Vol49,6,2006 1215 ANNALS OF GEOPHYSICS, VOL. 49, N. 6, December 2006 Key words intrinsic quality factor – stress drop – rise time – corrected Akaike information criterion 1. Introduction The intrinsic quality factor Qp of the com- pressional body waves is considered one of the geophysical parameters best correlated to the physical state of the rocks. This is because, as shown in laboratory studies (Kampfmann and Berckemer, 1985; Sato and Sacks, 1989) a type- Arrhenius exponential law relates Qp to the temperature T and the pressure P of the rocks. In volcanic areas, low Qp values are usually as- sociated with high temperature rocks (e.g., Sanders et al., 1995; de Lorenzo et al., 2001). However there are several reasons which make it difficult to determine the temperature of the rocks from estimates of Qp in the crust. First of all, Qp depends not only on the temperature but also on the percentage of fluid content in the rocks (Bourbiè et al., 1987). Morevoer, the Qp estimates are also dependent on the technique used to retrieve them (Tonn, 1989). These con- siderations lead to the conclusion that only a comparison between Qp and T in deep bore- holes can help us to locally calibrate a Qp-T re- lationship (e.g. de Lorenzo et al., 2001) and then to infer the temperature field from three- dimensional images. Intrinsic Qp at Mt. Etna from the inversion of rise times of 2002 microearthquake sequence Salvatore de Lorenzo (1)(2), Marilena Filippucci (1)(2), Elisabetta Giampiccolo (3) and Domenico Patanè (3) (1) Dipartimento di Geologia e Geofisica, Università degli Studi di Bari, Italy (2) Centro Interdipartimentale per la Valutazione e Mitigazione del Rischio Sismico e Vulcanico, Bari, Italy (3) Istituto Nazionale di Geofisica e Vulcanologia, Catania, Italy Abstract About three-hundred microearthquakes, preceeding and accompanying the 2002-2003 Mt. Etna flank eruption, were considered in this study. On the high-quality velocity seismograms, measurements of the first half cycle of the wave, the so-called rise time τ, were carried out. By using the rise time method, these data were inverted to infer an estimate of the intrinsic quality factor Qp of P waves and of the source rise time τ0 of the events, which represents an estimate of the duration of the rupture process. Two kind of inversions were carried out. In the first inversion τ0 was derived from the magnitude duration of the events, assuming a constant stress drop and Qp was inferred from the inversion of reduced rise times τ − τ0. In the second inversion both τ0 and Qp were inferred from the inversion of rise times. To determine the model parameters that realize the compromise between model sim- plicity and quality of the fit, the corrected Akaike information criterion was used. After this analysis we obtained Qp = 57 ± 42. The correlation among the inferred τ0 and Qp, which is caused by some events which concomitant- ly have high τ0 (>30 ms) and high Qp (>100) indicates that the technique used is able to model rise time versus travel time trend only for source dimensions less than about 80 m. Mailing address: Dr. Salvatore de Lorenzo, Diparti- mento di Geologia e Geofisica, Università degli Studi di Bari, Campus Universitario, Palazzo di Scienze della Ter- ra, Via E. Orabona, 4, 70125 Bari, Italy: e-mail: deloren- zo@geo.uniba.it 1216 Salvatore de Lorenzo, Marilena Filippucci, Elisabetta Giampiccolo and Domenico Patanè Some previous studies have examined the attenuation properties at Mt. Etna volcano by applying frequency domain techniques (Patanè et al., 1994, 1997). The results obtained by the above studies have shown that the effect of at- tenuation on seismic radiation at Mt. Etna is considerable. Patanè et al. (1994) reported sig- nificant variations of Q from P waves as a func- tion of depth. In particular, a drop of Q values for earthquakes located at depths less than 5 km was observed, supporting the idea that the up- per part of the crust and the shallow volcanic layers are characterized by low Q. Recently a P-wave attenuation tomography study was car- ried out at Mt. Etna, down to 15 km of depth, using high quality data recorded in the period 1994-2001 (De Gori et al., 2005). Results from this first tomography were refined in the shal- low layers (around 2 km) of the volcanic edifice by Martìnez-Arevalo et al. (2005), further con- firming significant variations of total Q as a function of depth and azimuth. In the frame of the European project named VOLUME (an acronym of «VOLcanoes Under- standing of Mass movEment») aimed to inves- tigate the movement of fluid masses inside Mt. Etna, a research line has been dedicated to the reconstruction of Qp images of the volcano with different techniques. This is because several studies have pointed out that the estimates of Q could depend on both the data to be inverted and on the technique used. The attenation pa- rameter Q−1 estimated by the inversion of seis- mic spectra at Mt. Etna by De Gori et al. (2005) and Martìnez-Arevalo et al. (2005) is the sum of two parameters, Qp−1 and Qc−1, which take in- to account two different and concomitant phys- ical phenomena. Qp−1 is the intrinsic attenuation of P waves, which is related to the losses of en- ergy due to the anelasticity of the Earth. Qc−1 is the scattering attenuation of the wave field and is related to the energy content of the secondary wave field, i.e. the wave field generated by