) are the same within the frequency resolution of the spectrum, Nf = 0.01 Hz. At both stations, the mean values of fP0 for all 15 tornillos are W(fP0 ) = 1.82 Hz with standard deviations of c(fP0 ) = 0.06 Hz. Since c(fP0 ) = 6 Nf, the £uctuation of the fundamental frequency fP0 , within the band (1.82 T 0.06) Hz from one tornillo to another is signi¢cant and must be incorporated into a model for the source. The mean values and standard deviations of fPn for the spectral peaks are summarized in Table 1. In each band the exponential decay measured from the envelope of X1 is also the same at both stations but the scatter is higher for Qn than for the frequency, due to the larger error in the mea-
surement of this parameter. The determination of Qn depends strongly on the time window selected for the least-squares ¢t of the slope to the logarithm of the Hilbert envelope. We estimate that the measurement error NQn is about 10%. Values of Qn for the tornillo shown in Fig. 1 are given in Figs. 3 and 4. In the scatterplot of Fig. 8 the correlation between the values of Qn measured at the two stations is clear. For the swarm of tornillos, the mean values of the damping factor for the fundamental Q0 at the two stations are W(Q0 ) = 99 at ANG and 100 at ACH, with standard deviations of c(Q0 ) of 41 and 46, respectively. As with the frequencies, c(Qn ) s NQ and the £uctuation of Qn must be related to changes at the source and cannot be caused by noise or errors in the analysis method. Table 1 also gives the mean values and standard deviations of Qn . From these observations of the frequencies of the tornillos’ peaks and their respective damping factors, we can draw several interesting conclusions. First, as the two parameters are the same
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VOLGEO 2612 19-6-03 Fig. 9. Basic characteristics of peak frequencies and polarization. (Left) The histograms of tornillo fundamental mode and overtone frequencies observed with the short-period network of OVP from 1992 to 1996 (top) as well as with the broadband station ANG from 1997 to 2000 (middle and bottom) reveal the persistence of various frequencies over the course of time and changes in the volcano’s states of activity. They imply that there is some relatively constant factor in the source process. (Right) Polarization of a tornillo recorded at station ACH. The polarization of tornillos recorded at the crater stations ANG and ACH is mostly linear in a nearly horizontal plane with small variations of azimuth Az, inclination In and rectilinearity Re in each frequency band.
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at the two stations for any given tornillo, we consider the peak frequencies fPn and the damping factors Qn to be source parameters and not characterizations of the propagation path. Second, their £uctuations from one tornillo to the next must be related to some small changes in the physical parameters at the source. Third, with many frequencies present in the 15 multichromatic tornillos recorded at the two crater rim stations, we can divide the frequency band between 0 Hz and 16 Hz into two bands and determine the distribution of frequencies in these two bands. Are they more or less evenly distributed ? Between 0 Hz and 8 Hz the tornillos from this swarm have peaks grouped around two mean frequencies, while between 8 Hz and 16 Hz there are four.
