Chapter 9

The Contingency Table:A Powerful Tool ofMultivariate Statistics

Margaret Hellweg and Dieter Seidl

As in other areas of geophysics or meteorology, the observations and datacollected at volcanoes are the result of experiments in which we cannot con-trol the variables we wish to study. Thus, statistical analysis is an extremelyimportant step in the data processing. Variations in the experimental pa-rameters must be controlled through the choice of samples and through thehypotheses chosen for testing. The evaluation of the samples is only pos-sible through the application of the proper statistical methods, especiallymultivariate statistics.

Volcanic activity is the manifestation of complex dynamic processesand interactions within the volcano. It depends on the movement of flu-ids as well as on the thermodynamics of the magma and gases within abranched network of conduits and cavities. The dynamic processes gener-ate various geophysical signals, as well as visible phenomena at the volcano’ssurface. State-of-the-art techniques for monitoring at volcanoes now includecontinuous and concurrent recording of a variety of both quantitative andqualitative observations using a multiparameter station in the near-field ofthe crater. Such a station has been installed at Galeras volcano in Colombia,and has been operating for several years (Seidl et al. 2003). Presumably,the signals and phenomena observed at the surface of a volcano have acommon source in terms of the strong interactions between various internalprocesses. Thus, data from different measurements should show a signifi-

1

cant correlation. Contingency tables are a powerful statistical method forinvestigating such multi-dimensional correlations between quantitative andqualitative data. The pattern of signals and phenomena, as well as the cor-relations between them, can then be used as input data for revealing thephysical processes occurring within the volcano.

9.1 The Contingency Table

The contingency table is a method for analysing the statistical relationshipbetween two or more variables which may be numerical values, qualitativedescriptions, or a combination of the two (Taubenheim, 1969, Chatfield,1983). As with all statistical tests, the result cannnot be an absolute proof,but rather a test whether a so-called “null hypothesis” can be excluded tosome high probability. For the contingency table, we consider N observa-tions, each characterized by two discrete random variables x = [xi, i = 1, k]and y = [yj, j = 1, l]. If one of the variables is continuous, its range ofvalues must be divided into finite intervals determined by the variance ofthe data and its resolution. The number of observations with x and y eachtaking on a specific value, x = xi and y = yj is given by nij . The variablesxi , yj and nij are arranged in a matrix called a contingency table for theobserved data (Figure 9.1). The probabilities for xi and yj are pi and qj ,respectively.

For the null hypothesis “The variables x and y are statistically in-dependent”, the probability for the simultaneous observation of xi and yj

is given by the product piqj . For N observations the expected numberof observations with x = xi and y = yj, is then given by the product,mij = Npiqj.

Replacing the observed numbers, nij , in Figure 9.1 with the expectednumbers, mij , gives the corresponding contingency table for the expectednumber of observations.

If the probabilities pi and qj are unknown, they can be estimated fromthe data using the maximum-likelihood values

pi =1

N

l∑

j=1

nij (9.1)

qj =1

N

k∑

i=1

nij (9.2)

The expected number of observations, mij , can then be approximated usingthe estimated values

mij = Npiqj . (9.3)

2

Since the probabilities are normalized to 1, there are k + l − 2 independentvalues. Replacing the numbers mij in the expected contingency table withthe estimated values mij the observations can be tested against the expectedcontingency table determined by the null hypothesis using a chi-square test.The chi-square test statistic u is given by the formula:

u =k

∑

i=1

l∑

j=1

(nij − mij)2

mij

(9.4)

The test function u approximates a chi-square distribution, χ2(fdeg, α), withfdeg degrees of freedom for a significance level α. fdeg is determined as fdeg =[number of intervals minus 1] - [number of estimated parameters minus 1]:

fdeg = [k × l − 1] − [k + l − 2] = (k − 1) × (l − 1) (9.5)

Thus, the chi-square test is used to accept or reject the null hypothesis. Thetest statistic is significant at level 1 − α, where α is usually 0.05 or 0.01, if

u > χ2(fdeg, α). (9.6)

If the condition in equation (9.6) is met, the null hypothesis of statisticalindependence must be rejected. In this case the alternative hypothesis ofstatistical dependence is concluded to be true, accepting a probability errorof the order of magnitude α. If

u < χ2(fdeg, α), (9.7)

the test statistic is not significant at the level α. In this case the conclusionmust be that the null hypothesis is not inconsistent with the test statistic,and the statistical independence of the variables or parameters cannot beruled out.

