Appendix

A.1 Magnitude

A.1.1 Definition of Magnitude

The scale of an earthquake was empirically evaluated using the intensity of groundmotion at a certain distance from the earthquake before the related theories wereestablished in seismology. Then, it was theoretically established that the origin ofan earthquake is faulting on a plane in the ground, and the seismic moment M0

generated from this faulting represents the scale of the earthquake (Sect. 2.1.1).However, before the CMT inversion was developed, it would take a long time tocalculate M0 because a variety of information and complicated processing wererequired. Empirical methods are therefore still used even today, and the scale of anearthquake so obtained is simply called magnitude (M for short).

The first M scale was conceived by Richter [27] for an earthquake in southernCalifornia, approximately 30 years before the theory of M0 was established [2].Therefore, media in USA and Europe often call M the Richter scale. Richter’s Mwas defined as the common logarithm of the maximum amplitude A in μm, in ahorizontal component of the Wood-Anderson seismograph (natural period 0.8 s,damping constant 0.8, static magnification 2,800) located at an epicentral distance� (Sect. 1.1) of 100 km. This is the local magnitude (ML for short). In reality, thereis not necessarily a seismograph at � = 100 km. Therefore, a correction term CL isgiven for � as

ML = log A + CL . (A.1)

Subsequent definitions of M follow thisway.However, in Richter [27], A ismeasuredin units of mm, and the sign of the correction term is inverted. Table I in this paper isrewritten for CL as in Table A.1. The paper also shows that the correction term from200 km< � < 600 km can be approximated by a linear function of log�. Whenthis is applied to Table A.1, we obtain the approximate formula

© Springer Nature Singapore Pte Ltd. 2021K. Koketsu, Ground Motion Seismology, Advances in Geological Science,https://doi.org/10.1007/978-981-15-8570-8

307

308 Appendix

Table A.1 Correction term given to ML for � in km (based on Utsu [37])

� (km) 30 50 100 150 200 250 300 400 500 600

CL −0.90 −0.37 0.00 0.29 0.53 0.79 1.02 1.46 1.74 1.94

CL = 3 log� − 6.37 . (A.2)

The approximate formula for the correction term is called a calibration function[37].

As can be seen fromTableA.1, the ML considers only relatively short-range earth-quakes with� less than 600 km. For teleseismic earthquakes (Sect. 3.1.11), Guten-berg [7] proposed another magnitude. He noticed that the surface waves (Sect. 1.2.4)with periods of around 20 s prevailed when a teleseismic earthquake was observed.He then defined the surface wave magnitude

Ms = log A + Cs , (A.3)

where A is the maximum amplitude of the surface wave horizontal ground motion.The groundmotionwas observed by the two horizontal components of a long-periodseismograph (Sect. 4.1.1) at that time, and the amplitudes measured on the twocomponents in μmwere just divided by the static magnification of the seismograph.A is obtained by taking the square root of the sum of squares, i.e., the vector sum,of the maximum amplitudes of the two components [37]. The correction term Cs isgiven in Table A.2 for � measured in degrees ( ◦ ), and this can be approximated bythe calibration function

Cs = 1.656 log� + 1.818 (A.4)

in the range 15◦ ≤ � ≤ 130◦ [7].However, the surface wave does not develop even for a teleseismic earthquake if

the depth h of the earthquake exceeds several ten kilometers; therefore, M must bedecided from body wave (Sect. 1.2.4) such as P and S waves. In the recording ofan intermediate-period seismograph (Sect. 4.1.1) with a natural period of severalto 10 s, the maximum amplitude A in μm and the period T in s of a body waveare obtained, both of which are measured on the recording paper. The maximumamplitude of an S wave is the vector sum of the two horizontal components, as forMs by surface waves. Using this resultant, Gutenberg [8] and Gutenberg and Richter

Table A.2 Correction term given to Ms for � in degrees ( ◦ ) (based on Gutenberg [7] and Utsu[37])

� (◦) 20 30 40 50 60 70 80 90 100 120 140 160

Cs 3.97 4.26 4.47 4.63 4.76 4.87 4.97 5.05 5.13 5.26 5.33 5.35

Appendix 309

hkm

Δ

Fig. A.1 The correction term q(�, h) of m B (based on Gutenberg and Richter [10], reprinted fromKoketsu [20] with permission of Kindai Kagaku)

[10] defined the body wave magnitude

m B = log

(A

T

)+ q(�, h) (A.5)

in addition to Ms . The correction term q(�, h) in (A.5) is a very complex functionof the epicentral distance � and the depth h, and is displayed in Fig. A.1 for theP wave part of a vertical ground motion. When horizontal ground motion by the Swave is used, PH written at the top of Fig. A.1 is added.