4. Constraints and implications for tornillo source modelling In investigations of individual tornillos and tremor, proposed sources have included such models as £uid-¢lled cracks (Chouet, 1996; GilCruz and Chouet, 1997; Kumagai and Chouet, 1999; Hagerty and Benites, 2003), lumped-parameter models (Julian, 1994) or self-excited eddy shedding and turbulent slug £ow oscillations (Hellweg, 2000b). None of these models explains all available observations, especially when quantitative aspects of the models are compared with amplitude measurements from the tornillos or tremor. Often, the models are based on the interpretation of data only from vertical component recordings. For example, the coupling of seismic energy between the £uids in the conduit system and the solid edi¢ce of the volcano and the propagating medium presents particular problems for such models. Although tremor is often considered to be a direct seismic window into the conduit system, the various tremor signals actually only illuminate those small elementary volumes within the complete conduit system containing fast, seismically active turbulent or oscillating £uid movement. Geometrically, such elementary tremor sources are likely to be found at some type of constriction in a branched conduit system such as a constriction between two tubes, a constriction
leading to a branch or a constriction at the opening of an otherwise closed volume which acts as a cavity resonator (Lighthill, 1996). Thus, tremor sources must be considered to be sensors monitoring the amplitude of £uid pressure variations rather than data for an accurate mapping of the total geometry and physics of the £uid conduit system. What role do the tornillos play in the inversion of the puzzle of tremor sources, especially the multichromatic tornillos observed recently at Galeras Volcano? We review the parameters characterizing the tornillos to determine some constraints for source models. 4.1. Frequencies Tornillos have been observed at Galeras Volcano since 1992. From 1992 to 1996, the activity at Galeras was high. Six explosion eruptions occurred between July 1992 and June 1993 (Narva¤ez et al., 1997). During this time, tornillos were recorded with the network of the OVP using shortperiod, vertical component seismometers. Since 1996, the tornillos have also been recorded by one or more broadband, three-component seismometers at the crater rim. The histograms in Fig. 9 summarize the peak frequencies of the fundamental modes and overtones observed in tornillos from 1992 to 2000. The fundamental mode and the ¢rst overtone are two separate, stable bands between 1 and 4 Hz. Their mean values and standard deviations are summarized in Table 2. The persistence of the bands is an indication that there is some relatively constant factor in the process generating the tornillos.
Table 2 Mean values and standard deviations of the fundamental and ¢rst overtone since 1992
OVP ANG Swarm
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Interval
f0 (Hz)
f1 (Hz)
1992^1996 1997^1998 1999^2000
1.40 T 0.10 1.90 T 0.10 1.82 T 0.06
3.10 T 0.60 2.80 T 0.70 3.70 T 0.06
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Fig. 10. Helmholtz cavity resonator. (Top left) Patterns of tremor signals observed at Galeras Volcano which can be explained as responses of a cavity source described by a linear di¡erential equation to excitation by several di¡erent types of impulsive or turbulent variations in pressure. (Top right) Helmholtz resonator, a £uid-¢lled cavity with a small opening which can be described approximately by a linear, second-order di¡erential equation with coe⁄cients related to the cavity’s geometrical parameters (volume V0 , Rayleigh conductivity, U = 2R for a circular opening and U = F/LP for a cylindrical neck with e¡ective length LP and area F) and physical parameters (velocity of sound in the £uid, c). (Bottom left) Relationships between the observed tornillo parameters, f0 and Q0 , and cavity parameters for the fundamental mode, under the assumption of acoustic emission. (Bottom right) Estimation of characteristic cavity dimensions for a Helmholtz resonator ¢lled with a gas of 80% H2 O+20% CO2 at a temperature of 723 K and for typical observed parameters f0 = 1.82 Hz and Q0 = 100.
4.2. Polarization The orientation and rectilinearity of the polarization ellipsoids change apparently randomly from one frequency band to another (see ¢g. 12 of Hellweg, 2003). This may be the result of a volume source with a complex shape and a nonuniform wall response. It is exacerbated by the fact that for low frequencies, the stations are in
the near-¢eld of the source. For high frequencies, scattering in the complex medium that is the volcanic edi¢ce is also important (Hellweg, 2003). Nevertheless, at least two qualitative conclusions may be drawn from the observed polarization parameters : (1) The mostly horizontal polarization (example in Fig. 9; Hellweg, 2003) for the fundamental modes indicates that the source is near the surface
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Fig. 11. Observed and synthetic tornillos as an onset transient for tremor oscillations. (Top left) Tornillo followed by a sequence of harmonic tremor, recorded at station ANG on 3 March 2000 at 21.20 UTC. The tornillo and the tremor have the same peak frequency f0 = 2.0 Hz, the tremor lasted about 1.5 h. (Top right) Details of tremor before (precursor) and after (coda) the tornillo, respectively, showing the increase in the amplitude of the tremor after the tornillo. The precursor has the same peak frequency f0 = 2.0 and polarization as the tornillo and the tremor. (Bottom left) Synthetic tornillo with tremor coda generated by numerical integration of the di¡erential equation for a self-excited, nonlinear Van der Pol oscillator for the parameters W = 0.25 and g = 2Zf with f = 1.5 Hz (Fig. 12). (Bottom right) Details of the synthetic harmonic tremor before (zero) and after the synthetic tornillo.
