Bulletin of the Seismological Society of America, Vol 73, No 6, pp 1835

Bulletin of the SeismologicalSocietyof America,Vol 73, No 6, pp 1835-1850,December1983

S H O R T - P E R I O D Lg MAGNITUDES: I N S T R U M E N T , ATTENUATION, AND SOURCE EFFECTS BY ROBERT B. HERRMANN AND ANDRZEJ KIJKO ABSTRACT The applicaton of the Nutli (1973) definition of the mbLg magnitude to instruments and wave periods other than the short-period WWSSN seismograph is examined. The basic conclusion is that the Nuttli (1973) definition is applicable to a wider range of seismic instruments if the Ioglo(A/T) term is replaced by IogloA. For consistency and precision, the notation mbLg should be applied only to magnitudes based upon 1.0 Hz observations. The mb,~ magnitude definition was constrained to be consistent with teleseismic P-wave mb estimates from four Central United States earthquakes. In general, for measurements made at a frequency f, the notation m,~(f) should be used, where m,g(f) = 2.94 + 0.833 Ioglo(r/lO) + 0.4342~,r + IogloA, and r is the epicentral distance in kilometers, -y is the coefficient of anelastic attenuation, and A is the reduced ground amplitude in microns. Given its stability when estimated from different instruments, the m,g(f) magnitude is an optimum choice for an easily applied, standard magnitude scale for use in regional seismic studies.

INTRODUCTION Nuttli (1973) introduced an mbLgmagnitude scale specifically designed for eastern North American earthquakes using Lg amplitudes recorded on short-period, verticalcomponent, W W S S N seismograms. The Lg wave travels with group velocities of 3.3 to 3.5 km/sec and is the largest arrival on the seismogram. Following the historical development of magnitude, he defined rnb ~ loglo(A/T), where A is the reduced Lg ground amplitude, and T is its period, which is taken to be near 1.0 sec. The use of the short-period vertical component Lg arrival to determine magnitude was quickly accepted as the basis for obtaining a short-period magnitude for small to moderate-sized earthquakes in eastern North America, being adopted by many of the regional networks. In a number of network bulletins, this magnitude is known as MN, where the N denotes Nuttli. Since this formula was the only one available, it was applied to situations outside its definition, e.g., to instruments other than the W W S S N short-period and to wave frequencies much greater than 1.0 Hz. Recent work has addressed the extension of the Nuttli (1973) mbL~ magnitude relation. Street and Herrmann (1976) tried to address the use of different frequencies for determining mbL~from the standpoint of spectral scaling, but they did not take into account the phase response of seismographs in relating spectral amplitude to time-domain amplitude measures. Ebel (1982) found that direct application of the Nuttli (1973) mbL~ relation to 5- to 10-Hz Lg waves led to a magnitude which is larger than ML, determined from a Wood-Anderson torsion seismometer, by about 0.4 magnitude units. Wetmiller and Drysdale (1982) applied the Nuttli (1973) relation to a set of eastern Canadian earthquakes covering a wide range in magnitude and epicentral distance. Using the teleseismic mb to define the magnitude, they then examined the form of the geometrical spreading used by Nuttli (1973). They 1835

1836

ROBERT B. HERRMANN AND ANDRZEJ KIJKO

proposed a substantially different geometrical spreading correction at distances less than 400 km to correct for the observation that the Nuttli (1973) relation would overestimate the mb value by up to 0.4 magnitude units at the shorter distances. There has also been discussion as to whether or not the logloA/T term in the Nuttli (1973) definition of mbLgshould be replaced by a logl0A term. Simply stated, the problem is that there are perceived problems with the definition of mbLg. On the other hand, a magnitude scale using the amplitude of the vertical component of the Lg wave is essential, given the lack of horizontal instruments in many modern regional microearthquake networks. The object of this paper is to numerically model the effects of earthquake source spectrum, seismic instrument response, anelastic attenuation, and the form of the amplitude term in order to properly define the mbL~magnitude formula and to present an mEg(f) formula that gives similar numerical values when the frequency of the maximum Lg amplitude is other than 1.0 Hz.

1,o

"\

flHz) FIG. 1. Amplitude response of the various instruments considered as a functton of frequency.