A quantitative measure of statistical independence between the ob-served and the expected contingency table can be determined by Pearson’scontingency coefficient C:

C =1

γxγy

√

u

u + N(9.8)

If the variables are completely independent, C = 0, and if they are com-pletely dependent, C = 1. The two variables, γx and γy are factors whichnormalize C to the interval [0,1]. They are determined from the numberof intervals or classes into which the variables x and y have been dividedor classified. Values for γ are given in Table 9.1 (Brooks and Carruthers1953).

The construction and analysis of a 2 × 2 contingency table for twovariables consists of five steps:

3

1. Categorize the two variables into a 2 × 2 matrix of a finite numberof intervals or cells. The limits chosen for each cell depend on therequired resolution and stability as well as on the number of observa-tions. The cell “boundaries” may be numerical values for quantitativeparameters or descriptions for nonparametric observations.

2. Determine the observed cell frequencies and estimate the column androw probabilities using Equations 9.1 and 9.2.

3. Estimate the expected cell frequencies using Equation 9.3 by assumingthat the null hypothesis of statistical independence is true,

4. Perform a chi-square test to determine the statistical independence bycomparing the observed and expected cell frequencies at a significancelevel α using Equations 9.4, 9.6 and 9.7.

5. Determine the contingency coefficient C as a quantitative measure ofindependence using Equation 9.8.

For more than two variables the 2× 2 contingency table can be gener-alized to more dimensions.

9.2 Tornillos at Galeras volcano, Colombia

When, after years of quiescence, activity resumed at Galeras Volcano, Colom-bia, in 1989, the Observatorio Vulcanologico y Sismologico de Pasto (OVP)was founded to monitor the volcano, and a network of short period, verti-cal component seismometers with analog telemetry was installed to observethe seismicity (Gil-Cruz and Chouet 1997). An unusual tremor wavelet wasfound in the seismic records. These wavelets, christened tornillos from theSpanish word for screw, have sinusoidal waveforms and screw-like envelopeprofiles that can last for several minutes (Narvaez et al. 1997). In therecordings of the vertical seismometers of the short period network of OVP,the tornillo signals appear simple and their spectra have only one, at mosttwo, narrow peaks.

During the early 1990s, when tornillos were first observed, several asheruptions occurred at Galeras Volcano. Using recordings from the shortpe-riod network, Narvaez et al. (1997) determined basic signal parameters foreach tornillo, such as its duration and the frequency of the main spectralpeak. They also collected other observations such as the number of tornil-los per day, and attempted to relate changes in these parameters to theoccurrence of eruptions. Based on their observations, Narvaez et al. (1997)concluded that tornillos were precursors to these explosions.

What causes these strange looking signals? Most models for the sourcemechanism are based on the tornillos’ general signal characteristics, onenarrow spectral peak at some frequency f0 and the length of its long coda,

4

described by a quality factor Q. In physics, narrow spectral peaks are pro-duced by resonators and oscillators, so it is natural to assume that somesubsystem within the volcano is a resonator, and that its occasional excita-tion produces the seismic waves we measure as tornillos. Initially, Gil-Cruzand Chouet (1997) proposed that the tornillos at Galeras might be generatedby the resonance of a fluid-filled crack or dyke within the volcano. Theyrepresent the crack as a two-dimensional resonator, with the length andbreadth determining the frequency of the resonance. The size of the crackand its contents were adjusted to give the observed frequency and Q value.More recently, Kumagai and Chouet (1999) have further developed thesemodels, suggesting that changes in these two parameters indicate changes inthe content of the crack. In contrast, Julian (1994) proposed that non-linearfluid flow in a dyke could produce volcanic tremor under some conditionsand tornillo-like signals under others. Seidl and Hellweg (2003) describetornillos, some having up to 15 narrow peaks in their spectra, recorded atGaleras Volcano by broadband, three-component seismometers. They showhow narrowband vibrations can be generated by a Helmholz resonator, thatis a volume resonator with a fundamental resonant frequency, f0, relatedto the volume of the cavity but not its shape. Overtones would be non-harmonic and which ones are present in the spectra would depend on thelocation within the cavitiy of the source exciting the resonance. An alter-native would be some other physical system which could be described by anonlinear Van-der-Pol differential equation (Seidl and Hellweg 2003). Thesestudies all share a shortcoming. The parameters describing the model areadapted until they reflect and reproduce the parameters measured from thetornillos, f0 and Q, and the model is “accepted” as being correct.

