Magnitude Scales and Their Relations for Taiwan Earthquakes: A Review

The magnitude scales, including ML, Mn, M8(GR), mB, M8, mB, MH, MJ and M1, applied to quantify earthquakes in the Taiwan region since 1900 are reviewed. Their relations studied by several authors are also discussed.


INTRODUCTION
Magnitude is essentially a directly measurable parameter to quantify earthquakes. Since Richter introduced local magnitude in 1935, numerous magnitude scales have been defined and widely used for scientific and practical purpose, for examples, the study on seismicity, the estimation of seismic risk, and earthquake prediction research. The magnitude scales are defined based on different types of seismic waves at different periods of oscillation.
Some magnitude scales are not used for the whole time period since 1900. It is necessary to understand the difference and relation between two magnitude scales for establishing a complete earthquake catalogue. Miyamura (1978), Bath (1981), Chung and Bemreuter (1981), and Utsu (1982b) reviewed various magnitude scales and their relations in detail.
In this paper, the magnitude scales used for quantifying Ta iwan earthquakes and their relations will be reviewed in detail. The materials are mainly from the papers published in numerous journals. Also included are a few current results done by the author.

MAGNITUDE SCALES
(1) Local Magnitude Richter (1935) defined the local magnitude ML based on the amplitudes recorded on the Wood-Anderson torsion seismographs with natural period of 0.8 sec, damping factor of 0.8 and magnifi cation of 2800. Richter defined the earthquake, for which the maximum trace amplitude at a distance of 100 km is 1 mm, to be the zero-magnitude earthquake. If A0 ( �) expresses the function of the maximum trace amplitude A0 of the zero-magnitude earthquake in terms of epicentral distance �.then ML is given by: where A is the maximum trace amplitude on the Wood-Anderson seismograph for the earth quake at a distance�. A table of -logA0 as a function of distance� (in kilometers) can be found in the text by Richter (1958). Eq.
(1) was originally determined only for the southern California earthquakes and for the maximum trace amplitudes with periods of between 0.0 and 0.5 sec, for which the magnification is 2800 for the Wood-Anderson seismograph. The attenuation of seismic waves in this period range is mainly caused by the absorptive prop erties of the upper layer of the earth;s crust. Hence, wide variation in the amplitude versus distance relations over the surface of the earth's crust must be remarkable. However, the ML scale has been widely used in other geological provinces without regional corrections.
In 1980, a Wood-Anderson seismograph with a magnification of 100, manufactured by Geotech Co., USA was operating at the Institute of Earth Sciences (IES), Academia Sinica.
Unfortunately, the seismograph was out of service after 1980. Since 1980, a simulated Wood Anderson seismograph from aL-4C sensor has been installed at the Institute ( Liu, 1981;Wang et aL, 1989). Liu (1981) measured the maximum amplitudes of six earthquakes recorded by the two seismographs at the same time. The ratios of the two maximum trace amplitudes change from 0.96 to 1.04 with the average of 1.0, thus indicating that the simulated one can work as a real one. Since 1980, the local magnitudes of Tai wan earthquakes with duration magnitude greater than 4 have been routinely determined based on the Richter's -logA0 values. However, Wang et al. (1989) stressed that the site effects from sediments beneath the station would amplify the short-period signal, thus inflating the ML value.