the elastic heterogeneities inside the medium trav- elled by the waves (e.g., Mitchell, 1995; Del Pezzo et al., 2001). It has also been shown (e.g., Liu et al., 1994) that the methods based on the inversion of pulse widths of first P waves are more appropriate to estimate the intrinsic Qp, in that only the start of the signal, which is rela- tively free of complicated effects, such as com- plexities at the source and/or scattering from heterogeneities, is used. Based on these reasonings, this article pres- ents the results of a first study aimed to obtain a first estimate of the average intrinsic Qp at Mt. Etna by using the classic rise time method (Gladwin and Stacey, 1974). The analysis con- cerns about three-hundred microearthquakes recorded at Mt. Etna preceeding and accompa- nying the 2002-2003 Mt. Etna flank eruption. 2. The Mt. Etna geological framework Mt. Etna is located in the complex geody- namic framework of Eastern Sicily, where ma- jor regional structural lineaments play a key role in the dynamic processes of the volcano (e.g., Patanè and Giampiccolo, 2003). Several years of structural and geophysical observations have revealed that the orientation of most of the eruptive systems coincides with two structural trends NNW-SSE and NE-SW, ob- served both in the volcanic area and in the region- al context (e.g., Azzaro and Neri, 1992). These alignments are hypothesized as the main volcano- genetic structures (e.g., Gresta et al., 1998) which control the evolution of Mt. Etna, as their interfer- ence establishes a weakness volume along which magma can rise from depth (Rasà et al., 1995). Over the last 30 years Mt. Etna has had a high rate of eruptive events and therefore it con- stitutes one of the most important natural labo- ratories for the understanding of eruptive processes and lava uprising in basaltic-type vol- canic environments. Seismic observations al- lowed us to carry out detailed investigations on major aspects of seismicity. In particular, the most recent effusive eruptions which occurred in 1989, 1991-1993, 1999, 2001 and 2002-2003 have offered good examples for quantitative analysis based on seismic data in digital format. We focus on the 2002-2003 flank eruption which started in the night between October 26 and 27, 2002 with a seismic swarm in the central and upper part of the volcano. Fissures on both the NE and S flanks were activated with a huge lava emission and powerful explosive activity from the southern fracture field. The eruption 1217 Intrinsic Qp at Mt. Etna from the inversion of rise times of 2002 microearthquake sequence ended on January 28, 2003 after 94 days. More than 800 events were recorded during this period by the permanent seismic network managed by INGV. Most of these events have magnitude less than 3.0 and only few earthquakes reached a magnitude duration Md = 4.4. 3. Technique The rise time method is based on early ex- perimental (Gladwin and Stacey, 1974) and the- oretical (Kjartannson, 1979) studies. More re- cently, Wu and Lees (1996) showed that the theoretical relationship on which the method is based remains valid if we consider a heteroge- neous anelastic structure. The most limiting as- sumption of this method is that it neglects the directivity effect of the seismic radiation gener- ated by a finite dimension seismic source (Zol- lo and de Lorenzo, 2001). On a velocity seismogram the rise time can be defined as the time interval between the on- set of the phase and its first zero crossing time (fig. 1). If the spatio-temporal finiteness of the seismic source process can be neglected, the variation of the rise time versus the travel time of the wave will be given by (3.1) where τ0 is the rise time at the source, Qp is the quality factor and T is the travel time. The con- stant C was found to be equal to 0.5 for a con- stant Q attenuation operator (Kjartansson, 1979). The methods based on the inversion of the rise times are expected to give the most reliable estimates of the intrinsic attenuation (Liu et al., 1994). In fact, since only a very limited portion of the seismogram is used, the effects of multi- ple waves generated in the thin layers around the recording site are usually minimized. Finally, as a consequence of the point-like source assumption, the Qp inferred by using eq. (3.1) can be considered as the minimum Qp es- timated from pulse width inversion (de Lorenzo et al., 2004). 4. Data analysis We used a data set of about 300 well locat- ed events (Erx, Ery and Erz <1 km, Gap < 100° and RMS < 0.20 s), with magnitude duration ranging between 1.4 and 4.4 and focal depth mainly concentrated in the first 5 km, recorded during an episode of intense seismic activity which preceded and accompanied the 2002- 2003 Mt. Etna flank eruption. The earthquakes were selected from the catalogue among those re-located by using the most recent 3D velocity model (Cocina et al., 2005). Data from stations of the Mt. Etna permanent seismic network run by Istituto Nazionale di Geofisica e Vulcanolo- gia of Catania (INGV-CT) were analysed. Most of the stations considered in this analysis are equipped with one-component short period (1 s) sensors, except 8 equipped with three-com- ponent sensors. Although other stations were operating during the occurrence of the analysed seismic sequence, data were not as well record- ed as required for accurate seismic attenuation analysis in time domain. We discarded the events for which the rise times are available at fewer than four stations. The remaining dataset was then composed of 147 events, all having typical tectonic-earth- Q C T p 0τ τ= + Fig. 1. Schematic picture of rise time τ as measured on a velocity seismogram. 