J Seismol DOI 10.1007/s10950-014-9453-0

ORIGINAL ARTICLE

A reappraisal of seismic Q evaluated at Mt. Etna volcano. Receipt for the application to risk analysis Edoardo Del Pezzo & Francesca Bianco & Elisabetta Giampiccolo & Giuseppina Tusa & Tiziana Tuvé

Received: 7 April 2014 / Accepted: 4 August 2014 # Springer Science+Business Media Dordrecht 2014

Abstract A new approach in dealing with seismic risk in the volcanic areas of Italy, by taking into account the possible occurrence of damaging pre- or syn-eruptive seismic events, is exciting the scientific interest and is actually the topic developed in several research projects funded by the European Community (e.g., UPStratMAFA, www.upstrat-mafa.ov.ingv.it/UPstrat/) and the Civil Defense Department of Italy. To achieve this goal, it is necessary to have a detailed knowledge of the local attenuation-distance relations. In the present paper, we make a survey of the estimates of the seismic quality factor of the medium reported in literature for the Etna area. In the framework of a similar paper published for the Campi Flegrei zone in Southern Italy, we first review the results on seismic attenuation already obtained for Etna and then apply a standard technique to separately measure intrinsic and scattering attenuation coefficients from passive seismic data recorded by the Etna seismological network. Indications are then given for the correct utilization of the attenuation parameters E. Del Pezzo (*) : F. Bianco Istituto Nazionale di Geofisica e Vulcanologia, sezione di Napoli “Osservatorio Vesuviano”, Naples, Italy e-mail: [email protected] E. Giampiccolo : G. Tusa : T. Tuvé Istituto Nazionale di Geofisica e Vulcanologia, sezione di Catania “Osservatorio Etneo”, Catania, Italy E. Del Pezzo Instituto Andaluz de Geofisica, Universidad de Granada, Granada, Spain

to obtain the best candidate quality factor Q to be used in this area for seismic risk purposes. Keywords Seismic attenuation . MLTWA method . Mt. Etna

1 Introduction The problem of correctly evaluating risk associated with volcanic eruptions is currently addressed to also include the effects produced by earthquakes accompanying the eruption. Moderate earthquakes may occur during eruptive cycles, producing local but severe damage. Several moderate earthquakes with epicenter in the Etna area, often producing local damage, have occurred with moment magnitude in the range of 4.2 to 5.3 (see Azzaro et al. 2006) in the time period 1865–2002, many of them accompanying Etna eruptions. This is one of the main reasons why the European Community (project UPStrat-MAFA, www.upstrat-mafa.ov.ingv.it/UPStrat/) has planned to refine the prevention policies for disaster occurrence, adding Etna in the zones under attention. For a complete revision of the seismic risk studies, a detailed knowledge of local (inside the volcanic area) attenuation-distance relationships becomes crucial, as volcanic areas behave quite differently from the tectonically active zones. Similar considerations have been done for the Campi Flegrei area (near Naples, Italy) by Del Pezzo and Bianco (2013). For risk purposes, the empirical amplitude (max displacement, velocity, or acceleration) decay curve with

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distance, calculated from experimental data, is commonly taken as the characteristic attenuation-distance curve in the area under study. In volcanoes, unfortunately, local seismicity is often highly space clustered, so that there is sometimes a hypocentral distance interval in the data available insufficiently small to calculate a stable empirical attenuation-distance curve; in addition, the local site effects may be severe in volcanic areas, due to the high geological heterogeneity, thus affecting the local estimates of max displacement (or amplitude or acceleration). To avoid these difficulties, an approach different from the empirical one can be used, based on the parametric relationship expressed in terms of quality factor of the earth medium, that, in turn, should be estimated taking into the necessary account all the possible site effects. Recently, Convertito and Zollo (2011) formulated the seismic hazard in the area of Campi Flegrei using this approach. The purpose of the present paper is to contribute to the formulation of the seismic hazard in the Etna area, with a new estimate of the quality factor, Q. From a physical point of view, the energy decay for wave cycle (−ΔE/E) is expressed in terms of the quality factor parameter, Q, in turn related to the attenuation coefficient η through the following equation: η¼

