Teleseismic magnitude relations

Markus Båth

Annali di Geofisica, Vol. 30, n. 3-4, 1977.

ANNALS OF GEOPHYSICS, VOL. 53, N. 1, FEBRUARY 2010

ABSTRACT

Using available sets of magnitude determinations, primarily fromUppsala seismological bulletin, various extensions are made of the Zurichmagnitude recommendations of 1967. Thus, body-wave magnitude (m)and surface-wave magnitudes (M) are related to each other for 12 differentearthquake regions as well as world-wide. Depth corrections for M arederived for all focal depths. Formulas are developed which permitcalculation of M also from vertical component long-period seismographs.Body-wave magnitudes from broad-band and narrow-band short-periodseismographs are compared and relations deduced. Applications are madeboth to underground nuclear explosions and to earthquakes. Thepossibilities of explosion-earthquake discrimination on the basis ofmagnitudes are examined, as well as the determination of explosive yieldfrom magnitudes. For earthquakes, relations between magnitudes of mainearthquakes and largest aftershocks are investigated. A world-wide stationnetwork for more homogeneous magnitude determinations is suggested inorder to provide the necessary reference system.

1. IntroductionAt the assembly of the International Association of

Seismology and Physics of the Earth's Interior (IASPEI) inZurich in 1967, certain recommendations were adopted inorder to create better homogeneity in seismic magnitudecalculations. In the past decade, the Zurich recommendationshave had both a stabilizing and a stimulating effect onmagnitude calculations and related research. In the pastyears, numerous sets of magnitude determinations havebeen collected which call for more detailed study, both inorder to revise earlier magnitude relations and to make anypossible extensions. In general, relations between body-wavemagnitudes (m) and surface-wave magnitudes (M) havehitherto been given as world-wide averages, a notableexception being Bormann and Wylegalla [1975]. It nowbecomes evident that significant regional deviations exist.Likewise, the focal-depth correction of surface-wavemagnitudes (M), hitherto only known approximately forfocal depths less than 100 km, could now be derived for anyfocal depth. The differences between body-wave magnitudes(m) as derived from different instruments, have also called formore detailed study. There are numerous applications of themagnitude concept which can be subjected to further

investigation and possible refinement, such as fordiscrimination between explosions and earthquakes, fordetermination of explosive yield, and in the study ofaftershock sequences.

Problems as those just mentioned and related ones,together with the present existence of homogeneous sets ofmagnitude determinations, prompted the present study. Inthis paper, I shall attempt to contribute to the solution of theproblems mentioned, using primarly observational materialfrom the Swedish seismograph network. Improved andestended magnitude relations will contribute to ourknowledge of the underlying causes, i. e. both source andpath properties. Naturally, not all magnitude problems havebeen dealt with. For example, the calibrating functions(distance and depth dependence, attenuation) have beenassumed as given in the Zurich recommendations. This willnot affect the methods used, nor the main results as these areprimarily based on relative measurements.

In the past decade, numerous valuable papers haveappeared in the literature with close relations to one sectionor another in the following. These papers would ratherdeserve a special detailed review than brief mentioning here.From such considerations, I have restricted the present paperessentially to my own contributions and limited thereferences accordingly. To a certain extent, the present paperfollows up and extends results derived earlier by me,especially in Båth [1956].

2. List of symbols and abbreviationsa, b, a´, b´ parameters in derived solutions;A ground amplitude, microns;AP, AR ground amplitude of Pwave and of Rayleigh

wave, respectively, microns;B Benioff instrument, used as subscript;c calculated, used as subscript;D standard deviation (subscripts are used to

specify which quantity the deviation refersto, e. g. Dm , DM);

E seismic wave energy, ergs;E, N, Z east-west, north-south and vertical

components, respectively, used as subscripts;g depth parameter in the calculation of M;

69

G Grenet instrument, used as subscript;h focal depth, km;H horizontal component, used as subscript;log decadic logarithm;L wavelength, km;m body-wave magnitude, average of UPP and

KIR;m (US) body-wave magnitude, as given by NEIS;M surface-wave magnitude, average of UPP

and KIR;M1 surface-wave magnitude of largest

aftershock, average of UPP and KIRN number of observations or

pairs of observations;o observed, used as subscript;p perpendicular distance of a straight line

from the origin;q (Δ, h) calibrating function for m;T wave period, sec;x,y Cartesian coordinates;Y yield of underground nuclear explosion, kt;α angle between abscissa and normal to

straight line;γ seismic efficiency;Δ epicentral distance, degrees;ISC International Seismological Centre;KIR Kiruna, Sweden: 67˚50.4´ N, 20˚25.0´ E;NEIS National Earthquake Information Service

(U. S. Geological Survey);NTS Nevada Test Site;

PAL Palisades, New York, U. S. A.: 41˚00.4´ N, 73˚54.5´ W;

UME Umeå (WWSSN), Sweden: 63˚48.9´ N,20˚ 14.2´ E;

UPP Uppsala, Sweden: 59˚51.5´ N, 17˚37.6´ E.