From the maximum amplitudes of the displacement seismograms synthesized from the strong-motion accelerograms of 10 events through the technique developed by Kanamori and Jenning (1918), Yeh et al. (1982) obtained an amplitude-distance curve for 0-100 km. Due to small number of data points for epicentral distance greater than 50 km, the. deviation of their curve from Richter's increases as the epicentral distance increases. Their amplitude-distance relation is used only by Yeh and his coauthors to establish their catalogues and not used in the routine work to determine local magnitude.
(2) Duration Magnitude Duration magnitude is a different magnitude estimated from the signal duration (F-P) in seconds by using an empirical formula in the general form: (2) where .6. is the epicentral distance in kilometers, h is the focal depth in kilometers and a1 -a4 are empirical constants.This magnitude was applied to quantify Russian earthquakes first by Bisztricsany (1958) from the duration of surface waves and by Solove'v (1965) from the total duration of seismogram. However, they used telemetered seismograms for determining magnitude. Tsumuta (1967) determined the duration magnitude from the total duration of oscillation from local earthquakes recorded by Wakayama, Japan microearthquake network.
His formulation for determining duration magnitude is still used today in Japan for local earthquakes. Lee et al. (1972) determined an empirical formula fo r estimating the duration magnitude for California earthquakes in the form: MD = -0.87 + 2.00logD + 0.0035.6.
( 3 ) Lee et al. determined this formula by using 351 central California earthquakes having local magnitude. They found that the ML 'of an earthquake can be estimated by Eq. (3) to within about ±0.25 unit.
(3) has been introduced to determine the duration magnitude of Taiwan earthquakes by the use of seismograms recorded by the TTSN (Wang, 1989). Since 1988, when a new short-period seismographic network was placed in operation by the CWB, this magnitude scale has also been used by this agency to determine the magnitude for Taiwan earthquakes. The signal duration used by Lee et al. in Eq. (3) was originally defined fr om the P arrival to the point in the coda where the largest peak-to-peak amplitude on a Geotech model 6585 film viewer (20X magnification) is less than 1 cm. Hence, it is impossible to compare the duration magnitudes determined from different instruments. In other words, earthquake magnitude determined from the total signal duration must be calibrated for each  Table 1. It can be seen that the Qc values of Taiwan are larger than those of southern California and almost equal to those of central California.
Since the formula by Lee et al. (1972) was deduced mainly from the earthquakes in central California, the direct use of their formula to determine duration magnitude for Taiwan earth- MD (Y L) = 0.632 5 88 + l.GG7354logD + 0.0005826.
But they did not clearly mention which magnitude scale listed in the PDE was used. After an examination of their data set, it is found that their calibration magnitude is the body wave magnitude. Comparison of Eq. (3) with Eq. (4) shows that the epicentral term is less important in the latter than in the former, and actually can be ignored in the practical calculation by us ing Eq. (4). However, . Yiu and Lin's formula has not been applied to determine the duration magnitude for Taiwan earthquakes.
According to the coda wave theory, Shin (1986) studied the station correction of Eq.
(3) for the TTSN. His revised formula is in the form: where R is the station correction and its value is in the range of from -0.01 to 0.45. He also related MD (Shin) to MD in the form: Essentially, there is only small difference between Mn and Mn (Shin) (3) Body-wave and Surface-wave Magnitudes

