Coda-derived source spectra, moment magnitudes and energy-moment scaling in the western Alps

Geophys. J. Int. (2005) 160, 263–275 doi: 10.1111/j.1365-246X.2005.02491.x

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Coda-derived source spectra, moment magnitudes andenergy-moment scaling in the western Alps

P. Morasca,1,∗ K. Mayeda,2 L. Malagnini3 and William R. Walter2

1Universita’ di Genova, Viale Benedetto XV, 5, Genova, Italy2Lawrence Livermore National Laboratory Ground-Based Nuclear Explosion Monitoring Program, L-205, Livemore, CA 94551, USA3Istituto Nazionale di Geofisica e Vulcanologia, via di Vigna Murata, 605, 00143, Rome, Italy

Accepted 2004 September 28. Received 2004 September 28; in original form 2003 October 23

S U M M A R YA stable estimate of the earthquake source spectra in the western Alps is obtained using anempirical method based on coda envelope amplitude measurements described by Mayedaet al. for events ranging between M W ∼ 1.0 and ∼5.0. Path corrections for consecutive narrowfrequency bands ranging between 0.3 and 25.0 Hz were included using a simple 1-D model forfive three-component stations of the Regional Seismic network of Northwestern Italy (RSNI).The 1-D assumption performs well, even though the region is characterized by a complexstructural setting involving strong lateral variations in the Moho depth.

For frequencies less than 1.0 Hz, we tied our dimensionless, distance-corrected coda ampli-tudes to an absolute scale in units of dyne cm by using independent moment magnitudes fromlong-period waveform modelling for three moderate magnitude events in the region. For thehigher frequencies, we used small events as empirical Green’s functions, with corner frequen-cies above 25.0 Hz. For each station, the procedure yields frequency-dependent correctionsthat account for site effects, including those related to f max, as well as to S-to-coda transferfunction effects. After the calibration was completed, the corrections were applied to the entiredata set composed of 957 events.

Our findings using the coda-derived source spectra are summarized as follows:

(i) we derived stable estimates of seismic moment, M 0, (and hence M W) as well as radiatedS-wave energy, (ES), from waveforms recorded by as few as one station, for events thatwere too small to be waveform modelled (i.e. events less than M W ∼ 3.5);

(ii) the source spectra were used to derive an equivalent local magnitude, M L(coda), that isin excellent agreement with the network averaged values using direct S waves;

(iii) scaled energy, e = ER/M0, where ER, the radiated seismic energy, is comparable toresults from other tectonically active regions (e.g. western USA, Japan) and supports theidea that there is a fundamental difference in rupture dynamics between small and largecrustal earthquakes in tectonically active regions.

Key words: coda, energy, energy-moment scaling, moment magnitude, source spectra.

1 I N T RO D U C T I O N

Virtually all source studies focus on the direct phases of the earth-quake such as P, S, or surface waves. The associated source pa-rameters are inferred from spectral or waveform modelling usu-ally adopting the omega-square source model (Brune 1970; Aki1967). Direct wave amplitude measurements, however, are affectedby source radiation pattern, directivity and heterogeneities along

�Corresponding author: DipTeRis – Settore Geofisica, Universita’ di Gen-ova, Viale Benedetto XV, 5, 16132 Genova, Italy.

the path, all of which can contribute to large amplitude variability(e.g. Favreau & Archuleta 2003). Some recent studies that com-pare direct waves with coda waves (e.g. Mayeda 1993; Mayeda &Walter 1996; Mayeda et al. 2003; Eken et al. 2004; Malagnini et al.2004) find that amplitude measurements of direct waves requiressignificant multistation averaging to achieve the same stability as asingle coda envelope measurement, because the coda averages overpath and source variability. In general, these studies find that thesource amplitude obtained from the coda envelope is a factor of3 to 4 times more stable than those derived from direct waves. Inaddition to source parameters such as focal mechanism and seismicmoment release, there has been renewed interest in quantifying the

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264 P. Morasca et al.

dynamics of the earthquake rupture process over a broad range ofevent sizes (e.g. 1.0 < M W < 8.0). In the past, there were many stud-ies that estimated the radiated energy for large magnitude events (e.g.Gutenberg & Richter 1956; Vassiliou & Kanamori 1982; Boatwright& Choy 1986; Kikuchi & Fukao 1988). More recently, with betterinstrumentation and denser networks, higher quality inversions forthe rupture dynamics of large earthquakes (M W > ∼6.0) have beenmade (e.g. Kikuchi & Kanamori 1991; Wald et al. 1996; Kaverinaet al. 2002; Dreger et al. 2004).