3. MethodsIn short, the Zurich recommendations can be stated as

follows [Båth 1969, p. 131]:Body-wave magnitudes m are calculated for P, PP, S

from:(1)

where q (Δ, h) is the calibrating function after Gutenberg andRichter [1956].

Surface-wave magnitudes M are calculated forhorizontal-component surface waves (Rayleigh) from:

(2)

i. e. the so-called Moscow-Prague formula [Vanek et al.1962].

The calculations in this paper will essentially consists inrelating various quantities two and two, like m and M. Ingraphical plots of such quantities, generally quite a largescatter is observed resulting in oval-shaped point clouds. Thisfact has been of dominating influence on our methods ofdealing with the data. As a general notation of the two

TELESEISMIC MAGNITUDE RELATIONS

70

Table 1. Observational material for m, M and m (US).

log / ,m A T q h= + D^ ^h h16c$D

log / 1.66 log 3.3M A THH= + +D^ h10 30 sec,T 20H = - c$D

71

quantities to be related we use y (ordinate) and x (abscissa).With a and b as constants, we can in our cases then bestdescribe the relation between y and x as a linear equation:

(3)

Putting a = 1, i. e. assuming a constant difference bbetween y and x, proves generally not to be an acceptableprocedure. On the other hand, it proves unnecessary toinclude second-degree terms in x. Actual tests on ourobservational material demonstrate that the scatter is suchthat a second-degree polynomial on the right-hand side ofequation (3) does not improve the accuracy.

With the straigh-line representation as in equation (3),there are three different possibilities to place such a line to fita given point cloud:

1. Regression of y on x, which we denote as . Thisis the common procedure, which by a least squares solution,i. e. minimizing the sum of the squares of the verticaldistances from individual points to the line, yields thefollowing expressions for the constants a and b:

(4)

where the sums are extended over all points, i. e. all pairs ofvalues y, x.

2. Regression of x on y, which we denote as . Thesame procedure applied to the horizontal distances (parallelto the x-axis), leads to a solution of the following equation:

(5)

3. Major axis solution or orthogonal regression, denoted, reproduces the major axis of the point cloud and is

obtained by minimizing the sum of the squares of thedistances perpendicular to the line. A least squares procedureleads to the following solution for the major axis:

with(6)

In the following we have applied all three methods, asthey have each their merits. Methods 1 and 2 are mainlyjustified for computational purposes, providing theminimum error of the required quantity. Thus, theregression of the form of equation (3), is used forcalculating y for a given x. Conversely the regression ,i. e. equation (5), is used for calculating x for a given y. On theother hand, the major axis solution, which best representsthe point cloud as such, is to be preferred for physical

TELESEISMIC MAGNITUDE RELATIONS

Table 2. Relations between m and M for h ≤ 50 km.

.y ax b= +

ax N x

x y N xy2 2=

R R

R R R

-

-

^ hb y a x N= R R-^ h

.x a y b= +l l

cos sin 0x y p+ =a a-

tan 22

x x N y y Nxy x y N

2 2 2 2=aR R R R

R R R

- - -

-

^ ^^h h

h6 6@ @

cos sinp x y N .= +a aR R^ hy x"

x y"

x y)

y x"x y"

reasons. While the major axis solution can be inverted, i. e.solved for either x or y, an inversion is generally notpermitted for the regressions. The relations between thethree solutions 1, 2 and 3 depend on the shape and slope ofthe point cloud. If the point cloud is very narrow,approaching a straight line, the three solutions will be closelysimilar, whereas they diverge more and more for a broadoval-shaped point cloud. The slope will also influence, anextreme case being a horizontal point cloud, for which theregression and the major axis solution will agree,whereas the regression will be at righ angle to theother two. In general, the three solutions show bestagreement with each other in the central parts of the pointclouds and they may diverge strongly towards theirextremities.

The scatter as observed in the point clouds is not due tomeasuring errors but to the circumstance that quite anumber of factors influences the measurements, such assource and path properties. To correct each reading for suchinfluences would be a practically hopeless task. Instead weapply statistical averaging and assume that by the mentionedstraight-line fittings to a sufficient sample, the various effectswill cancel each other. In the regressions, evidently onevariable is assumed to be correct and the deviation is assignedexclusively to the other variable, whereas in the major axissolution both variables are assumed to exhibit deviations.

In earthquake statistical research, where homogeneousmagnitude series over long periods of time are needed,relations of the types mentioned will have to be used.Therefore, a good rule for seismograph stations replacing oldequipment by new one is to have both running in parallel forsuch a long time as to permit reliable comparisons. Fromoverlapping series of records, it is possible to relate onemagnitude to another magnitude (from a differentinstrument or a different station, etc.), which will permit

reductions of long observational series to one homogeneousmagnitude scale. An example of such calculations can befound in a paper by Alsan et al. [1975]. It is then important touse the proper regressions, also to use only direct orimmediate relations and to avoid the consecutive use ofintermediary relations. The latter will rapidly lead toaccumulation of errors and becomes increasingly unreliable.

