On the Singular Values Decoupling in the Singular Spectrum Analysis of Volcanic Tremor at Stromboli

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Nat. Hazards Earth Syst. Sci., 6, 903–909, 2006 www.nat-hazards-earth-syst-sci.net/6/903/2006/ Natural Hazards © Author(s) 2006. This work is licensed and Earth under a Creative Commons License. System Sciences

On the singular values decoupling in the Singular Spectrum Analysis of volcanic tremor at Stromboli

R. Carniel1, F. Barazza1, M. Tarraga´ 2, and R. Ortiz2 1Dipartimento di Georisorse e Territorio, Universita` di Udine, 33100 Udine, Italy 2Departamento de Volcanolog´ıa, Museo Nacional de Ciencias Naturales, CSIC, Madrid, Spain Received: 6 December 2005 – Revised: 11 October 2006 – Accepted: 11 October 2006 – Published: 16 October 2006

Abstract. The well known strombolian activity at Strom- this kind of events, and therefore the possible appearance boli volcano is occasionally interrupted by rarer episodes of of recordable precursors. Talking about timescales, (Ripepe paroxysmal activity which can lead to considerable hazard et al., 2002) identified two degassing and explosive regimes for Stromboli inhabitants and tourists. On 5 April 2003 a at Stromboli, linked to the fresh gas-rich magma supply rate, powerful explosion, which can be compared in size with the that alternate on a 5–40 min timescale. At a longer timescale, latest one of 1930, covered with bombs a good part of the (Carniel and Di Cecca, 1999) identified days-to-weeks-long normally tourist-accessible summit area. This explosion was dynamical phases, sometimes separated by abrupt transi- not forecasted, although the island was by then effectively tions. Talking about possible precursors, (Carniel and Iacop, monitored by a dense deployment of instruments. After hav- 1996b) showed that paroxysmal phases are sometimes pre- ing tackled in a previous paper the problem of highlight- ceded by increasing lower frequency content in the tremor. ing the timescale of preparation of this event, we investigate Due to the complexity of the physical processes involved, here the possibility of highlighting precursors in the volcanic a stochastic (Jaquet and Carniel, 2001, 2003) or dynamical tremor continuously recorded by a short period summit seis- approach (Carniel and Di Cecca, 1999; Carniel et al., 2003) mic station. We show that a promising candidate is found is often a more appropriate choice for short-to-medium term by examining the degree of coupling between successive sin- forecasts aimed e.g. to schedule the tourist excursions when gular values that result from the Singular Spectrum Analysis volcanic hazard is lower. of the raw seismic data. We suggest therefore that possible On 28 December 2002, an effusive eruption (Alean anomalies in the time evolution of this parameter could be et al., 2006) marked the most significant effusive episode indicators of volcano instability to be taken into account e.g. since 1985–1986. This induced considerable morpholog- in a bayesian eruptive scenario evaluator. Obviously, further ical changes before its end on 21–22 July 2003 (Calvari, (and possibly forward) testing on other cases is needed to 2003) and showed the most energetic associated explosive confirm the usefulness of this parameter. event at 07:12 GMT of 5 April 2003. The event started with a reddish ash emission related to collapses within the craters, soon replaced by darker juvenile material from NE 1 Introduction Crater and followed by similar emissions from SW Crater which mushroom-shaped dark cloud rose about 1 km above Stromboli is the prototype for strombolian activity, observed the summit (Calvari, 2003). Bombs, ash, and blocks af- since at least 3–7 A.D. (Rosi et al., 2000). Occasionally, fected most the Western sector and the village of Ginostra; paroxysmal phases are observed (Barberi et al., 1993; Jaquet the volcano top above 700 m a.m.s.l. was completely covered and Carniel, 2003) involving an additional kind of magma, by bombs, including meter-sized ones (Alean et al., 2006), low-porphyritic and volatile-rich (Francalanci et al., 2004) and scientific equipment, including the seismic station of the that apparently plays a major role in the development of such University of Udine, was damaged or destroyed. In a previ- phases. This involvement of a different magma suggests that ous paper, Carniel et al. (2006) tackled the problem of high- there may be a sufficiently long timescale of preparation for lighting the timescale of preparation of this event, showing that the volcanic tremor recorded continuously by a short Correspondence to: R. Carniel period seismic station is sufficiently informative to derive ([email protected]) this kind of information. In fact, non-stationarity is a key

Published by Copernicus GmbH on behalf of the European Geosciences Union. 904 R. Carniel et al.: Singular values decoupling characteristic of the tremor (Konstantinou and Schlindwein, This is by definition a Hankel matrix, as it is Xi,j =xi+j−1. 2002), as it offers the possibility of monitoring changes in pa- The size of the embedding window m (number of columns rameters derived from an experimental time series and their of the trajectory matrix) should be sufficiently large to cap- possible use in forecasting (Carniel and Di Cecca, 1999) or ture the global behavior of the system. A common method at least to highlight different regimes (Ripepe et al., 2002; to determine m is to use the first zero of the linear correla- Carniel et al., 2003; Harris et al., 2005; Jones et al., 2005). tion between the first and the last column of the trajectory By the use of spectral and dynamical analysis (Carniel et al., matrix. Another method used to determine m (Shaw, 2000) 2006) showed that the paroxysmal phase of 5 April 2003 uses the point where mutual information between the first have built up over at least the previous 2.5 hours, but the ap- and the last column of the trajectory matrix reaches the first plication of the Material Failure Forecast Method during the minimum (Fraser, 1986; Fraser and Swinney, 1986). This is days before the event revealed a consistent trend that sug- in principle a better choice, since it takes into account also gests a preparation of the paroxysm further in the past, an the non-linear correlation within the time series. evolution which was possibly temporarily “paused” the day before, before finally accelerating during the consistent dy- 2.1.2 Decomposition of time series namical phase of the last few hours. In any case, there is sufficient “persistence” in the data to look for some kind of Once the time series are embedded into the trajectory matrix precursor as a hind-cast exercise. In particular, the Singular X, the singular value decomposition is performed on such Spectrum Analysis is the methodology that shows the most matrix. The singular values decomposition technique allows promising potential in the search for precursors. We there- to obtain the decomposition: fore start by presenting this technique in further detail. X = USVT (1)