A source scaling study by Kanamori et al. (1993) observed anincreasing trend in the energy-to-moment ratio (ER/M 0), or scaledenergy, e(= ER/M0) with increasing M 0, which they interpreted as adifference in frictional behaviour during rupture between small andlarge earthquakes (∼3.5 < M W < 7.2). However, difficulties ex-ist in developing scaling relationships with small events because oflimited resolution resulting from complex propagation in the crustand near-surface attenuation that strongly affect the seismic wavesat high frequency. Recently, Kanamori & Rivera (2004) discussedimplications on rupture dynamics if the generally accepted staticscaling relation (i.e. M 0 ∝ f −3

0 ) between M 0 and spectral cornerfrequency (f 0) is scale-dependent. They showed that the dynamicscaling relation between ER and M 0 and the static scaling relationare not independent. They point out the need to better constrain esti-mates of radiated energy (ER), static stress drop (�σ S) and rupturespeed (V ) for small events in future studies. Estimating the radi-ated energy of smaller magnitude events (i.e. less than M W ∼ 3.5)requires integration of source spectra to very high frequencies. Un-fortunately, surface stations, even under the best of circumstances(e.g. Steidl et al. 1996), have significant near-site attenuation (e.g.site kappa, κ) and corrections must be applied, especially when con-sidering high frequencies. In the most severe cases, such as soft soilsites, there are simply no measurable signals at high frequency. Toavoid the effects of near-surface attenuation, Abercrombie (1995)used high-frequency recordings from the 2.5-km-deep Cajon Passborehole in southern California and a similar study was performedin a 2-km borehole in Long Valley caldera in the eastern SierraNevada by Prejean & Ellsworth (2001). In both borehole studies,micro-earthquake apparent stress was found to increase with mo-ment, consistent with results of Kanamori et al. (1993) who consid-ered much larger events.

Mayeda & Walter (1996) used a completely different approachusing narrow-band coda amplitude measurements to reconstruct themoment-rate spectra for continental crustal earthquakes distributedthroughout the western USA. Though they used broad-band surfacesensors, their methodology accounts for near-site attenuation and/oramplification effects. They found that e, and hence the apparentstress, increased as M0.25

0 for events ranging between M W ∼ 3.0 and7.3. A similar scale dependence in e was also observed by Izutani& Kanamori (2001) for events in Japan.

Opposite conclusions have been drawn by Ide & Beroza (2001)and Perez-Campos & Beroza (2001). In their study, they argue thatthe different scaling purported between small and large earthquakesin other investigations was a result of improper computation of radi-ated energy at high frequency as well as to large scatter in the energy-moment ratio. It is clear that the existence of a scale-dependentenergy-to-moment ratio is still in question because, to obtain thesource spectra, one must effectively remove all the propagation andsite effects over a wide range of frequencies. In addition, earthquakesources are anisotropic and many of the larger events exhibit strongdirectivity thereby requiring good azimuthal averaging or a prioriinformation to correct the amplitude spectra at a particular azimuthand distance from the source. For smaller events, where observations

in e are sparse and often quite variable, the problem is compoundedbecause small wavelength structures in the Earth are poorly resolvedand may not be properly accounted for.

In the current study, we use the well-established coda stabilityto significantly reduce the measurement errors related to wholepath propagation effects. We use a combination of larger calibra-tion events with independent moments as well as empirical Green’sfunction events to correct for S-to-coda transfer function effects andsite response for frequencies ranging between 0.3 and 25.0 Hz.

2 S T RU C T U R A L S E T T I N G

The region of the western Alps is laterally complicated and pro-vides an excellent test of the coda method. In the following, wegive a brief overview of the tectonics of the region. In the simplestform, the alpine chain is the result of northwest–southeast conver-gence between Africa and Europe (Polino et al. 1990; Giglia et al.1996; Marchant & Stampfli 1997; Rosenbaum et al. 2002). Usuallyresearchers distinguish the western part of this chain in two belts(Eva et al. 1990; Sue et al. 1999): an external belt characterized byshallower earthquakes (<10 km) and an internal belt where eventsare located between 5 and 20 km (Paul et al. 2001). The crustal Pen-ninic front (CPF) (Sue et al. 1999) is the main crustal discontinuitybetween the internal and external zones (see Fig. 1). From a geolog-ical viewpoint, the area is characterized by crystalline massifs witha difference between the internal (Penninic) zone, showing high-grade metamorphic rocks, and the external zone that shows weaklydeformed and metamorphosed rocks. As observed by Morasca et al.(2004), the region is characterized by a relatively low attenuationcrust where QS( f ) = 310 f 0.2. The relatively high QS of the west-ern Alps indicates that the region may be considered more stablethan, for example, the Apennines, where the regional heat flow ishigher. However, this represents just a simplification because the tec-tonic processes that involve both the upper and lower crust are morecomplex, as confirmed by anomalous structures such as the Ivreabody (Kissling 1993; Di Stefano et al. 1999) and by the difficultiesin defining the depth of the Moho beneath the region. In fact, thedepth to the Moho in this area shows strong variations, ranging from10 km in the Ligurian sea to a maximum of 50 km beneath the Alps(Buness et al. 1990).

