Gasperini Et Al (2010) - BSSA Paper

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The Location and Sizing of Historical Earthquakes Using the Attenuation

of Macroseismic Intensity with Distance

by P. Gasperini, G. Vannucci, D. Tripone, and E. Boschi

Abstract We herein describe new methods for computing the quantitative param-eters of earthquakes using macroseismic data and the uncertainties associated withthese parameters. The methods allow for the location of epicenters that are offshoreor that have no intensities assigned to any points in the epicentral region by maximiz-ing the likelihood function of an attenuation equation with observed intensity data. Inthe most favorable cases, such an approach also allows the estimation of the sourcedepth and the local attenuation coefficients. We compute the parameter uncertaintiesin two ways: (1) using formal methods, such as the inversion of the Hessian of the log-likelihood function at its maximum, and (2) by using bootstrap simulations. We testedthe performance of our methods by comparison with reliable instrumental hypocentersof onshore earthquakes, and found a reasonable agreement with the epicentral loca-tions (within 10–15 km for more than 70% of cases) but not with the hypocentraldepths, for which our results are generally underestimated by a factor of 2 or moreand are poorly related to instrumental estimates. This finding indicates that the use ofmacroseismic depths in seismic hazard and seismotectonic investigations should betreated with caution. We nevertheless found good agreement (within 10°–15°)between the fault-trace orientations that were computed using the macroseismic dataand the associated focal mechanisms of earthquakes with Mw ≥5:7. The surprisingaccuracy of the macroseismic orientations obtained using this method could in somecases allow the true fault to be inferred between the two conjugate planes of a givenfocal mechanism.

Online Material: FORTRAN source code of key subroutines and functionscomputing parameters and relevant uncertainties in the Boxer 4.0 program.

Introduction

The determination of the quantitative parameters ofearthquakes that occurred before the installation of modernseismic networks (i.e., in the last few decades in most coun-tries) is of paramount importance to the study of the statis-tical properties of seismicity, as well as to the seismic hazardassessment. This is particularly true for countries such asItaly, for which a large amount of descriptive informationis available in terms of documentary evidence (e.g., newspa-per articles and contemporary surveys) for many earthquakesthat have occurred since Roman times and where the instru-mental location and sizing of earthquakes have only beenreliable for the past 20–30 years.

The problem of determining quantitative parameters ofpreinstrumental earthquakes is an interesting challenge inseismology because (1) the location of the epicenter mustbe based on amplitudes because the arrival times are notmeaningful for macroseismic data, (2) intensity is an ordinalquantity that is only weakly correlated with the amplitude of

the ground motion, and (3) the real paths traveled within theEarth’s crust and lithosphere by the seismic wave trains thatcarry the bulk of the seismic energy are not known.

Early attempts to locate earthquakes using macroseismicdata (see Cecic et al., 1996 for a thorough historical review)made use of the apparent direction of ground motion (Mallet,1862) or the time delays experienced between differentobservers (von Seebach, 1973), although it was later realizedthat such methods are not particularly reliable. More re-cently, a common approach used the practice of drawing iso-seismal lines and of taking the center of the innermost line asthe epicenter (e.g., Cecic et al., 1996). However, this proce-dure requires a considerable amount of subjective judgment,which may lead to the estimation of significantly differentlocations by different investigators, depending on theextension and shape of the innermost isoseismal used.Furthermore, this method takes no account of the inherentuncertainties involved, which are not estimated.

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Bulletin of the Seismological Society of America, Vol. 100, No. 5A, pp. 2035–2066, October 2010, doi: 10.1785/0120090330

The availability of extensive collections of macroseis-mic observations, and of modern computers, has led to thedevelopment of methods that use the spatial distribution ofintensity (i.e., the macroseismic field). Examples includethose proposed by Console et al. (1990), Gasperini andFerrari (1995; 2000), Bakun and Wentworth (1997), and Pet-tenati et al. (1999). The aim of such approaches is to identifythe macroseismic epicenter (or hypocenter) as the point on(or below) the Earth’s surface from which seismic energyappears to radiate. This location should approximatelycorrespond to the center of the seismogenic fault or its pro-jection on the surface, which is clearly at odds with the seis-mological definition of the focus of the earthquake as thepoint at which the seismic rupture initiates. Because thefocus is often located at one end of a seismogenic fault seg-ment, while the macroseismic hypocenter may reasonably bethought to correspond with its center (Richter, 1958), the twoapproaches are somewhat difficult to reconcile.

The methods of macroseismic location proposed byConsole et al. (1990), Bakun and Wentworth (1997), andPettenati et al. (1999) all use the best fit of a predictor ofground motion (an attenuation equation) against intensitydata obtained by varying the unknown parameters (e.g., epi-central coordinates, depth, epicentral intensity). Such meth-ods therefore yield the radiation center of the seismic energy.Gasperini and Ferrari (1995, 2000) instead computed thecenter of mass (or barycenter) of the spatial distribution ofthe sites that experienced the largest intensities by calculatingan average of the geographical coordinates of such sites,using the trimmed mean (i.e., the arithmetic average of thedata points that fall between the twentieth and eightieth per-centiles) to ensure the robustness of the method. Such anapproach mitigates against the bias introduced by the occur-rence of sites whose locations were wrongly reported, whichwas a rather common issue in the Italian macroseismicdatabase of the 1990s. Furthermore, this method is relativelystable and reliable, even when data are scarce (<5–10 sites).The barycenter method was used extensively to locate themacroseismic epicenters reported in the most recent Italianhistorical catalogs used for seismic hazard assessment(Catalogo Parametrico dei Terremoti Italiani [CPTI] WorkingGroup, 1999, 2004). In contrast with radiation center meth-ods, the barycenter approach may not be used for earth-quakes that are located at sea or where no intensity dataare available in the epicentral region. Furthermore, it cannotbe used to estimate the hypocentral depth.

In general, hypocentral depths are poorly known formost earthquakes that occurred in the first decades of theinstrumental era, when the coverage of the seismic networkswas relatively poor. In fact, in the absence of a dense networkof modern stations located at distances comparable with thesource depth, the determination of depth is an ill-posed prob-lem (Lomnitz, 2006), in which the uneven distribution ofreceivers (all of which are located at the Earth’s surface), thereading errors of the phase arrivals (particularly of S waves),and the uncertainties in the seismic velocity profile may

cause varying degrees of bias. This and the different physicalmeanings of the instrumental and macroseismic hypocentersimply that the calibration of macroseismic methods usinginstrumental estimates of focal depth might be arguable.Macroseismic depth might possibly be better correlated withthe depth of maximum earthquake fault slip, although theavailability of data is scarce. Furthermore, the very natureof intensity, which is the result of the superposition of dif-ferent wave trains generated by all the active portions of theseismogenic fault, allows the assumption of neither a simplewave path nor a unique originating point. Hence, the deter-mination of the hypocentral depth using macroseismic data isan even more ill-posed problem and must thus be approachedwith caution.

Sponheuer (1960) developed a method for computingthe depth from macroseismic data using an attenuation equa-tion originally proposed by von Köveslighety (1907). Themost recent form of this equation, as given by Burton et al.(1985), was adopted by Musson (1996) and implemented inhis Macdep program (see Data and Resources) as

I0 � Ii � γ�log10�IRi=h� � α log10�e��IRi � h��; (1)

where h is the hypocentral depth; I0 the epicentral intensity;Ii and IRi the intensity and the hypocentral (inclined) radiusof the i-th isoseismal, respectively; e the base of natural loga-rithms; α the attenuation coefficient (usually in the range0.003–0.006); and γ is a coefficient that relates intensity tothe physical ground motion (usually about 3). Musson(1996) suggested solving equation (1) for ranges of valuesof h and I0 by minimizing the root mean square (rms) of thedifferences between the observed and predicted intensitiesand using at least three isoseismals. The Macdep programyields contours of the values of the rms as a function of h andI0 and allows h to be obtained by picking the value of I0 thatbest corresponds to the lowest contour and is consistent withthe macroseismic observations.

The methods proposed in the literature for estimating themagnitude of earthquakes using macroseismic data mainlyrefer to three approaches. The first, which dates back to thepioneering work of Richter (1935), considers the maximum(epicentral) intensity as being representative of the strengthof the seismic source and relates it to the instrumental mag-nitude using an empirical relationship. The second approachrelates the magnitude either to the extent of the area wherethe earthquake was felt (i.e., Toppozada, 1975, hereafterreferred to as the felt area) or to the area within isoseismallines. The latter approach appears to be the more stable andreliable, but the former one can be applied even if, as oftenoccurs, only the maximum intensity is known. In considera-tion of the fact that the two methods could be biased withopposing signs due to the variability of the (unknown) sourcedepth, Galanopulos (1961) suggested using the product ofepicentral intensity and the felt area as an unbiased estimatorof magnitude. This concept was also adopted by Sibol et al.(1987), who found that a regression of the magnitudeM with

2036 P. Gasperini, G. Vannucci, D. Tripone, and E. Boschi

the square of the epicentral intensity I0 and the logarithmof the felt area FA was “the best among all possibleregressions”:

M � a� bI20 � c log2�FA�: (2)

Gasperini and Ferrari (1995, 2000) developed a modificationof the method of Sibol et al. (1987) that uses the areas withinall the available isoseismal lines. They define such areas asthose of circles with radii equal to the average epicentral dis-tance of sites with a given intensity and compute the coeffi-cients for separate regressions with instrumental magnitudesfor each isoseismal. Such empirical coefficients are then usedto estimate independently the magnitudes for each isoseis-mal, which are then averaged to estimate the magnitudeof the earthquake, thereby using almost all the available data.(See Appendix 1 for a description of Gasperini and Ferrari’s(1995, 2000) method, which has not been published else-where in such detail.) This method, in which a weight isassigned to each isoseismal as a function of both the uncer-tainty in the instrumental magnitude and of the number ofdata points and which uses a robust estimator of the averages(the trimmed mean), was shown to reproduce the entire dataset of magnitudes quite well (R2 � 0:74). It was particularlyeffective (R2 � 0:91) in reproducing the set of moment mag-nitudes that were computed directly using the inversion ofwaveforms (Gasperini and Ferrari, 2000).

Bakun and Wentworth (1997) proposed a third ap-proach, in which the attenuation relationship and the distancefrom the radiation center are used to estimate an Mi for eachpoint where the intensity is known. Following Richter’s(1935) definition of local magnitude, the preferred M is themean; that is, M � 1=N

PNi�1Mi. They also estimate the

confidence limits for the magnitude as a function of the num-ber of intensity data points.

Some researchers have also attempted to compute otherparameters of earthquakes using intensity data. Pettenati et al.(1999) and subsequent papers, proposed a method for theestimation not only of the epicenter, the depth, and the mag-nitude but also the strike, the dip, and the rake of the fault, inaddition to other parameters such as the velocity of the Swaves and the length of the seismogenic fault. Their approachused the hypothesis that the distribution of seismic intensityfollows the theoretical radiation pattern of SHwaves and useda tessellation of the study area by means of Voronoi polygonsto assign an appropriate weight to each intensity data point.Although the application to four relatively strong Californianearthquakes (M ≥5:9) yielded results that were fairly consis-tent with known fault plane solutions, an extensive testing ofthis method was not carried out.

Using a more conservative approach, Gasperini et al.(1999) computed only the axial orientation (in the range0°–180°) of the seismic fault as the weighted axial mean �αof the azimuthal distribution of sites where the intensitywas above a given threshold. They represented the surfaceprojection of the seismic source as a rectangle (i.e., a box, thus

naming their code Boxer), whose dimensions were computedas a function of the moment magnitude using Wells and Cop-persmith’s (1994) empirical formulas. Gasperini et al. (1999)tested their method on a set of earthquakes with known focalmechanisms, finding a satisfactory agreement betweenmacroseismic and instrumental orientations (with amaximumdeviation of the order of 20°–30°). They suggested that themacroseismic fault orientations obtained in this way werereliable only for earthquakes with Mw ≥5:5, whose sourcelength (about 5–10 km, according to Wells and Coppersmith,1994) is of the same order of magnitude as the spatial resolu-tion of the intensity data, and found a rather coherent patternin the orientations of the earthquakes with Mw ≥5:5 thatoccurred in central and southern Italy over the last fourcenturies. This method was used to characterize seismogenicfaults in geological and geodetic studies and contributed to thedefinition of a significant portion of the seismic sources in-cluded in the Italian Database of Individual SeismogenicSources (DISS) compiled by Basili et al. (2008).

We herein revisit the problem of the determination ofearthquake parameters using macroseismic data in the lightof recent findings and from our experience since the publica-tion of our earlier paper (Gasperini et al., 1999). In particular,we herein develop an original procedure (although still basedon the same general assumptions and criteria adopted by thepreviously cited papers) for locating the epicenters or thehypocenters of historical earthquakes by obtaining the bestfit between their intensity data and an attenuation relation-ship. We also apply an alternative method for computingearthquake magnitude as a function of the average intensitypredicted by the attenuation equation at a given distancefrom the hypocenter, following the approach proposed byBakun and Wentworth (1997). Our work takes advantageof a recently published attenuation equation for Italy (Paso-lini, Albarello, et al., 2008), which also includes the specifictreatment of intensity data that are uncertain between twodegrees (i.e., VII–VIII). We also estimate the parameteruncertainties using both formal methods and the bootstrapsimulation technique (Efron and Tbishirani, 1986; Hall,1992) and perform extensive tests to assess the efficiency ofvarious methods in reproducing instrumental estimates. Thefull procedure is implemented in a new version of the Boxercode (4.0) that is available for download from the site listedin the Data and Resources section.

New Location Algorithms

Pasolini, Albarello, et al. (2008) showed that, for crustalearthquakes in Italy, the intensity Ii at a site may be satisfac-torily predicted using a log-linear relationship with its dis-tance from the source

Ii � IE � a�Di � h� � b�ln�Di� � ln�h��; (3)

where h is a common average depth assumed for all theearthquakes; a and b are empirical attenuation coefficients;

Location and Sizing of Historical Earthquakes Using Attenuation of Macroseismic Intensity 2037

and Di ��R2i � h2

p. Rj is the distance to the epicenter of

the i-th site, which is computed along the geodetic (on aspherical Earth of radius R0) as

Ri � R0 arccos�sin�latE� sin�lati�� cos�latE� cos�lati� cos�lonE � loni��; (4)

in which latE and lonE are the epicentral coordinates, lati andloni are the geographical coordinates of the i-th site, and IEis the average intensity expected at the epicenter. Pasolini,Albarello, et al. (2008) also assumed that IE can be estimatedusing

IE � �I � a�h � �D� � bfln�h� � ln�D�g; (5)

where �I is the average of all of the intensities observed for thegiven earthquake and

�D � 1

N

XNi�1

Di ln�D� � 1

N

XNi�1

ln�Di�: (6)

Although the hypocentral distances Di do not correspond tothe real path traveled by any wave train, they are reasonablyconsistent with the corresponding seismic phases both forshort distances, where direct S waves dominate, and for longdistances, where most of the seismic energy is carried by sur-face waves.

