Journal of Seismology (2006) 10: 225236DOI: 10.1007/s10950-006-9012-4 C Springer 2006

Empirical global relations converting MS and mb to moment magnitude

E.M. ScordilisDepartment of Geophysics, School of Geology, Aristotle University, Thessaloniki 54124, Greece,e-mail: manolis@geo.auth.gr

Received 28 December 2005; accepted in revised form 12 January 2006

Key words: magnitude scales, moment magnitude, global empirical relations, homogeneous catalogs

Abstract

The existence of several magnitude scales used by seismological centers all over the world and the compilation ofearthquake catalogs by many authors have rendered globally valid relations connecting magnitude scales a necessity.This would allow the creation of a homogeneous global earthquake catalog, a useful tool for earthquake research.Of special interest is the definition of global relations converting different magnitude scales to the most reliableand useful scale of magnitude, the moment magnitude, MW. In order to accomplish this, a very large sample ofdata from international seismological sources (ISC, NEIC, HRVD, etc.) has been collected and processed. Themagnitude scales tested against MW are the surface wave magnitude, MS, the body wave magnitude, mb, and thelocal magnitude, ML. The moment magnitudes adopted have been taken from the CMT solutions of HRVD andUSGS. The data set used in this study contains 20,407 earthquakes, which occurred all over the world during thetime period 1.1.197631.5.2003, for which moment magnitudes are available. It is shown that well-defined relationshold between MW and mb and MS and that these relations can be reliably used for compiling homogeneous, withrespect to magnitude, earthquake catalogs.

Introduction

One of the most important parameters characterizingan earthquake is its size, which is a measure di-rectly related to the energy released. Since the firstwork of Richter (1935) when the local magnitudescale, ML, was initially defined using trace ampli-tudes of local earthquakes recorded on typical WoodAnderson seismographs (T0 = 0.8 s, critical damping0.8, V = 2, 800), the earthquake magnitude becamethe most common measure of the size of an earthquake.Its linear relation with the logarithm of physical quanti-ties characterizing the earthquake (seismic energy, seis-mic moment) turned it into a tool suitable for solvingseveral important problems of practical and theoreti-cal interest. In the course of time, new seismographswere constructed and different wave types, recordedat various distances, were used for magnitude estima-tion, which resulted in the definition of new magnitudescales.

Thus, Gutenberg (1945a) defined the surface wavemagnitude scale, MS, using the ground amplitudes ofsurface waves with period 1723 s measured at epi-central distances 15130. This magnitude could beestimated using the formula:

MS = log A + 1.656 log + 1.818 (1)

where A is the ground amplitude in m and theepicentral distance in degrees.

Gutenberg (1945b,c) and Gutenberg and Richter(1956) introduced the body wave magnitude scalebased on the recordings of P-waves with periods upto about 10 s by medium to long period instruments.It was denoted as mB and was originally determinedfrom the ratio of amplitude to period for P or S wavesaccording to the relation:

mB = log(

AT

)

+ q(, h) (2)

226

where A is the maximum amplitude observed, Tits respective period and q(, h) is a calibrationfunction, given in tables for shallow earthquakes(Gutenberg, 1945b) and in charts for all depths of earth-quake foci (Gutenberg, 1945c; Gutenberg and Richter,1956).

The unified magnitude, mb, included in the ISC andNEIC bulletins is estimated using the recordings of thefirst 5 s of short period (T 3 s) P waves by shortperiod instruments, following the procedure proposedby Gutenberg and Richter (1956), by applying the for-mula:

mb =n

i=1[

log( Ai

Ti

) + Q(i , hi )]

n 3 (3)

where Q(i , h) is the depthdistance factor, n is thenumber of stations (recordings) used and Ai and Ti arethe amplitude of the ith station in nm and its respec-tive period in s. This definition (the use of differentrecordings) resulted in differences between mB and mbscales that, in some cases (i.e. earthquakes produced bylarge faults or earthquakes with complicated ruptureprocess), can be remarkable (Abe, 1981; Kanamori,1983). Trying to quantify these deviations, Abe (1981)proposed the following relation:

mB = 1.5mb 2.2 (4)

connecting mB with mb estimated by ISC.The MS magnitudes reported in ISC and NEIC

bulletins are estimated using amplitudes and re-spective periods of Rayleigh waves with peri-ods ranging between 10 and 60 s at epicentraldistances 20160, applying the Prague formula(Vanek et al., 1962):

