Journal of Volcanology and Geothermal Research, 51 (1992) 185-198 185 Elsevier Science Publishers B.V., Amsterdam

Stochastic simulation of volcanic tremor from Ruapehu

A.W. Hurst DSIR Geology and Geophysics, P 0 Box 1320, Wellington, New Zealand

(Received September 30, 1991; revised version accepted January 3, 1992 )

ABSTRACT

Hunt, A.W., 1992. Stochastic simulation of volcanic tremor from Ruapehu. J. Volcanol. Geotherm. Res., 51: 185-198.

The most common volcanic tremor produced by Ruapehu is a continuous signal with a dominant frequency of about 2 Hz. This signal has a sharply peaked spectrum, and an autocorrelation function with a high degree of coherence, even for lags of over 20 seconds. These characteristics strongly suggest that the cause of this tremor is a single resonator, probably a fluid-filled cavity resonating in an "organ-pipe" mode.

The stochastic simulation of such a resonator uses the equation of motion of a Simple Harmonic Oscillator, which applies to an "organ-pipe" fundamental resonance, with either the characteristics of the oscillator, or the forcing function, containing a random element. A "white noise" forcing function, which would be appropriate for excitation of the cavity by a high pressure gas input, gave good agreement with the observed spectra and autocorrelation functions. Another pos- sible model used an oscillator with a damping factor which varied randomly, and was sometimes negative, so oscillations built up, rather than decayed. This also gave a reasonable simulation of Ruapehu tremor.

The third excitation model used a Poisson process, in which during each time interval there was a certain probability of applying a fixed impulse to the resonator. It was found that the impulses had to be frequent, i.e. several times a second, to match the characteristics of Ruapehu tremor.

It has been suggested that tremor is composed of a succession of low-frequency ("B-type") earthquakes. The results of this simulation show that at Ruapehu tremor could be produced by a resonator with positive feedback just sustaining oscillation, or by a resonator excited by external impulses. The most promising model for low-frequency earthquakes describes them as the result of a major external disturbance of the resonator.

Introduction

Volcanic tremor can be loosely defined as a quasi-continuous seismic signal produced by an active volcano. Various volcanoes produce dif- ferent varieties of tremor, ranging from that commonly called "harmonic", with a single frequency and little variation in amplitude, to much more variable and wide bandwidth sig- nals. Explanations for harmonic tremor con- centrate on possible resonators which can pro- duce such a sharply peaked signal. These are generally fluid-filled regions within the vol-

Correspondence to: A.W. Hurst, DSIR Geology and Geo- physics, P.O. Box 1320, Wellington, New Zealand.

cano, where the fluid may be volcanic gases, magma, or water. Various mechanisms have been proposed for the excitation of these reso- nators, such as high-speed gas flows (Steinberg and Steinberg, 1975; Seidl et al., 1981 ), the ex- pansion of cracks in the rock by high-pressure fluids (Aki et al., 1977; Chouet, 1985, 1988), "water-hammer" in moving liquids (St Law- rence and Qamar, 1979; Ferrick et at., 1982), and collapse of gas bubbles within fluids (Kieffer, 1984; Leet, 1988).

The composite volcano, Ruapehu, in the Taupo Volcanic Zone, New Zealand, often produces particularly monotonic tremor (Fig. 1 a). At the other extreme, some volcanoes such as Mount St. Helens normally produce rather

© Crown copyright

1 8 6 A.W. HURST

(a)

Power dB

O"

- 10 "

-20 "

-30 "

~40

.1

Ruapehu Tremor 1990 March ( b ) 0 -

- 1 0 -

- 2 0 -

- 3 0 -

. . . . . . . J , . . I . h . , , , - 4 0

I 1 0

Frequency Hz

Mount St Helens 1981

1 i

10

Fig. 1. (a) Normalised power spectrum of tremor from Ruapehu, recorded at station DRZ. (b) Normalised power spec- trum of tremor from Mount St. Helens, after Malone ( 1983 ).

discontinuous tremor with energy over a broad spectrum (Fig. 1 b). This latter kind of tremor has been explained (Fehler, 1983) as the re- sult of the superposition of a series of individ- ual small low-frequency ("B-type") earth- quakes, producing a quasi-continuous signal. This seems a reasonable explanation for this kind of signal, but it might be less confusing not to use the word "tremor" for the broad- band signal, restricting this term to the "har- monic tremor" with a sharply peaked spectrum.