Four short-period instrument responses are considered in this study. Figure 1 presents the normalized magnifications as used in this study. The first instrument considered is the WWSSN short-period, since the Nuttli (1973) definition of mbLg was developed using data from this instrument. The parameterized response is taken from Luh (1977). The Wood-Anderson response (Richter, 1958) is included to continue the comparison of the mbLgand ML magnitude scales (Herrmann and Nuttli, 1982). The Canadian Standard Seismograph Network (CSSN) is very useful for estimating magnitudes of eastern North American earthquakes. The response of the short-period instruments is that of an electromagnetic seismograph (Hagiwara, 1958) having seismometer and galvanometer natural frequencies of 1.0 and 5.0 Hz, respectively, with both instruments critically damped. The USGS response (Luh, 1977) is typical of the response of regional microearthquake network seismographs with 1.0 Hz seismometers and an overall instrument response flat to velocity between 2 and 15 Hz. This response is very similar to that of the digital Eastern Canadian Telemetered Network (ECTN). With the exception of a few networks having 2.0 Hz seismometers, these responses are typical of most shortperiod instruments used in North America.

SHORT-PERIOD Lg MAGNITUDES

1837

RANDOM NUMBER SEQUENCE LG MODEL Herrmann (1975) discussed a way to model the instrument effects on the shortperiod seismic coda. An envelope, representing the Lg wave and coda, is filled with a uniform distribution of random numbers. This time history can then be passed through various instrument responses using Fourier transform techniques. If r(t) is a random number sequence, such that - 1 =< r(t) to

(1)

Figure 2 shows 40.95 sec of time history generated using to = 12.5 sec, q = 1.8, and a sampling interval of 0.005 sec. These values are reasonable choices for modeling the Lg and Lg coda at a distance of 100 km in the Central United States. Figure 2 also shows the corresponding amplitude spectrum, which is flat enough to represent the medium response to a step dislocation earthquake source. I

I

t ' III ' 'rE I

1,o f(Hz)

?(sec)

FIG. 2. Time history and correspondingamphtude spectrum of the random number sequence. 40.95 sec of rime historyare displayed. In order to incorporate spectral content of the source, this time series was convolved with an Ohnaka pulse source time function with unit area (Ohnaka, 1973; Harkrider, 1976) defined as ( s(t)

J

0

/ aZte -~t

t < 0 t >- 0

(2)

which has a corner frequency of/c = a/27r between the fo and [-2 amplitude spectrum asymptotes. While this source time function is simple, the result of convolving it with the random number sequence is to yield a complex time history. Since it is difficult to separate the complexities of the source and the transmission medium in a real seismogram, the model used provides a method for incorporating gross spectral characteristics of the source into the synthetic time history. The amplitude spectrum of this source is just S(/, f~) = 1/(1 + (f/[~)e).

(3)

A Futterman (1962) causal anelastic attenuation operator was used to model the anelastic attenuation. This operator characterized in the frequency domain by exp(-~r/t* + i2[t*ln [),

(4)


1838

where f is frequency a n d t* = t / Q characterizes the extent of attenuation. T h e a t t e n u a t i o n o p e r a t o r is purposely simple. T h e object is not to model the seismic coda, but r a t h e r to generate an easily synthesizable t i m e series t h a t reflects source s p e c t r u m a n d anelastic a t t e n u a t i o n effects. Figures 3 a n d 4 show the results of convolving the t i m e history of Figure 2 with

OHNAKA DT = 0,, 005 FC 10.0

5. O

3.0 2- 0

1.0 0..

O,3

TSTA'R=0,,O0

PULSE .23g+02

,.

~NSSN SP 2r,glE+O I

USGS SP ,55E+02

J 12E+02 J .69E.1-01~! 1

CSSN SP . 2 1 Z+02

W - A .59E~01

o

. ~!5Et-01

]E~-lO

, ~f6E+OI

J ~tk_,23E'i-01 ~ O' Et-BE~O~

s..o, A a ~ 0

~0

•

BE÷O0

FIG. 3. Result of convolvmg Ohnaka pulse with corner frequencms FC with the different instrument responses and the modeled Lg time history for t* = 0.0. 5.11 sec of time history are displayed. The numbers above each trace indmate the peak amplitude. the four i n s t r u m e n t responses of Figure 1, source pulses with corner frequencies between 0.3 a n d 10.0 Hz, a n d a t t e n u a t i o n operators with t* = 0.0 a n d 0.2. In these figures, the 5.11 sec of t i m e history are plotted, starting 12.0 sec after the P time. T h e pulse displayed in the O h n a k a pulse convolved with the causal Q-filter. T h e anelastic a t t e n u a t i o n operator, e x p ( - ~ t * ) , can also be represented as e x p ( - ? r ) ,