Cs in Ms and q(�, h) in m B should have been given so that their values would beconsistent with ML (h = 18 km is assumed for m B) when the calculations were per-formed for a same earthquake. It is known, however, that they deviate systematicallydue to the difference in the period band of ground motions employed (see Fig. A.2in the next section).

In their book “Seismicity of the Earth” [9], Gutenberg and Richter give the M’sof major earthquakes around the world from 1904 to 1952. This is the de factostandard of M , and various subsequent formulas of magnitude are often adjusted togive values similar to those in “Seismicity of the Earth” for the same earthquakes. Itis not necessarily clear how M values in “Seismicity of the Earth” were determined.According toAbe [1], it is assumed that Ms was usedwhen the depth of an earthquakewas less than 40 km and m B was used when the depth was equal to or more than 40km. As described above, in the process of proposing various M’s, for example, bycombining (A.1) and (A.2) for ML , we obtain

310 Appendix

Fig. A.2 Relationships of various M’swith seismicmoment M0 andmomentmagnitude Mw (basedon Utsu [37], reprinted from Koketsu [20] with permission of Kindai Kagaku)

log A = a M − b log X + c , M = ML , X = �, a = 1 , b = 3 , c = 6.37 .

(A.6)Such equations as (A.6) are termed attenuation relation. Since then, many studieshave been performed (e.g., Si and Midorikawa [30]) using statistical approachesand sophisticated methodologies have been developed for various ground motionamplitudes A, magnitudes M , and distances X [24].

To classify earthquakes by M , the terms, which are large earthquake (M ≥ 7),moderate earthquake (5 ≤ M< 7), small earthquake (3 ≤ M < 5), andmicroearthquake (M < 3), as well as the recent term, which is great earthquake(M ≥ around 81), are used in Japan [36]. Utsu [36, 37] states that “they are notaccepted internationally”; however, generally similar terms and classifications areused internationally.

A.1.2 Recent Magnitudes

When “Seismicity of the Earth” was published, in Japan, the JMA (Sect. A.2.1) usedtheWiechert seismograph (natural period 5 s, damping constant 0.55, staticmagnifi-cation 80) and the Ichibai strong motion seismograph (Sect. 4.1.2) as displacementseismographs. Because these were different in characteristics from the seismographsused by Gutenberg and Richter [9], in 1954, Tsuboi [33] derived Tsuboi’s formula

M = log A + 1.73 log� − 0.83 (A.7)

1The reason why this range has “around” is to include the Kanto earthquake (1923, M 7.9).

Appendix 311

Table A.3 Correction term in (A.8) for � and h in km (based on Katsumata [18])

h\� 100 200 300 400 500 600 700 800 900 1000 1200 1400

50 2.58 3.14 3.40 3.69 3.90 4.08 4.23 4.29 4.41 4.54 4.68 4.83

100 2.65 3.19 3.38 3.73 3.99 4.18 4.38 4.41 4.55 4.74 4.83 5.04

150 2.85 3.31 3.43 3.77 4.01 4.18 4.40 4.45 4.58 4.76 4.85 5.07

200 3.11 3.47 3.54 3.83 4.01 4.15 4.35 4.43 4.53 4.78 4.79 4.98

250 3.39 3.64 3.68 3.89 4.01 4.10 4.27 4.38 4.44 4.56 4.70 4.85

300 3.67 3.80 3.85 3.97 4.03 4.08 4.21 4.33 4.36 4.44 4.61 4.71

350 3.90 3.95 4.02 4.07 4.07 4.10 4.18 4.29 4.31 4.36 4.55 4.60

400 4.09 4.08 4.17 4.19 4.16 4.18 4.21 4.36 4.30 4.33 4.53 4.55

450 4.22 4.20 4.30 4.32 4.29 4.30 4.29 4.27 4.35 4.37 4.56 4.57

500 4.30 4.34 4.39 4.48 4.46 4.45 4.41 4.31 4.44 4.47 4.64 4.65

550 4.35 4.51 4.44 4.65 4.66 4.61 4.54 4.38 4.57 4.61 4.74 4.78

600 4.41 4.77 4.42 4.84 4.87 4.74 4.64 4.46 4.72 4.77 4.83 4.93

to obtain similar M’s to theirs for the earthquakes around Japan in “Seismicity of theEarth”. For Ms , the amplitudes inμmaremeasured on the twohorizontal componentsof a seismograph above and divided by the static magnification of the seismograph.A in (A.7) is the vector sum of the maximum amplitudes of the two components.