1218 Fig. 2. Epicenters of the events (grey circles) and recording stations (black triangles) considered in this study. Table I. The dataset available for this study. Md = magnitude duration; Id = identification number of the event; Date = month-day-hour and minute of occurrence of the event (year: 2002). Id# Date Md Id# Date Md Id# Date Md Id# Date Md Id# Date Md 1 10262135 2.4 1 10262135 2.4 2 10262146 2.3 3 10262155 2.4 4 10262204 2.4 5 10262218 2.1 6 10262225 2.5 7 10262228 2.4 8 10262233 2.4 9 10262240 2.5 10 10262313 2.4 11 10262327 2.4 12 10262346 2.4 13 10262349 2.4 14 10270007 3.5 15 10270010 3.3 16 10270012 2.6 17 10270016 2.7 18 10270021A 2.4 19 10270021B 2.7 20 10270035 3 21 10270036 3.1 22 10270041 3.2 23 10270101 2.7 24 10270107 2.4 25 10270113 3.3 26 10270215 2.5 27 10270218 3.2 28 10270229 3.5 29 10270239 3.3 30 10270242 3.4 31 10270250 4.2 32 10270328 2.8 33 10270417 2.9 34 10270502 2.5 35 10270521 2.6 36 10270531 3.3 37 10270546 3.4 38 10270606 3.4 39 10270626 2.8 40 10270628 2.9 41 10270649 2.9 42 10270732 3.2 43 10271007 2.7 44 10271024 2.7 45 10271210 2.7 46 10271216 2.6 47 10271442 2.5 48 10271456 2.7 49 10271602 2.9 50 10271607 2.6 51 10280912 3.2 52 10281140 3.1 53 10281151 2.8 54 10281627 3 55 10282325 2.8 56 10290131 2.6 57 10290232 2.3 58 10290834 2.9 59 10290913 2.8 60 10290956 2 61 10291002 4.4 62 10291004 3.1 63 10291013 2.8 64 10291017 2.5 65 10291018 2.1 66 10291022 1.7 67 10291025 1.6 68 10291034 1.6 69 10291035 2.9 70 10291059 2.1 71 10291102 4 72 10291122 1.7 73 10291221 2.1 74 10291325 2.7 75 10291639 4 76 10291714 4.1 77 10291907 1.7 78 10292035 2 79 10292224 2.8 80 10300000 3.1 81 10300216 2 82 10300220 2.5 83 10300720 2.5 84 10301005 2.2 Salvatore de Lorenzo, Marilena Filippucci, Elisabetta Giampiccolo and Domenico Patanè 1219 Intrinsic Qp at Mt. Etna from the inversion of rise times of 2002 microearthquake sequence quake waveforms, whose localizations, deduced from the INGV catalogue, are shown in fig. 2, to- gether with the position of the recording stations. The origin time, the identification number and magnitude duration of the events used in the study, computed by INGV, are given in table I. A total of 1053 data was available for the study. The rise time was measured on each first ar- rival P-wave by computing the time interval be- tween the onset of the phase and its first zero crossing time. We discarded all data for which the P signal to noise ratio was lower than 10, and considered only the waveforms on which the onset of P-wave was clearly readable. We did not perform the deconvolution for the instrumental response for the following three reasons: 1) First of all, the technique we are using does not need amplitude information. 2) Second, the effect of filtering operated in the deconvolution for the instrumental response could cause the generation of unwanted artifi- cial signals (see Mulargia and Geller, 2003 and references therein) which could bias the rise time estimate. 3) Finally, if we exclude one da- tum, the observed rise times vary between 0.025 s and 0.3 s (fig. 3), to which correspond an average frequency content ranging from a minimum of about 3 Hz to a maximum of about 40 Hz. Consequently the frequency content of the analyzed signals is always contained in the frequency band (1-50 Hz) where the instrument response is flat and cannot distort the duration and shape of the observed signals. Only data not affected by multipathing dur- ing the first half-cycle of the wave were consid- ered in the analysis. The multipathing effects Fig. 3. Hystogram of the measured rise times. Table I (continued). Id# Date Md Id# Date Md Id# Date Md Id# Date Md Id# Date Md 85 10301006 2.5 86 10301047 2.6 87 10301321 2.8 88 10301525 3.2 89 10302113 1.9 90 10302114 2.5 91 10302117 1.7 92 10302118 1.5 93 10302312 1.6 94 10310640 2.4 95 10310651 1.9 96 10310734 1.9 97 10310919 1.9 98 10311041 3.2 99 10311122 2.4 100 10311150 1.9 101 10311224 1.5 102 10311319 1.9 103 10311325 1.9 104 10312002 1.9 105 10312108 2.3 106 10312209 2.3 107 11010042 2.4 108 11010638 2.1 109 11010857 2.7 110 11010921 3 111 11010929 1.7 112 11011301 1.5 113 11011532 3.1 114 11011814 1.6 115 11012219 1.6 116 11020901 3 117 11021006 2 118 11021033 1.4 119 11021527 1.5 120 11021539 1.7 121 11021709 2.8 122 11022308 2.5 123 11030022 2.3 124 11030535 2.1 125 11030536 2.5 126 11031343 2.2 127 11031344 2.4 128 11040529 1.9 129 11040826 1.9 130 11040847 2.6 131 11040956 1.9 132 11041048 2.4 133 11041052 3 134 11041054 3.1 135 11041117 1.3 136 11041221 1.5 137 11051646 2.6 138 11051900 2.8 139 11060628 2.1 140 11060927 1.9 141 11061640 3.2 142 11061642 2.7 143 11070616 2.7 144 11070618 2 145 11070903 2.3 146 11071507 2.4 147 11071635 2.6 1220 Salvatore de Lorenzo, Marilena Filippucci, Elisabetta Giampiccolo and Domenico Patanè can often be easily recognized as a sharp dis- continuity (e.g., de Lorenzo and Zollo, 2003) which breaks the approximately bipolar shape of the wave on the velocity seismogram (fig. 4) and are generally caused by the presence of thin layers below the recording site, where part of the energy remains trapped and is subjected to multiple reflection. The