1839

SHORT-PERIOD Lg MAGNITUDES

where ~ is the spatial anelastic a t t e n u a t i o n coefficient a n d r is epicentral distance. Since 7 = +rf/QU, where U is the wave group velocity a n d Q is the quality factor, we can equate r = t * Q U . For the Central U n i t e d States, with Q = 1500 at 1 Hz a n d U = 3.5 k m / s e c , t* = 0.2 corresponds to r = 1050 km. Q is a s s u m e d to be i n d e p e n d e n t of frequency. T h u s , the traces corresponding to different t* can be looked upon as

OHNAKA DT = 0 , 0 0 5 FC

10. 0

__.

TSTA~ = 0 . 2 0

PULSE

2"2E~-01

WWSSN~

.9P;~-OO

USG$ SP

.221;4-01

~

$P

o IeE÷Ol

W- A

,=,,~N~

i 5. 0

3.O

,_~:~'1"01

° 2.3E*01

2,0

_j~ 20E+OI

1,0

_,~lq'E+O1

0.5

-tr

g FIG. 4. Result of convolvmg Ohnaka pulse with corner frequencies FC with the different instrument responses and the modeled Lg hme history for t* = 0.20. traces expected in different Q regions or also as traces recorded at different epicentral distances, b u t because the r a n d o m sequence a(t) a n d the area of the source pulse are b o t h fixed, these same traces can also be considered to have been corrected for pure geometrical spreading because the zero frequency level of the ground s p e c t r u m is held fixed. T h e traces indicate t h a t there are significant

1840


differences in both the peak amplitude as well as the frequency content of each seismogram. The instrument responses behave as expected for low-frequency sources and attenuating media in that the WWSSN, USGS, and CSSN waveforms are very similar in appearance for such sources (Figure 4, fc = 1.0 to 0.3 Hz). This is not surprising since all three have velocity transducers and natural pendulum frequencies of 1.0 Hz and identical responses at low frequencies. The combined effect of low-frequency source and high attenuation is to minimize the differences in their high-frequency responses. In the Lg simulation, there are phases with frequencies as high as 2.5 Hz appearing in the WWSSN traces for high corner frequency sources. This agrees very well with Lg observations of small central Mississippi Valley earthquakes recorded at the nearby WWSSN station FVM. A 3-Hz Lg arrival is always readily apparent, so OHNAKA DT=OoO05 T~O. O00 ALG vs FC -2.01

I

I"Iiiii

WWSSN SP -1,0

I

I

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III~

CA3

USGS SP + C S S N SP X W - A

-0°0

--1.0

, , I,IIII1

, I I, iI

-3.0

-2,.0

-i.0

-0. 0 FIG. 5 (A) ALG versus FC for t* = 0,0 for the four instruments considered (B) common logamthm of ratio of reduced ground amplitude to the source spectrum excitation at the frequency measured.

much so that a special mbm~(3 Hz) magnitude formula was developed for this station (Stauder et al., 1981). Thus, the modeling is capable of producing this observed feature of small earthquakes. Since the object of this simulation is to decide upon the proper quantity to measure for defining Lg magnitude, three measures were considered: ALT, defined as the common logarithm of the peak trace amplitude; ALG, defined as the common logarithm of the peak trace amplitude reduced to ground amplitude using the instrument magnification at the frequency of the measured amplitude; and ALGT, defined as the common logarithm of the reduced ground amplitude divided by the measured period. Figures 5A through 8A present plots of the ALG and ALGT determined for each instrument, corner frequency, and t* combination. In these

SHORT-PERIOD

1841

Lg MAGNITUDES

figures, the "magnitude" is plotted versus the corner frequency. To compute these magnitudes, the original peak amplitudes were corrected for anelastic attenuation by multiplying by exp(+~rft*), where [ is the frequency of the peak amplitude. This is consistent with the Herrmann and Nuttli (1982) expression for the mbLgformula in which the anelastic attenuation correction is distinct from the geometrical spreading correction. Since this numerical experiment emulates the case when all four instruments are operating on the same seismograph pier, the choice of a proper magnitude scale is taken to be that one for which the "magnitudes" estimated from the different instruments agree with each other for sources with different spectral content (a change in seismic moment only will not affect the difference between the magnitude OHNAKA ])T =0° 0 0 5 T~O, 000 ALGT vs FC I

i

~ i llli

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f II~l

- z . o ~ VV~SN sP us6s sP

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-2°0

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,

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FIG. 6. (A) ALGT versus FC for t* = 0.0 for the four instruments considered. (B) common logarithm of ratio of reduced ground amplitude/period to the source spectrum excitation at the frequency measured.