The complex structure of the region has important implications inthe western alpine tectonic regime, resulting in an inhomogeneousstress field (Sue et al. 1999) characterized by a radial shortening ofthe front of the belt and extension in the core (Eva & Solarino 1998;Sue et al. 1999; Sue & Tricart 1999, 2002). Such complex tectonicregime imply variations in ground motion levels, as suggested bymany authors (Atkinson & Boore 1995; Atkinson & Silva 1997;Boore et al. 1997; Campell 1997; Sadigh et al. 1997; Spudich et al.1999).

3 DATA

In this study, we analysed 957 earthquakes distributed through-out the western Alps (northwestern Italy) recorded by five, three-component, broad-band sensors (Guralp CMG40), operated by theRegional Seismic network of Northwestern Italy (RSNI) between1996 and 2001 (see Fig. 1). Network-averaged local magnitudes(M L) were computed for all the events by using a scale calibratedfor northwestern Italy by Spallarossa et al. (2002). Our data set con-sisted mainly of small events, 0.5 < M L < 3.0, and there were alsoa few greater than M L = 3, of which three were large enough to be

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Energy-moment scaling in the western Alps 265

waveform modelled. The hypocentral distance of the events rangedbetween 5 and 160 km, with depths generally less than 15 km.

For the largest event, 2000 August 21, (M L = 5.1), an M W = 4.8was computed by MedNet (Istituto Nazionale di Geofisica e Vul-canologia, Italy). In addition, we performed 1-D waveform mod-elling for two other events, 1997 October 31, (M L = 4.1) and 1998April 11, (M L = 3.6), using the methodology of Randall et al.(1995), obtaining M W = 4.11 and 3.36, respectively. We used thesethree independent moment magnitudes as well as smaller eventsin order to calibrate and transform the non-dimensional, distance-corrected coda amplitudes into moment-rate spectra that are in unitsof dyne cm.

Figure 1. Stations (solid triangles) and events (circles) that were used in this study along with the major tectonic features of the region. Crustal Penninic front(CPF), the main boundary between external and internal (Penninic) zones; Tertiary Piemontese basin (TPB).

4 M E T H O D

4.1 Overview

Mayeda & Walter (1996) used the seismic coda to minimize theeffects of source radiation pattern, heterogeneous path effects andnear-site attenuation to produce stable estimates of moment-ratespectra, moment magnitude and radiated seismic energy. Theydemonstrated that it is possible to produce extremely stable moment-rate spectra for regional earthquakes, even from single-station obser-vations. For consecutive narrow frequency bands ranging between0.02 and 8.0 Hz, they formed coda envelopes for each horizontal

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component using the equation

E(t, f ) =√

v(t)2 + h(t)2, (1)

where f is the centre frequency, t is the time from the origin,ν(t) is the bandpass-filtered seismogram and h(t) is its correspondingHilbert transform. They used the average of the two horizontal com-ponents, together with a 2-D theoretical multiple-scattering model(Shang & Gao 1988) and produced synthetic coda envelopes as afunction of time and epicentral distance for each frequency bandconsidered. Calibrated synthetic envelopes were used to measurethe amplitude of the observed envelope as a function of frequencyby DC shifting the synthetics until they matched the observed en-velopes using an L-1 norm. In a recent study, Mayeda et al. (2003)improved upon this methodology and adopted a completely empir-ical approach for the computation of synthetic envelopes and pathcalibration applicable at both local and regional distances. Theyargued that scattering theories to date were still not sufficient to si-multaneously predict the local and regional distance scattering usinga single formulation. In fact, additional ad hoc distance calibrationhad to be used in the Mayeda & Walter (1996) study because themultiple-scattering model was too simplistic. In this paper, we usethe updated approach following Mayeda et al. (2003) where the codaamplitude at frequency fi, time t and distance r is written as

Ac( fi , t, r ) = Wo( fi ) · S( fi ) · T ( fi ) · P(r, fi ) · H

[t − r

v(r, fi )

]

·[t − r

v(r, fi )

]−γ (r, fi )

· exp

{b(r, fi ) ·

[t − r

v(r, fi )

]}, (2)

where Wo(fi) is the S-wave source amplitude, S(fi) is the site re-sponse, T(fi) is the S-to-coda transfer function resulting from scatter-ing conversion, P(r, fi) includes the effects of geometrical spreadingand attenuation (both scattering and absorption), H is the Heavisidestep function, v(r, fi) is the peak velocity of the S-wave arrival,γ (r, fi) and b(r, fi) control the coda envelope shape and t is the timein seconds from the origin time.