The observational scatter is in each case expressed as thestandard deviation of the respective formulas. For theregression solutions the standard deviations Dy and Dx arecalculated, respectively, while in the case of major axissolutions, we give three standard deviations, i. e. in additionto Dy and Dx also D, which refers to the perpendiculardistances from the points to the line. It is obtained from thefollowing equation:

(7)

In dealing with observations of this kind, it is importantto test that the sample used is large enough to berepresentative. An efficient method was found to usesuccessively 5, 10, 15, 20, ... observations, until the whole setis used up, and to calculate the parameters in each case.Plotting the respective parameters against the number ofobservations, we find that after some initial, sometimes largeoscillations, the parameters stay within the limitscorresponding to the standard deviation of the total sample.The number of observations at the point where theparameters begin to stay within the final limits, is theminimum number of observations needed for arepresentative sample.

4. Relations between m and MThe observational material consists of magnitudes

published in the Uppsala monthly seismological bulletin for

TELESEISMIC MAGNITUDE RELATIONS

72

Table 3. Correlations and regressions between the parameters a and b, a´ and b´ (earthquakes).

cos 90 sin 90D D Dy x= =c ca a- -^ ^h h

y x"x y"

73

the period from January 1970 to April 1975. The body-wavemagnitudes m are averages of short-period vertical-component P from Uppsala and Kiruna (Grenet), and thesurface-wave magnitudes M are averages of long-periodmeasurements from Uppsala (Benoff ) and Kiruna (Galitzin).These magnitudes ought to be rather homogeneous, havingall been measured and calculately by the present author. Thematerial has been divided into 12 regions according toepicenter location, as listed in Table 1. To avoid the influenceof local depth h on M, only earthquakes with h ≤ 50 km areused in the present section.

The regressions and the major axis solutions aresummarized in Table 2 for each region and for the totalmaterial, together with respective standard deviations. Wenote that the major axis solution for the total material, that is:

(8)

agrees almost exactly with the one included in the Zurichrecommendations:

(9)

An interesing result is that the parameters a and b arestrongly connected to each other, as evidenced by theircorrelation coefficients and regressions, summarized in Table3. Geometrically, this means that all the lines in each groupintersect in nearly one and the same point, but have differentslopes. Without loss of accuracy, it is thus possible tosubstitute these regressions into those between m and M,with the results given in the last column of Table 3. Thismeans that just one parameter, instead of two, is enough todefine each relation between m and M. This parameter bearsthe characteristics of a regional magnitude parameter. It isprobably a complicated function of source and path properties,

but it does not show any obvious dependence on M.Even though calculated magnitudes agree between the

different regions within error limits, at least in the centralparts of the point clouds, the parameters a and b differsignificantly from region to region (Db = Dm and Db´ = DM).This could not be caused by insufficient sampling, becausetests as describbed in Section 3 reveal that already 20 to 30observations per region represent a sufficient sample. Thedifferent parameters for different regions must reflect somephysical differences. A physical explanation, however, canonly be formulated in general terms. The differences of theregional magnitude parameters a, i. e. the slope Dm/DM,refer to unequal amplitude increase of P and of surfacewaves, as source energy increases, or in formula:

(10)

This ratio is less than 1, but significantly different indifferent regions. Source mechanism and radiation patterncould be suggested as factors of significance in thisconnection.

5. Focal depth correction of MAmplitudes of surface waves, and hence the calculated

M, decrease as focal depth h increases. So far only scantyinformation is available on this effect on M, and generallyrestricted to h ≤ 100 km. As a consequence, in nearly all cases,no depth correction is applied to M given in bulletins.

For the horizontal amplitude of Rayleigh waves in ahomogeneous medium, we have the following depth relation[Bullen 1963, p. 90]:

(11)

For h = 0, this expression is = 0.42; it becomes = 0 at

TELESEISMIC MAGNITUDE RELATIONS

Table 4. Depth dependence of surface-wave magnitudes M.

0.55 2.94m M= +

0.5 2.9 .m M6= +

log loga DM DM D A D A .P R= =

esp 5.34 0.58 esp 2.45A h L h LH + - - -^ ^h h

h/L = 0.19 and attains its (negative) maximum = 0.10 ath/L = 0.46. This result may be strongly modified by thecrustal layering. Partly due to this effect, partly due toinevitable inaccuracies of h, no clear-cut dependence is to beexpected for the depth range h = 0-50 km, as also verified byour observations. Below h = 50 km, the second term inequation (11) dominates more and more, which would meanthat the depth dependance would be more purelyexponential. The magnitudes, i. e. logarithms of amplitudes,could thus be expected to vary linearly with h for h > 50 km.

