An EGF technique to infer the rupture velocity history of a small

magnitude earthquake

S. de Lorenzo,1 M. Filippucci,1 and E. Boschi2

Received 12 November 2007; revised 15 April 2008; accepted 12 June 2008; published 22 October 2008.

[1] An empirical Green’s function (EGF) technique has been developed to detect therupture velocity history of a small earthquake. The assumed source model is acircular crack that is characterized by a single and unipolar moment rate function(MRF). The deconvolution is treated as an inverse problem in the time domain, whichinvolves an assumed form of the moment rate function (MRF). The source parameters ofthe MRF are determined by adopting a global nonlinear inversion scheme. A thoroughsynthetic study on both synthetic and real seismograms allowed us to evaluate the degreeof reliability of the retrieved model parameters. The technique was applied to four smallevents that occurred in the Umbria-Marche region (Italy) in 1997. To test the hypothesis ofa single rupture process, the inversion results were compared with those arising fromanother EGF technique, which assumes a multiple rupture process. For each event, thebest fit model was selected using the corrected Akaike Information Criterion.For all the considered events the most interesting result is that the selected best fit modelfavors the hypothesis of a single faulting process with a clear variability of the rupturevelocity during the process. For the studied events, the maximum rupture speed can evenapproach the P-wave velocity at the source, as theoretically foreseen in studiesof the physics of the rupture and recently observed for high-magnitude earthquakes.

Citation: de Lorenzo, S., M. Filippucci, and E. Boschi (2008), An EGF technique to infer the rupture velocity history of a small

magnitude earthquake, J. Geophys. Res., 113, B10314, doi:10.1029/2007JB005496.

1. Introduction

[2] The rupture velocity is essentially an unknownparameter in earthquake physics for both great and smallearthquakes [Kanamori and Rivera, 2004]. In particular,for small earthquakes, very few observations for rupturevelocity and stress drop are available, but rupture velocityand stress drop are key parameters for understanding thephysics of the rupture. As pointed out by Kanamori andRivera [2004], it is not yet clear if small earthquakes arecharacterized by high values of the stress drop, of theorder of tens of kbar, which in turn can accelerate therupture into super-shear velocities [Nadeau and Johnson,1998] or if they are characterized by small stress drop, upto some bars [Beeler et al., 2003]. Kanamori and Rivera[2004] highlighted that the resolution of this problemmust await accumulation of more reliable data on rupturevelocity and stress drop for small earthquakes. Thedifficulties in the inference of these fundamental sourceparameters when using kinematical crack models arelinked with the well-known trade-off among the source

parameters. In fact, any inference about the fault dimen-sion and the stress drop, based on kinematical sourcemodels with constant rupture velocity, needs assumptionson the rupture velocity [Boatwright, 1980]. In addition,Deichmann [1997], regarding kinematical crack models,pointed out that source models with constant rupturevelocity cannot be used to correctly reproduce the farfield P pulse because it is not possible to find a suitablecombination of Q and source parameters that reproducesboth the observed amplitudes and the pulse shapes.[3] In some cases, the difficulty of reproducing the

observed P pulses is due to the presence of a slow initialrise. This problem has been the object of deep andseveral studies and some controversies. In fact, someauthors [Sato and Mori, 2006a; Hiramatsu et al., 2002;Iio et al., 1999; Deichmann, 1997; Iio, 1995] attributedthe slow initial rise of P waves to a source effect,whereas other authors [Ellsworth and Beroza, 1998; Moriand Kanamori, 1996] attributed it to a path effect. Themost recent research activity is in favor of the interpre-tation of the gradual rise of the P wave as a source effect,even if it is more controversial for small earthquakes. Iio,[1995] concluded that, to reproduce the rising part of theP wave of microearthquakes it is necessary to assumesource models that account for the variability of therupture velocity during the rupture process. Deichmann[1997] showed that a variable rupture velocity model[Sato, 1994] is required to reproduce the first cycle ofthe P waveform, even if the trade off among source

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, B10314, doi:10.1029/2007JB005496, 2008ClickHere

for

FullArticle

1Dipartimento di Geologia e Geofisica and Centro Interdipartimentaleper la Valutazione e Mitigazione del Rischio Sismico e Vulcanico,Universita di Bari, Bari, Italy.

2Istituto Nazionale di Geofisica e Vulcanologia, Rome, Italy.

Copyright 2008 by the American Geophysical Union.0148-0227/08/2007JB005496$09.00

B10314 1 of 26

parameters and Q remain still severe. This is because theuncertainty on Q allows compensating the changes in theamplitude of the waveforms, which is also modulated bythe rupture velocity and the stress drop. This trade-offmakes the direct modeling of the waveforms nonsuitableto infer information about the rupture velocity history andthe other source parameters, as discussed in Deichmann[1997].[4] Difficulties in the modeling of the observed wave-

forms can be due also to their complexity, which in turncan be related to the complexity of the source process,i.e., the occurrence of multiple shocks [Kikuchi andKanamori, 1982; Thio and Kanamori, 1996; Singh etal., 1998; Sato and Mori, 2006b; Fischer, 2005]. Inparticular, studying earthquakes of different size, Satoand Mori [2006b] concluded that while large earthquakes(ML > 4–4.5) occur with complex rupture, small earth-quakes (ML < 4–4.5) can be due to a simple rupture.Singh et al. [1998] fed the controversy because theyfound that the observed simple seismograms for smallearthquakes, when cannot be explained by the attenuation,can be ascribed either to simple rupture process or tolocal site effects.[5] Studies on source complexity from waveform model-

ing require the signal to be free from (or corrected for)complications due to the propagation effect, i.e., to attenu-ation, scattering or multipathing [Boatwright, 1984; Ammonet al., 1994]. For this reason, one of the most powerfulapproaches in source studies is represented by the EmpiricalGreen’s Function (EGF) technique [Hartzell, 1978]. Clas-sically, it consists of isolating the source of an event(hereafter called MAIN) by performing a spectral ratiodeconvolution of its waveforms with that of a smaller event,which has approximately the same location and the samefocal mechanism of the MAIN and for which it is assumedthat the source time function is a Dirac delta function. Thespectral ratio is known to be an unstable process. In fact, itresults in negative, unphysical values of the source timefunction and in spurious oscillations [Zollo et al., 1995],which may impede to retrieve complexities of the ruptureprocess. To overcome these problems, many improvementsof the original technique have been proposed [Mueller,1985; Ammon et al., 1994; Zollo et al., 1995; Courboulexet al., 1996; Vallee et al., 2004], providing for increasingquality of the fitting of the observed waveforms. In partic-ular, Zollo et al. [1995] and Courboulex et al. [1996]developed two different nonlinear inversion techniques,using a priori constraints on the functional form of thesource time function of the MAIN. These approaches havethe advantage to implicitly include the positivity constraintof the source time function and then to stabilize theretrieved MRF.[6] Based on these grounds, assuming that the rupture

process can be described by a single rupture process with atime-dependent rupture velocity [Sato, 1994], we havedeveloped an EGF technique to study the rupture velocityhistory of a small earthquake. A time domain nonlinearglobal inversion scheme has been implemented to determinethe source model parameters and a new approach has beenused to estimate the errors on source parameters. We presenta thorough synthetic study on both synthetic and real

seismograms aimed to validate the reliability of the method.Finally, we apply the technique to some earthquakes thatoccurred in Sellano, during the 1997 (Umbria-Marche,Italy) seismic crisis. In order to establish if the ruptureprocess of the observed events is simple or complex, theinversion results are compared with those obtained assum-ing a multiple rupture process at the source. Both in thesynthetic study and in the application to real cases wehave used the correct Akaike information criterion (AICc)[Burnham and Anderson, 2002] to select the best fit modelamong models that have different degrees of complexity.

2. Method

[7] Let us consider two seismic events of differentmagnitude, recorded at the same seismic station. We indi-cate with MAIN the event of greater magnitude and withUobs(~r, t) its seismic recording. The assumption of the EGFtechnique [Hartzell, 1978] is that the source time functionof the smallest event, called EGF, is a Dirac delta function.As a consequence, the seismic recording EGF(~r, t) of thesmallest event will represent the joint effect of the wholepath attenuation, geometrical spreading, instrument and site.If MAIN and EGF have the same hypocenter and the samefocal mechanism, Uobs(~r, t) can be written as the convolu-tion of EGF(~r, t) with the true but unknown MRF:

Uobs ~r; tð Þ ¼ MRF ~mtrue; tð Þ *EGF ~r; tð Þ ð1Þ

where ~mtrue is the true and unknown source parametervector and * denotes the convolution operator.[8] In what follows, we will use the Sato [1994] model to

describe the source process of the MAIN. The modelconsists of a circular crack growing at a variable rupturevelocity. The analytical expression of the far-field displace-ment is obtained by imposing that the slip complies at eachtime with the Eshelby [1957] elastostatic solution. In the farfield regime and in the Fraunhofer approximation (i.e.,L2 � lr0/2, with L the crack radius, r0 the distance of theobserver from the Centre of the fault and l the maximumconsidered wavelength), Sato [1994] obtained the followingexpression for the MRF:

MRF tð Þ ¼ 12

7Ds

c

sin qL2a tð Þ � L2b tð Þ� �

ð2Þ

where Ds is the static stress drop, c is the phase velocity ofthe observed wave at the source, q is the take-off angle; thatis, the angle between the normal to the fault plane and thetangent to the ray leaving the source, La(t) and Lb(t)represent the distances between the center of the fault andthe points of the isochrones that are nearest and furthestfrom the observer, respectively. The isochrone is defined asa closed curve on the fault plane from which the radiation,generated at the rupture front, arrives to the observer at thesame time [Bernard and Madariaga, 1984]. The Sato[1994] MRF in equation (2) explicitly includes thepossibility for the take-off angle to vary. This implies that

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

2 of 26

B10314

the shape of the MRF will vary, varying the observationpoint of the fault and in this sense it can be referred to asAMRF. Therefore hereafter we will indicate with directivitythe dependence of the AMRF on the take-off angle fromwhich the fault is observed. This is to avoid confusion withthe directivity of the rupture that, in other cases, refers to aunidirectional or asymmetric propagation of the rupture.[9] Following Deichmann [1997], we considered that the

rupture velocity history of the Sato [1994] model consists ofthe following three stages:

Vr tð Þ ¼

Vrmax

t1t; 0 � t � t1ð Þ

Vrmax; t1 � t � t2ð ÞVrmax �

Vrmax

T � t2t � t2ð Þ; t2 � t � Tð Þ

8>>><>>>:

ð3Þ

where t1 is the time duration of the acceleration phase, t2 isthe time initiation of the deceleration phase, T is the totalduration of the rupture and Vrmax is the maximum rupturevelocity value (Figure 1). For t1 = 0 and t2 = T, theconsidered model reproduces the constant rupture velocitycrack of Sato and Hirasawa [1973]. To simplify theinversion problem, we assume that q and c in equation (2)are known from other observations. In this way, the vectorof the unknown model parameters is given by:

~mtrue ¼ L;t1

T;t2

T;Ds;Vrave

� �ð4Þ

where Vrave is the average rupture velocity.[10] The vector of the estimated model parameters ~mest

will be found by searching for the theoretical waveformsUest (~mest,~r, t) that best fit the observed seismograms Uobs

(~r, t) of the MAIN. Uest (~mest, ~r, t) is computed using therelationship:

Uest ~mest;~r; tð Þ ¼ AMRF ~mest; tð Þ *EGF ~r; tð Þ ð5Þ

Owing to the non linear dependence of Uest on the modelparameters equation (2), we decided to estimate ~mtrue byusing a nonlinear global optimization technique, based onthe search of the minimum of the cost function 1 � s:

1� s ¼ 1� 1

3sx þ sy þ sz� �

ð6Þ

where:

sj ¼

Xnpti¼1

Uestj tið Þ þ Uobs

j tið Þh i2

2Xnpti¼1

Uestj tið Þ

� �2

þ Uobsj tið Þ

� �2� � j ¼ x; y; z ð7Þ

is the semblance [Telford et al., 1990] between the observedUjobs and the theoretical Uj

est waveforms, npt is the numberof data points and the index j indicates the component of theground motion. The semblance between two signals rangesfrom 0 (when the two signals have opposite phases) to 1(when the two signals are equal). The main advantage inusing the semblance operator stands in its normalizedformulation that makes it independent on the amplitudescale of the signals, allowing an easy quantification of thesimilarity between different pairs of traces from differentseismic stations.[11] The search of the global minimum of the misfit

function has been performed using the commonly adopted[Abercrombie, 1995; Zollo and de Lorenzo, 2001] Simplexdownhill nonlinear optimization method [Press et al.,1989]. To overcome the problem of local minima, wecarried out many inversions in many subspaces of the wholemodel parameter space, by randomly initializing the Sim-plex in each subspace. This approach has been demonstratedto be robust in other seismological investigations [Zollo andde Lorenzo, 2001]. In this way, the best fit result ~mbest

corresponds to the absolute minimum (1 � s)min among thelocal minima of the misfit function in each subspace.[12] Owing to the nonlinearity of the problem, we pro-

pose to estimate the error D~mbest on ~mbest by computing theerror Ds on the semblance s and then by mapping Ds in themodel parameter space. It is worth to note that data errorsare not only due to the noise affecting the seismograms butalso to the error of q. Therefore using the error propagationformula, we have, for each component:

Dsð Þj ¼ @s

@Uobs

��������j

DUobsj

� �NOISE

þ @s

@Uest

��������j

�DUest

j

� �NOISE

þ DUestj

� �q

�; j ¼ x; y; z ð8Þ

Figure 1. Schematic cartoon of the three-stage rupturevelocity process. The acceleration stage (1) initiates at thetime zero and terminates at the time t1; the propagation stage(2) initiates at the time t1 and terminates at the time t2; thedeceleration stage (3) initiates at the time t2 and terminatesat the time T. Vrmax and Vrave are respectively the maximumand the average value of the rupture velocity.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

3 of 26

B10314

where:

@s

@Uobs

� �j

¼Xnpti¼1

2s Uobsj tið Þ þ Uest

j tið Þh i

Xnptk¼1

Uobsj tkð Þ þ Uest

j tkð Þh i2 þ

Xnpti¼1

2sUobsj tið Þ

Xnptk¼1

Uobsj tkð Þ

� �2

þ Uestj tkð Þ

� �2� �

@s

@Uest

� �j

¼Xnpti¼1

2s Uobsj tið Þ þ Uest

j tið Þh i

Xnptk¼1

Uobsj tkð Þ þ Uest

j tkð Þh i2 þ

Xnpti¼1

2sUestj tið Þ

Xnptk¼1

Uobsj tkð Þ

� �2

þ Uestj tkð Þ

� �2� �

j ¼ x; y; z

In equation (8) (DUobs)NOISE and (DUest)NOISE are the errorsof Uobs and of Uest due to the noise, and (DUest)q is the errorof Uest due to the error of q. To obtain a robust estimate ofthese errors we estimated them in L1 norm, using thefollowing formulas:

DUobs� �

NOISE¼ 1

Tn

ZTn0

Uobs tð Þ�� ��dt

DUestð ÞNOISE¼1

Tn

ZTn0

Uest tð Þj jdt

ð10Þ

where Tn is a selected time window before the P wavearrival (Figure 2A). (DUest)q is estimated by applying acentral finite difference approximation [Press et al., 1989]for the calculation of the first-order derivative, whichensures a second-order accuracy:

DUestð Þq¼1

2Ts

ZTs

0

Uest t; qþDqð Þ � Uest t; q�Dqð Þj jdt ð11Þ

where Dq is the error on q and Ts is the time interval of thefirst cycle of the P wave used in the inversion (Figure 2C).[13] Equation (8) allows us to define a threshold of

acceptability At of the semblance:

At ¼ 1� sð Þmin þDs; ð12Þ

where (1 � s)min is the misfit corresponding to the best fitsolution, as above described. All the solutions, which giverise to a misfit 1 � s that is below At, must be accepted. Themodel parameters and their errors are then estimated byaveraging all the accepted solutions. It is worth noting that(DUest)NOISE is due to the convolution of the noise of the EGFwith the AMRF, as it follows from equations (5) and (10). Asan effect of the low-pass filter operated by the convolution,this noise appears as high-amplitude oscillations with lowfrequency (Figure 2B) and with an amplitude that is inaverage higher than the original noise on the EGF.Consequently, Ds equation (8) may be also overestimated.However, since in seismological problems it is, in practice,very difficult to account for all the possible sources of error, anapproach that overestimates the error is preferable to one thatunderestimates it.

3. Testing the EGF Technique

3.1. Tests on Noise-Free Synthetic Seismograms

[14] Numerous tests on noise-free synthetic seismogramswere performed. All the synthetic seismograms were com-

puted using a sampling rate of 125 Hz. The Green’s functionwas computed considering a one-dimensional layered in-elastic structure, using the discrete wave number technique[Bouchon, 1979, 1981]. Synthetic waveforms Uobs werecomputed by the numerical convolution [Press et al., 1989]of the Green’s function with a known AMRF. In theinversion, the whole range of admissible values of themodel parameters has been chosen as follows: Ds isallowed to vary between 1 and 1000 bar, t1/T and t2/Tis allowed to vary between 0 and 1, with t2 t1. Sincewe are considering small magnitude earthquakes, weassume L to vary between 50 m and 700 m. This meansthat, considering a constant stress drop Ds = 30 bar, themaximum investigated magnitude is roughly Mw = 4[Kanamori, 1977].[15] The average rupture velocity Vrave is allowed to vary

from 0.6Vs to Vp, being the P wave velocity the maximumrupture velocity that is physically admissible for a shearcrack [Kostrov and Das, 1988] and for the MRF of P waves

ð9Þ

Figure 2. Schematization of the errors on the estimatedseismogram. (A) The vertical component of the P waveformand its early coda on an estimated seismogram Uest. Thetime windows highlighted in grey color are Tn, which is thewindow where the noise is computed, and Ts, whichincludes the signal used in the inversion. (B) The zoom onUest in the time window Tn. (C) The three seismograms inthe time window Ts represent the variations on Uest causedby the error Dq on the take off angle q. In this example q =25� and Dq = 10�.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

4 of 26

B10314

of the Sato [1994] model. The EGF was computed by thenumerical convolution of the Green’s function of the MAINwith a ramp-like source time function having an assignedrisetime t0,EGF at the source.[16] Since the EGF technique is based on the assumption

of a Dirac delta function, first of all we evaluated how muchthe source parameters can be biased by the use of a finitefrequency source of the EGF. To this aim, several synthetictests were carried out. An example is shown in Figure 3 andTable 1A. We computed seven synthetic EGFs, four ofwhich are shown in Figure 3, with t0,EGF ranging from 5 msto 35 ms. By comparing true and retrieved model parame-ters (Table 1A), we deduced that almost all the sourceparameters are generally very well recovered and do notdepend on t0,EGF. The only exception is represented by theparameter t1/T, whose misfit increases with increasingt0,EGF. As a consequence, the fit of the first part of theseismogram (P rise in Figure 3A) and of the AMRF(Figure 3B) tends to deteriorate for t0,EGF > 15 ms. Sincet0,EGF controls the duration DTEGF of the first cycle of theP wave of the EGF, by imposing that the maximum error

on t1/T has to be less than 0.1, we deduced the followingrule in selecting the EGF:

DTMAIN

DTEGF 2 ð13Þ

where DTMAIN is the duration of the first cycle of the Pwave of the MAIN.[17] As argued by Deichmann [1997], the most critical

problem in using variable rupture velocity model in anattenuating Earth is the severe trade-off between the sourceparameters and the attenuation. Therefore the second test wecarried out was aimed both to evaluate if the inversiontechnique is able to remove these trade-offs and to quantifythe intrinsic error on model parameters due to the inversionprocedure. To this aim, we computed 196 three-componentseismograms (MAIN) using 4 different layered structuresand 49 different AMRF. The errors on the retrieved modelparameters are plotted in Figure 4. Almost all the parame-ters are well resolved. We used the L1 norm to estimate theerrors on model parameters. The source radius L is affected

Figure 3. Results of the test on the frequency content of the EGF. (A) Left column: vertical componentsof four of the seven EGF in Table 1A, with the corresponding t0,EGF superimposed in each box; centercolumn: matching between the observed vertical seismogram (solid line) of the MAIN and the estimatedone (dashed line), using the corresponding EGF at left; in grey the Ts time window used in the inversion.(right column) Zoom on the P rise of the MAIN. (B) Comparison between the true AMRF (solid line) andthe AMRF retrieved from the inversion (dashed line) of the MAIN using the EGF reported on the samerow. Increasing t0,EGF, the fitting of the MAIN and of the AMRF, gets worse as well.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

5 of 26

B10314

Table 1. Results of the Tests on Synthetic Seismogramsa

A. The Effect of the Finite Frequency Content of the EGF

L (km) t1/T t2/T Vrave/Vs Ds (bar) (1 � s)min

True 0.30 0.20 0.80 0.90 80t0,EGF = 5 ms 0.30 0.21 0.75 0.87 87 1.6E�4t0,EGF = 10 ms 0.29 0.25 0.73 0.85 90 3.4E�4t0,EGF = 15 ms 0.30 0.28 0.77 0.89 84 2.4E�4t0,EGF = 20 ms 0.30 0.40 0.70 0.88 85 6.1E�4t0,EGF = 25 ms 0.31 0.50 0.88 1.00 76 4.6E�4t0,EGF = 30 ms 0.31 0.66 0.79 1.00 78 1.0E�3t0,EGF = 35 ms 0.31 0.75 0.75 0.99 84 1.8E�3

B. The Effect of a Different Localization Between the EGF and the MAIN

L(km) t1/T t2/T Vrave/Vs Ds (bar) (1 � s)min

True 0.4 0.3 0.8 0.9 80.Vertical distance D = 0.5 km 0.37 0.26 0.83 0.88 84.7 1.6E�3Horizontal distance D = 0.5 km 0.39 0.27 0.82 0.92 82.8 1.7E�4Horizontal distance D = 1 km 0.41 0.35 0.86 0.94 77.2 4.1E�4

C. The Random Deviates Technique on Noisy Signals

DL (%) Dt1/T Dt2/T D(Vrave/Vs) (%) D(Ds) (%)

(N/S)EGF = (N/S)MAIN = 1% 0.7 0.03 0.03 1.2 3.2(N/S)EGF = (N/S)MAIN = 5% 0.5 0.04 0.04 2.6 2.1(N/S)EGF(N/S)MAIN = 10% 1.6 0.07 0.03 3.4 3.6

D. The Shifting Procedure on Noisy Signals

DL (%) Dt1/T Dt2/T D(Vrave/Vs) (%) D(Ds) (%)

(N/S)EGF = (N/S)MAIN = 1% 1.65 0.07 0.02 1.04 6.7(N/S)EGF = (N/S)MAIN = 5% 3.53 0.21 0.06 6.22 6.5(N/S)EGF(N/S)MAIN = 10% 1.71 0.36 0.10 9.59 13.9

a(A) Test on the finite frequency content of the EGF (Figure 3). The retrieved source parameters and (1 � s)min, for the seven inversions with differentt0,EGF are compared with the true values. (B) The test of the effect of a different Green’s function of EGF and of MAIN for a vertical and horizontaldifference of the hypocenters. The retrieved source parameters and (1 � s)min are compared with the true values. (C) Error on the retrieved parametercomputed using the random deviates (Figure 6). (D) Error on the retrieved source parameters computed using the shifting procedure (Figure 7) on noisysignals for four different values of N/S.