( 6 )
From the definition of body-wave .and surface-wave magnitudes defined by Gute nberg and Richter in a series of papers, the two magnitude scales were very important for earthquake quantification before 1965. Gutenberg (1945a) defined the surface-wave magnitude in the form: Ms ( GR) = logA + l.656log6. + 1.818 + C

( 7 )
In this formula, A is the vector sum of the maximum amplitudes with period around 20 sec in mm along two horizontal components, 6. is the epicentral distance in degree, and C is the station correction. As only one component amplitude is avilable, A is the value of the maximum ampltide mult iplcd by J2 or 1.4. However, from empirical test, Lienkaemper (1984) showed 1.2 to be a better estimation of the vector sum than 1.4. This formula is mainly appropriate for epicentral distance in the range of from 15° to 130°. For very large earthquakes, the magnitude might be underestimated through Eq. (7). On the other hand, small earthquakes can not be accurately determined by using Eq. (7) due to limited number of data. Lienkaemper also reported that the two horizontal components of the maximum amplitude were not required to be simultaneous by Gutenberg and the periods of the maximum amplitude did not al ways lie between 18 to 22 sec, actually as low as 12 sec and as high as 23 sec for some cases. Gutenberg (1945b,c) also defined a body-wave magnitude to classify shallow and deep earthquakes based on P and S waves in the follo wing form: where T is the period related to the maximum amplitude A and q(�.h) is the correction term associated with epicentral distance (�) and focal depth (h). Gutenberg (1945a,b) also provided tabulations for the calculation of this term.The maximum amplitude wa8 selected in several ways: (a) the vertical or composite horizontal component of P phase; (b) the vertical or composite-horizontal component of PP phase; and (c) the composite horizontal component of S phase. As only one horizontal component seismogram is available, a value of the maximum amplitude multipled by -/2 or 1.4 is taken into account. Before 1950, the intermediate-period instn,unents were commonly operated, thus the medium-period wave motions were used for the determination of this body-wave magnitude. After careful exam ination, Abe and Kanamori (1980) stated that in the text of Gutenberg and Richter (1954), for IDB>6.9, the period of P waves used for the determination of magnitude is mainly of from 4 sec to 11 sec with a predominent period of about 7.8±2.3 sec for shallow events, 6.4±1.8 sec for intermediate-depth events and 5.5±1.4 sec for deep events. Gutenberg and Richter (1954) stated that the magnitude for well-observed earthquakes was assigned to the tenth of the unit, with an error less than two tenths, and for the majority of earthquakes, the magnitude was given to the nearest quarter unit. The M8(GR) and mB were originally adjusted to coincide near M=7, but were later found to be linearly divergent. Several lin ear relations were deduced for the two magnitudes by Gutenbetg and Richter in a series of papers. Finally, Gutenberg and Richter (1956a) related Ms(GR) to mn in the form: This formula was applied by them to calculate the mn from M8(GR) for the earthquakes whose IDB values could not be determined. From Gutenberg's original note, Abe and Kanamori (1980) found a sign error in the expression for IDB-Ms(GR) . They revised this error and deduced a new formula: mn = 0.5 7 Ms(GR) + 3.0 (10) However, both Eq. (9) and Eq. (10) can not fit the so-called class 'a' data for large earthquakes listed in Geller and Kanamori (1977). But, on the other hand, Gutenberg and Richter (1956a) showed that Eq. (9) fitted the data of IDB vs. Ms(GR) very well. A close examination of Gutenberg and Richter's original data, Abe and Kanamori (1980) stressed that the ms value (body-wave magnitude calculated from M8(GR) through Eq. (10)) used in their paper was actually a certain weighted average of IDB and Ms(GR) rather than the real m8 • Lienkaemper (1984) stated that MaR used in the text of Gutenberg and Richter was calculated in a form: MaR=f1M8(GR) + f2Mn, where Mn=l.33(mB-l.75). For some.
Ms and mn to compute Ma R held for all events. Hence, Eq. (9) as well as Eq. (10) is not a good fit to the data points of mn vs. M8(GR). Besides, Abe and Kanamori (1980) also pointed out that the two equations were determined from the data set which consists of events with Ms(GR) in a limited range of from 6 to 7.5. For large events, Abe and Kanomori (1980) deduced a new conversion formula for mn and M8(GR) in the form: Abe (1984) where A is the peak amplitude, T is the period of the peak amplitude and .6. Ms(Vertical) be greater than Ms(Horizontal), the observed differences are negligible (Hunter, 1972;Abe, 1981). The catalog by Gutenberg and Richter (1954) only includes events which occurred before 1954; while the WWSSN was installed after 1960. Hence, it is impossible to compare Ms(GR) and Ms directly. However, in the Rothe's catalog (1969), Ms(GR) was also used. Abe and Kanomori (1980) compared Ms and Ms(GR) from Rothe's catalog and concluded that Ms(GR) is higher than Ms by about 0.1 on the average. Lienkaemper (1984) stressed that Ms(GR) and Ms of PDE differ only slightly for shallow earthquakes (h<40 km) and one could treat PDE average Ms as directly comparable to MaR with correction. He also mentioned that adding 0.06 to Ms values published in Abe (1981) for events between 1910 to 1952, h<40 km would adjust them to a scale compatible with PDE Ms. Since the installation of the WWSSN in the early 1960's, the body-wave magnitude has been determined almost exclusively from the vertical component of the P wave ground motions at a period of approximately 1 sec through Eq. (8) and represented by mn. The difference between mn and mn has been studied by numerous authors. Guyton (1964) stated that the mn values for a single earthquake, determined from body waves at different seismographic stations, commonly vary by 0.5 or more, despite corrections for differences in epicentral distance among the stations. This variation, which is related to the differences in amplitudes of a factor of 3 or more, is generally due to azimuthal, instrumental and geological differences among the stations . Romney (1964) and Geller and Kanamori (1977) reported that the mn values are about 0.3-0.6 units higher than the mb values. Abe and Kanamori (1980) expressed that mn is systematically larger than mb by about 1.3 on the average for events with mn>7.
For 5.5<ms<7.8, Abe (1981) stated that mb is lower than ms by about 0.4-1.1 units. He also deduced a re lation for the two magnitudes in the form:

) Hsu's Magnitude
In order to determine the magnitude for Taiwan earthquakes, Hsu (1971) by Wang( 1985). As shown in Figure 1, for mb>5.56, Ms(Taiwan) is higher than Ms(Japan) and vice verse for m1i<5.56. Hence, the MH might be overestimated for mb<5.56 and underestimated for m1i>5.56. The magnitudes of earthquakes in Japan and some larger earthquakes in Taiwan are routinely detennined by the Japanese Meteorological Agency (JMA, formerly Central Mete orological Observatory) by using the formula obtained by Tsuboi (1951): 731log.6. -0.83 (17 ) where A is either the larger value of the maximum amplitudes along two horizontal comp� nents or the composite value of the two maximum amplitudes in µm and .6. is the epicentral distance in km. This magnitude was denoted as Mu in  and . Hayashi and Abe (1984) reported that the average period of wave motions used for detennining MJ is about 3 sec and this magnitude agrees very well with Ms· However, MJ deviates very systematically from Ms as Ms decreases, and MJ is overestimated by as much as 0.6 at Ms=4.
(6) Kawasumi's Intensity Magnitude Kawasumi (1943) defi ned a magnitude M1 (denoted by MK in his papers) based on the intensity value at an epicentral distance of 100 km. The intensity scale is the Japanese scale in 8 degrees from 0 to VII, which has been used in Ta iwan by combining VI and VII to be VI. The formula for the conversion of intensity of degree I and magnitude M1 as the epicentral distance (.6.) is not equal to 100 km is in the form: I= M 1 + 2ln(100/ .6.) -0.00183(.6. -100) .

(7) Moment Magnitude
The seismic moment M0=µAu, where µ is the shear modulus, A is the fault area and u is the spatial average slip on the fault during the earthquake occurrence, was first applied by Aki (1966) to quantify earthquake. The seismic moment can be related to the energy release in earthquakes. Aki (1966Aki ( , 1967 showed that the amplitude of very long period waves is proportional to M0 and Ben-Menahem et al. (1969) also stated that the far-field static-strain field is also proportional to M0• Besides, because M0 does not saturate, it is a good parameter to represent the size of great earthquakes and has been applied to define moment magnitude by Kanamori (1977) and Hanks and Kanamori (1979). Kanamori (1977) related the seismic energy {E8) given by M0/(2 x 104) to a moment magnitude using the formula by Gutenberg and Richter (1956b): logE s = l.5M s + 11.8 The moment magnitude (Mw) is defined as under an assumption that stress drop is constant. In Eq. (22), M0 is in the unit of dyne-cm. Hanks and Kanarnori (1979) stated that Eq. (22) is uniformly valid for 3<ML<7, 5<M8<7.5 and Mw>7.5. The M0 values for larger events can be found ' in the EDR. According to the method proposed by Bolt and Herraiz (1983), Li and Chiu (1989) estimated the seismic moment of Taiwan earthquakes from the simulated Wood-Anderson seismograrns. Their resultant formula is in the form: logM 0(LC) = (16.74±0.20) + (l.22±0.14)log( C x D x .6.)

(23)
where C is the peak-to-peak amplitude, Dis the duration between the S-arri val and the onset of the signal with amplitude of C/ d and .6. is the epicentral distance. They stated that the optimum estimation for seismic moment Clli1 be obtained as d=2.