2π f f ¼ − ΔE=E vQ v

ð1Þ

where v is the wave speed and f is the frequency of the wave motion. It is noteworthy that Q depends on the wave type (P, S, surface, coda) and from the physical mechanism of energy dissipation (intrinsic or scattering, see definitions in “Appendix. Methods for the evaluation of Q from seismograms”). Since each type of Q associated describes a different physical process, it is important to use the appropriate Q to correctly model the seismic energy attenuation with distance in seismic risk studies. In the present note, we first review several estimates of seismic quality factor, Q, already calculated (and reported in literature) for the Mt. Etna region. Then, we calculate a separate estimate of S-wave intrinsic and scattering quality factors from seismological data using the standard multiple lapse time window analysis (MLTWA) technique (e.g., Sato and Fehler 1998), based on the so-called coda normalization procedure that eliminates the site effects (see “Appendix. Methods for the evaluation of Q from seismograms”). When in a given area several, and sometimes contrasting, estimates of Q

have been reported in the literature, the problem is what is the Q parameter to choose for seismic risk applications. A wrong choice of Q may have severe consequences on the quality of the peak ground acceleration (PGA) estimate. We therefore calculate the pattern of the S-wave peak acceleration attenuation with distance for this area for both MLTWA and Spectral ratio estimates of S-wave total Q, showing how the different values of Q modify such a pattern.

2 A review of the Q measurements in Etna volcano In this section, we give a review of the results already published. For the sake of simplicity, definitions and methods are described in “Appendix. Methods for the evaluation of Q from seismograms,” where symbol definitions are also reported. The present review is essentially based on four papers: Patané et al. (1994), hereafter cited as PA94; Del Pezzo et al. (1995), hereafter cited as DP95; de Lorenzo et al. (2006), hereafter cited as DL04; and Giampiccolo et al. (2007), hereafter cited as GP07. In Table 1 and Fig. 1, a summary of the main results drawn by the above papers is reported. In Fig. 2, we show the source positions of the data used in each of the above-cited papers. PA94, using the spectral ratio technique, estimated the quality factors (total Q) for P- and S-waves and their ratio (Qα, Qβ, Qα/Qβ) in the frequency range between 2 and 20 Hz. These authors used a dataset composed of 85 local earthquakes with locations inside the Etna complex, with a depth range between the surface and 25 km and with local magnitudes between 1.7 and 3. Their results indicate a regular increase of Qα and Qβ with frequency; PA94, however, observed that Qα calculated for a subset of the events with the shallowest hypocenters (depth less than 5 km) is smaller than the average Qα, indicating a depth dependence of the P-wave quality factor. By applying the spectral ratio technique, GP07 reestimated the attenuation law for both compressional and shear waves in the Etna area using local shallow seismicity (depth less than 5 km, magnitudes between 2.6 and 4.2). Results indicate strong azimuthal variations in both Qα and Qβ. Marked variations of both Qα and Qβ with frequency were also measured. These results indicate a high heterogeneity of the shallow crust in Etna volcano. DP95 used the methodology developed by Wennerberg (1993) (see “Appendix. Methods for the

J Seismol Table 1 S-wave quality factor Qβ with its error; P-wave quality factor Qp with its error; intrinsic Q, Qi; scattering Q, Qs; total Q, QT found in the previous attenuation studies of Giampiccolo et al. 2007, Patané et al. 1994, de Lorenzo et al. 2006 Del Pezzo et al. 1995

Frequency (Hz)

Qβ

Qp

σ−Qp

Qi

Qs

QT

Reference

1

32

1

16

1

GP07

2

45

1

28

2

GP07

4

64

2

49

6

GP07

8

91

4

84

13

GP07

12

111

6

117

21

GP07

18

136

9

162

33

GP07

1

13

1

PA94

2

23

1

PA94

4

41

3

PA94

8

73

9

PA94

12

103

15

PA94

18

144

24

1–50

57

42

DL06

1

27

73

20

2

50

97

33

DP95

4

91

133

54

DP95

8

168

191

89

DP95

12

239

239

119

DP95

18

342

302

160

DP95

evaluation of Q from seismograms” for details) to separately estimate Qi and Qs from direct measurement of Qc (Q-coda) and Qβ. DP95 used the estimate of Qβ reported in PA94 and directly calculated Qc from the same dataset used by PA94. The results shown in the paper by DP95 show that the value of the total Q, which coincides with Qβ, is similar to the one by PA94 and GP07. Moreover, Qs is higher than Qi below 8 Hz, while the contrary occurs for higher frequency values. These results indicate the prevalence of intrinsic dissipation phenomena for S-waves at low frequency. Fig. 1 Quality factors estimated for Etna zone by DP95 (pink), PA94 (blue), and GP07 (black) as a function of frequency. The blue, gray, and black shaded areas define the error interval for Qβ estimates obtained by PA94, QP estimates obtained by GP07, and Qβ estimates obtained by GP07, respectively