The depth effect on M is calculated from the followingrelations:

(12)

where Mo = observed M and Mc = M calculated from given musing the regional regressions (Table 2,) which yieldthe highest accuracy. For the calculation of g we have usedthe material for h > 50 km (Table 1), and the results aresummarized in Table 4.

We find rather large and apparently irregular variationsof g for 50 < h < 100 km, with an average g = 0.0088 (in closeagreement with some earlier determinations by Båth, 1952,1956). On the other hand, for h > 100 km, g exhibits a steadydecrease with increasing depth, with the remarkable resultthat g (h – 50) is constant = 0.38 ± 0.03. For h > 100 km, thisimplies a constant correction to M. Depth corrections toapply to M can thus be summarized as in the right-handcolumn of Table 4. Certainly, regional variations of thisdepth dependence are to be expected, but our presentmaterial is insufficient to establish this reliably. It is of interestto note that a constant correction for h ≥ 100 km wasobtained already earlier for depths down to 300 km [Båth1952, figure 2], but that scarcity of data then precluded its

further study.The theory of Rayleigh waves in a homogeneous

medium is clearly insufficient to explain this result. It coulddo so for the depth range 50 < h < 100 km, correspondingapproximately to the lithosphere. But for greater depths, theRayleigh wave theory would require far greater Mcorrections than observed. A plausible explanation could bethat the asthenosphere occupies the range below 100 kmdepth and that the deeper earthquakes take place insubduction zones. The downgoing slab may act as a conveyerof the surface waves, leading to the observed effect. It is quitepronounced in deep earthquakes in the Japan and BoninIslands area, but it is apparently a general property of deepearthquakes.

6. Determination of M from vertical-component recordsWhere the Moscow-Prague formula for M, equation (2),

was developed only for the horizontal component of surfacewaves, it is obvious that it would be convenient to have acorresponding farmula for the vertical camponent. Partly,this would need measuring only one record instead of twofor the M calculation, partly and above all, the verticalcomponent shows pure Rayleigh waves with no interferenceof Love waves as the horizontal components. The problemcan be stated as to develop a formula for the verticalcomponent giving the closest possible fit in M to thehorizontal component, equation (2). This reduces todetermining the following regression:

(13)

where we admit a parameter an the basis of earlierexperience from Pasadena [Båth 1952]. On the other hand,we are assuming the same distance dependence (geametricalspreading, attenuation, dispersion) for the vertical as for the

TELESEISMIC MAGNITUDE RELATIONS

74

Table 5. Review of observational material for Sections 6 to 9.

50M M g hc o =- -^ hM a bmc = +l l

log / log /A T a A T bH H Z Z= +^ ^h h

M m"

1a !

75

horizontal component. The resulting horizontal amplitude is ineach case calculated as the vectorial sum af AE and AN, that is:

(14)

This will include also Love waves in AH, but this effect isfound to be relatively insignificant in the total material. TH istaken as the average of TE and TN.

Our observational material is reviewed in Table 5. In thiscase I have also included a set of long-period measurementswhich I made on Palisades records (Columbia Universityseismographs) on a visit in 1952. The depth range has beenrestricted to h ≤ 50 km, even thoug the relations between Hand Z components are expected to be valid for any depth.Least squares solutian of equation (13) yields the followingresults:

Uppsala:Kiruna:Palisades:

(15)

In these equations the standard deviations refer to eachstation separately, i. e. they are the deviations of MZ from MH

for Uppsala, Kiruna and Palisades, separately. In our bulletins,we report the average M for Uppsala and Kiruna, and for thisaverage the standard deviation is calculated as follows:

(16)where

We find DM= ± 0.11 (N = 232), which means that the fit ofthe vertical components to the horizontal ones has beenmade with a deviation well within errors of any singlemagnitude calculation.

The resulting vertical-component magnitude formulasbecome:

Uppsala:Kiruna: (17)Palisades:

Two conclusions are obvious from these results:1. The Pasadena result [Båth 1952] with in

equation (13) is not confirmed. The present result appearsmore agreeable also from a theoretical viewpoint.

2. Formulas for M from the vertical component can bewritten as for the horizontal component, equation (2), andthe deviation can be introduced as a station correction. Asthe ratio of the vertical to the horizontal component

depends on the structural layering around the station, this isa correct procedure also theoretically. A further refinementcould be to express this station correction as a function ofwave period and of azimuth, considering structuralvariations with depth and azimuth. However, this provessuperfluous, considering the small deviations alreadyachieved.

Focal depth corrections of M (Section 5) are assumed tobe the same as for the horizontal component.

