Source, attenuation, and site parameters of the 1997 Umbria-Marche seismic sequence from the...

Source, attenuation, and site parameters of the 1997Umbria‐Marche seismic sequence from the inversionof P wave spectra: A comparison between constant QP

and frequency‐dependent QP models

Salvatore de Lorenzo,1 Aldo Zollo,2 and Giammaria Zito1

Received 24 September 2009; revised 29 March 2010; accepted 23 April 2010; published 4 September 2010.

[1] The attenuation of P waves, the site responses, and the source parameters (seismicmoment, corner frequency, source dimension) of 490 seismic events that occurred duringthe 1997 Colfiorito, Umbria‐Marche seismic sequence have been inferred from theinversion of P wave velocity spectra. The Boatwright source model has been assumed tomodel the source spectra. A global nonlinear inversion scheme was developed to avoidany a priori selection of the initial QP and the corner frequency. To establish if afrequency‐dependent QP model fits the data better than a constant QP model, two inversionresults have been compared. Application of the Akaike information criterion indicates thatthe constant QP model represents the best compromise between model simplicity anddata misfit. The station QP values are small: in the range of 50 to 200. A one‐dimensionalQP model is obtained by back projecting the inverted t*. Our results indicate both well‐defined near‐site attenuation effects at some sites and heterogeneity in the inelasticproperties of the crust. With the exception of the amplification response at five seismicstations, most of the recording sites did not show amplification peaks at particularfrequencies. The stress drop clearly increases as a function of the seismic moment,which indicates a deviation from self‐similarity, whereas it does not show an increase withdepth, probably owing to the effects of fluid pressurization in the crust. A stress drop ofabout 39 MPa is inferred. The relationship between the seismic moment and the localmagnitude for P waves has been calibrated.

Citation: de Lorenzo, S., A. Zollo, and G. Zito (2010), Source, attenuation, and site parameters of the 1997 Umbria‐Marcheseismic sequence from the inversion of P wave spectra: A comparison between constant QP and frequency‐dependent QP models,J. Geophys. Res., 115, B09306, doi:10.1029/2009JB007004.

1. Introduction

[2] The 1997 Colfiorito, Umbria‐Marche (central Italy)seismic sequence was characterized by nine strong shockswith magnitudes greater than Mw = 5, and more than 2000aftershocks [Amato et al., 1998]. The focal mechanisms forthe main shocks indicate a normal faulting and NW–SEstriking fault planes, with tension axes oriented in the rangeof 40° to 60°, which is roughly perpendicular to the strike ofthe Apennine chain [Ekström et al., 1998]. During the crisis,the seismicity migrated from the north toward the south inthe direction of the Apennines, and the activity was con-centrated on a 40 km long area that was elongated in theNW–SE direction, at depths of less than 9 km [Amato et al.,1998].

[3] More than 1000 earthquakes were recorded during thiscrisis. Interpretations of these data have offered new insightsinto the relationships between the seismicity and tectonics ofthe Umbria‐Marche region. In particular, Vp and Vs three‐dimensional (3‐D) tomographic studies [e.g., Micheliniet al., 2000; Chiarabba and Amato, 2003, and referencestherein] have revealed a high degree of complexity of theupper crust in this area and the existence of clear lithologicaldiscontinuities that were interpreted as lateral variations ofmaterial properties along the faults that controlled the evo-lution of the rupture. For the rupture properties of the mainshocks, Capuano et al. [2000] indicated that the sharpchange with depth of the preexisting fault dip (from a lowangle at depth, to near vertical close to the surface) can actas a geometrical barrier to rupture propagation at relativelylow dynamic stresses. Moreover, the picture of the epi-centers, the focal mechanisms, and the fault plane orientationof the small magnitude earthquakes [Filippucci et al., 2006,and references therein] have indicated that the seismicityoccurred mainly along the Alto Tiberina fault [Boncio andLavecchia, 2000] and local systems of antithetic faults. Adominant normal faulting mechanism has usually been

1Dipartimento di Geologia e Geofisica and Centro Interdipartimentaleper il Rischio Sismico e Vulcanico, Università degli Studi di Bari AldoMoro, Bari, Italy.

2Dipartimento di Scienze Fisiche, Università degli Studi di NapoliFederico II, Naples, Italy.

Copyright 2010 by the American Geophysical Union.0148‐0227/10/2009JB007004

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, B09306, doi:10.1029/2009JB007004, 2010

B09306 1 of 23

http://dx.doi.org/10.1029/2009JB007004

inferred for almost all of the small earthquakes of thesequence [Chiaraluce et al., 2003].[4] The average Brune [1970] stress drop of the moderate‐

sized earthquakes (4.6 ≤ ML ≤ 5.9) [Bindi et al., 2004] is inthe range of 1–2 MPa, which is roughly in agreement withthe stress drop estimated from the inversion of the small‐magnitude earthquakes [Bindi et al., 2001]. In these studies,a frequency‐dependentQwas assumed, to correct the spectrafor the seismic attenuation effect. Bindi et al. [2004] showeda strong dependence of Q on frequency, Q( f ) = 49 f 0.9 for0.4 < f ≤ 8 Hz, consistent with the results of Bindi et al.[2001] and Castro et al. [2002]. On the other hand, forthe same region, Malagnini et al. [2000] and Malagnini andHerrmann [2000] showed a rather weak frequency depen-dence of Q (Q( f ) = 130 × f 0.1). Only a few studies, whichwere based on pulse width variations of small data sets, haveinvestigated the estimation of the constant QP of the first Pwaves [de Lorenzo and Zollo, 2003; Filippucci et al., 2006].[5] In the present study, we illustrate the results of a study

that was aimed at determining the quality factor of the Pwaves and the source parameters of the 1997 Umbria‐Marcheseismic events. It is worth noting that in a previous study ofsource parameters of the same events [Bindi et al., 2001], boththeP and S spectra were corrected for the frequency‐dependentQc estimated from the S-coda waves, assuming therelationship Qp = (5/4/) Qc. However, the theory ofinelastic absorption prescribes Qp less than Qs in the rocks ofthe Earth [Aki and Richards, 1980]. Moreover, many exper-imental studies have confirmed thatQp is different fromQs fordifferent tectonic regimes, as thoroughly debated by Tusaand Gresta [2008]. For these reasons, in the present study,we have used the information content of the P wave spectrato jointly infer Qp and the source parameters (corner fre-quency and seismic moment).[6] Several techniques have been proposed to estimate the