where U is a (n−m+1) × (n−m+1) real orthogonal matrix, 2 The Singular Spectrum Analysis (SSA) technique V is a m×m real orthogonal matrix and S is a (n−m+1) ×m diagonal real matrix such that its elements are the singular Time series analysis has recently profited from the appli- values of the trajectory matrix X. This is done by computing cation of spectral decomposition of matrices (e.g. (Marelli the lagged covariance matrix et al., 2002; Mineva and Popivanov, 1996; Pereira and Ma- ciel, 2001)), which has its roots mostly in chaos theory (Tak- C = XT X (2) ens, 1981; Broomhead and King, 1981). The SSA con- sists of several steps (Golyandina et al., 2001; Aldrich and so C is a symmetric semidefinite positive matrix and then a Barkhuizen, 2003). In the first step (embedding), the one unique spectral decomposition exists: dimensional time series is recast as an L-dimensional time C = 838T (3) series (trajectory matrix). In the second step (singular value decomposition), the trajectory matrix is decomposed into a where 8 is a real orthonormal matrix such that its columns sum of orthogonal matrices of rank one. These two steps are the eigenvectors of C, and 3 is a real diagonal matrix 2 constitute the decomposition stage of SSA. In the third and such that its elements σi are the eigenvalues of C in decreas- fourth steps, the components are grouped and the time series ing order. One can now observe that associated with the groups are reconstructed. C = XT X 2.1 Decomposition stage T = USVT USVT 2.1.1 Embedding of time series = VST UT USVT

Embedding expands the original time series into what is re- but U is an orthonormal matrix and S is a diagonal matrix, so ferred to as trajectory matrix of the system. This matrix is UT U=I and ST S=S2. Therefore associated with a certain window length that is the embed- 2 T ding dimension. The series is expanded by giving it a unitary C = VS V (4) lag and creating a certain number of lagged vectors. So, for Comparing Eqs. (3) and (4) one obtains that the time series x= {x1, x2, ... , xn} and for an embedding dimension m, we obtain the trajectory matrix: 8 = V   3 = S2 x1 x2 ··· xm    x2 x3 . . . xm+1  X = So one can deduce the important consequence that the right . . .. .  . . . .  eigenvectors of X are the eigenvectors of C while the singu-    xn−m+1 xn−m+2 . . . xn  lar values of X are the square root of the eigenvalues of C.

Nat. Hazards Earth Syst. Sci., 6, 903–909, 2006 www.nat-hazards-earth-syst-sci.net/6/903/2006/ R. Carniel et al.: Singular values decoupling 905

Therefore: and

  m σ1 0 ... 0   = X T  0 σ ... 0  noise X8i8i (10)  2   . . . .  i=q+1  . . .. .   . . .  S = 0 0 . . . σm , σ1 ≥ σ2 ≥ ... ≥ σm ≥ 0 The criteria for this separation are not completely formalized  0 0 ··· 0  and they depend on knowledge of the data, and obviously on    . . . .  σ ’s modules. For example if the first q singular values are  . . .. .  i   much greater than the others, then the choice can be straight-  0 0 ... 0  forward. Another way to write Eq. (1) is the so called spectral de- The new matrix Xq is not always a trajectory matrix (be- composition: cause in general it is not a Hankel matrix) and then it does not directly represent the noise-free x time series. However this m time series can be obtained by a diagonal average method. = X T X σiuivi (5) For example, if Xq is the matrix: i=1   x1,1 x1,2 x1,3 where ui and vi are respectively U’s and V’s matrix i-th    x2,1 x2,2 x2,3  columns. But we had demonstrated that V=8, and from Xq = x3,1 x3,2 x3,3 Eq. (1) it is not difficult to show that XV=US. So vi=8i    x4,1 x4,2 x4,3  and σiui=X8i where 8i is 8’s i-th column. Substituting this equality in Eq. (5) permits us to obtain a simpler and then we obtain: more useful form for X’s decomposition: n x + x x + x + x m = 2,1 1,2 3,1 2,2 1,3 X xq x1,1, , , X = X8 8T (6) 2 3 i i + + + i=1 x4,1 x3,2 x2,3 x4,2 x3,3 o , , x , 3 2 4 3 2.2 Reconstruction stage In conclusion, the noise-free time series xq depend on the The aim of this stage is to separate the additive components choice of the parameters m and q, so we can write: of the time series. It can be seen as separating the time series into two groups: “our signal” and the “noisy” components, xq = SSA (x; m, q) which are by definition the components we are not interested in. Obviously, the level of de-noising is higher when q is lower. The idea is to project the trajectory matrix over a q- (Hegger et al., 1999) suggest that q should be at least the cor- dimensional space. In fact, in Eq. (5), every term of the rect embedding dimension, and m considerably larger (e.g. sum has a lower importance with respect to the previous m=2q or larger). one in the construction of the signal. Such importance is given by the weight σi of each singular value of the base and σ1≥σ2≥... ≥ σm≥0 by construction. Then we can approxi- 3 Coupling of the singular values mate Eq. (5) and (6) with the following: The behavior of the singular values (also called the spectrum q X T of the singular values) can contain useful indications on the Xq = σiuivi (7) i=1 composition of the signal (i.e. Yang and Tse, 2003). In particular, for a sufficiently high embedding dimension some of and the singular values tend to show a certain degree of coupling, that is σ →σ , σ →σ and so on. An example is shown in m 2 1 4 3 X T Fig. 1. Note that in Fig. 1, the singular values are computed noise = σiuivi (8) i=q+1 using the following definition of the covariance matrix: 1 or, in a simpler form: C = XT X n − m q X T Xq = X8i8i (9) In this way the singular values are detrended, allowing a sim- i=1 pler visualization. www.nat-hazards-earth-syst-sci.net/6/903/2006/ Nat. Hazards Earth Syst. Sci., 6, 903–909, 2006 Roberto Carniel et al.: Singular values decoupling 3