within the active crater, and that it radiates longitudinal waves. (2) The nearly linear polarization and the £uctuations of the polarization parameters suggest the radiation comes from small elementary sources within a larger volume. The polarization depends on the complex three-dimensional eigenfunction of the source as well as the location
of the excitation pulse within the larger volume. 4.3. Tornillo source as a linear oscillator and the Helmholtz cavity resonator The multichromatic tornillos observed in the tornillo swarm from late 1999 and early 2000
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Fig. 12. Synthetic tremor generated by numerical integration of the di¡erential equation for a self-excited, nonlinear Van der Pol oscillator with g = 2Zf and f = 1.5 Hz, for di¡erent damping parameters W. (Top) Tornillo with tremor coda for two damping coe⁄cients W = 0.25 and W = 0.5, respectively. (Bottom left) Harmonic tremor as the result of the random £uctuations of the damping coe⁄cient around a mean value of W = 0.05 (parametric oscillator). (Bottom right) Cyclic tremor for a high damping coe⁄cient W = 20.
can be explained, at least qualitatively, as free vibrations of a £uid-¢lled cavity, excited by a pressure pulse either at the wall or within the cavity. The impulse response of a cavity of volume V, ¢lled with a £uid having sound velocity c,
is characterized by two basic theoretical results (Morse and Ingard, 1968): b The resulting pressure wave ¢eld is a superposition of the fundamental and higher-mode eigenvibrations pn = an (x)exp(ig n t). an (x) are the
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eigenfunctions (position vector x) and g n = 2Zfn +iqn are the complex eigenfrequencies of the cavity. The frequencies fn and the damping coef¢cients qn are related to the damping factor Qn and the bandwidth Bn of the mode’s amplitude spectrum by the formula : Qn = Zfn /qn = fn /Bn . Such behavior is observed in the example of mode decomposition given for the multichromatic tornillos in Figs. 3 and 4. b The number vN of eigenfrequencies fn in an interval vf increases with frequency and can be approximated for a cavity with a small opening by the formula vN = 4 V/(c3 f2 vf). Although the number of tornillos of the swarm which occurred during January and February 2000 may be too small for signi¢cant statistics, the values listed in Table 1 are in good agreement with this formula. We would like to mention an interesting implication of this formula. The mean separation Df between the frequencies fn is given by: Df = vf/ vN = c3 /(4 V f2 ). If Df 6 Bn = fn /Qn , the bands Bn of the modes will overlap and the discrete mode spectrum will converge to a quasi-continuum. This corresponds to a transition from successive tornillos or tremor packets to continuous tremor in the time domain or a transition from multichromatic peaked spectra to broadband spectra in the frequency domain. Such a transition can be achieved in several ways: by shifting the eigenfrequencies to lower values for ¢xed Qn , by decreasing the Qn (increasing the damping coe⁄cients qn ) for ¢xed fn , or by a combination of both e¡ects. One special case of a cavity resonator is the Helmholtz resonator. A wine bottle is a wellknown example of a Helmholtz resonator, a cavity with a small opening such as a circular hole or a cylindrical neck of length L (Fig. 10). When excited by an external source, the cavity responds like a precisely tuned system with a very small resonance impedance (Skudrzyk, 1971). Its impulse response is a sinusoidal signal with a sharp onset followed by exponentially decaying amplitude similar to the typical shape of a tornillo (Fig. 6). In his systematic, theoretical investigation of the Helmholtz resonator, Howe (1976) includes the transfer function of the resonator as well as excitation by a pressure pulse, a Karman vortex
street and a turbulent pressure ¢eld. For the fundamental mode, the transfer function can be approximated by a second-order, linear di¡erential equation with two parameters, a damping coef¢cient q0 and a natural angular frequency g0 = 2Zf0 with frequency f0 (Fig. 10). The frequency f0 of the fundamental mode depends mainly on the cavity volume V0 , the geometry and dimension of the opening, described by the Rayleigh conductivity U and the acoustic velocity c of the £uid. However, to ¢rst approximation f0 is independent of the shape of the cavity. This explains the stability of f0 during a single tornillo as well as its variation from one tornillo to the next over the course of hours or days. These