estimates for the same fc). This agreement of "magnitude" values obtained by different instruments is best for the ALG measure. The ALGT fails for t* = 0.0 at high corner frequencies because, as seen from Figure 3, there is a limit to the highest frequency that the W W S S N short-period seismogram will effectively pass. Thus, the use of (A/T) term leads to a discrepancy in mbLgestimated from the W W S S N and the other three instruments for high-frequency sources (small earthquakes). For t* = 0.20, ALGT works as well as ALG precisely because the Q-filter (anelastic attenuation for transmission to large distances) removes the high frequencies in the USGS, CSSN, and Wood-Anderson seismogram traces. A comparison of Figures 5A and 7A also indicates that the anelastic attenuation correction, as applied to the time-domain measurements, works well in that the t* = 0.0 and t* = 0.2 magnitude

1842


curves overlay to within one-tenth of a magnitude unit or so. Since differing values of t* can also be interpreted in terms of the effect of anelastic attenuation at different distances after the amplitudes are corrected for geometrical spreading, the conclusion is that the ALG measure works best for different instruments and also for different epicentral distances. The dashed line in Figure 7A indicates the ALT, peak trace amplitude measure, trend for the Wood-Anderson instrument for t* = 0.0. Since the peak trace amplitude on the Wood-Anderson is used to define local magnitude (Richter, 1935, 1958), this simulation study is in agreement with the conclusion of Herrmann and Nuttli (1982) that the mbLg and ML scales compare well to each other over wide ranges of earthquake source spectrum characteristics. Another question that must be addressed in the choice of a preferred magnitude OHNAKA DT =0o 0 0 5 T~Oo 2 0 0 ALG vs FC I

-2.0

I

I

I I I II

I

I

I

I

I I I I

CA]-

o ww~sN ~p A USGS SP + CSSN SP XW -A

-1,0 -0.0

- -1.0 -

I" I

I I

i I

I i I I III

i I

I

I I

I t

I I

-3,o

I I I II

CB]-

-2.0

-I,0 w m

O°0

° i

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I III

I

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III

FIG. 7. (A) ALG versus FC for t* = 0.2 for the four instrumentsconsidered, (B) commonlogarithm of ratio of reduced ground amplitudeto the source spectrum excitationat the frequencymeasured. All measured amphtudeshave been correctedfor anelastmattenuation. scale is "what do time domain magnitudes tell us about the source spectrum?" Aki (1967, 1972) developed the concept of seismic source spectrum scaling by attempting to find a scaling law consistent with seismic moment, Ms, and mb measurements. Seismic moment is a spectral parameter, but in order to use time-domain magnitudes, it was necessary to assume that the variation in Ms and mb is directly proportional to differences in the logarithms of spectral levels at frequencies of 0.05 and ] .0 Hz, respectively, between seismic events. This assumption is reasonable for the 20-sec surface wave measurement since the surface wave trains are usually sinusoidal in appearance. There is a problem in relating a P-wave time-domain amplitude directly to spectral amplitude, without first making some assumptions about the duration of the waveform with earthquake size (Aki, 1967; Hanks, 1979).

SHORT-PERIOD

1843

Lg MAGNITUDES

Since investigators are now considering the regional variation of spectral scaling laws (Nuttli, 1983), it is necessary to understand the relationship between spectral scaling and time-domain magnitudes for magnitudes determined using a wide variety of instruments. There is a monotonic increase in the trace log-amplitude with corner frequency in Figures 5A through 8A. Since the zero frequency level of the Fourier transform of the Ohnaka pulse was held fixed, this difference in amplitude reflects the difference in excitation of the wave periods actually measured. Given the corner frequency, [c, and the frequency, f, associated with the peak amplitude measured on the seismogram, the reduced seismogram amplitude, A, was divided by the source spectrum excitation at f, to yield the ratio A ([)/S(f, re). For ease of analysis, the OHNAKA BT=O. O05 T~Oo 2 0 0 ALGT vs FC I

-2~0

I

I

I IIII

I

I

I

I

I I III

CA]-

~ WWSSN SP

*" USGS SP

+ CSSN XW-A

SP

I

1,0

Oo 0

-Io0 -3°0

f

I I

I I

I I1! I IIII

I

I

I

ho I

I I

I |

I I

I III I I II

I

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I

I

-2.0

1o0

Oo 0 ° III

III

FIG. 8. (A) ALGT versus FC for t* = 0.20 for the four instruments considered. (B) common logarithm of ratio of reduced ground amphtude/period to the source spectrum excitation at the frequency measured. All measured amphtudes have been corrected for anelastlc attenuation.