The attenuation parameters estimated for Italy by Paso-lini, Albarello, et al. (2008) are reported in Table 1 (standardattenuation law), which also includes the coefficients of alinear orthogonal regression (Fuller, 1987) between IE andthe instrumental moment magnitude Mw

IE � c� dMw: (7)

Using an empirical relationship such as that of equation (7),any reference to the expected intensity at the epicenter IEmay be easily converted to yield the magnitude of the earth-quake because the orthogonal regression is invertible; that is

Mw � IE � c

d: (8)

It should be noted that the coefficient c for the standardattenuation law shown in Table 1 differs from that reportedin Table 3 of Pasolini, Albarello, et al. (2008) for the regres-sion with the instrumental moment magnitude (Msw) due to amisprint in the original paper.

Using equation (3), the problem of locating the earth-quake may now be posed in terms of finding the epicentralcoordinates latE and lonE, the depth h, the expected intensityat the epicenter IE, and the attenuation coefficients a and bthat minimize the sum of the squares of the residuals; that is,

SSR�latE; lonE; h; IE; a; b� �XNj�1

fIj � IE � a�Dj � h�

� b�ln�Dj� � ln�h��g2 (9)

This approach implies that intensity values are treated as realnumbers and that the uncertain estimates (e.g., VII–VIII)are either to be considered as semi-integers (e.g., 7.5), asassumed by Gasperini (2001), or discarded from the data set,as assumed by Albarello and D’Amico (2004).

Uncertain observations represent a significant portion(about 30%) of the Italian intensity data and are usuallyinterpreted as meaning that (1) the lower degree (e.g. VII)has certainly been reached at the site, but (2) there is alsoevidence (albeit weak) that the higher value (e.g. VIII) hasoccurred. This (epistemic) uncertainty between two intensityvalues differs substantially from the usual (statistical) uncer-tainty that affects any measurement. In this case, we do notknow if the (integer) value of the measurement, which is alsoaffected by statistical uncertainty, is actually VII or VIII.

Pasolini, Albarello, et al. (2008) proposed a rigorousstatistical approach to account for the discrete nature of in-tensity and the peculiarity of uncertain values. They assumedthat the two contiguous integer values (VII and VIII) are bothpossible outcomes of the process by which the intensity isassessed (see also Grunthal, 1998). Following the suggestionby Magri et al (1994), they assigned a probability p�I� toeach degree of the macroseismic scale, which represented adegree of confidence in the value. When using this approach,writing “VII–VIII” implies a probability density distributionof the form

p�I� � �0; 0; 0; 0; 0; 0; w1; w2; 0; 0; 0; 0�; (10)

with the condition that

X12I�1

p�I� � 1: (11)

In practice, Pasolini, Albarello, et al. (2008) assumed w1 �w2 � 0:5 and demonstrated that slightly different choices donot affect the results that much. Such a representation mayalso be applied to well-defined integer values of intensity. Insuch cases, the probability vector is 1 for the correspondingdegree and 0 for all the others. For degree VII, for example,

p�I� � �0; 0; 0; 0; 0; 0; 1; 0; 0; 0; 0; 0�: (12)

In order to deal consistently with such probabilistic defini-tions, an approach in terms of likelihood must be adopted. Ifthe intensity residuals follow a near-normal distribution (e.g.,Gasperini, 2001), the log-likelihood function for a data set ofN intensity observations with respect to a predictor functionμj with variance σ may be written as

L �XNj�1

ln�

1

σ��2π

pX12I�1

pj�I�Z

I�:05

I�0:5exp

�� i � μj�2

2σ2

�di

�:

(13)

In order to locate the hypocenter, the predictor of each j-thobservation may be written as:


μj�latE; lonE; h; IE; a; b� � IE � a�Dj�latE; lonE; h� � h�� bfln�Dj�latE; lonE; h�� ln�h�g: (14)

In principle, by maximizing L with respect to the sixunknown parameters of the predictor function, it should bepossible to determine their best-fit values using the macro-seismic field of a single earthquake. In practice, the intensitydata set from a single earthquake may not permit the reliableestimation of all six parameters because of a low numberand an uneven distribution of intensity sites (this might bereflected in the strong mutual correlation between the param-eters obtained). Hence, it may sometimes be useful to fixsome of the parameters using reasonable values determinedusing other means. We now outline some possible alternativemethods.

(1) All six parameters could be estimated jointly by maximiz-ing equation (13), using equation (14) as a predictor func-tion. This represents a rather optimistic approach thatcould be applied with confidence only to dense and well-distributed macroseismic fields.

(2) Only b is fixed (e.g., using estimates made by Pasolini,Albarello, et al., 2008), and the remaining five param-eters are computed using maximum likelihood estimation(MLE).

(3) Both a and b in equation (14) are fixed (using estimatesmade by Pasolini, Albarello, et al., 2008) while latE,lonE, h, and IE are determined jointly using the MLE ofequation (13). Even in this case, a high correlation be-tween h and IE may not allow the reliable joint estimationof the desired parameters.

(4) IE is assumed to be given by equation (5). In this case thepredictor function becomes

μj�latE; lonE; h� � �I � a�Dj�latE; lonE; h��D�latE; lonE; h�� bfln�Dj�latE; lonE; h�� ln�D�latE; lonE; h��g: (15)

Hence only latE, lonE, and h are estimated using MLE,while IE can be determined aposteriori using equation (5).

(5) For very poor data sets, it might be useful to fix h (e.g.,using the average value computed by Pasolini, Albarello,et al. 2008). In this case, we consider two possibilities:(a) the determination of IE using MLE together with latEand lonE, and (b) the assumption of IE using equation (5)and then the use of equation (15) (with fixed h) as a pre-dictor.

All these possibilities are considered in the following, inwhich we number the old and new location methods from 0to 6, according to Table 2. For methods 1–6, the standarddeviation σ of the normal distribution of the residuals in

equation (13) is another free parameter that must be esti-mated using MLE.

We perform the maximization of the likelihood functionusing a quasi-Newtonian algorithm with simple bounds,implemented in the Fortran subroutine BCONF/DBCONF ofthe International Mathematics and Statistics Library (IMSL,1991). The maximization begins using the epicenter com-puted using the barycenter method (0), the attenuation coef-ficients a and b, and the average depth h computed accordingto Pasolini, Albarello, et al. (2008), as well as the intensity atthe epicenter IE obtained using equation (5). The optimiza-tion performed by such a routine is usually accurate andrelatively fast.

In order to address problems associated with incomplete-ness of the data, we recall that Pasolini, Albarello, et al. (2008)discard all intensity data at epicentral distances where theexpected intensity is less than IV. In practice, this impliesthe limitation of the data set of intensity to epicentral distancesless than 200 km for large earthquakes (I0 � X) and to a fewtens of km for relatively small ones (I0 � VI). When theepicenter is located offshore and far away from the coast,this selection could prevent the location of the epicenter/hypocenter.

In order to avoid such problems and to locate the earth-quake in all cases, we could include even the data at longdistances in the computations. However, at long distances,the standard attenuation parameters of Pasolini, Albarello,et al. (2008) do not reproduce the average observed intensityvery well and tend to overestimate it by virtue of the fact thatsites that experience an intensity of III (“felt by a few peopleindoors”) or less are likely to be missed out in the macroseis-mic analyses (see the discussion in Gasperini, 2001 andPasolini, Gasperini, et al., 2008).

For the reason described previously, we computed anew set of extended attenuation parameters (also shown inTable 1), which we obtained without discarding the data forlong epicentral distances. We have verified that the extendedattenuation equation can be used with confidence to locateand estimate the size of an earthquake, yielding resultssimilar to the standard equation. Hence, we adopted it forall computations.

As previously noted, the barycenter method (0) does notestimate the hypocentral depth. However, for comparison,we applied the Musson (1996) method to estimate the depthas described in the Introduction, and the results obtained areherein referred to as location method 0, which is actuallyused to compute the epicenter and the average isoseismalradii required by the Musson (1996) procedure.

In order to estimate automatically a single value of h(and not a range of values as a function of I0, as in the Mac-dep program), we fixed I0 to the value determined accordingto the method by Gasperini and Ferrari (1995, 2000) and thenfound the value of h that maximized the log-likelihood func-tion (derived from the intensity residual sum of squares S2R,according to Main et al., 1999)


L�h� � �N

2ln�S2R�

� �N

2ln�XN

i�1

fI0 � Ii � γ�log10�IRi=h�

� α log10�e��IRi � h��g2�: (16)

Following Musson (1996), we estimated the depth only if atleast three isoseismal radii are available, but we include inthe count even those isoseismal radii that correspond to un-certain degrees (e.g., VII–VII). In order to calibrate the meth-od for the Italian case, we computed regional values of α andγ by maximizing equation (16) using a set of known hypo-central depths (see Appendix 2). We thereby obtained thevalues α � 0:006 and γ � 3:28. It follows that the attenua-tion of the Italian area is at the upper limit of the expectedrange of variation and that the amplitude of the groundmotion almost exactly doubles for an increase in intensityof one degree, as suggested by the seminal hypothesis ofCancani (1904), on which the Mercalli–Cancani–Sieberg(MCS) intensity scale is based (Sieberg, 1931). We have ver-ified that our computations correspond rather well with thosemade using the Macdep program and the same isoseismaldata and epicentral intensity.

Alternative Method for EstimatingMacroseismic Magnitude

Given the location of the hypocenter (either using theold or new location algorithms), the problem of computing

the magnitude can be treated similarly to the instrumentalRichter (1935) magnitude, as suggested by Bakun and Went-worth (1997). By combining attenuation equation (3) withthe relation between M and magnitude IE in equation (5), aseparate estimate of magnitude can be computed for eachpoint at which the intensity is known:

Mj �1

dfIj � a�Dj � h� � b�ln�Dj� � ln�h�� cg; (17)

where h is the average depth reported in Table 1 (for the ex-tended equation) and used to obtain best fits for the empiricalIE-M relationships and not to the actual value computed forthe given earthquake (which, however, is used to computehypocentral distances Dj). This implies that h in equa-tion (17) does not represent the computed source depth hbut instead concerns a hypocentral distance at which theIE values, used in an empirical regression with magnitude,are computed. That means that if h ≠ h, the IE value impliedby equation (17) does not correspond to the expected inten-sity at the epicenter but rather at a reference hypocentraldistance h that has been assumed in the regression with mag-nitude. Furthermore, following a scrutiny of our preliminaryresults, we decided to use the attenuation coefficients a andb, computed according to Pasolini, Albarello, et al. (2008)and shown in Table 1, even for methods 5 and 6 (which de-termine such parameters for the given earthquake) becausewe found that this yields a better agreement with the instru-mental magnitudes. The final magnitude of the earthquake isthe average of all the values of Mj, computed according toequation (17) using individual observations of intensity.

Table 2Estimated and Fixed Parameters for Different Location Methods

LocationMethod (Number) Location Method Estimated Parameters Fixed Parameters

0 Barycenter latE, lonE1 Attenuation equation latE, lonE h, a, b (according to Pasolini, Albarello, et al., 2008),

IE (according to equation 5)2 Attenuation equation latE, lonE, h a, b (according to Pasolini, Albarello, et al., 2008),

IE (according to equation 5)3 Attenuation equation latE, lonE, IE h, a, b (according to Pasolini, Albarello, et al., 2008)4 Attenuation equation latE, lonE, h, IE a, b (according to Pasolini, Albarello, et al., 2008)5 Attenuation equation latE, lonE, h, IE, a b (according to Pasolini, Albarello, et al., 2008)6 Attenuation equation latE, lonE, h, IE, a, b

Table 1Coefficients of Attenuation Equations and Magnitude-Intensity Relationships*

Attenuation Law a b h (km) c d σIE σMw

Standard 0:0086� 0:0005 1:037� 0:027 3:91� 0:27 �4:446� 0:645 2:210� 0:122 0.66 0.30Extended 0:0009� 0:0002 1:172� 0:014 4:49� 0:20 �5:368� 0:693 2:364� 0:133 0.74 0.31

*a, b, and h are the coefficients and average source depth of attenuation equation (3); c and d are the coefficients of generalorthogonal regression (7) between Mw and IE; σIE and σMw

are the standard deviations of regressions with dependent variables IEand Mw, respectively. Attenuation laws: standard as computed by Pasolini, Albarello, et al. (2008), extended as computed in thisstudy by the inclusion of intensity data at long epicentral distances (see text).


Estimation of Fault Orientation

Even though herein we did not develop a new method todetermine the orientation of the fault, we now recall somedetails of the previous method that were either omitted ornot sufficiently stressed by Gasperini et al. (1999). The un-derlying assumption behind the method is that it is ruptureextension, rather than source radiation pattern, that deter-mines the spatial distribution of the largest intensities inthe near field. In particular, the true strike of the fault shouldbe close to the azimuths of the sites that experience the lar-gest intensities with respect to the epicenter; and, the largerthe epicentral distance, the closer the strike. Hence, the ori-entation of the seismic fault is computed as the weightedaxial mean �α of the distribution of the axial orientationsof those sites that experience an intensity larger than a giventhreshold Imin:

�α � 1

2cos�1�C=Q� � 1

2sin�1�S=Q�; (18)

where

C �XNi�1

wi cos 2αi

�XNi�1

wi; S �XNi�1

wi sin 2αi

�XNi�1

wi;

and Q ��C2 � S2

p: (19)

Axial orientations (in the range 0°–180°) are computed as

αi ��sin�1�cos�lati� sin�loni � lonE�= sin�Ri=R0��; if sin�lati� ≥ cos�Ri=R0� sin�latE�� sin�1�cos�lati� sin�loni � lonE�= sin�Ri=R0��; if sin�lati� < cos�Ri=R0� sin�latE�

; (20)

where latE and lonE are the epicentral coordinates, lati andloni are the coordinates of the i-th site, Ri is the geodeticepicentral distance of the i-th site (equation 4), and R0 �6371 km is the radius of a spherical Earth.

The weights wi are defined as the ratio between theeffective epicentral distance of the site Ri and the predictedepicentral distance, which is obtained as a function of thedifference between the epicentral intensity I0 and the inten-sity at the site Ii, using a cubic attenuation equation (Berardiet al., 1993) fitted to the Italian data:

wi �Ri��

��I0 � Ii� � 0:46�=0:933p : (21)

The intensity threshold Imin is determined so that the result-ing average distance of the selected sites is the closest to thesubsurface rupture length (RLD) estimated by Wells andCoppersmith’s (1994) empirical relationships as a functionof moment magnitude.

The multiplication factors (2 for the orientations in equa-tion 19 and 1=2 for the mean in equation 18) were not

explicitly quoted in Appendix 3 of Gasperini et al. (1999)but were implemented correctly in all versions of the Boxercode. They are required to convert the axial orientations (in therange 0°–180°) to circular ones (in the range 0°–360°) and thenback to axial when computing the mean (Fisher, 1993, p. 37).

Because a uniform circular distribution of data has nosignificant central value, the circular (or axial) mean is gen-erally meaningless in such cases. Hence, Gasperini et al.(1999) tested the uniformity of the distribution of axial datausing the Rayleigh and Kuiper tests (Rock, 1988, Fish-er, 1993).