observations are both related to changes in the acoustic velocity of the £uid in the cavity due to £uctuations of the pffiffiffiffitemperature, as given by the relationship cV T . The eigenfunctions and eigenfrequencies of the higher modes depend very strongly on the shape of the cavity and the local geometry of the cavity wall. The e¡ects are even more pronounced for the damping factors which are very sensitive to the local acoustic impedance of the cavity wall and the frequency-dependent thermodynamic and viscous coupling of the £uid pressure £uctuations to the seismic motion in the solid medium. These problems cannot be treated analytically but only numerically for particular realizations of a conduit system. For acoustic radiation, the damping coe⁄cient q0 is related to the parameters c, V0 and U (Howe, 1976). Using this assumption, the Rayleigh conductivity U and the volume V0 can be estimated from the observed values of f0 and Q0 (Fig. 10). The example given in Fig. 10 assumes the cavity is ¢lled with a gas consisting of 80% H2 O and 20% CO2 at a temperature of 723 K. This gives a typical linear dimension of D = 15^20 m for a cavity which generates a tornillo with a frequency f0 = 1.82 Hz and a damping factor Q0 = 100. Although no cavity of such a size is known to exist at Galeras, the size is not unreasonable. 4.4. Tornillo source as a nonlinear oscillator and the Van der Pol equation For some strong tornillos at Galeras Volcano,
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the amplitude level of continuous tremor increases immediately following the tornillo, as though it had an extended coda. One example of a recording of such a combined tornillo^tremor is shown in Fig. 11. In a case like this, we may consider the tornillo to be the initial transient of a self-excited, nonlinear oscillator activated by an impulsive input signal. After the transient the process continues, feeding on heat or pressure energy from the volcano. Such a transient signal has both of the characteristic properties of a tornillo, a sinusoidal waveform with constant frequency and exponential amplitude decay. A synthetic signal with similar characteristics to the tornillo^tremor combination can be generated by integrating a nonlinear di¡erential equation, such as the Van der Pol equation (given in Fig. 12), for an impulsive input (Kirbani, 1983; Berge¤ et al., 1984). The Van der Pol equation, the most simple second-order, nonlinear di¡erential equation is analogous to the linear equation for the cavity shown in Fig. 10. Rather than the linear damping term there, the Van der Pol equation has a damping term which changes as a function of x. When x is greater than some critical or threshold value, the system is damped and it radiates energy. The signal amplitude decreases and disappears. If x is less than the critical value, the system absorbs energy from a heat or pressure reservoir by some internal feedback mechanism and the signal amplitude increases or remains constant. Thus, the threshold parameter controls the transition of the damping term between negative and positive values. In the Van der Pol equation shown in Fig. 12, we have normalized the variable x so that it is equal to 1 when the threshold parameter changes between negative and positive. There are many phenomena in volcanoes which could provide the mechanism needed for such a feedback system. A pressureactivated system which functions like a valve (Hellweg, 2000b) or bubble mixtures almost certainly present in magma or hydrothermal £uid systems (Martinelli, 1991) are two likely possibilities. Fig. 12 gives several examples of waveforms that can be generated by changing the value of the damping parameter, W, in the Van der Pol equation. The top two graphs are clearly tornillos
187
of various lengths with associated extended tremor codas. The bottom two graphs are similar to two other types of volcanic tremor, harmonic tremor (more or less sinusoidal, but varying amplitude) and cyclic tremor (typically has a spectrum with numerous peaks at harmonic frequencies). A model such as this one presents one major problem to interpretation. Unlike the case for the linear di¡erential equation describing the cavity resonator, it is di⁄cult to relate characteristics of the volcanic system, such as conduit geometry or the process providing the feedback mechanism, to the parameters determining the behavior of the nonlinear di¡erential equation, such as W. We are investigating this aspect of the model further to relate the parameters of the nonlinear di¡erential equation to physical reality.