Ruapehu tremor, with nearly all the energy in a single narrow frequency band, is hard to explain as being the superposition of earth- quakes. Figure 2a,b shows the quite different waveform character of tremor and low-fre- quency earthquakes at Ruapehu, and the cor- responding spectra are shown in Figures 1 a and 2c. The tremor spectrum has a much narrower peak than the low-frequency earthquake. If the tremor was made up by a series of earth- quakes, then the tremor spectra could not have a sharper peak than a typical low-frequency earthquake. The spectrum of a series of iden- tical impulses is wider than that of a single im- pulse, and any differences between the single impulses, or earthquakes, will produce an even wider spectrum (An example can be seen in Fig. 10 for artificial impulses). The dominant frequencies of low-frequency or "volcanic"

earthquakes at Ruapehu are the same as the most common tremor frequencies, with about 2 Hz most common, and about 3 Hz somewhat less common (Latter, 1981), so the tremor must be related to the low-frequency earth- quakes, even though it is not composed of earthquakes.

This paper compares the observed Ruapehu tremor with simulated tremor produced by mathematical models of harmonic oscillators, in which either the parameters or the exciting force involve random number series. These comparisons relate to the most common vari- ety of Ruapehu tremor, with a dominant fre- quency of approximately 2 Hz. The compari- sons also include some low-frequency earthquakes, also with a dominant frequency of approximately 2 Hz. The data on which these analyses were based was collected in 1989 and 1990, and is typical of the tremor pro- duced by Ruapehu in the last 10 years, during which time there have been no large eruptions. The tremor shown here was recorded at DRZ, about 1 km from the active vent of Ruapehu underneath Crater Lake, but very similar char- acteristics are seen in tremor recorded at sta- tion CNZ, about 10 km from Ruapehu, show- ing that these characteristics come from the tremor source. Particle motion measurements with a 3-component seismometer and mea- surements of the attenuation of the tremor sig-

S T O C H A S T I C S I M U L A T I O N O F V O L C A N I C T R E M O R F R O M R U A P E H U 18 7

: i i i ~ i i i ! i i i i ! ~" i ! i ~ i i ! T r e m o

i i ! i i i i i i i i i ! ! ! ! ! ! ! ! ! !

i i i i i.

Ruapehu B-type Earthquake 1990 June o

-10 Power

dB -20.

-30.

-40 . . . . . . . . . . . . . . . , .1 1 10

F r e q u e n c y H z

Fig. 2. (a) Waveform of volcanic tremor from Ruapehu, recorded at DRZ. (b) Waveform of low-frequency volcanic earthquake from Ruapehu, recorded at DRZ. (c) Normalised power spectrum of low-frequency volcanic earthquake from Ruapehu, recorded at DRZ.

nal with distance from the vent both show that at DRZ both body and surface waves are pres- ent, but at more than 2 km from the vent a sur- face Rayleigh wave is dominant.

The use of a second-order differential equa- tion to model volcanic tremor was originated by Kubotera (1963). The stochastic simula- tion method used here has some similarities to the maximum entropy spectral analysis used by Seidl et al. (1990) to analyse tremor from Etna and Merapi. They also used random numbers to simulate tremor, although the techniques used in this paper were derived independently of their methods.

The most likely cause of this 2 Hz resonance is a cavity filled by a compressible fluid. A sim- ple model would be an "organ-pipe", oscillat- ing at the frequency at which the pipe was a quarter-wavelength long. The quarter-wave- length resonance is inferred because only odd harmonics of 2 Hz have been observed on the digital recording system (Hurst, 1985) in- stalled in 1983 to record spectra of tremor from Ruapehu and Ngauruhoe. If the pipe was filled with superheated steam, a sound velocity of 520 m/s for steam at 20 bar and 210 °C would mean a pipe length of 65 m. An alternative is a magma-filled pipe, with a higher sound veloc-

188 A.W. HURST

ity of 1.6-2.7 k m / s (Ferrick et al., 1982; Chouet, 1985 ), and hence a length of 200-400 m .