quality factors of P and S waves. Each of these methods hasbeen based on specific assumptions, and each has its ownadvantages and limitations (for a thorough review, see Tonn[1989]). We have used an approach based on the well‐known representation of seismic spectra, in terms of source,site, and attenuation response, in the formulation proposedby Tsumura et al. [1996]. However, to establish whether afrequency‐dependent QP model can explain the inelasticabsorption observed in the P wave spectra better than aconstant QP model, we extended the technique of Tsumuraet al. [1996] to also include a frequency‐dependent QP. Inthe inversion we have assumed (according to Tsumura et al.[1996]) that the seismic radiation at the source obeys thefunctional form proposed by Boatwright [1980].[7] The technique has been applied here to the whole data

set of P waves recorded in the Umbria‐Marche area by amixed (temporary and permanent) seismic array [Govoniet al., 1999; Bindi et al., 2001].

2. Data Analysis

[8] The data set includes the recordings of 621 aftershocksthat occurred between 18 October 1997 and 3 November1997. The seismic traces were recorded by a network of23 three‐component seismic stations deployed in centralItaly (15 temporary, 8 permanent; Figure 1). The temporaryarray was placed in the area during a survey that was run by

several Italian institutions (University of Genova, Universityof Trieste, and Protezione Civile). Ten of these stations(Figure 1, black triangles) consisted of four Lennartz LE‐3D/5s (flat velocity response between 0.2 and 40 Hz) and sixLennartz LE‐3D/1s (flat velocity response between 1 and40 Hz) seismometers equipped with MarsLite data loggersrecording onto 230 Mb optical disks. The stations wereequipped with three‐component sensors, and these stationsrecorded in continuous mode at 125 samples per s. The per-manent stations were operated by the RSM (OsservatorioGeofisico Sperimentale di Macerata) and RESIL (RegioneUmbria) networks. These recorded in continuous mode at62.5 samples per second. These stations were equippedwith Mark L4C‐3D seismometers (flat response between 1and 40 Hz) and MARS‐88/FD data loggers.[9] The velocity waveforms have been corrected for the

instrumental responses, which were computed in terms oftheir poles and zeros by Govoni et al. [1999]. The analysisconcerned the whole data set of early P signals. For eachevent, only the recordings that showed clear vertical P waveonset were considered, and the events for which there wereless than six P readings were discarded. This selection rulereduced the number of available events to 490. Despite thesmall magnitudes (1.4 ≤ ML ≤ 4.4) of these events, most ofthem were characterized by high‐quality recordings at agreat number of stations. An example of an event clearlydetected at almost all of the seismic stations is shown inFigure 2.[10] For each recording (Figure 3a), a 1.28 s time win-

dow was selected, which included the first P wave andstarted 0.14 s before the picked P wave arrival (Figure 3b).To reduce distortions due to the finite length of the signals,a cosine taper window with a 20% fraction of tapering[Stein and Wysession, 2003] was applied to the P wavetime window (Figure 3d) before computing its amplitudespectrum (Figure 3e). The analysis presented in the nextsections provides an estimation of QP and the source para-meters in the range 0.5 ≤ f ≤ 31 Hz.[11] To estimate the noise in the data, the fast Fourier

transformwas computed for a signal that was a timewindowof1.28 s (Figure 3c) preceding the P wave window. Finally, anaverage moving windowwith a full width of five neighboringpoints [Press et al., 1989] was used to smooth the spectra.[12] An analysis of 10% of randomly sampled events

allowed us to establish that the vertical components of thefirst P wave are characterized by a signal‐to‐noise ratiousually at least 1–2 orders of magnitude greater than isseen for the horizontal motion. For this reason, we decidedto only use vertical P waves in the inference of the modelparameters. A total number of 7194 P spectra was thenselected by disregarding those spectra for which the averagesignal‐to‐noise ratio was less than 5 (Figure 4). With thischoice, none of the spectra considered were dominated bynoise, and therefore we decided to use their entire spectralcontents in the inversions.

3. Technique

[13] The velocity spectrum Uij( f ) of the ith event observedat the jth station can be expressed as [e.g., Scherbaum, 1990]

Uij fð Þ ¼ Sij fð ÞBij fð ÞRj fð Þ ð1Þ

DE LORENZO ET AL.: QP IN THE UMBRIA‐MARCHE REGION (ITALY) B09306B09306

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In equation (1), Sij( f ) is the source spectrum, Bij( f ) is theattenuation spectrum, andRj( f ) is the site response. In the far‐field approximation, the source spectrum can be written as

Si;j fð Þ ¼ 2�fW0;i;j

1þ f =fc;i� ��n� �1=n ð2Þ

Using the geometrical spreading assumption, the low‐levelspectral amplitude in equation (2) is related to the seismicmoment by the relationship

W0;i; j ¼ M0;iR �; �ð Þ4��ri; jc3

ð3Þ

where R(�, �) is the radiation pattern, r is the density, c isthe velocity of the considered (P or S) wave, ri,j is thehypocenter distance, fc,i is the corner frequency, g is thehigh‐frequency spectral falloff, and n is a constant [e.g.,Zollo and Iannaccone, 1996].[14] The source spectrum is generally described by the

Brune [1970] source model, which corresponds to the so‐called “omega square” model (i.e., n = 1 and g = 2 inequation (2)). As it is not uncommon to find recordings of

several earthquakes that show a corner sharper than theoriginal Brune model [e.g., Boatwright, 1980; Abercrombie,1995; Tsumura et al., 1996], Boatwright [1980] proposed adifferent approximation to the source model that corre-sponds to g = 2 and n = 2 in equation (2). On the basis ofthis reasoning, in the following, we use the Boatwright[1980] source model to fit the spectra.[15] For the attenuation spectrum, its shape depends on the

value of the quality factor Q, according to the relationship

Bij fð Þ ¼ exp ��fri; jcQ

� �¼ exp ��ft*i; j

� ð4Þ

where

t*i; j ¼ri; jcQ

: ð5Þ

Several theoretical and experimental studies have focusedon the problem of the dependence of Q on frequency (for athorough discussion, see Aki and Richards [1980]). Asdemonstrated by Kjartansson [1979], this problem does nothave a unique solution, in that a constant Q model in a finite

Figure 1. The seismic stations of the permanent and temporary arrays located in the Umbria‐Marcheregion during the 1997 crisis. Squares, diamonds, and triangles represent stations run by the differentinstitutions (as detailed by Filippucci et al. [2006]). Circles represent epicenters of the events recordedduring the crisis.