7. Relations between m and m (US)Although the type of short-period seismograph to be

used in calculating m is not stated in the Zurichrecommendations, it soon became clear that there aresignificant differences in m from narrow-band instruments,like Benioff, compared to even slightly more broad-bandedseismographs, like Grenet. In our bulletins, m from Uppsalaand Kiruna is based on Grenet instruments. On the otherhand, m (US), closely agreeing with m (ISC), is based onBenioff instruments. As both magnitudes are in current use,it appears desirable to have clear relations between the two.

Regressions and as well asmajor axis solutions have been derived usingthe same material as in Table 1, but this time includingearthquakes of any depth, as the relations between m and m(US) are expected to be depth-independent. The results aresummarized in Table 6.

As in Section 4, it is found that the parameters a and b,a´ and b´ are strongly correlated to each other, as evidencedby the results summarized in Table 3. Again, we are thereforein a possibility to replace the two parameters by just oneparameter in each case, as given in the last column of Table3. As in Section 4, these results imply that the straight linesin each group intersect in nearly one and the same point, buthave different slopes. As in Section 4, we also find that theresults in terms of m and m (US), respectively, generally agreebetween regions, at least in the central parts of the pointclouds, but that the parameters a and b´, a´ and b´ showsignificant deviations between regions. On the other hand,there are no relations between a, b, a´, b´ for m – m (US) andthose for m – M (Section 4), which means that the underlyingfactors are different in the two cases.

The significant difference between m and m (US) makesit necessary to specify in each case (bulletins, publications,etc.) which kind of m is used or which type of instruments isused. Judging from the worldwide relation between m andM, equation (8), it is suggested that m is in better agreementwith the definition of Gutenberg and Richter [1956] than ism (US) or m (ISC).

It is instructive to compare the different standarddeviations for the m – M relations (Table 2) and the m – m(US) relations (Table 6). Due to the larger slope of the pointclouds in the latter case, with α approximating 135˚, all four

TELESEISMIC MAGNITUDE RELATIONS

A A A .2 2H E N

12= +^ h

log / 0.97 log / 0.05 0.14A T A TH H Z Z= !-^ ^h hlog / 0.9 log / 0. 0.14A T A T9 18H H Z Z= + !^ ^h hlog / 0.9 log / 0.0 0.1A T A T9 2 2H H Z Z= !-^ ^h h

2D M M N2ZM

12= ! R - -^ ^h h6 @

21 log / log / UPPM M a A T b A TZ Z Z H H= + +- -^ ^h h6 @

21 log / log / .a A T b A T KIRZ Z H H+ + -^ ^h h6 @

log / 1.66 log 3.2M A TZ Z= + +D^ hlog / 1.66 log 3.M A T 5Z Z= + +D^ hlog / 1.66 log 3.M A T 3Z Z= + +D^ h

1a !

(US)m m" (US)m m"(US)m m)

standard deviations Dm and Dm (US) are approximately equal.But in the m – M case, the slopes are smaller with theconsequence that DM > Dm both for regressions and for majoraxis solutions. Comparing the two latter, we find that DM isabout equal but that DM is larger for the major axis solutionsthan for the regressions.

Proceeding into more detail and comparing the differentregions, we find similar relations between standarddeviations and point cloud slopes. Estimating the averagedeviation of any single magnitude determination as around±0.3 units, we find that all D are within or equal to this limit,except DM.

While m is derived from records at UPP and KIR, m (US)is based on a number of various stations reporting to NEIS.Therefore, it would appear desirable to have not only m butalso m (US) under more direct control. For this purpose, weoperated at Umea (UME) for a few months in 1966 a Grenetseismograph side by side with the Benioff instruments. Withthe material as specified in Table 5, measurements on thetwo instruments could be related to each other, with thefollowing regressions as result:

(18)

The corresponding relation between the magnitudes is thus:

(19)

with sufficient accuracy. This means that there is a constantdifference between these two magnitudes. Similar results arealso found from parallel recordings at Uppsala.

Identifying mG with our m, and mB with m (US), wefind approximate agreement with results in Table 6 onlyfor the major axis solutions for Region 5 and the regression

for Region 12. But a significant difference is thatthe correlation between our G and B is much stronger, withconsiderably smaller standard deviation in the results. Wehave anticipated the differences between G and B to be dueto different bandwidths of their response curves. Eventhough this is at least a partial explanation, it is possibly notthe whole explanation. Partly do the response curves for Gand B not deviate considerably from each other, partly arethe recorded periods identical in all the 159 cases investigatedat Umeå. Other instrumental differences, not fully clarified,may contribute to the difference.