σ−Qβ

DP95

DL04 estimated total Q for P-waves, Qα, under the assumption that it is constant in the frequency range analyzed (1–50 Hz). It is noteworthy that the Qα estimates of GP07 are inside the error bar associated with the estimate of DL04 (see Table 1).

3 Dataset In the present paper, we benefit of the use of data acquired by the high-quality permanent seismic

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Fig. 2 a Epicentral distribution (black dots) and stations (triangles) for the data used by PA94 and by GP07. b The same for data used by DP95 (no hypocentral section available) and by DL04. The figure has been redrawn from data

network managed by the Istituto Nazionale di Geofisica e Vulcanologia, Osservatorio Etneo, in Catania (INGV-OE). We use a total number of more than 1,500 waveforms, selected from a set of 136 high-quality seismic events with ML>2.5 recorded at Mt. Etna in the period 2005–2013. All the stations considered for the present analysis are 24-bit digital stations, equipped with broadband three-component Nanometrics Trillium seismometers, having a flat response within the 0.01–40-s period range. At the INGV monitoring center in

Catania, data are stored with a sampling interval of 0.01 s over consecutive, 2-min-long digital archives. Earthquake data have been selected from the INGV-CT catalogue (Gruppo Analisi Dati Sismici, 2014) on the basis of the following: (i) minimum signal to noise ratio higher than 3, (ii) absence of spikes and other disturbances in the waveforms, and (iii) absence of secondary events in the coda. The earthquakes located using the Hypoellipse code (Lahr 1999) and the 1-D velocity model derived from Hirn et al. (1991). The spatial

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Fig. 2 (continued)

distribution of the seismic stations of the INGV-OE seismic network, the source locations of the earthquakes analyzed, and a sketch of the main tectonic features of Mt. Etna are shown in Fig. 3. During the study period, seismicity was spatially distributed at depths spanning from 2 km a.s.l. to 15 km b.s.l. in the southern and eastern sectors of the volcano. In the western sector, seismicity spreads over a wide volume from the sea level until 30 km b.s.l. We organized the selected data in three different subsets: the first containing the seismograms with hypocentral distances between 0 and 20 km, the second with distances between 20 and 50 km, and the third with distances in the whole range (0–50 km).

4 Separate estimate of Qi and Qs The dataset described above has been used in the present paper to separately estimate the intrinsic and scattering quality factor for Mt. Etna following two different approaches: (a) applying the Wennerberg method and (b) through the classical MLTWA approach. 4.1 Separate estimate of Qi and Qs using the Wennerberg method We used the Wennerberg technique (for the details of the method, see “Appendix. Methods for the evaluation of Q from seismograms”) to retrieve the Qi and Qs values

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Fig. 3 Upper left panel: Map of Mt. Etna and epicenters of the earthquakes used (black circles). White triangles indicate the three component broadband seismic stations of INGV-OE network. The inset map on the lower left shows the location of the study area. Right panels: West–east and north–south sections. The black dots

are the source locations. Lower panel: Simplified structure of Mt. Etna, reporting the main rifts and the main fault zones. TFS Timpe Fault System, RFS Ragalna Fault System, PFS Pernicana Fault System (redrawn after Norini and Acocella (2011))

for the new dataset, also with the aim to do a comparison with the results obtained by DP95. In order to achieve this goal, we first reevaluated Q c for our data. Introducing the estimate of Qβ obtained by GP07 (already described in the previous section) in Eqs. 7 and 8 reported in the “Appendix. Methods for the evaluation of Q from seismograms,” we separately obtained results for Qi and Qs that are consistent with those described in DP95. In Fig. 4, we plot the estimates obtained using the Wennerberg method applied to both the present dataset and the old one, while in Table 2, we report only the present estimates (the old ones by DP95 are in Table 1). The two estimates (present data and old dataset)

are slightly different. As reported in “Appendix. Methods for the evaluation of Q from seismograms,” this is due to the different (and independent) estimates of Qc and Qβ utilized. 4.2 Separate estimate of Qi and Qs using MLTWA technique Multiple lapse time analysis (see “Appendix. Methods for the evaluation of Q from seismograms” for any detail on the method) is probably the most powerful, stable, and robust technique to calculate Qi and Qs from the seismogram energy envelopes. Since the pioneering