Figure 2. Representative narrow-band envelopes for the 1997 October 31 event at station STV2. The strong frequency dependence in the coda envelopesshape requires that we parametrize as a function of frequency.

In this study, waveforms were narrow-band filtered in 13 con-secutive frequency bands ranging between 0.3 and 25.0 Hz andprocessed as outlined in Mayeda et al. (2003). Fig. 2 shows repre-sentative narrow-band envelopes for the M W = 4.11 1997 October31 event at station STV2.

4.2 Coda shape and velocity

The parameters, v(r, fi), b(r, fi) and γ (r, fi), were determinedempirically for each station and then averaged over all five sta-tions. For each narrow frequency band we describe v(r, fi), b(r,fi), and γ (r, fi) by fitting the data with a simple form of a hyper-bola as a function of distance r following Mayeda et al. (2003).For the group velocities of the envelope peak S arrival, v(r, fi), weused

v(r, fi ) = v0i − v1i

v2i + r, (3)

where v0i, v1i and v2i are the three parameters to describe the veloc-ities as a function of distance that were determined through a simplegrid search in each frequency band fi. Likewise, for the envelopeshape parameters, b(r, fi) and γ (r, fi), we used

b(r, fi ) = b0i − b1i

b2i + r, (4)

γ (r, fi ) = γ0i − γ1i

γ2i + r, (5)

where b0i, b1i, b2i and γ 0i , γ 1i , γ 2i were also obtained through agrid-search routine. We note that γ (r, fi) controls the shape of theearly coda immediately following the direct waves, whereas b(r, fi)dominates the coda envelope shape at larger times well beyond thedirect phase.

Figs 3(a), (b) and (c) show data examples and best-fitting curvesfor the three parameters γ (r, fi), b(r, fi) and v(r, fi) for the 4.0–6.0 Hz band. In general, most of the data in this study could be

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Figure 3. (a) The coda shape parameter γ is plotted as a function of distancefor the 4.0–6.0 Hz case. Notice that at short distances γ is large and thendecreases with increasing distance. From eq. (2), we see that this parametercontrols the shape of the early coda immediately following the direct Swave. (b) The coda shape parameter, b, dominates the coda envelope shapeat larger times. (c) Peak velocity of the observed coda envelopes is plottedversus distance. In all figures the lines represent the best-fitting hyperbolaand the parameters for all frequency bands and stations are listed in Table 1.

fit with a line, but for flexibility and consistency with past studies,we are using the form of a hyperbola. Once γ (r , fi), b(r , fi) andv(r, fi) are obtained, synthetic envelopes can be formed and used tomeasure each observed, narrow-band envelope (see Mayeda et al.

2003). Table 1 lists best-fitting average parameters for each narrowfrequency band.

4.3 Synthetic envelopes

As for measuring the coda envelope amplitude, we modified ourprocedure from that of Mayeda et al. (2003) by normalizing thesynthetics at a fixed lapse time (120 s) as measured from the origintime. This normalization procedure has no effect on the final coda-derived source spectra, however it makes the interpretation of ourpath attenuation parameters more consistent with local and regionalcoda observations. Specifically, observations of local coda energyappear to be homogeneously distributed in time and space behindthe expanding direct wave front (e.g. see review by Sato & Fehler1998). To illustrate this, Figs 4(a) and (b) show coda envelopes fortwo events recorded at stations STV2 and NEGI for two narrowfrequency bands (e.g. 2.0–3.0 and 20.0–25.0 Hz). For both events,independent of distance from either station, we see that the codaenvelopes attain roughly the same level and shape, whereas theircorresponding direct S waves decrease rapidly as a function of dis-tance as a result of geometrical spreading and attenuation. For the2.0–3.0 Hz case, we note that the early coda immediately follow-ing the direct S wave, as the closer recordings are steep relative tothe more distant recordings, but this is accounted for by having adistance-dependent γ . To generate the synthetics, we first set W , S,P and T to unity in eq. (2), then normalized each synthetic to 120 spast the origin time. Fig. 4(c) shows normalized synthetics for the2.0–3.0 Hz case generated at 20-km intervals, ranging between 20and 200 km, and Fig. 4(d) is for the 20.0–25.0 Hz case. The char-acter of these synthetics is consistent with what we observe in Figs4(a) and (b), which exhibit steep early coda at short distances, butsimilar shapes and levels at larger lapse times and distances.