8. Underground nuclear explosionsMagnitudes have found extensive application in

explaining dynamic properties of seismic events. We shallillustrate this by two examples, in this and the followingsections. Due to the special source mechanism ofunderground nuclear explosions, the relation between m andM is generally different from the relation for earthquakes(Section 4). For any given M, the corresponding m isgenerally much higher for explosions than for earthquakes.In order to investigate the m – M and m – m (US) relations forexplosions we collected material from the Uppsala

TELESEISMIC MAGNITUDE RELATIONS

76

Table 6. Relations between m and m (US) for all h.

log / . 6 log / 0. 0.A T A T0 9 47 05B G= !-^ ^h hlog / . log / 0. 0. .A T A T1 03 47 05G B= + !^ ^h h

0.47 0.05m mG B = !-

(US)m m"

77

seismological bulletins and from the PreliminaryDetermination of Epicenter monthly listing of NEIS (Table5). The results are summarized on the bottom lines in Tables2 and 6. A certain trouble arises from the fact that in someinstances the period or the distance is below the rangestipulated for the Moscow-Prague M-formula, equation (2).

There are quite clear deviations between results fordifferent shot locations. For both the regressions and

, the residuals mo – mc are generally negative forNevada, positive for Semipalatinsk and the Aleutians andaround zero for Novaya Zemlya and the Caspian Sea area.

The relations between m and m (US) have remarkablydifferent slopes compared to earthquakes, but they do notcontribute anything to the discrimination problem. On theother hand, the relations between m and M are of interest fordiscrimination between underground explosions andearthquakes. Our results indicate that discrimination basedon magnitudes is not so fully reliable as has sometimes beensuspected in the past. Comparing the major axis solutions,as best representing the point clouds, we find, including thestandard deviations, that separation is best for lowmagnitudes and can be trusted up to about M = 6.3 forearthquakes with h ≤ 50 km but only up to M = 5.6 forearthquakes of greater depth. This holds even thoughsurface waves are usually more important in earthquakes ofany depth than in explosions. Discrimination on this basis isgenerally more favourable for Semipalatinsk and theAleutians than for Nevada. If instead of comparing majoraxes and respective standard deviations, we compare thepoint clouds directly, we find even worse results. The pointclouds overlap partially from M about 5.5 and up, whencemore reliable discrimination on the basis of magnitudesalone would be possible only for magnitudes below thisvalue. This is an argument in favour of the installation ofmore high-sensitive long-period seismographs.

Another application of magnitudes to undergroundexplosions is to calculate their yield. The needed informationon yields for the derivation of such relations is available onlyfor American shots [Bolt 1976]. We find a good relationbetween our m (average of Uppsala and Kiruna) and theyields Y in kiloton for NTS shots (Table 5), as follows:

Correlation between m and log Y = +0.95 ± 0.03

Regression : m = 0.53 log Y + 4.95 ± 0.08Regression : log Y = 1.70 m – 8.20 ± 0.16Major axis solution : m = 0.54 log Y + 4.92 ± 0.08

log Y = 1.84 m – 9.05 ± 0.16Resulting D = ± 0.08 (20)

The regression with a standard deviation of±0.16 in log Y, implies that Y can be calculated with an errorof 37 percent. This error compares favourably with results

of spectral calculations, and in fact, by subdividing the Y-range, it is possible to decrease the error to 18 percent of theyield for Y ≤ 200 kt for NTS shots from magnitudes only.Tests on other explosions with approximately given yields,not used in deriving equations (20), give in general acceptableagreement. But this is true only for NTS shots. Caution isrequired in using these relations for other shot areas. Forinstance, shots in the Aleutians and probably also in theSemipalatinsk area give too high magnitudes at our stations.In other words, yields calculated from equation (20) by ourmagnitudes are too high. This depends probably onstructural properties near the source, especially downgoingasthenospheric slab in case of the Aleutians and exceptionallygood propagation properties in the Semipalatinsk case.

The regression , equation (20), permits an estimateof the seismic efficiency γ, i. e. the fraction of the explosiveenergy that is converted into seismic wave energy. From theE – M relation [Båth, 1958, 1966]:

(21)

in combination with the world-wide major axis solutionfor earthquakes (Table 2):

(22)and the energy relation:

(23)we find:

(24)

Numerical results are given in Table 7. Values of γ liegenerally between 10–2 and 10–1 and exhibit a slight increasewith increasing yield. This could be due to a positivecorrelation between yield and depth of explosion, the greaterdepths implying better seismic efficiency.

9. Main earthquakes and largest aftershocksA dynamic property of aftershock sequences is that the

difference in M between the main shock and the largestaftershock averages 1.2 units (the so-called Båth's law, cf.

TELESEISMIC MAGNITUDE RELATIONS

Table 7. Seismic efficiency γ of underground NTS explosions.

log 12.24 1.44 ME = +

1.80 .M m 5 29= -

4.2 10E Y19= $ c

Y 1800.4=c

(US)m m"m M"

m Y"

m Y"

Y m"

Y m"

m Y)

m M)

Richter 1958, p. 69). In other words, there is a finitemagnitude difference between these two events. When firstformulated, it was based on relatively few sequences, but ithas got additional confirmation in course of time and beenstudied by several authors both from statistical and physicalsides. With the greater abundance of reliable magnitudesnow available, it appears desirable to test this formula on alarger material.

