Characterization of Strombolian events by using independent component analysis

Nonlinear Processes in Geophysics (2004) 11: 453–461SRef-ID: 1607-7946/npg/2004-11-453 Nonlinear Processes

in Geophysics© European Geosciences Union 2004

Characterization of Strombolian events by using independentcomponent analysis

A. Ciaramella1, E. De Lauro2, S. De Martino2, 3, 4, B. Di Lieto2, M. Falanga2, 3, 4, and R. Tagliaferri1

1Dipartimento di Matematica ed Informatica, Salerno University, via S. Allende, I-84081 Baronissi (SA), Italy2Dipartimento di Fisica, Salerno University, via S. Allende, I-84081 Baronissi (SA), Italy3INFM, unita di Salerno, via S. Allende, I-84081 Baronissi (SA), Italy4INFN, gruppo collegato di Salerno, via S. Allende, I-84081 Baronissi (SA), Italy

Received: 16 June 2004 – Revised: 13 October 2004 – Accepted: 14 October 2004 – Published: 21 October 2004

Part of Special Issue “Nonlinear analysis of multivariate geoscientific data – advanced methods, theory and application”

Abstract. We apply Independent Component Analysis(ICA) to seismic signals recorded at Stromboli volcano.Firstly, we show how ICA works considering synthetic sig-nals, which are generated by dynamical systems. We provethat Strombolian signals, both tremor and explosions, in thehigh frequency band (>0.5 Hz), are similar in time domain.This seems to give some insights to the organ pipe modelgeneration for the source of these events. Moreover, we areable to recognize in the tremor signals a low frequency com-ponent (<0.5 Hz), with a well defined peak corresponding to30 s.

1 Introduction

The classical procedure to construct physical models is sim-ple and well established. Firstly, rough but explanatory mo-dels are inferred from phenomenology. At this preliminarystage, many possible intuitive pictures live together. Theyare used to reproduce, on laboratory scale, all the relevantobservational aspects. The experiments are, in general, re-producible and some physical laws can be established. Theseseminal models are improved and discriminated accordingto their capability to provide new phenomena to test expe-rimentally. There are, however, physical systems for whichit is impossible to proceed in this way. In fact, even simplephenomenological aspects cannot be reproduced on labora-tory scale, both for practical and conceptual reasons. Thesesystems are characterized by strong nonlinear behaviour thatinvolves many time and spatial length scales.

Geophysical systems belong to this class. In this and simi-lar cases, people speak about observational data rather thanexperimental data. Sequences, signals, messages, texts, con-figurations are examples of observational data. Among them,

Correspondence to:M. Falanga([email protected])

an important role is played by the observations (measure-ments) made in the course of the time, i.e. scalar time series.We can associate, in a natural way, the concepts of complexi-ty, statistics, ergodicity to these series in order to distinguishthem, quantitatively. All the sequences can be described, ina formal way, by the unifying concept of Dynamical System(DS). Many theorems and mathematical devices have beenelaborated to study DSs. However, they give powerful me-thods to study asymptotic properties. In the real experimentalcases, we start from finite sequences, sometimes very long,and we want to obtain the model. The problem is to extractrelevant properties from one or more of the available scalarfinite sequences. Then, numerical analysis arises to under-stand all important parts of information included within se-ries. A review of many numerical methods of nonlinear sig-nal processing developed in the recent years can be found inAbarbanel(1996).

In many cases, sequences can contain information relativeto different DSs or sources or signals: therefore a preliminarystep is to recognize independent components. The methodsbased on information theory are natural to use in this researchfield. The concept of entropy plays a particular role: it is verygeneral and gives a powerful methodology to distinguish thecomplexities associated to different DSs.

Independent Component Analysis (ICA) is an entropybased technique, useful to separate mixtures of signals (formore details seeHyvarinen et al.(2001) and many paperstherein cited). ICA was introduced in the early 1980s in aneurophysiological setting (Herault, 1984). The introductionof FastICA algorithm (Hyvarinen and Oja, 1997) contributedto the application of ICA to large scale problems due to itscomputational efficiency.

In the next section, we shall give the mathematical set-ting of ICA. Here, we consider the problem to understandthe behaviour of ICA method with respect to scalar time se-ries generated by geophysical systems. Since they can be

454 A. Ciaramella et al.: Strombolian events and ICA

interpreted as time evolution of a suitable dynamical system,we start clarifying the performance of ICA when it is appliedto known DSs. We select different DSs equations representa-tive of large classes, namely linear, nonlinear in the regime oflimit cycle, and stochastic DSs, and we use ICA to make sep-aration of the generated synthetic signals. After this section,we show the application of ICA to observational recordedtime series of Stromboli volcano.

2 Independent component analysis

ICA is a method to find underlying factors or componentsfrom multivariate (multidimensional) statistical data.

It is a well defined method in speech context, to solve theclassical problem of cocktail party. Imagine to have somepeople speaking in a room and some microphones recordingtheir voice. The goal of ICA is to extract, from the mixturesof voices, step by step, each independent speaker.

Let us explain in brief the mathematical setting on whichICA is based. We can suppose to havem different recordedtime seriesx, that we hypothesize to be the linear superpo-sition of n mutually independent unknown sourcess, due todifferent mixing, represented by a constant unknownm× nmatrix A. This mixing is essentially due to path, noise,instrumental transfer-functions, etc. The hypothesis is tohave instantaneous linear mixtures of some independent dy-namical systems. If the mixing has to be linear, nothing isassumed with respect to the sources, which can be linear ornonlinear.

Formally, the mixing model is written as

x = As+ µ. (1)

The termµ takes into account the presence of an additivenoise, often omitted in Eq. (1), because it can be incorporatedin the sum as one of the source signals. In addition to thesource independence request, ICA assumes that the numberof available different mixturesm is at least as large as thenumber of sourcesn. Under these hypothesis, the ICA goalis to obtain a separating matrixW = (w1, ..., wm)T , inverseof A, in such a way that the vector

y = Wx (2)

is an estimatey ' s of the original independent source sig-nals. We have to remark that the hypothesis of instantaneousmixtures means that we are able to extract “modes” havingthe same velocity. As a consequence, no delay among therecordings is allowed. In presence of delay, as we will see inthe Sect.4, it is required to align the recorded signals.

We stress that ICA is a statistical model. In other words,from the central limit theorem, we know that the distribu-tion of a sum of independent random variables tends towardsa gaussian distribution, under certain conditions. The mainidea of the ICA model is to maximize the non-Gaussianity(super-Gaussianity or sub-Gaussianity) to extract the inde-pendent components. The separation is achieved on the basis

of their statistical independence, evaluated by using 4th orderstatistical properties.

In the following, we shall use the fixed-point algorithm,namely FastICA (Hyvarinen and Oja, 1997). The FastICAlearning rule finds a direction, i.e. a unit vectorw such thatthe projectionwT x maximizes independence of the single es-timated sourcey. Independence is here measured by the ap-proximation of the negentropy, that is a measure of nonGaus-sianity, given by

JG(w) = [EG(wT x) − EG(ν)]2, (3)

wherew is anm-dimensional (weight) vector,x representsour mixture of signals,E· indicates the expectation value,E(wT x)2

= 1 means variance set to 1,G is a suitable ap-proximating contrast function,ν is a standardized Gaussianrandom variable.

Maximizing JG allows to find “one” independent compo-nent, or projection pursuit direction. The algorithm requiresa preliminary whitening of the data: the observed variablexis linearly transformed to a new variablev with zero-meanand unit variance. Whitening can always be accomplishedby e.g. Principal Component Analysis (PCA). This methodinvolves a mathematical procedure that transforms a numberof (possibly) correlated variables into a (smaller) number ofuncorrelated variables called principal components. The firstprincipal component accounts for as much of the variabilityin the data as possible, and each succeeding component ac-counts for as much of the remaining variability as possible.Traditionally, principal component analysis is performed oncovariance matrix (Bishop, 1995).

The one-unit “fixed-point” algorithm for finding a rowvectorw is

wi∗

= E[vg(wTi v)] − E[g′(wT

i v)]wi

wi = w∗

i /‖w∗

i ‖, (4)

where g(·) is a suitable nonlinear function, in our caseg(y) = tanh(y), andg′(y) is its derivative with respect toy. This nonlinear function is introduced to analyse nonlineartime series by means of high-order statistics. The cited al-gorithm can be generalized to estimate several independentcomponents, which have to be orthonormal one to each other.

ICA is considered a generative model. This means that itdescribes how the observed data are generated by a process ofmixing the components. The basic ICA model, summarizing,requires the following assumptions:

– all the independent components, with the possible ex-ception of one component, must be non-Gaussian (Belland Sejnowski, 1995; Karhunen, 1996);

– the number of the observed linear mixtures must be atleast as large as the number of independent components;

– the matrixA must be of full column rank.

A. Ciaramella et al.: Strombolian events and ICA 455A. Ciaramella et al.: Strombolian events and ICA 3

We have used the basic model of ICA. There are, how-ever, many generalizations born along the time. Several ap-proaches have been proposed to apply ICA when the numberof mixtures is less than the number of sources (Roweis, 2000;Jang and Leen, 2003). Other approaches are introduced toaccomplish separation of convolved signals (signals with de-lays and echoes). Moreover, another interesting extension ofthe ICA could be used in the case of nonlinear mixing (Pa-junen and Karhunen, 1997; Hyvarinen and Pajunen, 1999).In the next section, to show the effectiveness of basic ICAfor linear mixing, we consider the ICA application to someclasses of DSs.