5. Conclusions Recordings of a swarm of tornillos occurring in January and February 2000, at two three-component, broadband stations installed on the crater rim of Galeras Volcano, Colombia, a¡ord a new opportunity. With the insights from a detailed investigation of the characteristics of these tornillos at the two stations, we have developed new constraints for modelling their source. The tornillos of this swarm were multichromatic, having between two and 10 narrow spectral peaks between 1 and 20 Hz. Some tornillos at Galeras were followed by a coda consisting of tremor with amplitudes higher than those before the tornillo. We used mode decomposition to investigate the signal characteristics of the individual spectral peaks and parameterize them individually. We treated these tornillos of the swarm as members of an ensemble, and determined distribution and correlation functions of the parameters. They were then used as additional constraints for describing and classifying the basic source and excitation processes. Tremor is a collective term for a large variety of impulsive or continuous seismic signals with random, harmonic or cyclic signature generated in the £uid conduit system of the volcano. The variety of signal forms may re£ect a corresponding
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variety of possibilities either for the source process, such as forced or self-excited oscillating and resonating vibrations, or for the excitation process, such as shock waves, turbulent £ow or convective transport phenomena. Data from the myriad forms that tremor takes in the seismic record provide necessary, but far from su⁄cient information for constraining unique physical and morphological models of the £uid dynamics describing these source processes. Individual models for the source of tremor are still very much a topic of discussion, especially with regard to their quantitative aspects and their uniqueness. There seems to be general agreement, however, that nonlinear and two-phase phenomena in the £uid ¢lling the conduit system of the volcano play a fundamental role in the processes generating tremor (Martinelli, 1991; Julian, 2000; Ferrick et al., 1982). Nonlinearities can be caused by many e¡ects. These include phenomena such as £uid^£uid or £uid^solid coupling and feedbacktype boundary conditions in the conduit geometry. Other possible sources of nonlinearity are £ow transients and instabilities, transitions to turbulence or critical thresholds of material parameters such as the strong dependence of the sound velocity on the content of free gases in two-phase £uids (Martinelli, 1991). The similarity of polarization and characteristic frequencies for tremor and tornillo signals at Galeras Volcano suggests that they have a common source with di¡erent states of oscillation and excitation. The observations of multichromatic tornillos as well as of tornillos followed by an increased level of tremor suggest two possible models for the source: a linear model in which the tornillo is the free vibration response of a £uid-¢lled cavity to a pressure pulse; or a nonlinear model in which the tornillo is an initial transient leading into a tremor sequence generated by a nonlinear, self-excited oscillator. A Helmholtz resonator is a simple realization of the linear model (Fig. 10). It is supported by the waveforms and the distribution of frequencies observed in the multichromatic tornillos of the swarm. For such a model, the parameters of the di¡erential equation are related to the geometry of the resonator or conduit and the velocity of sound in the medium