From the Uppsala seismological bulletins, 116aftershock sequenze were listed (Table 5). The aftershockshave occurred within a few hours up to several months aftertheir respective main earthquakes. Sometimes there may betwo or several main earthquakes of nearly equal magnitude,likewise there may be two or several about equally largeaftershocks. In such cases the main earthquake is assigned amagnitude M corresponding to the total energy release in allmain earthquakes, and similarly the largest aftershock isassigned a magnitude corresponding to the total energy ofthese aftershocks, using equation (21) and following aprinciple suggested by Båth [1965]. If M´, M˝, ... , (M´ ≥M˝ ≥... ) is the sequence of nearly equally large main shocks (oraftershocks), then the magnitude for the main shock (orlargest aftershock, respectively) is obtained from thefollowing formula:

(25)

In spite of this procedure, it is unavoidable that some biascreeps into this work, as it is not always obvious if anearthquake is to be considered as a main earthquake or as anaftershock. Cases of more swarm-like characteristics have tobe excluded. In a few cases, even large main shocks are notfollowed by any aftershocks recorded by our long-periodseismographs, and these have naturally also to be excluded.Still, it is believed that such effects will have only a minorinfluence on the result.

As an average of all magnitude differences M – M1 wefind:

(26)

which is in perfect agreement with the original formulation[Richter 1958, p. 69]. In combination with our relationwe then get the corresponding relation for m:

(27)

Expressed in energy, this means that the energy of the largestaftershock is only about 2 percent of the energy of the mainearthquake.

We have further investigated possible relations betweenM – M1 and other parameters, with the following results.

1. Magnitude M. Plotting M – M1 versus M we find in

spite of large scatter some suggestion for a dependence ofthe type:

(28)

As seen, the standard deviation is practically as large as inequation (26) and no improvement in accuracy has beenachieved.

2. Epicentral region. Listing M – M1 for different regions,as defined in Table 1, we find clear regional variations, asfollows for regions with a sufficient number of observations:

Region 3 M – M1 = 1.2 Normal N = 134 1.2 Normal 195 0.8 Low 77 1.2 Normal 148 1.0 Normal 18

11 1.6 High 13Kermadec-New Zeland 1.0 Normal 6New Hebrides 0.9 Low 7

3. Focal depth. Our present material has been restrictedto h ≤ 70 km, and within this depth range no dependenceexists between M – M1 and h. Observations are too scanty todeduce any relation between M – M1 and h for deeper events,even though a dependence is expected, as tentativelysuggested by Båth [1965].

Probably the most extensive use that magnitudes havefound in earthquake statistics is represented by magnitude-frequency relations of the type:

(29)

Our bulletin material does not prove very suitable forthis kind of study. As a world-wide average for M = 5.5 – 8.0and h ≤ 50 km, we find the value of b to be as low as around0.7. It is not excluted that this may be influenced by a gradualtapering off of our material towards the lower magnitudes inthe range studied.

10. Homogeneous magnitudesComplete magnitude formulas should include both

source correction and station correction [Båth 1973, p. 110].Theoretically this is naturally a step in the right direction, i.e. to correct for radiation patterns from the source, for pathproperties and for receiver structural influences, etc. But inpractice we are quite handicapped in determining reliablecorrections because of the lack of acceptable referencemagnitudes. Usually this is circumvented by usingdeterminations at one or a few stations, considered to bereliable. However, even if they may be very reliable, theirreadings are also subjected to the same influences as at anyother station. Such reductions can therefore hardly be

TELESEISMIC MAGNITUDE RELATIONS

78

1.441 log 1 10 ...M M 1.44 M M= + + +-l m l6 6 @@

1.15 0.42M M1 = !-

0. .m m 631 =-

. 0. 0.4M M M0 28 93 01 = !- -

log .N a bM= -

m M)

79

considered to be more than a relative calibration.One way to approach a solution of this problem would

be to use a world-wide net of magnitude reference stations.I have earlier [Båth 1969, p. 80] suggested such a worldsystem of magnitude stations, numbering about 15, welllocated and well distributed around the earth. By an evendistribution we would eliminate influences of sourcemechanism, path and station properties, which are unavoidablefor individual stations, however good they may be.

Instead of forming a simple average magnitude fromthese stations, I like to discuss an alternative procedure. Theunderlying principle is to deduce a magnitude whichcorresponds to the energy radiated from the source. In otherwords, the adopted magnitude should correspond to theaverage energy. Due to the relation between E and M,equation (21), the average energy is not equivalent to theaverage magnitude. The magnitude corresponding to theaverage energy is calculated from the following equation:

(30)

obtained by applying equation (21) and where M´, M˝, ... arethe individual station magnitudes. As an example, with onlyN = 2, M´ = 6.5, M˝ = 7.5, the magnitude average is 7.0, whilethe magnitude which corresponds to the average energy is7.3 from equation (30). The latter magnitude would then beconsidered more representative than the simple average.