3 Dynamical Systems

DSs can be considered a general representation of large classof sequences. Then, it is very interesting and explicative asa first step to apply ICA to DSs. We consider, in particular,three types of DSs: linear, nonlinear in the regime of limitcycle, and stochastic systems, taking into account both DSswith few and infinite degrees of freedom. The linear casemakes clear how ICA works. Limit cycle and stochastic sys-tems have characteristic behaviour that can be found in manygeophysical systems. We have restricted our analysis to lin-ear/nonlinear systems which are linearly superposed. We ex-clude nonlinear superpositions from our analysis. They haveto be treated using a different methodology (Hyvarinen andPajunen, 1999). In fact, we have not considered nonlinearsystems developing chaotic behaviour as for example Rosslerattractor, in which the z-component is not well separated be-cause of the nonlinear contribution of the xz term (Ciaramellaet al., 2002).

Generally, DSs are described by a first order equation sys-tem:

dx(t)

dt= F(x(t)) (5)

where F(x(t)) is a linear, nonlinear or stochastic field.In particular for stochastic systems, we consider diffusive

stochastic processes described by the following equation:

dx = [v(x, t)]dt + εdW (6)

where v(x, t) is a field (drift) and dW is a Wiener process.

3.1 Linear dynamical systems

The linear systems that we analyse are single and coupledharmonic oscillators. It is enough to consider two coupledoscillators, because the behaviour of a system with many os-cillators is completely equivalent to the previous one. Wehave made many different experiments on mixtures gener-ated by these systems. We illustrate:

– the separation of an harmonic oscillator and an additiveGaussian noise;

– the separation of coupled oscillators, in beating regime,one harmonic oscillator and a Gaussian noise.

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Fig. 1. Separation of a mixture of an harmonic oscillator (0.01Hz

frequency) and Gaussian noise (SNR is −20db); time (in numberof points) is on x-axis, dimensionless amplitude is on y-axis: a)original sources; b) mixed signals; c) separated signals.

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Fig. 2. Separation of the mixture of two coupled harmonic oscilla-tors, one harmonic oscillator and Gaussian noise: a) source signals;b) mixed signals; c)separated signals;

We can observe in (Fig.1) how ICA is able to extract theharmonic oscillator from Gaussian noise with a very lowsignal to noise ratio (SNR = −20db). Similar results areachieved even considering different noise.

In Fig.2 we report the results related to the second exper-iment. We easily recognize as extracted signals the standardnormal modes, a single harmonic oscillator and the additivenoise, though the low SNR.

3.2 Nonlinear dynamical systems

We consider a particular nonlinear oscillator, i.e. the An-dronov oscillator (Andronov et al., 1966). This oscillator isa nonlinear system that generates, with suitable parameters,a limit cycle which is approached asymptotically by all otherphase paths. The limit cycle is dynamically stable. It is rep-

Fig. 1. Separation of a mixture of an harmonic oscillator (0.01 Hzfrequency) and Gaussian noise (SNR is−20 db); time (in numberof points) is on x-axis, dimensionless amplitude is on y-axis:(a)original sources;(b) mixed signals;(c) separated signals.

Moreover, ICA contains the following ambiguities or in-determinacies:

– we cannot determine the variances (proportional toenergies) of the independent components;

– we cannot determine the order of the independent com-ponents.


3 Dynamical systems

DSs can be considered a general representation of large classof sequences. Then, it is very interesting and explicative asa first step to apply ICA to DSs. We consider, in particular,three types of DSs: linear, nonlinear in the regime of limitcycle, and stochastic systems, taking into account both DSswith few and infinite degrees of freedom. The linear casemakes clear how ICA works. Limit cycle and stochastic sys-tems have characteristic behaviour that can be found in manygeophysical systems. We have restricted our analysis to li-near/nonlinear systems which are linearly superposed. Weexclude nonlinear superpositions from our analysis. Theyhave to be treated using a different methodology (Hyvarinen

A. Ciaramella et al.: Strombolian events and ICA 3


3 Dynamical Systems

DSs can be considered a general representation of large classof sequences. Then, it is very interesting and explicative asa first step to apply ICA to DSs. We consider, in particular,three types of DSs: linear, nonlinear in the regime of limitcycle, and stochastic systems, taking into account both DSswith few and infinite degrees of freedom. The linear casemakes clear how ICA works. Limit cycle and stochastic sys-tems have characteristic behaviour that can be found in manygeophysical systems. We have restricted our analysis to lin-ear/nonlinear systems which are linearly superposed. We ex-clude nonlinear superpositions from our analysis. They haveto be treated using a different methodology (Hyvarinen andPajunen, 1999). In fact, we have not considered nonlinearsystems developing chaotic behaviour as for example Rosslerattractor, in which the z-component is not well separated be-cause of the nonlinear contribution of the xz term (Ciaramellaet al., 2002).

Generally, DSs are described by a first order equation sys-tem:

dx(t)

dt= F(x(t)) (5)

where F(x(t)) is a linear, nonlinear or stochastic field.In particular for stochastic systems, we consider diffusive


dx = [v(x, t)]dt + εdW (6)

where v(x, t) is a field (drift) and dW is a Wiener process.


The linear systems that we analyse are single and coupledharmonic oscillators. It is enough to consider two coupledoscillators, because the behaviour of a system with many os-cillators is completely equivalent to the previous one. Wehave made many different experiments on mixtures gener-ated by these systems. We illustrate:



a)

0 1000 2000 3000 4000 5000−10

−5

0

5

10


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plitu

de

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500

1500

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plitu

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1000


c)

0 1000 2000 3000 4000 5000−2

−1

0

1

2


Am

plitu

de

0 1000 2000 3000 4000 5000−4

−2

0

2

4

Fig. 1. Separation of a mixture of an harmonic oscillator (0.01Hz

frequency) and Gaussian noise (SNR is −20db); time (in numberof points) is on x-axis, dimensionless amplitude is on y-axis: a)original sources; b) mixed signals; c) separated signals.

a)

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plitu

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plitu

de

c)

Fig. 2. Separation of the mixture of two coupled harmonic oscilla-tors, one harmonic oscillator and Gaussian noise: a) source signals;b) mixed signals; c)separated signals;

We can observe in (Fig.1) how ICA is able to extract theharmonic oscillator from Gaussian noise with a very lowsignal to noise ratio (SNR = −20db). Similar results areachieved even considering different noise.

In Fig.2 we report the results related to the second exper-iment. We easily recognize as extracted signals the standardnormal modes, a single harmonic oscillator and the additivenoise, though the low SNR.


We consider a particular nonlinear oscillator, i.e. the An-dronov oscillator (Andronov et al., 1966). This oscillator isa nonlinear system that generates, with suitable parameters,a limit cycle which is approached asymptotically by all otherphase paths. The limit cycle is dynamically stable. It is rep-

Fig. 2. Separation of the mixture of two coupled harmonic oscilla-tors, one harmonic oscillator and Gaussian noise:(a)source signals;(b) mixed signals;(c) separated signals.

and Pajunen, 1999). In fact, we have not considered non-linear systems developing chaotic behaviour as for exampleRossler attractor, in which the z-component is not well se-parated because of the nonlinear contribution of the xz term(Ciaramella et al., 20021).

Generally, DSs are described by a first order equation sy-stem:

dx(t)

dt= F(x(t)), (5)

whereF(x(t)) is a linear, nonlinear or stochastic field.In particular for stochastic systems, we consider diffusive


dx = [v(x, t)]dt + εdW, (6)

wherev(x, t) is a field (drift),dW is a Wiener process andεis the diffusion coefficient.


The linear systems that we analyse are single and coupledharmonic oscillators. It is enough to consider two coupledoscillators, because the behaviour of a system with manyoscillators is completely equivalent to the previous one. Wehave made many different experiments on mixtures genera-ted by these systems. We illustrate:



1Ciaramella, A., De Lauro, E., De Martino, S., Falanga, M., andTagliaferri, R.: Independent Component Analysis and DynamicalSystems, Salerno University, J. Mach. Learn. Res., preprint sub-mitted, 2002.

456 A. Ciaramella et al.: Strombolian events and ICA4 A. Ciaramella et al.: Strombolian events and ICA

resentative of many nonlinear systems with feedback. As wesee in the following, this behaviour can be associated to someinteresting natural systems. The equations of the Andronovoscillator with natural frequency ω0, in the suitable form, are:

x + 2h1x + ω2

0x = 0 if x < b (7)

x− 2h2x + ω2

0x = 0 if x > b. (8)

The nonlinearity is contained in a suitable threshold b rulingthe self-coupling. We fix b = −1; h1 and h2 are respectivelydissipative and constructive parameters.By using different parameters and threshold, the Andronovoscillator has different behaviours. In fact, if the thresholdis negative, then we can have a limit cycle or forcing oscil-lations, while, if the threshold is positive, no limit cycle isobtained. In the other cases, for the other parameters, wehave:

– 0 < h1 < h2 < 1: it does not have a limit cycle for anythreshold;

– b = −1 and 0 < h2 < h1 < 1: it does have a limitcycle for any initial conditions;

– b = −1 e 0 < h2 h1 > 1: it does have a limit cycle forany initial conditions.

We show a representative example to illustrate the inter-esting properties of ICA when it is applied to self-oscillatingnonlinear systems. In this experiment, we consider two lin-early coupled Andronov oscillators with frequency respec-tively of 0.93Hz and 1.1Hz, one Andronov oscillator withfrequency of 0.9Hz and Gaussian noise. The SNR in thiscase is −20 db. Applying the ICA we obtain four separatedsignals (Fig.3).

In conclusion linearly coupled Andronov oscillators arewell separated by ICA both among them and from superim-posed noise. This experiment shows the power of ICA that isable, as in the linear case regarding the normal modes, to ex-tract the independent limit cycles in time domain. We stressthat it is not trivial because the nonlinear differential equa-tions cannot be solved and FFT, due to the nonlinearity ofthe problem, looses its efficacy. If we lower SNR under −20db the separation of noise is not so optimal as in linear case.