Tremor simulation

Methods

The sharply peaked tremor spectra could be produced by a high-Q resonator in two ways, either by positive feedback sustaining an oscil- lation of the resonator, or by the resonator ef- fectively filtering an input signal, i.e. the en- ergy source is external to the resonator.

The mathematical models used for tremor simulation are all based on the equation of mo- tion of a simple harmonic oscillator with a forcing function F( t ) , namely:

£+&+ f lx=F(t) (1)

For these parameters, the undamped fre- quency (i.e. when o l=0) is x/fl/2rt, while for small positive values of a, any initial oscilla- tory motion decays according to e -'~t/2. This occurs when there is negative feedback to the resonator, whereas when there is positive feed- back, a has a negative value, giving an increas- ing amplitude with time. All iterations were calculated with 20-ms steps, to correspond to the 50 samples/s of the observed data, and cal- culations were made using the Runge-Kut ta method.

Stochastic simulations use random number series to provide some aspect of the tremor simulation. In the first model, the damping factor ot is varied randomly. This model is, therefore, referred to as the random damping model. Each iteration ofEq. 1 uses a new value of a, from a random distribution evenly dis- tributed in the interval ( - ao, oto). This means that the mean value of ot is zero, and its range is from - ceo to Ceo. Negative values of a cor- respond to positive feedback to the resonator, and hence increasing amplitude. Physically, this requires an energy input to the resonant system. The forcing function for this model is

zero, but an arbitrary initial motion is applied at the initial time. The output of this simula- tion is an oscillation with variable amplitude. If ao is small, the output waveform is similar to a pure sinewave, larger values of ao give more variable amplitudes, and the power spec- tra of the output waveform shows a broader peak.

For the second type of model, the damping factor is a positive constant. In the "white noise" model, the forcing function F is a ran- domly produced signal with a flat spectrum in the range of interest, namely near the resonant frequency of the oscillator. The release of high- pressure gas through an orifice is a well-known source of white noise, a spectacular case being the discharge of geothermal bores to the at- mosphere. The white noise generator uses the mean value of 12 random numbers for each signal value. 50 values were calculated each second, corresponding to the 50 samples/s of real data, and the signal has a sensibly flat spectra in the 0-10 Hz band. In the "Poisson" model, a standard impulse is applied to the os- cillator (i.e. F(t) is normally zero, but takes a large value for one 20-ms step), at random times. This model represented earthquake triggering of the resonator, and variation of the probability of an event in unit time alters from cases in which there were several impulses per second, right down to a single impulse. Figure 3 shows "cartoons" of these three models.

For all models, a standard random number generator was used (VAX Fortran RAN) with an initial "seed" value, so that runs could be repeated or independent runs made.

The tremor simulations were compared with sections of real tremor, selected for high am- plitude and hence good signal/noise ratio. The sections used included those with the narrow- est bandwidth spectra, to see what resonator model was required to produce such a signal. "Typical" sections of tremor, showing the characteristic increases and decreases in am- plitude, were also selected, to see whether the simulations could reproduce this appearance.

STOCHASTIC SIMULATION OF VOLCANIC TREMOR FROM RUAPEHU 189

Energy Steam Source Input

Impulse

Fig. 3. Cartoon depicting possible tremor mechanisms. (a) "Random Damping"; energy supplied to the resonator provides positive feedback. (b) "White Noise"; the reso- nator is driven by a broadband source, such as a gas es- caping a vent at high pressure. (c) "Poisson"; the reso- nator is excited by a series of impulses.

Comparisons were made of the visual appear- ance of the signal, especially the ampli tude changes, and the spectra and autocorrelation functions of the signal.