J Seismol Fig. 4 Present (blue) and previous (pink) estimates of intrinsic Q, scattering Q, and total Q obtained using Wennerberg’s method

papers of Hoshiba et al. (1991) and Mayeda et al. (1992), a number of papers have been published on this technique (see Sato et al. 2012), all showing the importance of the scattering phenomena in determining the total attenuation amount. The three time integrals of the energy envelopes (see “Appendix. Methods for the evaluation of Q from seismograms”) are estimated in the present paper by squaring the Fourier transform of the seismogram instead of (as ordinarily carried out in literature) filtering in the selected frequency band and then calculating the mean squared (MS) energy envelope of the filtered signal. We Table 2 Intrinsic Q, Qi; scattering Q, Qs; total Q, QT, estimates obtained using both the MLTWA and Wennerberg methods

use three consecutive adjacent time windows, respectively, starting at tS (the travel time of the S-waves), tS +12 and tS +24 s. The normalization window starts at 45 s from origin time, with a duration of 12 s. For each of the three time windows, the dataset composed of the integrals of the energy envelopes as a function of source-receiver distance was smoothed with a moving window containing 10 data points and sliding each step of 5 points. We have fit both unsmoothed and smoothed data to Eq. 9 in “Appendix. Methods for the evaluation of Q from seismograms” using the procedure described in this section. As the results obtained are

Frequency (Hz)

Qi

Qs

QT

Distance range (km)

Method

1.5

144

16

14

0–20

MLTWA

3.0

177

38

31

0–20

MLTWA

6.0

262

190

110

0–20

MLTWA

12.0

823

2,740

633

0–20

MLTWA

1.5

156

52

39

0–50

MLTWA

3.0

188

126

75

0–50

MLTWA

6.0

266

298

141

0–50

MLTWA

12.0

808

2,500

611

0–50

MLTWA

1.5

161

33

27

0–50

MLTWA

3.0

191

80

56

0–50

MLTWA

6.0

266

245

127

0–50

MLTWA

12.0

819

2,650

625

0–50

MLTWA

1.0

38

190

32

0–50

Wennerberg

2.0

63

158

45

0–50

Wennerberg

4.0

105

163`

64

0–50

Wennerberg

8.0

177

186

90

0–50

Wennerberg

12.0

240

206

111

0–50

Wennerberg

18.0

326

233

136

0–50

Wennerberg

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Fig. 5 I1, I2, and I3 (Eq. 13) as a function of source–receiver distance (blue, violet, and yellow dots, respectively), smoothed with an average of 10 points, sliding of 5 points. In the same plots, the corresponding theoretical quantities

T1, T2, and T3 (blue, violet, and yellow continuous lines, respectively) calculated using the best fit values of B0 and L e − 1 (continuous lines, corresponding colors) are superimposed

almost coincident for smoothed and unsmoothed data, we report here only those for smoothed data (Fig. 5). The estimates of the parameters B0 and Le−1, respectively the seismic albedo and the extinction length inverse (see definition in “Appendix. Methods for the evaluation of Q from seismograms”), with their uncertainties, are used to calculate (see Eqs. 10 and 11 in the “Appendix. Methods for the evaluation of Q from seismograms”) Qi and Qs that are reported in Table 2. In the plots of Fig. 6, we represent the frequency pattern for Qi, Qs, and QT (1/QT =1/Qi +1/Qs) obtained for the three data subsets at different hypocentral distances described above. The uncertainty estimates are plotted as a gray strip only for the estimate of total Q, QT, in order to avoid superpositions. The errors for Qi and Qs are of the same order as those for QT. In Fig. 6, Qi and Qs obtained with MLTWA show almost the same patterns for the three data subsets. In particular, for frequencies lower than 8 Hz, scattering attenuation slightly prevails on the intrinsic one, while the