¢lling the resonator. The simplest description of a nonlinear oscillator is the Van der Pol equation. We have used this equation to generate synthetic waveforms which, under some conditions, mimic the tornillos as initial transients to sequences of harmonic tremor (Fig. 12). It is, however, di⁄cult to relate the parameters of the nonlinear di¡erential equation to parameters describing physical characteristics of some elements of the volcanic system. However, Fig. 12 demonstrates that an oscillator which can be described by the Van der Pol equation can generate tornillos as initial transients, continuous tremor as parametric oscillator signals when the damping £uctuates and cyclic tremor, such as that observed at Lascar Volcano, Chile (Hellweg, 2000b). The tornillos are only one member of the wide range of impulsive, harmonic or random tremor signals observed at active volcanoes. While such signals are clearly di¡erent from one volcano to the next and even from one occurrence to the next, they nevertheless also exhibit some similarities on which to base classes of models. The variation among these di¡erent forms of signals must be related to di¡erences in the geometry of channels, conduits and cavities, as well as to the variety of £ow phenomena possible when the moving £uid is a mixture of liquids and gases. Many different classes of mechanisms are possible as sources of tremor in such branched conduit systems. They include free, forced, resonant, parametric and self-excited motion in systems with linear or nonlinear responses. Using tremor data with low seismic amplitudes from an interval at Galeras Volcano with relatively low activity, we have only been able to derive a few possibilities for basic models for the source of tornillos and tremor. The mechanisms which may be described by these models suggest that seismic signals only offer us a small window into the seismically active portion of the volcano’s conduit system. In order to improve our understanding of the physics of volcanoes, we hope to combine the tremor observations and analysis with other data, such as gaschemical or electromagnetic signals, in a multiparameter approach (Seidl et al., 2003; Faber et al., 2003).
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Acknowledgements The Multiparameter Station at Galeras Volcano is a cooperative project of the Federal Institute of Geosciences and Natural Resources (BGR) in Hannover (Germany) and the Instituto de Investigacio¤n e Informacio¤n Geocient|¤¢ca, MineroAmbiental y Nuclear (INGEOMINAS) in Bogota¤ (Colombia). We would like to express our gratitude to the Deutsches Zentrum fu«r Luft- und Raumfahrt (DLR) for providing ¢nancial assistance for travel and transport expenses. This work would not have been possible without the data collected through the hard work and system maintenance of our Colombian colleagues at the Observatorio Vulcanolo¤gico y Sismolo¤gico de Pasto. We would like to thank Guiseppe Lombardo and an anonymous reviewer for their comments and suggestions for improving this presentation of our work. References Berge¤, P., Pomeau, Y., Vidal, Ch., 1984. Order within Chaos: Towards a Deterministic Approach to Turbulence. John Wiley and Sons, New York, 329 pp. Chouet, B.A., 1996. New methods and future trends in seismological volcano monitoring. In: Scarpa, R., Tilling, R.I. (Eds.), Monitoring and Mitigation of Volcano Hazards. Springer-Verlag, Berlin, pp. 23^97. Faber, E., Moran, C., Garzon, G., Poggenburg, J., Teschner, M., 2003. Continuous gas monitoring at Galeras Volcano, Colombia: ¢rst evidence J. Volcanol. Geotherm. Res. 125 (this issue). doi: 10.1016/S0377-0273(03)00086-3 Ferrick, M.G., Qamar, A., Lawrence, W.F.St., 1982. Source mechanism of volcanic tremor. J. Geophys. Res. 87 (B10), 8675^8683. Gil-Cruz, F., Chouet, B., 1997. Long-period events, the most characteristic seismicity accompanying the emplacement and extrusion of a lava dome in Galeras Volcano, Colombia, in 1991. J. Volcanol. Geotherm. Res. 77, 121^158. Go¤mez, D.M., Torres, R.A., 1997. Unusual low-frequency volcanic seismic events with slowly decaying coda waves observed at Galeras and other volcanoes. J. Volcanol. Geotherm. Res. 77, 173^193. Go¤mez, D.M., Torres, R.A., Seidl, D., Hellweg, M., Rademacher, H., 1999. Tornillo seismic events at Galeras Volcano, Colombia: a summary and new information from broadband three-component measurements. Ann. Geo¢s. 42, 365^378. Hagerty, M., Benites, R., 2003. Tornillos beneath Tongariro Volcano, New Zealand, J. Volcanol. Geotherm. Res. 125 (this issue) doi: 10.1016/S0377-0273(03)00094-5.
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