An autocorrelation function with an enve- lope that decays slowly with t ime is an indica- tion of a consistent signal. The autocorrelation function is particularly sensitive to phase changes and small changes in frequency that would not be distinguished in spectral mea- surements. Seidl et al. (1990) calculated au- tocorrelation functions for tremor from Etna and Merapi, in which the autocorrelation functions decay fairly rapidly, suggesting that there is not a single high-Q resonator at those volcanoes.

Results

Random damping

For small values of Oto, the random damping model produced a signal that was a sine-wave

with slight variations in amplitude. Figure 4 shows that the waveform, spectrum and auto- correlation function of a section of this simu- lated tremor with a o = 7.0 are similar to a sec- tion of real tremor. In Figure 5 a similar comparison is shown, but with a o = 14.0, giv- ing much more amplitude variation, but again being similar to a different section of real tremor. When a is greater than about 15, the amplitude variations become extreme, with sudden increases and decreases. In Figure 6, a section of record with a o = 20.0 is compared with a low-frequency volcanic earthquake ("B- type") . There is a considerable resemblance in waveform, but the real earthquakes have a wider spectrum, and their autocorrelation functions decay much more quickly. Both these differences are due to the low-frequency earth- quakes having variations in dominant fre- quency during the event, as compared to the tremor of Figures 4 and 5, which has a con- stant dominant frequency.

A physical interpretation of this could be that the dominant resonator is normally in a state in which a small degree of positive feedback balances the losses and produces tremor of constant frequency, with some variation in amplitude. At other times, the same oscillator is triggered into oscillation, but the triggering pulse disturbs its normal conditions, produc- ing variations in the resonator frequency, and rapid variations of amplitude. Figure 6a shows a high-frequency arrival five or six seconds be- fore the low-frequency oscillation starts, and such arrivals, interpreted as from small vol- cano-tectonic earthquakes at shallower depths, known as "roof-rock" events (Latter, 1981 ), are often seen before low-frequency earth- quakes at Ruapehu. Since the record is from seismic station DRZ, only about 1 km from the source, it is not an earlier arrival from the same event. The high-frequency wave may have come from a fracture event which sent a pres- sure wave towards the resonant cavity, pro- ducing the low-frequency earthquake some seconds later.

190 A.W. HURST

i

D R Z 0 0 B u f f e r = 1 8 6

> 1 second

b

i i " i i • • i • ~ i ~ " i

i " i i i , • • i i i • i

~ i m V ~ = O i i . i i . i , i i B u f ' f e r = OZ Z i Z " i i

D R Z B u f f e r = 1 8 6

" ~ S i m V B u t l e r = 0

b o 20

Z

40 r ' j . . . . . . . . . . . ~

0 1 2 5 4 5 6 8 9 1 0 1 1 1 2 a z

B u f f e r - - 1 ~ 6 : . : :

8 1.o D R Z

a g o u _

g - 1 . 0 . . . . . . : . . . . . . . . - " . . . . . . . . . . . . . : : : : : ".~ ~,~ :

o £ b gc

f

lag in seconds

Fig. 4. W a v e f o r m s , powerspec tra and autocorre la t ion funct ions . ( a ) T r e m o r recorded at D R Z , 1990 O c t o b e r 12 0001 U T . ( b ) R a n d o m d a m p i n g s imula t ion , 2 .0 Hz , cons tant (O~o) = 7.0, s eed = 6.

STOCHASTIC SIMULATION OF VOLCANIC TREMOR FROM RUAPEHU 191

a

DRZ C=O i i : : Z : : i : : : : i i i : : : i Z : : i ; : Z i i Z i : : i Z i : i Z : i : i : : : : i ; i

E : i i i i i ! i i ; ! i i ! ! i i i ! i i i i ! i i i i i i i i i i i i

> 1 s e c o n d : i i ! i ! i i i ! i i i i i i i ! i i i i i i i i i i ! i ! i i i ! i i ! ! : i i i i i

i i i i i i ; i ! i i i i i ! ; i i ! i i i i i i i i i i i i i ! i ! i i i i i i i i i i i i ! i

~ ; I n :~VG=O i i : i ; : i : : : : i : : i : ; i : i : : ; i : : : i i i i : : : : : ; : i i : : : 1 . . . . .

• . . . . . . , . . . . , . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . .

} } DRZ I~u f f e r = g8 - :

a ~ 20 ¢L u )

~ d B 4 0 . .

(~. :

"~ S imv Bu f f e r = 10

o , . . . . ~ . . . . . . . . . ~ . . . . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~ ~ ~ - - - ~ . . . . . . i . . . . . ; . . . . . . . . . i - - - - . . . . . ; . . . . . . . . . ; . . . . i . . . . . i-- z 2 o i . . . . ~ . . . . . } . . . . . ; . . . . - - - - i i .

L . . . . . .

40

o 2 s . s ~ l o ~ 1 2 H z

i i i . i i i i i i i : i : : : : : : : : . . : . : : : : : : : : 1 : B ~ , . . ; = 6 8 : : : : : : : : : : i : i i i i i : : : : : : : : : : : : :

. 0 - ~ - - ~ - + ' - ' - ' - - ' - - ' - - ' - - , - , - , - , - , - , - , - , _ _ , _ _ , _ _ , _ _ , _ , _ , _ ' _ ' _ ' _ ' _ , _ _ ' _ _ ' _ _ , _ _ , _ , i - -

a :i b "5

1

l a g i n s e c o n d s

Fig. 5. Waveforms, powerspectra and autocorrelation functions. (a) Tremor recorded at DRZ, 1990 April 5 0624 UT. (b) Random damping simulation, 2.0 Hz, constant (O~o) = 14.0, seed = 6.

1 9 2 A.W. HURS'I

b

o

" ; . i : : . • ' • :

DRZ: G~O: Bu f i l r ~ 1 2 0

I I 1 second

S imV G=O Blaf f e r ~ ' 0

a " 6 (I) ¢-j

0

"0

b ~ 0

Z

DRZ Bu f f e r= 120

0 - - -

40 . . . . . . . . . . . . . . . . . . . . .

S imV 'Bu f f e r =

0 . . . . . . . . . . q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 0

0 I 2 3 4 5 6 • B 9 ~0 11 12 Hz

• : i i 8~ f f e r : = 1]20 : : . . . .

E

0 1 2 3 4 5 6 7 8 9 I 0 11 12 13 14 15 16 17 18 19 20 21 22 23 2 25 26 27 28 79 30 31 32 3.3 35 .

lag in seconds Fig. 6. Waveforms, powerspectra and autocorrelation functions. (a) Low-frequency volcanic earthquake recorded at DRZ, 1990 August 5 0308UT. (b) Random damping simulation, 1.95 Hz, constant (ao) = 20.0, seed = 12347.

STOCHASTIC SIMULATION OF VOLCANIC TREMOR FROM RUAPEHU 193

b

E ,2o > ¢u

¢ 1

o Q. "1o

U )

b E o Z

D R 7 G = O : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : B u f f e T = ~ 8 6 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ; : : : : : : : : : : : : : : :

I I i : : i ! i l i l i l i l ; i i i i i i i i i i i i i i i ! i i i ) i i i i i i i ; i i i ! : i i : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

, ooi::i i i:: i i ii ~ : : i~: i i i i i i i ;

DRZ ~ u f f e r = 8 6

2(1

d g 4 0 . . .

SimW Buf fe r = : 0 i . . . . .

2 0

: : i 40 . . . . ~ ~ ~ ~

0 I 2 3 4 6 8 10 11 12 Hz

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

1,0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D R Z

a g 0 u_

g -i .0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . ~_~_~_ _ _ _ _ _~_~ _ : ~ 1.0-;--~---:-~--~--~--i--i--?-:: ~ - ~ ~ - ~ ~ :: :: i i : . : : ? ~ - ! i : : --:--- : : -~ - o ~ o

"9'0 is~ b ~

lag in seconds

Fig. 7. Waveforms, powerspectra and autocorrelation functions. (a) Tremor recorded at DRZ, 1990 April 5 1917 UT. (b) White noise simulation, 1.8 Hz, damping (ol) =0 .03 , s eed= 34567.