contrary occurs for higher frequencies; this is true for the whole hypocenter distance range. The Qi and Qs behavior is in substantial disagreement with the results of the Wennerberg technique (Fig. 4) that shows intrinsic attenuation prevailing on the scattering one, while the contrary occurs for MLTWA results. This discrepancy is difficult to explain, but it is noteworthy that, differently from the Wennerberg technique, MLTWA takes implicitly into account the site effects (Sato et al. 2012), due to the coda normalization procedure (see “Appendix. Methods for the evaluation of Q from seismograms”).

5 Discussion and conclusions We have reviewed the results about the estimation of the seismic attenuation, both for P- and S-waves in the area of Mt. Etna volcano, and checked, using the present

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Fig. 6 MLTWA estimates of Qi, Qs, and QT in the three selected distance ranges (0–20 km at the top, 20–50 km at the middle, 0–50 km at the bottom)

dataset, the results obtained by DP95 which utilized the Wennerberg technique to separately estimate Qi and Qs. We have finally applied the MLTWA technique to a large dataset of local earthquakes in order to separately calculate intrinsic and scattering Q (Qi and Qs) and hence to obtain a new estimate of total Q, QT, for Swaves. For the sake of clarity, note that QT obtained with MLTWA should coincide with the direct total quality factor of S-waves Qβ measured with other techniques (PA94, GP07). The present MLTWA results suggest that the values of Qβ for S-waves are almost comparable with those already calculated (and described above), except in the frequency band between 8 and 16 Hz, where the MLTWA estimate gives QT greater than Qβ showing the greatest error bar. The wide error bar associated with the estimate of QT in the highest frequency band may be explained as the effect of increasing noise

power in this band, which could produce a bias in the estimate of coda energy at long lapse times. This in turn may produce a bias in the estimate of I1, I2, and I3 (see point 2 in the “Appendix. Methods for the evaluation of Q from seismograms”) and hence in the fit. The MLTWA estimates obtained by considering three different distance ranges (0–20, 20–50, and 0– 50 km) suggest that attenuation parameters do not depend on distance, being the Qi, Qs, and QT values approximately of the same order. However, scattering attenuation prevails over the intrinsic absorption at frequencies lower than 8 Hz, whereas the contrary occurs above 8 Hz (Fig. 6). The present result evidences that beneath Mt. Etna, the heterogeneities which generate the scattering phenomena play an important role in determining the attenuation of the seismic waves notwithstanding the presence of molten materials under the

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volcanic complex. The obtained result is in agreement with the findings retrieved on other volcanoes such as Mt. Merapi (Wegler and Luhr 2001) and Mt. Vesuvius (Bianco et al. 1999; Del Pezzo et al. 2006). Applying the Wennerberg (1993) method, we found that Qs dominates over Qi (i.e., intrinsic attenuation dominating over scattering attenuation) below 8 Hz (Fig. 4). For frequencies larger than 8 Hz, scattering attenuation slightly prevails over intrinsic attenuation. A similar result has previously been obtained at Mt. Etna by DP95 and by Del Pezzo et al. (1996) that already attempted to separate Qi and Qs by applying the Wennerberg method. The values and the frequency pattern for Qi and Qs obtained using MLTWA approach are thus in disagreement with the results obtained through the Wennerberg method. Correctly addressing the source of the observed discrepancy is not a simple task; however, the Wennerberg method has been applied using Qβ and Qc values that were separately and independently obtained. In particular, Qβ is that obtained by GP07 for a dataset characterized by sources embedded in the first 5 km of the crust, while Qc is that calculated for the new dataset with sources deeper than those of GP07. Consequently, the adopted values of Qβ and Qc are not strictly characteristic of the same earth volume, and this may result in a bias for the results obtained following the Wennerberg approach. A second important bias in the estimates done with the Wennerberg method may come from possible site effects affecting the estimate of Qβ. Conversely, the MLTWA method uses a homogeneous dataset providing self-consistent estimates. We are therefore confident that the MLTWA estimates of

