PAGEOPH, Vol. 12 L 5/6 (1983) 0033-4553/83/050835-1851.50+0.20/0 © 1983 Birkh/iuser Verlag, Basel

Erupt ion Patterns o f the Chi lean Vo lcanoes Villarrica, Llaima, and Tupungat i to

MIGUEL M u i q o z ~

A b s t r a c t - The historical eruption records of three Chilean volcanoes have been subjected to many statistical tests, and none have been found to differ significantly from random, or Poissonian, behaviour. The statistical analysis shows rough conformity with the descriptions determined from the eruption rate functions. It is possible that a constant eruption rate describes the activity of Villarrica; Llaima and Tupungatito present complex eruption rate patterns that appear, however, to have no statistical significance. Questions related to loading and extinction processes and to the existence of shallow secondary magma chambers to which magma is supplied from a deeper system are also addressed.

The analysis and the computation of the serial correlation coefficients indicate that the three series may be regarded as stationary renewal processes. None of the test statistics indicates rejection of the Poisson hypothesis at a level less than 5%, but the coefficient of variation for the eruption series at Llaima is significantly different from the value expected for a Poisson process. Also, the estimates of the normalized spectrum of the counting process for the three series suggest a departure from the random model, but the deviations are not found to be significant at the 5% level.

Kolmogorov-Smirnov and chi-squared test statistics, applied directly to ascertaining to which probability P the random Poisson model fits the data, indicate that there is significant agreement in the case of Villarrica (P = 0.59) and Tupungatito (P = 0.3). Even though the P-value for Llaima is a marginally significant 0.1 (which is equivalent to rejecting the Poisson model at the 90% confidence level), the series suggests that nonrandom features are possibly present in the eruptive activity of this volcano.

Key words: Eruption patterns; Statistical analysis; Stochastic processes.

Introduction

The analysis of eruption patterns of volcanoes presents numerous difficulties related both to the complex interactions which result in triggering an eruption and to the historic records, where the possibility exists of missing or mistakenly added events. The restrictions imposed are larger in the case of American volcanoes, where the records are usually shorter than those representing the eruptive activity of volcanoes located on other continents.

The search for stochastic models that fit the observed eruption

~) Departamento de Geofisica, Universidad de Chile, Casilla 2777, Santiago, Chile.

836 Miguel Mut~oz

patterns of volcanoes can start with the investigation of the survivor function and the eruption rate resulting from those patterns, and advance into a more detailed description of the parameters involved (WICKMAN, 1976). Also, strict statistical tests may be applied to analyse those patterns of eruption (REYMENT, 1969). The acceptance or rejection of stochastic models revealed by this approach may lead to distinguishing between random and deterministic phenomena (tectonic processes).

Villarrica and Llaima are the most active volcanoes in Chile; nevertheless, the available records of their eruptive activity are short. Much of this work will therefore be unavoidably approximate in character.

The observations of Villarrica, Llaima, and Tupungatito volcanoes were extracted from the work by CASERTANO (1963), edited by the International Volcanological Association, and from the Bulletin of Volcanic Eruptions (Volcanologicai Society of Japan). Further infor- mation was obtained from Report No. 1 of the Volcanological Service at our Department of Geophysics, prepared by GONZALEZ-FERRXN (1980), and from newspapers giving diverse opinions on the outbreaks.

Villarrica, Llaima, and Tupungatito volcanoes

Villarrica volcano (39 ° 25'S, 71 o 57'W) is located in the southwestern extreme of the Andean Cordillera. It forms part of the volcanic chain that starts in the southeast with the Lanin volcano, situated on the border of Argentina. The volcanic chain is controlled by a NW- SE-striking fracture system whose volcanic activity has migrated from SE to NW. The only active centre on this chain is ViUarrica volcano, a stratovolcano with a great somma displaced towards SE. The main active centre is surrounded by several pyroclastic cones. A recent fracture can be distinguished in the sides of the somma. At present the principal crater is plugged. Lavas of high fluidity and pyroclastic material of andesite-basaltic composition characterize the effusions from the main crater of Villarrica volcano and from fractures around it (GONZ,~LEZ-FERR,~N and KAUSEL, 1980).