MS = log(

AT

)

max

+ 1.66 log + 3.3 (5)

where A is the maximum ground amplitude, in m,observed on horizontal components, T its respectiveperiod and the epicentral distance in degrees. Thefocal depths of the earthquakes for which the MS isestimated must not exceed 60 km.

Gutenberg and Richter (1956) defined the followingrelations connecting the ML, mB and MS magnitudescales:

mB = 0.63MS + 2.5 (6)MS = 1.27(ML 1) 0.016M2L (7)

The relation between mb magnitudes (published byISC) and MS was studied by Karnik (1973). He usedearthquakes with mb,ISC 4.5 but with MS 6.5,to avoid the saturation effect, finally suggesting therelation:

mb,ISC = 0.46MS + 2.74 (8)

Recent works (i.e., Murphy et al., 2001; Murphyand Barker, 2003) deal with the reliability of mb esti-mated by ISC and/or NEIC. Murphy and Barker (2003)re-estimated the body wave magnitudes for a largenumber of earthquakes recorded by stations of the In-ternational Monitoring System (IMS) at epicentral dis-tances ranging from 23 up to 180 using short periodrecordings and the corrections for epicentral distanceand depth proposed by Veith and Clawson (1972). Theyalso found that the new magnitudes estimated deviatedsignificantly from the mb magnitudes of ISC and NEIC.

The main problem of all the above magnitude scalesis that they do not behave uniformly for all magnituderanges. Another problem is that the ML, MS and mbscales exhibit saturation effects at different levels forlarge earthquakes. Both these limitations could resultin under- or over-estimation of earthquake magnitudes.These limitations led Kanamori (1977) and Hanks andKanamori (1979) to propose a new magnitude scale,namely moment magnitude, MW, defined by:

MW = 23 log M0 10.7 (9)

where M0 is the seismic moment in dyn.cm. From atheoretical point of view, this scale is reasonably re-liable since it is controlled by the fault size and thedislocation. The fact that seismic moment estimationis based on spectral amplitudes ensures the robustnessof the MW estimation. MW does not saturate, since it isdirectly proportional to the logarithm of seismic mo-ment, resulting in a uniform behavior for all magni-tude ranges. For these reasons, MW is considered asthe most reliable magnitude accurately describing thesize of earthquakes. However, since it was initiallydefined for earthquakes of magnitudes MS 7.5 itis of great interest to examine its behavior for weakearthquakes. Recent works revealed possible limita-tions in the seismic moment magnitude estimation. Forexample, Patton and Randal (2002) pointed out thatfor earthquakes of central Asia the seismic moments,M0, included in the Centroid Moment Tensor (CMT)catalog of Harvard Seismology (2004), HRVD, exhibit

227

Figure 1. Spatial distribution of 20,407 earthquakes globally, for which MW values in the range 3.18.4 are available.

remarkable deviations from the M0 estimated from re-gional surface waves.

Since the MW scale was first introduced, manystudies have been carried out in different regions andseismotectonic environments to establish relations con-necting other magnitude scales to MW (Heaton et al.,1986; Johnston, 1996; Shedlock, 1999; Papazachoset al., 2002; among many others). The most importantreason for this was to compile earthquake catalogs withall magnitudes expressed in one common scale (MW) tosolve important practical problems (i.e., seismic hazardassessment), as well as theoretical ones (calculation ofcrustal deformation, etc.).