The Wiechert seismographs were then abandoned, so that the Ichibai strongmotion seismographs and intermediate-period seismographs with similar frequencycharacteristicswere used. Since 1994, in JMA, M has been determined by (A.7) usingthe D93-type seismograph, which are acceleration seismographs with displacementoutput by digital integration. Their natural periods are 10 s, and they are equippedwith a filter that blocks signals with frequencies higher than 10 Hz. This magnitudeis called JMA magnitude and sometimes abbreviated as MJ . As the natural periodof the Wiechert seismograph is 5 s while that of the D93-type seismograph is 10 s,maximum amplitudes should be read after removing components with periods longerthan 5 s from records [36]; however, this was not considered until 2000.

Furthermore, if the depth h of an earthquake exceeds 60 km, the formula byKatsumata [18]:

M = log A + K (�, h) (A.8)

is used. The correction term K (�, h) is given in Table A.3. (A.7) and (A.8) shouldbe determined to match their M’s with M’s in “Seismicity of the Earth”, which werecalculated by (A.3) and (A.5). However, the correction terms in (A.7) and (A.8) aresmaller by about two than those in (A.3) and (A.5). The reason for this is that �’s inthe former are given in km whereas those in the latter are given in degrees ( ◦ ).

In addition, short-period components are predominant in ground motions whenthe scale of an earthquake decreases (Sect. 2.3.5). Therefore, if the above M usingintermediate-period seismographs such as the Wiechert seismograph is smaller than5.5, the maximum amplitude AV in 10−5 m/s of vertical ground motion by short-

312 Appendix

period seismographs is used and M is calculated by the formula of Kanbayashi andIchikawa [15]:

M = log AV + 1.64 log� + C (A.9)

where C is 0.22 or 0.44 depending on the type of short-period velocity seismograph.However, if M using the intermediate-period seismographs is larger than M usingthe short-period seismograph by 0.5 or more, the former is adopted as it is [25].

The problem of the direct use of records of the D93-type seismograph mentionedabove became apparent during the Tottori earthquake (2000, Mw 6.7). The MJ ofthis earthquake calculated from the records was announced to be 7.3; however, thescale of the damage and the spread of aftershocks were less than those of the Kobeearthquake (1995, Mw 6.9, MJ 7.2 at that time). Thus, the validity of the value ofMJ was questioned, and a review committee was established by the JMA. By thedetailed examination in the committee, it was proven that MJ was larger by 0.06 onaverage due to the difference in the natural period of a used seismograph. In addition,what was more important than this difference was the difference in the installationplacement of a seismograph.While the previous seismographs were installed at JMAoffices in urban areas, the D93-type seismographs are installed in suburban areas.An urban area is often located in the center of a sedimentary basin, and groundmotion amplification frequently occurs. However, this is suppressed in a suburbanarea so that MJ is smaller by 0.22 on average. Based on these results, the filter inthe D93-type seismograph was modified and 0.22 has been added to MJ since then.For significant earthquakes from 1994 to then, the MJ ’s were recalculated usingrecords observed by seismic intensity meters (Sect. A.2.3) installed at JMA offices.According to this recalculation, the MJ of the Kobe earthquake was revised from 7.2to 7.3, while the MJ of the Tottori earthquake remained unchanged 2 [14].

ISC andUSGS determine Ms andm B of earthquakes in theworld by using recordssuch as observed by the WWSSN (World-Wide Standard Seismograph Network),which was developed for the purpose of detecting nuclear tests. In accordance withthe IASPEI (International Association of Seismology and Physics of the Earth’sInterior) recommendation in 1967, the determination is performed using the formulaof Vanek et al. [38]:

Ms = log

(A

T

)+ 1.66 log� + 3.30 , 20◦ ≤ � ≤ 160◦ (A.10)

where A and T are the vector-sum maximum amplitude in μm and the period in s ofa surface wave record observed by the Press-Ewing seismograph (natural period 15∼ 30 s) etc. The above A is the vector sum of the two horizontal components. How-ever, the vertical component of the record can be used instead, because A becomessmaller while T also becomes smaller on the vertical component. Comparing Ms

in (A.10) with Ms in (A.3) by Gutenberg [7], it should be noted that Vanek’s Ms

2There should remain something different because the Mw 6.7 of the Tottori earthquake is smallerby 0.2 than the Mw 6.9 of the Kobe earthquake.