Although the chronicles record eruptions in 1558, 1640, and 1775, only the record starting with the 1806 eruption will be used in this work. Through the last fifty years the observation of eruptive activity has been more satisfactory; the information regarding earlier centuries is inconstant and unreliable.

Llaima volcano (38°42'S, 71°42'W) is a stratovolcano located at the east of Temuco. The main cone has constant fumarolic activity and is surrounded by several pyroclastic cones. The eruptive activity is controlled by fracture systems striking N 140°E and N 15°E, the lavas

Eruption Patterns of Chilean Volcanoes 83 7

being generally characterized as olivine-bearing basalts. Only the eruptions starting with that of 1852 are considered here.

Tupungatito volcano (33 °24'S, 69°48'W) is a stratovolcano located east of Santiago, on the border of Argentina. It is at the base of the extinct volcano Tupungato, on the rim of an old, wide crater. The eruptive activity of Tupungatito is controlled by a N-S-striking fracture and fault system. The andesitic flows that constitute the volcanic structure are fresh and not divided by erosion. Fumarolic activity continues at Tupungatito. Only the record beginning with the 1829 eruption of Tupungatito will be used in this study. Because of the inaccessibility of this volcano, eruptions are poorly described. Over the last twenty years, the events may have been pyroclastic eruptions, but there is no definitive information with respect to this.

Further information on these volcanoes is contained in the works by CASERTArqO (1963), TI-UELE and KATSUI (1969), and GONZALEZ- FERRAN (1972).

Survival number and eruption rate

The six assumptions presented by WICKMAN (1966a) constitute the basis for the study of eruptions and repose periods of volcanoes on the bests of the renewal theory discussed by Cox and LEWIS (1966) .

I use the survival number N and eruption rate ~t as presented in the work by WICKMAN (1966a). The survival number Nis defined as

N(X) = NoF(X),

where N is the total number of repose periods of the volcano during the period of study and F(X) is the survival function which gives the probability that a repose is longer than the time interval X.

The age-dependent eruption rate ~(X) is established according to

d [InN(X)]= d dX ~ [ln F( X)] = -(,( X).

The observed step function of the survival numbers, the assumed survival function N(X) and the eruption rate #(X) are all derived from the observed durations X of the repose periods and are shown in Figures 1, 2 and 3. Some uncertain eruption dates contribute to uncertainty in the survival number N(X). In the case of ViUarrica volcano (Figure 1) two curves have been drawn. When only the year of the outbreak is known, the eruption has been dated July 1 of that year.

The diagram of #(X) is the graphical derivative of the smoothed N(X) and is very approximate. Curves 1 and 2 for 4(X) in Figure 1 may lead to completely diverse physical interpretations of the behaviour of

838

N

20

15

10

Miguel Mufioz

B0 ~60 240 320

VILLARRICA

~.00 L,80

5

i ~F

\

0 B0 T60 240 320 400 4B0 X(months)

Figure 1. Upper figure shows the step function of the observed number of reposes with duration longer than X months for Villarrica volcano; the heavy line is the graphically drawn survival number function, N. Lower figure shows the corresponding age-dependent eruption rate, ¢, measured in 10 -3 months-L

Villarrica volcano. The curve 1 approximation indicates a roughly constant eruption rate equal to 0.0145 per month. This represents a simple Poisson eruption process with no memory effects (WICKMAN, 1966b). Then the probability P , of n eruptions within a period of time t (the durations of the eruptions are neglected, by comparison with those

Eruption Patterns of Chilean Volcanoes 839

of the reposes) is given by the Poisson distribution

(¢t)" P . = exp (--#t) (1)

n!