Figure 4. Test on the reliability of the inversion. (Top) Plots of the errors on L, Vrave/Vs and Ds versusthe identification number of the inversion. (Bottom) Plots of the absolute errors on t1/T and t2/T versus theidentification number of the inversion. The subscript ‘‘est’’ means ‘‘estimated’’ and refers to the values ofthe parameters estimated from the inversion.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

6 of 26

B10314

by an average error of about 0.4 %; Vrave/Vs is affected byan average error of 1.1 %; the stress drop is affected by anaverage error of 0.9 %. Since t1 and t2 are normalized on T,the errors on t1 and on t2 are computed as errors on t1/T andt2/T. This means that, if Dt1/T = j(t1/T)est � (t1/T)truej = 0.1,we have that the duration of the acceleration stage isretrieved with an error equal to the 10% of the total durationof the rupture. The same occurs for t2/T. The inferredaverage errors are 0.004 on t1/T (0.4% of T) and 0.01 ont2/T (1% of T) respectively. These results indicate that theinversion technique allows to remove the trade-offs amongthe model parameters.[18] The third problem we investigated is the bias on

model parameters caused by a difference in the Green’sfunction between the MAIN and the EGF, which is un-avoidable when using real data. We simulated this effect byintroducing a hypocentral difference D between MAIN andEGF. An example is reported in Figure 5 for a gradientlayered crustal model and a vertical hypocentral difference

D = 0.5 km (Figure 5A). After numerous tests, we foundthat, for a different location D of the MAIN and the EGF,the inversion results are always unreliable. In order toovercome this problem, we propose to shift the EGFseismogram along the time axis and to carry out aninversion for each point of shifting. This is because theinversion is carried out in the time domain and then it is offundamental importance to find the ideal alignment of the Ppulses of EGF and MAIN, i.e., the alignment, which resultsin the absolute minimum of the cost function. The shiftingprocedure uses a time step equal to the sampling interval ofthe seismograms. We decided to start the shifting procedurepositioning the P pulse of the EGF on the onset of the Ppulse of the MAIN and continuing with the procedure untilthe P pulse of the EGF is completely behind the P pulse ofthe MAIN. This means that we moved the EGF P pulsefrom the left to the right on the time axis, passing throughthe P pulse of the MAIN. This choice allows us to span thewhole time interval where the best fit position can lie. We

Figure 5. Test of the hypocentral difference D between the MAIN and the EGF. The assumed modelparameters are given in Table 1B. (A) The crustal model (density r, quality factor Q, and P velocity) usedin this test. The hypocenters of the MAIN (open star), of the EGF (solid star) and the position of therecording station (triangle) are reported. In this example, the distance of the epicenter from the station is10 km and D is 0.5 km. (B) Plot of 1 � s versus nshift, i.e., the number of shifting points of the EGF. Thebest fit point is the point of minimum of 1 � s. The points 1 and 2 correspond to the positions ofmaximum cross correlation between the signals of MAIN and EGF (see the text). (C) Top box, theideal EGF; middle box, the EGF having the hypocenter in the solid star (solid line) and the EGFshifted (dashed line) in the best fit position; lower box, the fitting of the MAIN in the best fit point.(D) Comparison between the true AMRF (solid line) with the AMRF computed (dashed line) in the"Best fit" point.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

7 of 26

B10314

will see later that this procedure is fundamental when theseismograms are affected by the noise. The plot of thesemblance versus the shifts of the EGF, nshift, allows us toinfer the shifted position which gives rise to the minimum of1 � s (Figure 5B). In this point, the shifted EGF (dashedline in Figure 5C) is aligned with the ideal EGF, i.e., theEGF having the same hypocenter of the MAIN. This couldseem an obvious result, since the convolution of the EGFwith the AMRF produces a shifting in the MAIN aboutequal to the risetime of the AMRF (i.e., the time interval inwhich the AMRF reaches its maximum). The duration of therisetime of the AMRF, in turn, is controlled by threeparameters: the duration of the deceleration stage, thesource radius and the average rupture velocity. Thisexample shows that the shifting procedure allows us toretrieve the best fit position of the EGF without introducingany hypothesis on the source parameters. Moreover, itallows us to retrieve the best AMRF also when the MAINand the EGF have a different Green’s function (Table 1B andFigure 5D). As we will show in the next section, thisprocedure is even more important when the noise on seismo-grams impedes to rightly position the EGF respect to theMAIN and an objective criterion has to be chosen to alignthe traces. The same results were obtained also using thecrustal model in Figure 5A but a horizontal distance ofhypocenters D = 0.5 km and D = 1 km (Table 1B). Manyother tests, using different crustal models, indicated us thatthe reliability of the results is preserved for differences

between the MAIN and the EGF hypocenters up to a10% of the source to receiver distance, assuming amaximum error on the retrieved model parameters equalto 10%.[19] An important result of this test is that the best fit shift

of the EGF (Figure 5B) does not correspond to any of thetwo positions that maximize the cross-correlation betweenthe P pulse of the MAIN and the EGF; in Figure 5B, point 1maximizes the cross-correlation between the first half cyclesof the P pulse of the MAIN and the EGF and point 2maximizes the cross-correlation between the second halfcycles of the P pulse of the MAIN and the EGF. Thisimplies that we cannot use the cross-correlation to align theseismograms as it is usually done in the inversion of theslip distribution on the fault plane using the EGF [e.g.,Courboulex et al., 1996]. This discrepancy is probably dueto the difference between the assumed source models. Infact, when the problem consists of determining the slipdistribution on the fault plane [Courboulex et al., 1996;Abdel-Fattah and Badawy, 2002], the rupture velocity isassumed to be a constant. In our case, on the contrary, theintroduction of a variable rupture velocity and, in particular,of an acceleration stage, strongly affects the shape of the Ppulse and could cause an apparent time delay of the earlypart of P pulse. Depending on the duration of the accelera-tion process, this could give rise to a finite time intervalduring which the P pulse slowly increases without clearlyemerging. This problem has been the subject of several

Figure 6. Test on noisy seismograms of the MAIN using the random deviates technique. Plots of theretrieved parameters (averaged on the Nrand = 100 inversions) varying the noise to signal ratio (N/S) ofthe MAIN and of the EGF and varying the parameter t1/T versus the true value of the parameter t1/T.Colored error bars represent the standard deviation of the averaged parameters. Black points are the truevalue of the parameters.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

8 of 26

B10314

studies and has led to a controversy among some researcherswhich attributed the slow initial phase to a source effect [Iio,1995; Iio et al., 1999; Deichmann, 1997; Hiramatsu et al.,2002] and others which attributed it to an attenuation effect

[Mori and Kanamori, 1996; Ellsworth and Beroza, 1998].The above tests led us to use the procedure of time shifting asa general rule to perform the inversion.

Figure 7. Test of the shifting procedure applied to noisy seismograms. The true source parameters are:L = 0.35 km; t1 = 0.3 T; t2 = 0.8T; Vr = 0.9Vs; Ds = 6 bar. (Top) Plots of 1 � s versus nshift for the threeconsidered noise to signal ratio (N/S)MAIN,EGF. The solid line corresponds to At, whose value issuperimposed. (Center) Comparison among the theoretical AMRF with the AMRFs retrieved from theinversion and corresponding to the accepted solutions below At. (Bottom) Comparison between thesynthetic observed seismograms of the MAIN (solid line) and the best fit seismograms (dashed line)retrieved from the inversion, for the three considered noise-to-signal ratio (N/S)MAIN,EGF.

Table 2. Events Used for the Tests on Real Seismogramsa

Event Origin Time ML Longitude (�) Latitude (�) Depth (km) (d, f)1 (�) (d, f)2 (�)

#1 (EGF) 9711011946 2.4 12.8755 42.9825 3.8 (60, 300) (50, 60)#2 (Green’s function) 9711012343 2.4 12.8775 42.9813 3.5 (60, 300) (50, 60)

aOrigin time (year month day hour min), local magnitude (ML), and hypocenter location for the two selected events are from Govoni et al. [1999]. Thetrue and auxiliary fault planes (d, f)1,2 are from Filippucci et al. [2006].

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

9 of 26

B10314

3.2. Tests on Noisy Synthetic Seismograms

[20] In order to evaluate how the noise on data affects themodel parameter estimates, several synthetic tests on noisyseismograms have been performed. The seismograms of theEGF were computed considering t0,EGF = 10 ms at thesource. The noise on the synthetic seismograms was com-puted using three linear congruential generators of normallydistributed random numbers, taking care that there be nosequential correlations [Press et al., 1989]. As in theprevious paragraph, the synthetics were computed using asampling rate of 125 Hz.[21] We evaluated the level of bias on the retrieved source

parameters due to the noise, by varying the noise to signalratio (N/S) of the seismograms of the MAIN and of the EGF.The main result of these tests is that the central value of allthe parameters is always well recovered, with the exceptionof t1/T. The reason why t1/T is biased can be that the noiseon the P pulse tends to hide the rising part of the P waveform, which is controlled by this parameter.[22] In order to study how to treat noisy signals, we used

two different approaches and compared the results. First, weused a classical statistical approach based on the randomdeviates [Vasco and Johnson, 1998]. It can be summarizedin the following way: let us suppose that a seismogram is

characterized by a noise to signal ratio (N/S). Then, we cancompute Nrand different seismograms, by adding to eachpoint of the seismogram a random quantity which variesbetween �(N/S)Amax and +(N/S) Amax, where Amax is themaximum absolute amplitude of the P pulse. Nrand differentestimates of the source parameters can then be obtainedfrom the Nrand inversions of these seismograms. The resultsof the Nrand inversions, having the same probability torepresent the true source time function, can be averagedto obtain an estimate of the source parameters. We consid-ered five different durations of the acceleration stage, t1/T =(0.1, 0.2, 0.3, 0.4, 0.5), Nrand = 100 and three values of thenoise to signal ratio: (N/S)MAIN = (N/S)EGF = (1%, 5%,10%). In Figure 6 we plotted the estimated source param-eters with the standard deviation and in Table 1C wesummarized the errors on the retrieved model parameters.[23] As a second approach, we applied the shifting proce-

dure to noisy EGFs and MAINs with different noise to signalratio, considering the following three cases: (N/S)MAIN =(N/S)EGF = (1%, 5%, 10%) and varying the sourceparameters. This second approach was aimed to test theshifting procedure on noisy signal and to evaluate if it ispossible to use it also to compute how the noise on dataaffects the model parameter estimates. The results are listed

Table 3. Test on Real Seismogramsa

q ± Dq (�) f sampl (Hz)

(N/S)MAIN (%) (N/S)EGF (%)

EW NS Z EW NS Z

CANC 25.4 ± 19.3 62.5 0.01 0.30 0.15 0.21 0.42 0.11CMR 31.3 ± 22.3 125 1.43 0.08 1.19 0.86 0.48 0.31TREV 30.8 ± 19.8 62.5 0.33 1.19 0.82 0.36 0.55 0.27

aTake-off angle q, sampling frequency f sampl, and percentage of noise-to-signal ratio (N/S)MAIN and (N/S)EGF of the east–west (EW), north–south (NS),and vertical (Z) components at the stations CANC, CMR, and TREV of the MAIN and of the EGF, respectively.