The Rayleigh test is the standard parametric test used fora von Mises–type circular distribution. It is based on themean resultant length Q of a group of data and their numberN. The probability pR of rejecting the null H0 hypothesis ofrandom uniform distribution when it is actually true can becomputed as

pR � exp��Z��1� �2Z � Z2�=�4N�� 24Z � 123Z2 � 76Z3 � 9Z4�=�288N2��; (22)

where Z � NQ2 (Fisher, 1993, p. 70). In previous versionsof the Boxer code, Z was instead computed as Z � 2NQ2,according to Rock (1988, p. 233). After a review of therelevant literature, we decided to adopt the Fisher (1993)formulation here, as well as in the new version of Boxercode. This change has the effect of rejecting the uniformityhypothesis for larger Q values, thereby reducing the number

of cases for which the average orientation can be computedconsistently.

The (nonparametric) Kuiper test is the circular analog ofthe linear one-sample Kolmogorov–Smirnov test. It uses themaximum deviation of the data from a uniform distribution.For axial data, we compute

VN � max�2αi

360∘ �i

N

��max

�i � 1

N� 2αi

360∘�; (23)

where max indicates the maximum, for i varying from 1 toN, when the values of αi are arranged in increasing numer-ical order. The critical values of VN above which the H0

hypothesis of uniform distribution can be rejected are givenby the formula (Mardia, 1972)

VC � VN=�

��N

p� 0:155� 0:24=

��N

p�; (24)

where VN � 1:620, 1.747, 1.862, and 2.001 correspond

to significance levels pK � 0:1, 0.05, 0.025, and 0.01,respectively.


We recently discovered that previous versions of theBoxer code were not properly treating orientations as axialdata in the Kuiper test (the factors of 2 in equation 23 weremissing). We corrected this mistake, which may have ledto incorrect assessments of the nonuniformity of axial distri-bution of orientations in a significant number of cases.However, we also realized that the Kuiper test does not takeinto account the weights (equation 21) assigned by the algo-rithm to the intensity data, and that it could (erroneously) notreject the uniformity hypothesis for data sets with weightsvery different from each other. We therefore conservativelydecided to adopt the result of the Kuiper test only in thosecases where the Rayleigh test does not reject the uniformityhypothesis. In summary, we discarded the orientation onlywhen both tests did not reject the H0 hypothesis (when bothpR and pK were larger than 0.1).

Estimation of Uncertainties

The term “uncertainty” denotes an index of the disper-sion of the measures of a given parameter that is determinedquantitatively in terms of a range of values within which agiven fraction of measures (i.e., the confidence level) areexpected to fall. The uncertainty gives a reliable estimate ofthe accuracy, which represents the closeness of the measureto the true value (which is usually unknown) of the param-eter, assuming that this measure is not affected by systematicerrors or bias. The uncertainties that we compute and reportherein represent an estimate of the dispersion of our macro-seismic estimates, which themselves rely on assumptionsthat may be wrong or inaccurate due to the fact that seismicintensity has no clear physical definition. These uncertaintiesmay therefore largely underestimate the accuracy of thereproductions of the instrumental estimates, in particularfor earthquake magnitude, which is empirical in nature andlacks a solid physical description, even if it is computedusing instrumental data.

Gasperini et al. (1999) estimated formal uncertaintiesonly for magnitude and fault orientation. They also computeda kind of uncertainty indicator for the epicentral coordinates asthe standard deviation of the coordinates of the sites actuallyused to compute the trimmed mean. Gasperini et al. (1999)argued that this could not be considered to be a true estimateof the uncertainty of the location because it also depended onthe size of the mezoseismal area concerned. We recently rea-lized that, even ignoring such an objection, the standard de-viation of the coordinates overestimates the uncertainty oftheir mean because it represents the variability of the sample(and not of the mean). We herein correct this overestimationby dividing the standard deviation by the square root of thenumber of coordinates used to compute the trimmed mean.For the new location methods (1–6), all of which makeuse of a maximum-likelihood approach, a formal variance/covariancematrix can be computed as the inverse of the finite-difference Hessian matrix of the log-likelihood function at itsmaximum (Guo and Ogata, 1997). In Appendix 3, we give

details of the estimation of the uncertainties for all the param-eters concerned. In the case of fault orientation, it should benoted that the values differ significantly (by a factor of 1=2)from those computed byGasperini et al. (1999) due to a bug inthe original Boxer code.

We also describe herein our empirical estimation ofuncertainties, which makes use of the bootstrap method(Efron and Tbishirani, 1986; Hall, 1992). The underlyingprinciple of the bootstrap method is that the empirical fre-quency distribution of the data provides an optimal empiricalestimate (in the sense of maximum likelihood) of the prob-ability distribution that characterizes the unknown parentpopulation. This hypothesis implies that any new data sets(usually called bootstrap samples or paradata sets) that areobtained by randomly resampling the original set (withreplacement) preserve the statistical features of the parentpopulation. Paradata sets can be used to evaluate (using adistribution-free approach) the properties of a sample of agiven population from the analysis of the empirical valuesof the parameter computed for each paradata set. Comparedwith the jackknife technique, which makes use of reduceddata samples, the bootstrap method has the advantage ofbeing asymptotically consistent because the size of replicatesamples is the same as that of the real data set.

In our present study, the number of generated paradatasets is established using an automatic criterion, the aim ofwhich is to achieve abalancebetweenaccuracyandcomputingefficiency.Wegeneraten � 50� N=2paradata sets,whereNis the number of intensity data points for the given earthquake.

It is important to note that for programming convenience,we recompute all the parameters for each paradata set usingthe same procedure as used for the real data set. This impliesthat for some parameters, like the magnitude and orientationof the fault, which in principle are computed separately fromthe epicentral location, bootstrap variability depends also onthe particular location of the epicenter computed for each gi-ven paradata set. In turn, this implies that the uncertaintiesestimated by the bootstrap for such parameters will be largerthan the formal ones, which are instead computed using thedistances and/or angles relative to the epicenter of the real dataset. A bootstrap estimate that is more consistent with the for-mal uncertainties could be obtained by resampling only thederived quantities (distances or angles) that are actually usedto compute the given parameter. The implementation of such aprocedure may be considered in the future, although the pre-sent approach has the advantage of describing parametervariability in a similar manner to the case where some ofthe data are neglected or lost, which can actually occur inmacroseismic analyses. The details of the computation ofbootstrap uncertainties for all of the parameters are describedin Appendix 3.

Testing the Accuracy of Estimated Parameters

Although the accuracy of a measure cannot be assessedprecisely because its true value can never be known, we may


estimate its upper limit by comparing different estimationmethods. In this section we compare our macroseismic esti-mates of the parameters with the instrumental ones that areavailable for the same earthquakes. Even though, in somecases (e.g., the epicentral coordinates and the depth), themacroseismic and instrumental estimates do not representprecisely the same physical parameter (see the discussion inthe Introduction), in practice (e.g., in seismic hazard assess-ment) they are often assumed to be equivalent. This analysistherefore allows us also to evaluate the overall homogeneityof the integrated (macroseismic plus instrumental) earth-quake catalog.

We applied all seven methods of location to the samedata set used in Pasolini, Albarello, et al. (2008), which con-sists of the most recently published versions of the Param-etric Catalog of Italian Earthquakes (CPTI04; CPTI WorkingGroup, 2004; see also the Data and Resources section) and ofthe related database of macroseismic intensity observationsin Italy (DBMI04, Stucchi et al., 2007; see also the Data andResources section). For most of these tests, we excluded(according to Pasolini, Albarello, et al. (2008)) all thoseearthquakes that were offshore, close to the coast, or withinthe volcanic areas of Mt. Etna and Ischia island, as well as allearthquakes that had less than 20 intensity data points. Wethus aim to show how well our methods can reproduce theparameters derived from instrumental data, provided that weuse good data sets, and we therefore defer the extensive test-ing of the performance of the method using poor data sets toa future study.

Epicenter

We consider the distances between the locations of theepicenters computed using the seven methods and the instru-mental ones available in the literature (see Table A1 ofAppendix 2 for a list of the 77 instrumental epicenters usedfor the comparison). In Figure 1, we show the frequencydistribution of the distances between the macroseismic andinstrumental epicenters using fixed distance ranges (in km).The results are similar for all the methods used; only about20%–25% of the macroseismic epicenters are at 5 km or lessfrom the corresponding instrumental epicenter, but about55%–60% of them are at 10 km or less. In very few cases(less than 5%), the distances are greater than 30 km.

In order to understand the efficiency of the new and oldmethods, we consider in detail the cases of two earthquakeswhere the distance between the barycenter of largest inten-sities (method 0) and the instrumental epicenter are the great-est (about 50 km).

For the Passo della Cisa earthquake (25 October 1972;Fig. 2), the largest intensities (V, green dots) were observedmainly along the shoreline of the Ligurian Sea, but the in-strumental hypocenter (asterisk) was located close to thecrest of the Apennines, at a depth of 76 km (InternationalSeismological Centre On-line Bulletin [ISC]; see Data andResources section). Clearly, the barycenter of the largest in-

tensities (cross) is attracted towards the coast, 50 km awayfrom the instrumental location. This discrepancy also occursfor three of the new methods (4, 5, and 6), while methods 1,2, and 3 locate the epicenter at a distance of less than 20 kmfrom the instrumental one. The large ellipses of uncertainty(computed using the formal Hessian approach at the 90%confidence level) indicate that the location of this earthquakeis poorly constrained owing to the peculiar distribution of theintensity data.

A somewhat similar case pertains to the earthquake inPotentino, southern Italy (5 May 1990; Fig. 3), whose hypo-center (asterisk) was located using the Catalog of ItalianSeismicity (CSI; see Data and Resources section) at a depthof about 23 km. In this case, the distribution of the largestintensities VI–VII (yellow-orange) and VII (orange) is ratheruniformly distributed over a wide area. The barycenter meth-od locates the epicenter (cross) at the center of this distribu-tion and about 40 km to the west of the instrumental one,although all the epicenters determined using the new meth-ods are located closer to the instrumental epicenter, at a dis-tance of about 10–15 km from it.

The analysis of both these earthquakes seems to showthat a relatively deep hypocenter produces widely distrib-uted macroseimic fields (perhaps due to the more efficientpropagation of seismic waves in the lower crust and in thelithosphere), and that only the new location methods can sa-tisfactorily identify the point from which the seismic wavesappear to radiate.

In Figure 4, we show the comparison between the resultsobtained using the old barycenter method and those obtainedusing the new methods (the comparison uses 329 earth-quakes). The differences between the locations are less than5 km for about 55% of the computed epicenters. In only afew cases (about 5%) do we show differences greater than30 km. Hence, the new methods (1–6) do not appear to give

Figure 1. Percentages of macroseismic epicenters within givendistance ranges (in km) from instrumental epicenters for the differ-ent methods of location used. All earthquakes have onshore epicen-ters and at least 20 intensity data points.


very different results from the old one (0) for the greatmajority of the earthquakes studied.

We earlier stated that the location of the macroseismichypocenter is likely to correspond to the center of the seismicsource. We therefore compared the macroseismic locationswith the surface projections of the moment tensor centroidsof the 32 earthquakes in our database that had a CentroidMoment Tensor (CMT) solution (see Table A1 in Appen-dix 2). Figure 5 illustrates that the correspondence is worsethan for the instrumental epicenters (Fig. 1), thus indicatingthat the CMT centroids are probably not very accurate esti-mators of the center of the seismic source.

As noted previously, the new location algorithms allowthe location of earthquakes in the sea. We consider the caseof the earthquake that occurred in the Ligurian Sea (19 July1963; Fig. 6). For this earthquake, we used the intensity dataprovided in version 4 of the Catalogue of strong earthquakes

in Italy (see Data and Resources section) because in this dataset the intensity data are much more numerous and moreuniformly distributed than those included in the DBMI04database.

While the old method locates the epicenter on the coast(cross), all the new methods give locations that correspondreasonably well (within 20–30 km) with the instrumentallocation. For this particular earthquake, some data from theIsland of Corsica (in the southeastern corner of Fig. 6) allowsa significant reduction of the azimuthal gap and affords agood degree of constraint to the macroseismic epicenterscomputed using the new methods.

Another interesting example is that of the earthquakethat occurred close to Palermo on 6 September 2002 (Fig. 7).In this case (included in the DBMI04 database but not con-sidered in the extensive tests shown previously), the barycen-ter method locates the epicenter on the coast (cross), but all

Figure 2. Intensity distribution, instrumental (asterisk) and macroseismic (symbols) epicenters and ellipses of 90% uncertainty (com-puted using the Hessian approach) of the earthquake of 25 October 1972 (Passo della Cisa, Northern Italy).


the new algorithms locate it offshore and, in most cases (i.e.,using methods 2–6), very close to the instrumental epicenter.However, the very large extent of the ellipses of uncertaintyindicates that the location is not very well constrained asa result of the scarcity of data points to the north of theepicenter.

We may deduce that the locations given by the newmethods are reasonably accurate and are particularly appro-

priate for cases in which the hypocenter of the earthquake isdeep or offshore.

Depth

We now compare the depths computed by the new meth-ods (2, 4, 5, and 6) and by the Musson (1996) method (0)with a set of 41 instrumental estimates (listed in Table A1 of

Figure 3. Intensity distribution (from DBMI04 database), instrumental (asterisk) and macroseismic (symbols) epicenters, andellipses of 90% uncertainty (computed using the Hessian approach) of the earthquake of 5 May 1990 (Potentino, southern Italy). Theintensity at Paupisi (orange dot close to NW corner) was reduced from VII–VIII to VII, according to R. Camassi (personal comm.).


Appendix 2), all of which can be considered to be reliable;that is, depths that are computed in specific studies or that arefound to be consistent between the different catalogs (CSI,ISC, etc.).

In order to ensure a degree of consistency in our com-parisons, we consider the ratio between the macroseismicand instrumental depths. Figure 8 shows histograms of thenumber of earthquakes that occur within various bands ofthe ratio; it is apparent that, for all the methods (and parti-cularly for method 2), the macroseismic depths are verywidely dispersed and greatly underestimated (by a factor2–5). Only for a very few earthquakes are the values of depthsimilar to or greater than the instrumental ones. Using theMusson (1996) method (0), the degree of underestimationis slightly less but the degree of dispersion is comparable.For method 2, which most underestimates the depths, thedepths are all confined within the range of 0–10 km, even

for those earthquakes that are well known to be much deeper(e.g., the Umbria-Marche event at a depth of 45 km that tookplace on 26 March 1998) and for which methods 0, 4, 5, and6 all yield fairly consistent values. The particularly poor per-formance of method 2 could be related to the assumption thatthe intensity at the epicenter is given by equation (5), whichmight not be entirely appropriate for deep earthquakes.

The underestimation of depth is consistent with theaverage value of macroseismic depth (3.9 km) computedby Pasolini, Albarello, et al. (2008) in comparison with theaverage instrumental depth of Italian earthquakes (10–15 km).These authors have argued that their shallow average depth,which is also consistent with the similar values obtained bySabetta and Pugliese (1987, 1996) using the instrumental dataobtained from strong motions, could be a result of the fact“that the shallower (and closer) portion of the seismogenicfault contributes more than the deeper one in determining theground motion level at close sites” (Pasolini, Albarello, et al.,2008, p. 707). However, the discrepancy obtained between themacroseismic and instrumental depths cannot merely be aconsequence of the different definitions of instrumental andmacroseismic hypocenters but might rather reflect a defi-ciency in the assumptions made in the macroseismic methodsused. In our approach (and even in other similar ones), it isassumed, for example, that (1) the dimensions of the sourceare negligible with respect to the length of the wave path,(2) the properties of the attenuation are homogeneousthroughout the whole of the Earth’s crust, and (3) the spread-ing exponent does not vary as a function of the distance fromthe source. Such assumptions might reasonably apply to thefar field but not to the near field, where Pasolini, Albarello, etal. (2008) showed that the data control the macroseismicdepth. In the near field, the dimensions of the source, or otherfactors, could influence the effective amplitude of the groundmotion, thereby rendering it unpredictable using a generalattenuation equation.