Fig. 7 Intrinsic, scattering, and total Q obtained in this study (black), compared with the Qi, Qs, and QT values retrieved for the Messina Strait (Tuvé et al. 2006, blue dashed line) and the Hyblean Foreland (Giampiccolo et al. 2006, red line)

Qi, Qs, and hence QT are less biased than those from the Wennerberg method. An interesting comparison of the present MLTWA results can be carried out with scattering and intrinsic Q−1 obtained, using MLTWA (with the same parameters as those used in the present approach), in the adjacent tectonic areas of Hyblean Foreland (Giampiccolo et al. 2006) and Messina Strait (Tuve’ et al. 2006). Looking at Fig. 7, we observe that at Mt. Etna, attenuation due to scattering is higher than in the Hyblean Foreland and in the Messina Strait, in the first three frequency bands centered at 1.5, 3, and 6 Hz. This can be interpreted as due to a higher heterogeneity of the Etna upper crust with respect to the two adjacent zones. Moreover, QT−1 is closer to Qi−1 in both tectonic areas whereas it is closer to Qs−1 at Mt. Etna. Below 3 Hz intrinsic absorption at Mt. Etna is similar to that of Hyblean Foreland, whereas above 3 Hz, it is higher than that in the Hyblean Foreland and approximately of the same order of Q−1 estimated in the Messina Strait. Our results show that heterogeneity, which generates the scattering phenomena (explained with the presence of different directions of faults, medium with strong layering, magma intrusions, source of heat flows, etc.), plays a key role in the attenuation mechanism at Mt. Etna with respect to that in the two adjacent tectonic environments. MLTWA estimates of QT is, in our opinion, the best estimate of the total S-wave seismic attenuation at Etna, as it represents the average of all the QT values over the volume which encompasses the propagation of the shear waves forming the whole seismogram (Sato et al. 2012);

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this may be one of the reasons why it could be considered the most adaptable to be used for risk analysis, as it is a stable estimate of the total Q characteristic of the whole zone under study. To demonstrate with a simple example how the different estimates of Qβ affect the prediction of the ground motion, we use the method first developed by Boore (1983) and then widely used throughout the world to simulate the ground motion in two different attenuation media, the first characterized by a quality factor given by the estimate obtained by GP07 and the second by the present MLTWA estimate. We simulate the peak ground acceleration (PGA) for a given earthquake magnitude using the random vibration theory (RVT) as described also in Boore (2003). We followed the approach described in Galluzzo et al. (2004) based on the Parseval-Plancherel theorem which relates the root mean square (rms) of the time series to its power spectrum. From the velocity or acceleration theoretical spectrum, corresponding to the Brune model, we calculate the rms. Then, we generate a set of N=104 Gaussian random numbers with σ=rms and take the maximum of this set as an estimate of the max acceleration (PGA) or velocity (PGV). We used typical values of stress drop for the area of Etna, 2·106 Pa, a source with a moment of 2.8 · 1014 Nm, corresponding to a M = 3.6 earthquake, and a corner frequency of 3 Hz. We run the simulation twice, one for each attenuation relationship, for a set of source-to-receiver distance. In Fig. 8, we plot the pattern of PGA values as a function of distance, together with their fit to the following relationships (see Kramer 1996, page 88

Fig. 8 Log PGA as a function of distance (meters) synthesized with the method (Boore) described in the text (dots) for total Q obtained through the MLTWA approach in the present study (red line) and Qβ estimates obtained by GP07 (blue line)

for a wide discussion on the empirical relationship describing peak acceleration decay with distance) Log½PGAŠ ¼ a þ bM þ cM 2 þ dR þ f Log½RŠ

ð2Þ

where a, b, c, d, and f are the parameters to be determined by the fit; M is the earthquake magnitude; and R is the distance. As it can be seen by the plot, the patterns obtained by the simulations in different attenuation conditions are quite different, indicating the importance of correctly measuring the quality factor for the area under study.

6 Appendix. Methods for the evaluation of Q from seismograms 6.1 Definitions We briefly give the definitions and then the methods currently adopted to estimate Q from short-period seismograms. For the sake of simplicity, the description is essential. Part of this appendix is taken from Del Pezzo and Bianco (2013). Total quality factor, QT, is defined as the total energy fractional loss for wave cycle (Aki and Richards 1980): ΔE=E ¼ 2π=QT

ð3Þ

Hereafter, QT will be denoted as Qα or Qβ indicating, respectively, the total quality factor for P- or S-waves.