quality of the onset varies with varying the level of noise of the recorded traces. For this reason we divided the dataset into three cate- gories. A maximum error on rise time equal to 5% was estimated for data which have the high- est quality. For data of intermediate quality a maximum error of 10% was estimated. Finally, a maximum error of 20% was associated to da- ta having the worst quality. As a consequence an average error on rise time equal to 10 ms was estimated. The plot of the seismic rays under the as- sumption of a homogeneous velocity model is shown in fig. 5a,b. We infer that the central- eastern part of the array is very well illuminat- ed by the seismic rays, until to a depth of about 3-4 km. The Qp estimates we present in the next section will be then particularly representative of the average Qp of this volume. 5. Results We carried out two different kinds of data in- versions. In the first inversion we assumed that the stress drop of the studied earthquakes is a Fig. 4. A velocigram of a microearthquake occurred at Mt. Etna. The discontinuity occurring during the first half-cycle of the first P-wave, due to a secondary ar- rival, impedes the measurement of the rise time on this seismogram. Fig. 5a,b. Plot of seismic rays available for this study: a) plot in the horizontal plane; b) plot in the vertical latitude-depth plane a b 1221 Intrinsic Qp at Mt. Etna from the inversion of rise times of 2002 microearthquake sequence constant, so that a self-similar behaviour of earth- quakes with a different energy content is expect- ed. Under this assumption the source rise time τ0 of each event was computed by using the seismic moment versus magnitude duration relationship calibrated for the Etnean area (Patanè et al., 1993) and the laws which relate source radius, seismic moment, stress drop, rupture velocity and source rise time for a circular crack model. In the second inversion, source rise times were directly estimated from data using the relationship (3.1). 5.1. Qp estimates under the assumption of a constant stress drop For each of the 147 considered events, seis- mic moment M0 was inferred from the magni- tude duration Md (table I), using the relation- ship which relates Md to M0 for the Etnean area (Patanè et al., 1993) (5.1) with (5.2) . (5.3) For a circular crack, the source rise time τ0, which represents the time duration of the slip- ping on the fault, is related to the source radius L and the rupture velocity Vr of a circular crack by the equation (Brune, 1970) . (5.4) By combining eq. (5.4) with the relationship which combines stress drop ∆σ, M0 and L (Keilis-Borok, 1959) (5.5) we obtain . (5.6) In order to compute τ0, we need to estimate the rupture velocity Vr of the earthquakes. Unfortu- nately, this parameter is poorly known for small V M1 16 7 r 0 03τ σ∆ = L M 16 7 3 0σ∆ = V L r 0τ = . .b 0 9 0 1!= . .a 17 8 1 9!= Log M a bM0 d= + magnitude earthquakes, owing to its correlation with the other source parameters (e.g., Deich- mann, 1997). Theoretical and laboratory studies (Madariaga, 1976) indicate that Vr ranges be- tween 0.6 Vs and 0.9 Vs. Starting from the ob- servation of a great dispersal of rise time versus travel time for the considered events, which is often attributed to significant directivity effects (e.g., de Lorenzo et al., 2004), we have as- sumed a high value of the rupture velocity (Vr = 0.9 Vs = 2.2 km/s). For the Mt. Etna ∆σ is known to range from a few MPa to about 100 MPa (e.g., Patanè et al., 1994, 1997; Patanè and Giampiccolo, 2003). For this reason, first of all we carried out five different inversions by using five possible val- ues of ∆σ (∆σ = 1, 5, 10, 50, 100 MPa), and then we compared the results of the inversions. To this aim, for each value of ∆σ, source rise times τ0 of the events were computed using eq. (5.6). Then, for each event, the reduced rise times τ − τ0 were inferred from eq. (3.1) and in- verted to estimate Qp. At the first step of the inversion we assumed a constant Q for the entire area. The inversion of the entire data set of rise times was per- formed using a weighted linear inversion scheme with weigths equal to the inverse of the variance of data. The results are summarized in fig. 6a-c. It is worth noting that about the same quality of the fit is obtained for ∆σ = 10 MPa (an average residual of 42 ms in L1 norm and an average residual of 62 ms in L2 norm) and for ∆σ = 50 MPa (an average residual of 42 ms in L1 norm and an average residual of 61 ms in L2 norm). The coefficient of correlation for the two cases, as computed using the relationship for weighted data (Green and Margerison, 1978) is also about the same (R2 = 0.24). For this reason, Qp was estimated by averaging the two Qp esti- mates for ∆σ = 10 MPa (Qp∼33) and ∆σ = 50 MPa (Qp∼30), obtaining . (5.7) Owing to the difficulties in accurately account- ing for both the errors on data and the tradeoff among source parameters, instead of fixing the stress drop to one of the values which produce the comparable fit to data (∆σ = 10 or ∆σ = 50 Q 32 2p != 1222 Fig. 6a-c. a) Results of the inversions of rise times for different values of ∆σ, assuming a constant ∆σ and a homogeneous Qp. On the top-left, the plot of the average residual in L2 norm versus ∆σ is shown. On the top- right the plot of the average residual in L1 norm versus ∆σ is shown. On the bottom-left the plot of average Qp versus ∆σ is shown. On the