It may be concluded that the macroseismic depth (parti-cularly when computed using method 2) does not provide areliable estimate of the hypocentral depth. However, whencompiling our reference instrumental data set, we verified thateven the instrumental depths of some significant earthquakes(Mw >4) are somewhat unreliable, given that we foundmanycases (not used in the comparisons presented here) in whichdifferent agencies published rather different estimates ofdepth, even though data from a large number of relativelyclose stationswere available. The poor reliability of the instru-mental depths of moderate and strong earthquakes could beattributed to the difficulty of determining the onset of the Sphases at short epicentral distances with any clarity, due tothe saturation of the seismic traces recorded using the analogand narrow-band digital equipment that was in use in Italy andthe surrounding area until about a decade ago. In addition, thesource depth was almost completely neglected in recentItalian historical catalogs (CPTI Working Group, 1999,2004). In the near future, we therefore plan to improve thereliability of our data set of instrumental depths by the careful

Figure 4. For new location methods (see text), percentages ofepicenters within given distance ranges (in km) from epicenterscomputed using the barycenter method.

Figure 5. For different location methods, percentages of macro-seismic epicenters lying within given distance ranges (in km) fromCMT centroid surface projections.


evaluation of the degree of uncertainty of the locations, a testof stability of the location procedures used, and, in somecases, the reanalysis of the original phase data.

Figure 9 shows the same comparison as Figure 8 butinstead uses the centroid depths determined by moment ten-sor inversion for the 32 earthquakes of our instrumental dataset (Table A1 in Appendix 2). Perhaps surprisingly, and indisagreement with the results obtained using the comparison

between the epicenters (Fig. 5), the agreement is slightlybetter than for the case of the hypocentral depths. Althoughfor many of the earthquakes listed in Table A1 of Appendix 2,the centroid depths appear to have been assigned toconventional values (e.g., 10 km), it may be argued that theyare more consistent with the macroseismic estimates than thehypocentral depths, although the agreement neverthelessremains poor.

Figure 6. Intensity distribution, instrumental (asterisk) and macroseismic (symbols) epicenters, and ellipses of 90% uncertainty(computed using the Hessian approach) of the earthquake of 19 July 1963 (Ligurian Sea, northern Italy).


Magnitude

We compared the macroseismic magnitudes obtainedusing the old and alternative methods with a set of 201instrumental moment magnitudes determined using momenttensor inversion or deduced by empirical regressions fromother types of instrumental magnitude (Ms, Ml, and mb) re-ported by national and international agencies. Such a set wascollected during the compilation of the latest release of theCPTI catalog (CPTI Working Group, 2004) and is the resultof the careful scrutiny of the procedures used by differentagencies. In particular, local magnitudesMl were consideredto be reliable only if they were determined using real or syn-thetic Wood–Anderson waveforms, while all the estimates of

Figure 7. Intensity distribution, instrumental (asterisk) and macroseismic (symbols) epicenters, and ellipses of 90% uncertainty(computed using the Hessian approach) of the earthquake of 6 September 2002 that occurred in the southern Tyrrhenian Sea (southernItaly), close to Palermo.

Figure 8. Histograms of the ratio between the depth determinedusing various macroseismic methods and the instrumental depth.


Ml that were derived from empirical formulas using othertypes of magnitude (e.g., Md) were discarded.

In Table 3, we summarize the statistics obtained from acomparison between estimates of macroseismic and instru-mental magnitude in which the epicenter was computed usingdifferent macroseismic methods. The old method slightly un-derestimates the instrumentalmagnitude,while the alternativemethod slightly overestimates it. Although very small, suchdiscrepancies could reflect small differences between the dataset used for the comparison and that used for the calibration, inaddition to (in the case of the old method) the use of an equalweight for all the earthquakes concerned.

The average discrepancies obtained for both methodsbetween the macroseismic and instrumental magnitudesare all lower than 0.1, with standard deviations of the orderof 0.3 magnitude units in most cases. For the alternative mag-nitude method, the average differences and standard devia-tions tend to increase with the number of parameters jointlyestimated using different location methods.

It may be seen from Table 3 and Figure 10 that the dis-tribution of differences in magnitude, although slightlyskewed to low values and less peaked, are close to being nor-mally distributed in most cases. The alternative magnitudemethod yields lower average differences and, for location

methods 0 and 1, even yields lower standard deviations.Hence, the alternative method can be confidently employedin place of the old one, particularly when a recalibration for adifferent region is necessary, because such a calibration issimpler and requires the use of fewer data.

Orientation of the Seismogenic Fault

In this case, the comparison with instrumental data is notas obvious as for the other parameters because orientationsare expressed as axial data (ranging from 0° to 180°), whilefault strikes are circular (ranging from 0° to 360°). Further-more, the focal mechanisms provide two alternative planes,with the inherent impossibility of distinguishing between thereal fault plane and the auxiliary one (at least when usingonly seismological data). For our comparisons, we considerthe minimum angular difference obtained between themacroseismic orientation and those of the two nodal planes(thereby reducing the strikes to axial orientations in the range0°–180°). We used a data set comprised of 38 focal mecha-nisms (listed in Table A2 of Appendix 2), which was com-piled from the literature, checked for consistency, and storedin the database of Earthquake Mechanisms of MediterraneanArea (see Data and Resources section) by Vannucci and Gas-perini (2003, 2004). For some of them (depending on thelocation method), the macroseismic orientations obtainedmight be not considered to be reliable, according to the Ray-leigh and Kuiper test; hence they may not have been countedin some of our comparisons.

It is clear that the maximum possible angular differencebetween two orientations is 90°, while the expected averagedifference between two sets of independent orientations is45°. However, in considering the minimum difference be-tween the orientations of two conjugate fault traces, boththe maximum possible difference and the expected averagedifference also depend on the relative angular differencebetween the two faults concerned. This is 0° for pure dip-slip (normal or inverse) mechanisms and 90° for pure strike-slip mechanisms. The maximum and expected average

Figure 9. Histograms of the ratio between the depth determinedusing various macroseismic methods and the depth of the momenttensor centroid.

Table 4Summary Statistics of Differences Obtained between Instrumental

and Macroseismic Magnitudes

Location Algorithm 0 1 2 3 4 5 6

Old Magnitude MethodMean 0.095 0.088 0.079 0.074 0.078 0.077 0.072Standard Deviation 0.314 0.311 0.308 0.310 0.306 0.306 0.313Skewness �0:224 �0:234 �0:222 �0:207 �0:251 �0:240 �0:237Kurtosis 0.001 0.056 0.049 0.007 0.086 0.089 0.063

New Magnitude MethodMean �0:034 �0:050 �0:061 �0:067 �0:066 �0:083 �0:100Standard Deviation 0.307 0.306 0.311 0.321 0.323 0.338 0.358Skewness �0:194 �0:132 �0:120 �0:219 �0:143 �0:290 �0:582Kurtosis �0:113 �0:126 �0:168 0.011 �0:219 0.213 0.971


differences therefore vary from 90° to 45°, and from 45° to22.5°, respectively.

In order to evaluate the efficiency of our algorithm inreproducing the instrumental orientations, we must thereforecompare the observed distribution of differences with thatexpected in the case of two sets of uncorrelated orientations.Hence, we generated 1000 random macroseismic orienta-tions for each of the focal mechanisms included in our dataset and computed the mean and median differences that are

compared with those deduced from the real macroseismicorientations.

The results in Table 4 show that, for the full data set offocal mechanisms, the average differences computed usingthe real data do not differ much from the random case. Thisfinding can be shown rather better by comparing the histo-grams in Figure 11 of the occurrences of real data (black) andof random simulations (gray), using location method 0. Onlyfor differences less than 10° do the occurrences of the real

Figure 10. Histograms (using bins of 0.2 magnitude units) of the differences between the macroseismic magnitude estimates computedusing the old (gray symbols) and alternative (black symbols) methods and instrumental estimates, when the epicenter is determined usingdifferent methods. The lines indicate the expected frequencies of the normal distributions with means and standard deviations as reported inTable 3.


data set prevail over those of the random case; the occur-rences of the large differences are similar for both. Wededuce that, for the full set of mechanisms concerned, themacroseismic orientation computed using intensity datayields a poor estimate of fault strike.

We already mentioned that Gasperini et al. (1999) havenoted that fault orientations are likely to be unreliable forsmall earthquakes because of the poor spatial resolutionof macroseismic data. In fact, they suggested computing azi-muths only for earthquakes withMw ≥5:5, for which subsur-face fault length, of the order of 5 km or more (Wells andCoppersmith, 1994), may be resolved well by the typicalspatial distribution of sites.

In Table 4, a noticeable improvement in the reliability ofthe orientations obtained using real data compared with thatof random ones results when considering only those earth-quakes having Mw ≥5:5. The results are particularly goodfor method 0 (barycenter), for which the average differencefor real data is about 14°. We may also note that by furtherincreasing the magnitude threshold, the average differencecontinues to decrease and assumes values of about 9° forMw ≥5:7 and 7° for Mw ≥5:9. Furthermore, the lower medi-ans (8° for Mw ≥5:5, 7° for Mw ≥5:7, and 5° for Mw ≥5:9)compared to the averages indicate that for most earthquakesthe agreement is even better and that it is only the presence ofa few outliers that determines a larger average difference.The results for the other location methods (1–6) are not asgood as that of the barycenter. It may be argued that the algo-rithm used for the computation of fault orientation is moreconsistent with the barycenter method of computing the lo-cation because both use only part of the data set (data havingthe largest intensity) and sometimes use the very same data.

Previous evidence indicates that the orientations arereliable only for strong earthquakes and that a reasonablelower threshold of reliability is given by Mw � 5:7. In fact,Figure 12 shows that, for the 10 earthquakes above thisthreshold, the frequency of differences lower than 10° is

clearly higher than that of the random set and that the dif-ference in orientation is greater than 30° (actually 33°) onlyfor the earthquake of 18 October 1936 at Bosco Cansiglio,for which the focal mechanism, computed using the polari-ties of the first arrivals recorded by mechanical seismo-meters, could be inaccurate.

In Figure 13, we show a visual representation of thecorrespondence between the instrumental and macroseismicorientations using the superposition of the macroseismic azi-muths (black lines) on the focal mechanisms of earthquakeshaving Mw ≥5:5. If we exclude the two earthquakes withMw <5:7 (gray shaded boxes), the agreement appears to besurprisingly good, even with first pulse mechanisms of theearthquakes of the first half of the twentieth century (toprow), for which the accuracy might be somewhat arguable.For the more reliable CMT solutions (bottom row), thedeviations are about 5°–6° in most cases and larger than10°–15° only for the smallest (Mw � 5:64) earthquake of

Table 4Mean and Median Differences between Macroseismic

and Instrumental Orientations

Location Method 0 1 2 3 4 5 6 Random Set

Entire Data Set (28)Mean 19.6 22.5 21.5 21.7 24.7 23.1 23.8 27.0Median 16.2 22.1 20.2 20.2 20.8 24.4 25.2 25.6

Mw ≤5:5 (12)

Mean 13.8 19.7 21.7 21.9 26.1 19.3 19.8 29.2Median 8.4 19.5 20.2 20.2 22.3 19.1 19.1 27.4

Mw ≤5:7 (10)

Mean 9.1 14.4 16.1 16.3 20.5 13.9 15.7 27.6Median 6.7 15.5 18.9 18.9 19.5 14.2 19.0 25.3

Mw ≤5:9 (5)

Mean 7.2 10.4 13.6 13.6 23.0 10.4 13.3 31.0Median 5.1 10.1 18.9 18.9 19.5 11.1 19.0 28.1

Figure 11. Histograms of the axial differences between themacroseismic and instrumental orientations (the lowest value be-tween the two nodal planes) for real data using location method0 (black) and randomly generated orientations (gray).


Calabria-Lucania (9 September 1998), showing a relativelylarge compensated linear vector dipole component that isusually an indication of noisy data.

It is also apparent that the macroseismic orientations insome cases allow discrimination between the two conjugate

nodal planes, thus adding further information to that given bythe instrumental focal mechanisms. Although deeper seismo-logical and geological analyses of all the earthquakes wouldbe required to test this hypothesis properly, it may be notedthat in fact, in the case of the earthquake of Umbria-Marche(26 September 1997), the macroseismic orientation corre-sponds fairly well with the southwestward dipping plane thatChiaraluce et al. (2004) identified as the real fault plane, fol-lowing their analysis of the three-dimensional distribution ofthe aftershocks. In addition, for the Molise earthquake (31October 2002), the macroseismic orientation is close to thatof the east–west dextral fault identified by Chiarabba et al.(2005) as the source of the main shock. Even for the Irpiniaearthquake of 23 November 1980, the conjugate plane closerto the macroseismic orientation (4°) is the one that dipsnortheastwards and most closely corresponds to the complexfault system proposed by Pantosti and Valensise (1990) forsuch an earthquake. In this case, however, the closeness ofthe macroseismic orientation to the other conjugate plane(11°) and the complexity of the source might suggest thatthe correct choice of plane may simply be a matter of chance.

Figure 12. As Figure 11 but for earthquakes with Mw ≥5:7.

Figure 13. Reliable (see text) macroseismic orientations (black lines) superimposed on the focal mechanisms computed using the firstpulses (top row) and using moment tensor inversion (bottom), for earthquakes withMw ≥5:5. Inst1, Inst2, and Macr indicate the orientations(clockwise from north) of the two instrumental nodal planes and of the macroseismic estimate, respectively. The instrumental orientation thatbest corresponds to the macroseimic one is typed in boldface. Earthquakes with Mw <5:7 are highlighted in gray.


Statistics of Uncertainties

We now summarize some of the evidence that may beinferred from the analysis of the frequency distribution ofthe uncertainties computed using the Italian macroseismicdatabase and reported in detail in Appendix 3. In this anal-ysis, we also compare the distribution of parameter uncer-tainties with the distribution of the absolute differencesbetween the instrumental and macroseismic determinationsin order to evaluate how well the estimated uncertaintiesmight represent the accuracy with which the instrumentalestimates are reproduced.

For the epicentral coordinates, the formal uncertainties(which may clearly be obtained more quickly) correspondwell to the bootstrap estimates (which require some verytime-consuming simulations); hence, the former may beadopted with some confidence in practice. This finding alsoapplies to method 0 and thus indicates, despite the assertionof Gasperini et al. (1999), that the standard deviation of thetrimmed mean of the coordinates of the sites used is not in-fluenced very much by the actual density of sites within theepicentral area. The uncertainties in the location are clearlylower than the average distances between the instrumentaland macroseismic epicenters but are of the same order ofmagnitude as the distances between the macroseismic epi-centers computed using method 0 and those computed usingthe other methods. By considering the different meanings ofthe instrumental and macroseismic epicenters, we contendthat the computed uncertainties represent the accuracy ofthe coordinates of the macroseismic center rather well.