9 4 A.W, H U R S T

a ~ ~ ~ ~ ~ V ~ ~ : ' i : i : i i : i : i : : : " : ! : i ¸ : i ¸ : i : :

: : ; i l ; ! i : : i ; ! i : : i i i ~ i : ; i ~ i D R ~ C~O: : . : : : . . . . : : : " " Z Buf fe r := :52 : : : : : :

o = I I : : : : : : : : II) . . . . . . . . > 1 s e c o n d : : : : : : : : i

: : i : . . . .

b

0RZ

4 0 0 (3-

" ~ S i r n W B u f f e r = 0 .

.~ 0

b o 20 7

0 I 2 3 4 5 6 7 8 9 %0 11 12 az

B~,f'~,,,:= sz : : : : : : : i : g 1.0

" ~ O R Z

a g 0 U_

._~ - 1 . 0 ~ ~ - ~ - - - ~ - ~ - - ~ : : ! ! : ! : i

i B u ' f f e r ~ = 0 : : : : : : : ; i

b ~ O: i . . . . . . . . i : ~ " } ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-1.0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 t7 18 19 20 ~1 22 23 24 25 26 27 28 29 30 31 32 33 34 35 3£

lag in seconds

Fig. 8. Waveforms, powerspectra and autocorrelation functions. (a) Tremor recorded at DRZ, 1990 November 15 1221 UT. (b) White noise simulation, 2.0 Hz, damping (o0 =0.4, seed= 12345.

STOCHASTIC SIMULATION OF VOLCANIC TREMOR FROM RUAPEHU 195

b

E o

n

" 0

b E 0 z

i i i i i i i i i i ! i i i i i i i ! i i i i i i i i i i i i i i i i ~ i i i ~ i i ~ ! i ~ i i i ! ~ ! o , z i c ; o ; i : : i : : : : : : : : : : : : : : : : : ! : i i i : i ; i i i i i i i i i ! i i i B u f f e r = 8 7 . . . . . . : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

S l m O = ~ : i ; : : : : : : : : : : ; . . . . . . . . . . . . . . . . . . . . 2 :

DRZ B u f t l r = 87

20

dB

4o ; i N i m P Bur f e r = ~ i

20

40 . . . . .

0 1 2 3 4 10 I I 12 Hz

1 . 0 - ; - ; - T - ~ - - ~ - - ~ - - : : - - i - - : : - - ~ - - ~ - - T - T - ~ - ~ - ; i i i_ : : := i _ : : : ; :: :: ! - ~

Q) : Bu ; f f o~= 0 i : : i : ', '. : : : : : : ', : ', : : i : : : : h

L- o o 11- ! .~`~-;~.~;~.~.;~;~.~:~-:~;.~.~;~?~-~?~i~i~i~.~ {

lag in seconds

Fig. 9. Waveforms, powerspectra and autocorrelation functions. (a) Tremor recorded at DRZ, 1990 April 5, 1924 UI" (b) Poisson simulation, 2.0 Hz, damping ( a ) = 0.1, seed = 12345, event probability 0.1 in each 20-ms interval.

196 A.W. HURS'I

b

E .9o

a

0 D .

" 0

u )

b E 0

z

I I 1 s e c o n d

: i i i : .

I " : i i i i $ i m P i i i . ,

i

0 1 2 ~ 4 5 6 8 10 11 12

5 i m P

2 O

4 0 . . . . . . . . . . . H

0 1 2 3 4 5 6 7 8 9 10 11 12 H Z

. . . . . . i

u_ 1 0 . . . . . . . . . . C i 2 : : i 2 : : : : 2 i : : i 2 2 : :

N : : : : : :: : ) : " " o 1 . 0 " - - * - ~ - 2 - 2 - - i - i ? ~ - ~ - f - f - ~ - = - - : - : - - - - ~ - ~ - - f - ~ ~ ~

SimP. ' ' _ _ ; : ; ; : : ; ' i . . . . } } ;

1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - _2_Z_ 2__ i ; - -Z_2_2_2 _2_2 ± 2 Z__ I__ :_ ~_Z_L_ I

lag in s e c o n d s

Fig. 10. W a v e f o r m s , powerspec tra and autocorre la t ion funct ions . ( a ) P o i s s o n s imula t ion , 2 .0 Hz , d a m p i n g ( a ) = 0 . 1 , seed = 12345 , event probabi l i ty 0.01 in each 2 0 - m s interval . ( b ) Single event s i m u l a t i o n 2 .0 Hz , d a m p i n g (o~) = 0.1.