Figure 8. Test on real seismograms. The three component (vertical, east–west and north–south)seismograms recorded at the stations CANC, CMR and TREV. (A) The focal mechanism and therecorded signals of the event 9711012343A used as EGF. (B) The focal mechanism and the recordedsignals of the event 9711011946 used as Green’s function of the MAIN. (C) The signals of the syntheticMAIN obtained by the convolution of the Green’s function with a ‘‘true’’ AMRF (source parameters inTable 4). On each box the duration of the time window (in seconds) is reported.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

10 of 26

B10314

in Table 1D. In Figure 7 we presented an example for (N/S) =(1%, 5%, 10%).We can observe that, as expected, the numberof solutions satisfying the criterion of acceptability increaseswith increasing the noise to signal ratio (plots of (1 � s) andof AMRF in Figures 7A, 7B, and 7C). It is worth also tonote that, with increasing (N/S)EGF, the spurious oscillationson the estimated MAIN due to the convolution of the noiseof the EGF with the AMRF increase as well (plots of thevelocity seismograms in Figures 7A, 7B, and 7C).[24] Finally, we compared the results from these two

approaches (Tables 1C and 1D). We can observe that theerrors on data computed from the shifting procedure are abit higher than the errors computed from the randomdeviates approach. In particular, the parameter t1/T isaffected by errors that are higher than that of t2/T and higherthan that of the random deviates approach. These testsindicated us that the shifting procedure overestimates theerrors on parameters due to the noise on data respect to thestatistical approach based on the random deviates and then

can be considered more realistic to account for the effect ofthe noise on seismograms. Another advantage of the shift-ing procedure is that the random deviates approach has alower computational cost. If we consider, in addiction, thatwe need a criterion to align the traces in the time domain,we can affirm that the shifting procedure is a fundamentaltool in the EGF inversions.

3.3. Testing the Inversion Technique on RealSeismograms

[25] The tests discussed in the previous sections werecarried out considering data from a single station. In thisparagraph, we will show how to treat data recorded at morethan one station. Usually, the EGF inversions are carried outindividually for each station assuming nondirective sourcemodels and the directivity of the rupture is a posterioriestimated by the analysis of the AMRFs retrieved at eachavailable station [e.g., Mori and Frankel, 1990; Singh et al.,2004; Vallee, 2007]. In this work, since the Sato [1994]

Table 4. Results of the Test on Real Seismogramsa

True

L (km) t1/T t2/T Vrave/Vs Ds (bar)

(1 � s)min ± Ds AICc npt0.35 0.3 0.7 0.7 7

CANC 0.29 ± 0.1 0.50 ± 0.49 0.92 ± 0.07 0.61 ± 0.01 19.7 ± 16.5 6.5E�4 ± 3E�5 �171.2 39CMR 0.36 ± 0.06 0.42 ± 0.31 0.73 ± 0.15 0.74 ± 0.10 6.9 ± 5.4 6.1E�4 ± 8E�5 �169.1 93TREV 0.48 ± 0.08 0.40 ± 0.23 0.51 ± 0.23 0.92 ± 0.05 3.8 ± 1.8 1.4E�3 ± 2E�4 �192.2 57JOINT 0.34 ± 0.04 0.42 ± 0.20 0.73 ± 0.17 0.73 ± 0.09 7.8 ± 2.4 6.0E�3 ± 1.1E�2 �316.6 189

aAveraged source parameters (L, t1/T, t2/T, Vrave/Vs, Ds) compared with the true source parameters at the three stations (CANC, CMR, and TREV)separately and the results of the JOINT inversion. (1 � s)min ± Ds is the cost function of the inversions. AICc is the Corrected Akaike Information Criterionwith k = 5, npt is the number of data points in the inversion.

Figure 9. Test on real seismograms. (A) Plot of the misfit function 1 � s versus nshift for the stationCANC. On the plot, the number corresponding to the solid line indicates the value (1 � s) = (1 � s)min +Ds. (B) The same as (A) for the station CMR. (C) The same as (A) for the station TREV. (D) Plot of themisfit function h1 � si versus ncomb for the joint inversion. On the plot, the number on the solid line isAt = h(1 � s)imin + hDsi. (E) Comparison among the true AMRF (source parameters in Table 4) with theAMRFs retrieved from the joint inversion and corresponding to the 12 accepted solutions below At.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

11 of 26

B10314

model incorporates the effect of the variability of the take-off angle, we can use this source model in a joint inversionof the waveforms at several stations with the aim ofobtaining a unique MRF.[26] We used a two-step inversion scheme: at the first

step, the inversions are carried out one for each station. Thisallows us to find, for each station, the shift compatible withthe error on semblance, as discussed in the sections 3.1 and3.2. At the second step, the joint inversion of the modelparameters is performed using all the available seismogramsand combining all the shifts found at the first step. The jointinversion has to account for the different signal-to-noiseratio of each seismogram and their different samplinginterval. For this reason, we use the following weightedformula for the calculation of the semblance:

1� sð Þh i ¼

Xj;m

1� sð Þj;m w S=Nð ÞEGFj;m w S=Nð ÞMAIN

j;m w f sampl

mXj;m

wS=Nð ÞEGFj;m w S=Nð ÞMAIN

j;m w f sampl

m

;

1 � j � 3; 1 � m � nstð Þ ð14Þ

where wj,m(S/N)EGF = (S/N)j,m

EGF and wj,m(S/N)MAIN

= (S/N)j,mMAIN are

respectively the weights that account for the signal-to-noiseratio of the EGF and the MAIN at the j-th component ofthe m-th station, wm f sampl = f m

sampl is the weight due to thesampling frequency f m

sampl at the m-th station and nst is thenumber of recording stations. The error on semblance Dsmat the m-th station is computed as a weighted average ofthe errors on the semblance Ds in equation (8).

[27] Once performed the joint inversion, we need todecide on its results. In particular, we need to find the bestfit result between the joint inversion and the single stationinversion. We used the corrected Akaike information crite-rion (AICc) [Akaike, 1973; Sugiura, 1978; Hurvich andTsai, 1995; Burnham and Anderson, 2002] to select thebest-compromise among different models and different datavectors, according to the Occam’s Razor principle [Chouetet al., 2005; Emolo and Zollo, 2005; Nakano and Kumagai,2005].[28] The AICc is given by:

AICc ¼ npt log s2� �

þ 2k þ 2k k þ 1ð Þnpt � k � 1

ð15Þ

where npt is the number of points of the seismograminvolved in the inversion and k is the number of modelparameters. In equation (15) s2 is a weighted sum ofresidual squares given by:

s2 ¼

Xnstm¼1

X3j¼1

Xnpti¼1

Uobsmji � Uest

mji

� �2

wS=Nð ÞEGFm; j w S=Nð ÞMAIN

m; j w f sampl

m

Xnstm¼1

X3j¼1

wS=Nð ÞEGFm; j w S=Nð ÞMAIN

m; j w f sampl

m

ð16Þ

where Uobs and Uest are respectively the observed and theestimated MAIN, nst is the number of recording stations, j isthe component of the seismogram ( j = x, y, z). Weights arethe same as for equation (14). According to the correctedAkaike information criterion, the best fit model correspondsto the minimum value of AICc.[29] The joint inversion has been tested, following the

approach suggested by Vallee [2004], on real data. In thisway, it is possible to evaluate how unavoidable unmodeledthree-dimensional path effects, due to different hypocenters,and unavoidable differences in the focal mechanism canalter the retrieved model parameters. The synthetic seismo-gram of the MAIN was computed by the convolution of theseismogram of a small real earthquake with a syntheticAMRF. In the inversion, we used as EGF the seismogram ofanother small real event having a slightly different hypo-center location and approximately the same fault planeorientation. Two small earthquakes, occurred on 1 Novem-ber 1997 during the Umbria-Marche seismic crisis andlocated below the town of Sellano (Umbria, Italy) [Govoniet al., 1999], were selected for this test (Table 2). The event#1 was used as EGF and the event #2 was used as theGreen’s function for the computation of the synthetic MAIN(Figure 8). For both events, fault dip d and fault strike f areavailable [Filippucci et al., 2006] (Table 2). When usingreal events, to account for the fact that the seismogram ofthe EGF will approximate a Green’s function only forfrequencies for which the displacement amplitude spectrumis flat [e.g., Mueller, 1985] and this holds only below thecorner frequency f0. Therefore the seismograms of the EGFmust be low-pass filtered below f0. Since we are working inthe time domain, it will be necessary to low-pass filterbelow f0 also the seismograms of the MAIN, if we want toavoid the introduction of meaningless high-frequency data

Figure 10. Test on real seismograms. Matching of thesynthetic observed MAIN (solid line) with the MAINcomputed by the joint inversion (dashed line). Symbols areexplained in Figure 8.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

12 of 26

B10314

in the inversion. f0 has been estimated to be about 9 Hz, byfitting the EGF spectrum with a theoretical spectrum [e.g.,Scherbaum, 1990] computed by convolving a w2 Brune[1970] source model with a constant Qp attenuation oper-ator. Qp was set equal to 354 as inferred from Filippucci etal. [2006] in this area. Based on the results of the tests onnoisy synthetic signals, we selected only the seismogramshaving a noise to signal ratio less than about 1% (Table 3).The take-off angle q was computed using the equation[Zollo and de Lorenzo, 2001] that relates q both to (d, f)and to the tangent to the ray leaving the source. The seismicrays were traced using the one-dimensional velocity modelof the Umbria-Marche region [Cattaneo et al., 2000]. Theerror on q (Table 3) was estimated by taking into accountthe errors on location, on d and on f [Filippucci et al.,2006].