determination of the frequency, f, departed from usual seismological practice in that manual period measurements were not taken from individual traces. Rather, the frequency of zero crossings of the waveform was used. The common logarithm of this ratio is plotted in Figures 5B and 7B. Figures 6B and 8B show the logarithms of the ratio A ( / ) / T S ( f , fc). For ALG, the plots for each instrument are nearly horizontal, indicating that there is a direct relation between source spectrum excitation and time-domain amplitudes. This means, for example, that the 2.0-Hz amplitude recorded on a W W S S N instrument tells us something about the source excitation at a frequency of 2.0 Hz. To really understand source spectrum scaling, we must study earthquakes of different size. To explore the relation of source spectrum scaling to magnitude, the

1844


data used to generate Figures 5 through 8 were used to compare two earthquakes, one with a corner frequency of 10 Hz and a unit area for the source time function, and the other with a corner frequency of 1.0 Hz and a source time function having an area 10,000 times greater than the first. That is, the seismic moment of the second earthquake is 10,000 times greater than that of the first. Table 1 presents the anelastic attenuation-corrected ALG for the WWSSN and USGS instruments for the two sources. Three values of t* are used, 0.0, 0.06, and 0.20. The frequency observed in the seismogram is also given. The far right column gives the common logarithm of the source spectrum level at the measured frequency. To check the stability of the "magnitude" estimates, it is noted that the magnitudes are 0.49 + 0.13 for the 10-Hz source and 3.65 _+ 0.11 for the 1.0-Hz source. The ALG "magnitude" estimate is reasonably cgnsistent for the various instruments. To compare the relationship between magnitude differences and spectral scaling, consider the measurements based on the 2.00- to 2.05-Hz frequencies. The differences in magnitudes between the events (column 5 of Table 1) is 3.67 - 0.31 = 3.36. The TABLE 1 EXAMPLE OF MAGNITUDE AND SPECTRA SCALING [c

Instrument

t*

f

ALG

10 10 10 10 10 10

WWSSN WWSSN WWSSN USGS USGS USGS

0.00 0 06 0.20 0.00 0.06 0.20

2 37 2.34 2.02 12.99 6 64 2.71

0 46 0.43 0.31 0.57 0.70 0.47

- 0 02 - 0 02 -0.02 -0.43 - 0 16 -0.03

1 1 1 1 1 1

WWSSN WWSSN WWSSN USGS USGS USGS

0.00 0 06 0.20 0.00 0 06 0.20

2 05 1.37 1 02 5.42 2.33 2.01

3 67 3.66 3.75 3.44 3.69 3.67

3.28 3 54 3.69 2.51 3.19 3.29

logloS([, /c)

corresponding difference in the logarithm of the spectral levels (column 6 of Table 1) is 3.28 - (-0.02) = 3.30. For the measurements in the 5.4- to 6.7-Hz range, the difference in magnitudes is 3.44 - 0.7 = 2.74, while the corresponding difference in the logarithm of the spectral levels is 2.51 - (-0.16) = 2.67. At low frequencies, the difference in the logarithm of the zero frequency levels of the source spectra is, by assumption, 4.0 units. We must conclude that even though we may obtain numerically similar "magnitude" estimates from different instruments at different distances by using the ALG measure, if we wish to use magnitude differences between events for spectral scaling studies, we must be very careful to specify the frequency upon which the magnitude estimate is based. NORMAL MODE LG

MODEL

Herrmann and Kijko (1983) demonstrated that normal mode surface wave theory can be used to model some observed empirical Lg relations. Following up on their work, a series of synthetic seismograms were generated at distances of 100, 200 and 1000 km for a point source dislocation buried at a depth of 5.0 km in the earth model of Table 2. The dislocation source was chosen to be a pure 45 ° dip-slip reverse fault striking north, and the seismograms were generated along an azimuth of 45 °.

SHORT-PERIOD

Lg

1845

MAGNITUDES

A sampling interval of 0.025 sec was used. The Q values given in Table 2 are the 1.0-Hz values of a frequency-dependent Q model, where Q(f) = Q(1 Hz)/°3. The Q values and their frequency dependence are compatible with Central United States observations on Lg and LgQ (Mitchell, 1980, 1981; Singh and Herrmann, 1983). In addition, a simple source spectrum scaling law based on the work of Street et al. (1975) in the Central United States was used to relate corner frequencies to seismic moment and is given in Table 3. Figures 9 and 10 present the vertical component Lg seismograms so generated for the WWSSN and USGS responses, respectively, as given in Figure 1. The differences in instrument response and the effect of source spectral content are again very apparent. To provide some interpretation of these traces, an rnbLgmagnitude was computed' for each trace in the same manner as is traditionally done with actual data. The individual traces were displayed on an interactive graphics terminal, and peak TABLE 2 CENTRALU.S. EARTHMODEL H (kin)