STOCHASTIC SIMULATION OF VOLCANIC TREMOR FROM RUAPEHU 197

White noise

Figures 7 and 8 show the waveforms, spectra and autocorrelation functions for white noise signals filtered by a harmonic oscillator, com- pared to real tremor. In Figure 7, a period of comparatively constant tremor is approxi- mated by the effect of white noise on a lightly damped (or=0.03) resonator, whilst Figure 8 shows more variable tremor which is approxi- mated by a similar white noise signal and a heavily damped resonator (or=0.4). These correspond to a resonator Q of 210 and 16 re- spectively. A greater amount of damping pro- duces a wider spectrum, and a more rapidly decaying autocorrelation function. This model also seems able to approximate tremor, just as the "'random damping" model can, but the "white noise" models cannot easily simulate the low-frequency earthquake of Figure 6a, as these models cannot reproduce the rapid changes in level at the start and end of such events.

Poisson

This model excites the resonator by a series of impulses. The probability of an impulse in each time interval ( 1/50 s), is constant, so the impulses will be distributed according to a Poisson distribution. The variation in charac- ter is here seen in Figures 9 and 10. In Figure 9, the probability of an impulse in each 1/50 of a second is 10%, so there would be several impulses per second. The result is similar to the "white noise" model, and is shown compared to a section of fairly steady tremor. In Figure 10a, there is a probability of only 0.5% of an impulse in each interval, and the waveform shows obvious step changes in amplitude, whilst the autocorrelation function decays more rapidly. This leads to the extreme case of Figure 10b, in which there is only one impulse. Note that all these models use the same damp- ing factor, a = 0.1. These cases still show fairly narrow spectra, and neither Figure 10a nor

Figure 10b gives a close approximation to the low-frequency earthquake of Figure 6a. It is particularly noticeable that, when a single im- pulse is applied to the resonator, the resultant waveform has a well maintained autocorrela- tion function, showing that the rapidly decay- ing autocorrelation of the earthquake in Figure 6a must be a resonator characteristic.

Conclusions

Simulations run using all three models seem to give reasonable approximations of typical Ruapehu "2 Hz" tremor. The more variable character of the low-frequency earthquakes, which have the same dominant frequency as the tremor, and are often associated with tremor, was better modelled by the "random damping" model.

The overall conclusion would have to be, however, that the resonator producing the 2 Hz Ruapehu tremor has a fairly high Q, and that the filtering effect of such a resonator makes it very hard to actually derive any of the charac- teristics of the energy source exciting it.

The resonator is most likely to be a quarter wavelength "organ-pipe", as there is no 4 Hz (second harmonic) tremor on these records, but a small amount of 6 Hz (third harmonic) which sometimes reaches a level about 15 dB below the 2 Hz fundamental. The high values of resonator Q that are needed to simulate the observed tremor also mean that the resonator must have little interaction with the surround- ing material. The Crater Lake model of Hurst et al. (1991), has a two-phase (steam and water) heat-pipe as the immediate heat source of Ruapehu Crater Lake. The resonator is un- likely to be this two-phase region, as steam water mixtures have very low velocities and considerable attenuation for sound waves, so it is most likely below the heat-pipe, either in superheated steam or magma. The mechanism ofLeet ( 1988 ), that the tremor is produced by bubble collapse in or immediately under Cra- ter Lake, is also a possibility. More accurate es-

198 A.W. HURST

timates of the depth of origin of the tremor are needed to distinguish between these possible sOurces .

Work is continuing on other aspects of vol- canic tremor at Ruapehu. A further aspect, suggested by this study, is that the current real- time recording of tremor spectra has been ex- tended to include the calculation of the damp- ing factor, and hence the Q of the resonator, to see how this relates to the state of the volcano.