where ~b is constant; eruptions can occur at any time and are random. Even if a volcano is Poissonian, it may change its eruption rate after a

fairly consistent pattern Of some hundreds of years (F. E. WICKMAN, personal communication, 1983). The complex character of the common part of curves i and 2 in Figure 1 indicates a more intrincate pattern of the eruption rate function. The complexity may be due to short and/or unreliable records and also to the uncertainty in drawing N(X). Nevertheless, it is interesting to comment on the more general form of the curve ~b(X). The part of curve 1 for ~(X) in Figure 1 for repose periods shorter than 60 months indicates that an outbreak makes a new eruption in the immediate future less probable. As pointed out by WICKMAN (1966b), this is a characteristic of volcanoes requiring a loading time after a great eruption in which the magma available in the conduits and in the shallow reservoirs is exhausted. On the other hand, a similar behaviour is expected if the conduit is closed by a vent plug between the eruptions, thus generating a resistance to the emergent magma. Later, the increasing gas pressure acting on the vent plug and/or tectonic movements resulting in its failure should augment the probability of a new eruption. This explanation does not seem to fit the volcanoes studied, since plugs do not form in the craters between eruptions. A vent plug was observed in ViUarrica volcano only after the 1972 eruption, when fractures were produced along the crater; also, secondary emanation centres became active (O. GONZALEZ-FERR~,N, personal communication, 1980). If the requirement of a loading time for Villarrica volcano is accepted, it should be related to the emptying of conduits and hypothetical secondary magma chambers; according to curve i of #(X), the eruption rate should become constant when those conduits and chambers are replenished. Following WICKMAN (1966b), curve 2 of ~ (X) indicates the possibility of magma cooling phenomena in the conduits and chambers or gas migration through the host rock, for long repose periods.

As shown in Figures 2 and 3, ¢(X) is more complex in the cases of the Llaima and Tupungatito volcanoes. Both cases are generally similar to a step-type O(X) curve. A large eruption rate of ~ = 0.2 per month is seen for long repose periods of Llaima. The eruption rate patterns of Llaima and Tupungatito may correspond to a Poisson process with several intermediate states. WICKMAN (1966b) ascribe Poisson vol- canoes with several states to the interaction between magma chambers at different depths. In this model a shallower secondary chamber is

840 Miguel Muhoz

40 80 120 160 N

T5

I0

7

5

3"

2~

I Ii I

3O0

200 [

Figure 2.

/ /

/ /

/

40 80 I20 160 x (month5)

Llaima volcano. For explanation see Fig. 1.

refilled from a primary one having an eruption rate 4; the secondary chamber is assumed to conform to a simple Poisson process with the rate v, where v > 4. This implies a lowering of the overall eruption rate during periods when the shallow chamber is empty. This model also

Eruption Patterns of Chilean Volcanoes 841

Figure 3.

N O 15

80 160 240 320 400

TUPUNGATI TO

13

;0

7

5

80 160 2t, O 320 aO0

x ( m o n t h s )

Tupungatito volcano. For explanation see Figure I .

requires a parameter/t that gives the probability of extinction of the secondary chamber within the next unit of time. The form of the ¢ ( X ) curve of Llaima and Tupungatito may indicate a more complex process than one corresponding to two magma chambers. A more detailed study of the fluctuations in the eruption rate is not worth pursuing, because the fluctuations are not statistically significant and because short and

842 Miguel Mufioz

imprecise records may have induced some of the patterns in the eruptive activity.

Even though the study of the eruption patterns of Villarrica, Llaima, Tupungatito is not conclusive, it may be reconsidered when detailed geological, geophysical, and magma composition data become available. Gravity, magnetism, and self-potential studies are now just starting at Villarrica and Llaima volcanoes. Only more detailed data can make possible a more accurate approach to the problem, as in MARTIN and ROSE (1981) and KLEI~ (1982).

Statistical analysis o f eruptions regarded as a series o f point events

General considerations and results of the statistical analysis of the eruption patterns of Villarrica, Llaima, and Tupungatito will be given here in concise form. The method is described by Cox and LEwis (1966) and LEWIS (1966). The approach follows the analysis of volcanological data performed by REYMENT (1966).

As before, X designates the duration of repose periods, Xg being the repose period following the eruption i. The time t at which the eruption i occurred is denoted by Ti:

0 < T , < T 2 < . . . < T n.