Another way to write Eq. (1) is the so called spectral de- then we obtain: composition: n x2,1 + x1,2 x3,1 + x2,2 + x1,3 m xq = x1,1, , , X T 2 3 X = σiuivi (5) x4,1 + x3,2 + x2,3 x4,2 + x3,3 o i=1 , , x 3 2 4,3 where ui and vi are respectively U’s and V’s matrix i-th columns. But we had demonstrated that V = Φ, and from In conclusion, the noise-free time series xq depend on the choice of the parameters m and q, so we can write: Eq. (1) it is not difficult to show that XV = US. So vi = Φi and σiui = XΦi where Φi is Φ’s i-th column. Substituting xq = SSA (x; m, q) this equality in Eq. (5) permits us to obtain a simpler and more useful form for X’s decomposition: Obviously, the level of de-noising is higher when q is lower. (Hegger et al., 1999) suggest that q should be at least the cor- m X rect embedding dimension, and m considerably larger (e.g. X = XΦ ΦT (6) i i m = 2q or larger). i=1 2.2 Reconstruction stage 3 Coupling of the singular values The aim of this stage is to separate the additive components of the time series. It can be seen as separating the time series The behavior of the singular values (also called the spectrum into two groups: “our signal” and the “noisy” components, of the singular values) can contain useful indications on the which are by definition the components we are not interested composition of the signal (i.e. Yang and Tse (2003)). In par- in. ticular, for a sufficiently high embedding dimension some of The idea is to project the trajectory matrix over a q- the singular values tend to show a certain degree of coupling, dimensional space. In fact, in Eq. (5), every term of the that is σ2 → σ1, σ4 → σ3 and so on. An example is shown in sum has a lower importance with respect to the previous Fig. 1. Note that in Fig. 1, the singular values are computed one in the construction of the signal. Such importance is using the following definition of the covariance matrix: given by the weight σi of each singular value of the base and 1 C = XT X σ1 ≥4σ2 ≥ ... ≥ σm ≥ 0 by construction. Then we can Roberto Carniel etn al.:− m Singular values decoupling approximate Eq. (5) and (6) with the following: In this way the singular values are detrended, allowing a sim- 4 The data q 906very fast analysis). The first two singular values do not R. Carniel et al.: Singular values decoupling X T pler visualization. Xq = σiuivi (7) show any anomalous behaviour in their ratio ρ1, as can be One of the longest seismic time series available at Strom- i=1 seenσ in Fig. 2. However, if we plot the same time evolu- 3 Willmore MKIII/A seismometer (f0=0.5 Hz), is located at boli comes from an automatic station installed in 1989 by and tion but we take into account the “degree of coupling” be- 800 m a.m.s.l. and at about 300 m from the craters (Beinat the Dipartimento di Georisorsem e Territorio of the University tween0.6 the singular values σ3 and σ4, i.e. ρ2, we observe a et al., 1994). During this last effusive phase, the hardware of Udine (Beinat et al., 1994)X with theT purpose of studying noise = σiuivi (8) very clear anomalous decrease of the parameter, that starts and software of the receiving station was upgraded in col- the long-term evolution of the strombolian activity (Carniel i=q+1 to0.5 diverge from the "normal" value - that theoretically, as laboration with CSIC, Madrid, for a continuous acquisition and Iacop, 1996a). The summit station, based on 3 Will- or, in a simpler form: we have shown, shouldσ1 be close to unity - already about 7 and internet data transmission. Data are now sampled with more MKIII/A seismometer (f0 = 0.5Hz), is located at 800 hours0.4 before the explosion (Fig. 3). Note that it is impor- q 16 bits (96 dB) at 50 Hz (Ortiz et al., 2001; Carniel et al., m a.m.s.l. and at about 300X m from theT craters (Beinat et al., tant to observe the relative change in the behaviour of the Xq = XΦiΦi (9) 0.3 2006). Although at the time of 5 April 2003 paroxysmal 1994). During this last effusive phase, the hardware and soft- ρk rather than the absolute change. Inσ fact, the coupling is i=1 2 ware of the receiving station was upgraded in collaboration observed moreσ3 easily for singular values of smaller indexes, event a dense network of monitoring equipment installed by andwith CSIC, Madrid, for a continuous acquisition and internet 0.2 the Civil Defence was operating, the explosion was not fore- m while the phenomenon tends to be less evident as the index data transmission. Data areX now sampledT with 16 bits (96dB) is increased. Moreover, the physical explanation of the cou- casted by any evident change in the data. In the following noise = XΦiΦ (10) σ4 at 50 Hz (Ortiz et al., 2001; Carniel et al.,i 2006). Although at pling0.1 is still subject of research. we will examine with the SSA methodology described above i=q+1 the time of 5 April 2003 paroxysmal event a dense network If we assume the lowest 5% percentile of the coupling pa-m the seismic data recorded by our station during the days be- The criteria for this separation are not completely formalized 0 of monitoring equipment installed by the Civil Defence was rameter 0 range 50 as 100 a threshold 150 200 for an alert,250 in 300 this case 350 the 400 ab- fore this paroxysmal event, in particular from 25 March to and they depend on knowledge of the data, and obviously on operating, the explosion was not forecasted by any evident solute value of the threshold is about 0.45. This threshold 5 April 2003. The confidence in the method of analysis goes σ ’s modules. For example if the first q singular values are Fig. 1. Example of coupling between singular values for the sig- i change in the data. In the following we will examine with the Fig.generates 1. Example two short-lastingof coupling between false alerts singular at about values 115 for and the signal152 back again to Carniel et al. (2006), where another promising nal sin (20 · 2πt) + 0.7 sin (57 · 2πt) (sampled at 1000Hz) in an muchSSA greater methodology than the described others, then above the the choice seismic can data be straight- recorded sinhours(20·2 beforeπt) +0 the.7 sin explosion,(57·2πt) but(sampled gives a at "warning 1000 Hz) time" in an before increas- parameter was studied, derived from the SSA methodology: increasing embedding dimension m = 10 to 400 forward.by our station during the days before this paroxysmal event, ingthe embedding real explosion dimension of morem= than10 to6 400.hours k Thein particular new matrix fromX 25q is March not always to 5 April a trajectory 2003. The matrix confidence (be- X causein in the general method it ofis not analysis a Hankel goes matrix) back again and then to Carniel it does et not al. We can define the “degree of coupling” as: σi ratio ρ1 directly(2006), represent where another the noise-free promisingx time parameter series. was However studied, this de- σ2 = i=1 ρ1 = rk m timerived series from can the be SSA obtained methodology: by a diagonal average method. 0.8 σ1 X σ σi For example, if Xq is the matrix: 4 k ρ2 = i=k+1  X  0.7 σ3 x x σxi  1,1 1,2 1,3  = ... The idea behind this parameter was to monitor the time evo-  x xi=1 x  rk 2=,1 m2,2 2,3 0.6 Xq = σ2k lution of the relative weight of the first k SSA components in x3,1 xX3,2 x3,3 ρ =  σi  k the construction of the full tremor signal.   σ2k−1 x4,1 ix=4k,+12 x4,3 0.5