[30] At the first inversion step, we performed the single-station inversions by applying the shifting procedure abovedescribed. Then, we performed the joint inversions bycombining all the previously retrieved shifts lying belowthe threshold of acceptability At equation (12) of eachstation. Even if this approach has a high computationalcost, it allows to globally explore all the possible solutionscompatible with the data. Results are summarized in Table 4.In Figures 9A–9C the plots of (1 � s) vs. the number ofshifting point nshift for the single station inversion areshown. In Figure 9D, the results of the joint inversionsare shown as a plot of h(1 � s)i versus ncomb. ncomb is aprogressive identification number of the joint inversion.Since, from the inversions at each single station we extractonly a finite number of shifts that satisfy the criterion ofacceptability, we must carry out the joint inversion for each

Figure 11. Test on real seismograms. Comparison among the theoretical AMRF (solid line) with theAMRF retrieved from the spectral division (dashed line) and with the AMRF retrieved from the jointinversion (dotted line) for the three stations CANC, CMR, and TREV, for both normalized and non-normalized functions. For each station, the water level used for the correction of the spectral division issuperimposed.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

13 of 26

B10314

combination of these acceptable shifts at all the stations.ncomb is then a number which is uniquely associated to agiven combination of the shifts at all the stations. Then itwill vary from 1 to a maximum number that is given by theproduct of the number of shifts selected at each station. Asan example, in the case of Figure 9D, four shifts wereselected for the EGF at the station CANC, 5 shifts wereselected for the EGF at the station CMR and 4 shiftswere selected for the EGF at the station TREV. In total, wecarried out 4 5 4 = 80 joint inversions and ncomb willvary between 1 and 80.[31] From Figure 9D, taking into account the errors due

both to the noise on signal and to the uncertainty on q, wededuce that 12 solutions are acceptable after the jointinversion. The AMRFs corresponding to these 12 solutionsare plotted in Figure 9E and compared with the true AMRF,fixing q = 0.[32] Using the AICc equation (15) we found out that the

joint inversion gives rise to the best fit model (Table 4). It isworth also to note that the parameters inferred from the jointinversion are the closest to the true ones. Moreover, theerrors on the parameters retrieved from the joint inversionare the smallest although the higher number of solutionscompatible with the error on semblance (Figure 9D). This isbecause the different solutions, inferred using the jointinversion, are close one to each other and to the truesolution, independently on the shifting position. This does

not occur for the single station inversion, where, in somecases (e.g., station CANC), the true value seems to bebiased by the small number of admissible models. This isone of the reasons why we developed the joint inversion. Infact, by using the joint inversion, we can realize a compro-mise between the waveforms which better constrain themodel parameters and those which are less sensitive to thevariations in the model parameters.[33] The excellent quality of the fit of the waveforms after

the joint inversion is shown in Figure 10. In Figure 11 weshowed the match of the true AMRF with the AMRFretrieved from the joint inversion and, for comparison, alsowith the AMRF obtained using the classical spectral ratiodeconvolution, with the water level criterion [Clayton andWiggins, 1976]. The spectral division was performed usingthe program PITSA [Scherbaum and Johnson, 1992]. It isworth to note that unavoidable spurious oscillations andnegative values make the AMRF retrieved from the spectraldivision hardly comparable with the theoretical one.

4. Application to Real Earthquakes

[34] In this section, we present the application of thetechnique to a data set of small earthquakes that occurredin Sellano during the Umbria-Marche (Italy) 1997 seismicsequence. These events were densely recorded after the firststrong shock occurred in the town of Colfiorito in Septem-

Figure 12. Map of the Umbria-Marche Network that recorded the 1997, 18 October–3 November,seismic crisis. The events selected for this study were localized below the town of Sellano [from Govoniet al., 1999].

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

14 of 26

B10314

ber 1997. The network of stations consisted of 15 temporaryand 8 permanent stations (Figure 12). All the instrumentshave flat velocity response between 1 and 40 Hz and thesensors have three components (vertical, north–south andeast–west). All these stations have recorded in continuousmode with high quality and with three different samplingfrequencies (62.5 Hz, 100 Hz and 125 Hz). In a previousstudy [Filippucci et al., 2006], the fault plane orientations of22 events of the Sellano sequence were inferred.[35] In this study, a careful analysis of the seismograms of

these events has been carried out to select the waveformsthat satisfy the criteria arising from the above-describedsynthetic study. First, considering the results of the tests onnoisy signals, we have selected only those waveforms forwhich the signal-to-noise ratio is less than 3% for both theMAIN and the EGF. Second, to ensure that the signals ofthe MAIN and the EGF have traveled approximately thesame path, we have selected the seismograms having asimilar P coda waveform. Third, to ensure that the finitefrequency content of the EGF does not bias the results, wehave discarded all the seismograms that do not satisfyequation (13). Finally, we have taken care that the seismo-grams of the couples EGF-MAIN have the same samplingfrequency, for avoiding the resampling of the traces. In fact,since the rupture velocity history strongly affects the shape ofthe waveforms, we cannot be sure that the resampling proce-dure of the P pulse shape does not affect the inversion results.[36] After the data selection, we collected four couples

MAIN-EGF, listed in Table 5 together with their local magni-tude ML and hypocenter localization as inferred by Govoniet al. [1999]. For three couples of events, MAIN and EGFhave roughly the same fault plane orientation [Filippucciet al., 2006]. For the fourth couple, MAIN and EGF haveroughly the same focal mechanism, as inferred from theinversion of P polarities, using the FPFIT code [Reasenbergand Oppheneimer, 1985] (Table 5 and Figure 13D).[37] Since the take off angle q is a datum of the inversion

problem, we estimated q (Table 5) at each station selected foreach event. To this aim, we used a one-dimensional velocitymodel of the Umbria-Marche region [Cattaneo et al., 2000],the hypocenter location and the values of d and f given inTable 5. The error Dq on the take-off angle (Table 5) wascomputed using the error propagation formula, consideringthe errors on dip d and strike f (Table 5) and the error in thehypocenter location. The error on dip and strike for the events#1, #2 and #3 was previously computed by Filippucci et al.[2006], while for the event #4 it was estimated from theFPFIT inversion of polarities for the focal mechanism solu-tion (Figure 13). We considered an error in the hypocenterlocalization of 1 km3, which is the same magnitude order ofthe error inferred by Govoni et al. [1999]. In this way, we canreasonably account also for possible lateral variations of theelastic properties of the crust respect to the one-dimensionalconsidered velocity structure. We can observe that, for eachconsidered event,Dq strongly varies from a station to anotherone (e.g., in Table 5, for the 2nd event, Dq varies from 8� atthe station RA1 to 20� at the station CMR).[38] The selected waveforms are shown in Figure 13. It is

worth to observe the similarity of the P wave and its coda ateach station for each MAIN-EGF couple.

Table

5.TheFourSelectedEventsPairsa

Event

Date,

(y.m

.d.h.m

)M

LLat

(km)

Lon(km)

Depth

(km)

d±Dd(�)

f±Df(�)

f 0EGF,Hz

Station

Nam

eq±Dq(�)

fsampl

(Hz)

(N/S)E

GF(%

)(N/S)M

AIN

(%)

EW

NS

ZEW

NS

Z

1MAIN

34

9710201028

3.3

4760.3

327.2

3.6

70±10

115±8

APPE

72±9

125

0.2

0.1

0.1

1.32

0.56

0.52

EGF37

9710202117

24759.9

327.4

3.4

50±10

150±5

12

CMR

42±18

125

0.1

0.1

0.1

0.16

0.13

0.05

TREV

42±20

125

0.6

2.5

0.3

0.22

0.39

0.19

2MAIN

18

9710190852

2.9

327.3

4760.5

3.5

75±3

120±10

CMR

28±20

100

0.6

0.7

0.3

0.65

0.36

0.52

EGF11

9710182048

2.2

327.

4761.2

3.4

60±8

158±5

10

RA1

20±8

62.5

0.4

0.7

0.3

0.67

0.60

0.24

TREV

26±16

125

0.2

0.2

0.1

0.50

0.31

0.32

3MAIN

25

9710200609

3.9

327.6

4760.3

3.7

65±20

320±8

APPE

44±19

125

3.2

2.1

1.5

0.51

1.13

0.13

EGF44

9710241448

2.1

328.

4761.6

3.1

30±5

310±23

15

CMR

37±17

125

0.9

1.0

0.7

0.01

0.02

0.03

4MAIN

(4)

9711020414

3.2

326.9

4760.8

3.3

(1)65±3

(1)25±8

CMR

1)89±10

125

1.9

0.7

0.3

0.5

0.16

0.05

(2)30±3

(2)165±8

.TREV

1)66±10

62.5

2.9

2.7

0.5

0.29

0.23

0.20

EGF(31)

9710200714

2.1

327.2

4760.7

5.2

(1)50±33

(1)0±20

10

CMR

2)79±21

""

""

""

"(2)50±33

(2)210±20

TREV

2)28±21

aFrom

theleft:Theidentificationdateoftheevent(yearmonthday

hourmin).ThelocalmagnitudeM

L.Thelatitude,thelongitude,andthedepth

from

Govonietal.[1999].Thefaultdip

dandthefaultstrikefofthe

events#1,#2,and#3arefrom

Filippuccietal.[2006];fortheevent#4,dandfofthetrue(1)andtheauxiliary

(2)faultplaneareinferred

from

theFPFIT

inversion.Thecorner

frequency

f 0fortheEGFevent.The

nam

eoftherecordingstation.Thetake-offangleqcomputedusingdandf(fortheevent#4,theqwas

computedusingthetrue(1)andtheauxiliary

(2)faultplane).Thesamplingfrequency

fsamplofthestationand

thepercentageofnoiseto

signal

ratio(N/S)ofeach

component(EW,NS,andZ)ofthegroundvelocity

oftheMAIN

andEGF.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

15 of 26

B10314

Figure

13.

Threecomponentvelocity

seismogramsofthefourstudiedMAIN

eventsandthecorrespondingselected

EGF

events,recorded

atdifferentstations.Oneach

box,theam

plitudescaleandthetimewindowofthetraces

isindicated.In

the

boxD,also

thefocalmechanismsofthetwoevents,as

inferred

from

FPFIT

inversion,areplotted.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

16 of 26

B10314

Figure

14.

Resultsoftheinversionsforthefourevents.In

each

box,theplots

of1�

sversusnshiftcorrespondto

the

singlestationinversionandtheplotofh(1�

s)iv

ersusncombcorrespondsto

thejointinversion.(A

)Resultsfortheevent

#1.(B)Resultsfortheevent#2.(C)Resultsfortheevent#3.(D

)Resultsfortheevent#4,where(d

1,f1)and(d

2,f2)

arethetwofaultplaneorientationsoftheMAIN

retrieved

from

thefocalmechanism

solution.Oneach

plot,thenumbers

superim

posedonthesolidlinerepresentthethreshold

ofacceptabilityAtofthesolution.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

17 of 26

B10314

4.1. The EGF Inversion With the Sato [1994] SourceModel

[39] The inversions were performed following the guide-lines described in the synthetic tests on real seismograms.As discussed in section 3.3, the validity of the EGFapproximation requires the seismograms to be low-passfiltered with a cut-off frequency equal to the corner fre-quency f0 of the EGF, computed by convolving a w�2

source function with a constant Qp attenuation operator[Azimi et al., 1968]. The estimated f0 ranges between 10 Hzand 15 Hz (Table 5).[40] At the first step, the inversion was performed sepa-

rately for each station, by using the shifting procedure. Thecorresponding plots of the semblance versus the number ofshifting point nshift are in Figure 14, grouped for event. Thesolid line on the plots represents the threshold of accept-ability At equation (12). The black points lying below thesolid line in Figure 14 are the acceptable solutions, i.e., thesolutions compatible with the errors on data.[41] At the second step, the joint inversion for each event

was performed. Figure 14 shows the plots of h(1 � s)iversus ncomb for each event. In these plots, the solid linerepresents the threshold of acceptability At of the solutionscomputed as h(1 � s)i +Ds. Only for the event #4, since wedo not have any knowledge about the true fault planeorientation, we carried out both the single station and thejoint inversion for each of the fault planes inferred from thefocal mechanism solution (Figure 14 and Table 6).[42] The values of AICc equation (15) for the single

station and for the joint inversions with the Sato [1994]source model, the number npt of data points and the numberk of degrees of freedom are reported in Table 6. For theevents #1 and #2, the joint inversion results in the minimumAICc value, indicating that the model parameters are con-strained by the whole set of waveforms. For the event #4,the minimum of AICc results in correspondence of the jointinversion using the fault plane (d, f)1 = (65�, 25�). For theevent #3, the minimum of AICc is inferred for the inversionat the station CMR, not for the joint inversion. This resultindicates that we cannot find, for this event, a unique best fitsource model from the joint inversion, so that the Sato[1994] model cannot be considered suitable to model theobserved waveforms at all the available stations.