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The energy losses are due to many factors, mainly scattering and intrinsic dissipation, both (in different combinations) contributing to the decrease of the energy as a function of distance. To account for these two factors, Eq. 3 can be explicitly written as
 1 1 ΔE=E ¼ 2π þ ð4Þ Qi Qs where Qi is the intrinsic and Qs is the scattering quality factor. To correctly take into account all the energy losses associated with the direct phases, Qα or Qβ needs to be calculated. The time envelope of the S-coda filtered at a given frequency band centered in f is characterized by the following exponential decaying shape: Aðt∣ f Þ∝

 1 exp −πf tQc −1 ½ f ŠŠ n t

where Qc is the coda quality factor, f is the frequency, t is the lapse time (from the origin time), and n is equal to (i) 0.5 (in case of predominance of surface waves), (ii) 0.75 (in case of predominance of diffusive waves), or (iii) 1 (in case of predominance of body waves). 6.2 Methods In the present paper, we review results obtained with the “spectral ratio” method (PA94 and GP07), the Wennerberg method (Wennerberg 1993), and pulse broadening or “rise time” (Gladwin and Stacey 1974; Kjartansson 1979). A more complete survey of methods and techniques is reported in the Appendix of Del Pezzo and Bianco (2013). 6.3 Spectral ratio Taking the ratio between the amplitude spectrum at ith station and the amplitude spectrum at a rth (reference) station, after correcting for instrument and site and calculating the logarithms, it results, for the jth earthquake, that Ln½Rir Š j ¼ Ln

  rr j −πf t i j −t r j =Q½ f Š ri j

ð5Þ

where Rir represents the spectral ratio between receivers r and i, tij is the travel time measured at station i for event j, and trj is the travel time measure at reference station r for event j. Measuring the spectral ratios at a number of stations and averaging in successive

frequency bands centered at f, one can invert for the total attenuation Q−1 for each frequency band. 6.4 Wennerberg method Wennerberg (1993) compared the time pattern of coda amplitude, AZ, predicted by the multiple scattering model (Zeng 1991) with that of the coda amplitude, AAC, predicted by the single scattering model of Aki and Chouet (1975). This author found that the ratio between the coda amplitude decay curves, AAC/AZ, plotted as a function of τ can be best fit to a straight line in a wide range of τ values. Here, τ indicates the mean free time (τ=ωt/Qs, where t is the lapse time measured from the origin of the earthquake). Wennerberg (1993) thus established a mathematical relationship between AAC/ AZ and τ, valid in a wide interval of mean free times, from which he derived the following formula linking the observed value of Qc associated with the single scattering model, to the intrinsic and scattering Q, Qi and Qs, describing the coda pattern in the multiple scattering model: 1 1 1−2δ½τ Š ¼ þ Qc Qi Qs

ð6Þ

where 1−2δ(τ)=−1/(4.44+0.738τ) and ω is the angular frequency. As 1/Qβ =1/Qi +1/Qs, as already written in Eq. 4, from Eq. 6, it can be easily derived that ! 1 1 1 1 ¼ − ð7Þ Qs 2δ½τ Š Qβ Qc½τ Š

1 1 ¼ Qi 2δ½τ Š

1 2δ½τ Š−1 þ Q c ½τ Š Qβ

! ð8Þ

In order to obtain separate estimates of Qi and Qs, independent estimates of Qc and Qβ, both measured as a function of frequency, f, are necessary. It is noteworthy that in this method, the correct estimate of Qc and Qβ is critical: Both parameters should be estimated for the same earth volumes (Wennerberg 1993). 6.5 Multiple lapse time windows analysis The multiple scattering process may be formally described through the radiative transfer or transport equation (hereafter TE) that is an integral equation whose analytical solution in three dimensions is not already

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known (Sato et al. 2012). An approximate analytical solution in 3-D was found by Paasschens (1997) in case of uniform half space with constant

E ij ½r; t Š ≈

velocity. It describes the pattern of the seismogram energy envelope, Eij [r, t], as a function of lapse time, t, and distance, r, as