The idea behind this parameter was to monitor the time evo- 0.4 5 From tremor to singular values coupling lution of the relative weight of the first k SSA components in the construction of the full tremor signal. 0.3 t[h] −250 −200 −150 −100 −50 0 First of all we subdivide the raw data in windows of 60 s (i.e., 3000 sample points at 50 Hz). Both the methods of autocor- σ2 Fig. 2. ρ1 =σ2 in an embedding space of dimension m = 10, for relation function and of mutual information (see Sect. 2.1.1) 5 From tremor to singular values coupling Fig. 2. ρ = σ1 in an embedding space of dimension m=10, for 1 σ1 StromboliStromboli volcanic volcanic tremor. tremor supply an embedding dimension estimate m between 7 and First of all we subdivide the raw data in windows of 60 s 10, so m=10 seems a correct minimal choice for the con- (i.e., 3000 sample points at 50 Hz). Both the methods of au- struction of the trajectory matrix. The singular values are tocorrelation function and of mutual information (see 2.1.1) Weratio can define the “degreeρ of2 coupling” as: computed on each of the one-minute time windows. In or- 0.55 supply an embedding dimension estimate m between 7 and der to avoid amplitude-dependent effects, before the calcula- σ 10, so m = 10 seems a correct minimal choice for the con- ρ = 2 tion of the singular values, the data in each time window is 0.5 1 struction of the trajectory matrix. The singular values are σ1 normalized to zero mean and unitary variance. We start our computed on each of the one-minute time windows. In or- σ4 ρ2 = analysis with the minimum embedding dimension m=10. der to avoid amplitude-dependent effects, before the calcula- 0.45 σ3 threshold 5% tion of the singular values, the data in each time window is = ... 5.1 Behaviour of ρk parameter for m=10 normalized to zero mean and unitary variance. We start our 0.4 σ2k ρ =false positive analysis with the minimum embedding dimension m = 10. k We analyze the time evolution of ρk approaching the parox- σ2k−1 0.35 ysmal event, that will be denoted by the hour “zero”, at the 5.1 Behaviour of ρk parameter for m = 10 extreme right of our graphs, for the embedding dimension t[h] 4 0.3 The data m=10 (This embedding dimension is probably too small to We analyze the time evolution of ρk approaching the parox- −250 −200 −150 −100 −50 0 ysmal event, that will be denoted by the hour "zero", at capture the behaviour of the signal but allowed a very fast σ4 OneFig. of3. ρ the2 = longestin an seismic embedding time space series of dimension availablem = at 10 Strom-, for analysis). The first two singular values do not show any the extreme right of our graphs, for the embedding dimen- σ3 sion m = 10 (This embedding dimension is probably too boliStromboli comes volcanic from antremor automatic station installed in 1989 by anomalous behaviour in their ratio ρ1, as can be seen in small to capture the behaviour of the signal but allowed a the Dipartimento di Georisorse e Territorio of the Univer- Fig. 2. However, if we plot the same time evolution but sity of Udine (Beinat et al., 1994) with the purpose of we take into account the “degree of coupling” between the studying the long-term evolution of the strombolian activ- singular values σ3 and σ4, i.e. ρ2, we observe a very clear ity (Carniel and Iacop, 1996a). The summit station, based on anomalous decrease of the parameter, that starts to diverge