3.3 Stochastic systems

The third class of systems are stochastic diffusive processes.The archetype of models is represented by a simple symmet-ric bistable potential (double-well) driven by both an addi-tive random noise, i.e. white and Gaussian, and an exter-nal periodic bias. Given these features, the response of thesystem undergoes resonance-like behaviour as a function ofthe noise level; hence the name stochastic resonance (Gam-maitoni et al., 1998). Formally, if we consider the Langevinequation with a small periodic forcing, we obtain:

dx = [x(a− x2) + Acos(Ωt)]dt + εdW (9)

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Fig. 3. Separation of the mixtures of two linearly coupled Andronovoscillators, one Andronov oscillator and Gaussian noise: a) sourcesignals; b) mixed signals; c)separated signals.

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Fig. 4. Power Spectrum Density: a) a generic story of stochasticprocess described by eq. 9; b) z(t) described by eq. 10.

where dW is a Wiener process, i.e. a Gaussian process withzero mean and unitary variance; x is a dimensionless physi-cal variable; a is a parameter related to the double-well po-tential, A is the amplitude, Ω the angular frequency of theexternal periodical forcing, and ε is the diffusion coefficient.

This system is compared with another that displays a sim-ilar frequency content. The latter, denoted as z(t), is de-scribed by the following equations:

x = −Ω2x

dy = νdW (10)z(t) = Ax(t) + By(t)

where the first equation is a simple harmonic oscillator withangular frequency Ω; the second is a Wiener process; thethird is the superposition of the two, according to the coeffi-cient A, B. If we choose Ω equal to the angular frequency ofthe stochastic process, in the regime of resonance, we obtainthat z(t) has a similar frequency content as x(t) (see Fig.4).

As one can see in Fig.5, this case is very impressive,namely the separation is optimal. ICA separates low-dimensional and high-dimensional systems, i.e. harmonicoscillator and both stochastic resonance and Gaussian noise,

Fig. 3. Separation of the mixtures of two linearly coupled Andronovoscillators, one Andronov oscillator and Gaussian noise:(a) sourcesignals;(b) mixed signals;(c) separated signals.

We can observe in (Fig.1) how ICA is able to extractthe harmonic oscillator from Gaussian noise with a very lowsignal to noise ratio (SNR= −20 db). Similar results areachieved even considering different noise.

In Fig. 2, we report the results related to the second expe-riment. We easily recognize as extracted signals the standardnormal modes, a single harmonic oscillator and the additivenoise, though the low SNR.


We consider a particular nonlinear oscillator, i.e. the An-dronov oscillator (Andronov et al., 1966). This oscillator isa nonlinear system that generates, with suitable parameters,a limit cycle which is approached asymptotically by all otherphase paths. The limit cycle is dynamically stable. It is re-presentative of many nonlinear systems with feedback. Aswe see in the following, this behaviour can be associated tosome interesting natural systems. The equations of the An-dronov oscillator with natural frequencyω0, in the suitableform, are:

x + 2h1x + ω20x = 0 if x < b

x − 2h2x + ω20x = 0 if x > b. (7)

The nonlinearity is contained in a suitable thresholdb rulingthe self-coupling. We fixb = −1; h1 andh2 are respectivelydissipative and constructive parameters.

By using different parameters and threshold, the Andronovoscillator has different behaviours. In fact, if the thresholdis negative, then we can have a limit cycle or forcing oscil-lations, while, if the threshold is positive, no limit cycle isobtained. In the other cases, for the other parameters, wehave:



resentative of many nonlinear systems with feedback. As wesee in the following, this behaviour can be associated to someinteresting natural systems. The equations of the Andronovoscillator with natural frequency ω0, in the suitable form, are:

x + 2h1x + ω2

0x = 0 if x < b (7)

x− 2h2x + ω2

0x = 0 if x > b. (8)

The nonlinearity is contained in a suitable threshold b rulingthe self-coupling. We fix b = −1; h1 and h2 are respectivelydissipative and constructive parameters.By using different parameters and threshold, the Andronovoscillator has different behaviours. In fact, if the thresholdis negative, then we can have a limit cycle or forcing oscil-lations, while, if the threshold is positive, no limit cycle isobtained. In the other cases, for the other parameters, wehave:


– b = −1 and 0 < h2 < h1 < 1: it does have a limitcycle for any initial conditions;

– b = −1 e 0 < h2 h1 > 1: it does have a limit cycle forany initial conditions.

We show a representative example to illustrate the inter-esting properties of ICA when it is applied to self-oscillatingnonlinear systems. In this experiment, we consider two lin-early coupled Andronov oscillators with frequency respec-tively of 0.93Hz and 1.1Hz, one Andronov oscillator withfrequency of 0.9Hz and Gaussian noise. The SNR in thiscase is −20 db. Applying the ICA we obtain four separatedsignals (Fig.3).

In conclusion linearly coupled Andronov oscillators arewell separated by ICA both among them and from superim-posed noise. This experiment shows the power of ICA that isable, as in the linear case regarding the normal modes, to ex-tract the independent limit cycles in time domain. We stressthat it is not trivial because the nonlinear differential equa-tions cannot be solved and FFT, due to the nonlinearity ofthe problem, looses its efficacy. If we lower SNR under −20db the separation of noise is not so optimal as in linear case.


The third class of systems are stochastic diffusive processes.The archetype of models is represented by a simple symmet-ric bistable potential (double-well) driven by both an addi-tive random noise, i.e. white and Gaussian, and an exter-nal periodic bias. Given these features, the response of thesystem undergoes resonance-like behaviour as a function ofthe noise level; hence the name stochastic resonance (Gam-maitoni et al., 1998). Formally, if we consider the Langevinequation with a small periodic forcing, we obtain:

dx = [x(a− x2) + Acos(Ωt)]dt + εdW (9)

a)

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100

0 2000 4000 6000 8000 10000−200

0

200

0 2000 4000 6000 8000 10000−500

0

500


Am

plitu

de

b)

0 2000 4000 6000 8000 10000−2

0

2

0 2000 4000 6000 8000 10000−2

0

2

0 2000 4000 6000 8000 10000−2

0

2

0 2000 4000 6000 8000 10000−5

0

5


Am

plitu

de

c)

Fig. 3. Separation of the mixtures of two linearly coupled Andronovoscillators, one Andronov oscillator and Gaussian noise: a) sourcesignals; b) mixed signals; c)separated signals.

10−3

10−2

10−1

100

−60

−40

−20

0

20

40

60

Frequency (Hz)

Pow

er S

pect

rum

Mag

nitu

de (d

B)

10−3

10−2

10−1

100

−60

−40

−20

0

20

40

60

Frequency (Hz)

Fig. 4. Power Spectrum Density: a) a generic story of stochasticprocess described by eq. 9; b) z(t) described by eq. 10.

where dW is a Wiener process, i.e. a Gaussian process withzero mean and unitary variance; x is a dimensionless physi-cal variable; a is a parameter related to the double-well po-tential, A is the amplitude, Ω the angular frequency of theexternal periodical forcing, and ε is the diffusion coefficient.

This system is compared with another that displays a sim-ilar frequency content. The latter, denoted as z(t), is de-scribed by the following equations:

x = −Ω2x

dy = νdW (10)z(t) = Ax(t) + By(t)

where the first equation is a simple harmonic oscillator withangular frequency Ω; the second is a Wiener process; thethird is the superposition of the two, according to the coeffi-cient A, B. If we choose Ω equal to the angular frequency ofthe stochastic process, in the regime of resonance, we obtainthat z(t) has a similar frequency content as x(t) (see Fig.4).

As one can see in Fig.5, this case is very impressive,namely the separation is optimal. ICA separates low-dimensional and high-dimensional systems, i.e. harmonicoscillator and both stochastic resonance and Gaussian noise,

Fig. 4. Power Spectrum Density:(a) a generic story of stochasticprocess described by Eq. (8); (b) z(t) described by Eq. (9).

– b = −1 and 0< h2 < h1 < 1: it does have a limitcycle for any initial conditions;

– b = −1 e 0< h2 h1 > 1: it does have a limit cycle forany initial conditions.

We show a representative example to illustrate the inte-resting properties of ICA when it is applied to self-oscillatingnonlinear systems. In this experiment, we consider two li-nearly coupled Andronov oscillators with frequency respec-tively of 0.93Hz and 1.1 Hz, one Andronov oscillator withfrequency of 0.9 Hz and Gaussian noise. The SNR in thiscase is−20 db. Applying the ICA we obtain four separatedsignals (Fig.3).

In conclusion linearly coupled Andronov oscillators arewell separated by ICA both among them and from superim-posed noise. This experiment shows the power of ICA that isable, as in the linear case regarding the normal modes, to ex-tract the independent limit cycles in time domain. We stressthat it is not trivial because the nonlinear differential equa-tions cannot be solved and FFT, due to the nonlinearity ofthe problem, looses its efficacy. If we lower SNR under−20db the separation of noise is not so optimal as in linear case.


The third class of systems are stochastic diffusive processes.The archetype of models is represented by a simple symme-tric bistable potential (double-well) driven by both an ad-ditive random noise, i.e. white and Gaussian, and an exter-nal periodic bias. Given these features, the response of thesystem undergoes resonance-like behaviour as a function ofthe noise level; hence the name stochastic resonance (Gam-maitoni et al., 1998). Formally, if we consider the Langevinequation with a small periodic forcing, we obtain:

dx = [x(a − x2) + Acos(t)]dt + εdW, (8)

wheredW is a Wiener process, i.e. a Gaussian process withzero mean and unitary variance;x is a dimensionless physi-cal variable;a is a parameter related to the double-well po-tential, A is the amplitude, the angular frequency of theexternal periodical forcing, andε is the diffusion coefficient.