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Figure 2. Vertical component of the velocity ground motion of a seismic event recorded at the array.The signals are of high quality at almost all of the seismic stations.


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frequency band also satisfies the causality principle. On thebasis of these findings, along with tomographic attenuationstudies, several studies of source parameters and Q havebeen carried out by assuming a constant Q [e.g., Scherbaum,1990; Tsumura et al., 1996; de Lorenzo et al., 2001;Haberland and Rietbrock, 2001; Rietbrock, 2001], whereasin other studies a frequency‐dependent Q has been assumed.In these latter cases, the most commonly used relationship[e.g., Tusa and Gresta, 2008; Drouet et al., 2008, andreferences therein] was

Q fð Þ ¼ Q0 f =f0ð Þ� ð6Þ

where Q0 = Q(f = f0) and f0 is usually assumed to be 1.0 Hz.In the following, we have compared the model parameterresults obtained from the assumption of a constant QP with

those arising from the assumption of a frequency‐dependentQP described by equation (6). This is because there is noway to establish a priori if a constant QP model can repro-duce the whole path attenuation of P waves better than afrequency‐dependent QP model; this can only be evaluated aposteriori on the basis of the quality of the fit.[16] For the site response, Rj( f ) has to account for both

the near‐site attenuation Kj( f ) and the local site amplifica-tion Aj( f ) [e.g., Scherbaum, 1990]. Therefore, the followingcan be written

Rj fð Þ ¼ Aj fð ÞKj fð Þ ð7Þ

The near‐site attenuation is usually described above a limit-ing frequency known as fmax [Hanks, 1982] in terms of the kj

Figure 3. Calculation of a seismic spectrum. (a) Vertical recording, (b) signal window of 1.28 s, includ-ing the P wave arrival and starting 0.14 s before of the P wave arrival, and an adjacent window of 1.28 sof noise. A 20% cosine taper was applied to both the (c) noise and the (d) signal window before comput-ing the (e) amplitude spectra.


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attenuation factor, which was first introduced by Andersonand Hough [1984]:

Kj fð Þ ¼ exp ��f kj� � ð8Þ

The local site amplification is not described by any partic-ular mathematical relationship, and it depends on the elasticand geometrical properties of the rocks near the recordingsite [e.g., Tsumura et al., 1996].[17] Following Tsumura et al. [1996], to determine the

source parameters (W0,i, j, fc,i), the quality factor QP, and thesite response, we applied the following two‐step iterativeapproach. At the first step, the theoretical spectrum given byequation (1) was approximated by

Uij fð Þ ¼ Sij fð ÞBij fð Þ ð9Þ

and a function quantifying the misfit between the observedand the theoretical spectra was minimized. Edwards et al.[2008] carried out detailed synthetic tests, and they estab-lished that the L2 norm is the most appropriate metric inspectral inversions. For this reason, we quantified the misfitbetween the observed and the theoretical spectrum using thevariance in the L2 norm:

�2 ¼ 1

N

XNj¼1

Aobsj � Ateo

j

2 ð10Þ

At the second step, the residual

Resij fnð Þ ¼ Uobsij fnð Þ � U teo

ij fnð Þ ð11Þ

are used to estimate the site response Rj( f ) at station j. Thisis obtained by minimizing the following quantity at eachstation [Tsumura et al., 1996]:

XNi

i¼1

Resij fnð Þ � lnRj fnð Þ ð12Þ

where Ni is the number of spectra available for the event i.This procedure is repeated as long as the misfit functiondecreases.[18] A synthetic study using noisy spectra has allowed us

to establish that the results obtained using the misfit functionof equation (10) are more accurate than those inferred byminimizing the misfit function that uses the natural loga-rithms of the amplitudes.[19] For the optimization strategy in the search for the best

fit model parameters, many approaches have been proposed.The main difficulty in these studies has been the trade‐offbetween QP and fc when these parameters are estimatedfrom a single spectrum [e.g., Scherbaum, 1990]. As a con-sequence, different strategies have been proposed so as tominimize these correlations. As a general rule, the inversionproblem is based on a preliminary estimation of the cornerfrequency arising from a subjective, and sometimes difficult,visual inspection of the spectrum. This can become moredifficult when the site amplification response is frequency‐dependent. Owing to the nonlinear dependence of the the-oretical spectrum on the model parameters, which arisesfrom equations (1) to (4), the best fit model is then obtainedby linearizing the inverse problem around an initial choiceof model parameters. In these cases, the final model candepend on the initial model. To provide a more exhaustiveexploration of the model parameter space and to remove thepart of the spectrum that was more affected by noise, Tusaet al. [2006a, 2006b] developed a grid search methodaround an initial model. Edwards et al. [2008] showed thatthe inversion problem becomes very well behaved if a gridsearch approach is used to infer only one variable (in theircase, fc) and a linearized Powell approach is used to inferthe remaining two unknowns. Their approach allowed thebypassing of the problem of both the visual inspection of thespectra and the correlation between Q and fc. On the basis ofthese recent studies, in the present study we handle theinversion problem with an algorithm that follows the samestrategy described by Edwards et al. [2008]. The only dif-ference is that we use a full grid search approach, i.e., a gridsearch over fc, W0, and Q.[20] The optimization strategy is essentially the same as

that proposed by Lomax et al. [2000] for the problem oflocating an earthquake using a nonlinear approach. Thisconsists of a two‐step procedure. First, the whole physically

Figure 5. Contoured plots of the misfit functions normalized to the data error for each station of a studied event. The plotsare shown in the three distinct subspaces of the model parameter space: [fc, log10(W0)], [fc, log10(t*)], [log10(t*), log10(W0)].White points represent the model parameters obtained from the single station inversion for each plot. Vertical lines representthe event fc obtained after the complete inversion procedure. The stars indicate the t* values obtained after the completeinversion procedure.