Even for depth, there is substantial agreement betweenthe formal and bootstrap uncertainties, which indicates thatfor all the hypocentral parameters concerned, the faster for-mal methods yield consistent estimates. However, compari-son with the distribution of the absolute differences obtainedbetween the instrumental and macroseismic depths indicatesthat the computed uncertainties clearly underestimate theaccuracy of the simpler methods (0, 2, and 4). In such meth-ods, we make assumptions that constrain the hypocentralsolution, thereby reducing the statistical uncertainty butincreasing the epistemic uncertainty due to the possibleinappropriateness of such assumptions for the given earth-quake. For methods 5 and 6, which make fewer assumptions,the degree of underestimation is less but still apparent. Thesediscrepancies confirm that the accuracy of the estimates ofmacroseismic depth is arguable; hence the use of estimatesof macroseismic depth in investigations of seismic hazardand seismotectonics should be undertaken with caution.

The analysis of the distribution of the uncertaintiesof magnitude indicates that the bootstrap estimates are gen-erally larger than the estimates computed using formal meth-ods, even if they are generally consistent in terms of theirorder of magnitude. However, apart from the type-I uncer-tainties, which are described in Appendix 1 for the oldmagnitude method that also considers regression errors,uncertainties of magnitude largely underestimate the accu-

racy of reproducing the instrumental magnitudes, which maybe evaluated from Figure 10 and Table 3 as being of the orderof 0.3 magnitude units. For both the old and the alternativemagnitude methods, it may be recalled that both the standarddeviations of the mean values and the bootstrap uncertaintiesrepresent only the statistical uncertainties associated with theaveraging procedure and neglect the uncertainties due to theregressions.

The frequency distribution of the uncertainties asso-ciated with the macroseismic orientations indicate that, formethod 0 (the one that gives the best fit with the instrumentaldata), the formal uncertainties reproduce the distribution ofthe absolute differences between the instrumental and macro-seismic orientations rather better than the bootstrap ones do.For methods 1–6, neither of the two approaches seems to beconsistent with the observed differences. This also confirmsthat location method 0 is the most appropriate for estimatingthe orientation of the seismogenic fault.

In summary, we contend that formal uncertainties repre-sent the variability of the hypocentral parameters and theorientation of the fault fairly well, while they clearly under-estimate the uncertainties in magnitude. For the old magni-tude method, the approach that uses a combination ofregression uncertainties and the number of data used to com-pute the isoseismals seems to provide the most reliableestimate. The development of a similar approach for thealternative method will be undertaken in a future study.

Calibration for Different Seismic Areas

The old methods for determining the epicenter and theorientation of the fault do not require a specific calibrationfor areas outside Italy, unlike the old method for computingmagnitude does. In fact, the Boxer code also includes a pro-cedure for computing the relevant regression coefficients ofmagnitude from a set of intensity data relating to earthquakeswith known magnitudes. As a general rule, at least 20–30earthquakes of various epicentral intensities are necessaryfor a reliable calibration; however, this is not always possiblein countries where seismicity is relatively weak, and hencesignificant earthquakes with known instrumental magnitudesare few and far between. In such cases, the only possibility isto calibrate a posteriori the magnitudes computed by theBoxer code, using the Italian default coefficients, via anempirical regression with the instrumental ones.

A future version of Boxer code will also include thecomplete procedure for fitting the attenuation equationsand the M-IE relationships required by the new methodsusing a calibration with intensity data and magnitudes,following exactly the same procedure as Pasolini, Albarello,et al. (2008). However, a fairly reliable calibration could bemade using an external user code, provided that the sameattenuation and M-IE equations (3) and (7) are used, even ifthe code does not deal with uncertain data as rigorously aswe do here. In fact, Pasolini, Albarello, et al. (2008) demon-strated that the attenuation parameters are not very different


if the uncertain data are treated as half integers (e.g., 7.5) ordiscarded altogether.

Conclusions

We have herein described the development of new meth-ods to locate and size earthquakes using macroseismic data,by fitting an equation of intensity attenuation for Italy, whichwas recently published by Pasolini, Albarello, et al. (2008).In principle, the approach is analogous to other methodspreviously proposed (Console et al., 1990; Bakun and Went-worth, 1997) but differs from them in terms of the method ofestimation of the parameters (maximum likelihood) thataffords the rigorous treatment of uncertain intensity data(e.g., VII–VIII), and to compute the formal uncertainties ofthe estimates of the parameters in a consistent manner.

Our new methods allow the estimation of the sourcedepth and can locate earthquakes offshore or in uninhabitedareas. To improve their performance in such cases, we haveadopted an extended version of the attenuation equation thatcan reproduce the apparent attenuation over long distances.Although such an equation cannot be used to predict theactual intensity at a site because it is biased by the incom-pleteness of data below the level of diffuse perceptibility (seethe discussion in Pasolini, Gasperini, et al., 2008), it is usefulfor expanding the data set of intensity, which can then beused to locate and size earthquakes.

We found that about 50% of the epicentral locationscomputed using the new methods lay at 5 km or closer tothose computed using the Boxer algorithm (the barycenterof the largest intensities) and about 90% of them lay within15–20 km. This implies that the two approaches are consis-tent for the great majority of earthquakes.

We also compared in detail the epicenters obtained usingboth the new and old methods with a set of reliable instru-mental locations available from national and internationalagencies and from the literature. About 20% of the macro-seismic epicenters obtained were very close (distance ≤5 km) to the instrumental ones, and only about 5% werefurther than 30 km away. In particular, the new locationmethods were more accurate than the old one for some rel-atively deep events (h > 20–25 km). In such cases, whichare characterized by very flat distributions of intensity, thenew macroseismic methods of location can reproduce the in-strumental locations quite well. The new methods were alsoshown to compute locations consistent with instrumentalones for some earthquakes with epicenters offshore.

The comparison of macroseismic source depths with aset of reliable instrumental estimates shows very poor agree-ment. In fact, the macroseismic depths appear generally to beunderestimated of a factor of 2–5 and are rather poorlyrelated to the corresponding instrumental values. This under-estimation is consistent with the average depth computed byPasolini, Albarello, et al. (2008) and by analogous computa-tions made by Sabetta and Pugliese (1987, 1996) usinginstrumental strong-motion data. The reasons for such a dis-

crepancy remain unclear, but it can be argued that the (un-modeled) effects of source size and of heterogeneities inwave propagation in the vicinity of a seismogenic fault couldbe significant. Similar comparisons made with respect to aset of moment tensor centroids indicated a worse correspon-dence between epicenters but a better agreement with depths.This might indicate that CMT depth is more indicative of theseismic source than the corresponding epicenter is.

The alternative method for computing magnitude wasshown to be as accurate as the old one for cases where thesimplest methods to compute the epicenter were used. Thedifference obtained between the macroseismic and instru-mental magnitudes was on average less than 0.05–0.10 mag-nitude units, and the standard deviation was about 0.3 unitswith a statistical distribution of residuals that was quite closeto being normal.

The comparison between the macroseismic estimates ofseismogenic fault orientation and the fault strikes deducedfrom a reliable set of focal mechanisms illustrated thatthe macroseismic orientations showed fairly good agreementwith the instrumental ones for earthquakes with momentmagnitudes Mw ≥5:7 where the epicenter was computedusing the old barycenter method. In fact, while the averageangular difference obtained between the macroseismic orien-tation and the closest fault orientation of the traces of the twoinstrumental nodal planes for the entire set of mechanismswas very similar to the average angular difference computedusing a random set of orientations (about 27°), the averagedifference was shown to reduce to about 9° for the set ofearthquakes having Mw ≥5:7. The macroseismic and instru-mental orientations showed a greater consistency when usingthe old location method, perhaps because the barycenter sub-stantially refers to the same set of intensity data (with thehighest values) used to compute the orientation.

The surprising accuracy of some of the macroseismicorientations could allow the correct determination of the faultplane from the two possible conjugate planes given by thefocal mechanisms concerned. This is the case for the earth-quake of 26 September 1997 of Umbria-Marche, for whichthe computed macroseismic orientation corresponds to thefault plane that dips in a southwesterly direction, rather thanto the conjugate one, as indicated by the distribution of after-shocks. A similar (correct) inference may also be made forthe Molise earthquake of 31 October 2002 and for the Irpiniaearthquake of 23 November 1980, although for the latterearthquake we cannot exclude the possibility that the corre-spondence might be fortuitous.

Parameter uncertainties were computed using two differ-ent approaches, namely (1) a formal one in which we computea variance/covariance matrix by inverting the finite-differenceHessian of the log-likelihood function at its maximum or astandard deviation of the parameter averages and (2) anempirical one based on bootstrap simulations.We showed thatthe formal uncertainties, which are much faster to computethan the bootstrap ones, represent the variability of theepicentral coordinates and of the fault orientation fairly well


and so may confidently be adopted for use in commonpractice. Furthermore, we found that standard deviations ofthe single magnitude estimates and bootstrap uncertaintiesboth appear to underestimate the uncertainties in magnitude.For the old magnitude method, the previous approach, basedon the combination of the regression uncertainties and of thenumber of data points used to compute the isoseismals, stillseems to be the most reliable option.

The new release of theBoxer code,which includes a user-friendly MATLAB interface for the selection of data, theparameter setting, and the graphical postprocessing of the re-sults, will be made freely available for download from the sitelisted in the Data and Resources section. ⒺThe FORTRANsource codes of key subroutines and functions for computingthe parameters and uncertainties in the Boxer 4.0 program areprovided in the electronic supplement to this paper.

Data and Resources

Some plots were made using the Generic MappingTools, version 4.2.1 (www.soest.hawaii.edu/gmt; Wessel andSmith, 1998).

The Boxer code (Gasperini et al., 1999) is availableat: http://gaspy.df.unibo.it/paolo/boxer/boxer.html (last ac-cessed 27 April 2010).

The database of Earthquake Mechanisms of Mediterra-nean Area, version 2.2 (Vannucci and Gasperini, 2003,2004), is available at http://gaspy.df.unibo.it/paolo/ATLAS/pages/EMMA_description.htm (last accessed 27 April2010).

Macdep, a program for calculating depth of an earth-quake from macroseismic data (Musson, 1996) was searchedusing ftp://ftp.nmh.ac.uk/pub/gsrg/musson/Macdep.zip (lastaccessed 10 October 2009).

The Catalogue of Strong Earthquakes in Italy 461 B.C.–1977 and the Mediterranean Area 760 B.C.–1500 (CFTI 4Med.) by Guidoboni et al. (2007) was searched using http://storing.ingv.it/cfti4med/ (last accessed 10 October 2009).

The Parametric Catalog of Italian earthquakes (CPTI04;CPTI Working Group, 2004) was searched using http://emidius.mi.ingv.it/CPTI04/ (last accessed 10 October 2009).

The database of macroseismic intensity observations inItaly, used for the compilation of CPTI04 (DBMI04; Stucchiet al., 2007) was searched using http://emidius.mi.ingv.it/DBMI04/ (last accessed 10 October 2009).

The Regional Centroid Moment Tensor data by Pondrel-li et al. (1999, 2001, 2002, 2004, 2006) were searched fromthe European-Mediterranean Regional Centroid MomentTensors catalog using http://www.bo.ingv.it/RCMT/ (lastaccessed 10 October 2009).

The Global (formerly Harvard) Centroid Moment Ten-sor project database was searched using www.globalcmt.org/CMTsearch.html (last accessed 10 October 2009).

The Catalog of Italian Seismicity, version 1.1., wassearched using http://csi.rm.ingv.it/ (last accessed 10 October2009).

The International Seismological Centre On-line Bulletinwas searched using http://www.isc.ac.uk/search/bulletin/index.html (last accessed 10 October 2009).

The International Seismological Summary catalog re-vised by Villaseñor and Engdahl (2005), was searched usinghttp://earthquake.usgs.gov/research/data/iss_summ.php (lastaccessed 10 October 2009).

The data from the Bollettino Strumentale dell’IstitutoNazionale di Geofisica (in raster scan format) was searchedusing http://sismos.rm.ingv.it/bulletins (last accessed 10October 2009).

Acknowledgments

We thank Bill Bakun and Marco Mucciarelli, whose thoughtful re-views helped much to improve this manuscript. We also thank Dario Albar-ello for fruitful discussions. This research has benefited from fundingprovided by the Italian Ministero dell’Istruzione, Università e Ricerca,which also supported the Ph.D. grant of one of the authors (D.T.) withinthe Airplane Project of the Istituto Nazionale di Geofisica e Vulcanologia,and by the Italian Presidenza del Consiglio dei Ministri–Dipartimento dellaProtezione Civile (DPC). Scientific papers funded by DPC do not representits official opinion and policies.

References

Anderson, H., and J. Jackson (1987). Active tectonics of the AdriaticRegion. Geophys. J. R. Astr. Soc. 91, 937–983.

Albarello, D., and V. D’Amico (2004). Attenuation relationship of macro-seismic intensity in Italy for probabilistic seismic hazard assessment,Boll. Geofis. Teor. Appl. 45, 271–284.

Bakun, W. H., and C. M. Wentworth (1997). Estimating earthquake locationand magnitude from seismic intensity data, Bull. Seismol. Soc. Am. 87,1502–1521.

Basili, A., and G. Valensise (1991). Contributo alla Caratterizzazione dellaSismicità dell’area Marsicano-Fucense, in Aree Sismogenetiche eRischio Sismico in Italia, E. Boschi and M. Dragoni (Editors), IstitutoNazionale di Geofisica, Rome, Italy, 197–214 (in Italian).

Basili, R., G. Valensise, P. Vannoli, P. Burrato, U. Fracassi, S. Mariano,M. M. Tiberti, and E. Boschi (2008). The Database of IndividualSeismogenic Sources (DISS), version 3: Summarizing 20 years ofresearch on Italy’s earthquake geology, Tectonophysics 453, 20–43,doi 10.1016/j.tecto.2007.04.014.

Barba, S., and R. Basili (2000). Analysis of seismological and geologicalobservations for moderate size earthquakes: The Colfiorito fault sys-tem (central Apennines, Italy), Geophys. J. Int. 141, no. 1, 241–252.

Berardi, R., C. Petrungaro, L. Zonetti, L. Magri, and M. Mucciarelli (1993).Mappe di sismicità per l’area italiana, Istituto Sperimentale Modelli eStrutture (ISMES), Bergamo, Italy, 52 pp. (in Italian).

Burton,P.W.,R.McGonicle,G.Nelson, andE.M.W.Musson (1985).Macro-seismic focal depth and intensity attenuation for British earthquakes, inEarthquake Engineering in Britain, Telford, London, 91–110.

Cancani, A. (1904). Sur l’emploi d’une double echelle sismique des inten-sités, empirique et absolute, Gerland Beitr. Geophys., Ergänzungs-band 2, 281–283 (in French).

Cecic, I., R. Musson, and M. Stucchi (1996). Do seismologists agree uponepicentre determination from macroseismic data? A survey of theESC “macroseismology” working group, Ann. Geofis. 39, no. 5,1013–1027.

Cheeney, R. F. (1983). Statistical Methods in Geology for Field and Labora-tory Decisions, Allen and Unwin, London, 169 pp.