Nat. Hazards Earth Syst. Sci., 6, 903–909, 2006 www.nat-hazards-earth-syst-sci.net/6/903/2006/ 4 Roberto Carniel et al.: Singular values decoupling

4 The data very fast analysis). The first two singular values do not show any anomalous behaviour in their ratio ρ1, as can be One of the longest seismic time series available at Strom- seen in Fig. 2. However, if we plot the same time evolu- boli comes from an automatic station installed in 1989 by tion but we take into account the “degree of coupling” be- the Dipartimento di Georisorse e Territorio of the University tween the singular values σ3 and σ4, i.e. ρ2, we observe a of Udine (Beinat et al., 1994) with the purpose of studying very clear anomalous decrease of the parameter, that starts the long-term evolution of the strombolian activity (Carniel to diverge from the "normal" value - that theoretically, as and Iacop, 1996a). The summit station, based on 3 Will- we have shown, should be close to unity - already about 7 more MKIII/A seismometer (f0 = 0.5Hz), is located at 800 hours before the explosion (Fig. 3). Note that it is impor- m a.m.s.l. and at about 300 m from the craters (Beinat et al., tant to observe the relative change in the behaviour of the 1994). During this last effusive phase, the hardware and soft- ρk rather than the absolute change. In fact, the coupling is ware of the receiving station was upgraded in collaboration observed more easily for singular values of smaller indexes, with CSIC, Madrid, for a continuous acquisition and internet while the phenomenon tends to be less evident as the index data transmission. Data are now sampled with 16 bits (96dB) is increased. Moreover, the physical explanation of the cou- at 50 Hz (Ortiz et al., 2001; Carniel et al., 2006). Although at pling is still subject of research. the time of 5 April 2003 paroxysmal event a dense network If we assume the lowest 5% percentile of the coupling pa- Roberto Carniel et al.: Singular values decoupling 5 of monitoring equipment installed by the Civil Defence was rameter range as a threshold for an alert, in this case the ab- Roberto Carniel et al.: Singular values decoupling 5 operating, the explosion was not forecasted by any evident solute value of the threshold is about 0.45. This threshold 5.2 Behaviour of ρk parameter for m > 10 6 Practical issues on the use of the ρk parameter change in the data. In the following we will examine with the generates two short-lasting false alerts at about 115 and 152 5.2 Behaviour of ρk parameter for m > 10 6 Practical issues on the use of the ρk parameter SSA methodology described above the seismic data recorded hours before the explosion, but gives a "warning time" before In order to verify that the anomalous behaviour is not strictly Close to the paroxysmal event, as we have shown, only by our station during the days before this paroxysmal event, In order to verify that the anomalous behaviour is not strictly Close to the paroxysmal event, as we have shown, only the real explosion of more than 6 hours dependent on the particular choice of the embedding dimen- some of the ρk parameters show an anomalous decreasing in particular from 25 March to 5 April 2003. The confidence dependent on the particular choice of the embedding dimen- some of the ρk parameters show an anomalous decreasing sion m = 10, we performed a similar analysis also in higher behaviour. Other ρk on the contrary don’t show anything in the method of analysis goes back again to Carniel et al. sion m = 10, we performed a similar analysis also in higher behaviour. Other ρk on the contrary don’t show anything ratio ρ dimensions, in particular for m = 20 and m = 100. In interesting. Consequently, a problem arises regarding the (2006), where another promising parameter was studied, de- 1 dimensions, in particular for m = 20 and m = 100. In interesting. Consequently, a problem arises regarding the both cases we observe again that some of the "degree of cou- number of parameters to be monitored in order to observe rived from the SSA methodology: both cases we observe again that some of the "degree of cou- number of parameters to be monitored in order to observe 0.8 pling" ratios show a strong decrease approaching the parox- possible anomalous behavior. We propose here a solution to pling" ratios show a strong decrease approaching the parox- possible anomalous behavior. We propose here a solution to ysm. Fig. 4 shows for instance the anomalous time behaviour this problem, with the definition of a "summarizing" param- k ysm. Fig. 4 shows for instance the anomalous time behaviour this problem, with the definition of a "summarizing" param- X 0.7 σ6 eter defined as the minimum of all ρ computed in a given σi of the ratio ρ3 = σσ6 in the embedding dimension m = 20, k of the ratio ρ3 = 5 in the embedding dimension m =σ 20, eter defined as the minimum of all ρk computed in a given σ5 30 embedding dimension. Care should be taken however in or- i=1 while Fig. 5 shows the anomalous decrease of ρ15 = σ30 in rk = 