Having this, other stations could calibrate their ownmagnitudes against this standard. Calibration for a sufficientnumber of earthquakes, well distributed around the earth,would permit calculation of a station correction. Then, afurther improvement would be to calculate corrections foreach seismic region separately, i. e. source corrections. Thetwo corrections can naturally be combined into one value,for each region. Along these lines, magnitude residuals couldbe made equally unbiased as travel-time residuals. A relatedsuggestion for the Eurasian region has been given byChristoskov et al. [1974].

Availability of homogeneous magnitudes would alsoassist considerably in further development of the formulasfor m and M (Section 3), especially to bridge the gap betweenabout 10˚ to where local scales (ML) may be applicable, andthe lower limits of 16˚ and 20˚ for m and M, respectively.

11. ConclusionsMain results of general validity can be summarized in

the following points.1. Relations between body-wave magnitude (m) and

surfacewave magnitude (M) depend on the earthquakeregion concerned and can be expressed in terms of oneregional magnitude parameter.

2. The focal deph correcfon to M incereases from zero at50 km depth to +0.4 at 100 km depth and after that it remains

constant = + 0.4 for all greater depths.3. Surface-wave magnitudes (M) can be calculated from

the vertical component by the same formula as for thehorizontal component, only by including a minor stationcorrection.

4. Body-wave magnitudes (m) determined from broad-band short-period seismographs show consistent differencesfrom those determined from narrow-band seismographs.This is illustrated by differences between Grenet and Benioffinstruments.

5. Relations between m and M for underground nuclearexplosions demonstrate that discrimination fromearthquakes on the basis of magnitudes alone is trustworthyonly for M less than about 5.5 to 6.0.

6. Relations between m and the yield of undergroundnuclear explosions in Nevada suggest a seismic efficiency ofthe order of 10–2 to 10–1, increasing slightly with yield (ordepth).

7. Main earthquakes and largest aftershocks show anaverage difference of 1.15 in surface-wave magnitude (M).The difference increases slightly with magnitude of the mainshock, it also varies from region to region, but it isindependent of focal depth at least down to h = 70 km.

8. A homogeneous world-wide network is suggested forestablishing more relable magnitudes, particularly fordeducing corrections to individual station determinationsand for further development of magnitude scales.

ReferencesAlsan, E., L. Tezuçan and M. Båth (1975). An earthquake

catalogue for Turkey for the interval 1913-1970, Seismol.Inst., Uppsala, Rep. No. 7-75, 166.

Båth, M. (1952). Earthquake magnitude determination fromthe vertical component of surface waves, Trans. Am.Geophys. Un., 33, 81.

Båth, M. (1956). The problem of earthquake magnitudedetermination, Publ. Bur. Centr. Séism. Int., A 19, 5.

Båth, M. (1958). The energies of seismic body waves andsurface waves, Contr. in Geophys. (Gutenberg Vol.), 1, 1.

Båth, M. (1965). Lateral inhomogeneities of the uppermantle, Tectonophysics, 2, 483.

Båth, M. (1966). Earthquake energy and magnitude, Phys.and Chem. Earth., 2, 115.

Båth, M. (1969). Handbook on Earthquake MagnitudeDeterminations, Seismol. Inst., Uppsala, 158.

Båth, M. (1973). Introduction to Seismology, Basel.Bolt, B. A. (1976). Nuclear Explosions and Earthquakes,

San Francisco.Bormann, P. and K. Wylegalla (1975). Untersuchung der

Korrelationsbeziehungen zwischen verschiedenen Artender Magnitudenbestimmung der Station Moxa inAbhängigkeit vom Gerdtetyp und vom Herdgebiet, Publ.Inst. Geophys., Polish Acad. Sci., 93, 159.

TELESEISMIC MAGNITUDE RELATIONS

1.441 log 10 10 ...M N1.44 1.44M M= + +l m^ h6 @

Bullen, K. E. (1963). An Introduction to the Theory of Seismology,Cambridge.

Christoskov, L., I. V. Fedorova, N. V. Kondorskaya andJ. Vanek (1974). On a homogeneous system of selectedEurasian stations for magnitude determinations, PureAppl. Geophys., 112, 619.

Gutenberg, B. and C. F. Richter (1956). Magnitude andenergy of earthquakes, Ann. Geofis., 9, 1.

Richter, C. F. (1958). Elementary Seismology, San Francisco.Vanek, T., A. Zátopek, V. Kárník, N. V. Kondorskaya, Yu. V.

Riznichenko, E. F. Savarensky, S. L. Solovyev and N. V.Shebalin (1962). Standardization of magnitude scales, Izv.Akad Nauk SSR, Ser. Geofiz., 2.

© 2010 by the Istituto Nazionale di Geofisica e Vulcanologia. All rightsreserved.

TELESEISMIC MAGNITUDE RELATIONS

80