Chiarabba, C., P. De Gori, L. Chiaraluce, P. Bordoni, M. Cattaneo,M. De Martin, A. Frepoli, A. Michelini, A. Monachesi, M. Moretti,G. P. Augliera, E. D’Alema, M. Frapiccini, A. Gassi, S. Marzorati,


http://www.soest.hawaii.edu/gmt




http://gaspy.df.unibo.it/paolo/boxer/boxer.html





http://gaspy.df.unibo.it/paolo/ATLAS/pages/EMMA_description.htm






ftp://ftp.nmh.ac.uk/pub/gsrg/musson/Macdep.zip





http://storing.ingv.it/cfti4med/




http://emidius.mi.ingv.it/CPTI04/





http://emidius.mi.ingv.it/DBMI04/





http://www.bo.ingv.it/RCMT/




http://www.globalcmt.org/CMTsearch.html





http://csi.rm.ingv.it/




http://www.isc.ac.uk/search/bulletin/index.html






http://earthquake.usgs.gov/research/data/iss_summ.php




http://sismos.rm.ingv.it/bulletins




http://dx.doi.org/10.1016/j.tecto.2007.04.014

P. Di Bartolomeo, S. Gentile, A. Govoni, L. Lovisa, M. Romanelli,G. Ferretti, M. Pasta, D. Spallarossa, and E. Zunino (2005).Mainshocks and aftershocks of the 2002 Molise seismic sequence,southern Italy, J. Seismol. 9, 487–494.

Chiaraluce, L., A. Amato, M. Cocco, C. Chiarabba, G. Selvaggi,M. Di Bona, D. Piccinini, A. Deschamps, L. Margheriti, F. Courbou-lex, and M. Ripepe (2004). Complex normal faulting in the Apenninesthrust-and-fold belt: The 1997 seismic sequence in central Italy, Bull.Seismol. Soc. Am. 94, 99–116.

Chiodo, G., M. F. Currà, A. Moretti, and I. Guerra (1994). OsservazioniSulla Sismicità del Mar Ionio Occidentale, Atti del XIII Convegno,Gruppo Nazionale di Geofisica della Terra Solida, Rome, Italy,28–30 November 1994, 915–926 (in Italian).

Console, R., C. Gasparini, and M. Murru (1990). Un nuovo metodo per ilcalcolo macrosismico dei parametric ipocentrali, Atti del VII Conveg-no, Gruppo Nazionale di Geofisica della Terra Solida , Rome, Italy, 20November–2 December 1988, 63–78 (in Italian).

Catalogo Parametrico dei Terremoti Italiani (CPTI) Working Group (1999).Catalogo Parametrico dei Terremoti Italiani, Editrice Compositori,Bologna, Italy, 88 pp. (in Italian).

CPTI Working Group (2004). Catalogo Parametrico dei Terremoti Italiani(CPTI04), on-line publication available at http://emidius.mi.ingv.it/CPTI04/presentazione.html (last accessed October 2009; in Italian).

Delacou, B., C. Sue, J.-D. Champagnac, and M. Burkhard (2004). Present-day geodynamics in the bend of the western and central Alps asconstrained by earthquake analysis, Geophys. J. Int. 158, 753–774,doi: 10.1111/j.1365-246X.2004.02320.x.

Efron, B., and R. J. Tbishirani (1986). Bootstrap methods for standard errors,confidence intervals and other measures of statistical accuracy, Stat.Sci. 1, 54–77.

Eva, E., and S. Pastore (1993). Revisione dei Meccanismi Focali dell’Ap-pennino Settentrionale, in Atti del XII Convegno, Gruppo Nazionale diGeofisica della Terra Solida, Rome, Italy, 24–26 November 1993,147–159.

Eva, E., and S. Solarino (1998). Variations of stress directions in the westernAlpine arc, Geophys. J. Int. 135, 438–448.

Fisher, N. I. (1993). Statistical Analysis of Circular Data, CambridgeUniversity Press, Cambridge, United Kingdom, 277 pp..

Fuller, W. A. (1987).Measurement Error Models, Wiley, New York, 440 pp.Galanopulos, A. G. (1961). On magnitude determination by using macro-

seismic data, Ann. Geofis. 14, 225–253.Gasparini, C. (1974). Attività sismica in Italia nel 1969, Ann. Geofis. 27,

107–150.Gasperini, P. (2001). The attenuation of seismic intensity in Italy: A bilinear

shape indicates dominance of deep phases at epicentral distanceslonger than 45 km, Bull. Seismol. Soc. Am. 91, 826–841.

Gasperini, P., and G. Ferrari (1995). Stima dei parametric sintetici,in Catalogo dei forti terremoti in Italia daql 461 a.C. al 1980,E. Boschi, G. Ferrari, P. Gasperini, E. Guidoboni, G. Smriglio, andG. Valensise (Editors) , ING-SGA Bologna/Rome, Italy, 96–111(in Italian).

Gasperini, P., and G. Ferrari (2000). Deriving numerical estimates fromdescriptive information: The computation of earthquake parameters,Ann. Geofis. 43, 729–746.

Gasperini, P., F. Bernardini, G. Valensise, and E. Boschi (1999). Definingseismogenic sources from historical earthquake felt reports, Bull. Seis-mol. Soc. Am. 89, 94–110.

Gasparini, C., G. Iannaccone, and R. Scarpa (1985). Fault-plane solutionsand seismicity of the Italian peninsula, Tectonophysics 117, 59–78.

Grünthal, G. (1998). European macroseismic scale 1998, Cah. Cent. Eur.Géodyn. Séism. 13, 1–99.

Grünthal, G., and D. Stromeyer (1992). The recent crustal stress field incentral Europe: Trajectories and finite element modeling, J. Geophys.Res. 97, no. B8, 11805–11820.

Guidoboni, E., G. Ferrari, D. Mariotti, A. Comastri, G. Tarabusi, andG. Valensise (2007). CFTI4Med, Catalogue of Strong Earthquakesin Italy (461 B.C.–1997) and Mediterranean Area (760 B.C.–1500),

INGV-SGA. Available from http://storing.ingv.it/cfti4med/ (last ac-cessed October 2009).

Guo, Z., and Y. Ogata (1997). Statistical relations between the parametersof aftershock in time, space and magnitude, J. Geophys. Res. 102,2857–2873.

Hall, P. (1992). The Bootstrap and Edgeworth Expansion, Springer, NewYork, 372 pp.

International Mathematics and Statistics Library (IMSL), (1991). MATHLi-brary User’s Manual, Version 2.0, IMSL Inc., Houston, Texas,1313 pp.

Jiménez, E. (1991). Focal mechanisms of some European earthquakesfrom the analysis of single station long-periods record, PublicaciónTécnica n. 8, Serie Monografia, Instituto Geográfico Nacional,Madrid, 87–96.

Lomnitz, C. (2006). Three theorems of earthquake location, Bull. Seismol.Soc. Am. 96, 306–312.

Main, I. G., T. Leonard, O. Papasouliotis, C. G. Hatton, and P. G. Meredith(1999). One slope or two? Detecting statistically significant breaks ofslope in geophysical data, with application to fracture scaling relation-ships, Geophys. Res. Lett. 26, 2801–2804.

Mallet, R. (1862). Great Neapolitan earthquake of 1857, the first principlesof observational seismology, in Mallet’s Macroseismic Survey on theNeapolitan Earthquake of 1957, E. Guidoboni and G. Ferrari (Editors),ING-SGA, Roma-Bologna, Italy (anastatic reprint).

Magri, L., M. Mucciarelli, and D. Albarello (1994). Estimates of site seis-micity rates using ill-defined macroseismic data, Pure Appl. Geophys.143, 617–632.

Mardia, K. V. (1972). Statistics of Directional Data, Academic Press, NewYork, 357 pp.

Musson, R. M. W. (1996). Determination of parameters for historical Britishearthquakes, Ann. Geofis. 39, 1041–1047.

Pantosti, D., and G. Valensise (1990). Faulting mechanism and complexityof the November 23, 1980, Campania-Lucania earthquake, inferredfrom surface observations, J. Geophys. Res. 95, 15319–15341.

Pasolini, C., D. Albarello, P. Gasperini, V. D’Amico, and B. Lolli (2008b).The attenuation of seismic intensity in Italy, Part II: Modeling andvalidation, Bull. Seismol. Soc. Am. 98,692–708.

Pasolini, C., P. Gasperini, D. Albarello, B. Lolli, and V. D’Amico (2008a).The attenuation of seismic intensity in Italy, Part I: Theoretical andempirical backgrounds, Bull. Seismol. Soc. Am. 98,682–691.

Pettenati, F., L. Sirovich, and F. Cavallini (1999). Objective treatment andsynthesis of macroseismic intensity data sets using tessellation, Bull.Seismol. Soc. Am. 89, 1203–1213.

Pino, N. A., B. Palombo, G. Ventura, B. Perniola, and G. Ferrari (2008).Waveform modeling of historical seismograms of the 1930 Irpiniaearthquake provides insight on “blind” faulting in southern Apennines(Italy), J. Geophys. Res. 113, no. B05303, doi 10.1029/2007JB005211.

Pondrelli, S., G. Ekström, A. Morelli, and S. Primerano (1999). Study ofsource geometry for tsunamigenic events of the Euro-Mediterraneanarea, in International Conference on Tsunamis, UNESCO Books,Paris, 297–307.

Pondrelli, S., G. Ekström, and A. Morelli (2001). Seismotectonic re-evaluation of the 1976 Friuli, Italy, seismic sequence, J. Seismol. 5,73–83.

Pondrelli, S., A. Morelli, G. Ekström, S. Mazza, E. Boschi, and A. M. Dzie-wonski (2002). European-Mediterranean Regional Centroid MomentTensors: 1997–2000, Phys. Earth Planet. In. 130, 71–101.

Pondrelli, S., A. Morelli, and G. Ekström (2004). European-MediterraneanRegional Centroid Moment Tensor catalog: Solutions for the years2001 and 2002, Phys. Earth Planet. In. 145,127–147.

Pondrelli, S., S. Salimbeni, G. Ekström, A. Morelli, P. Gasperini, andG. Vannucci (2006). The Italian CMT dataset from 1977 to the present:An extended view of the seismotectonics of this region, Phys. EarthPlanet. In. 159, 286–303, doi 10.1016/j.pepi.2006.07.008.

Reinecker, J., and W. A. Lenhardt (1999). Present-day stress field anddeformation in eastern Austria, Int. J. Earth Sci. 88, 532–550.


http://emidius.mi.ingv.it/CPTI04/presentazione.html






http://dx.doi.org/10.1111/j.1365-246X.2004.02320.x




http://dx.doi.org/10.1029/2007JB005211

http://dx.doi.org/10.1029/2007JB005211

http://dx.doi.org/10.1016/j.pepi.2006.07.008

Richter, C. (1935). An earthquake magnitude scale, Bull. SeismolSoc. Am.25, 1–32.

Richter, C. (1958). Elementary Seismology, W. H. Freeman, San Francisco,768 pp.

Rock, N. M. S. (1988). Numerical Geology, Springer-Verlag, Berlin, 427 pp.Sabetta, F., and A. Pugliese (1987). Attenuation of peak horizontal accel-

eration and velocity from Italian strong-motion records, Bull. Seismol-Soc. Am. 77, 1491–1513.

Sabetta, F., and A. Pugliese (1996). Estimation of response spectra andsimulation of nonstationary earthquake ground motions, Bull. Seismol-Soc. Am. 86, 337–352.

Seebach, K. V. (1973). Das Mittlesdeutsch Erdebeben von 6 Marz 1872Haessel, Leipzig, Germany, 132–175 (in German).

Sibol, M. S., G. A. Bollinger, and J. B. Birch (1987). Estimations ofmagnitudes in central and eastern North America using intensityand felt area, Bull. Seismol. Soc. Am. 77, 1635–1654.

Sieberg, A. (1931). Erdebeben, in Handbuch der Geophysik, B. Gutenberg(Editor), Vol. 4, Gebrüder Borntraeger, Berlin552–554 (in German).

Sleiko, D., G. Neri, I. Orozova, G. Renner, and M. Wyss (1999):Stress field in Friuli (NE Italy) from fault plane solutions ofactivity following the 1976 main shock, Bull. Seismol. Soc. Am. 89,1037–1052.

Sponheuer, W. (1960). Methoden zur Herdtiefenbestimmung in derMakroseismik, Freiberger Forschungsthefte C88, Akademie Verlag,Berlin, 117 pp.(in German).

Stucchi, M., R. Camassi, A. Rovida, M. Locati, E. Ercolani, C. Meletti,P. Migliavacca, F. Bernardini, and R. Azzaro (2007). DBMI04 ildatabase delle osservazioni macrosismiche dei terremoti italianiutilizzate per la compilazione del catalogo parametrico CPTI04,Quaderni di Geofisica 49, ISSN 1590-2595, 38 pp. (in Italian).

Toppozada, T. R. (1975). Earthquake magnitude as a function of intensitydata in California and western Nevada, Bull. Seismol. Soc. Am. 65,1223–1238.

Vannucci, G., and P. Gasperini (2003). A database of revised fault planesolutions for Italy and surrounding regions, Comput. Geosciences29, 903–909.

Vannucci, G., and P. Gasperini (2004). The new release of the database ofearthquake mechanisms of the Mediterranean area (EMMAVersion 2),Ann. Geophys. 47, 307–334.

Villaseñor, A., and E. R. Engdahl (2005). A digital hypocenter catalogfor the International Seismological Summary, Seism. Res. Lett. 76,554–559.

von Kövesligethy, R. (1907). Seismicher starkegrad und intensitat der beben,Gerland, Beitr. Geophys. 8, 363–36 (in German).

Wells, D. L., and K. J. Coppersmith (1994). New empirical relationshipsamong magnitude, rapture length, rupture width, rupture area, andsurface displacement, Bull. Seismol. Soc. Am. 84, 974–1002.

Wessel, P., andW. H. F. Smith (1998). New, improved version of the GenericMapping Tools Released, EOS Trans. AGU 79, 579.

Westaway, R. (1993). Quaternary uplift of southern Italy, J. Geophys. Res.97, 15437–15464.

Appendix 1

Old Method of Computing the Magnitudeand Corresponding Uncertainties

Gasperini et al. (1999) used a weighting scheme totake into account both the uncertainties associated withthe instrumental magnitudes and the number of intensity dataused to compute the average radii of the isoseismals. Usingsuch an approach, the natural weight, which is inverselyproportional to the square of the standard deviation of eachobservation, is corrected by a factor proportional to the num-

ber of intensity data. The procedure involves the use of acalibration phase to compute the coefficients of the empiricalregressions, as well as other information required to estimatethe uncertainties. For a reliable calibration, a minimum of20–30 earthquakes with known instrumental magnitudes isusually required.