Since 1996, a joint project between the Bundesanstalt furGeowissenschaften und Rohstoffe (BGR, Germany) and the Instituto Colom-biano de Geologıa y Minerıa (INGEOMINAS, Colombia) has supplementedthe shortperiod network of the OVP with a multiparameter (MP) monitor-ing station (Seidl et al. 2003). Most of the sensors for this station are inor near the crater (Figure 9.2). Goals for the MP station include reducingthe exposure of OVP personnel and other scientists studying the volcanoto danger, and improving the quality and information content of the data.The data from the station are digital, available continuously and in realtimeto improve monitoring capability. To encourage comparison between datafrom different instruments and joint analysis, the data from all sensors, ex-cept those of the gas system, have a single format and flow into a singleviewing and recording system. Taken together, the seismometers, the infra-sound and weather sensors, the gas chemistry sensors, the electrodes andmagnetometers, and the occasional thermographic images provide a broadview into the volcanic processes at Galeras, a basis for investigating the

5

physical and chemical processes which generate various tremor waveforms,and their significance in the sequence leading to ash eruptions.

The first element of the MP station, a three-component, broadbandseismometer was installed on the crater rim of Galeras in March 1996,at CR2 (Figure 9.2). Over the course of the following years, three fur-ther broadband, three-component seismometers were added, two along thecrater rim (ANG and ACH) and one as a reference station five kilometersfrom the summit (OBO). Initial attempts to monitor fumaroles with a gaschromatograph and a mass spectrometer proved to use too much energy.Finally, a system of small, portable gas sensors was developed and deployedon Galeras’ active cone (Faber et al. 2003). The sensors at and in Deformesfumarole (Figure 9.2) monitor pressures and temperatures in and near thefumarole, as well as various components of the emissions, such as CO2, SO2

and 222Rn. Pressure and temperature are also monitored at Chavas fuma-role on the west slope of the cone. The value at each sensor is sampled every6 s, digitized in the field and transmitted by radio to OVP, where the valuesreceive a time stamp and are recorded. The array of electromagnetic sensors(EM) are deployed within the crater close to the active cone (Figure 9.2).Two perpendicular, 100 m long sensor lines with non-polarizable electrodesat each end measure the electric field at the surface in the east-west andnorth-south directions, while a horizontal loop measures the time derivativeof the vertical magnetic field. At the same time a fluxgate magnetometermeasures the three components of the magnetic field with a sensitivity ofabout 0.1 nT (Seidl et al. 2003). These components of the MP station areaugmented by an infrasound sensor and a weather station, both located atCR2 (Seidl et al. 2003).

Figure 9.3 shows the recording of a tornillo from the broadband, three-component seismometer located at the station ANG, along with a spectro-gram from the E component. The noise with periods between 10 s and 20 sand particularly large amplitudes on the N component is most likely dueto the wind, while the smaller signals with periods of about 5 s on all threecomponents are ocean microseisms. For this recording of the tornillo atANG, the characteristic screw-like shape is most clearly apparent on therecording of the E component. In the digital, high dynamic range record-ings with the broadband seismometers such as this, the tornillos appear tobe more complex than in those from the short period network. They aremultichromatic having narrow spectral peaks at up to ten, or even more,not necessarily harmonic frequencies.

Following the installation of the MP station, tornillos only occurredsporadically. Finally, a swarm of tornillos occurred at Galeras Volcanobetween 08 December 1999 and 12 February 2000 (Seidl and Hellweg 2003).Most of these tornillos had a narrow spectral peak between 1.4 Hz and

6

2.5 Hz, which we consider to be the fundamental frequency, f0. In addition,nearly all had one, or in many cases more, narrow-band spectral peaks athigher frequencies which are higher modes. These tornillos were recordedwell at ANG and, after 05 January 2000, also at ACH. This swarm oftornillos offers an opportunity for statistical parameter analysis. One of thegoals of this analysis is to investigate differences in the tornillos, in termsof which spectral peaks are present and determine whether their presenceor absence is consistent with the hypothesis that the tornillos are producedby the resonance of a cavity in response to excitation by impulsive sourceslocated at different positions within it.