In this study, an attempt is made to use a verylarge sample of data (much larger and extended tobroader magnitude ranges than previous works) in or-der to define new empirical relations connecting mb,MS and ML magnitudes with MW. In particular, the mband MS magnitudes, included in the catalogs of theInternational Seismological Centre (2004), ISC, andNational Earthquake Information Center (2004), NEICas well as MS magnitudes included in a broadly usedEuropean earthquake catalogue (Karnik, 1996), arecorrelated with MW values reported in the CMT catalogof Harvard Seismology (2004), HRVD (Dziewonskiet al., 1981 and subsequent papers appeared quarterlyon Phys. Earth Plan. Int.) and in the United States Ge-ological Survey Source Parameter Database (2004),USGS SOPAR, catalog.

The data

To perform this study it was necessary to createan earthquake catalog with information on all earth-quakes, for which magnitudes expressed in the scalesunder examination, estimated by several agencies, wereavailable, and which occurred during the last fewdecades.

As a reference magnitude we used the moment mag-nitude estimated by HRVD (CMT solutions 20,196events from 1976 up to the end of May, 2003). Momentmagnitudes for 212 additional events were taken fromUSGS. Therefore, the total number of earthquakes withestimated MW, ranging between 3.1 and 8.4, reached20,407. Their spatial distribution is shown in the mapof Figure 1.

To check the consistency of MW given by HRVDwith MW given by USGS Figure 2 shows the relation be-tween these two moment magnitudes. 3,756 events withmoment magnitudes reported by both sources (time pe-riod 1.1.198031.5.2003) were used. The relation de-rived is:

MW,HRVD = 1.00( 0.01)MW,USGS+ 0.04( 0.09),5.0 MW,USGS 8.2,

R2 = 0.95, = 0.11, n = 3, 756 (10)

This result shows that the differences across thewhole magnitude range are negligible (less than 0.05),

228

Figure 2. Correlation between MW given by HRVD and by USGS; 3,756 events. The dashed line is the bisector and the straight line is the bestfit. The same line symbols apply to Figures 37.

allowing to practically consider them as equivalent.Anyhow, moment magnitudes from USGS were usedonly for less than 1% of the earthquakes.

To examine the behavior of the body wave magni-tude scale, mb, 284,157 values from ISC and 229,375from NEIC (for earthquakes occurred during the pe-riod 1.1.196531.5.2003) have been collected whilefor the surface wave magnitude scale, MS, the re-spective numbers are 56,184 from ISC (time period1.1.197831.5.2003) and 32,464 from NEIC (time pe-riod 16.5.196831.5.2003).

Surface wave magnitude scale (MS)

MS magnitudes reported in the bulletins of ISC andNEIC are all estimated using the Prague formula ex-pressed by Equation (5) (Utsu, 2002). Comparison ofrelations (1) and (5) shows that they are slightly differ-ent. Utsu (2002) noticed that for T = 20 s Equation (5)gives MS values larger by about 0.2 than Equation (1).The MS overestimation by the Prague formula com-pared to relation (1) has also been observed by several

other authors (Nuttli and Kim, 1975; Thomas et al.,1978; Christoskov et al., 1985; Panza et al., 1989; Herakand Herak, 1993; Rezapour and Pearce, 1998 amongothers). However, this bias is compensated by the bene-fit of Equation (5) of using seismic waves with periodsbetween 10 and 60 s recorded at epicentral distances20160, significantly increasing the number of earth-quakes for which MS estimation is possible.

Since both ISC and NEIC estimate MS using thesame technique, it is expected that the magnitudesshould be more or less equivalent. To verify this as-sumption we have plotted MS given by NEIC versusMS estimated by ISC for events in 19782003. The re-lations that express the best-fit lines in the least squaressense are:

MS,NEIC = 0.99( 0.003)MS,ISC + 0.05( 0.02),h < 70 km,

R2 = 0.95, = 0.16, n = 25,960 (11)MS,NEIC = 0.98( 0.05)MS,ISC + 0.07( 0.24),

70 km h 640 km,R2 = 0.96, = 0.17, n = 65 (12)

229

Figure 3. Correlation between MS values given by NEIC and by ISC for shallow earthquakes (h < 70 km); 25,960 events. Similar results areobtained for 65 events in the depth range 70640 km and for all 26,025 events taken together.