4.4 Amplitude measurements and path correction

For each observed envelope, we measured its relative amplitude byDC shifting the normalized synthetic envelope to match the observedenvelope using an L-1 fit. For multiple pairs of stations, we selectedcommon events that had good signal-to-noise ratio at both stations(calibration events) and performed a grid search based on a relationthat was used previously by Mayeda et al. (2003) for Dead Sea riftearthquakes:

P(r, fi ) =[

1 +(

r

p2i

)p1i]−1

, (6)

where f is the centre frequency, r the epicentral distance, and p1i andp2i are the path parameters from the grid search for each frequencyband fi. The path correction P is approximately constant for valuesof r � p2i whereas, for values of r � p2i , it decays proportionallyto the p1i power. The parameter p2i can be interpreted as the crit-ical distance beyond which the coda is no longer homogeneouslydistributed in space and the parameter p1i controls how rapidly thecoda envelope level decreases with increasing distance.

Other empirical functions could be used, but the functional formof eq. (6) appears to be perfectly adequate and agrees well with localand regional coda observations throughout the world (e.g. Sato &Fehler 1998). In general, a correction function could be determinedfor each station, but we have assumed the simplest case of a laterallyhomogeneous region, therefore requiring that the two parameters bethe same for all stations, for each frequency band. For each fre-quency band, the parameters p1i and p2i were chosen based on the

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Table 1. Calibration parameters for velocity, coda shape and path corrections used for all stations. v0i, v1i, v2i, are the parameters used in the hyperbolaeq. (3) describing group velocity of the peak S-wave arrival; b0i, b1i, b2i and γ 0i , γ 1i , γ 2i are the coda envelope shape parameters used in the hyperbolaeqs. (4) and (5) respectively; p1i and p2i are the path parameters from eq. (6).

Freq. (Hz) v0i v1i v2i b0i b1i b2i γ 0ι γ 1ι γ 2ι p1i p2i

0.3–0.5 2.65 6 4 −0.0032 0.00 10−4 0.1 0 1 0.75 1000.5–0.7 2.75 8 4 −0.0047 0.14 143 0.1 −13 7 3.50 3000.7–1.0 2.85 8 4 −0.0058 0.18 500 0.1 −35 34 1.50 2001.0–1.5 3.00 4 4 −0.0073 0.10 500 0.1 −25 17 2.25 2001.5–2.0 3.05 2 1 −0.0145 0.00 10−4 0.1 −31 19 2.50 1502.0–3.0 3.15 2 1 −0.0131 0.00 10−4 0.1 64 101 1.25 503.0–4.0 3.15 2 1 −0.0151 0.06 500 0.1 −51 52 2.00 1004.0–6.0 3.15 2 1 −0.0162 0.02 2 0.1 −62 75 2.25 1006.0–8.0 3.20 2 1 −0.0209 0.14 487 0.1 −41 45 1.75 508.0–10.0 3.15 2 1 −0.0115 4.00 484 0.4 −9 6 3.00 10010.0–15.0 3.20 2 1 −0.0115 1.06 53 0.6 0 2.4 4.25 15015.0–20.0 3.15 2 1 −0.0115 3.26 163 0.5 0 1.9 2.25 10020.0–25.0 3.15 2 1 −0.0230 0.68 68 0.4 0 1.4 1.50 50

lowest average interstation standard deviation (see Table 1). Fig. 5shows examples of direct S and coda waves before and after distancecorrections were applied. For the correction of direct S waves, weused a path correction according to Morasca et al. (2004), who de-termined attenuation specific to this region. For nearly all frequencybands, the coda interstation data standard deviation is usually below0.1 whereas the direct S-wave scatter is a factor of 3 to 4 larger,consistent with past coda studies. We note that the raw or uncor-rected coda amplitudes have very small interstation scatter. This isbecause our initial normalized synthetics are a very good first ap-proximation to the actual coda attenuation. Had we not normalizedour synthetics first, this would have severely increased the scatter,comparable to those of the direct waves. Finally, the constant shiftof the data points observed in Fig. 5 with respect to the line y = x ,represents the relative site effect for the specific station pair, at thespecific frequency (e.g. see also fig. 6 of Mayeda & Walter 1996).