9.3 Contingency table case studies

We apply contingency table analysis to 35 tornillo events from this swarm.For each of these tornillos, the recorded signals are superpositions of a funda-mental mode having a frequency between 1.0 Hz and 2.5 Hz and one or, moreoften, more higher modes with frequencies up to 30 Hz. The starting pointfor this analysis using contingency tables is the observation that the fre-quencies of the higher modes which are present in two tornillos with nearlythe same fundamental mode frequency can be very different. This observa-tion is demonstrated in Figure 9.4, where seismograms, spectrograms andmoving-window amplitude spectra for three tornillos are displayed. Thesetornillos occurred on 16 December 1999, 08 January 2000 and 17 January2000. The fundamental frequencies, f0, for these three tornillos are 1.80 Hz,1.85 Hz, and 1.79 Hz, respectively. The persistent lines present at higherfrequencies can be attributed to different higher modes. For example, thetornillo on 16 December 1999 has only one strong spectral line higher thanthe fundamental. It is in the band of the first mode, f1 < 5 Hz, and hasa frequency of 3.70 Hz. The second tornillo shown, recorded on 08 Jan-uary 2000, has several higher mode peaks. One is in the band of the firstmode, at 4.61 Hz, and two spectral lines are at higher frequencies, namelyat 9.42 Hz and 10.71 Hz. The third tornillo example has no line in thefirst mode band, but does have spectral lines at 9.07 Hz and 10.07 Hz. Weare interested in the correlations between the higher frequency peaks andthe fundamental, because the way the excitation probabilities for differenthigher modes depend on the various bands of the fundamental frequencycan give clues which will help in modelling the tornillo source as well as theexcitation processes.

Table 9.2 is the contingency table for the simultaneous excitation ofthe fundamental mode and at least one higher mode in the band f1 < 5 Hzfor N = 35 tornillos. This is the band of the first higher mode. Basedon a histogram of the fundamental frequencies the frequency range of the

7

fundamental mode, f0 = [1.4, 2.5] Hz, has been divided into three bands.These bands form the three rows of the table. The two columns in the leftpart of the table represent a “Yes” and a “No” column. For each bandfor the fundamental, the count in the “Yes” column gives the number oftornillos with at least one spectral line in the f1-band, and the count in the“No” column gives the number of tornillos without such a line. The rightpart ot the table contains the corresponding expected tornillo counts forthe simultaneous occurrence of the fundamental and a line in the first modeband. These counts have been estimated based on the null hypothesis, thatthe excitations of the fundamental and the first higher mode are statisticallyindependent. This hypothesis is tested by applying a chi-square test for allobserved - expected cell pairs for two levels of significance, α = 0.01 andα = 0.05 using equation (9.4). The number of degrees of freedom fdeg = 2,according to equation (9.5). The estimated chi-square value is u = 9.4. Forthe significance level α = 0.05, u > χ2

0.05 = 5.99. At this level the nullhypothesis can be rejected and a considerable correlation can be assumedto exist between the simultaneous excitation of one of the fundamentals anda first higher mode. This result agrees with the value of the contingencycoefficient C = 0.67. For the level α = 0.01 the estimated value u = 9.4is close to χ2

0.01 = 9.21. If u ≫ χ2

0.01, we would clearly reject the nullhypothesis, and accept the assumption that the probability of simultaneousexcitation in these two bands is significant. However, u is only an estimateand we cannot be sure of its standard deviation. In this case more datacould be useful for accepting or rejecting the null hypothesis at the level0.01.

The chi-square test for Table 9.2 has been performed for all cells of thetable. It is a characteristic of contingency tables, compared to the usualtotal measures of ensemble statistics, that the influence of the various cellson the final result of the chi-square test can be estimated. In Table 9.2, thefirst and the second row show a low joint-probability of excitation for thefundamental and the first higher mode, in contrast to the third band forwhich the two modes are clearly connected.