0.6 while Fig. 5 shows the anomalous decrease of ρ15 = 29 in embedding dimension. Care should be taken however in or- m an embedding with m = 100. σ29 der to obtain meaningful results: the last ratios should be X an embedding with m = 100. der to obtain meaningful results: the last ratios should be σi Using again the 5% percentile criterion, for m = 20 we excluded from this minimum calculation. The last (i.e. the Using again the 5% percentile criterion, for m = 20 we excluded from this minimum calculation. The last (i.e. the i=k+1 0.5 have a threshold of about 0.75. As for the case m = 10, least important) components of the SSA decomposition are have a threshold of about 0.75. As for the case m = 10, least important) components of the SSA decomposition are we obtain false alerts at about 115 and 152 hours before the in fact associated to what is essentially a noise signal (8), The idea behind this parameter was to monitor the time evo- we obtain false alerts at about 115 and 152 hours before the in fact associated to what is essentially a noise signal (8), 0.4 true explosion. Also in this case however, the explosion is and should therefore be removed. We can then finally define: lution of the relative weight of the first k SSA components in true explosion. Also in this case however, the explosion is and should therefore be removed. We can then finally define: preceded by a warning more than 6 hours in advance. the construction of the full tremor signal. 0.3 t[h] preceded by a warning more than 6 hours in advance. ρˆq = min {ρk − µk} q < m −250 −200 −150 −100 −50 0 For m = 100 the 5% percentile threshold corresponds to ρˆq =k min=1..q{ρk − µk} q < m For m = 100 the 5% percentile threshold corresponds to k=1..q an absolute value of about 0.89. Also in this case we have σ2 an absolute value of about 0.89. Also in this case we have where µk is the average of the ratio ρk that can be estimated Fig. 2. ρ1 = σ in an embedding space of dimension m = 10, for false alerts at about 115 and 152 hours before the true explo- where µk is the average of the ratio ρk that can be estimated 5 From tremor to singular values coupling 1 false alerts at about 115 and 152 hours before the true explo- during a normal period of activity (in our case the first 4 days Stromboli volcanic tremor sion, and an additional one about 48 hours before the explo- during a normal period of activity (in our case the first 4 days R. Carniel et al.: Singular values decouplingsion, and an additional one about 48 hours before the explo- 907 in the graph). A suitable value of q should be chosen; a rule First of all we subdivide the raw data in windows of 60 s in the graph). A suitable value of q should be chosen; a rule sion.sion. of thumb is to take (i.e., 3000 sample points at 50 Hz). Both the methods of au- of thumb is to take 1 ρ ρ q ' 1 m tocorrelation function and of mutual information (see 2.1.1) ratio 2 ratio ρ3 q ' 3m 0.55 0.95 ratio 3 3 supply an embedding dimension estimate m between 7 and 0.95 In Fig. 6 the time evolution of the parameter ρˆq is shown for 0.9 In Fig. 6 the time evolution of the parameter ρˆq is shown for 10, so m = 10 seems a correct minimal choice for the con- 0.9 q = 4 in an embedding embedding dimension dimension mm == 20 20.. The The decreasing decreasing struction of the trajectory matrix. The singular values are 0.5 0.85 0.85 anomalous behaviour is is again again very very evident evident before before the the parox- parox- computed on each of the one-minute time windows. In or- 0.8 0.8 ysmal event. 0.45 ysmal event. der to avoid amplitude-dependent effects, before the calcula- 0.75 threshold 5% 0.75 threshold 5% ρˆ tion of the singular values, the data in each time window is threshold 5% ρˆ4 0.7 ratio 4 0.7 0.05 ratio normalized to zero mean and unitary variance. We start our 0.4 0.05 analysis with the minimum embedding dimension m = 10. false positive 0.65 0.65 false positive false positive 0.00 0.6 0.00 0.35 0.6 5.1 Behaviour of ρ parameter for m = 10 k 0.55 0.55 −0.05 t[h] t[h] −0.05 0.3 0.5 t[h] We analyze the time evolution of ρk approaching the parox- −250 −200 −150 −100 −50 0 0.5−250 −200 −150 −100 −50 0 −250 −200 −150 −100 −50 0 −0.10 ysmal event, that will be denoted by the hour "zero", at −0.10 σ4 σ6 the extreme right of our graphs, for the embedding dimen- Fig. 3. ρ2 = in an embedding space of dimension m = 10, for Fig. 4. ρ3 = σ6 in an embedding space of dimension m = 20, for σσ43 Fig. 4. ρ3 =σσ65 in an embedding space of dimension m = 20, for −0.15 Fig. 3. ρ2= in an embedding space of dimension m=10, for Fig. 4. ρ3= σ5in an embedding space of dimension m=20, for −0.15 sion m = 10 (This embedding dimension is probably too Stromboli volcanicσ3 tremor StromboliStromboli volcanic volcanicσ5 tremor tremor threshold 5%5% Stromboli volcanic tremor. Stromboli volcanic tremor. −0.20 small to capture the behaviour of the signal but allowed a −0.20

ρρ15 −0.25 ratioratio 15 −0.25 from the “normal” value – that theoretically, as we have 0.96 0.96