4.5 Transformation to moment-rate spectra

Fig. 6(a) shows distance-corrected coda spectra in dimensionlessunits for our three calibration events along with two smaller eventsthat we are using as empirical Green’s functions. In order to trans-form these spectra into an absolute scale, we need to correct forfrequency-dependent S-to-coda transfer functions and site effects,which we call site-transfer corrections. Following Mayeda & Wal-ter (1996) and Mayeda et al. (2003), we tie the lower frequenciesto independent moments determined from long-period waveformmodelling. Fig. 6(b) shows an example of 1-D waveform modellingfor one of the calibration earthquakes and the corresponding seis-mic moment. Assuming that the smallest of our calibration events(1998 April 11, M W = 3.36), has a corner frequency above 1 Hz, wecan find the frequency-dependent corrections that, in a least-squaressense, match our three independent moments, thereby giving us thecorrections between 0.3 ≤ f ≤ 1.0 Hz that we will use for all eventsin that frequency band. For frequencies above 1 Hz, we use smallevents as empirical Green’s functions, noting that these events arearound 1.0 ≤ M W ≤ 1.6 based upon their corrected 1 Hz amplitudes.Next, assuming that these small events have flat source spectra atleast up to ∼25.0 Hz, we flatten each spectrum to be consistentwith its previously corrected lower frequency amplitude at 1 Hz.Because we use many Green’s function events (not just the twoshown in Fig. 6a), we obtain the average correction for each stationfor 1.0 < f ≤ 25.0 Hz. This assumption is consistent with high-

quality source spectra observed by Abercrombie (1995) and Prejean& Ellsworth (2001) for comparable sized events from their respec-tive borehole studies. Finally, Fig. 6(c) shows the transformation ofthe non-dimensional spectra in Fig. 6(a) to absolute source spectraor moment-rate spectra, M(ω), that are now in units of dyne cm. Toreiterate, this procedure accounts for all frequency-dependent path,S-to-coda transfer function and site effects. These corrections arethen applied to the entire data set to form source moment-rate spec-tra, then consequently used to estimate seismic moment and radiatedS-wave energy. The frequency-dependent site-transfer correctionsfor each station are listed in Table 2.

5 R E S U LT S

5.1 Magnitudes

After applying the path and site-transfer corrections to all the data,we averaged the source spectra over all available stations. For eachsource spectrum, we averaged the amplitudes of the two lowest fre-quencies to estimate the seismic moment and the moment magni-tude, M W(coda), was computed using the relation, M W = 2/3 log10 M 0

− 10.73 by Hanks & Kanamori (1979). Fig. 7(a) shows good agree-ment between M W(coda) and M W derived by Morasca et al. (2004)using direct waves. Next, we compared the coda-derived M W againstthe network-averaged M L that were defined using a local magnitudescale calibrated for northwestern Italy by Spallarossa et al. (2002;Fig. 7b). From Fig. 7(b), we see that M W(coda) scales with a 2

3 slopefor small events (i.e. M L less then ∼ 3) and scales roughly 1-to-1 forthe intermediate magnitude events. This scaling is expected basedupon the definition of Kanamori (1977), because M W is propor-tional to 2

3 log10 M0 whereas M L is proportional to log amplitude.Finally, to obtain a stable equivalent M L, we regressed the 2.0–3.0 Hz source spectral amplitudes with the network M L and foundthe following relation:

ML(coda) = (log10 Ac − 17.5)/0.93, (7)

where Ac is the coda-derived source amplitude at 2.0–3.0 Hz in dynecm. We see from Fig. 7(c) that we have good agreement with thenetwork results with the added advantage that a stable estimate ofM L(coda) can be obtained from as few as one station as opposed to thetraditional M L, which requires network averaging to reduce scatter.

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Figure 4. To illustrate coda homogeneity and differences with direct S waves, we show examples of two events recorded at stations STV2 and NEGI fortwo frequency bands: (a) 2.0–3.0 Hz and (b) 20.0–25.0 Hz. Event 972480856 is only 5 km from NEGI and 55 km from STV2. Though the direct waves aredecaying as one would expect based upon geometrical spreading and Q, the coda envelopes as measured from the origin time have converged to the same level.Event 990660401 is the reciprocal case that is 51 km from NEGI and only 8 km from STV2. (c) Example of synthetics, normalized at 120 s, were generated at20-km intervals for the 2.0–3.0 Hz case. (d) Same as (c), but for the 20.0–25.0 Hz case. These represent our initial assumptions on the spatial distribution ofthe coda. We then use these to measure raw coda envelope amplitudes by DC shifting the synthetics to match the observed envelopes. We then use a two-stationgrid-searching technique using eq. (6) to find the best average model for the region using all possible pairs of stations.