MS,NEIC = 0.99( 0.003)MS,ISC + 0.05( 0.02),all h,

R2 = 0.95, = 0.16, n = 26,025 (13)

where h is the focal depth. Equation (11) is shown inFigure 3. It is obvious that throughout a wide range(2.6 MS,ISC 8.3), MS estimated by ISC andby NEIC are practically identical irrespective of focaldepth, allowing their consideration as a unified data set.

The distribution of MW versus MS for shallow earth-quakes, h < 70 km, is given in Figure 4 (13,591 pointsfrom ISC and 12,714 points from NEIC). Bubbles withsize related to the number of points give a clearer pic-ture of the distribution. These plots exhibit a bilin-ear correlation between MW and MS expressed by theequations:

MW = 0.67( 0.005)MS + 2.07( 0.03),3.0 MS 6.1,

R2 = 0.77, = 0.17, n = 23,921 (14)

MW = 0.99( 0.02)MS + 0.08( 0.13),6.2 MS 8.2,

R2 = 0.81, = 0.20, n = 2,382 (15)

For MS < 4.0 the data are rather poor (Figure 4). How-ever, the relation can give, at least, indicative results forearthquakes of that range of magnitudes.

Karnik (1968, 1971, 1973, 1996) made a signifi-cant attempt to compile an accurate, homogeneous, andcomplete catalog of earthquakes that occurred duringthe last two centuries in Europe. In his latest catalog(Karnik, 1996) earthquakes that occurred from 1800 to1990 in Europe and surrounding areas are included. Theconverted, or re-estimated, magnitudes are in an MSscale consistent with the Prague formula (Vanek et al.,1962). Since this catalog is widely used for earthquakesin Europe, it is of interest to see how its magnitudes arerelated to MW. The available sample of earthquakes forwhich both MW and Karnik magnitude, MSK, are avail-able is rather small (about 280 shocks) and does notinclude earthquakes with MW 4.8. For this reason,

230

Figure 4. Relation between MW and MS for shallow earthquakes; 26,305 points. The bubble size corresponds to the number of values. The samesymbols apply to Figures 5 and 7. A break at MS = 6.2 is obvious.

MSK was compared with MS (from ISC and NEIC)providing a larger sample, 2,149 events, and coveringa wider magnitude range, 2.9 MS 8.0. Figure 5shows that for a broad range of magnitudes MSK and MSare almost identical. Particularly, for the larger earth-quakes the relation is:

MS = 1.05( 0.05)MSK 0.41( 0.31),5.4 MSK 8.1,

R2 = 0.82, = 0.27, n = 266 (16)

and for the smaller earthquakes the relation is:

MS = 1.19( 0.06)MSK 1.14( 0.26),4.0 MSK 5.3,

R2 = 0.53, = 0.37, n = 1,730 (17)

The scatter of points for earthquakes with MSK < 5.4shows loose (indicative) correlation between the twomagnitude scales. The data are not enough to extendthe relation for MSK < 4.0.

Comparing the formulae (14), (15), (16), (17) wecan extract new relations connecting MSK to MW. Theserelations are:

MW = 0.80MSK + 1.31, 4.0 MSK 5.3, = 0.41 (18)

MW = 0.70MSK + 1.80, 5.4 MSK 6.2, = 0.29 (19)

MW = 1.04MSK 0.33, 6.3 MSK 8.1, = 0.31 (20)

Body wave magnitude scale (mb)

mb is one of the most widely used magnitude scales. Inthe mB scale definition in Gutenbergs original work(Gutenberg, 1945a,b), intermediate period displace-ment sensors were used giving peak amplitudes in the612 s period range while a linear attenuation modelwas adopted. In the present study the magnitudes cal-ibrated are in the mb scale, as they were reportedby ISC and/or NEIC. These centers estimate the mb

231

Figure 5. Correlation between MS (from ISC and NEIC) and MSK (from Karnik, 1996) for earthquakes which occurred in the broader area ofEurope between 1965 and 1990; 2,149 events.

magnitudes of the earthquakes which occurred sincethe early 60s using the first 5 s of P-waves recorded onshort period instruments.