Vp (km/sec)

V~(kin/see)

p (gm/cm3)

Qp

Qs

1 9 10 20

5.00 6 10 6.40 6.70 8.15

2.89 3.52 3.70 3.87 4.70

2.5 2.7 29 30 3.4

1200 1200 1200 8000 8000

600 600 600 4000 4000

TABLE 3 SOURCE SPECTRUM SCALING Model

Selsnuc Moment (dyne-cm)

Corner Frequency (Hz)

a b c d e f g h 1 j k 1

2.0E+18 1.0E+19 2.0E+19 1.0E+20 2.0E+20 1.0E+21 2.0E+21 1.0E+22 2.0E+22 1.0E+23 2.0E+23 1.0E+24

10 00 6.68 5.62 3.76 3.16 2.11 1.78 1.19 1.00 0.67 0.56 0 37

amplitudes and associate periods were manually picked in the same manner as done with analog seismograms. The mbLg formula of Herrmann and Nuttli (1982) for vertical component Lg waves was next applied to the data set. This relation is mbLg = 2.94 + 0.833 logl0(r/10) + 0.4342"yr + lOgl0A,

(5)

where r is the epicentral distance in kilometers, A is the ground amplitude in microns, and % the coefficient of anelastic attenuation is taken to be a function of frequency, - / = 0.001 [o 7 km-1. The 1-Hz value of ~, = 0.001 km -1 and its frequency dependence are realistic values for this normal mode Lg model and for the Central

1846


and E a s t e r n United States ( H e r r m a n n and Kijko, 1983; Singh and H e r r m a n n , 1983). Figure 11 presents a plot of the computed Lg magnitudes using equation (5) versus seismic moment, Mo. T h e different symbols represent data t a k e n from the W W S S N and U S G S short-period instruments. In addition, the data were taken from synthetics generated at 100 and 200 km, open symbols, and for models b and 1 of Table 3, at 1000 km, solid symbols. For the purpose of comparing the ALG and A L G T measures of magnitude, the coefficient A in equation (5) was replaced by ( A / T ) to obtain A L G T . T h e results of this set of computations are given in Figure 12. Comparing these figures, we again conclude t h a t the ALG measure is internally more consistent when applied to different i n s t r u m e n t s t h a n is the A L G T measure. ° 170

e

-06

Ca] o8 9 6 e - 0 6

[b] ° 179e-05 It]

•mmh.V-~.v-~...

o 818e-05 (d) o

15~e-O~

[e]

o 107 e - 0 3 C9 )

° 37~e-03 [hi o618e-03

[j ] CK) °613e-02 [

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~

~

FiG 9. Vertical compoment Lg seismograms synthesized at a distance of 100 km for each of the source spectrum models of Table 3. The traces start 20.0 sec followingthe origin time. A total of 25.58 sec of time history is displayed. The letter to the left of each trace corresponds to the partmular model of Table 3. The numbers to the right of each trace are the peak amplitudes in units of centimeters. Again, the reason is t h a t the high-frequency i n s t r u m e n t has high-frequency seismograms at short distances. Even though it may be difficult to see in the figures, the U S G S m e a s u r e m e n t at 1000 k m for the seismic m o m e n t 1.0E + 19 dyne-cm, fits in with the W W S S N measurements, even for the A L G T measure, again demonstrating the lowpass filtering effect of propagation over large distances. T h e A L G T measure yields values about 0.1 to 0.2 magnitude units larger t h a n the ALG measure. T h e modeling also matches observed Central United States mbLg -- Mo

Lg MAGNITUDES

SHORT-PERIOD ~

1847

. 135 e-05

=@

,577e-05

[b] [ c]

:':'--==~~

~

,~,

~

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o 295e-0~

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° 774e-03 ° 121 e - 0 2

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° 326e-02 e-O2

, ~83

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Cl] FIG. 10 Same as Figure 9, but for the USGS instrument.

24

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WWSS SPUSGS SP I

5°0

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MAG FIG. 11 Plot of computed mbLgmagnitudes versus semmlc moment using the A term in equation (4).

1848


scaling (Street and Herrmann, 1976), in that an mbLg = 5.0 earthquake has M0 = 1.0E + 23 dyne-cm. This results, in part, from the azimuths of the receive~ and from the focal mechanism used in the example, but does support the internal consistency of the parameters used as input to the model and the observational data from the Central United States.