This system is compared with another that displays a si-milar frequency content. The latter, denoted asz(t), is de-scribed by the following equations:


0 1 2 3 4 5

x 104

−10

0

10

0 1 2 3 4 5

x 104

−2

0

2

0 1 2 3 4 5

x 104

−5

0

5


Am

plitu

de

a)

0 1 2 3 4 5

x 104

−10

0

10

0 1 2 3 4 5

x 104

−10

0

10

0 1 2 3 4 5

x 104

−10

0

10


Am

plitu

de

b)

0 1 2 3 4 5

x 104

−2

0

2

0 500 1000 1500 2000−2

0

2

0 1 2 3 4 5

x 104

−5

0

5


Am

plitu

de

c)

Fig. 5. Separation of the mixture of linear oscillator with frequencyof 0.003Hz with an additive Gaussian noise and stochastic reso-nance with the same resonance frequency: a) source signals; b)mixed signals; c) extracted components.

also in presence of a very similar frequency content. Sameresults are achieved using different kinds of noise (e.g. uni-form noise). Obviously, ICA is not able to recognize thenumber of degrees of freedom and if we consider real sys-tems for which we have not ”a priori” knowledge, we mustadd to ICA other independent methods (De Martino et al.,2002c).

We should note that linear methods based on FFT fail,because they do not distinguish the systems underlying ourmixtures, describing the observed spectra as due to the sameDS. In that framework, a stochastic resonance is not at alldifferent from a simple oscillator with noise. The ICA welldistinguishes the case in which the harmonic oscillator is lin-early superposed to noise from the case in which the FFTpeak is generated by a stochastic behaviour, i.e. stochasticresonance.

Now we can draw our conclusions about the explicativesynthetic examples. We have applied ICA to some dynami-cal systems; firstly to linear and nonlinear systems with fewdegrees of freedom, and then to infinite degrees of freedomsystems, namely diffusion processes in the regime of stochas-tic resonance.Regarding the linear systems, we have obtained optimal sep-aration from very high superposed noise. Furthermore, theICA acts as Fast Fourier Transform but in time domain sinceit gives us the normal modes of the system.The performance of ICA is valuable also in the case of non-linear systems where we separate coupled Andronov oscil-lators, not trivial from the point of view of dynamical sys-tems. Also in these experiments the separation from noise iswell made. The experiments with stochastic resonance arevery impressive: the superposed periodic and stochastic sig-nals are completely separated, i.e. ICA perfectly recognizesthe different superposed dynamical systems also when theFourier Transform is irresolute (insensitive). As a conclusive

0 100 200 300 400 500 600 700 800 900−4

−3

−2

−1

0x 10

4

0 100 200 300 400 500 600 700 800 900

−5000

0

5000

Time (s)

Ampli

tude

(cou

nts)

Fig. 6. In the upper figure, 15 minutes of the vertical component ofbroadband event is plotted (amplitudes in adimensional unit). It wasrecorded at T1 station on September 19

th at 22:00, during the ex-periment performed in 1997, where 21 three-component broadbandseismometers were deployed on the flanks of the volcano, groupedinto three rings at different levels: Top, Middle, Bottom stations. Inthe lower figure, the same event is filtered in the band 0.02−0.5Hz,in order to make evident the waveform of the explosions superposedto background tremor.

remark we can say that ICA is a method to apply a priori asa pre-analysis to scalar experimental series, namely it allowsto recognize if the scalar series contain one or more indepen-dent signals.

4 Application of ICA to Strombolian events

We study the seismic signals recorded at Stromboli volcano.This volcano is characterized by basaltic eruptions. In theseeruptions, the relative motion of gas with respect to the fluidproduces either an annular flow (Hawaian Fire Fountains)or a Slug flow (Strombolian explosions). In fact, the typi-cal seismic signature of Stromboli is the continuous volcanictremor (continuous vibration of the ground around the vol-cano) due to degassing to which repeating explosion quakesare superposed (see Fig.6).Tremor and explosions have a very similar frequency con-tent. In fact, both the behaviours are generated by complexprocesses of magma flow and turbulent degassing.

Despite many studies (e.g. Chouet et al., 1997, 1999,2003), the dynamics underlying the generation of these be-haviours is not yet well understood.

In our studies, we have applied ICA to Strombolian eventsconsidering both short-period (0.5 − 50Hz) and broadband(0.02 − 50Hz) recorded seismograms. Strombolian signals,due to their stationarity, are suitable to apply ICA. The aimis to decompose, if it is possible, recorded series into statis-tically independent components. In this way, we get infor-mation about the ”modes” involved in the full dynamics andconstrain source geometry and mechanism. The analysis willbe carried on explosions and tremor, separately.In order to avoid any delay among recording stations locatedin different places, the seismic traces are aligned using the

Fig. 5. Separation of the mixture of linear oscillator with frequencyof 0.003 Hz with an additive Gaussian noise and stochastic reso-nance with the same resonance frequency:(a) source signals;(b)mixed signals;(c) extracted components.

x = −2x

dy = εdW (9)

z(t) = Ax(t) + By(t),

where the first equation is a simple harmonic oscillator withangular frequency; the second is a Wiener process; thethird is the superposition of the two, according to the coeffi-cientA, B. If we choose equal to the angular frequency ofthe stochastic process, in the regime of resonance, we obtainthatz(t) has a similar frequency content asx(t) (see Fig.4).

As one can see in Fig.5, this case is very impres-sive, namely the separation is optimal. ICA separates low-dimensional and high-dimensional systems, i.e. harmonic os-cillator and both stochastic resonance and Gaussian noise,also in presence of a very similar frequency content. Sameresults are achieved using different kinds of noise (e.g. uni-form noise). Obviously, ICA is not able to recognize thenumber of degrees of freedom and if we consider real sy-stems for which we have not “a priori” knowledge, we mustadd to ICA other independent methods (De Martino et al.,2002c).

We should note that linear methods based on FFT fail,because they do not distinguish the systems underlying ourmixtures, describing the observed spectra as due to the sameDS. In that framework, a stochastic resonance is not at alldifferent from a simple oscillator with noise. The ICA welldistinguishes the case in which the harmonic oscillator is li-nearly superposed to noise from the case in which the FFTpeak is generated by a stochastic behaviour, i.e. stochasticresonance.

Now we can draw our conclusions about the explicativesynthetic examples. We have applied ICA to some dynami-cal systems; firstly to linear and nonlinear systems with fewdegrees of freedom, and then to infinite degrees of freedom


0 1 2 3 4 5

x 104

−10

0

10

0 1 2 3 4 5

x 104

−2

0

2

0 1 2 3 4 5

x 104

−5

0

5


Am

plitu

de

a)

0 1 2 3 4 5

x 104

−10

0

10

0 1 2 3 4 5

x 104

−10

0

10

0 1 2 3 4 5

x 104

−10

0

10


Am

plitu

de

b)

0 1 2 3 4 5

x 104

−2

0

2

0 500 1000 1500 2000−2

0

2

0 1 2 3 4 5

x 104

−5

0

5


Am

plitu

de

c)

Fig. 5. Separation of the mixture of linear oscillator with frequencyof 0.003Hz with an additive Gaussian noise and stochastic reso-nance with the same resonance frequency: a) source signals; b)mixed signals; c) extracted components.

also in presence of a very similar frequency content. Sameresults are achieved using different kinds of noise (e.g. uni-form noise). Obviously, ICA is not able to recognize thenumber of degrees of freedom and if we consider real sys-tems for which we have not ”a priori” knowledge, we mustadd to ICA other independent methods (De Martino et al.,2002c).

We should note that linear methods based on FFT fail,because they do not distinguish the systems underlying ourmixtures, describing the observed spectra as due to the sameDS. In that framework, a stochastic resonance is not at alldifferent from a simple oscillator with noise. The ICA welldistinguishes the case in which the harmonic oscillator is lin-early superposed to noise from the case in which the FFTpeak is generated by a stochastic behaviour, i.e. stochasticresonance.

Now we can draw our conclusions about the explicativesynthetic examples. We have applied ICA to some dynami-cal systems; firstly to linear and nonlinear systems with fewdegrees of freedom, and then to infinite degrees of freedomsystems, namely diffusion processes in the regime of stochas-tic resonance.Regarding the linear systems, we have obtained optimal sep-aration from very high superposed noise. Furthermore, theICA acts as Fast Fourier Transform but in time domain sinceit gives us the normal modes of the system.The performance of ICA is valuable also in the case of non-linear systems where we separate coupled Andronov oscil-lators, not trivial from the point of view of dynamical sys-tems. Also in these experiments the separation from noise iswell made. The experiments with stochastic resonance arevery impressive: the superposed periodic and stochastic sig-nals are completely separated, i.e. ICA perfectly recognizesthe different superposed dynamical systems also when theFourier Transform is irresolute (insensitive). As a conclusive

0 100 200 300 400 500 600 700 800 900−4

−3

−2

−1

0x 10

4

0 100 200 300 400 500 600 700 800 900

−5000

0

5000

Time (s)

Ampli

tude

(cou

nts)

Fig. 6. In the upper figure, 15 minutes of the vertical component ofbroadband event is plotted (amplitudes in adimensional unit). It wasrecorded at T1 station on September 19

th at 22:00, during the ex-periment performed in 1997, where 21 three-component broadbandseismometers were deployed on the flanks of the volcano, groupedinto three rings at different levels: Top, Middle, Bottom stations. Inthe lower figure, the same event is filtered in the band 0.02−0.5Hz,in order to make evident the waveform of the explosions superposedto background tremor.

remark we can say that ICA is a method to apply a priori asa pre-analysis to scalar experimental series, namely it allowsto recognize if the scalar series contain one or more indepen-dent signals.


We study the seismic signals recorded at Stromboli volcano.This volcano is characterized by basaltic eruptions. In theseeruptions, the relative motion of gas with respect to the fluidproduces either an annular flow (Hawaian Fire Fountains)or a Slug flow (Strombolian explosions). In fact, the typi-cal seismic signature of Stromboli is the continuous volcanictremor (continuous vibration of the ground around the vol-cano) due to degassing to which repeating explosion quakesare superposed (see Fig.6).Tremor and explosions have a very similar frequency con-tent. In fact, both the behaviours are generated by complexprocesses of magma flow and turbulent degassing.