Figure 4. The average signal‐to‐noise ratios for the 7194 Pwave spectra considered in this study. Their average value isrepresented by the horizontal line.


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Figure

5


7 of 23

Figure

5.(contin

ued)


8 of 23

admissible model parameter space is subdivided into acoarse grid. The range of model parameter values in thecoarse grid is selected on the basis of the a priori analysis ofthe whole data set. At each point of this grid, the misfitfunction between the observed and the theoretical spectrumis computed. For the case of constant QP, the misfit functionis given by

� fc; t*;W0

� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N � 1

XNm¼1

Sobs fmð Þ � Si;j fm; fc; t*;W0

� � 2

vuut

ð13Þ

where t* = r/cQP.[21] For the case where a frequency‐dependent QP is

assumed, the misfit function is given by

� fc; t*;W0; �

� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N � 1

XNm¼1

Sobs fmð Þ � Si;j fm; fc; t*;W0; �� 2

vuut

ð14Þ

where t* = r/(cQ0). This allows us to quantitatively estimatean initial model mbest_ini = ( fc, t*, W0)

best_in or mbest_ini =( fc, t*, W0, a)

best_ini without there being subjective criteriabased on the visualization of the spectra.[22] At the second step, a refined grid is built around

mbest_ini and the misfit function is computed at each point ofthis grid, allowing us to estimate the best fit model para-meters mbest for each station.[23] Figure 5 shows the contoured plots of the misfit

function for equation (13) (the case of a constant Q) for eachof the parameter combinations of the model for a studiedevent. These plots indicate that the low‐level amplitude

spectra are tightly constrained by the inversion. For most ofour stations, the t* value is not influenced by the cornerfrequency, although the corner frequency and the t* areweakly correlated for some stations. In other data sets,stronger correlations between fc and t* have been observed[Scherbaum, 1990].[24] Following Menke [1984], the shape of the misfit

function around the absolute minimum is used to estimatethe errors on the model parameters. Let us consider, forinstance, the error on the corner frequency. If we indicatewith ( fc

b, t*b, W0b) the point corresponding to the absolute

minimum of the misfit function, as inferred from theinversion of the spectrum at a given station, at the secondorder, the value of the misfit function in the point ( fc

b + dfc,t*b, W0

b) will be given by

�2 f bc þ dfc; t*b;Wb

0

� ¼ �2 f bc ; t

*b;Wb0

� þ 1

2

@2�2

@f 2c

f bc ;t

*b;Wb0

� �dfc:ð15Þ

From equation (15), it follows that the error on the cornerfrequency at a given station can be estimated as

dfc ¼E f bc þ dfc; t*b;Wb

0

� � E f bc ; t

*b;Wb0

� 1

2

@2E

@f 2c

f bc ;t

*b;Wb0

� �: ð16Þ

The second derivative can be computed by analyzing theshape of the cost function around its absolute minimum,using the finite difference central approximation [Press et al.,1989]:

@2E

@f 2c

¼ E fc �Dð Þ � 2E fcð Þ þ E fc þDð ÞD2 ð17Þ

with D as the step considered in the inversion on the refinedgrid.[25] Once the error on the corner frequency at all of the

stations is computed, an average corner frequency can beestimated as the weighted average of the corner frequencies atall of the available stations. The same procedure is followedto compute the errors on station t* and stationW0. The seismicmoment is estimated from the low‐frequency amplitudelevel of the displacement spectrum using equation (3). Weused R(�, �) = 0.52, c = 6.5 km/s, r = 2800 kg/m3; i.e., thesame values used by Bindi et al. [2001]. The seismicmoment of the event is obtained by the logarithmicallyweighted average of the station values. Only the spectra forwhich the misfit is less than the data error Ed have beenconsidered in the calculation of the event corner frequency

Table 1. Comparisons Between the Frequency‐Dependent QModel Inferred From the Inversion of P Waves (This Study) andthe Frequency‐Dependent Model Inferred From the Inversion ofS Coda Waves [Bindi et al., 2001]

This Study Bindi et al. [2001]

Q0 a Number of Spectra Q0 a Station

19 ± 7 0.61 ± 0.16 477 83 0.63 APPE44 ± 4 0.37 ± 0.15 211 AQ120 ± 7 0.65 ± 0.16 298 96 0.69 ARM140 ± 4 0.47 ± 0.16 172 BETT6 ± 5 0.57 ± 0.15 254 CANC11 ± 4 0.59 ± 0.13 364 87 0.67 CAS141 ± 3 0.34 ± 0.15 209 CASC8 ± 7 0.66 ± 0.15 461 CMR13 ± 5 0.67 ± 0.13 198 64 0.84 COLL24 ± 4 0.35 ± 0.16 419 ETZZ59 ± 3 0.29 ± 0.13 207 GUA247 ± 5 0.42 ± 0.17 62 GUBB51 ± 5 0.44 ± 0.18 320 93 0.65 LAVE12 ± 4 0.49 ± 0.15 270 MC19 ± 8 0.63 ± 0.17 440 MVL17 ± 6 0.56 ± 0.16 386 NRC21 ± 5 0.55 ± 0.17 268 PIED13 ± 4 0.50 ± 0.15 398 RA112 ± 8 0.65 ± 0.17 452 68 0.73 RASE19 ± 6 0.56 ± 0.16 439 SERR18 ± 7 0.73 ± 0.14 278 81 0.73 SPRE15 ± 5 0.53 ± 0.15 415 74 0.73 TREV15 ± 6 0.52 ± 0.17 440 TRP

Table 2. Comparisons Using the AIC and the AICc, Between theConstant Q model and the Frequency‐Dependent QModel InferredFrom the Inversion of P Wave Spectra

Constant QModel (× 106)

Frequency‐DependentQ Model (× 106)

AIC (E1) 1990 2470AIC (E2) 1410 1870AICc (E1) 13.1 13.3AICc (E2) 2.31 2.34


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and seismic moment. The error Ed has been estimated as theaverage fluctuation in L1 norm of the data:

Ed ¼ 1

N

XN�1

i¼1

A fiþ1ð Þ � A fið Þj j: ð18Þ

[26] It is well known that a correlation between fc and QP

can occur from single spectrum inversion [Scherbaum,1990]. Therefore, the use of spectra at different stationscan help us to obtain more stable estimates of these para-meters, as has been discussed in many studies [e.g.,Edwards et al., 2008, and references therein]. For this rea-son, we have carried out two additional inversion steps tominimize the correlation between QP and fc. In the first step,for each event, the t* parameters were inferred from theinversion of the station spectra, by fixing the event fc to theweighted average of station fc values. At the second step, foreach event, all of the spectra were inverted to estimate theevent fc, with the t* values fixed to the values inferred atthe previous step.