bottom-right the plot of the squared correlation coefficient versus ∆σ is shown. b) Plot of reduced rise times versus travel times for ∆σ=10 MPa and ∆σ=50 MPa. c) Plot of rise time residuals versus travel times for ∆σ=10 MPa and for ∆σ=50 MPa under the assumption of a homogeneous Qp. a b c Salvatore de Lorenzo, Marilena Filippucci, Elisabetta Giampiccolo and Domenico Patanè 1223 Intrinsic Qp at Mt. Etna from the inversion of rise times of 2002 microearthquake sequence MPa), we preferred to consider, in the calcula- tion of Qp of each event, both the cases ∆σ = 10 MPa and ∆σ = 50 MPa, choosing, for each event, the Qp value which gives rise to the best fit result. Results are summarized in table II. Only two events show a negative non physical Qp value: event #85, for which we have Qp=−37 and event #100, for which we have Qp=−100. Table II. Qp estimates under the assumption of a constant stress drop. Id N data Q ∆Q L1 residual L2 residual ∆σ Id N data Q ∆Q L1 residual L2 residual ∆σ (s) (s) (Mpa) (s) (s) (Mpa) 1 14 21 1 43 47 50 2 15 24 1 15 27 10 3 9 30 1 12 14 50 4 7 26 2 18 24 10 5 7 24 1 26 38 10 6 10 25 1 27 33 10 7 14 31 1 28 32 10 8 15 37 3 53 61 50 9 14 31 1 23 27 10 10 19 28 1 18 24 10 11 6 19 1 32 39 50 12 6 28 2 38 55 10 13 14 26 1 14 19 50 14 14 21 1 27 34 50 15 4 14 1 21 33 50 16 7 80 15 21 30 50 17 12 22 1 85 92 10 18 9 27 1 40 44 10 19 4 25 1 7 9 10 20 6 20 2 18 26 50 21 12 1230 100 61 74 50 22 6 14 1 52 66 50 23 4 27 3 15 19 10 24 5 35 4 27 39 10 25 8 35 6 156 180 10 26 4 37 3 63 86 10 27 10 19 1 36 45 50 28 5 19 1 30 52 50 29 6 23 2 76 96 50 30 6 26 3 41 74 50 31 9 96 2 124 173 50 32 7 28 3 21 33 50 33 7 12 1 28 41 10 34 7 18 1 65 88 10 35 7 28 2 20 23 50 36 9 20 2 39 47 10 37 13 25 2 68 89 50 38 9 24 1 67 91 10 39 6 25 1 43 69 50 40 7 15 1 23 32 50 41 5 16 1 172 200 10 42 4 14 1 21 39 50 43 4 −37 5 86 114 10 44 4 30 3 32 55 10 45 5 24 2 30 42 50 46 4 32 5 39 53 50 47 5 33 4 60 82 50 48 8 15 0 85 106 10 49 5 18 1 74 94 10 50 4 69 18 55 73 10 51 4 21 6 56 69 10 52 6 16 1 26 32 50 53 11 38 2 17 23 10 54 17 29 1 65 95 10 55 5 25 1 16 21 50 56 9 28 2 48 63 10 57 6 32 2 10 12 50 58 10 −100 −20 147 170 50 59 11 23 1 31 41 10 60 6 21 1 18 41 10 61 12 29 1 58 70 50 62 14 23 1 77 88 50 63 5 34 2 25 40 10 64 4 35 2 9 12 50 65 11 31 1 17 24 50 66 5 35 2 14 18 50 67 5 41 2 9 13 10 68 4 18 1 17 22 10 1224 Table II (continued). Id N data Q ∆Q L1 residual L2 residual ∆σ Id N data Q ∆Q L1 residual L2 residual ∆σ (s) (s) (Mpa) (s) (s) (Mpa) 69 11 27 1 11 17 10 70 10 25 1 19 24 10 71 7 16 1 46 62 50 72 8 33 1 7 9 50 73 4 29 1 3 3 50 74 5 23 2 53 77 10 75 8 32 3 100 111 50 76 4 25 1 50 99 50 77 6 43 2 5 7 10 78 7 31 1 37 45 50 79 10 35 1 13 17 50 80 9 12 1 17 26 50 81 7 36 2 21 27 50 82 5 33 4 76 97 10 83 9 18 1 7 9 10 84 5 47 2 32 66 50 85 7 53 4 24.0 30.8 50 86 5 64 12 37.3 44.4 10 87 4 27 2 10.7 16.8 10 88 9 26 2 48.0 58.5 10 89 6 38 3 9.6 15 50 90 13 26 1 59.2 65.8 10 91 4 25 1 6.8 13.5 50 92 5 31 2 9.0 14.4 50 93 6 36 4 8.3 10.6 10 94 6 32 1 14.8 19.3 10 95 6 24 1 7.2 12.1 10 96 5 32 2 9.2 13.2 50 97 6 53 4 13.6 19.1 50 98 13 24 1 51.6 81 10 99 10 32 1 27.1 37 50 100 4 30 2 3.0 5.1 10 101 6 28 1 6.2 8.5 50 102 5 35 3 7.7 11 50 103 5 15 1 37.6 56.1 10 104 5 23 1 9.7 18.7 50 105 4 27 2 21.7 29.9 10 106 7 30 1 7.5 10.3 10 107 5 19 1 9.6 12.5 50 108 5 32 2 12.1 15.2 10 109 7 39 2 7.4 9.3 50 110 11 31 1 15.2 20.1 50 111 4 47 5 34.9 48.6 10 112 4 44 4 21.2 28.8 10 113 5 22 1 28.3 35.9 10 114 5 45 3 5.8 9 10 115 4 25 2 19.0 25.6 50 116 4 20 2 39.1 47.3 50 117 8 44 2 11 15.3 10 118 5 45 2 4.3 6.1 50 119 5 26 2 32.900 38.2 10 120 4 31 3 41.900 50.6 10 121 12 31 1 17.700 23.6 10 122 15 35 1 9.700 12.3 50 123 5 31 2 12.600 20.1 50 124 5 21 1 15.500 23.3 10 125 8 19 1 34.600 49.4 50 126 4 18 2 12.300 20 10 127 4 36 6 57 101 50 128 4 48 4 12 15 50 129 5 23 1 25 31 10 130 8 34 1 10 13 10 131 4 26 5 4 6 10 132 4 30 2 3 4 10 133 12 22 1 15 23 10 134 12 22 1 31 47 10 135 5 33 2 17 24 10 136 6 21 1 45 59 50 137 4 26 5 21 25 10 138 5 19 2 55 90 10 139 4 17 1 12 17 50 140 4 46 5 11 15 10 141 4 9 1 50 63 10 142 5 10 1 36 47 10 143 13 32 1 25 32 10 144 7 29 2 13 15 10 145 5 18 1 11 13 10 146 7 33 2 23 30 10 147 4 16 1 11 14 50 Salvatore de Lorenzo, Marilena Filippucci, Elisabetta Giampiccolo and Domenico Patanè 1225 Intrinsic Qp at Mt. Etna from the inversion of rise times of 2002 microearthquake sequence Moreover, only event #21 shows a very high value (Qp=1230). For the remaining events the Qp variations are more limited, with a minimum Qp=9 and a maximum Qp=96. The weighted av- erage of Qp, using the results of all the 147 in- versions and as weights the inverse of the resid- uals in L1 norm is exactly the same as the Qp obtained at the first inversion step . (5.8) However, with respect to the previous result, based on the assumption of a homogeneous Qp, the assumption of a heterogenous Q structure gives rise to a further variance reduction. In fact the average residual is now 34 ms in L1 norm (a residual reduction of 19%) and 53 ms in L2 norm (a residual reduction of about 13%). The Q 32 2p != trend of residuals versus travel time is shown in fig. 7a. 5.2. Joint estimation of τ0 and Qp In another attempt we inverted the rise times to jointly infer τ0 and Qp of each event. In a first inversion run, we used a linear weighted inversion scheme, with weights equal to the inverse of the variance of data. Unfortu- nately this approach does not allow us to im- pose