Despite many studies (e.g. Chouet et al., 1997, 1999,2003), the dynamics underlying the generation of these be-haviours is not yet well understood.

In our studies, we have applied ICA to Strombolian eventsconsidering both short-period (0.5 − 50Hz) and broadband(0.02 − 50Hz) recorded seismograms. Strombolian signals,due to their stationarity, are suitable to apply ICA. The aimis to decompose, if it is possible, recorded series into statis-tically independent components. In this way, we get infor-mation about the ”modes” involved in the full dynamics andconstrain source geometry and mechanism. The analysis willbe carried on explosions and tremor, separately.In order to avoid any delay among recording stations locatedin different places, the seismic traces are aligned using the

Fig. 6. In the upper figure, 15 minutes of the vertical componentof broadband event is plotted (amplitudes in adimensional unit). Itwas recorded at T1 station on 19 September at 22:00, during the ex-periment performed in 1997, where 21 three-component broadbandseismometers were deployed on the flanks of the volcano, groupedinto three rings at different levels: Top, Middle, Bottom stations. Inthe lower figure, the same event is filtered in the band 0.02−0.5 Hz,in order to make evident the waveform of the explosions superposedto background tremor.

systems, namely diffusion processes in the regime of stochas-tic resonance.

Regarding the linear systems, we have obtained optimalseparation from very high superposed noise. Furthermore,the ICA acts as Fast Fourier Transform but in time domainsince it gives us the normal modes of the system.

The performance of ICA is valuable also in the case ofnonlinear systems where we separate coupled Andronov os-cillators, not trivial from the point of view of dynamical sy-stems. Also in these experiments, the separation from noiseis well made. The experiments with stochastic resonance arevery impressive: the superposed periodic and stochastic sig-nals are completely separated, i.e. ICA perfectly recognizesthe different superposed dynamical systems also when theFourier Transform is irresolute (insensitive). As a conclusiveremark, we can say that ICA is a method to apply “a pri-ori” as a pre-analysis to scalar experimental series, namely itallows to recognize if the scalar series contain one or moreindependent signals.


We study the seismic signals recorded at Stromboli volcano.This volcano is characterized by basaltic eruptions. In theseeruptions, the relative motion of gas with respect to the fluidproduces either an annular flow (Hawaian Fire Fountains)or a Slug flow (Strombolian explosions). In fact, the typi-cal seismic signature of Stromboli is the continuous volcanictremor (continuous vibration of the ground around the vol-cano) due to degassing to which repeating explosion quakesare superposed (see Fig.6).

458 A. Ciaramella et al.: Strombolian events and ICA6 A. Ciaramella et al.: Strombolian events and ICA

cross-correlation function. This satisfies the ICA request ofinstantaneous mixing.

To get other information, we have also adopted differ-ent techniques. They consist in techniques generally usedin nonlinear signal processing, and well-known or innova-tive methods to investigate seismological signals. In partic-ular, we have applied parametric and non parametric spec-tral analysis; nonlinear denoising techniques (Kostelich andSchreiber, 1993); particle motion and polarization filtering(Kanasewich, 1981); methods to reconstruct phase spacestarting from scalar time series (estimate of the dimension(Grassberger and Procaccia, 1983), Average Mutual Infor-mation (Fraser and Swinney, 1986), False Nearest Neighbors(Kennel et al., 1992)); trajectory space analysis to estimatethe variety of dynamical systems presents in the data (Pal-adin and Vulpiani, 1987).

As regards explosions at high frequency, we report the re-sults obtained decomposing signals recorded by using short-period seismometers (Chouet et al., 1998). ICA has been ap-plied to explosions recorded by seismometers along the threeorthonormal directions of motion, i.e. radial, transverse, ver-tical with respect to the crater area.

We display in Fig.7 the results of the radial direction; theother directions show a similar behaviour (Acernese et al.,2003). As one can see (Fig.7), the wavefield is the linearsuperposition in time domain of three independent compo-nents, characterized by well defined and separate frequencybands (respectively 0.8− 1.2, 2.4− 3.0, 3.2− 4.5Hz).

The first two bands present wavefield mainly composedof body waves with radial polarization, pointing towards thecrater area. In the last band, the very low SNR, together withthe corresponding short wavelengths, does not allow to indi-viduate a defined direction (Acernese et al., 2004).

Similar results are achieved analysing broadband explo-sions. In addition, in this case, we extract also a componentcorresponding to the VLP signal (Falanga, 2003) as alreadyobserved by Chouet et al. (2003).

The reconstruction of the phase space establishes that ex-plosions are associated to a low-dimensional dynamical sys-tem characterized by dimensions in the range [2 − 3] (DeMartino et al., 2002a).

Then, trajectory space analysis, performed on broadbandsignals, states that explosions are generated by an unique dy-namical system (De Martino et al., 2004), though Chouetet al. (2003) have found two distinct kinds of explosions,which have been associated to the two distinct vents atStromboli in 1997. The differences between the two typesof events are related more to slight variations in conduit ge-ometries rather than differences in the dynamics generationof the phenomena.

We have also analysed, as already said, the tremor. In thiscase, it is convenient to consider separately two frequencybands (> 0.5Hz and < 0.5Hz). Namely, the low frequencyband can contain waves travelling with different velocitywith respect to the high frequency wavefield. In Fig.8, as youcan see, tremor shows ICA extracted components similar toexplosion quakes, in waveform and frequency content. Of

0 2 4 60

2

4

6

PSD

0 2 4 60

1

2

Frequency (Hz)

0 2 4 60

20

40

60

0 1000 2000 3000−1

0

1

Ampli

tude

0 1000 2000 3000−0.5

0

0.5


0 1000 2000 3000−2

0

2

Fig. 7. Explosions in the band greater than 0.5Hz: Denoised inde-pendent components of radial direction of motion and their spectra(sampling frequency equal to 125Hz; amplitude in adimensionalunit.

0 1 2 3 4 5 6 7 80

20

40

60

80

20 40 60 80

−1

0

1

0 1 2 3 4 5 6 7 80

20

40

PDS

20 40 60 80

−1

0

1

Ampli

tude

(a.u

.)

0 1 2 3 4 5 6 7 80

20

40

Frequency(Hz)20 40 60 80

−1

0

1

Time(s)

Fig. 8. Denoised independent components of tremor, by ICA, in therange > 0.5Hz related to the vertical direction and their spectra;amplitude in adimensional unit.

course, regarding tremor, the individuation of clear bands ismore difficult due to very low SNR. Polarization analysis ontremor has already been performed on short-period recordedsignals Chouet et al. (1997). We are extending the analysis tobroadband signals as extracted by ICA. This will be matterof a forthcoming paper.

All the results persuade us to think that, regarding highfrequency content, the superficial source is stationary andnot destructive. The wavefield is generated by the excita-tion of only feww degrees of freedom of the complex fluid-dynamical source system.

Possible models of the production of these oscillationshave been postulated by Julian (1994), Ida (1996) and Jameset al. (2004). In particular, Julian (1994) suggested an or-gan pipe model, with a constant rate supply of fluids inside acylinder conduit for a variety of almost periodic signals ob-served on volcanoes.

The independent component analysis of organ pipe acous-tic emission by Bottiglieri et al. (2004) seems to support thismodel. Namely, in an organ pipe, a constant rate supplyof pressure produces self-sustained sounds and ICA is able

Fig. 7. Explosions in the band greater than 0.5 Hz: Denoised inde-pendent components of radial direction of motion and their spectra(sampling frequency equal to 125 Hz; amplitude in adimensionalunit).

Tremor and explosions have a very similar frequency con-tent. In fact, both the behaviours are generated by complexprocesses of magma flow and turbulent degassing.

Despite many studies (e.g.Chouet et al., 1997, 1999,2003), the dynamics underlying the generation of these be-haviours is not yet well understood.

In our studies, we have applied ICA to Strombolian eventsconsidering both short-period (0.5−50 Hz) and broadband(0.02−50 Hz) recorded seismograms. Strombolian signals,due to their stationarity, are suitable to apply ICA. The aimis to decompose, if it is possible, recorded series into stati-stically independent components. In this way, we get infor-mation about the “modes” involved in the full dynamics andconstrain source geometry and mechanism. The analysis willbe carried on explosions and tremor, separately.

In order to avoid any delay among recording stations lo-cated in different places, the seismic traces are aligned usingthe cross-correlation function. This satisfies the ICA requestof instantaneous mixing.

To get other information, we have also adopted differenttechniques. They consist in techniques generally used innonlinear signal processing, and well-known or innovativemethods to investigate seismological signals. In particu-lar, we have applied parametric and non parametric spec-tral analysis; nonlinear denoising techniques (Kostelich andSchreiber, 1993); particle motion and polarization filtering(Kanasewich, 1981); methods to reconstruct phase spacestarting from scalar time series (estimate of the dimension(Grassberger and Procaccia, 1983), Average Mutual Infor-mation (Fraser and Swinney, 1986), False Nearest Neighbors(Kennel et al., 1992)); trajectory space analysis to estimatethe variety of dynamical systems presents in the data (Pal-adin and Vulpiani, 1987).

As regards explosions at high frequency, we report the re-sults obtained decomposing signals recorded by using short-period seismometers (Chouet et al., 1998). ICA has been ap-plied to explosions recorded by seismometers along the three


cross-correlation function. This satisfies the ICA request ofinstantaneous mixing.

To get other information, we have also adopted differ-ent techniques. They consist in techniques generally usedin nonlinear signal processing, and well-known or innova-tive methods to investigate seismological signals. In partic-ular, we have applied parametric and non parametric spec-tral analysis; nonlinear denoising techniques (Kostelich andSchreiber, 1993); particle motion and polarization filtering(Kanasewich, 1981); methods to reconstruct phase spacestarting from scalar time series (estimate of the dimension(Grassberger and Procaccia, 1983), Average Mutual Infor-mation (Fraser and Swinney, 1986), False Nearest Neighbors(Kennel et al., 1992)); trajectory space analysis to estimatethe variety of dynamical systems presents in the data (Pal-adin and Vulpiani, 1987).