4.2. Comparison With a Multiple Source Model

[43] Before accepting as reliable the model parametersinferred using the variable rupture velocity model of Sato[1994], we have to evaluate if a multiple rupture modelbetter approximates the source process of the studied events.In fact, the Sato [1994] model, which has been assumed asrepresentative of the source process of our events, ischaracterized by a variable rupture velocity history but itis a single rupture model with simple geometry. To establishif a multiple rupture process better fits the consideredevents, following Zollo et al. [1995], we considered amoment rate function S(t) consisting of a sum of Npseudotriangular (Hanning) functions:

S tð Þ ¼XNi¼1

Ai

21� cos

2p t � tið Þdi

� �ð17Þ

Table 6. Results of the Application of the Akaike Information

Criterion to the Inversions at the Single Station and to the Joint Inversion

Using Both the Sato [1994] and the Hanning as Source Time Functiona

Event InversionSource TimeFunction AICc k

1 APPE (npt = 90) Sato �335 5Hanning (N = 1) �365.8 2Hanning (N = 2) �353.9 4

CMR (npt = 132) Sato �595.2 5Hanning (N = 1) �621.7 2Hanning (N = 2) �639.5 4Hanning (N = 3) �644.7 6

TREV (n = 89) Hanning (N = 4) �644. 8Sato �371.6 5

Hanning (N = 1) �315.6 2Hanning (N = 2) �473.2 4Hanning (N = 3) �439.1 6

JOINT (npt = 327) Sato �888.9 5Hanning (N = 1) �585.1 2Hanning (N = 2) �619.7 4Hanning (N = 3) �610.6 6

2 CMR (npt = 73) Sato �342.5 5Hanning (N = 1) 332. 2Hanning (N = 2) 337.3 4Hanning (N = 3) �335.9 6

RA1 (npt = 63) Sato �332.3 5Hanning (N = 1) �336.0 2Hanning (N = 2) �323.1 4

TREV (npt = 91) Sato �269.2 5Hanning (N = 1) �283.9 2Hanning (N = 2) �275.1 4

JOINT (npt = 227) Sato �736.0 5Hanning (N = 1) �510.1 2Hanning (N = 2) �596.1 4

3 APPE (npt = 54) Sato �241. 5Hanning (N = 1) �255. 2Hanning (N = 2) �251.4 4

CMR (npt = 101) Sato �531.8 5Hanning (N = 1) �500 2Hanning (N = 2) �497 4

JOINT (npt = 155) Sato �386.5 5Hanning (N = 1) �259.3 2Hanning (N = 2) �325.4 4Hanning (N = 3) �297.7 6

4 (d, f)1 = (65,25) CMR (npt = 107) Sato �607.9 5Hanning (N = 1) �535.5 2Hanning (N = 2) �556.8 4Hanning (N = 3) �545.6 6

TREV (npt = 48) Sato �225.5 5Hanning (N = 1) �217.4 2Hanning (N = 2) �261.3 4Hanning (N = 3) �232 6

JOINT (npt = 155) Sato �706.9 5Hanning (N = 1) �321.5 2Hanning (N = 2) �356.6 4Hanning (N = 3) �463.7 6Hanning (N = 4) �428.6 8

4 (d, f)2 = (30,165) CMR (npt = 107) Sato �605.9 5Hanning (N = 1) �535.5 2Hanning (N = 2) �556.8 4Hanning (N = 3) �545.6 6

TREV (npt = 48) Sato �229.1 5Hanning (N = 1) �217.4 2Hanning (N = 2) �261.3 4Hanning (N = 3) �232 6

JOINT (npt = 155) Sato �669.4 5Hanning (N = 1) �302.6 2Hanning (N = 2) �328.6 4Hanning (N = 3) �280.3 6

aN is the number of Hanning pulses. For the event #4, (d, f)1 and (d, f)2are the two fault plane orientations obtained from the FPFIT inversion.AICc is the corrected Akaike Information Criterion number, npt is thenumber of data points, k is the number of free model parameters. In bold thebest-fit model of each event.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

18 of 26

B10314

where Ai is the amplitude, di is the time duration and ti isthe time delay of the i-th pulse (t1 = 0). For i > 1, the timedelay is related to the time durations by:

ti ¼XNi¼2

di�1 ð18Þ

To take into account the directivity effect, we can relate thetotal duration di of the i-th pulse to the source radius Li of acrack with circular geometry and with constant rupturevelocity [Boatwrigth, 1980]:

di ¼Li

Vrþ Li sin q

cð19Þ

where q is the take off angle, c is the P wave velocity at thesource and Vr the rupture velocity (Vr = 0.9 Vs). Theamplitude Ai of the i-th pulse is related to the stress dropDsi and the seismic moment M0i of the i-th pulse by theEshelby [1957] relation:

M0i ¼16

7DsiL

3i ¼

1

2Aidi ð20Þ

Therefore the model parameter vector is ~m = (Ai, di), 1 �i � N. The inversion has been performed following theguidelines given in Zollo et al. [1995], but using timedomain waveforms instead of their spectra and the abovedescribed shifting procedure. This choice allows us todouble the number of data available in the amplitudespectrum, and therefore it should result in a more robustestimation of the model parameters. The inversion of theHanning functions involves the same first P pulse of theseismograms of the MAIN used in the previous inversion.We performed both the single station and the jointinversions by progressively increasing the number ofsources. The best fit source model was selected by meansof the AICc in equation (15).[44] By using the AICc, we compared the best fit Han-

ning source models with the best fit Sato source model. InTable 6 we listed the number N of Hanning pulses, thenumber n of data points, the number k of degree of freedomand the values of the AICc, corresponding both to the singlestation and to the joint inversion. For the event #4, werepeated the procedure for the two possible values of q, i.e.,for each fault plane.[45] Comparing the results in Table 6, we can observe

that, for the events #1 and #2, the minimum of AICc occursfor the joint inversion with the Sato [1994] source andtherefore we can affirm that the Sato [1994] source modelcan be considered the best fit model. For the event #4, theminimum of AICc occurs in correspondence of the jointinversion with the Sato [1994] model fixing the fault plane

(d, f)1 = (65�, 25�), which can be then considered the mostprobable fault plane orientation. For the event #3, weobtained that the joint inversion neither with the Hanningnor with the Sato source time function is able to retrieve thebest fit model. In fact, for this event, the minimum of AICcoccurs in correspondence of the inversion at the singlestation CMR with the Sato [1994] model.[46] For the three resolved events (event #1, event #2 and

event #4), the source parameters with their errors aresummarized in Table 7. For the event #1, the sourceparameters are affected by small errors, while for the event#4 the parameter t1/T is affected by a large error, maybebecause of the noise to signal ratio of the EGF, which is thegreatest among all the events (about 3% for the horizontalcomponents, Table 5). For these events, also the seismicmoment M0 and its error was computed (Table 7) by usingequation (20) and the error propagation formula.[47] In Figures 15A, 15B, and 15C, we plotted the match

between the observed and the estimated seismograms forthe resolved events. In each box, on the left we plotted theseismograms on a time window that includes the first Pcycle and a part of the P coda. On the right, we plotted thezoom of the P pulse of the seismograms. From these plots,we can observe that the match is very good for almost allthe events. As an unexpected result, we found out that alsopart of the P coda of the seismograms was well reproduced,even if it was not involved in the inversion. In Figure 15Dwe plotted, for each resolved event, all the AMRFs, whichcorrespond to the points below At in Figure 14, fixing q = 0.[48] Figures 16 and 17 show, for the resolved events, the

directivity effect on the best fit AMRF. As we specified insection 1, differences in the AMRF at different stations forthe same event are due to the variations of the take-off angleat the source. Figure 16 also shows the retrieved rupturevelocity histories for the resolved events. On each plot,three curves are drawn: they represent the range of vari-ability of the rupture velocity history according to the errorson the retrieved source parameters (t1, t2, Vrave) listed inTable 7.

5. Discussion and Conclusions

[49] In this article, an EGF technique aimed to infer therupture velocity history of a small earthquake has beenpresented and applied to some small earthquakes thatoccurred in the Umbria-Marche (Italy) region in 1997.The idea for this work was due to the observation[Deichmann, 1997] that the variability of the velocity ofthe rupture process can represent the key for reproducing thewaveform of the first P cycle of the wave both in duration andin amplitude. The robust synthetic study led us to concludethat the present availability of high-quality seismogramsrecorded at local scale can be used to carefully detail thevariability of the rupture process at the source of a small

Table 7. Inversion Results for the Resolved Eventsa

Event Inversion L (km) t1/T t2/T Vrave/Vs Ds (bar) (1 � s)min ± Ds M0 (1014 N m)

1 JOINT 0.78 ± 0.03 0.18 ± 0.07 0.55 ± 0.11 1.19 ± 0.01 3.6 ± 0.2 0.025 ± 0.002 3.9 ± 0.72 JOINT 0.32 ± 0.04 0.61 ± 0.23 0.87 ± 0.09 0.67 ± 0.05 21.4 ± 6.2 0.013 ± 0.008 1.5 ± 0.94 (d, f)1 = (65,25) JOINT 0.51 ± 0.1 0.33 ± 0.23 0.57 ± 0.14 0.87 ± 0.07 6.2 ± 2.4 0.005 ± 0.015 1.8 ± 4.9

aSource parameters and seismic moment M0 with their error retrieved from the joint inversion and the minimum of (1 � s) are shown with their errors.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

19 of 26

B10314

Figure

15.

(A)Matchingbetweentheobserved

velocity

seismograms(solidline)

andtheseismogramsretrieved

from

the

jointinversion(dashed

line)

oftheevent#1.In

each

box,ontheleft,thetimewindowincludingthePpulseandpartofits

coda;ontheright,thezoom

ofthePpulse.(B)Sam

eas

(A)fortheevent#2.(C)Sam

eas

(A)fortheevent#4.(D

)AMRFs

(dashed

lines)correspondingto

theacceptedsolutionsbelow

Atas

from

thejointinversionplots(Figures14A,14B,and

14D)forevent#1(3

solutions),event#2(13solutions),andevent#4(20solutions),together

withtheaverageAMRF

(sourceparam

etersin

Table

7).

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

20 of 26

B10314

Figure

15.