4. Inversion Results

[27] Two different inversions were carried out. In the first,a constant QP was assumed. In the second, a frequency‐dependent QP model was estimated from the inversion ofthe P wave spectra.[28] Let us first consider the inversion with a constant QP

model. In this case, there is a trade‐off between Q−1 and kfor the determination of the attenuation spectrum. Therefore,a high k can compensate for a small Q−1, and vice versa. Asa consequence, it is impossible to retrieve QP and k fromsingle station inversion. To allow for this trade‐off, we usedthe following shape of the theoretical spectrum in the firstinversion step:

Uij fð Þ ¼ Sij fð ÞBij0 fð Þ ð19Þ

where

Bij0 fð Þ ¼ Bij fð ÞKj fð Þ ¼ exp ��ft*i;j

� ð20Þ

In equation (20), t*i,j includes both the whole path and thenear‐site attenuation effects. Therefore, the inverted qualityfactor includes both the whole path and the near‐siteattenuation.[29] In the second inversion scheme, we assume the

frequency‐dependent QP model described in equation (6),with Q0 and a to be determined from the inversion of theP wave spectra. The Q0 and a values at each station wereaveraged to obtain the results summarized in Table 1. Wenote here that the Q0 values estimated from the inversion ofthe P spectra are consistently smaller than those obtained byBindi et al. [2001]. This is easy to explain since we did notcorrect the spectra for the near‐site attenuation, and there-fore the estimated Q0 values also partially include the kcorrection. In contrast, the retrieved a values (Table 1) arecomparable to those inferred from the S coda waves (Table 1),thus indicating that the dependence of QP on frequency forthe P waves is approximately the same as that previously

Figure 6. (a) Station QP as a function of the station corner frequency. (b) Station QP as a function of theevent corner frequency. (c) Station QP as a function of the source to receiver distance.

Figure 7. Distribution of the level of significance of thenull hypothesis H0 (H0: fc and QP are not correlated) usingthe station and event fc.


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Figure

8.Distributionof

QPvalues

ateach

station.

(a)Statio

nswith

anapproxim

atelyunim

odal

QPdistributio

n.(b)Sta-

tions

with

amultim

odal

oruniform

QPdistributio

n.


11 of 23

Figure

8.(contin

ued)


12 of 23

inferred by Bindi et al. [2001]. Since this kind of inversionimplicitly includes for a = 0 the inversion for the constantQP case, this model always has a reduced variance withrespect to the constant QP model. Therefore, to establishwhether the frequency‐dependent QP model inferred fromthe data is to be preferred to the constant QP model, it is notpossible to simply compare the fits of the two models to thedata, owing to the different number of model parameters

used in the single‐station inversions (four unknown modelparameters in the frequency‐dependent QP model [Q0, a, fc,W0] versus three unknown model parameters in the constantQP model [t*, fc, W0]). Therefore, to establish which of thesephysical models best reproduces the data, we carried out astatistical comparison based on the Occam’s razor principle,as described by the Akaike information criterion (AIC)[Akaike, 1974]. This criterion can be used to select the bestfit model among different models that have differentnumbers of model parameters. In the present cases, thegenerally observed variance reduction that arose from theincrease in the number of model parameters is not sufficientto guarantee the significance of the underlying, more com-plex, physical model. For this reason, the AIC has beenwidely used in many seismological studies in recent years[e.g., Chouet et al., 2005; Emolo and Zollo, 2005; deLorenzo et al., 2008]. The AIC can be written as

AIC ¼ NtNs lnE þ 2NmNt ð21Þ

where Nt is the whole number of spectra for which the misfitis less than the error on the data, Ns is the number of pointsof each spectrum, Nm is the number of unknown modelparameters, and E is the variance. Moreover, a correctedAkaike information criterion (AICc) has also been used

Table 3. Station QP Estimates Under the Assumption of aConstant QP Model

Q DQ Station

152 4 BETT54 6 CANC56 5 CAS187 4 CASC59 5 CMRZ181 4 COLL40 7 ETZZ165 4 GUA2165 3 GUBB35 6 MC1Z68 5 PIED64 5 RA1Z76 5 TREV53 6 TRPZ

Figure 9. Ray tracing in the regional one‐dimensional velocity model of the Umbria‐Marche region.


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[Burnham and Anderson, 2002] by determining the AICcfor the two models in the following:

AICc ¼ NtNs lnE þ 2Nm þ 2Nm Nm þ 1ð ÞNtNs � Nm � 1

: ð22Þ

To allow for possible discrepancies caused by error nor-malization, two different expressions of the variance weretaken into consideration:

E ¼ E1 ¼ 1

Ns

XNt

i¼1

1

Nt;i

XNt;i

j¼1

Aobsi;j � Ateo

i;j

2 ð23Þ

E ¼ E2 ¼ 1

Ns

XNt

i¼1

1

Nt;iEd;i

XNt;i

j¼1

Aobsi;j � Ateo

i;j

2 ð24Þ

where Nt,i is the number of points in the ith spectrum and Ed,i

is the average data error in the ith spectrum, according toequation (18). In the first case (equation (23)) the variancedoes not account for the error in the data, whereas in thesecond case (equation (24)) the variance is normalized todata errors. In the calculation, we used the averaged seismicmoments and corner frequencies of each event.[30] The results given in Table 2 indicate that the constant

QPmodel always provides the minimum of both the AIC andthe AICc, and therefore this has to be considered as the bestcompromise betweenmodel simplicity and adherence to data.

5. Discussion

[31] In this section, we focus on the results derived fromthe constant QP model, which we have demonstrated to bethe preferred attenuation model for P wave spectra.