To check how the mb reported by ISC is cor-related with the mb reported by NEIC, 215,163earthquakes which occurred globally between 1.1.1965and 31.5.2003 with mb magnitudes ranging from 2.5 upto 7.3 were used. The diagram of Figure 6 shows thevariation of mb ISC versus mb NEIC (least-squaresfit). The relation is:

mb,ISC = 1.02( 0.003)mb,NEIC 0.18( 0.01),2.5 mb,NEIC 7.3,

R2 = 0.99, = 0.20, n = 215, 163 (21)

This relation indicates that the mb magnitudes given byISC and NEIC are, practically, equivalent. The slightbias between them has been also observed by otherresearchers (e.g., Utsu, 2002).

Considering mb given from ISC and NEIC as a uni-fied magnitude scale it is of great interest to examineits behavior against MW. For this reason a data set con-

sisting of 20,870 earthquakes with both mb (from ISCand/or NEIC) and MW values available (40,580 pairs)was prepared, covering the time period 19652003.

The plot of MW against mb (Figure 7) clearly showsthat mb values are consistently lower than those of MW,as has been shown in several previous studies (e.g.,Nuttli, 1983, 1985; Giardini, 1984; Kiratzi et al., 1985;Heaton et al., 1986; Patton and Walter, 1993, 1994;Johnston, 1996; Papazachos et al., 1997 among others).The data show an approximate linear distribution up toan mb value of about 6.2 which is expressed by therelation:

MW = 0.85( 0.04)mb + 1.03( 0.23),3.5 mb 6.2,

R2 = 0.53 = 0.29, n = 39, 784 (22)

For mb < 4.5 the data are rather poor (Figure 7). How-ever, relation (22) can give, at least, indicative re-sults for earthquakes of that range of magnitudes.For mb > 6.2 (approximately) the relation increasesits slope showing an unstable behavior that could be

232

Figure 6. Correlation of mb from ISC and from NEIC for earthquakes which occurred globally from 1965 up to the end of May, 2003 with mbmagnitudes ranging from 2.5 to 7.3; 215,163 events.

considered as saturation. This is expected because themb magnitudes are estimated from the amplitudes ofthe first 5 s of short period recordings. Consequently,in many cases of strong earthquakes the maximum am-plitudes occur later, a fact that leads to underestimationof magnitudes.

Local magnitude scale (ML)

Several authors have defined relations between MW andML using data from earthquakes occurring in differentregions of the world (e.g., Kim et al., 1989; Uhrhammeret al., 1996; Papazachos et al., 1997, 2002; Wahlstromand Grunthal, 2000; Grunthal and Wahlstrom, 2003among others). These relations show a linear, bilinearor quadratic connection between these two magnitudescales.

However, there is confusion regarding the magni-fication of the Wood Anderson seismographs whichwere used to estimate the original ML magnitudes. Ac-cording to Richter (1935), the typical Wood Ander-son (WA) seismograph used in the definition of theML scale had a magnification of 2,800 (Anderson and

Wood, 1924, 1925). More recent studies by Uhrham-mer and Collins (1990) and Uhrhammer et al. (1996)have pointed out that the effective magnification of thetypical WA seismograph is around 2,080, leading tosystematic errors in ML estimations. The magnifica-tion is often different for different WA instruments.For instance, Papazachos et al. (1997) and Margarisand Papazachos (1999) showed that the Wood Ander-son seismograph, still operating at the National Ob-servatory of Athens, Greece, has an even lower mag-nification (1,000) resulting in systematic underesti-mation of ML, which has systematically affected localmagnitude estimations for the southern Balkan area.Moreover, equivalent ML magnitudes are also cal-culated by using recordings of several short-period in-struments (i.e., Kiratzi, 1984; Kiratzi and Papazachos,1984; Scordilis, 1985; Papanastasiou, 1989; Uhrham-mer and Collins, 1990; Uhrhammer et al., 1996) cal-ibrated against (possibly incongruous magnification)Wood Anderson seismographs.