RELATIONS BETWEEN MAGNITUDE SCALES
The relations between magnitudes obtained by numerous authors will be described as follows. It is noted that the data points of mb-MH clli1 not be described by a single regression equation due to high dispersion (Wlli1g lli1d ), thus will not be discussed further. Basically six groups of relations are discussed.
(1) Relations of ML vs. Mn, ML vs. MH and ML vs. fib Three reltions of ML vs. Mn were studied by three groups of authors. They are by Liaw and Tsai (1981);  Hsu (1971Hsu ( , 1980Hsu ( , and 1985 contains the most complete instrumentally-determined seismic data during 1900-1978. It is necessary to compare local magnitude ML, which has been used since 1973, with MH before the establishment of a complete catalog for Taiwan earthquakes. Yeh et al. (1982) first related MH to their local magnitude ML (Yeh) in the form: A relation between the two magnitudes was. determined by Yeh and Hsu (1985) in the following form: �( 2 8) Cheng and Yeh (1989) obtained a slightly different form for the relation between the two magnitudes: The three equations are shown in Figure 3. It is evident that for MH>6, the ML values  Yeh and Hsu (1985) in dotted line and Cheng and Yeh (1989) in solid line. by Cheng and Yeh (1989). The three regression equations are shown in Figure 4. (2) Relations of Mv vs. mb and Mv vs. Ms Wang and Chiang (1987) compared Mv with mb and Ms for shallow earthquakes with focal depth less than 40 km and deep ones with focal larger than 40 km. The data points for The two regression equations are very similar. As the previous mention, the MH was originally defined based on the surface-wave magnitude M8• A comparison between the two magnitudes is significant. Figure 6  Although the MH was determined from local seismic data, it is like the surface-wave mag nitude Ms. However, from Eq. (38), as MH>6.3, MH<Ms and as MH<6.3, MH>Ms.
It is also interesting to compare MH with mb because the determination of MH was actually originally from mb through a conversion formula of Ms and rub. Figure 7 shows Although  suggested that M1 is not an appropriate magnitude to quantify Taiwan earthquakes, for the purpose of reference, the relation between M1 and MH for M1<8 is presented as: (4) Relation of M8(GR) vs. IDB Ms(GR) and mB are two magnitudes scales used by Gutenberg and Richter to quantify earthquakes before 1954. Their relation for Taiwan earthquakes is in the form: by . This equation is different from that obtained by Gutenberg and Richter (1954): for global earthquakes.
by  The two regression equations for Taiwan earthquakes agree closely with the average seismic moment-magnitude relations for the Pacific plate margin earthquakes obtained by Nuttli {1983). But the Ms-mb relation for Taiwan earthquakes is different from that for the Pacific plate margin ones by Nuttli (1983). Seismic moment calculated from very long-period surface waves is associated with static property of the fault; while the Ms determined from surface waves with period of about 20 sec and the mb determined from body waves with period of about 1 sec are both related to kinematic rupture on the fault. Given results might show the tectonics in the Taiwan region are similar to that in the whole Pacific plate margin but the rupture process of earthquake in the former might be different from the average one in the latter. The relation between M0 and ML is in the form:   Li and Chiu (1989) in dashed line and Wang et al.

463
The selection of the "d" value in Eq. (23) is questionable. The optimum value chosen by Bolt and Herraiz (1983) was the time between the S (with amplitude C) onset and the point having an amplitude c/C=l /3, while Li and Chiu's result is c/C=l/2. A physically reasonable interpretation about the d value is needed before the use of the Bolt and Herraiz's technique (1977) to determine M0 from local seismograms.

CONCLUSIONS
From the above discussion, several points can be derived as follows: 1. The ML-MD relations obtained by three groups of authors are similar.
2. The three ML -MH re lations obtained by Yeh and his coauthors are essentially the same.
3. The ML-mb re lations obtained by Shin (1986) and Wang et al. (1989) are close to each other even their data sets are different. But they are remarkably different from that obtained by Cheng and Yeh (1989). 4. The MD-mb relations obtained by Shin (1986) and Wang and Chiang (1987) are almost the same. 5. Although the formula to determine the MH by Hsu (1971) was originally defined based on a Japanese M8-mb conversion relation and determined from local seismograms, the correction between MH and M8(GR) as well as MH and Ms for Taiwan earthquakes is good enough. Consequently, although Hsu's magnitude was determined from local seismograms, it is like a surface-wave magnitude in the practical use.