(continued)

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

21 of 26

B10314

earthquake. However, as shown in this paper, to obtain anaccurate description of the rupture process it is of fundamen-tal importance to develop an inversion technique thatexplores the complete model parameter space.[50] The developed technique uses time domain data and

this implies that the results of the EGF deconvolutionstrongly depend on the position of the traces of the MAINand the EGF. The problem of how positioning the traces hasbeen poorly discussed in literature. However, our resultsindicate that this is a key aspect in the inversion of therupture velocity history. In fact, only by using the timeshifting of the traces we can take into account the slowinitial phase of the P wave that could be due to the presenceof an acceleration phase of the rupture. We have demon-strated that, since we do not have a priori information on thecorrect position of the seismograms to carry out the inver-sion in the time domain, a plot of the misfit function versusthe time shifting will be always necessary.[51] The main limit of the presented technique is repre-

sented by the high computational cost, mainly because wehave shown that the reliability of model parameters is aconsequence of the use, in real cases, of the shiftingprocedure. It is worth to note that, using the shiftingprocedure, we can treat the problem of a slightly different

radiation pattern and/or a slightly different value of the take-off angle of MAIN and EGF exactly as we treated theproblem of a different location, since both affect the seismicwaveforms.[52] From the results of the synthetic tests on synthetic

(Table 1) and real (Table 4) seismograms, we can observethat the stress drop is always well recovered. This resultindicates that the stress drop estimates can be consideredreliable within the bounds of the assumptions of thetechnique. We think that this result is the consequence ofthe joint use of the quasidynamical circular crack model ofSato [1994] and of the adopted global nonlinear inversionstrategy on seismograms recorded at several stations. Thisapproach allows us to reasonably reduce the trade-offsamong source parameters. In a previous work, Hiramatsuet al. [2002], modeling the initial rise of the P pulse with asource model that describes only the initial part of therupture [Sato and Kanamori, 1999], found out that thestress drop acts as a scaling factor for the amplitudes andits estimation was unreliable.[53] From the synthetic study on noisy synthetic seismo-

grams, we can observe that the greatest bias on the retrievedsource parameters, due to the noise on data, is in corre-spondence of the highest value of the parameter t1/T and the

Figure 16. (Top) Normalized AMRF retrieved for the resolved events from the joint inversion. On eachcurve, the value of the take-off angle q of the station is indicated. (Bottom) Rupture velocity history forthe resolved events retrieved from the joint inversion. Curve 1, rupture velocity history considering themaximum possible value of the average rupture velocity and the minimum duration of the propagationstage; Curve 2, rupture velocity history considering the average source parameters; Curve 3, rupturevelocity history considering the minimum admissible value of the average rupture velocity and themaximum duration of the propagation stage. On the plots the following are indicated: the value of the Pwave velocity Vp at the source, S wave velocity Vs at the source, and average rupture velocity Vrave.(A) Results for event #1. (B) Results for event #22. (C) Results for event #4.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

22 of 26

B10314

problem is proportional to the noise to signal ratio. Thismeans that very slow accelerating phases of the rupturevelocity risk being hidden by the oscillations of the noise.We showed that the problem of the noise on the signal couldbe approached with the statistical approach based on therandom deviates technique. The limitation of this approachconsists in the computational cost of running the Nrand = 100inversions, for each time shift, that is very expensive. Thetest of the shifting procedure on noisy seismograms showedthat the shifting procedure is able to retrieve reliableestimates of the source parameters, up to a signal ratio ofthe order of about 5%. When N/S reaches the 5%, weobserved that the parameter t1/T, which is related to the riseof the P pulse, shows a high error, of the order of 20% of T(Table 1D). Therefore in the application to real events, weavoided as much as possible noisy signals, selecting theseismograms to be as clear as possible, being preferable touse a smaller number of high-quality waveforms instead ofa higher number of seismograms having a high noise-to-signal ratio.[54] From a theoretical point of view, an interesting result

of our analysis on synthetic data stands in the use of a globalnonlinear inversion method, which allows us to reduce thetrade-off among the rupture velocity, the source radius andthe stress drop, since it leads to very small errors on theseparameters (Figure 4).[55] Owing to the formulation of the Sato [1994] model

and to the accounted variability of the take-off angle, thesignals at several stations can be jointly inverted. In this waythe inversion, keeping constant the number of unknowns, can

benefit of a larger data set and gives rise to more robustresults. From the point of view of the applicability of thetechnique, the joint inversion of several stations improvesthe reliability of the results and helps to reduce the trade-offs among the source parameters. Moreover, theaccounted variability of the take-off at more than onestation allowed us to retrieve not only the AMRF but alsothe MRF.[56] We decided to discard S waves mainly because of the

noise on real S waveforms, which is usually much higherthan the error on P pulse. In fact, in addition to the noisewhich affects both P- and S-waves, in the case of S waveswe have to account for the interference of S wave with Pcoda waves which introduces another source of error andcould strongly bias the results. As above discussed, noise onseismogram tends to hide the rising part of the body wave,causing high errors on the source parameters. This couldinvalidate the inversion results based on the S wave pulse.[57] Theoretically, this inversion technique can be applied

also to the S wave pulses only if we restrict our analysis to asubshear rupture velocity. This is because the Sato [1994]model in the Deichmann [1997] version is based on theassumption that the rupture velocity cannot exceed thepropagation velocity c of the considered wave. Under theseassumptions it follows, for instance, that the source risetimeis given by [Deichmann, 1997]:

t1=2 ¼L

Vrave1� Vrave

csin q

� �

Figure 17. Three component velocity seismograms of the pair #1 of MAIN and EGF recorded atdifferent stations. The time window is centered on the S wave arrivals. On each box, the amplitude scaleand the time window of the traces is indicated.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

23 of 26

B10314

Clearly this relation holds for Vravec

sinq � 1. It can be easilyshown that, when rupture velocity results in Vrave

csinq > 1 the

source risetime will be given by:

t1=2 ¼L

Vrave

Vrave

csin q� 1

� �

[58] This is because the signal radiated by the edge of thecircular fault, which is the point nearest to the receiver, willprecede the signal coming from the center of the fault. Inthis case, the times at which discontinuous changes in theacceleration of the rupture appear in the seismograms [Ta1,Ta2 and Ta3 in the article of Deichmann, 1997] are equal andopposite to the same times computed for the subshear rupturevelocity (equations which follow equation (6) in Deichmann[1997]). As a consequence, the time corresponding to thearrival from the isochrone point La, has now to be computedthrough the relationship:

t Lað Þ ¼ Lasin qc

� t Lað Þ

Even if the discussion of the consequence of these newequations is beyond the aim of this paper, the Sato [1994]model, in the formulation proposed by Deichmann [1997]seems to be very suitable to allow, in a future work, thecomputation of the MRF also in the supershear regime, bysimply accounting for the changing of the sign in thecharacteristic times.[59] We applied the technique to four small earthquakes

that occurred in the Umbria-Marche region and comparedthe inversion results with those arising from the assumptionof a complex rupture process, using the AICc criterion to

select the best fit model. We found that three of the fourselected earthquakes are characterized by a single rupturemechanism and a clear variability of the rupture speed. Theexcellent fit of the waveforms in Figure 15 indicates that theP pulse and its early coda can be reproduced by hypothe-sizing a single crack with a realistic rupture velocity history,without assuming a multiple rupture process. The inferredrupture velocity histories (Figure 16) show that the maxi-mum rupture velocity can reach very high values and evenapproach the P wave velocity at the source, as also foreseenby Kostrov and Das [1988] in studies of the physics of therupture process. For two events (Figure 16), the averagerupture velocity approaches the shear wave velocity and inone case is higher. This result is in agreement with veryrecent studies of fracture mechanics both in the laboratoryand in field, which, supported by the improvement inquality and quantity of seismometers worldwide, led tonew measurements of supershear rupture speeds [Das,2007, and reference therein]. Another important feature ofthe studied earthquakes is the presence of very long accel-eration and deceleration stages. In particular, for the events#1 and #4, the duration of the deceleration stage exceeds thehalf of the total duration of the rupture, while the duration ofthe constant velocity stage is less than the 20% of the totalduration of the process. The application of the technique tothe event #4 allowed us also to discriminate the true faultplane (d, f) = (65�, 25�) from the auxiliary one. Thematching among the observed and the estimated first Pcycle of seismograms and a consistent part of their coda(Figures 15A, 15B, and 15C) is excellent for all the events.Our results then validate the results of previous works [Iio,1995; Iio et al., 1999; Deichmann, 1997; Hiramatsu et al.,2002], which attributed the nucleation of the rupture to a

Figure 18. Figure modified from Imanishi et al. [2004]. Comparison of our results with the results ofother authors in other areas. (A) Plot of Vr/b (where b is the S wave velocity) versus the seismic momentM0. (B) Plot of the seismic moment M0 versus the source dimension L.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

24 of 26

B10314

source effect rather than to a path effect, being the rise of theP wave modeled assuming a variable rupture velocity. Theestimated source parameters (including the seismic mo-ment) are characterized in general by small errors, maybethanks to a very careful selection of the seismograms. Theretrieved stress drop and the seismic moment are in goodagreement with the estimates by Bindi et al. [2001] for theUmbria-Marche 1997 earthquakes.[60] Finally, we compared the results of this study with

other works on small earthquakes. In particular, we consid-ered the results summarized in the article of Imanishi et al.[2004]. The authors analyzed the seismic swarm occurred inWestern Nagano (Japan, 1984) using an empirical Green’sfunction technique in the frequency domain. The analyzedearthquakes have Mw between 1.3 and 2.7. Imanishi et al.[2004] compared their results with Abercrombie [1995],Prejean et al. [2002], Boatwright [1981, 1984], Frankel etal. [1986], and Li et al. [1995] for earthquakes smaller thanMw = 5. In Figure 18 we modified two plots from Imanishiet al. [2004] adding the results of our study. We can see thatthe model parameter results of our technique (stress dropand average rupture velocity) are comparable with thoseinferred in other areas.

[61] Acknowledgments. We thank S. Gresta, G. Tusa, M. Vallee, andA. Zollo for their reading of the manuscript and for their comments. We aregrateful to M. Belardinelli for having improved the results of the EGFtechnique with focused suggestions on the stress drop. We also thank N.Deichmann for an enlightening comment on the source model and thedetailed list of comments which significantly improved the manuscript. Ananonymous reviewer is acknowledged for a careful check of the quality ofthe paper, which significantly improved the manuscript. This research wassupported by University of Bari funds.

ReferencesAbdel-Fattah, A. K., and A. Badawy (2002), Source process of the south-east Beni-Suef, northern Egypt earthquake using empirical Green’s func-tion technique, J. Seismol., 6, 153–161.

Abercrombie, R. E. (1995), Earthquake source scaling relationships from�1 to 5 ML using seismograms recorded at 2.5 km depth, J. Geophys.Res., 100(B12), 24,015–24,036.

Akaike, H. (1973), Information theory as an extension of the maximumlikelihood principle, in Second International Symposium on InformationTheory, edited by B. N. Petrov and F. Csaki, pp. 267–281, AkademiaiKiado, Budapest.

Ammon, C. J., T. Lay, A. A. Velasco, and J. E. Vidale (1994), Routineestimation of earthquake source complexity: The 18 October 1992Colombian earthquake, Bull. Seismol. Soc. Am., 84(4), 1266–1271.

Azimi, S. A., A. V. Kalinin, V. V. Kalinin, and B. L. Pivovarov (1968),Impulse and transient characteristic of media with linear and quadraticabsorption laws, Izv. Russ. Acad. Sci. Phys. Solid Earth, Engl. Transl., 2,88–93.

Beeler, N. M., T.-F. Wong, and S. H. Hickman (2003), On the expectedrelationships among apparent stress, static stress drop, effective shearfracture energy, and efficiency, Bull. Seismol. Soc. Am., 93(3), 1381–1389.

Bernard, P., and R. Madariaga (1984), A new asymptotic method for themodelling of near-field accelerograms, Bull. Seismol. Soc. Am., 74, 539–557.

Bindi, D., D. Spallarossa, P. Augliera, and M. Cattaneo (2001), Sourceparameters estimated from the aftershocks of the 1997 Umbria-Marcheseismic sequence, Bull. Seismol. Soc. Am., 91, 448–455.

Boatwright, J. (1980), A spectral theory for circular seismic sources: Simpleestimates of source dimension, dynamic stress drop and radiated seismicenergy, Bull. Seismol. Soc. Am., 70, 1–28.

Boatwright, J. (1981), Quasi-dynamic models of simple earthquakes: Ap-plication to an aftershock of the 1975 Oroville, California, earthquake,Bull. Seismol. Soc. Am., 71, 69–94.

Boatwright, J. (1984), The effect of rupture complexity in estimate ofsource size, J. Geophys. Res., 89(B2), 1132–1146.

Bouchon, M. (1979), Discrete wave number representation of elastic waveields in three-space dimensions, J. Geophys. Res., 84(B7), 3609–3614.