positivity constraints on τ0 and Qp . For this reason, after the inversions, only for some events we inferred positive values of both τ0 and Qp, reported in table III. To overcome the problem of negative values of τ0 and/or Qp for the remaining events, we used a non linear in- Fig. 7a,b. a) Plot of rise time residuals versus travel times under the assumption of a constant ∆σ and a het- erogeneous Qp. b) Plot of rise time residuals versus travel times after the joint inversion of τ0 and Qp. a b Table III. τ0 and Qp estimates of the events; the lines with bold characters indicates the events for which the Simplex method has been used to overcome the problem of negative τ0 and Qp values. Id N data τ0 ∆τ0 Q ∆Q L1 residual L2 residual Id N data τ0 ∆τ0 Q ∆Q L1 residual L2 residual (s) (s) (s) (s) (s) (s) (s) (s) 1 14 33 3 22 1 20 28 2 15 51 2 53 9 13 17 3 9 43 3 74 19 12 15 4 7 33 1 23 2 17 19 1226 Table III (continued). Id N data τ0 ∆τ0 Q ∆Q L1 residual L2 residual Id N data τ0 ∆τ0 Q ∆Q L1 residual L2 residual (s) (s) (s) (s) (s) (s) (s) (s) 5 7 69 2 393 36 13 17 6 10 76 4 571 77 18 20 7 14 33 1 25 57 14 21 8 15 83 3 194 99 17 26 9 14 71 3 147 42 14 21 10 19 59 3 77 17 20 25 11 6 12 5 14 1 25 33 12 6 33 4 25 2 19 32 13 14 33 2 31 10 16 22 14 14 83 4 35 4 24 31 15 4 62 1 17 0 34 53 16 7 38 1 100 2 12 16 17 12 19 9 15 3 42 55 18 9 33 3 25 5 22 28 19 4 15 1 16 0 6 8 20 6 59 1 59 5 13 19 21 12 65 2 349 136 12 17 22 6 90 3 42 3 17 25 23 4 32 4 25 1 12 23 24 5 33 6 25 1 30 41 25 8 62 9 20 0 68 84 26 4 36 17 29 1 43 81 27 10 58 7 17 2 30 45 28 5 62 10 22 2 29 54 29 6 62 14 21 2 53 65 30 6 67 36 25 3 48 89 31 9 116 27 100 0 44 64 32 7 44 10 30 5 34 49 33 7 47 13 17 4 74 93 34 7 36 13 17 29 54 71 35 7 38 6 29 1 32 41 36 9 62 6 14 4 29 35 37 13 67 22 22 4 60 78 38 9 67 6 18 0 47 59 39 6 22 15 18 2 35 60 40 7 47 10 17 0 68 81 41 5 47 27 17 1 84 100 42 4 58 19 17 1 70 98 43 4 41 10 50 0 55 96 44 4 41 16 29 0 56 87 45 5 41 19 30 0 58 70 46 4 38 14 41 4 33 44 47 5 36 9 17 1 49 64 48 8 44 7 31 3 35 44 49 5 47 4 17 1 50 66 50 4 38 6 25 24 29 50 51 4 26 2 6 0 10 18 52 6 78 1 112 18 18 38 53 11 30 4 30 3 21 30 54 17 51 8 20 2 49 68 55 5 43 1 55 6 8 10 56 9 53 3 32 3 20 28 57 6 31 2 100 1 17 22 58 10 47 2 50 10 41 57 59 11 44 2 25 5 23 27 60 6 17 2 17 1 25 52 61 12 133 3 100 21 41 53 62 14 63 2 20 1 27 35 63 5 15 2 20 1 15 22 64 4 36 2 100 2 12 17 65 11 9 4 18 1 25 39 66 5 21 6 32 2 17 23 67 5 4 4 24 2 7 14 68 4 51 3 45 5 13 23 69 11 88 2 241 30 11 14 70 10 44 2 37 3 13 23 71 7 10 7 12 0 31 48 72 8 27 1 56 5 7 8 73 4 14 1 22 1 5 6 74 5 41 3 23 5 36 56 75 8 101 11 18 1 71 79 76 4 51 13 26 2 53 103 77 6 21 1 54 22 8 10 78 7 25 3 31 3 14 20 79 10 20 2 28 1 11 13 80 9 160 6 42 9 19 26 81 7 32 4 45 9 21 31 82 5 56 2 20 1 14 21 83 9 60 2 29 3 12 17 84 5 29 25 100 3 42 80 Salvatore de Lorenzo, Marilena Filippucci, Elisabetta Giampiccolo and Domenico Patanè 1227 Intrinsic Qp at Mt. Etna from the inversion of rise times of 2002 microearthquake sequence version scheme based on the use of the Simplex Downhill method (Press et al., 1989) which al- lowed us to impose positivity constraints on both τ0 and Qp. To estimate the error on the in- ferred model parameters we applied a statistical approach based on the use of the random devi- ates (Vasco and Johnson, 1998). This consists of computing Nrand datasets by adding to each data a random quantity selected in the range of the error affecting them. The inversion results of the Nrand datasets can then be used to estimate the average values of the model parameters and Table III (continued). Id N data τ0 ∆τ0 Q ∆Q L1 residual L2 residual Id N data τ0 ∆τ0 Q ∆Q L1 residual L2 residual (s) (s) (s) (s) (s) (s) (s) (s) 85 7 24 2 34 3 14 17 86 5 38 1 50 0 21 29 87 4 24 3 29 2 11 13 88 9 64 1 46 4 31 44 89 6 24 2 30 85 13 18 90 13 17 3 18 1 24 36 91 4 21 5 30 1 17 24 92 5 0 2 19 1 10 14 93 6 19 1 34 0 18 22 94 6 33 5 35 0 21 27 95 6 46 0 162 22 3 4 96 5 24 2 37 0 13 18 97 6 24 2 100 0 19 25 98 13 58 2 25 1 21 30 99 10 20 4 19 1 22 35 100 4 36 1 119 9 4 6 101 6 38 1 101 23 6 8 102 5 24 2 38 3 6 9 103 5 51 1 137 39 12 17 104 5 24 1 20 1 18 21 105 4 31 5 25 6 22 27 106 7 47 2 37 3 9 12 107 5 7 1 14 0 11 22 108 5 27 4 35 0 17 24 109 7 41 1 100 21 7 8 110 11 47 2 75 12 16 19 111 4 21 3 40 10 17 28 112 4 41 2 141 18 11 18 113 5 24 6 25 2 25 40 114 5 19 1 50 1 7 12 115 4 19 1 35 1 14 19 116 4 116 3 213 41 18 34 117 8 25 1 50 1 10 18 118 5 39 2 130 30 5 7 119 5 18 2 18 1 11 17 120 4 21 4 17 2 12 20 121 12 49 1 70 9 19 28 122 15 24 2 26 1 9 14 123 5 31 2 41 0 20 31 124 5 60 1 52 3 7 9 125 8 55 2 43 5 18 32 126 4 53 0 26 1 10 13 127 4 33 2 100 0 46 91 128 4 24 1 36 3 11 13 129 5 24 1 22 6 9 14 130 8 38 1 50 0 18 22 131 4 33 1 29 1 6 9 132 4 23 1 32 1 5 7 133 12 93 2 102 15 10 14 134 12 106 4 129 49 20 26 135 5 39 3 263 112 18 35 136 6 55 3 335 45 22 40 137 4 28 1 17 0 13 18 138 5 44 6 17 31 48 79 139 4 38 1 35 1 12 19 140 4 29 1 57 4 13 18 141 4 27 2 7 0 34 61 142 5 114 1 50 4 29 45 143 13 53 4 79 17 24 31 144 7 25 1 30 1 19 25 145 5 73 2 104 21 9 12 146 7 33 3 42 9 22 33 147 4 72 4 116 32 16 22 1228 Salvatore de Lorenzo, Marilena Filippucci, Elisabetta Giampiccolo and Domenico Patanè their standard deviations. The inversion results are summarized in table III. After this analysis we inferred Qp values ranging from a minimum of 6 to a maximum of 570. The weighted aver- age Qp, with weights equal to the inverse of the average residual in L1 