As regards explosions at high frequency, we report the re-sults obtained decomposing signals recorded by using short-period seismometers (Chouet et al., 1998). ICA has been ap-plied to explosions recorded by seismometers along the threeorthonormal directions of motion, i.e. radial, transverse, ver-tical with respect to the crater area.

We display in Fig.7 the results of the radial direction; theother directions show a similar behaviour (Acernese et al.,2003). As one can see (Fig.7), the wavefield is the linearsuperposition in time domain of three independent compo-nents, characterized by well defined and separate frequencybands (respectively 0.8− 1.2, 2.4− 3.0, 3.2− 4.5Hz).


Similar results are achieved analysing broadband explo-sions. In addition, in this case, we extract also a componentcorresponding to the VLP signal (Falanga, 2003) as alreadyobserved by Chouet et al. (2003).

The reconstruction of the phase space establishes that ex-plosions are associated to a low-dimensional dynamical sys-tem characterized by dimensions in the range [2 − 3] (DeMartino et al., 2002a).

Then, trajectory space analysis, performed on broadbandsignals, states that explosions are generated by an unique dy-namical system (De Martino et al., 2004), though Chouetet al. (2003) have found two distinct kinds of explosions,which have been associated to the two distinct vents atStromboli in 1997. The differences between the two typesof events are related more to slight variations in conduit ge-ometries rather than differences in the dynamics generationof the phenomena.

We have also analysed, as already said, the tremor. In thiscase, it is convenient to consider separately two frequencybands (> 0.5Hz and < 0.5Hz). Namely, the low frequencyband can contain waves travelling with different velocitywith respect to the high frequency wavefield. In Fig.8, as youcan see, tremor shows ICA extracted components similar toexplosion quakes, in waveform and frequency content. Of

0 2 4 60

2

4

6

PSD

0 2 4 60

1

2

Frequency (Hz)

0 2 4 60

20

40

60

0 1000 2000 3000−1

0

1

Ampli

tude

0 1000 2000 3000−0.5

0

0.5


0 1000 2000 3000−2

0

2

Fig. 7. Explosions in the band greater than 0.5Hz: Denoised inde-pendent components of radial direction of motion and their spectra(sampling frequency equal to 125Hz; amplitude in adimensionalunit.

0 1 2 3 4 5 6 7 80

20

40

60

80

20 40 60 80

−1

0

1

0 1 2 3 4 5 6 7 80

20

40

PDS

20 40 60 80

−1

0

1

Ampli

tude

(a.u

.)

0 1 2 3 4 5 6 7 80

20

40

Frequency(Hz)20 40 60 80

−1

0

1

Time(s)

Fig. 8. Denoised independent components of tremor, by ICA, in therange > 0.5Hz related to the vertical direction and their spectra;amplitude in adimensional unit.

course, regarding tremor, the individuation of clear bands ismore difficult due to very low SNR. Polarization analysis ontremor has already been performed on short-period recordedsignals Chouet et al. (1997). We are extending the analysis tobroadband signals as extracted by ICA. This will be matterof a forthcoming paper.

All the results persuade us to think that, regarding highfrequency content, the superficial source is stationary andnot destructive. The wavefield is generated by the excita-tion of only feww degrees of freedom of the complex fluid-dynamical source system.

Possible models of the production of these oscillationshave been postulated by Julian (1994), Ida (1996) and Jameset al. (2004). In particular, Julian (1994) suggested an or-gan pipe model, with a constant rate supply of fluids inside acylinder conduit for a variety of almost periodic signals ob-served on volcanoes.

The independent component analysis of organ pipe acous-tic emission by Bottiglieri et al. (2004) seems to support thismodel. Namely, in an organ pipe, a constant rate supplyof pressure produces self-sustained sounds and ICA is able

Fig. 8. Denoised independent components of tremor, by ICA, inthe range>0.5 Hz related to the vertical direction and their spectra(amplitude in adimensional unit).

orthonormal directions of motion, i.e. radial, transverse, ver-tical with respect to the crater area.

We display in Fig.7 the results of the radial direction; theother directions show a similar behaviour (Acernese et al.,2003). As one can see (Fig.7), the wavefield is the linearsuperposition in time domain of three independent compo-nents, characterized by well defined and separate frequencybands (respectively 0.8−1.2, 2.4−3.0, 3.2−4.5 Hz).


Similar results are achieved analysing broadband explo-sions. In addition, in this case, we extract also a componentcorresponding to the VLP signal (Falanga, 2003) as alreadyobserved byChouet et al.(2003).

The reconstruction of the phase space establishes that ex-plosions are associated to a low-dimensional dynamical sy-stem characterized by dimensions in the range[2−3] (DeMartino et al., 2002a).

Then, trajectory space analysis, performed on broadbandsignals, states that explosions are generated by an unique dy-namical system (De Martino et al., 2004), thoughChouetet al. (2003) have found two distinct kinds of explosions,which have been associated to the two distinct vents atStromboli in 1997. The differences between the two typesof events are related more to slight variations in conduit geo-metries rather than differences in the dynamics generation ofthe phenomena.

We have also analysed, as already said, the tremor. In thiscase, it is convenient to consider separately two frequencybands (> 0.5 Hz and< 0.5 Hz). Namely, the low frequencyband can contain waves travelling with different velocitywith respect to the high frequency wavefield. In Fig.8, asyou can see, tremor shows ICA extracted components similarto explosion quakes, in waveform and frequency content. Ofcourse, regarding tremor, the individuation of clear bands is


10−2

100

0

10

20~ 42 s

10−2

100

0

10

20

~ 37 s

PSD

10−2

100

0

10

20

Frequency(Hz)

0 50 100 150 200 250 300−2

0

2

0 50 100 150 200 250 300−5

0

5

Ampl

itude

0 50 100 150 200 250 300−2

0

2

Time (s)

~ 30 s

Fig. 9. Denoised independent components by ICA in the range[0.02 − 0.5Hz] of vertical direction of motion and their spectra;amplitude in adimensional unit.

to recognize exactly three independent components in limitcycle regime, corresponding to a fundamental Landau modeand two excited ones. In this case, due to the cylindrical sym-metry of the pipe the waveforms are simpler than in Strom-boli case.

In the scheme of organ pipe model for Stromboli volcano,the tremor is the basic signal; explosions are generated byan enhancing in amplitude of tremor. The enhancing may bedue to the formation of slug within the shallow plumbing sys-tem. In order to get information about the generative processof slugs, i.e. coalescence phenomenon, we focalise our atten-tion on tremor at very low frequency band (0.02−0.5Hz). Wereport in Fig.9 the extracted components by means of ICA re-lated to windows containing obviously only tremor. As wecan see, three components in the band 30− 42s appear, witha fundamental peak corresponding to a 30s periodicity (DeMartino et al., 2002b, 2003; Falanga, 2003).

Polarization analysis, performed on few tremor signals, fil-tered in a very low frequency band (< 0.1Hz) in order toinvestigate the nature of the 30− 40s extracted components,shows, as a preliminary result, a wavefield polarized in trans-verse direction with respect to crater area. Further investiga-tion, that takes into account all the statistics, is under study.

Finally, we have estimated the apparent velocity of thiscomponent. We obtain values of velocity very low of theorder of 100m/s. The latter suggests that we are dealingwith a subsonic slow wave originating in crater area.

Oscillations, in this low frequencies regime, have been ob-served already on other volcanoes (e.g. Aster et al., 2003).They have been associated to the formation of slug within themagmatic system. So, they have been recognized as gravitywaves although the sizes of slug are not well estimated (e.g.Aster et al., 2003).

5 Conclusions

We have applied ICA to Strombolian seismological signals asan ”a priori” tool. Combining ICA with other methods, we

have obtained many constrains to model Strombolian activitysource. ICA gives five ”modes” directly in the time domain.This means that, though in a suitable coupling limit, Strom-boli recorded signals are the linear superposition of these fivemodes (three at high frequency, VLP and 30 − 40s mode).Then, the physical model must reproduce these ”modes”.

Our approach clarifies the situation of high frequencies(> 0.5Hz) of Stromboli: in this range, we are in presenceof vibrations of volcanic conduits induced by permanent de-gassing. We prove that tremor and explosions display notonly a similar frequency content, but also similar waveformsin time domain. A pregnant difference is the amplitude en-hancing.

Analogies with organ pipe model are possible once hav-ing shown that strombolian high frequency wavefield can bedecomposed into three separate frequency bands. In fact thedegassing is almost stationary as the pumping in the organpipe, although the wavefield produced by the vibration ofconduit propagates in a more complex medium with respectto the air (as in the case of acoustic field produced by or-gan pipe). The waveforms are more complex, namely we aredealing with a pipe (conduit) having a more complex geom-etry than a cylindrical pipe. But, in both cases, we find threeself-sustained ”modes” produced by a dynamical system inlimit cycle regime.

The heart of sound production in organ pipe is the edge.The same function for the generation of ground vibration (thecoupling between degassing and the conduit) can be ruled bythe variable geometry of the conduit (James et al., 2004).

The ICA has evidenced, also, component at 30s, that canfurnish the means by which the permanent tremor becomesenhanced in the explosion.

We can consider a very simple model to clarify this point.Take one dimensional limit cycle (Eq.7) with a threshold bdepending on time (see Fig.10). The threshold in the limitcycle acts as a sort of potential ruling the contributes com-ing from pumping and dissipation, such a way it representsthe stationary degassing and all the dissipative effects surelypresent in a volcanic structure.

When the formation of slug occurs, there is a time varia-tion of pumping so that the system goes in a excited level.Due to the fact that time duration of slug formation is veryshort, the system immediately decays towards the basic en-ergetic state.