DISCUSSION This study representsextensivemodelingto distinguishthe effectsof instrument, anelastic attenuation, and functionalform of a short-periodmagnituderelation. We conclude that direct application of equation (5) is appropriate for the class of instruments generallyused in eastern North America, including those with seismometer periods of 1.0 sec and velocitytransducers and also the Wood-Anderson short-period torsion seismometer.The discrepancybetweenMLand MN,as pointed 2~

23

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( ~

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E 22 u I o c 21 -'(3 ~-~ 20 o (~(~ ~I) A

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WWSS SPU S G S SP I 5.0

6.0

MAG FIG. 12 Plot of the mb~gmagnitude versus seismic moment obtained by replacing the A term m equation (4) by (A/T). out by Ebel (1982) can be removed by changing the manner in which the magnitudes of his network are computed, i.e., logloA rather than lOglo(A/T) should be used. This conclusion was reached a number of years ago for the Saint Louis University regional seismic network (Dwyer et al., 1979; Stauder et al., 1981), but it remained for the present study to provide convincing demonstration of the reason for not using ( A / T ) . Wetmiller and Drysdale reported on the relationship of mbLs and teleseismic mb for six earthquakes in eastern Canada. They investigated the applicability of the Nuttli (1973) definition of mbL~ as m~L~ = 3.75 + 0.90 1ogl0A + loglo(A/T)

(6a)

mbL~ = 3.30 + 1.66 logl0A + logi, o(A/T)

(6b)

where (6a) is valid for 0.5 ° ~< A ~< 4.0 ° and (6b) is valid for 4.0 ° ~< A ~< 30 °. A is in degrees, and A is the instrument corrected ground amplitude in mickons. Using data with periods between 0.1 and 0.5 sec, they found that direct use of equation (6a)

SHORT-PERIOD

Lg MAGNITUDES

1849

would lead to overestimates of mb by 0.3 units. They also found that equation (6b) yielded magnitudes quite consistent with teleseismic mb even at distances as short as 25 km. They further proposed the use of equation (6b) for distances as short as 25 km in lieu of equation (6a). Wetmiller and Drysdale (1982) have succeeded in defining an Lg magnitude scale which yields a value compatible with teleseismic mb estimates. It is disturbing because it calls for a change in the nature of geometrical spreading of the Lg wave at short distances, which was inherent in the Nuttli (1973) development. In our mind, this goes against the basic physics of wave propagation. The reason that they require this different attenuation relation is that the Nuttli (1973) definition's use of (A/T) will overestimate mbg~when used with short-period data from Canadian broadband instruments, both the CSSN and ECTN. It is our conclusion that they could still use the distance-attenuation relation of Nuttli (1973) if they use logloA in their formulas rather than loglo(A/T). Obviously further study is required. To use mbLg from earthquake catalogs for spectral scaling studies, we must be precise in stating how it was obtained. It is essential that not only the magnitude be given, but also the frequency upon which it is based. We propose that the magnitude be given as mL~(f), where [ denotes the frequency. We may thus see mL~(10 Hz), for example. To clarify Nuttli (1973), we define mbLg= mLg(1 Hz). We have purposely dropped the subscript b because it indicates a quantity observed at only 1.0 Hz. We feel that we have done substantially more than clarify the usage of the mbL~ magnitude relation. Equation (5) is really the basis for defining a magnitude scale for use with crustal continental earthquakes anywhere, not just eastern North America. As long as the instrument responses are similar to those investigated in this study, earthquake sizes can be compared. The regional bias in the definition of maglaitude scales is eliminated. All that is required is the determination of the correct value of ~, the estimation of which is discussed by Singh and Herrmann (1983). The expanded mLg(f) scale is recommended as a standard. An anonymous reviewer criticized this article because it did not provide a direct tie or calibration between mL~([) and fundamental earthquake source measurements such as mb o r ML. In addition, the reviewer felt that quantification of the relationship between mL~(f) and seismic moment would be even more useful in that this magnitude would then be related to a fundamental source parameter. Both of these criticisms are valid, but are also somewhat naive. Magnitude is in use because it is an easily obtained measure of earthquake size. By definition, it does not define source parameters, rather it is a composite effect of source characteristics convolved in a complex manner with wave propagation and instrumental effects. One magnitude scale is not inherently preferable over another since each measure reflects certain characteristics of the source. We should be able to infer more about the seismic source by using a number of magnitude scales rather than just one. Even though the teleseismic P-wave mb and the Lg mLg(1 Hz) are nominally 1-Hz magnitudes, focal mechanism and rupture characteristics can cause these magnitudes to differ (Herrmann and Kijko, 1983). This points out one of the concerns of this study that a magnitude scale be well defined and understood. The extensive computations performed in this study do indicate a relation between corner frequency and the period of the Lg wave and also a relation between the peak timedomain amplitude and seismic source spectrum (Table 1). A discussion of the relation of mLg(f) to seismic moment is possible, but spectral studies seem better suited to this than time-domain measures.