Calibration

First, the average epicentral distances ARkl are computed

as the trimmed means of the epicentral distances of the siteswith given intensity (by considering only the distances be-tween the twentieth and eightieth percentiles), for each k-th isoseismal and l-th earthquake of the calibration set.Uncertain intensities (e.g., VII–VIII) are usually consideredas separate isoseismals. The epicentral intensities I0 of all theearthquakes are also computed using the algorithm describedby Gasperini and Ferrari (1995, 2000). The procedureexcludes those isoseismals computed using less than fourintensity data points and those with intensities not lower thanthe epicentral intensity. The Sibol et al. (1987) formula isfitted separately for each isoseismal by minimizing theweighted sum of the squares

WSSk �XNk

l�1

fMl � a�log10�πARkl ��2 � b��I0�l�2 � cg2wk

l ;

(A1.1)

where a, b, and c are empirical coefficients. The weights wkl

are directly proportional to the number Lkl of data points used

to compute the k-th isoseismal of the l-th earthquake and areinversely proportional to the square of the standard deviationσl of the instrumental magnitude of the l-th earthquake

wkl �

Lkl

Kk

��σk

σl

�2

: (A1.2)

Kk and �σk are normalization factors (for the number of datapoints and the magnitude of the standard deviation, respec-tively) that are computed as

Kk � 1

Nk

XNk

l�1

Lkl (A1.3)

and

�σk ��P

Nk

l�1 LklP

Nk

l�1

Lkl

�σl�2

vuut ; (A1.4)

whereNk is the total number of earthquakes in the calibrationset that contain data for the k-th isoseismal. The normaliza-tion factors are chosen such that the sum of the weightsequals the number of earthquakes:


XNk

l�1

wkl �

XNk

l�1

Lkl

Kk

��σk

σl

�2

� Nk: (A1.5)

The significance of coefficient b is tested against the H0

hypothesis that it is actually 0 using the Student’s t-test. Ifthe null hypothesis cannot be rejected at a significance levellower than 0.05, the coefficient b is assumed to be 0, and theregression coefficients a and c are recomputed by minimiz-ing a different weighted sum of squares:

WSSk �XNk

l�1

fMl � a�log10�πARkl ��2 � cg2wk

l : (A1.6)

At the end of the procedure, the coefficients for each isoseis-mal are saved, as well as the relevant (unbiased) standarddeviations of the regressions sk �

��WSSk=�Nk � 3�

pand

normalization factors Kk for use in the computations ofmagnitude.

Magnitude Computation

The average epicentral distances ARk for each k-th iso-seismal and epicentral intensity I0 are computed first. Then,the magnitudes from each isoseismal are computed as

Mk � a�log10�πARk��2 � b�I0�2 � c: (A1.7)

The magnitude of the earthquakes is computed as theweighted trimmed mean (discarding the highest and lowestestimates) of the magnitudes obtained from each isoseismal:

M �P

Lk�1k�2 MkwkPLk�1k�2 wk

; (A1.8)

where theMk values are arranged in numerical order and theweights are

wk � Lk

Kk�sk�2 ; (A1.9)

Lk being the number of data used to compute the k-th iso-seismal for the given earthquake. This scheme is chosen inorder that the average of weights over all of the earthquakesof the calibration set equals the inverse of the standarddeviation of the squared regression

1

Nk

XNk

l�1

�wk�l �1Nk

PNk

l�1�Lk�lKk�sk�2 � 1

�sk�2 : (A1.10)

Hence, each weight can be assumed to correspond tothe inverse of the square of the standard deviation of the mag-

nitude computed using the k-th isoseismal for the givenearthquake.

Computation of Uncertainties

Using equation (A1.10), the magnitude uncertainty forthe given earthquake is computed as the square root of theinverse of the sum of the weights of the isoseismals used inthe trimmed mean:

σi ��

1PLk�1k�2 wk

s; (A1.11)

where the values wk are arranged according to the numericalorder of the estimates of Mk. The value computed usingequation (A1.11) is referred to in the main text as type-I mag-nitude uncertainty.

The previous version of the Boxer code also computedthe weighted standard deviations of the single magnitudeestimates made using different isoseismals, which representsthe empirical dispersion of the sample of the estimates ofmagnitude and not of their mean. For a consistent estimateof uncertainty of the mean magnitude, one must divide by thesquare root of the number of isoseismals. In this study, wetherefore compute it as

σii ��P

Lk�1k�2 �M �Mk�2wk

�Lk � 3�PLk�1k�2 wk

s: (A1.12)

Such a value is referred to in the text as type-II magnitudeuncertainty.

Appendix 2

Instrumental Data Used for Comparisons withthe Macroseismic Estimates

Table A1 lists the 77 instrumental epicenters used for thecomparison with macroseismic estimates.

Table A2 consists of the 38 focal mechanisms compris-ing the data set used in determining the orientation of theseismogenic fault.

Appendix 3

Computation of Parameter Uncertainties

In the following section, we describe the computation ofthe standard (i.e., 1σ) uncertainties that, for a single param-eter, approximately correspond to the 68% confidence inter-val. For two-dimensional data (e.g., epicentral coordinates)the confidence bounds have an elliptical shape, while in threedimensions (coordinates and depth), they are ellipsoidalsurfaces. Standard (1σ) ellipses and ellipsoids correspond


Table A1Instrumental Hypocenters and Moment Tensor Centroids Used for Comparisons

Hypocenter Centroid

Date(dd/mm/yyyy) Time Latitude Longitude Depth Reference* Latitude Longitude Depth Reference*

13/01/1915 06:52:41 41.9755 13.6046 BAS91

23/07/1930 00:08:43 41.0700 15.3600 14.60 PIN0808/06/1934 03:16:59 46.3000 12.5000 ISS17/07/1937 17:11:05 41.7000 15.1667 BSING11/02/1939 11:16:54 44.0670 11.6500 BSING15/10/1939 14:05:00 44.2330 10.2030 27.00 EVA9324/07/1943 01:43:55 46.0000 11.9000 BSING29/06/1945 15:37:13 44.8000 9.2000 BSING05/09/1950 04:08:57 42.5167 13.3500 BSING08/08/1951 19:56:31 42.6000 13.5000 BSING12/05/1955 14:16:00 44.5000 7.3000 BSING27/08/1957 11:54:40 44.3000 10.9500 BSING26/04/1959 14:45:13 46.4670 13.0170 BSING19/02/1960 02:30:00 45.5200 10.5900 ISS29/10/1960 00:08:39 44.0000 11.3000 BSING23/01/1962 17:31:00 44.1600 12.8700 ISS31/10/1967 21:08:08 37.8400 14.6000 38.00 ISC30/12/1967 04:19:20 44.6300 12.0100 ISC15/01/1968 02:01:04 37.7800 13.0300 ISC18/06/1968 05:27:33 45.7300 7.9600 ISC17/04/1969 09:12:34 41.4000 13.6000 40.00 ISC02/07/1969 07:55:43 42.1833 12.0000 GAS7411/08/1969 13:55:09 43.1100 12.2100 ISC19/08/1970 12:19:55 43.2500 10.7700 ISC06/02/1971 18:09:08 42.3124 11.7550 ISC15/07/1971 01:33:23 44.7819 10.2918 ISC04/10/1971 16:43:33 42.8169 13.0580 ISC18/01/1972 23:26:12 44.2181 8.1713 ISC25/10/1972 21:56:11 44.5038 9.8651 76.00 ISC26/11/1972 16:03:08 42.9732 13.4004 ISC05/11/1973 08:40:47 41.6820 13.7630 PDE16/01/1975 00:09:48 38.1378 15.7111 21.40 ISC06/05/1976 20:00:13 46.2620 13.3000 5.71 SLE99 46.3600 13.2700 11.70 PON9915/09/1976 09:21:19 46.3000 13.1740 11.26 SLE99 46.3200 13.1700 12.00 PON0113/12/1976 05:24:00 45.8338 10.8301 ISC05/06/1977 13:59:22 37.8430 14.4645 11.30 ISC11/03/1978 19:20:48 38.0554 16.0733 26.10 ISC 38.1000 16.0300 33.00 HCMT19/09/1979 21:35:37 42.7300 12.9560 6.00 BSING 42.8100 13.0600 16.00 HCMT23/01/1980 21:21:06 36.7851 15.0765 ISC23/11/1980 18:34:52 40.7240 15.4140 12.00 WES93 40.9100 15.3700 10.00 PON0614/02/1981 17:27:46 41.0605 14.7940 CSI 41.0500 14.6000 10.00 PON0615/08/1982 15:09:52 40.8188 15.3952 28.50 ISC 40.8100 15.3600 10.00 PON0617/10/1982 06:45:37 43.1625 12.7142 5.94 CSI 43.1200 12.5900 13.00 PON0609/11/1983 16:29:51 44.6487 10.3665 28.13 CSI 44.6900 10.3200 37.00 HCMT29/04/1984 05:03:00 43.2082 12.5680 5.97 CSI 43.2700 12.5700 14.00 HCMT07/05/1984 17:49:42 41.7007 13.8628 20.50 CSI 41.7700 13.8900 10.00 HCMT23/01/1985 10:10:17 44.0642 10.4148 24.06 CSI23/07/1986 08:19:51 40.6422 15.6993 24.65 CSI06/12/1986 17:07:20 44.9475 11.4448 CSI02/05/1987 20:43:53 44.7940 10.6780 CSI 44.8200 10.7200 10.00 PON0624/05/1987 10:23:26 45.7065 10.7143 CSI05/07/1987 13:12:37 43.7587 12.2083 15.49 CSI 43.7800 12.2300 11.00 PON0608/01/1988 13:05:48 40.0287 15.9447 19.22 CSI 40.0800 16.0100 10.00 PON0601/02/1988 14:21:38 46.3595 13.0745 3.10 CSI13/09/1989 21:54:01 45.8363 11.1893 17.23 CSI 45.8000 11.2100 10.00 PON0611/02/1990 07:00:37 44.9435 7.6130 24.01 CSI05/05/1990 07:21:22 40.6400 15.8600 22.54 CSI 40.7500 15.8500 26.00 HCMT26/05/1991 12:26:01 40.6842 15.7918 CSI 40.7300 15.7700 8.00 PON06

(continued)


to 39% and 20% confidence limits, respectively. Larger con-fidence limits, corresponding to higher confidence levels,can also be computed. For example, the 90% confidence el-lipses plotted in Figures 2, 3, 6, and 7 correspond to 2:146σ,while in three dimensions, 90% confidence ellipsoids wouldcorrespond to 2:4477σ.

Hypocentral Location

For the old location method (0), epicentral uncertaintiesare computed as the standard deviations of the mean of the Ncoordinates xj used to compute the trimmed means:

σformal ��P

Nj�1�xj � �x�2N�N � 1�

s: (A3.1)

For the new location methods (1–6), a formal variance/cov-ariance matrix is computed as the inverse of the finite-differ-ence Hessian matrix of the log-likelihood function at itsmaximum (Guo and Ogata, 1997)

varHij � inv��Hij� � inv�� ∂2L�θ�∂θi∂θj

�; (A3.2)

where the function inv indicates the inverse matrix, L thelog-likelihood function, and θj the free parameters. Finite-difference partial derivatives are computed numerically inthe neighborhood of the parameter values that maximize L.

For the old and new location methods, a distribution-freeempirical variance/covariance matrix can be obtained frombootstrap replicate sets as

varBij �P

nk�1��θi � θki ��θj � θkj�

n; (A3.3)

where θki is the estimate of the i-th parameter for the k-thbootstrap replicate set and �θi is the average value of the i-th parameter computed from the estimates of the n bootstrapreplicate sets.

Parameter uncertainties are computed as the square rootsof the diagonal elements of the variance/covariance matrices(A3.2) and (A3.3), while the amplitudes and orientations ofthe semiaxes of the uncertainty ellipses and ellipsoids arecomputed as the eigenvalues and eigenvectors, respectively,of such matrices.

Depth Using the Musson (1996) Method

A formal uncertainty of the estimates of depthmade usingtheMusson (1996)method is computed numerically using thelog-likelihood function Hessian that, for the case of a singleparameter, corresponds to the inverse of the finite-differencesecond derivative of equation (16) with respect to h:

σformal �� 1

∂2L�h�∂h2

s: (A3.4)

Table A1 (Continued)Hypocenter Centroid

Date(dd/mm/yyyy) Time Latitude Longitude Depth Reference* Latitude Longitude Depth Reference*

05/06/1993 19:16:17 43.1413 12.6018 9.91 CSI 43.1200 12.6800 8.00 PON0620/04/1994 21:25:26 46.3395 12.5437 7.11 CSI 46.3000 12.5700 10.00 PON0624/08/1995 17:27:34 44.1392 10.7187 17.73 CSI 44.1300 10.7600 34.00 PON0630/09/1995 10:14:34 41.8135 15.9143 27.40 CSI 41.9000 15.9700 10.00 HCMT10/10/1995 06:54:23 44.1328 10.0183 8.23 CSI 44.1800 10.0100 10.00 PON0629/10/1995 13:00:26 45.6483 9.8762 23.86 CSI03/04/1996 13:04:36 40.6547 15.4418 11.43 CSI 40.7600 15.4900 10.00 PON0613/04/1996 13:00:23 46.3135 12.5713 7.15 CSI27/04/1996 00:38:27 39.4970 16.4748 20.59 CSI15/10/1996 09:56:02 44.7632 10.6048 CSI 44.7900 10.7800 10.00 HCMT19/03/1997 23:10:50 41.3920 14.6308 CSI 41.4000 14.6300 10.00 PON0212/05/1997 13:50:15 42.7632 12.5260 1.44 CSI26/09/1997 09:40:27 43.0297 12.8348 8.00 BAR00 43.0300 12.8500 10.00 PON0226/03/1998 16:26:17 43.1458 12.8090 44.82 CSI 43.1900 12.8400 40.00 PON0215/08/1998 05:18:09 42.3622 13.0562 2.91 CSI 42.4100 12.9800 10.00 PON0209/09/1998 11:28:00 40.0600 15.9490 29.21 CSI 40.0300 15.9800 10.00 HCMT07/07/1999 17:16:13 44.2928 10.8522 11.54 CSI 44.2900 10.9000 10.00 PON0229/12/1999 20:42:34 46.6108 10.2213 4.65 CSI 46.6000 10.3100 10.00 PON0231/10/2002 10:32:59 41.7167 14.8932 25.15 CSI 41.7900 14.8700 10.00 PON0419/07/1963 05:45:28 43.1500 8.0833 BSING06/09/2002 01:21:29 38.3813 13.6543 27.01 CSI

*BAR00, Barba and Basili (2000); BAS91, Basili and Valensise (1991); BSING, Bollettino Strumentale dell’Istituto Nazionale diGeofisica (see Data and Resources); CSI, Catalog of Italian Seismicity (CSI) Versione 1.1 (see Data and Resources); HCMT, GlobalCentroid Moment Tensor project catalog (see Data and Resources); GAS74, Gasparini (1974); ISC, International SeismologicalCentre On-line Bulletin (see Data and Resources); ISS, International Seismological Summary catalog (see Data and Resources);PIN08, Pino et al. (2008) ; PON99, Pondrelli et al. (1999); PON01, Pondrelli et al. (2001); PON02, Pondrelli et al. (2002); PON04,Pondrelli et al.(2004); PON06, Pondrelli et al. (2006); SLE99, Slejko et al. (1999); WES93, Westaway (1993);


Bootstrap uncertainties are computed from replicate sets, asthe standard deviations of the estimates of depth.

Magnitude

The formal uncertainties of the oldmagnitudemethod arecomputed in twoways: type I as the square root of the inverseof the sumof the inverses of the squared standard deviations ofregressions for each isoseismal, computed as a function of the

standard deviations of the magnitude–intensity regressionsand of the number of data used (equation A1.11 inAppendix 1), and type II as the weighted standard deviationsof the mean of the magnitude estimates made using differentisoseismals (equation A1.12 in Appendix 1). The firstapproach takes into account the uncertainties inherent in theinstrumental magnitudes and of the regressions and was pre-viously used for the compilation of Italian catalogs for seismichazard assessment (CPTIWorkingGroup, 1999, 2004), whilethe second represents only the empirical dispersion of themean of the estimates of magnitude.