This very rough model is constructed to show a possiblemechanism of self-interaction to obtain the seismic recordassociated to explosion, as enhancing of tremor.

Namely, the high frequency amplitude enhancing is the ef-fect of the density time variation due to coalescence, i.e. tothe formation of slug.

Obviously, the simulation of tremor as a simple limit cycleat the frequency of 1Hz and the simulation of low frequencymode as harmonic oscillator with a period of 30s are very farfrom the reality.

For example, we have not considered that many ”modes”come within the real system, that there is a neglected intrin-sic stochasticity. Moreover the threshold signal, which rep-

Fig. 9. Denoised independent components by ICA in the range(0.02−0.5 Hz) of vertical direction of motion and their spectra (am-plitude in adimensional unit).

more difficult due to very low SNR. Polarization analysis ontremor has already been performed on short-period recordedsignals (Chouet et al., 1997). We are extending the analysisto broadband signals as extracted by ICA. This will be matterof a forthcoming paper.

All the results persuade us to think that, regarding highfrequency content, the superficial source is stationary and notdestructive. The wavefield is generated by the excitation ofonly few degrees of freedom of the complex fluid-dynamicalsource system.

Possible models of the production of these oscillationshave been postulated byJulian(1994), Ida (1996) andJameset al. (2004). In particular,Julian (1994) suggested an or-gan pipe model, with a constant rate supply of fluids inside acylinder conduit for a variety of almost periodic signals ob-served on volcanoes.

The independent component analysis of organ pipe acou-stic emission byBottiglieri et al. (2004) seems to supportthis model. Namely, in an organ pipe, a constant rate sup-ply of pressure produces self-sustained sounds and ICA isable to recognize exactly three independent components inlimit cycle regime, corresponding to a fundamental Landaumode and two excited ones. In this case, due to the cylindri-cal symmetry of the pipe the waveforms are simpler than inStromboli case.

In the scheme of organ pipe model for Stromboli volcano,the tremor is the basic signal; explosions are generated by anenhancing in amplitude of tremor. The enhancing may be dueto the formation of slug within the shallow pumping system.In order to get information about the generative process ofslugs, i.e. coalescence phenomenon, we focalise our attentionon tremor at very low frequency band (0.02−0.5 Hz). Wereport in Fig.9 the extracted components by means of ICArelated to windows containing obviously only tremor. As wecan see, three components in the band 30−42 s appear, witha fundamental peak corresponding to a 30 s periodicity (DeMartino et al., 2002b, 2003; Falanga, 2003).


0 50 100 150 200 250 300 350 400 450−400

−300

−200

−100

0

100

200

300

400

Time (s)

Ampl

itude

(a.u

.)

Fig. 10. Toy-model: time evolution of a limit cycle with time vari-able threshold; in the same figure, time evolution of threshold isalso plotted (bold line describing a limit cycle).

resents in a rough way the low frequency mode, is by itselfnonlinear and stochastic. Finally, the recorded waveformsare due to a complex geometry of conduit and the real sys-tem is not at all one-dimensional. Nevertheless, we think thatthis toy model could be a starting point that deserves furtherstudies to be implemented to become the physical model ofStromboli in the regime of its standard activity.

References

Abarbanel, H. D. I.: Analysis of observed chaotic data, Springer-Verlag, New York Inc., 1996.

Acernese, F., Ciaramella, A., De Martino, S., Falanga, M., andTagliaferri, R.: Neural Networks for blind sources separation ofStomboli explosion quakes, IEEE Trans. Neural Netws., 14, 1,2003.

Acernese, F., Ciaramella, A., De Martino, S., Falanga, M., Godano,C., and Tagliaferri, R.: Polarisation analysis of the independentcomponents of low frequency events at Stromboli volcano (Eo-lian Island, Italy), J. Volc. Geotherm. Res., Special Issue: an in-troduction, Eds. A. Harris, M. Hort, M. Ripepe, 137, 1-3, 153-168, doi:10.1016/j.jvolgeores.2004.05.005, 2004.

Andronov, A. A., Vitt, A. A., and Khaikin, S. E.: Theory of oscilla-tors, Dover Publication, Inc, New York, 1966.

Aster, R., Mah, S., Kyle, P., McIntosh, W., Dunbar, N., Jonson,J., Ruiz, M., and McNamara, S.: Very long period oscillationsof Mount Erebus Volcano, J. Geophys. Res., 108, B11, 2522,doi:10.1029/2002JB002101, 2003.

Bell, A. J., and Sejnowski, T. J.: An information-maximisation ap-proach to blind separation and blind deconvolution, Neural Com-putation, 7, 1129–1159, 1995.

Bishop, M.C.: Neural Networks for Pattern Recognition, ClarendonPress-Oxford, 1995.

Bottiglieri, M., De Lauro, E., De Martino, S., and Falanga, M.:Analogical model inferred from time domain analysis and gener-ating organ pipes self sustained-tone, Int. J. of Mod. Phys. B, 18,4-5 509–518, doi:10.1142/S0217979204024124, 2004.

Chouet, B. A., Saccorotti, G., Martini, M., Dawson, P., De Luca, G.,Milana, G., and Scarpa, R.: Source and path effect in wavefieldof tremor and explosions at Stromboli Volcano, Italy, J. Geophys.Res., 102, 15129–15150, 1997.

Chouet, B. A., Dawson, P., De Luca, G., Martini, M., Milana, G.,Saccorotti, G., and Scarpa, R.: Array analyses of seismic wave-fields radiated by eruptive activity at Stromboli volcano, Italy,CNR-Gruppo Nazionale per la Vulcanologia, Felici editore, Os-pedaletto, Pisa, 1998.

Chouet, B. A., Saccorotti, G., Dawson, M., Martini, M., Scarpa,R., De Luca, G., Milana, G., and Cattaneo, M.: Broadband mea-surements of sources of explosions at Stromboli Volcano, Italy,Geophys. Res. Lett., 26, 13, 1937–1940, 1999.

Chouet, B. A., Dawson, P., Ohminato, T., Martini, M., Saccorotti,G., Giudicepietro, F., De Luca, G., Milana G., and Scarpa, R.:Source mechanism of explosions at Stromboli determined frommoment tensor inversions of very-long-period data, J. Geophys.Res., 108, B1, 2019, doi:10.1029/2002JB001919, 2003.

Ciaramella, A., De Lauro, E., De Martino, S., Falanga, M., andTagliaferri, R.: Independent Component Analysis and Dynami-cal Systems, Salerno University, preprint submitted to J. Mach.Learn. Res., 2002.

De Martino, S., Godano, C., and Falanga, M.: Self-sustained os-cillator as a model for explosion quakes at Stromboli Volcano,Nonlinear Processes in Geophysics, 9, 31–35, 2002a.

De Martino, S., Falanga, M., Godano, C., Scarpa, R., and Tronca,F.: Caratteristiche dei segnali sismici a larga banda di Stromboli,21 Convegno Nazionale del Gruppo Nazionale di Geofisica dellaTerra Solida (GNGTS), Roma, 19-21 novembre, 2002b.

De Martino, S., Falanga, M., Mona, L.: Stochastic ResonanceMechanism in Aerosol Index Dynamics, Phys. Rev. Lett., 89, 12,2002c.

De Martino, S., Falanga, M., Godano, C., Scarpa, R., and Tronca,F.: Characteristics of broadband events at Stromboli volcano,EGS-AGU-EGU 2003 Joint Assembly, Nice, France, 6-11 April,2003.

De Martino, S., Falanga, M., and Godano, C.: Dynamical similar-ity of explosions at Stromboli volcano, Geophys. J. Int., 157, 3,1247-1254, doi:10.1111/j.1365-246X.2004.02263.x, 2004.

Falanga, M.: Costruzione di modelli fisici a partire da segnali os-servazionali, PhD thesis, Salerno University, Febbraio 2003.

Fraser, A. M., and Swinney, H. L.: Independent coordinates forstrange attractors from mutual information, Phys. Rev. A, 33,1134–1139, 1986.

Gammaitoni, L., Hanggi, P., Jung, P., and Marchesoni, F.: Stochas-tic resonance, Rev. Mod. Phys., 70, 1, 223, 1998.

Grassberger, P., and Procaccia, I.: Measuring the strangeness ofstrange attractors, Physica D, 9, 189–208, 1983.

Herault, J., and Ans, B.: Circuits neuronaux a synapses modifiables:decodage de messages composites par apprentissage non super-vise, C. -R. de l’Accademie des Sciences, 299(III-13), 525–528,1984.

Hyvarinen, and Oja, E.: A fast fixed-point algorithm for indepen-dent component analysis, Neural Computation, 9, 7, 1483-1492,1997.

Hyvarinen, A., Karhunen, J., and Oja, E.: Independent ComponentAnalysis, John Wiley & Sons, New York, 2001.

Hyvarinen, A., and Pajunen, P.: Nonlinear independent componentanalysis: existence and uniqueness results, Neural Networks, 12,3,429-439, 1999.

Ida, Y.: Cyclic fluid effusion accompanied by pressure change: im-plication for volcanic eruptions and tremor, Geophys. Res. Lett.,23, 1457–1460, 1996.

James, M. R., Lane, S. J., Chouet, B., Gilbert, J. S.: Pressurechanges associated with the ascent and bursting of gas slugs inliquid-filled vertical and inclined conduits, J. Volc. Geoth. Res.,

Fig. 10. Toy-model: time evolution of a limit cycle with time vari-able threshold; in the same figure, time evolution of threshold isalso plotted (bold line describing a limit cycle).

Polarization analysis, performed on few tremor signals, fil-tered in a very low frequency band (< 0.1 Hz) in order toinvestigate the nature of the 30−40 s extracted components,shows, as a preliminary result, a wavefield polarized in trans-verse direction with respect to crater area. Further investiga-tion, that takes into account all the statistics, is under study.