1850

ROBERT B. HERRMANN

AND ANDRZEJ KIJKO

ACKNOWLEDGMENTS The many frmtful dlscussmns with Otto W. Nutth are gratefully acknowledged The beneflcml discussions with James Zollweg, John Dwyer, and Crmg Nmholson are also acknowledged. This research was sponsored m part by the U.S. Nuclear Regulatory Commissmn under Contract NRC-04-81-195-03, by the U.S. Geological Survey under Contract 14-08-0001-21262 and by the National Scmnce Foundatmn under Grant CEE-8204222. REFERENCES Aki, K (1967). Scaling law of seismic spectrum, J Geophys Res. 72, 1217-1231. Aki, K. (1972). Scahng law of earthquake source time-function, Geophys J 31, 3-25. Dwyer, J J., R. B. Herrmann, C. C. Nlcholson, and O. W. Nuttli (1979) Time-domain scaling and magnitude relations m the central Umted States (abstract), EOS 60, 875. Ebel, J. E. (1982). ML measurements for northeastern United States earthquakes, Bull. Se~sm. Soc Am 72, 1367-1378. Futterman, W. I. (1962). Dispersive body waves, J Geophys Res 34, 251-263. Hag~wara, T (1958). A note on the theory of electromagnetic seismograph, Bull Earthquake Res. Inst., Tokyo Unw 36, 139-164. Hanks, T. C. (1979). b values and ~-~ seismic source models implications for tectomc stress variations along active crustal fault zones and the estimation of high-frequency strong ground motion, J Geophys Res 84, 2235-2242 Harknder, D. G. (1976) Potentials and displacements for two theoretical seismic sources, Geophys. J 47, 97-133. Herrmann, R. B. (1975). The use of duration as a measure of seismic moment and magnitude, Bull Se~sm Soc Am 65,899-913. Herrmann, R. B. and O. W. Nuttli (1982). Magmtude: the relation of ML to mbL~,Bull Sezsm Soc Am 72,389-397. Herrmann, R. B. and A. Kijko (1983). Modeling some empirical Lg relations, Bull Selsm Soc Am. 73, 157-171 Luh, P. C. (1977). A scheme for expressing instrumental responses parametrically, Bull. Se~sm Soc Am 67, 957-969. Mitchell, B. J. (1980). Frequency dependence of shear wave internal friction in the continental crust of eastern North America, J. Geophys. Res 85, 5212-5218. Mitchell, B. J. (1981) Regional variation and frequency dependence of Q~ m the crust of the Umted States, Bull Se~sm. Soc Am. 71, 1531-1538. Nuttli, O. W. (1973). Seismic wave attenuation and magnitude relations for eastern North America, J Geophys. Res 78, 876-885. Nuttli, O. W. (1983). Average seismic source-parameter relations for mid-plate earthquakes, Bull Se~sm Soc Am 73,519-535. Ohnaka, M. {1973) A physmal understanding of the earthquake source mechanism, J Phys. Earth 21, 39-59. Rmhter, C. F. (1935) An instrumental magnitude scale, Bull. Se~sm Soc Am 25, 1-32. Rmhter, C. F. (1958). Elementary Setsmology, W. H. Freeman and Company, San Francmco, 344-345. Smgh, S. and R B. Herrmann {1983). Regmnalization of crustal Q m the continental Umted States, J Geophys Res 88, 527-538. Stauder, W., R. Herrmann, S. Smgh, C. Nicholson, D. Reidy, R. Perry, S. Morrissey, and E. Haug (1981). Central Mississippi Valley earthquakes--1979, Earthquake Notes 52, 26-31. Street, R. L. and R. B. Herrmann (1976). Some problems using magnitude scales for eastern North American earthquakes, Earthquake Notes 47, 37-45. Street, R. L., R. B. Herrmann, and O. W. Nuttli {1975). Spectral characteristms of the Lg wave generated by central United States earthquakes, Geophys. J 41, 51-63. Wetmfller, R. J and J. A Drysdale (1982). Local magnitudes of eastern Canadian earthquakes by an extended Mb(Lg) scale (abstract), Earthquake Notes 53, 40. DEPARTMENTOF EARTHAND ATMOSPHERICSCIENCES SAINT LOUIS UNIVERSITY 3507 LACLEDEAVENUE SAINT LOUIS,MISSOURI63103 (R B.H.) Manuscript received 18 April 1983

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