5.2 Radiated energy

We now focus our attention on quantifying the dynamic stress dropscaling in the region by computing the radiated S-wave energy (ES)from the observed source spectra. As previously done by Mayeda &Walter (1996), we extrapolated the observed coda-derived moment-rate spectra, M(ω), to both high ( f = ∞ Hz) and low frequency

( f = 0 Hz) limits, converted to velocity by multiplying by omega(ω = 2π f ), squared the spectra and then integrated to obtain ES.Assuming that the P-wave contribution is 7 per cent of the totalradiated elastic wave energy, we multiplied ES by 1.07 to obtain thetotal radiated energy, where ES is

ES = I

4π 2ρβ5

∫ ∞

0|ω · M(ω)|2 dω, (8)

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Figure 5. Plots of amplitudes at station STV2 versus amplitudes at station MONE for two frequency bands, 3.0–4.0 and 10.0–15.0 Hz. The panels onthe left show raw, uncorrected S and coda waves along with their associated interstation standard deviations. The panels on the right are the correspondingdistance-corrected amplitudes and here we see that the direct S-wave amplitudes are 3 to 4 times more scattered than the coda-derived amplitudes. Notice thatthe uncorrected coda amplitudes in the left panels also show very little variance, indicating that our initial normalized synthetics (see Figs 4c and d) were veryclose to the final, best-fitting model.

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Figure 6. (a) Distance-corrected coda amplitudes for a range of event sizes that are still in dimensionless units. Using independent moments for the threelargest events, we can tie the low-frequency measurements to an absolute scale. We also use smaller events as empirical Green’s function events to derivecorrections for the higher frequencies. (b) Example of waveform modelling for seismic moment for one of the larger calibration events. (c) Source spectra indyne cm after all corrections for site effect and S-to-coda transfer function have been applied.

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Table 2. Site and S-to-coda transfer function corrections for GENL,MONE, NEGI, STV2 and TRAV. We note that, for a given frequency band,the variation between stations represents the relative site response.

Freq. (Hz) GENL MONE NEGI STV2 TRAV

0.3–0.5 20.2231 20.2595 20.0143 20.4184 20.47090.5–0.7 21.2562 20.2595 21.1804 21.4607 21.41340.7–1.0 20.1421 20.2443 20.0583 20.3967 20.29281.0–1.5 19.7647 19.8311 19.7819 20.0358 19.85681.5–2.0 20.5172 20.5181 20.5032 20.7069 20.59032.0–3.0 19.5517 19.2838 19.4993 19.6804 19.48173.0–4.0 19.9319 19.4793 19.8276 20.0450 19.78694.0–6.0 19.9240 19.5344 19.8898 20.0932 19.89636.0–8.0 20.1220 19.5848 19.9790 20.1852 20.06848.0–10.0 20.3995 19.7861 20.1634 20.3681 20.234410.0–15.0 20.5252 20.3450 20.2612 20.5109 20.277115.0–20.0 21.1161 20.8756 20.7326 20.8510 20.614820.0–25.0 21.0222 21.1060 20.9756 20.9646 20.9160

where the rms radiation pattern for S waves I = 25 , the density

ρ = 2700 kg m−3 and the S-wave velocity β = 3.5 km s−1.Fig. 8 shows the western Alps energy estimates for two subsets of

data compared against previous results from the western USA. Thefirst are events where we extrapolated less than 30 per cent of thetotal S-wave energy and the other set are events where we extrapo-lated between 30 to 45 per cent. In all cases, the energy extrapolatedat the low frequencies ( f = 0 Hz) is minimal, contributing less thana few per cent. Even for the second set of events, where the ex-trapolation was the largest, all of these events had observed cornerfrequencies. In contrast, smaller events (M W < 2.0) such as thoseused as empirical Green’s functions had flat source spectra at leastup to 25.0 Hz (e.g. see Fig. 6c) and were not used in the energyestimation. From Fig. 8, we see that both subsets of data generallyfollow the scaling that was previously observed in the western USAby Mayeda & Walter (1996) and are in good agreement with thosefrom the Cajon Pass borehole study of Abercrombie (1995). For thelarger events (M W > 3.5), there appears to be evidence for slightlyelevated apparent stress (>3 MPa), which might reflect differencesin the state of stress in this region. This same elevated level is alsoobserved for comparable sized earthquakes in the Sierra Nevadabatholith, when compared against earthquakes distributed through-out northern California (Mayeda et al. 2004).

Our results show that with good, high-frequency data and thecorrections for attenuation and site effect, it is possible to obtainstable energy estimates for events with M W ∼ 2.5 or larger usingsurface sensors. This assertion is supported by borehole studies,which also observe S-wave corner frequencies between ∼10 and20 Hz for events near M W = 2.5 (e.g. Abercrombie 1995; Prejean& Ellsworth 2001). For this reason, we cannot make reliable energyestimates below M W ∼ 2.5 for this data set, however for eventsabove this there is clearly an increasing trend in scaled energy, e,with increasing moment.