0.94 −0.30 t[h] shown, should be close to unity – already about 7 h before 0.94 −0.30 t[h] −250 −200 −150 −100 −50 −0 −250 −200 −150 −100 −50 −0 the explosion (Fig. 3). Note that it is important to observe 0.92 0.92 the relative change in the behaviour of the ρ rather than the 0.9 k 0.9 Fig. 6. ρˆ4 inin an an embedding embedding space space of of dimension dimensionmm== 20 20,, for for Strom- Strom- thresholdthreshold5%5% 4 absolute change. In fact, the coupling is observed more easily 0.88 0.88 boli volcanic tremor for singular values of smaller indexes, while the phenomenon 0.86 0.86 tends to be less evident as the index is increased. Moreover, In Fig. 4 a phase is is evident evident with with a a peculiar peculiar behaviour behaviour start- start- 0.84 0.84 the physical explanation of the coupling is still subject of re- false positive ing about 40 hours hours before before the the explosion. explosion. This This is is the the time time at at 0.82 0.82 false positive search. which Carniel et al. (2006) observe observe the the ﬁrst ﬁrst successful successful ap- ap- 0.8 0.8 If we assume the lowest 5% percentile of the coupling pa- plication of the Failure Forecast Forecast Method Method (FFM), (FFM), that that gives gives a a 0.78 0.78 forecast for 5 April April 2003 2003 at at 15 15 GMT GMT (see (see line line 1 1 in in Fig. Fig. 9 9 of of rameter range as a threshold for an alert, in this case the ab- t[h] 0.76 0.76 t[h] Carniel et al. (2006)). More than than 6 6 hours hours before before the the explo- explo- solute value of the threshold is about 0.45. This threshold −250−250 −200 −200 −150 −150 −100 −100 −50 0 sion, the ρ2,, ρ4,, ρ1515 andand ρˆρˆ44 parametersparameters (see (see Fig. Fig. 3, 3, 4, 4, 5 5 and and 6 6 generates two short-lasting false alerts at about 115 and 152 h σ σ30σ3030 Fig.Fig.Fig. 5. 5. 5.ρρρ1515=== σ ininin an an an embedding embedding embedding space space space of of of dimension dimension dimension m=100,= 100 for, respectively) go below below the the percentile percentile threshold threshold of of 5%. 5%. At At this this before the explosion, but gives a “warning time” before the 15 σ29σ2929 Stromboliforfor Stromboli Stromboli volcanic volcanic volcanic tremor. tremor tremor time the FFM (see (see line line 2 2 in in Fig. Fig. 9 9 in in Carniel Carniel et et al. al. (2006)), (2006)), af- af- real explosion of more than 6 hours ter an apparent recovery recovery of of the the stability, stability, is is able able again again to to give give 5.2 Behaviour of ρ parameter for m>10 k 6 Practical issues on the use of the ρk parameter

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5.2 Behaviour of ρk parameter for m > 10 6 Practical issues on the use of the ρk parameter