Table 9.3 shows the contingency table for the simultaneous excitationof the fundamental mode and at least one mode in the band 5 Hz < f2 < 10.We assign this band to the second higher mode. For the significance levelα = 0.01, u = 1.7 ≪ χ2

0.01 = 9.21. Therefore, the null hypothesis cannot berejected by the test, suggesting that the independence between the excita-tion of the fundamental and of the second higher mode is significant. Thisresult agrees with the low value of the contingency coefficient C = 0.31.

The main result of these two contingency tables, Table 9.2 and Ta-ble 9.3, is the inferrence of excitation coupling between the fundamentaland the first two higher modes. This trend continues for increasing mode

8

numbers. Figure 9.5 shows the contingency table for the fundamental modeand all higher modes. In each cell, the counts of the observed (upper left)and the expected (lower right) number of spectral peaks are given. In themajority of the cells, both numbers are close together indicating an inherentindependence between the various cells. The number of degrees of freedomis fdeg = 9. The estimated chi-square has the value u = 16.0 For the signif-icance level α = 0.01, u = 16.0 < χ2

0.01 = 21.7, and the null hypothesis cannot be rejected. In this case, the hypothesis cannot even be rejected at theweaker significance level α = 0.05, where u = 16.0 < χ2

0.05 = 16.92. Thus,significant independence between the higher modes in indicated. This resultagrees with the low value of the contingency coefficient C = 0.40.

9.4 Concluding Remarks

The contingency table is a very effective tool of multivariate statistics. Itis a method which can be applied when it is necessary to test statisticalrelationships between variables. The method is therefore particularly suit-able for the analysis of multiparameter data recorded at active volcanoes byan array of various geophysical and geochemical sensors, such as the mul-tiparameter station operating at Galeras Volcano in Colombia (Seidl et al.

2003). This modern concept of volcano monitoring requires the correlationof nonparametric descriptions of the visible activity with the quantitativemultiparameter data.

The particularly useful characteristics of analysis using contingency

tables are: 1) the ability to simultaneously analyse correlations betweenvariables, 2) the possibility of generalizing from the most common, and rel-atively simple 2-by-2 contingency table to larger and more complex cases,and 3) the way that the use of contingency tables reveals incremental corre-lations between the individual cells of the table, and how these correlationscontribute to the total measures of ensemble statistics.

We have demonstrated this method for a swarm of seismic tornillosignals recorded at Galeras Volcano. These signals are superpositions ofdifferent oscillating modes with narrow-band spectral peaks in characteristicfrequency bands. The particular statistical problem to which contingencytables has been applied is the estimation of the probabilities for mode-coupling effects as they depend on the order of the mode. The analysisshows that the joint probabilities for mode-coupling decrease with increasingmode number. This result confirms that source model assuming tornillos tobe eigenvibrations of a fluid cavity resonator excited by internal randomlydistributed pressure pulses is consistent with the observed frequencies (Seidland Hellweg 2003).

The application of contingency tables is particularly suitable to cases

9

where both qualitative and quantitative data must be analysed, and tocases where the correlation of qualitatively different quantitative parame-ters must be explored. This is true for the analysis of data from a systemof instruments such as the multiparameter station at Galeras (Seidl et al.

2003). Recently, the level of activity at Galeras has increased (Gomez et

al. 2004). Strong changes in the concentrations and temperature of gasesat the fumaroles preceded an increase in the seismicity and a series of ex-plosions and ash emissions. We intend to apply contingency table analysisto investigate the relationships between the various measured parametersin the time interval leading up to the explosions.

Further Reading

Both Chatfield (1983) and Taubenheim (1969) describe the use of contingencytables for statistical analysis.

Acknowledgements

The Multiparameter Station at Galeras volcano is a cooperative project of the

Federal Institute of Geosciences and Natural Resources (BGR) in Hannover (Ger-

many) and the Instituto Colombiano de Geologıa y Minerıa (INGEOMINAS) in

Bogota (Colombia). Financial assistance for travel and transport expenses are

provided by the Deutsches Zentrum fur Luft- und Raumfahrt (DLR). 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 Vul-

canologico y Sismologico de Pasto. MH received support for travel to the work-

shop on “Statistics in Volcanology” from the Environmental Mathematics and

Statistics Programme funded jointly by NERC/EPSRC, UK; and from the U.C.