6 D I S C U S S I O N

We successfully transported the coda methodology to the region ofthe western Alps, measured stable, absolute source parameters suchas ES and M 0 and derived band-limited magnitudes such as M L

from as few as one station. We believe that application of the codamethodology to the western Alps data set overcomes the difficultiesassociated with properly accounting for high frequency attenu-ation and site response. Our moment-rate spectra appear to be free of

Figure 7. (a) M W(coda) from this study is in excellent agreement with M W

derived from a previous study of Morasca et al. (2004) using direct S waves.(b) For events greater than M W ∼ 2.5, we see that M W(coda) scales roughly1-to-1 with the network M L, however for smaller events M W scales with aslope of 2

3 with M L. This break in slope is expected and is because M L isbeing computed below the corner frequency of the event where the amplitudeis directly proportional to seismic moment. (c) Using the 2.0–3.0 Hz sourceamplitude from the coda spectra, we have derived our own local magnitude,M L(coda), using eq. (7). A single-station coda measurement can provide thesame stability as a network-averaged M L using direct S waves.

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Figure 8. Total radiated energy versus seismic moment. Solid diamonds represent events where the extrapolated S-wave energy was less than 30 per cent ofthe total and the circles represent events where we extrapolated 30 to 45 per cent of the energy. Results are compared with the energy estimates from the westernUSA (Mayeda & Walter 1996) and the Cajon Pass borehole (Abercrombie 1995).

source radiation pattern and virtually insensitive to lateral crustalheterogeneity. Our coda-derived estimates of M L and M W are inexcellent agreement with network-averaged results but are signif-icantly more stable. In addition, we examined the dynamic stressdrop scaling in the region by computing the radiated S-wave energy(ES) from the source spectra of events M W ∼ 2.5 and larger. Weshowed that the scaled energy, e, and hence the dynamic stress-dropor apparent stress (e.g. Aki 1966; Wyss & Brune 1968) is not con-stant in this region and increases with increasing seismic moment, ingood agreement with results from the western USA (e.g. Kanamoriet al. 1993; Abercrombie 1995; Mayeda & Walter 1996; Prejean &Ellsworth 2001) and Japan (Izutani & Kanamori 2001). We note thatthe apparent stress for the moderate sized events is systematicallylarger by a factor of 2 than those from the western USA, whichmight reflect a higher state of stress in the current study region.

Although Ide & Beroza (2001) suggested that the reported scalingof radiated energy of small events may be the result of band lim-itations, considering events for which we observed at least 55 percent of total radiated S-wave energy, we do observe a difference in efor events above M W ∼ 2.5. As Kanamori & Heaton (2000) argue,we also believe that even if we allow for deficiency in energy esti-mates, the stress drop variation is too significant to be discounted.Attenuation and propagation effects may introduce errors in energyestimates (Ide & Beroza 2001; Izutani & Kanamori 2001) but we feelconfident that we accounted for all frequency-dependent path atten-uation and site effects by a careful empirical calibration analysis thatyielded stable source spectra. Our results support the idea that dif-

ferences exist between the dynamics of small and large earthquakes(Kanamori & Heaton 2000; Brodsky & Kanamori 2001; Izutani &Kanamori 2001). Various mechanisms are suggested to justify therelative increase of radiated energy with earthquake size. Non-linearprocesses such as fault melting or thermal fluidization can lower thedynamic frictional stress for large slip, increasing the amount ofradiated seismic energy (Kanamori & Heaton 2000). The elastody-namic theory proposed by Brodsky & Kanamori (2001) argues thatthere is a reduction in friction for large events relative to small eventsindependent of any thermal effects. They contend that the presenceof fluids in a fault under certain circumstances can produce a viscousstress balanced by dynamic pressure that reduces friction. Anotherinteresting possibility is that large earthquakes tend to occur on well-developed mature faults, so that the fracture energy could be verysmall. For small earthquakes, the fracture energy is relatively large,because they often occur on smaller cracks that need a lot of energyto grow (Kanamori & Brodsky 2001). We believe that our stable re-sults focusing on small earthquakes can be a useful contribution tohighlight the way earthquake radiated energy scales with increasingmagnitude.

A C K N O W L E D G M E N T S

The authors wish to thank Professor Claudio Eva and the RSNInetwork staff (G. Carenzo, M. Pasta and E. Zunino) for the care-ful network management and data collection that made this workpossible. We are also indebted to the reviewers, N. Deichmann and

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K. Koch, who provided valuable comments that significantly im-proved the original manuscript. KM’s work was performed under theauspices of the US Department of Energy by the University of Cali-fornia, Lawrence Livermore National Laboratory under contract No.W-7405-Eng-48. This is LLNL contribution UCRL-JRNL-202169.

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Coda-derived source spectra, moment magnitudes and energy-moment scaling in the western Alps