For the alternative magnitude method, we compute thestandard deviation of the mean of the estimates of magnitudemade using equation (17) for each value of intensity:

σformal ��P

Nj�1�Mj � �M�2N�N � 1�

s: (A3.5)

We also compute the magnitude uncertainties for the old andthe alternative methods using the bootstrap approach, as thestandard deviations of the magnitudes computed from thereplicate sets.

It must be stressed that apart from the type-I uncertain-ties, all other estimators neglect the uncertainties due to theregressions with the instrumental magnitudes; hence, theydescribe only the reproducibility of the computational pro-cedure and not the accuracy of the macroseismic magnitudein reproducing the instrumental one. Such an accuracy wouldbe better represented using the standard deviation of the re-gression between IE and Mw (Table 1) or using the standarddeviation of the differences between the instrumental andmacroseismic magnitudes (Table 3), which are both of theorder of 0.3 units in most cases.

Fault Orientation

Gasperini et al. (1999) computed the formal uncertaintyof the orientations as a function of the concentration para-meter k of the von Mises distribution and of the mean resul-tant lengthQ of the data vectors (equation 19) by the formula(not reported explicitly by Gasperini et al., 1999)

σformal ��1

NkQ

s; (A3.6)

where N is the number of data points used. We recently rea-lized that this formula is missing a multiplication factor of1=2, which is required to convert back from circular to axialdata (see the discussion of the method in the Estimation ofFault Orientation section). Furthermore, because equa-tion (A3.6) requires the computation of k using an approx-imate formula (Cheeney, 1983; also reported in Appendix 3of Gasperini et al., 1999), we moved to the use of a differentapproach described by Fisher (1993, p. 34) that defines thecircular standard deviation of the sample as

Table A2Strikes of the Conjugate Fault Planes Used in the Comparisons

with the Macroseismic Orientations

Date(dd/mm/yyyy) Time Mw Strike 1 Strike 2 Reference*

13/01/1915 06:52:00 7.04 241.0 138.0 BAS9127/03/1928 08:32:00 5.71 202.5 112.5 REI9923/07/1930 00:08:43 6.65 344.8 108.3 JIM9108/06/1934 03:16:59 5.07 205.0 115.0 GRU9218/10/1936 03:10:12 5.81 201.7 294.0 GRU9211/02/1939 11:16:54 5.16 315.5 166.3 GAS8511/05/1947 06:32:17 5.71 247.4 359.8 CHI9405/09/1950 04:08:57 5.68 207.0 36.1 GAS8515/05/1951 22:54:00 5.24 125.0 224.0 EVA9308/08/1951 19:56:31 5.30 42.4 308.0 GAS8505/11/1956 19:45:48 5.16 184.0 94.0 REI9926/04/1959 14:45:13 5.23 210.0 305.0 REI9919/02/1960 02:30:00 4.91 84.5 350.5 EVA9331/10/1967 21:08:08 5.24 8.8 273.7 GAS8530/12/1967 04:19:20 5.42 134.6 322.1 GAS8518/06/1968 05:27:33 5.14 240.0 149.8 DEL0419/08/1970 12:19:55 5.09 35.3 291.4 GAS8515/05/1971 01:33:23 5.38 250.0 346.0 AND8725/10/1972 21:56:11 5.11 0.5 262.7 EVA9311/01/1975 15:54:37 4.45 17.2 252.5 EVA9316/01/1975 00:09:48 4.80 63.2 330.3 GAS8519/06/1975 10:11:14 5.03 14.8 281.1 GAS8506/05/1976 20:00:13 6.46 282.3 70.6 PON9915/09/1976 09:21:19 5.98 271.7 63.5 PON0119/09/1979 21:35:37 5.86 183.1 341.3 HCMT23/11/1980 18:34:52 6.89 135.3 302.6 PON0614/02/1981 17:27:46 4.90 254.4 1.0 PON0605/07/1987 13:12:37 4.47 297.7 107.6 PON0608/01/1988 13:05:48 4.73 147.9 323.5 PON0605/05/1990 07:21:22 5.80 184.0 90.1 HCMT30/09/1995 10:14:34 5.18 196.7 53.4 HCMT10/10/1995 06:54:23 4.85 89.3 180.0 PON0603/04/1996 13:04:36 4.93 123.0 325.2 PON0615/10/1996 09:56:02 5.41 216.8 94.2 HCMT26/09/1997 09:40:27 6.01 144.5 312.0 PON0226/03/1998 16:26:17 5.29 204.0 307.7 PON0209/09/1998 11:28:00 5.64 139.0 310.6 HCMT31/10/2002 10:32:59 5.74 174.0 267.4 PON04

*AND87, Anderson and Jackson (1987); BAS91, Basili and Valensise(1991); CHI94, Chiodo et al. (1994); DEL04, Delacou et al. (2004);EVA93, Eva and Pastore (1993); EVA98, Eva and Solarino (1998);HCMT, Global Centroid Moment Tensors project catalog (see Data andResources); JIM91, Jiménez (1991); GAS85, Gasparini et al. (1985);GRU92, Grünthal and Stromeyer (1992); PON99, Pondrelli et al. (1999);PON01, Pondrelli et al. (2001); PON02, Pondrelli et al. (2002); PON04,Pondrelli et al. (2004); PON06, Pondrelli et al. (2006); REI99,Reinecker and Lenhardt (1999).


σsample ��1 � ρ2�2Q2

s; (A3.7)

where ρ2 is the second central trigonometric moment ofcircular data distribution, which, in our weighted approach,may be computed as

ρ2 �XNi�1

wi cos 2�2αi � 2 �α��XN

i�1

wi; (A3.8)

where αi are the orientations, �α their axial mean, and wi theweights assigned to each data point (equation 21). Fromequation (A3.7), the uncertainty of the mean axial orientation(in the range from 0° to 180°) is computed as

σformal �1

2

��1 � ρ2�2NQ2

s: (A3.9)

We verified that the old (A3.5) (when corrected for the factor1=2) and the new (A3.9) formulations give consistent valuesin most cases.

Figure A1. Percentages of location uncertainties within given ranges (in km) computed (a_c) using formal methods and (b_d) using thebootstrap approach. Percentages of distances (in km) (e) between macroseismic and instrumental epicenters and (f) between epicenterscomputed using method 0 (barycenter) and other methods.


We also computed the uncertainty of the orienta-tion using the bootstrap approach as the axial standarddeviation of the fault orientations computed from the repli-cate sets:

σbootstrap �1

2

��1 � 1

M

PMi�1 cos 2�2θi � 2�θ��

2P2

s; (A3.10)

where θi are the fault orientations computed from theM replicate sets, �θ is the axial mean of the replicatefault orientations, and P is the mean resultant length of re-plicate mean orientation vectors. Because of the insignifi-cance of the mean orientation for a uniform distribution,we discarded from the computations those replicate setsfor which both the Rayleigh and Kuiper tests do not rejectthe uniformity hypothesis. It is possible that such a strategycould somehow reduce the statistical rigor of the bootstrapapproach.

Statistics on Parameter Uncertainties

We herein show the frequency distributions of the un-certainties estimated both using formal methods and thebootstrap approach for the full data set of the 329 earth-quakes with reliable macroseismic locations in the Italiandata set. In order to appreciate how well such uncertaintiesrepresent the accuracy of reproducing instrumental data, wealso report the distributions of the absolute distances betweenthe instrumental and macroseismic estimates of various pa-rameters, using the same display format in order to facilitatethe comparison.

It may be noted that, for the new location methods (1–6),the frequency distributions of the uncertainties of latitude(Fig. A1a,b) and longitude (Fig. A1c,d) are very similar. Areasonable agreement between the uncertainties estimatedusing the formal Hessian (Fig. A1a,c) and bootstrap(Fig. A1b,d) approaches may also be seen. Uncertainties ofless than 5 km are computed in about 80% of cases usingthe Hessian approach and in 70% of cases using the bootstrapapproach, while uncertainties greater than 20 kmmake up lessthan 10% using the Hessian and around 15% using the boot-strap approaches, respectively. This means that, for epicentralcoordinates, the formal uncertainties (whose computation isclearly faster than the bootstrap ones) are reasonably accurateand can be confidently adopted in most cases. Figure A1a,b,c,d also shows that the formal and bootstrap uncertainties aregenerally consistent, even for method 0. This implies that,despite the consideration of Gasperini et al. (1999), the stan-dard deviation of the trimmed mean of the coordinates of thesites used is not greatly influenced by the actual density oflocalities in the epicentral area and represents a reasonablyunbiased estimate of epicentral uncertainty.

From the comparison with the distribution of distancesbetween the instrumental and macroseismic epicenters(Fig. A1e) and with the distribution of the distances betweenepicenters computed using method 0 (barycenter) and other

methods (Fig. A1f), we deduce that the epicentral uncertain-ties represent the relative accuracy of the macroseismic meth-ods rather well, while clearly underestimating the accuracyof macroseismic methods in reproducing the instrumentalepicenters. This may easily be explained by the differentphysical meanings of the macroseismic and instrumentalepicenters.

Figure A2 shows the frequency distribution of the uncer-tainties of depth for our new location methods (2, 4, 5, 6) and

Figure A2. Percentages of depth uncertainties within givenranges (in km) for methods 0 (Musson, 1996), 2, 4, 5, and 6, com-puted (a) using formal methods and (b) using the bootstrap method.(c) Percentages of absolute differences obtained between the macro-seismic and instrumental depths.


for theMusson (1996)method (0). Even in this case, we note agood agreement between the formal (Fig. A2a) and bootstrap(Fig.A2b) uncertainties, whichmight permit the assertion thatthe faster formal approach affords reasonably consistentestimates of all the hypocentral parameters. In addition, theuncertainties in the simpler methods are smaller, particularlyfor method 2 (less than 1 km for more than 90% of earth-quakes), which only computes the coordinates and depth.It must be stressed that uncertainties in the depth should beconsidered with great care because they only represent thestatistical uncertainties of the location procedure and donot reflect the epistemic uncertainty related to the assumptionsmade. In the case of location method 2, we make some addi-tional assumptions with respect to other methods, namelythat (1) the intensity expected at the epicenter IE is givenby equation (5) and (2) the attenuation coefficients are thosecomputed according to Pasolini, Albarello,et al. (2008) for thewhole data set. These assumptions significantly constrain thehypocentral solution, thereby reducing the statistical uncer-tainty but increasing the epistemic uncertainty in terms of theirpossible inappropriateness for the given earthquake. Thecomparison with the distribution of the absolute differencesbetween the instrumental andmacroseismic depths (Fig. A2c)

indicates that the uncertainties largely underestimate theaccuracy in methods 0, 2, and 4, while the underestimationis lower for methods 5 and 6, which make fewer assumptionsabout the parameters of the attenuation equation.

In Figure A3, we compare the uncertainties determinedfor the old magnitude method, using the three differentapproaches described in the Magnitude section of this appen-dix. As expected, the bootstrap uncertainties (Fig. A3b) arelarger than the type-II formal ones (Fig. A3a) computedusing the weighted standard deviations of the magnitudeestimates made using different isoseismals because of thevariability of epicenters among the bootstrap sets. The dis-tributions of the formal uncertainties are rather similar for thedifferent location methods and are lower than 0.1 magnitudeunits for the great majority of earthquakes, while the boot-strap uncertainties are more variable and particularly largefor location method 0. This could be explained by the highersensitivity of such methods to the possible absence from thebootstrap paradata set of some or even all of the sites in thevicinity of the epicenter.

At the same time, it may be seen that the type-I uncer-tainties (Fig. A3c), which were computed using the combi-nation of the standard deviations of each isoseismal obtained

Figure A3. Percentages of magnitude uncertainties within given ranges (in magnitude units) using the method of Gasperini et al. (1999):(a) computed as the weighted standard deviation of the estimates made using different isoseismals, (c) as the square root of the inverse of thesum of the inverses of the squared uncertainties for each isoseismal and (b) by the bootstrap method. (d) Percentages of absolute differencesbetween macroseismic (old method) and instrumental magnitudes.


by regression, are almost independent of the locationmethod used because they depend only on the isoseismalsused and are clearly larger than the type-II and bootstrap onesbecause they also reflect the uncertainties of instrumentalmagnitudes.

A comparison of the distribution of the absolute differ-ences obtained between the instrumental and macroseismicmagnitudes calculated using the old method (Fig. A3d) in-dicates that the type-I uncertainties (Fig. A3c) are reasonablyrepresentative of the accuracy of the instrumental magni-tudes, while the type-II and bootstrap ones (Fig. A3a,b)largely underestimate that accuracy.

In Figure A4a and Figure A4b, for the alternative mag-nitude method, we show the frequency distributions of theformal standard deviation of the mean of the estimates fromsingle intensity observations and the result of the bootstrapapproach, respectively. As expected (see Estimation ofParameter Uncertainties section), the bootstrap uncertaintiesare generally higher than the standard deviations, but bothlargely underestimate the absolute differences between theinstrumental and macroseismic magnitudes (Fig. A4c). Infact, as for the type-II and bootstrap uncertainties of theold magnitude method, such estimates do not take intoaccount the uncertainty related to the empirical regressionin equation (8).

Figure A4. Percentages of magnitude uncertainties within gi-ven ranges (in magnitude units) for the alternative method imple-mented in this work according to Bakun and Wentworth (1997),computed (a) as the standard deviation of the estimates obtainedfrom single observations and (b) using the bootstrap method. (c) Per-centages of absolute differences obtained between macroseismic(alternative method) and instrumental magnitudes.

Figure A5. Percentages of orientation uncertainties within gi-ven ranges (in degrees) computed (a) as the standard deviation ofthe axial mean and (b) using the bootstrap method. (c) Percentagesof the absolute differences between macroseismic and instrumentalfault orientations.


Figure A5a and Figure A5b show the comparison be-tween the axial standard deviations of the mean orientationsand the results of bootstrap simulations, respectively. As inthe case of the magnitudes, the bootstrap uncertainties over-estimate the formal ones because, in our implementation, theresampling of the intensity data also implies a recalculationof the epicentral distances and orientations. It may also beseen that, for method 0 (i.e., the one that yields the betterfit with the instrumental data, see Testing the Accuracy ofEstimated Parameters section) the formal uncertainties(Fig. A5a) reproduce the distribution of absolute differencesbetween instrumental and macroseismic orientations(Fig. A5c) rather better than the bootstrap ones do (Fig. A5b).For methods 1–6, neither of the two approaches seems to beconsistent with the observed differences. This also confirmsthat location method 0 is the most appropriate for estimatingthe orientation of the seismogenic fault.

Dipartimento di FisicaUniversità di BolognaViale Berti Pichat, 8I-40127 Bologna, [email protected]@unibo.it.

(P.G., E.B.)

Istituto Nazionale di Geofisica e VulcanologiaSezione di BolognaVia Donato Creti, 12I-40128 Bologna, [email protected]@bo.ingv.it

(G.V., D.T.)

Manuscript received 12 October 2009


Gasperini Et Al (2010) - BSSA Paper

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