5.1. Attenuation and Site Parameters

[32] The plots of station Q versus station fc and the plot ofstation Q versus event fc are shown in Figures 6a and 6b,respectively. The plot of Q with the source to receiver dis-tance R is given in Figure 6c. No trend between Q and R isinferred.[33] A statistical Pearson’s test [Press et al., 1989] was

carried out to evaluate the level of correlation between Qand station fc (after the first two steps of the inversionprocedure) and the level of correlation betweenQ and event fc(after the two last steps of the inversion procedure). Figure 7shows how the level of significance of the null hypothesisH0 (H0: Q and fc are not correlated) increases when weconsider the results with the event corner frequency. Thisrepresents a posteriori confirmation of the ability of the twoadditional inversion steps to stabilize the results.[34] To better focus on the reasons for the great variability

of the station QP values (Figure 6), we first computed theaveraged station Q. Figure 8 shows the histograms of stationQP using the width of the bins as a quarter of a logarithmicdecade. In particular, an approximately unimodal distributionof log10(QP) is inferred at 14 stations (Figure 8a). For thesestations, we can therefore assume a lognormal distribution toestimate the average station QP and its error (Table 3). Thestation QP values are generally small, as they range from aminimum of 53 to a maximum of 181. For the remainingnine stations, the log10(QP) distributions are far from beingunimodal (Figure 8b), and an estimation of an average sta-tion QP is meaningless.[35] The histograms in Figure 8b and the dispersed values

of station QP versus R in Figure 6c indicate that the crust inthe Umbria‐Marche region is characterized by local‐scalevariations in the anelastic properties of the medium. Althoughthe inference of a tomographic Qp image is beyond the

Figure 10. (a) Three one‐dimensional QP models obtained by back projection and corresponding to the(b) three considered one‐dimensional velocity models, together with the parameters hti and hDti/htirelated to the geometrical ray resolution in each layer.


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scope of this article, an initial one‐dimensional Qp modelfor the area can be estimated from the spectral measure-ments of the parameters t* (t/Qp) by back projecting themin the regional one‐dimensional velocity model of thearea. We plotted the seismic rays (Figure 9) in the one‐dimensional layered VP velocity model of Cattaneo et al.[2000] (Figure 10b) using the algorithm proposed by Akiet al. [1977]. Each t* value previously inferred from theinversion was then back projected along the seismic ray, andthe whole path QP was partitioned along the ray aftercomputing the travel time of the ray in each layer. We deducethat QP progressively increases with depth (Figure 10a),from a minimum of 210 in the first 2 km of the crust, to amaximum of 371 at depths greater than 6 km. Figure 10bsummarizes both the velocity model used in the inversionand the estimation of the actual resolution, in terms of theratio between the average travel time hti in each layer andthe average standard deviation of the travel time normalizedto the average travel time hDt/ti; since this latter parameteris proportional to the azimuthal ray deviation used by Ponkoand Sanders [1994], it provides a relative measure of thelevel of ray crossing in each layer. Therefore, the resolutionhas a maximum in the second layer, where hDt/ti reaches itsmaximum. To study the sensitivity of the retrieved attenu-ation (QP) model to the assumed velocity (VP) model, weperformed two other back projections after relocating theevents in the minimum and maximum one‐dimensionalvelocity models of Cattaneo et al. [2000] (Figure 10b). Thecorresponding one‐dimensional QP models (Figure 10a)confirm the increase in QP with depth, and they allow us toestimate an average one‐dimensional QP model and realisticerrors on QP in each layer (Table 4). The inferred QP values(Table 4) are of the same order of magnitude as those seen inprevious source and attenuation studies [de Lorenzo andZollo, 2003; Filippucci et al., 2006].[36] When compared with the layered one‐dimensional

QP model, higher (QP > 400) and lower (QP < 200) QP

values clearly indicate the existence of crustal volumescharacterized by anomalous attenuating properties, and/orresidual site attenuation/site amplification effects. To betterquantify the site attenuation or site amplification effects, wecomputed the residuals between the t* values inferred fromthe inversion (t*obs) and the theoretical t* values computedin the estimated one‐dimensional Qp model (t*est):

Dt* ¼ t*obs � t*est ð25ÞThese results are plotted in Figure 11. If we assume thatsystematic site attenuation (Dt* > 0) or site amplification

(Dt* < 0) occurs if the same sign of Dt* repeats in at least75% of cases, we deduce that there is a site attenuationeffect for 13 stations, while there is a site amplificationeffect for only three stations (ARM1, LAVE and SPRE)(Figure 11, Table 5).[37] Interestingly, almost all of the sites where a positive

Dt* is inferred (i.e., where additional site attenuation effectsoccur) represent stations for which a nearly unimodal stationQP distribution (Figure 8a) has also been inferred, i.e., sta-tions for which it is significant to assume a homogeneous Q.Therefore, these positive Dt* stations represent additionalsite attenuation effects that are not taken into account by theone‐dimensional attenuation model.[38] On the other hand, almost all of the stations where

Dt* does not have a well‐defined sign are also stationswhere the station QP is uniformly distributed or has amultimodal distribution (Figure 8b). Since the variability ofthe QP values at these stations cannot be simply attributed tounsolvable site amplification or site attenuation effects, thevariability of QP has to be related to the 3‐D anomaliesdistributed in the volume sampled by the seismic rays.[39] Figure 12 shows the P wave site responses (Aj in

equation (7)) at the seismic stations when grouped in termsof similar functions. The seismic responses at stationsgrouped in Figure 12a do not show particular trends withfrequency. The fluctuations of these spectra around unitymight be an effect of random fluctuations in the data due tothe noise. The site responses grouped in Figure 12b showsignificant site attenuation effects at high frequency (around30 Hz), which can be attributed to near‐surface attenuatinglayers. A well‐resolved amplification peak at a frequencybetween 25 and 30 Hz is inferred for the stations grouped inFigure 12c. A clear peak at a lower frequency, of around10 Hz, is inferred for the stations grouped in Figure 12d. Asthe site responses are obtained from the averages of a highnumber of residuals (equation (12)), it is difficult for them toinclude nonmodeled source effects. In particular, as direc-tivity source effects show a dependence on the source‐to‐receiver azimuth [e.g., Zollo and de Lorenzo, 2001],they tend to be removed when averaged over a largeamount of data. Therefore, we favor the hypothesis thatthese peaks are due to local resonance phenomena that arerelated to the geological properties of the rocks near therecording site or to complicated 3‐Dpropagation effects. It isworth noting that these peaks can strongly affect the initialestimate of the corner frequency in a classical linearizedinversion approach.[40] Figure 13 shows the matching of the model to the

data for an event of the sequence. The theoretical spectracan, for the most part, reproduce some peculiar properties ofthe observed spectra. For instance, the abrupt decay around30 Hz observed at stations RA1, CANC, and MC1 is verywell reproduced by the model.