In order to verify that the anomalous behaviour is not strictly Close to the paroxysmal event, as we have shown, only dependent on the particular choice of the embedding dimen- some of the ρk parameters show an anomalous decreasing sion m = 10, we performed a similar analysis also in higher behaviour. Other ρk on the contrary don’t show anything dimensions, in particular for m = 20 and m = 100. In interesting. Consequently, a problem arises regarding the both cases we observe again that some of the "degree of cou- number of parameters to be monitored in order to observe pling" ratios show a strong decrease approaching the parox- possible anomalous behavior. We propose here a solution to ysm. Fig. 4 shows for instance the anomalous time behaviour this problem, with the definition of a "summarizing" param- σ6 of the ratio ρ3 = in the embedding dimension m = 20, eter defined as the minimum of all ρk computed in a given σ5 σ30 while Fig. 5 shows the anomalous decrease of ρ15 = in embedding dimension. Care should be taken however in or- σ29 an embedding with m = 100. der to obtain meaningful results: the last ratios should be Using again the 5% percentile criterion, for m = 20 we excluded from this minimum calculation. The last (i.e. the have a threshold of about 0.75. As for the case m = 10, least important) components of the SSA decomposition are we obtain false alerts at about 115 and 152 hours before the in fact associated to what is essentially a noise signal (8), true explosion. Also in this case however, the explosion is and should therefore be removed. We can then finally define: preceded by a warning more than 6 hours in advance. ρˆq = min {ρk − µk} q < m For m = 100 the 5% percentile threshold corresponds to k=1..q an absolute value of about 0.89. Also in this case we have where µk is the average of the ratio ρk that can be estimated false alerts at about 115 and 152 hours before the true explo- during a normal period of activity (in our case the first 4 days sion, and an additional one about 48 hours before the explo- in the graph). A suitable value of q should be chosen; a rule sion. of thumb is to take 1 ratio ρ3 q ' m 0.95 3 In Fig. 6 the time evolution of the parameter ρˆq is shown for 0.9 q = 4 in an embedding dimension m = 20. The decreasing 0.85 anomalous behaviour is again very evident before the parox- 908 R. Carniel et al.: Singular values decoupling 0.8 ysmal event. 0.75 threshold 5% ρˆ4 7 Conclusions 0.7 ratio 0.05 0.65 false positive 0.6 0.00 Forecasting paroxysmal events such as the one recorded on Stromboli on 5 April 2003 is of paramount importance, es- 0.55 −0.05 t[h] pecially at a volcano like Stromboli, where tourism is a ma- 0.5 −250 −200 −150 −100 −50 0 jor economical resource. This is undoubtly a difficult task −0.10 if we consider the forecast in a deterministic sense. How- σ6 Fig. 4. ρ3 = in an embedding space of dimension m = 20, for σ5 −0.15 ever, if we consider the risk of a volcano in a probabilistic threshold 5% Stromboli volcanic tremor sense (Aspinall et al., 2003, 2005) the issuing of a Tempo- −0.20 rary Increase in Probability of a paroxysmal event would be ρ already a considerable success. In order to do this, a num- ratio 15 −0.25 0.96 ber of parameters should be monitored and their anomalies t[h] 0.94 −0.30 highlighted and weighted in a bayesian sense. Different use- −250 −200 −150 −100 −50 −0 0.92 ful parameters can be derived from the Singular Spectrum 0.9 Fig. 6. ρˆ in an embedding space of dimension m = 20, for Strom- Analysis, one of which – the rk parameter weighting the rel- threshold 5% Fig. 6. ρˆ44in an embedding space of dimension m=20, for Stromboli 0.88 volcanicboli volcanic tremor. tremor ative importance of the first k SSA components in the con- 0.86 struction of a geophsyical signal – was proposed by Carniel In Fig. 4 a phase is evident with a peculiar behaviour start- 0.84 et al. (2006). In this paper we have shown the potential of ing about 40 hours before the explosion. This is the time at 0.82 false positive another family of parameters, ρk, which measure the degree inwhich the graph). Carniel A et suitable al. (2006) value observe of q should the first be successful chosen; a ap- rule 0.8 of coupling between successive singular values. A practical ofplication thumb is of to the take Failure Forecast Method (FFM), that gives a way of getting rid of the problem of how to choose which 0.78 forecast for 5 April 2003 at 15 GMT (see line 1 in Fig. 9 of of these couplings to monitor was also proposed, with the 0.76 t[h] Carniel et al. (2006)). More than 6 hours before the explo- −250 −200 −150 −100 −50 0 1 parameter ρˆq , which looks for the (significant) minimum of sion, the ρ2, ρ4, ρ15 and ρˆ4qparameters' m (see Fig. 3, 4, 5 and 6 σ30 3 these degrees of coupling. An important brand new class of Fig. 5. ρ15 = in an embedding space of dimension m = 100, respectively) go below the percentile threshold of 5%. At this σ29 for Stromboli volcanic tremor time the FFM (see line 2 in Fig. 9 in Carniel et al. (2006)), af- monitoring parameters are therefore available for inclusion in bayesian eruptive scenario evaluators. Obviously, further Inter Fig. an apparent6 the time recovery evolution of the of stability, the parameter is able againρˆq is to shown give for q=4 in an embedding dimension m=20. The decreasing (and possibly forward) testing on other cases is needed for anomalous behaviour is again very evident before the parox- testing the effective forecasting capabilities of these new pa- ysmal event. rameters. In Fig. 4 a phase is evident with a peculiar behaviour start- ing about 40 hours before the explosion. This is the time at which Carniel et al. (2006) observe the first successful ap- Acknowledgements. The methodologies developed and applied in plication of the Failure Forecast Method (FFM), that gives a this work are partially supported by the projects “V4 – Conception, verification and application of innovative techniques for studying forecast for 5 April 2003 at 15 GMT (see line 1 in Fig. 9 of volcanoes” by the Istituto Nazionale di Geofisica e Vulcanologia Carniel et al. (2006)). More than 6 h before the explosion, the – Dipartimento Protezione Civile, Italy, “TEGETEIDE – Tecnicas´ ρ2, ρ4, ρ15 and ρˆ4 parameters (see Fig. 3, 4, 5 and 6, respec- geofísicas y geodesicas´ para el estudio de la zona volcanica´ tively) go below the percentile threshold of 5%. At this time activa del complejo Teide – Pico Viejo (Tenerife)”, Spain, TEI- the FFM (see line 2 in Fig. 9 in Carniel et al. (2006)), after an DEVS (Spain, CGL2004-05744), INGV-DPC V4 “Conception, apparent recovery of the stability, is able again to give a (even verification, and application of innovative techniques to study more reliable) forecast, for 5 April 2003 at 10:20 GMT, with active volcanoes” (Italy) and PRIN 2004131177 “Numerical and a less than 3 h difference with respect to the real occurrence graphical methods for the analysis of time series data” (Italy). The of the explosion. SCILAB package (see http://www.scilab.org) was used to integrate the different analysis routines. Italian Civil Defence provided A practical problem is that there are some embedding di- logistic support when access to the summit area was forbidden. mension ranges for which the anomalous behaviour disap- The seismic station was originally installed under a research grant pears. One can think of a minimization procedure similar to of the Italian Gruppo Nazionale di Vulcanologia. The authors the one adopted for the index of the ratio, but exploring e.g. gratefully acknowledge the collaboration of J. Alean, S. Ballaro,` every embedding dimension in the range m=10 to 500, al- S. Calvari, C. Cardaci, M. Fulle, A. García, A. Llinares, P. Malisan, though theoretically possible, can lead to unacceptable com- V. Perin, M. Ripepe and three anonymous referees in the various puting times. stages of data acquisition, logistic help, data analysis, interpretation and manuscript reviewing. Also the choice of the minimum embedding dimension to use is not straightforward. The value suggested by the first Edited by: J. Marti zero of the autocorrelation function or by the first minimum Reviewed by: three referees of the mutual information (i.e. m=10) seems in fact too low.

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