norm, was . (5.9) A further variance reduction was obtained with respect to the previous case of a heterogeneous Qp but a constant stress drop ∆σ. In fact the av- erage residual in L1 norm is now equal to 23 ms (a residual reduction of 33%) and the average residual in L2 norm is equal to 37 (a residual re- duction of 30%). The trend of residuals versus travel times is shown in fig. 7b. 5.3. Application of the «Occam razor principle» One point that needs to be addressed concerns the evaluation of the statistical significance of the two models used to fit data. Since the two models are characterized by a different number of model 68 51Qp != parameters it is important establish, for each event, what of them satisfy the so called Occam’s razor principle, i.e. realizes the best compromise between the quality of the fit and the simplicity of the model. To solve this problem we used the cor- rected Akaike Information Criterion (AICc) (Ca- vanaugh, 1997; Cavanaugh and Shimway, 1998), which represents a modified version of the Akaike (1974) criterion, to account for incom- pleteness of data. This consists of comparing, for the two models, the value of the parameter (5.10) where k is the number of model parameters, n the number of data and L represents the likeli- hood function, given by . (5.11) The model which gives rise to the minimum AICc has then to be considered the most signif- icant from a statistical point of view. ( ) expL 2 1 2 obs teo ii n i 1 2 2 πσ σ τ τ = − − = ; E% 2 2 ( ) ( ) AICc log L k n k k k 1 1 = − + − − − + Table IV. Application of the corrected Akaike information criterion to the two considered models. AICc1 = val- ue of the Akaike information criterion for the model with one model parameter (only Qp); AICc2 = value of the Akaike information criterion for the model with two model parameters (τ0 and Qp); ∆AICc = AICc1-AICc2. Id# AICc1 AICc2 ∆AICc Id# AICc1 AICc2 ∆AICc Id# AICc1 AICc2 ∆AICc Id# AICc1 AICc2 ∆AICc event event event event 1 391 18 373 2 −9 −13 4 3 12 56 −45 4 1 −6 7 5 214 3 211 6 196 −6 202 7 339 1 338 8 816 26 789 9 161 −5 167 10 132 92 40 11 119 −16 135 12 59 40 19 13 −30 −24 −6 14 34 46 −12 15 −20 −12 −8 16 35 −19 53 17 2379 137 2242 18 505 177 328 19 −28 −36 9 20 −29 −36 7 21 777 −63 841 22 7 −31 38 23 −23 −31 8 24 9 6 4 25 643 49 594 26 222 22 200 27 56 66 −10 28 −9 −10 1 29 212 69 144 30 −23 −18 −6 31 2653 5832 −3178 32 −31 0 −31 33 −23 30 −53 34 160 87 73 35 −20 40 −60 36 4 −28 32 37 213 229 −17 38 342 66 275 39 39 9 30 40 −11 121 −132 41 1362 195 1167 42 −23 21 −43 43 194 42 153 44 −19 7 −26 45 0 7 −7 46 18 148 −129 47 20 98 −78 48 318 179 139 49 123 34 89 50 115 81 34 51 3 586 −583 52 72 461 −389 53 80 354 −274 54 924 942 −18 55 123 34 89 56 518 935 −417 1229 Intrinsic Qp at Mt. Etna from the inversion of rise times of 2002 microearthquake sequence Table IV summarizes the comparison be- tween the AICc values for the two considered models. In this way we were able to infer that, for 59 events the above choice of retrieving source rise times assuming a constant stress drop produces a better fit of data, whereas for the remaining 68 events the source rise times are better constrained from the inversion of rise times. By averaging the Qp estimates obtained with this analysis we finally obtained . (5.12) Figure 8 shows the trend of the Qp of each event estimated with the AICc as a function of τ0. It is worth noting that a residual correlation among τ0 and Qp is inferred. 4257Qp != 6. Discussion and conclusions As thoroughly debated in Wu and Lees (1996) and de Lorenzo (1998), the slope C of the straight line interpolating τ versus Qp-1 de- pends on the frequency content of the source; by calibrating eq. (3.1) with different source time functions, Wu and Lees (1996) showed that C = 0.5 has to be adopted if it is assumed that the the signal generated at the source has a Gaussian shape, which is a smooth representa- tion of the unipolar source time function gener- ated by a circular crack. By taking into account that a more realistic source cannot have the same low-frequency content as the Gaussian function, Wu and Lees (1996) showed that the C could be slightly higher. They estimated as Table IV (continued). Id# AICc1 AICc2 ∆AICc Id# AICc1 AICc2 ∆AICc Id# AICc1 AICc2 ∆AICc Id# AICc1 AICc2 ∆AICc event event event event 57 47 512 −465 58 3974 126 3848 59 189 173 16 60 −41 651 −692 61 1053 494 559 62 1096 252 845 63 7 212 −205 64 18 98 −80 65 14 117 −102 66 16 −1 17 67 −23 561 −584 68 74 75 −1 69 23 449 −426 70 173 93 80 71 −5 1055 −1060 72 45 46 −2 73 −33 203 −236 74 25 −7 32 75 514 246 269 76 −2 −8 6 77 −18 17 −35 78 771 126 645 79 143 22 121 80 −40 −41 1 81 34 41 −6 82 125 −27 152 83 −53 −51 −2 84 −7 71 −78 85 299 58 241 86 25 −4 29 87 −20 −22 2 88 424 94 330 89 −4 30 −34 90 1718 44 1674 91 −33 −4 −29 92 −33 −13 −20 93 −31 −18 −13 94 54 129 −75 95 −38 −55 17 96 −10 76 −85 97 10 100 −90 98 204 −2 206 99 222 90 132 100 −33 −39 6 101 −5 4 −9 102 −31 −38 7 103 200 −10 210 104 −29 4 −32 105 −3 3 −5 106 −47 −44 −2 107 −33 −33 0 108 22 71 −49 109 11 −15 27 110 112 121 −9 111 31 −11 43 112 66 −33 98 113 1 −25 27 114 −37 −32 −5 115 57 61 −4 116 62 −30 92 117 114 112 2 118 −27 −24 −3 119 96 −22 118 120 233 6 227 121 237 81 157 122 −40 −19 −21 123 44 167 −123 124 −23 −28 5 125 46 −31 77 126 −25 −27 2 127 4 −16 19 128 28 −4 32 129 125 −22 147 130 15 123 −109 131 −27 −31 5 132 −30 −34 3 133 −45 −67 22 134 65 −23 88 135 28 3 25 136 303 −17 320 137 −17 −29 13 138 23 9 14 139 13 −26 38 140 −25 −32 7 141 0 −25 26 142 29 −26 55 143 130 187 −56 144 −37 −11 −27 145 1 −30 30 146 68 57 11 147 −12 10 −22 Fig. 9. Plot of τ0 versus the identification number of the event and of Qp versus the identification number of the event. 