The total time of observation during which n eruptions are observed is denoted by T. The statistical analysis of eruptions regarded as a series of point events assumes the equality

i--I 7", = E xj , (2)

j=O

where X o denotes the time between the beginning of the observation series and the first eruption considered. Nevertheless, T; will always be somewhat greater than the figure indicated by equation (2), because an eruption cannot be regarded, in a strict sense, as a point event. The former should probably not introduce considerable deviations in the following analysis.

Trend analysis

The analysis for trend in the series studied is performed to investigate whether the density of volcanic activity is independent of arbitrary intervals considered in the observed series. If it is independent, the series can be handled as a stationary point process in the statistical sense (Cox and LEWIS, t966, pp. 59-60).

Instead of a constant eruption rate parameter in a Poisson process over the time interval available, it is assumed it has the functional form

Eruption Patterns of Chilean Volcanoes 843

¢(t) = exp (or + /30, a being essentially a nuisance parameter (LEWIS, 1966). Locally, near fl = 0, ¢(t) is equivalent to a linear trend. The test for fl = 0 against /? :~ 0 is based on the distribution of the random variable

1

ni=l 2 U = T(1/12 n) x/2 (3)

that tends rapidly to the standardized normal form as n increases. Thus the test procedure is to compute U given by equation (3), that is, the centroid of the observed times-to-events, T,., is compared with the mid-point of the period of observation (T/2). Positive values of U indicate that fl > 0, implying a trend towards an increasing rate of occurrence of eruptions. Physically, this should mean that more favorable tectonic conditions and of availability of magma for triggering eruptions are encountered in the volcanic system over the recent stages of its development. Because of complex eruption phenomena it is not possible to determine the nature of these conditions.

The results of the tests based on the variable U are shown in Table 1. If U is greater than 1.96, it will be an indication of trend at the 5% significance level.

There is no indication of trend (at least at the 5% level) for the three series studied. Llaima seems to be more free of trend than the other two volcanoes. The value U = 1.41 for Tupungatito may reproduce the fact of some mistakenly added eruptions--as noted previously--in the last section of the record.

Test for renewal processes The tests are concerned with the computation of quantities which

characterize the second-order, joint properties of the intervals between events (Cox and LEWIS, 1966, Chap. 5). These quantities are the serial correlation coefficients p~, which are the Fourier coefficients of the spectral density function f÷ (09):

f+ (09) = - I + 2 pj cos (J'O~) , 0 ~< co ~< zc. (4) ~rC j = l

Table 1

Tests for Trend

Volcanoes n U

Villarrica 20 1.44 Llaima 16 0.31 Tupungatito 14 1.41

844 Miguel Mufioz

If the correlation between the successive times between events is not found to be significant, it may be taken that the repose periods between successive eruptions are independent and identically distributed; in this case, there is agreement between the data and a renewal process model. If the distribution is one of the exponential type, the special case of a Poisson process is to be considered.

A renewal process will have a constant spectrum of intervalsf+(o2) = 1/z~ (Cox and LEwis, 1966, p. 99). The estimates ~j of the serial correlation coefficient pj in (4) are computed from the standard expression:

(Xl Xf+j) ( n - j y X~ Xg+ i (n-j) i : ' : -i:~ -

[ 1 ,~_:J l / " - i \~] '~2

I 1 2 i/2 Z x~+.i xi÷.i l=l (n _j)2 -i=l -

The tests of the hypothesis R/-- 0 , j = 1, 2 . . . . . is made by examining Pi (n --j)~/z, which has, approximately, a unit normal distribution if n > 100 and a renewal process is assumed. In the case studied, n is always much less than 100, so the usefulness of the serial correlations is limited. Owing to the short series of data, only the first eight serial correlation coefficients are computed. They are shown in Table 2.

There are two significant values ([~i (n _j)l/Z > 1.96), one for Llaima and one.for Tupungatito, which may be considered significant at the 5% level. This can occur just by random chance. Therefore it will be tentatively assumed that the three series studied correspond, roughly, to renewal processes.