As a result of this confusion, the ML magni-tudes reported by several seismological stations cannotbe considered as equivalent and, therefore, regional

233

Figure 7. Correlation between MW and mb (from ISC and NEIC) for earthquakes which globally occurred since 1965 up to the end of May,2003; 40,580 points.

relations connecting them with MW are required. Forthis reason it is not possible to define unique globalrelations connecting ML to MW or to other magnitudescales.

Conclusions

The main target of the present work is to derive glob-all valid empirical relations converting magnitudes ex-pressed in widely used magnitude scales to equivalentmoment magnitudes. Such relations could become avery useful tool in compiling homogeneous earthquakecatalogs.

The MS magnitudes estimated by ISC and NEICapplying the Prague formula (Vanek et al., 1962) areequivalent throughout a wide magnitude range (MS =2.68.3). New relations connecting MS with MW havebeen defined for earthquakes with foci not exceeding adepth of 70 km. It has been shown that for strong earth-quakes (6.2 MS 8.2) these magnitude scales arepractically equivalent (relation 15), while for weakerevents (3.0 MS 6.1) the MS values are signifi-cantly lower than MW (relation 14).

The magnitude MSK reported in Karnik (1996) es-timated for earthquakes covering the broader area ofEurope, is, according to the author, equivalent to MS.However, its comparison with MS estimated by ISCand NEIC shows a clear bilinear correlation (relations16, 17).

The consistency between mb magnitudes estimatedby ISC and NEIC has been demonstrated throughouta wide magnitude range (2.5 mb 7.3), althoughthere is a slight bias observed (Figure 6, relation 21).The relation between mb and MW clearly reveals lineardependency up to mb 6.2 expressed by relation (22).The mb magnitude scale exhibits an unstable behaviorthat could be considered as saturation for earthquakeswith mb > 6.2 (or its equivalent MW > 6.3) seeFigure 7.

The main reasons for the inconsistency of ML es-timated by several seismological centers are: (a) theyare calculated based on Wood Anderson seismographs(or their simulated) with different effective magnifica-tion, usually smaller than the nominal one (2,800),(b) the distance corrections applied are often adoptedfrom the original ML definition and not estimated for

234

Figure 8. Summary plot of the final results for the comparison between mb and MS, and MW, derived using the global earthquake cataloguedeveloped in the present work.

the local region. For these reasons, no general globallyvalid relation between ML and MW magnitudes can beproposed and such relations must be derived for eachgroup of Wood-Anderson seismographs and possiblyfor each seismotectonic region.

The results of the present work are summarized inthe diagram of Figure 8 showing the variation of MSand mb with MW. It is important to notice that no mag-nitude other than MW is capable to express the size ofearthquakes with MW > 8.2 and that the available datado not permit for weak earthquakes (MW < 4.0) theimplicit estimation of moment magnitudes from mag-nitude measurements in other scales. The MS-MW rela-tion is bilinear changing slope at MS = 6.2 and the un-certainties of both branches are reasonable ( 0.19).The mb-MW relation is linear for 3.5 mb 6.2 but theuncertainties are rather high ( = 0.29).

It must be clearly stressed that all the relations de-rived in the present work are valid only for large-scalestudies. For regional studies new and more detailed re-lations connecting MW or MS or mb with magnitudesfrom local agencies should be derived.

Acknowledgements

The author would like to thank B. Papazachos for hisscientific support and encouragement during all thestages of this work as well as G. Karakaisis and C.Papazachos for critically reading the manuscript. Spe-cial thanks are also due to the two anonymous review-ers for their constructive criticism and their efforts toimprove this work. The maps have been produced us-ing the GMT software (Wessel and Smith, 1995). Thiswork was supported by the project Pythagoras fundedby the EPEAEK. Dept. of Geophysics contribution:#657/2006.

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