Bouchon, M. (1981), A simple method to calculate Green’s functions forelastic layered media, Bull. Seismol. Soc. Am., 71, 959–971.

Brune, J. (1970), Tectonic stress and the spectra of seismic shear wavesfrom earthquakes, J. Geophys. Res., 75(26), 4997–5009.

Burnham, K. P., and D. R. Anderson (2002), Model Selection and Multi-model Inference: A Practical Information-Theoretical Approach, 2nd ed.,Springer-Verlag, New York.

Cattaneo, M., et al. (2000), The 1997 Umbria-Marche (Italy) earthquakesequence: Analysis of data recorded by the local and temporary network,J. Seismol., 4, 401–414.

Chouet, B., P. Dawson, and A. Arciniega-Ceballos (2005), Source mechan-ism of Vulcanian degassing at Popocate’petl Volcano, Mexico, determinedfrom waveform inversions of very long period signals, J. Geophys. Res.,110, B07301, doi:10.1029/2004JB003524.

Clayton, R. W., and R. A. Wiggins (1976), Source shape estimation anddeconvolutionof teleseismic body waves, Geophys. J. R. Astron. Soc., 47,151–177.

Courboulex, F., J. Virieux, A. Deshamps, D. Gibert, and A. Zollo (1996),Source investigation of small event using empirical Green’s function andsimulated annealing, Geophys. J. Int., 125, 768–780.

Das, S. (2007), The need to study speed, Science, 317(5840), 905–906.Deichmann, N. (1997), Far-field pulse shape from circular sources withvariable rupture velocities, Bull. Seismol. Soc. Am., 87(5), 1288–1296.

Ellsworth, W. L., and G. C. Beroza (1998), Observation of the seismicnucleation phase in the Ridgecrest, California, earthquake sequence, Geo-phys. Res. Lett., 25(2), 401–404.

Emolo, A., and A. Zollo (2005), Kinematic source parameters for the 1989Loma Prieta earthquake from the nonlinear inversion of accelerograms.,Bull. Seismol. Soc. Am., 95(3), 981–994, doi:10.1785/0120030193.

Eshelby, J. D. (1957), The determination of the elastic field of an ellipsoidalinclusion, and related problems, Proc. R. Soc. Lond., 241, 376–396.

Filippucci, M., S. de Lorenzo, and E. Boschi (2006), Fault plane orienta-tions of small earthquakes of the 1997 Umbria-Marche (Italy) seismicsequence from P-wave polarities and rise times, Geophys. J. Int., 166(1),322–338, doi:10.1111/j.1365-246X.2006.02998.x.

Fischer, T. (2005), Modelling of multiple events using empirical Green’sfunctions: method, application to swarm earthquakes and implications fortheir rupture propagation, Geophys. J. Int., 163(3), 991 – 1005,doi:10.1111/j.1365-246X.2005.02739.x.

Frankel, A., J. Fletcher, F. Vernon, L. Haar, J. Berger, T. Hanks, and J. Brune(1986), Rupture characteristics and tomographic source imaging of ML_3earthquakes near Anza, Southern California, J. Geophys. Res., 91(B12),12,633–12,650.

Govoni, A., D. Spallarossa, P. Augliera, and L. Troiani (1999), The 1997Umbria-Marche earthquake sequence: The combined data set of theGNDT/SSN temporary and the RESIL/RSM permanent seismic networks(Oct. 18–Nov. 3, 1997), Project GNDT-CNR: PROGETTO ESECUTIVO1998, 6a1, ‘‘Struttura e sorgente della sequenza’’ (CoordinatorM. Cattaneo)(On electronic CD-ROM Support).

Hartzell, S. (1978), Earthquake aftershocks as Green’s functions, Geophys.Res. Lett., 5(1), 1–4.

Hiramatsu, Y., M. Furumoto, K. Nishigami, and S. Ohmi (2002), Initialrupture process of microearthquakes recorded by high sampling boreholeseismographs at the Nojima fault, central Japan, Phys. Earth Planet. Int.,132, 269–279.

Hurvich, C. M., and C.-L. Tsai (1995), Model selection for extended quasi-likelihood models in small samples, Biometrics, 51, 1077–1084.

Iio, Y. (1995), Observation of the slow initial phase generated by micro-earthquakes: Implications for earthquake nucleation and propagation,J. Geophys. Res., 100(B8), 15,333–15,349.

Iio, Y., S. Ohmi, R. Ikeda, E. Yamamoto, H. Ito, H. Sato, Y. Kuwahara,T. Ohminato, B. Shibazaki, and M. Ando (1999), Slow initial phase gen-erated by microearthquakes occurring in the Western Nagano prefecture,Japan - the source effect, Geophys. Res. Lett., 26(13), 1969 –1972,10.1029/1999GL900404.

Imanishi, K., M. Takeo,W. L. Ellsworth, H. Ito, T. Matsuzawa, Y. Kuwahara,Y. Iio, S. Horiuchi, and S. Ohmi (2004), Source parameters and rupturevelocities of microearthquakes in Western Nagano, Japan, determinedusing stopping phases, Bull. Seismol. Soc. Am., 94(5), 1762–1780.

Li, Y., C. Doll Jr., and M. N. Toksoz (1995), Source characterization andfault plane determinations for MbLg_1.2 to 4.4 earthquakes in the Char-levoix seismic zone, Quebec, Canada, Bull. Seismol. Soc. Am., 85, 1604–1621.

Kanamori, H. (1977), The energy release in great earthquakes, J. Geophys.Res., 82(20), 2981–2987.

Kanamori, H., and L. Rivera (2004), Static and dynamic scaling relationsfor earthquakes and their implications for rupture speed and stress drop,Bull. Seismol. Soc. Am., 94(1), 314–319.

Kikuchi, M., and H. Kanamori (1982), Inversion of complex body waves,Bull. Seismol. Soc. Am., 72(2), 491–506.

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

25 of 26

B10314

Kostrov, B. V., and S. Das (1988), Principles of Earthquake Source Me-chanics, 286 pp., Cambridge Univ. Press, New York.

Mori, J., and A. Frankel (1990), Source parameters for small events asso-ciated with the 1996 North Palm Springs, California, earthquake deter-mined using empirical Green’s functions, Bull. Seismol. Soc. Am., 80(2),278–295.

Mori, J., and H. Kanamori (1996), Initial rupture of earthquakes in the 1995Ridgecrest, California sequence, Geophys. Res. Lett., 23(18), 2437–2440.

Mueller, C. S. (1985), Source pulse enhancement by deconvolution of anempirical Green’s function, Geophys. Res. Lett., 12(1), 33–36.

Nadeau, M., and L. R. Johnson (1998), Seismological studies at Parkfield:VI. Moment release rates and estimates of source parameters for smallrepeating earthquakes, Bull. Seismol. Soc. Am., 88(3), 790–814.

Nakano, M., and H. Kumagai (2005), Waveform inversion of volcano-seismic signals assuming possible source geometries, Geophys. Res. Lett.,32, L12302, doi:10.1029/2005GL022666.

Prejean, S. G., W. L. Ellsworth, M. Zoback, and F. Waldhauser (2002), Faultstructure and kinematics of the Long Valley Caldera region, California,revealed by high-accuracy earthquake hypocenters and focal mechanismstress inversions, J. Geophys. Res., 107(B12), 2355, doi:10.1029/2001JB001168.

Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling (1989),Numerical Recipes, The Art of Scientific Computing (Fortran Version),Cambridge Univ. Press, Cambridge.

Reasenberg, P., and D. H. Oppheneimer (1985), FPFIT, FPPLOT andFPPAGE: FORTRAN computer program for calculating and displayingearthquake fault-plane solutions, U.S. Geol. Surv. Rep., 85–739.

Sato, T. (1994), Seismic radiation from circular cracks growing at variablerupture velocity, Bull. Seismol. Soc. Am., 84(4), 1199–1215.

Sato, T., and T. Hirasawa (1973), Body wave spectra from propagatingshear cracks, J. Phys. Earth, 21, 415–431.

Sato, T., and H. Kanamori (1999), Beginning of earthquakes modeled withthe Griffith’s fracture criterion, Bull. Seismol. Soc. Am., 89, 80–93.

Sato, K., and J. Mori (2006a), Relation between rupture complexity andearthquake size for two shallow earthquake sequences in Japan, J. Geo-phys. Res., 111, B05307, doi:10.1029/2005JB003614.

Sato, K., and J. Mori (2006b), Scaling relationship of initiations for mod-erate to large earthquakes, J. Geophys. Res., 111, B05306, doi:10.1029/2005JB003613.

Scherbaum, F. (1990), Combined inversion for the three-dimensional Qstructure and source parameters using microearthquake spectra, J. Geo-phys. Res., 95(B8), 12,423–12,438.

Scherbaum, F., and J. Johnson (1992), PITSA, programmable interactivetoolbox for seismological analysis, IASPEI Software Library, 5.

Singh, S. K., M. Ordaz, T. Mikumo, J. Pacheco, C. Valdes, and P. Mandal(1998), Implications of a composite source model and seismic-waveattenuation for the observed simplicity of small earthquakes and reportedduration of earthquake initiation phase, Bull. Seismol. Soc. Am., 88(5),1171–1181.

Singh, S. K., J. F. Pacheco, B. K. Bansal, X. Perez-Campos, R. S.Dattatrayam,and G. Suresh (2004), A source study of the Bhuj, India, earthquake of26 January 2001 (Mw 7.6), Bull. Seismol. Soc. Am., 94(4), 1195–1206.

Sugiura, N. (1978), Further analysis of the data by Akaike’s informationcriterion and the finite corrections, Commun. Stat. - Theory Methods, A7,13–26.

Telford, W. M., L. P. Geldart, R. E. Sheriff, and D. A. Keys (1990), AppliedGeophysics, 2nd ed. Cambridge Univ. Press, Cambridge.

Thio, H. K., and H. Kanamori (1996), Source complexity of the 1994Northridge earthquake and its relation to aftershock mechanisms, Bull.Seismol. Soc. Am., 86(1B), 84–92.

Vallee, M. (2004), Stabilizing the empirical green function analysis: Devel-opment of the projected Landweber method, Bull. Seismol. Soc. Am.,94(2), 394–409.

Vallee, M. (2007), Rupture properties of the Giant Sumatra earthquakeimaged by empirical, Bull. Seismol. Soc. Am., 97(1A), S103–S114,doi: 10.1785/0120050616.

Vasco, D. W., and L. R. Johnson (1998), Whole earth structure estimatedfrom seismic arrival times, J. Geophys. Res., 103(B2), 2633–2672.

Zollo, A., and S. de Lorenzo (2001), Source parameters and three-dimen-sional attenuation structure from the inversion of microearthquake pulsewidth data: Method and synthetic tests, J. Geophys. Res., 106(B8),16,287–16,306.

Zollo, A., P. Capuano, and S. K. Sing (1995), Use of small earthquakerecords to determine the source time functions of larger earthquakes: Analternative method and an application, Bull. Seismol. Soc. Am., 85(4),1249–1256.

�����������������������E. Boschi, Istituto Nazionale di Geofisica e Vulcanologia, Via di Vigna

Murata 605-00143 Rome, Italy.S. de Lorenzo and M. Filippucci, Dipartimento di Geologia e Geofisica

and Centro Interdipartimentale per la Valutazione e Mitigazione del RischioSismico e Vulcanico, Universita di Bari, Via Orabona 4-70125 Bari, Italy.(marilena.filippucci@geo.uniba.it)

B10314 DE LORENZO ET AL.: EGF METHOD TO INFER RUPTURE VELOCITY

26 of 26

B10314