Tests for Poisson processes

Tests for a Poisson process (a special kind of renewal process) are now performed. The first three sample moments of the intervals between eruptions and three related quantities are computed. The coefficient of variation, which has a value around unity for a Poisson process, has a particular significance in the computational results shown in Table 3.

Villarrica and Tupungatito have coefficients of variation around unity and therefore accord with what would be expected for a simple Poisson process. The sample moments and related quantities for Llaima indicate that eruptions cluster significantly between certain repose times, result- ing in a departure from a random, toward a more periodic, mode of eruption.

Tab

le 2

Est

imat

ed S

eria

l C

orre

lati

on C

oeff

icie

nts

r..

j 1

2 3

4 5

6 7

8

~i(

n-j

) 1'

2

~o

o

Vill

arri

ca

0.13

4 0.

019

-0.2

46

-0

.25

9

0.30

5 -0

.25

8

0,34

4 -0

.28

9

0.56

9 0.

078

-0.9

84

-1

.00

3

1.14

1 --

0.93

0 1.

192

--0.

959

Lla

ima

0.30

6 --

0.33

7 --

0.28

0 --

0.65

7*

--0.

429

0.54

8 0.

625

--0.

098

1.14

5 --

1.21

5 --

0.97

0 --

2.17

9 --

1.35

7 1.

644

1.76

8 --

0.25

9 T

up

un

gat

ito

0,

223

--0.

411

--0,

477

0.01

9 0.

770*

0.

366

--0.

514

--0.

690

0.77

2 --

1.36

3 --

1.50

8 0.

057

2.17

8 0.

968

--1.

259

--1.

543

O

~r

ga

o o

* V

alue

s si

gnif

ican

t at

at

leas

t th

e 5%

lev

el.

O0

846 Miguel Mufioz

Table 3

Sample Moments for the Intervals between Eruptions

Villarrica Llaima Tupungatito

Mean 104.32 101.47 138.46 Variance 10584.29 2843.88 17343.31 Standard deviation 102.88 53.33 131.69 Coefficient of variation 0.99 0.53 0.95 Measure of skewness 2.61 0.28 1.05

The next tests for Poisson processes are based on the conditional distribution of reposes (Cox and LEWIS, 1966, pp. 142--158). The quantities

7", yi=--~, i = 1 , 2 . . . . . n

are, under the null hypothesis, the order statistics from a random sample of size n from a population uniformly distributed over (0, 1). A test for the Poisson hypothesis based on testing the uniform distribution of the yi's is called a uniform conditional test. This is the canonical form of all distribution-free tests of goodness-of-fit, as the one-sided (D, + and D7,) and two-sided (D,) Kolmogorov-Smirnov statistics and the Anderson- Darling statistics (W2). These tests are possible even on small samples and do not require arbitrary groupings of the data, but are not consistent against certain stationary alternatives, and are most sensible to trend alternatives. Modified tests (Durbin's modification), however, give relatively powerful tests of the Poisson hypotheis (Cox and LEWIS, pp. 151-152; Lewis, 1966). The intervals between eruptions are denoted by X 1, X 2 . . . . . . X,, where n is the observed number of reposes, and the interval between the last observed eruption and the end of the observation period is X,+~ = T - T n. The (n + 1) quantities X; are ordered by magnitude to obtain the observed order statistics:

The quantities

o < x', ~ x [ ~<... ~ x ' . ~ x ' + ~.

x', x [ x~_, x" w i = - - + - - + . - . + - - + (n+ 2-- i ) ..... ,

T T T T

have the same distributional properties as the y~'s. The interval X ' ÷ ~ is not used in the computation.

Critical values related to the distribution-free tests for rejection of the Poisson hypothesis were taken from OWEY (1962). The values obtained for the tests, D +, D~, and W~,, and for the same employing the transformed quantities, equation (5), are shown in Table 4.