Finally, we have estimated the apparent velocity of thiscomponent. We obtain values of velocity very low of theorder of 100 m/s. The latter suggests that we are dealing witha subsonic slow wave originating in crater area.

Oscillations, in this low frequencies regime, have been ob-served already on other volcanoes (e.g.Aster et al., 2003).They have been associated to the formation of slug within themagmatic system. So, they have been recognized as gravitywaves although the sizes of slug are not well estimated (e.g.Aster et al., 2003).

5 Conclusions

We have applied ICA to Strombolian seismological signals asan “a priori” tool. Combining ICA with other methods, wehave obtained many constrains to model Strombolian activitysource. ICA gives five “modes” directly in the time domain.This means that, though in a suitable coupling limit, Strom-boli recorded signals are the linear superposition of these fivemodes (three at high frequency, VLP and 30−40 s mode).Then, the physical model must reproduce these “modes”.

Our approach clarifies the situation of high frequencies(>0.5 Hz) of Stromboli: in this range, we are in presenceof vibrations of volcanic conduit induced by permanent de-gassing. We prove that tremor and explosions display notonly a similar frequency content, but also similar waveformsin time domain. A pregnant difference is the amplitude en-hancing.

Analogies with organ pipe model are possible once havingshown that strombolian high frequency wavefield can be de-composed into three separated frequency bands. In fact thedegassing is almost stationary as the pumping in the organ


pipe, although the wavefield produced by the vibration ofconduit propagates in a more complex medium with respectto the air (as in the case of acoustic field produced by or-gan pipe). The waveforms are more complex, namely we aredealing with a pipe (conduit) having a more complex geo-metry than a cylindrical pipe. But, in both cases, we findthree self-sustained “modes” produced by a dynamical sys-tem in limit cycle regime.

The heart of sound production in organ pipe is the edge.The same function for the generation of ground vibration (thecoupling between degassing and the conduit) can be ruled bythe variable geometry of the conduit (James et al., 2004).

The ICA has evidenced, also, component at 30 s, that canfurnish the means by which the permanent tremor becomesenhanced in the explosion.

We can consider a very simple model to clarify this point.Take one dimensional limit cycle (Eq.7) with a thresholdb depending on time (see Fig.10). The threshold in thelimit cycle acts as a sort of potential ruling the contributescoming from pumping and dissipation, such a way it repre-sents the stationary degassing and all the dissipative effectssurely present in a volcanic structure. When the formation ofslug occurs, there is a time variation of pumping so that thesystem goes in a excited level. Due to the fact that time dura-tion of slug formation is very short, the system immediatelydecays towards the basic energetic state.

This very rough model is constructed to show a possiblemechanism of self-interaction to obtain the seismic recordassociated to explosion, as enhancing of tremor. Namely, thehigh frequency amplitude enhancing is the effect of the den-sity time variation due to coalescence, i.e. to the formation ofslug.

Obviously, the simulation of tremor as a simple limit cycleat the frequency of 1 Hz and the simulation of low frequencymode as harmonic oscillator with a period of 30 s are very farfrom the reality.

For example, we have not considered that many “modes”come within the real system, that there is a neglectedintrinsic stochasticity. Moreover the threshold signal, whichrepresents in a rough way the low frequency mode, isby itself nonlinear and stochastic. Finally, the recordedwaveforms are due to a complex geometry of conduit andthe real system is not at all one-dimensional. Nevertheless,we think that this toy model could be a starting point thatdeserves further studies to be implemented to become thephysical model of Stromboli in the regime of its standardactivity.

Edited by: M. ThielReviewed by: two referees

References

Abarbanel, H. D. I.: Analysis of observed chaotic data, Springer-Verlag, New York Inc., 1996.

Acernese, F., Ciaramella, A., De Martino, S., Falanga, M., andTagliaferri, R.: Neural Networks for blind sources separation ofStomboli explosion quakes, IEEE Neural., 14, 1, 2003.

Acernese, F., Ciaramella, A., De Martino, S., Falanga, M., Go-dano, C., and Tagliaferri, R.: Polarisation analysis of the inde-pendent components of low frequency events at Stromboli vol-cano (Eolian Island, Italy), J. Volcanol., Special Issue, editedby: Harris, A., Hort, M., and Ripepe, M., 137, 1–3, 153–168,doi:10.1016/j.jvolgeores.2004.05.005, 2004.

Andronov, A. A., Vitt, A. A., and Khaikin, S. E.: Theory of oscilla-tors, Dover Publication Inc., New York, 1966.

Aster, R., Mah, S., Kyle, P., McIntosh, W., Dunbar, N., Jonson, J.,Ruiz, M., and McNamara, S.: Very long period oscillations ofMount Erebus Volcano, J. Geophys. Res.-Sol., 108, B11, 2522,doi:10.1029/2002JB002101, 2003.

Bell, A. J. and Sejnowski, T. J.: An information-maximisationapproach to blind separation and blind deconvolution, NeuralComp., 7, 1129–1159, 1995.

Bishop, M. C.: Neural Networks for Pattern Recognition, Claren-don Press-Oxford, 1995.

Bottiglieri, M., De Lauro, E., De Martino, S., and Falanga, M.:Analogical model inferred from time domain analysis and gen-erating organ pipes self sustained-tone, Int. J. Mod. B, 18, 4-5509–518, doi:10.1142/S0217979204024124, 2004.

Chouet, B. A., Saccorotti, G., Martini, M., Dawson, P., De Luca, G.,Milana, G., and Scarpa, R.: Source and path effect in wavefieldof tremor and explosions at Stromboli Volcano, Italy, J. Geophys.Res.-Sol., 102, 15129–15150, 1997.

Chouet, B. A., Dawson, P., De Luca, G., Martini, M., Milana, G.,Saccorotti, G., and Scarpa, R.: Array analyses of seismic wave-fields radiated by eruptive activity at Stromboli volcano, Italy,CNR-Gruppo Nazionale per la Vulcanologia, Felici editore, Os-pedaletto, Pisa, 1998.

Chouet, B. A., Saccorotti, G., Dawson, M., Martini, M., Scarpa,R., De Luca, G., Milana, G., and Cattaneo, M.: Broadband mea-surements of sources of explosions at Stromboli Volcano, Italy,Geophys. Res. Lett., 26, 13, 1937–1940, 1999.

Chouet, B. A., Dawson, P., Ohminato, T., Martini, M., Saccorotti,G., Giudicepietro, F., De Luca, G., Milana G., and Scarpa, R.:Source mechanism of explosions at Stromboli determined frommoment tensor inversions of very-long-period data, J. Geophys.Res.-Sol., 108, B1, 2019, doi:10.1029/2002JB001919, 2003.

De Martino, S., Godano, C., and Falanga, M.: Self-sustained os-cillator as a model for explosion quakes at Stromboli Volcano,Nonl. Proc. Geophys., 9, 31–35, 2002a,SRef-ID: 1607-7946/npg/2002-9-31.

De Martino, S., Falanga, M., Godano, C., Scarpa, R., and Tronca,F.: Caratteristiche dei segnali sismici a larga banda di Strom-boli, 21 Convegno Nazionale del Gruppo Nazionale di Ge-ofisica della Terra Solida (GNGTS), Roma, 19–21 Novembre,2002b.

De Martino, S., Falanga, M., and Mona, L.: Stochastic ResonanceMechanism in Aerosol Index Dynamics, Phys. Rev. L., 89, 12,2002c.

http://direct.sref.org/1607-7946/npg/2002-9-31


De Martino, S., Falanga, M., Godano, C., Scarpa, R., and Tronca,F.: Characteristics of broadband events at Stromboli volcano,EGS-AGU-EGU 2003 Joint Assembly, Nice, France, 6–11 April,2003.

De Martino, S., Falanga, M., and Godano, C.: Dynamical similarityof explosions at Stromboli volcano, Geophys. J. I., 157, 3, 1247–1254, doi:10.1111/j.1365-246X.2004.02263.x, 2004.

Falanga, M.: Costruzione di modelli fisici a partire da segnali os-servazionali, PhD thesis, Salerno University, Febbraio 2003.

Fraser, A. M. and Swinney, H. L.: Independent coordinates forstrange attractors from mutual information, Phys. Rev. A, 33,1134–1139, 1986.

Gammaitoni, L., Hanggi, P., Jung, P., and Marchesoni, F.: Stochas-tic resonance, Rev. Mod. Phys., 70, 1, 223, 1998.

Grassberger, P., and Procaccia, I.: Measuring the strangeness ofstrange attractors, Physica D, 9, 189–208, 1983.

Herault, J., and Ans, B.: Circuits neuronauxa synapses modifiables:decodage de messages composites par apprentissage non super-vise, C.-R. de l’Accademie des Sciences, 299(III-13), 525–528,1984.

Hyvarinen, and Oja, E.: A fast fixed-point algorithm for indepen-dent component analysis, Neural Comp., 9, 7, 1483–1492, 1997.

Hyvarinen, A., Karhunen, J., and Oja, E.: Independent ComponentAnalysis, John Wiley & Sons, New York, 2001.

Hyvarinen, A. and Pajunen, P.: Nonlinear independent componentanalysis: existence and uniqueness results, Neural Netw., 12, 3,429–439, 1999.

Ida, Y.: Cyclic fluid effusion accompanied by pressure change: im-plication for volcanic eruptions and tremor, Geophys. Res. Lett.,23, 1457–1460, 1996.

James, M. R., Lane, S. J., Chouet, B., Gilbert, J. S.: Pressurechanges associated with the ascent and bursting of gas slugs inliquid-filled vertical and inclined conduits, J. Volcanol., 129, 61–82, 2004.

Jang, G. J. and Leen, T. W.: A probabilistic Approach to SingleChannel Blind Signal Separation, MIPS 2003, 15, MIT Press,2003.

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Characterization of Strombolian events by using independent component analysis

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