W 0 exp½−Le−1 vtŠ h r i δ t− þ 4πr2 v v

  2 !3=4 3 r2 1=8 2 h r i 1− v2ijt2 r  ij ij −1 −1 5 W 0 H t− ⋅  1− 2 2 3=2 exp −Le vt F 4vtB0 Le v vt 4πvt

ð9Þ

3B0 Le−1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where F ½xŠ ¼ ex 1 þ 2:026=x ; W0 is the energy at source, v is the wave speed in the half space, H is the Heaviside function, δ is the Dirac delta. ηin and B0 and Le−1 represent, respectively, the seismic albedo and the extinction length inverse, expressed in terms of Q by B0 ¼ Qr =Qs

ð10Þ

and Le−1 ¼

2πf v




1 1 þ QS Qi

 ð11Þ

The MLTWA method consists in the following operations: 1. Selection of three consecutive time windows in the seismogram. The three windows span, respectively, the intervals [tS, tS +Δt], [tS, tS +2Δt], and [tS, tS + 3Δt], where tS is the onset time of the S-waves. 2. Selection of the coda normalization time interval: [tc, tc +Δt] where tc is the coda reference time (the start time with respect to origin time). In the present analysis, Δt=12 s is chosen, also for point 1. 3. Evaluation of the seismogram energy envelope integrals in the four above-mentioned time windows, as a function of frequency and source-station distance. Differently from the common use (see Del Pezzo and Bianco, 2010a, as an example), we estimate such integrals Fourier transforming the seismogram portion contained in each time intervals, taking the module and squaring the result, thus obtaining an energy spectrum. Finally, we take the energy spectrum averaged over the two horizontal components and over frequency in the following frequency bands: 1–2, 2–4, 4–8, and 8–16 Hz, hence obtaining I1, I2, and I3, i.e., the energy

integrals in the three successive windows (see points 3 and 1). 4. Normalization of the energy spectra in the first three time windows by the fourth (coda) window. This is commonly named “coda normalization.” 5. The logarithm of the three normalized energy integrals, corrected for geometrical spreading factor (4πr2), is plotted as a function of source-receiver distance and fit to the corresponding theoretical values for each of the frequency bands described in point 2 above. The theoretical values are calculated integrating Eij [r, t] defined in Eq. 9 in the same time windows defined in point 1, normalizing for the integral of Eij [r, t] in the coda window, multiplying for 4πr2, and taking logarithms. In Del Pezzo and Bianco (2010a, b) and references therein, a complete discussion of this method is reported, together with the corresponding Mathematica-7 code. The unique difference with the present paper is point 2. The best fit values are obtained minimizing the misfit function Ndati       2    X 1  M B0; Le 1 ¼ I 1 ri B0; Le −T 1 I 1 ri B0Le −1 i¼1

     þ I 2 r1 jB0; Le −1 −T 2 ri jB0; Le −1 2         2  þ I 3 ri B0; Le −1 −T 3 r B0; Le −1 −T 3 ri jB0Le −1 i

ð12Þ where I1, I2, and I3 are the quantities calculated as described in point 2 and T1, T2, and T3 are the corresponding theoretical values (point 4). In the present paper, the couple of B0 and Le−1 corresponding to the

J Seismol

minimum of the misfit function is estimated using Mathematica-9 routine “FindMinimum” which utilizes the conjugate gradient method. Errors on B0 and Le−1 are estimated grid searching the B0 and Le−1 intervals for which the values of the misfit function are statistically equal at a given confidence (see Del Pezzo and Bianco 2010a, b for details). Acknowledgments This work has been partially supported by the following projects: V2-Precursori di Eruzioni in Vulcani Quiescenti: Campi Flegrei e Vulcano-Convenzione INGV-DPC (2012–2013); EPHESTOS CGL2011-2949-C02-01 (University of Granada, Instituto Andaluz de Geofisica); MEDiterranean SUper-site Volcanoes (MED-SUV) FP7 ENV.2012.6.4-2 grant agreement no. 308665 (European Community); and Urban Disaster Prevention Strategies Using MAcroseimic Fields and Fault Sources (UPStrat-MAFA), http://upstratmafa.ov.ingv.it/ UPStrat/ (European Community). Jens Havskov is gratefully acknowledged for his helpful revision.

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