( i : 1 , . . . ,n) (5)

Eruption Patterns of Chilean Volcanoes

Table 4

Tests for Poisson Processes

847

Untransformed data Transformed data

Volcano n D +" D~ D,, W~ n D+~ D~ D, W~ Villarrica 20 0.19 0.24 0.24 1.33 19 0.14 0.14 0.14 0.81 Llaima 16 0.10 0.12 0.12 0.29 15 0.22 0.29 0.29 1.44 Tupungatito 14 0.22 0.29 0.29 1.50 13 0.25 0.17 0.25 1.04

None of the values in Table 4 indicates rejection of the Poisson hypothesis at a level less than 5% when both the transformed and untransformed data are considered. Nevertheless, the transformed data regarding Llaima give much higher values than the untransformed data, conforming to the former analysis of the eruption pattern of this volcano.

Another test for a Poisson hypothesis will now be performed. A characterization of the second-order point properties of the counting process N t is the spectrum thereof, designated g+(09) (Cox and LEwm, 1966, pp. 124-130). If m = n / T denotes the mean rate of occurrence of events, the normalized spectrum :rcg+(09)/m has the value 1 for all 09 for a Poisson process. The estimate I ( j ) of the normalized spectrum is computed (LEwis, 1966) as follows:

2 I ( j ) = - - {[A(j)] 2 + [B(j)] 2 }

n - 1

where

A(j) = ~-~ cos jB i=2

B(j) = ~ sin jB i=2

q being equal to (T , -- Tl)/(n -- 1). The computation is performed for j = 1, 2, .. . , P. Here P and B are input parameters. If B is taken to be 27ff(n -- 1), as is normally the case, then j = 09T/2r~. The input parameter P should be greater than n; in most cases, all the salient features of the spectrum will be shown by computations of I ( j ) for j up to P = 2n.

The estimate I ( j ) is smoothed by using a uniform weight scheme to obtain a consistent estimate of g+(09). The smoothed estimate of the normalized spectrum of the observations from Villarrica, Llaima, and Tupungatito are shown in Figures 4, 5, and 6, respectively.

The series are too short to obtain much resolution. The deviation of

848 Miguel Mufioz

the estimates from the expected values for a Poisson process have a surfeit of low or high values in the series for Llaima and Tupungatito, respectively. The expected value--which is around 2 for the three series--has been calculated from P/r, where P is the quantity defined above and r = T/~ is the total observation time with the mean interval as a unit (BARTLETT, 1963). The 5% significance bands are also shown (dashed lines in Figures 4, 5, and 6). The significance levels for individual values in the uniform weighting case are proportional to Z 2 quantities

tj

S Vi l larr ica

4

2

1

t I 1 I

10 20 30 ~0 j--> Figure 4. Villarrica volcano. Estimate of the normalized spectrum of the counting process. The value expected under the assumption that the series is a Poisson process and the 5% significance bands (dashed lines) are also shown.

Figure 5.

.

J 5

/,

3

2

I

Llaima

1 t i

10 20 30 40

Llaima volcano. Estimate of the normalized spectrum of the counting process. See also Figure 4.

Figure 6.

Eruption Patterns of Chilean Volcanoes 849

.

J 5

3

2

1

Tupungotito ] . . . . . . . . . . . . . . . . . .

I

I 1

10 20 30 j - - , , .

Tupungatito volcano. Estimate of the normalized spectrum of the counting process. See also Figure 4.

with 8 degrees of freedom. Only one estimate for Villarrica falls slightly below these bands, and is not statistically significant.

A random eruption model tofit the data

If the hypothesis of a renewal process corresponding to each one of the eruption patterns is accepted ab initio, without prescribing a certain confidence level for its rejection, we may ask at what probability P the Poisson model fits the data. This procedure tends to eliminate the difficulties of the previous analysis, where more restricting descriptions of the eruption patterns were successively investigated. P-values are given by the Kolmogorov-Smirnov (D~) and chi-squared (Z 2) tests of goodness-of-fit.