5.2. Source Parameters

[41] The seismic moment ranges from 1.25· × 1012 N·m to2.02 × 1016 N·m. The corner frequencies of the events range

Table 4. Qp One‐Dimensional Models Inferred by BackProjections and Their Averaged Q Model

Ztop (km) Q Model 1 Q Model 2 Q Model 3 Averaged Q

0 210 219 191 207 ± 142 263 243 242 249 ± 124 333 266 331 310 ± 386 372 321 373 355 ± 29

Figure 11. Distribution of residual station Dt* with respect to the one‐dimensional Qp model. (a) Stations with systematicsite attenuation (Dt* > 0) or site amplification (Dt* < 0) effects. (b) Stations without systematic site attenuation (Dt* > 0) orsite amplification (Dt* < 0) effects.


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Figure

11


16 of 23

Figure

11.(contin

ued)


17 of 23

from 5.0 Hz to 23.5 Hz. All of the source parameters areshown in Figure 14a together with the error bars and arecomputed according to the approach described in the tech-nique section.[42] We used 1/r as the approximation for the geometrical

spreading coefficient in the calculation of the seismicmoment (equation (3)). In other studies, different approx-imations for the geometrical spreading factor have beenused, depending on the investigated source‐to‐receiver dis-tance. Some studies [e.g., Edwards et al., 2008; Edwardsand Rietbrock, 2009] have proposed the use of the piece-wise linear function (1/rli) after the study of Atkinson and

Mereu [1992], who analyzed the behavior of the geometri-cal spreading coefficient as a function of the hypocentraldistance. It is worth noting that our spectral data are recordedat quite a short epicentral distance (R < 40 km, for most ofthe data), for which the approximation 1/R for the geomet-rical spreading is generally considered. Atkinson and Mereu[1992] showed that for source‐to‐receiver distances lessthan 70 km, the approximation 1/r for the geometricalspreading is satisfactory.[43] The source radius was computed using the Madariaga

crack model [Madariaga, 1976]:

L ¼ 0:32Vs

fcð26Þ

The plot of the seismic moment versus the source radius isshown in Figure 14b. Here, the straight lines of constantstress drop were computed according to the relationship[Keilis‐Borok, 1959]

D� ¼ 7

16

M0

L3ð27Þ

The seismic moment slowly decreases with increasing cornerfrequency (Figure 15).[44] An average stress drop, of Ds = 39 MPa, was esti-

mated using equation (27). The stress drop clearly increaseswith the seismic moment (Figure 16a), which is evidence forthe violation of self‐similarity in the explored momentrange. The trend of the seismic moment versus the stressdrop is approximately the same as that inferred in manyother regions of the Earth, as shown in Figure 16a, where

Figure 12. (a–d) Site responses estimated from the inversion and grouped in terms of similar functions.

Table 5. Residual Site Attenuation (Dt* > 0) or Site AmplificationEffects (Dt* < 0)

Dt*(s) Station

0.069 MC10.040 RA10.014 BETT0.018 GUBB0.021 CANC0.023 CASC−0.006 ARM1−0.008 LAVE0.031 PIED−0.013 SPRE0.026 ETZ0.024 TRP0.042 CAS10.023 TREV0.020 CMR


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Figure 13. Comparisons between observed (black lines) and retrieved (red lines) source spectra for anevent considered in this study.


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the obtained results are compared with the results describedby De Natale et al. [1987] and Tusa and Gresta [2008].Since the original plots of De Natale et al. [1987] and Tusaand Gresta [2008] used the Brune [1970] stress drop Dss,their data have been appropriately scaled to obtain theseismic moment as a function of the stress drop using theMadariaga [1976] formula for the source radius. It can beeasily shown that the scaling factor is

D�=D�s ¼ 5:47: ð28Þ

Figure 16b shows the differences between the invertedseismic moment and the seismic moment computed usingequation (27) and a stress drop Ds = 39 MPa. This infersthat the deviation from self‐similarity occurs around M0 =4· × 1014 Nm. However, to confirm the deviation from self‐similarity, we first derived the relationship between theseismic moment and the stress drop:

log10 M0 ¼ 0:7 log10 D�� 13:3: ð29Þ

To determine whether the model described by equation (29)explains the data better than a constant stress drop model,

we then compared the AIC values obtained under these twoassumptions. We used equation (21) to compute the AICvalues with

E ¼ 1

Nevents

XNevents

i¼1

log10 Mtrue0 � log10 M

est0

2 ð30Þ

where Nevents is the number of seismic events, M0true is the

inferred seismic moment, and M0est is the seismic moment

estimated under each of these two assumptions. Weinferred here that the stress‐dependent seismic moment(equation (29)) relationship has to be preferred, in that itresults in an AIC (AIC = −1.27 × 106) that is smaller thanthat obtained under the assumption of a constant stress drop(AIC = −7.52 × 105).[45] There is a controversy in the seismological commu-

nity on the origin of the worldwide stress‐drop variationwith the seismic moment that is seen. Many studies havetended to favor the hypothesis of a constant stress drop [e.g.,Abercrombie, 1995; Ide and Beroza, 2001; Allmann andShearer, 2007, and references therein], which was firstsuggested by Aki [1967] in explaining the departure fromself‐similarity as an unmodeled near‐surface attenuationeffect, which would prevent an accurate estimation of theevent corner frequency for small magnitude earthquakes.Other studies have argued in favor of the hypothesis of aphysical dependence of stress drop on seismic moment (Aki[1987], De Natale et al. [1987], Dysart et al. [1988],Centamore et al. [1997], Tusa and Gresta [2008], amongothers). This issue is far from being resolved, both becauseof poor data resolution and of model assumptions [Scholz,1982; Papageorgiou and Aki, 1983b; Aki, 1984]. Indeed,the departure from a constant stress‐drop model is not nec-essarily caused by propagation [e.g., Frankel, 1982] or site[Cranswick et al., 1985] effects, although it might also becaused by source effects [Papageorgiou and Aki, 1983a;Aki, 1987]. As in our spectral fitting procedure, we haveaccounted for both site effects and raypath attenuation, andwe tend to favor the hypothesis that the scaling of theseismic moment versus the stress drop is a source effect. In

Figure 14. (a) Corner frequency (fc) and seismic moment(M0) of the studied events and (b) Seismic moment (M0) ver-sus source radius (L).