1230 Fig. 8. Plot of Qp versus τ0 of the studied events after the application of the AICc. The points enclosed in the grey rectangle are mainly responsible for the observed correlation. upper limit the value C = 0.65. If we assume C = 0.65, instead of Qp = 57±42 we obtain a slightly higher Qp value (Qp = 74±55). Let us consider now the residual correlation between Qp and τ0. First, we note that the Qp es- timate obtained using the AICc (Qp = 57±42) is intermediate between the Qp estimate obtained under the assumption of a constant stress drop and that obtained by jointly retrieving τ0 and Qp. However, this result does not necessarily im- Salvatore de Lorenzo, Marilena Filippucci, Elisabetta Giampiccolo and Domenico Patanè 1231 Intrinsic Qp at Mt. Etna from the inversion of rise times of 2002 microearthquake sequence ply a departure from the constancy of the ener- gy release per seismic moment, nor is it neces- sarily caused by a great heterogeneity in the at- tenuating properties of the area, as one can sup- pose by taking into account the high value of the standard deviation on Qp. A possible interpreta- tion of the correlation between Qp and τ0 may be that, in many cases, the data available are not sufficient to correctly estimate the two parame- ters. As an example, it can be easily demonstrat- Fig. 10. Plot of rise time versus travel time for some studied events. On each plot the best fitting straight line, obtained from the inversion of rise times, is superimposed. 1232 Salvatore de Lorenzo, Marilena Filippucci, Elisabetta Giampiccolo and Domenico Patanè ed that a strong clustering of empirical data in a small range of values for both the considered variables induces a correlation between the re- gression parameters. Thus the uneven and insuf- ficient sampling for many events could be re- sponsible for the observed phenomenon. Alternatively, we have to account for the limitations of the hypothesis on which the used technique is based. In fact the adopted tech- nique uses a very simplified source model, i.e. an isotropic seismic source. It is known that, for a finite dimension seismic source, the directivi- ty source effect on the rise time of first P waves can be described through the relationship (Zol- lo and de Lorenzo, 2001) (6.1) where θ is the takeoff angle (the angle formed by the tangent to the ray leaving the source and the normal to the fault plane) and Vp the P-wave velocity of at the source. From this equation it immediately follows that the variations of the source rise time around the non-directive source rise time given in eq. (5.4) increase with increasing the source dimension L. Moreover, for a directive source time function, the equa- tion which describes the relationship between τ and Q−1 is non linear, as demonstrated in Zollo and de Lorenzo (2001) and de Lorenzo et al. (2004). In particular Zollo and de Lorenzo (2001) showed that the higher the source di- mension is, the higher will be the difference be- tween the rise times predicted by the directive source model and those predicted by eq. (3.1). In our case, all the data which are responsible for the observed correlation among τ0 and Qp (the data shown in the grey square of fig. 8) have been selected by AICc among the model parameters obtained from the joint inversion of τ0 and Qp, i.e. without imposing a constant stress drop for the area (fig. 9). This could im- ply that, above a given threshold (τ0 ~ 30-40 ms) the directivity source effect tends to be so high to produce a non linear trend of rise times ver- sus the travel time, which may result in a bad estimation of the estimated τ0 and Qp if one us- es the linear interpolating scheme given by eq. (3.1). This result is also supported by the visual sin V L V V 1 r p r 0τ θ= −c m inspection of the quality of the matching of the model to data, as shown in fig. 10 for some events. It is quite evident that, even if, in some cases, an increase in rise time versus travel time can be observed, the dispersal of data around the best fit straight line is significantly higher than the error on data. A possible bias in Q estimates could be caused by the different instrumental responses. It has to be noted that twenty-two of the avail- able stations are equipped with Mark L4C, hav- ing an eigenfrequency of 1 Hz. The remaining two stations are equipped with Lennartz Le3D, having an eigenfrequency of 0.05 Hz. For both the two kind of stations the cutoff frequency is around 50 Hz. Since the average frequency con- tent of the pulses ranges from 3 to 40 Hz, i.e. lies in the linear part of the amplitude and phase response of the instrument, we do not expect a significant distortion in amplitude of signals and phase shifting, even if the signals having the higher average frequency content (around 40 Hz) could be more sensitive to the non linear part of the instrumental response. However, on- ly a dozen of data (fig. 3) have an average fre- quency higher than 30 Hz and then this problem affects only a very negligible portion of the dataset and cannot affect the obtained average Q estimate. 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