Acknowledgments

Thanks to Dr Peter Thompson, Victoria University of Wellington, for helpful discus- sions on methods of analysing the simulations, and to Dr R.R. Dibble, Victoria University of Wellington, Dr J.H. Latter, DSIR Geology and Geophysics, Dr S. Gresta, University of Ca- tania and Prof. R. Schick, University of Stutt- gart for their helpful reviews. S. Sherburn as- sisted in the recording of Ruapehu tremor.

References

Aki, K., Fehler, M. and Das, S., 1977. Source mechanism of volcanic tremor: fluid-driven crack models and their application to the 1963 Kilauea eruption. J. Volcanol. Geotherm. Res., 2: 259-287.

Chouet, B., 1985. Excitation of a buried magmatic pipe: a seismic source model for volcanic tremor. J. Geophys. Res., 90: 1881-1893.

Chouet, B., 1988. Resonance of a fluid-driven crack: ra- diation properties and implications for the source of long-period events and harmonic tremor. J. Geophys. Res., 93: 4335-4400.

Fehler, M., 1983. Observations of volcanic tremor at Mt. St. Helens Volcano. J. Geophys. Res., 88: 3476-3484.

Ferrick, M.G., Qamar, A. and St. Lawrence, W., 1982. Source mechanism of volcanic tremor. J. Geophys. Res., 87: 8675-8683.

Gordeev, E.I., Saltykov, V.A., Sinitsyn, V.I. and Chebrov, V.N., 1990. Temporal and spatial characteristics of volcanic tremor wave fields. J. Volcanol. Geotherm. Res., 40: 89-101.

Hurst, A.W., t 985. A volcanic tremor monitoring system. J. Volcanol. Geotherm. Res., 26:181-187.

Hurst, A.W., Bibby, H.M., Scott, B.J. and McGuinness, M.J., 1991. The heat source of Ruapehu Crater Lake; deductions from the energy and mass balances. J. Vol- canol. Geotherm. Res., 46: 1-20.

Kieffer, S.W., 1984. Seismicity at Old Faithful Geyser: an isolated source of geothermal noise and possible ana- logue of volcanic seismicity. J. Volcanol. Geotherm. Res., 22: 59-95.

Kubotera, A., 1963. Volcanic micro tremor of the second kind-nature of its wave generation and source condi- tion. Spec. Contrib. Geophys. Inst., Kyoto University. 1: 81-91.

Latter, J.H., 1981. Volcanic earthquakes, and their rela- tion to eruptions at Ruapehu and Ngauruhoe volcan- oes. J. Volcanol. Geotherm. Res., 9: 293-309.

Leer, R.C., 1988. saturated and subcooled hydrothermal boiling in groundwater flow channels as a source of harmonic tremor. J. Geophys. Res., 93: 4835-4849.

Malone, S., 1983. Volcanic earthquakes: Examples from Mount St. Helens. In: H. Kanamori and E. Boschi (Editors), Earthquakes: Observations, Theory and Interpretation. North-Holland, Amsterdam, pp. 436- 455.

McNutt, S., 1987. Volcanic tremor at Pavlof Volcano, Alaska, October 1973-April 1986. Pure Appl. Geo- phys., 125: 1051-1077.

Seidl, D., Kirbani, S.B. and Brustle, W., 1990. Maximum entropy spectral analysis of volcanic tremor using data from Etna (Sicily) and Merapi (central Java). Bull. Volcanol., 52: 460-474.

Seidl, D., Schick, R. and Riuscetti, M., 1981. Volcanic tremors at Etna: a model for hydraulic origin. Bull. Volcanol., 44: 43-56.

St. Lawrence, W. and Qamar, A., 1979. Hydraulic tran- sients: a seismic source in volcanoes and glaciers. Sci- ence, 203: 654-456.

Steinberg, G.S. and Steinberg, A.S., 1975. On possible causes of volcanic tremor. J. Geophys. Res., 80:1600- 1604.