Figure 15. Seismic moment (M0) versus corner frequency(fc).


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particular, as has been more recently shown by Robinsonet al. [2006] and Das [2007], a difference in rupture veloc-ity between small and large events might be the cause forthe observed stress drop increase with seismic moment.Although this hypothesis would be consistent with theevidence for the supershear rupture velocities for some ofthe most energetic events of the 1997 Umbria‐Marcheseismic sequence [de Lorenzo et al., 2008], the difficultyin estimating the rupture velocity of the small magnitudeevents does not allow us to draw definitive conclusions.[46] Using the Brune formula to compute the source

radius, we inferred an average static stress drop (6 MPa) thatis comparable to that given by Bindi et al. [2001] (2 MPa),even if we used a constant QP model. This is an a posterioriindication that the data set of P waves cannot solve thedependence of QP on the frequency.[47] By comparing our results with those reported in

Figure 10 of Bindi et al. [2001], we can deduce that oursource size estimates are generally shifted toward valueshigher than those inferred by Bindi et al. [2001]. However,this difference is compensated for by the generally highervalues of our seismic moment estimates, which result inapproximately the same average stress drop.[48] Some studies have pointed out a dependence of the

stress drop on the depth of the seismic events [e.g.,Hardebeck and Aron, 2009], which can be explained interms of the increase in the applied shear stress with depth,along with increasing overburden pressure. For this reason,Figure 16c shows the stress drop as a function of the eventdepth. We do not infer any clear increase in the stress dropversus the depth. A possible explanation for this issue can befound considering the role played by fluid overpressure inthe Umbria‐Marche region, which can reduce the shearstress rate in the crust, with respect to its value in a dry crust.This hypothesis is supported by Miller et al. [2004], whoindicated that the 1997 Umbria‐Marche crisis cannot bewell‐interpreted using models of elastic stress transfer.

They proposed that the aftershocks of the sequence weredriven by the coseismic release of trapped high‐pressurefluids that propagated through damaged zones that werecreated by the main shocks. By applying a model of porepressure relaxation,Miller et al. [2004] estimated an averagepermeability of 4 × 10−11 m2 around the fault zones, which ismuch higher than the typical permeability value (10−18 m2)around most of the known cataclastic fault zones [Rice,2006].[49] In Figure 17 the estimated seismic moment is plotted

as a function of the local magnitude determined by Govoni

Figure 16. (a) Seismic moment versus stress drop. Results from the present study are superimposed onthose reported in two previous studies of De Natale et al. [1987] and Tusa and Gresta [2008].(b) Differences between estimated and theoretical seismic moments versus estimated seismic moment.(c) Stress drop versus depth. Straight line represents superimposed least squares analysis of stress dropversus depth.

Figure 17. Seismic moment (M0) versus local magnitude(ML). Straight line represents superimposed inferred leastsquare analysis (equation (31)).


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et al. [1999]. The following moment magnitude relationshipis therefore inferred for P waves:

Log10M0 ¼ 0:81 �0:04ð ÞML þ 11:85 �0:12ð Þ ð31Þ

where the seismic moment is measured in N·m.

6. Conclusions

[50] In the present study, estimation of the source, atten-uation, and site parameters has been carried out from theinversion of microearthquake P wave data recorded incentral Italy during the aftershock sequence of the 1997Colfiorito, Umbria‐Marche event. Comparisons betweendifferent Qp models have allowed us to establish that aconstant QP provides the best compromise between datafitting and model simplicity. On the other hand, the inferredone‐dimensional Qp model showed a clear increase withdepth, which is evidence for a vertically heterogeneousdistribution of anelastic properties in the shallow crust.[51] An average stress drop Ds = 39 MPa was estimated

using the Madariaga [1976] source model, with a clearincrease in the seismic moment versus the stress drop. Ourdata cannot define the cause for the observed violation ofself‐similarity, but we are in favor of a cause that is relatedto an unmodeled source effect. However, this calls intoquestion the use of constant stress drop models for groundmotion simulation and for the determination of attenuationrelationships in the central Apennines. Future studies will benecessary to establish whether unmodeled rupture velocityeffects are responsible for the apparent deviation fromself‐similarity.[52] The absence of a clear trend in stress drop versus

depth strengthens the hypothesis that fluid pressurizationeffects had an important role around the Umbria‐Marchefault system. The variability in the estimates of QP at someof the stations, which appears to be significant when com-pared with the inferred one‐dimensional QP model, providesa clear indication of the heterogeneity of the crust in theUmbria‐Marche region, which can be ascribed to the tec-tonic regime in the area, as inferred in other studies[Rietbrock, 2001]. This heterogeneity might be the maincause for the inapplicability of the power law describing thedependence of QP on frequency.[53] Overall, our data confirm the previously inferred

complexities of the upper crust in the Umbria‐Marcheregion [Chiarabba and Amato, 2003]. The variation of theapparent station QP estimates and the residual Dt* indicatethat a further 3‐D tomographic study is needed to determinethe position and the geometry of the crustal bodies that areresponsible for the anomalous P wave attenuation.

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S. de Lorenzo and G. Zito, Dipartimento di Geologia e Geofisica,Università degli Studi di Bari Aldo Moro, Via Orabona 4, I‐70125 Bari,Italy. ([email protected]; [email protected])A. Zollo, Dipartimento di Scienze Fisiche, Università degli Studi di

Napoli Federico II, Via Diocleziano 328, I‐80124 Napoli, Italy. ([email protected])


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