Berkeley - Los Alamos National Laboratory collaborative Institute for Geophysics

and Planetary Physics Project number 04-1407.

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Bibliography

[1] Brooks, C.E.W. & Carruthers, N. 1953. Handbook of statistical methods

in meteorology, Her Majesty’s Stationary Office, London.

[2] Chatfield, C. 1983. Statistics for technology. Chapman and Hall, Lon-don.

[3] Faber, E., Moran, C., Poggenburg, J., Garzon, G. & Teschner, M.2003. Continuous gas monitoring at Galeras Volcano,Colombia: Firstevidence. Journal of Volcanology and Geothermal Research, 125, 13-23.

[4] Gil-Cruz, F. & Chouet, B. 1997. Long-period events, the most charac-teristic seismicity accompanying the emplacement and extrusion of alava dome in Galeras Volcano, Colombia, in 1991. Journal of Volcanol-

ogy and Geothermal Research, 77, 121-158.

[5] Gomez, D., Hellweg, M., Buttkus, B., Bker, F., Calvache, M.L., Cortes,G. , Faber, E., Gil Cruz, F., Greinwald, S. , Laverde, C. , Narvaez, L., Ortega, A., Rademacher, H., Sandmann, G., Seidl, D., Silva, B. &Torres, R. 2004. A volcano reawakens: Multiparameter observationsof activity transition at Galeras Volcano (Colombia), 2004 Fall Meet-ing, American Geophysical Union, San Francisco, CA, December 13-17,2004.

[6] Julian, B.R. 1994. Volcanic tremor: nonlinear excitation by fluid flow.Journal of Geophysical Research, 99(B6), 11859-11877.

[7] Kumagai, H. & Chouet, B. A., 1999. The complex frequencies of long-period seismic events as probes of fluid composition beneath volcanoes.Geophysical Journal International, 138, F7-F12.

[8] Narvaez, M.L., Torres, R.A., Gomez, D.M., Corts, G.P.J., Cepeda,H.V. & Stix, J. 1997. Tornillo-type seismic signals at Galeras vol-cano, Colombia, 1992-1993. Journal of Volcanology and Geothermal

Research, 77, 159-171.

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[9] Seidl, D., Hellweg, M., Calvache, M., Gomez, D., Ortega, A., Tor-res, R., Bker, F., Buttkus, B., Faber E. & Greinwald, S. 2003. Themultiparameter-station at Galeras Volcano (Colombia): Concept andrealization. Journal of Volcanology and Geothermal Research, 125, 1-12.

[10] Seidl, D. & Hellweg, M. 2003. Parametrization of multichromatictornillo signals observed at Galeras Volcano (Colombia). Journal of

Volcanology and Geothermal Research, 125, 171-189.

[11] Taubenheim, J. 1969. Statistische Auswertung Geophysikalischer und

Meteorologischer Daten. Akademische Verlagsgesellschaft Geest & Por-tig K.-G., Leipzig.

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k,l 2 3 4 5 6 8 10 12 15 ∞

γ1, γ2 .798 .859 .915 .943 .959 .976 .985 .989 .993 1.0

Table 9.1: Contingency coefficients

ObservedFirst Higher Mode Band

f1 ≤ 5.0 Hz Sum ExpectedFundamentalMode Bands Yes No Yes No1.4 ≤ f0 < 1.8 Hz 9 2 11 9 21.8 ≤ f0 < 1.9 Hz 13 0 13 10 31.9 ≤ f0 < 2.5 Hz 6 5 11 9 2Sum 28 7 N = 35

u = 9.4χ2(α = 0.01, fdeg = 2) = 9.21χ2(α = 0.05, fdeg = 2) = 5.99C = 0.67

Table 9.2: First mode

ObservedFirst Higher Mode Band

5.0 > f2 ≤ 10 Hz Sum ExpectedFundamentalMode Bands Yes No Yes No1.4 ≤ f0 < 1.8 Hz 7 4 11 7 41.8 ≤ f0 < 1.9 Hz 6 7 13 8 51.9 ≤ f0 < 2.5 Hz 8 3 11 7 4Sum 21 14 N = 35

u = 1.7χ2(α = 0.01, fdeg = 2) = 9.21χ2(α = 0.05, fdeg = 2) = 5.99C = 0.31

Table 9.3: Second mode

13

klkkk

l

l

l

nnnx

nnnx

nnnx

yyy

...