Values for the Kolmogorov-Smirnov test are taken from those corresponding to transformed data in Table 4. The chi-squared test compares the numbers of observed and expected reposes longer than X; the expected number of reposes is E = N O exp ( -X /~ ) , where N O is the total number of reposes and ~ is the mean repose time. The repose-time axis of Figures 1, 2, and 3 is divided into the maximum value k of intervals, such that the number of reposes in each interval expected from the Poisson distribution is equal to or greater than 5, which is the least number of expected events in each interval required to perform the chi- squared test. The well-known chi-squared test statistic is

k ( 0 i __ E.~2 z~= Y

i = 1 E i

850 Miguel Mufioz

Table 5

Goodness-of-Fit Results

Number of Po Px ~ Volcano reposes D (see footnote) X2 v (see footnote)

Villarrica 19 0.14 >0.4 0.31 1 0.59 Llaima 15 0.29 0.13 2.80 I 0.10 Tupungatito 13 0.25 0.3 - - - - - -

Po and Px, are the probabilities that the random model fits the data as follows from the Kolmogorov-Smirnov (D) and chi-squared (X r) test statistics.

where Oi is the number of reposes observed in an interval, Ei is the number of reposes expected from the random model, and the sum is over the k intervals. The number v of degrees of freedom is k - 1 minus the number of parameters of the expected distribution determined from the data (in this case, 1, the mean repose). Since v must be a positive integer, at least 3 data intervals and 15 reposes are required to perform the test.

The probability P that the Poisson model fits the data may be obtained from most handbooks of statistical tables (e.g., OWEN, 1962; WEAST, 1964). The application of these tests to volcanological data is extensively explained in the work by KLEIN (1982). Goodness-of-fit results are shown in Table 5. Tables for the Kolmogorov-Smirnov test are less complete, so the P-values obtained from this test are approximate.

P-values for ViUarrica and Tupungatito indicate a significant agreement between the data and the Poisson model. In the case of Llaima, P-values of around 0.1 obtained from both tests may imply only marginal agreement between data and model.

Conclusions

The investigation of the eruption rate # from the series of eruptions of Villarrica indicates the possibility of a constant rate describing its particular pattern. Complex patterns of the eruption rate are obtained for Llaima and Tupungatito; this possibly indicates eruption phenomena related to several intermediate states with an added loading-time characteristic. Nevertheless, in all the cases, the description is subjected to shortcomings owing to the length of the series available. Models based on the eruption rate function have to be accepted or rejected on the grounds of geologic, geophysical, and chemical studies. The generally complex eruption rate functions of the volcanoes studied may indicate some interesting geologic situations and significant interaction processes,

Eruption Patterns of Chilean Volcanoes 85 1

but a more detailed stochastic modelling is not possible in the case of Chilean volcanoes.

The statistical analysis shows rough conformity with the in- terpretations derived from the description of the eruption rate functions. There is no trend in the data for the three volcanoes studied at the 5% level adopted to decide the level of significant trend. Also, the estimated serial correlation coefficients are not significant enough to reject the hypothesis of a (random) renewal process. ViUarrica and Tupungatito have coefficients of variation in agreement with what would be expected for a Poisson process, and the tests based on distribution-free statistics corroborate that hypothesis. Durbin's modification applied to the data results in more significant values (closer to the rejection of the Poisson hypothesis) in the case of Llaima. The deviations from the expected value of the estimates of the normatized spectra indicate that a misfit between the data and the Poisson random model is not significant at the 5% level. Moreover, these estimates much depend on the weighting scheme used when the series are too short, as in this case.

The crude probability of the Poisson model's fitting the data is 0.59 for Villarrica, around 0.1 for Llaima, and 0.3 for Tupungatito. There is significant agreement between the data from the three volcanoes and the random Poisson model. Only the series regarding Llaima may imply a departure from the random model if the analysis throughout this work is considered.

REFERENCES

BARTLETT, M. S. (1963), The spectral analysis of point processes, J. Royal Statist. Soc. B 25, 264-296.

CASERTANO, L. (1963), Catalogue of the Active Volcanoes and Solfatara Fields of the Chilean Continent. Catalogue of the Active Volcanoes of the World including Solfatara Fields, International Volcanological Association, Part XV, 55 pp.

Cox, D. R., and LEWIS, P. A. W., The Statistical Analysis of Series of Events (Methuen, London 1966) 285 pp.

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(Received 15th February 1983, revised 13th June 1984, accepted 21st June 1984)