.....

.....

.....

...

...

...

21

222212

112111

21

Figure 9.1: Scheme of a 2-by-2 Contingency Table for incremental intervalsof the variables x and y. The cell counts nij are the observed or expectedjoint frequencies for simultaneous observations of xi and yj, respectively.

ACH

ANG

CR2

SummitRoad

Deformes

fumarole

activecone

EM

craterrim

PoliceStation

GAS

Telenariño

N

0 300m

Telenariño

Police station

Figure 9.2: The multiparameter system at Galeras Volcano, Colombia. (a)The view of Galeras from the city of Pasto showing the locations of thePolice station and Telenarino on the crater rim. (b) Map of the craterregion showing the locations of the seismic stations, ACH, ANG and CR2;the electromagnetic sensors (EM); and the fumarole gas sensors (GAS). Theweather station and infrasound sensor are located at CR2.

14

Time

Gro

un

dV

elo

cit

y(μ

m/s

)

ANG 05 Jan 2002 01:50 UTC

-6-4-20246

-6-4-20246

01:58:20 01:59:00 01:59:40

-6-4-20246

Z

N

E

02:00:20

Time

Fre

qu

en

cy [

Hz]

0

5

10

15

20

25

30

35

40ANG-E 05 Jan 2002 01:50 UTC

01:58:20 01:59:00 01:59:40 02:00:20

(a)

(b)

Figure 9.3: A typical tornillo. (a) Three-component velocity records. (b)Spectrogram of the east component.

15

-0.20

0.20.4

-0.20

0.20.4

5 10 15 20 25 30

-0.20

0.20.4

Time(s)

ANG1999 12161410

Z

0 10 200

5

10

15

20

25

0 10 20

E

0 10 20

N

5

10

15

20

25

Fre

qu

en

cy

(H

z)

Dis

pla

cem

en

t (µ

)

Interval(s)

2

0

2

2

0

5 10 15 20 25 30-2

0

Time(s)

ANG2000 01080500

Z

0 10 200

5

10

15

20

25N

0 10 20

E

0 10 20

5

10

15

20

25

0.2

0

0.2

0.2

0

5 10 15 20 25 30-0.2

0

Time(s)

ANG2000 01170025

Z

0 10 200

5

10

15

20

25N

0 10 20

E

0 10 20

5

10

15

20

25

6 10

Time(s) Time(s) Time(s)

Interval(s) Interval(s)

11987506 10 1198750 6 10 1198750

Fre

qu

en

cy

(H

z)

(a)

(b)

(c)

Figure 9.4: Three examples of tornillos from the December 1999 - February2000 swarm. (a) Three-component displacement waveforms for the first 25 sof the tornillos which occurred on 16 December 1999 at 14:10, on 08 Jan-uary 2000 at 05:00 and on 17 January 2000 at 00:25 (all times UTC). Theinstrument response has been removed and the data have been bandpassfiltered between 1 Hz and 40 Hz. (b) Spectrograms for each of the threecomponents for each of the tornillos. (c): Moving window spectra for theeast component for each of the tornillos to help discriminate between tran-sient spectral peaks at the start of the tornillo and the long-lasting spectralpeaks from which the tornillo (screw) takes its name. For each trace, the30 s time window has been divided into two at the time given on the hor-izontal axis. The black trace shows the spectrum for the waveform to theright of the dividing time, and the grey trace the spectrum for the waveformto its left. For the first trace, the dividing time is 0 s, and there is no traceto the left and therefore no grey spectrum.

16

30

8 5 7 11

4 9 9 9 15

2 5 11 10

5 11 11 11 10

3 5 6 10

3 7 7 7 5

4 12 13 5 Hig

he

r M

od

e B

an

ds (

Hz)

5 10 10 10 2

1.4 1.7 1.8 1.9 2.5

Fundamental Mode Bands (Hz) u = 16.0

x2(α = .01, fdeg = 9) = 21.67

x2(α = .05, fdeg = 9) = 16.92

c = .40

Figure 9.5: The contingency table for the fundamental mode and all highermodes.

17