Appendix 313

is approximately 0.2 larger than Gutenberg’s Ms in the case of T = 20 s, which isthe definition of the surface wave for Gutenberg’s Ms [25]. In addition, Utsu [36]mentioned that the maximum A/T should be searched in Vanek et al. [38] while inreality the above measurement is performed.

For body wave magnitudes, ISC and USGS also apply the formula in (A.5)for Gutenberg’s m B to body wave records observed by short-period seismograph(Sect. 4.1.1) such as Benioff seismograph (natural period 1 s). However, as A ismeasured only within 5 s from the initial motion of the P wave, the result is oftenquite different from m B due to this and the difference in the natural period of theseismograph used. It is therefore abbreviated as mb to distinguish it from m B . Inparticular, for large earthquakes, the short-period components of ground motions arerelatively small (Sect. 2.3.5) so that mb is much smaller than m B [37].

M is basically determined from the amplitudes of groundmotion records, althoughthe frequency characteristics of ground motions differ depending on the scale ofan earthquake. M can therefore be deviated if the frequency characteristics of aseismograph used or ground motions covered by M are different from the frequencycharacteristics associated with the scale of an earthquake. In general, as the scaleof an earthquake increases, the long-period component of ground motion increases,but the short-period component does not increase as much. For example, when anM 8 earthquake is compared with an M 6.2 earthquake, according to the scalinglaw of the ground motion spectrum (Sect. 2.3.5), the long-period component is 500times larger, while the short-period component is only 10 times larger. Thus, m B

and mb, which are determined from body waves mainly composed of short-periodcomponents and observed by a short-period seismograph, are not appropriate forrepresenting the magnitude of an M 8 class earthquake, as they saturate aroundM 6. Conversely, Ms , which is determined from surface waves mainly composed oflong-period components and observed by a long-period seismograph, gives a smallermagnitude for an M 5 class earthquake. Saturation also occurs even in Ms for a greatearthquake exceeding M 8.3 ∼ 8.5. The JMA magnitude MJ , which uses either anintermediate-period seismograph or a short-period seismograph depending on themagnitude of an earthquake, has a wider application range than Ms , m B , and mb

(Fig. A.2).In any case, all M’s calculated from the amplitudes of seismograms have limita-

tions. Therefore, Kanamori [16] returned to the seismic moment M0 (2.1.2), whichdirectly represents the magnitude of an earthquake, and proposed the moment mag-nitude Mw

3

log M0 = 1.5 Mw + 16.1 (A.11)

so that this matches the conventional magnitudes, especially Ms of a plate-boundaryearthquake (Sect. 2.1.1). Here, M0 takes a value in dyne·cm, but if it is given inN·m, 16.1 is replaced with 9.1. By the advance of analysis methods such as the CMT

3Although (A.11) and the symbol Mw first appeared in Kanamori [16], the phrase “moment mag-nitude” first appeared in Kanamori [17]. Kanamori [16] used the subscript in italic as in this book;however, Kanamori [17] used that in roman.

314 Appendix

inversion, M0 would be determined daily and Mw would be mainly used. However,as even for the determination of M0 seismograms are mostly used, saturation mayoccur unless records of very long period seismographs such as gravimeter andcrustal deformation data are used additionally. Correct Mw cannot be obtained forsmall earthquakes unless seismographs sensitive to short periods are used.

In the case of earthquakes in Japan, the M with the most complete dataset isof course the JMA magnitude MJ . Therefore, a relationship between MJ and M0

is often required. Takemura [32] obtained log M0 = 1.17 MJ + 17.72 for crustalearthquake (Sect. 2.1.1) in Japan. This was rounded up to the first decimal place as

log M0 = 1.2 MJ + 17.7 , (A.12)

which is more often used. For earthquakes along the Japan Trench and the easternmargin of the Sea of Japan, Sato [29] obtained

log M0 = 1.5 MJ + 16.2 . (A.13)

(A.13) is almost identical to (A.11), so that MJ ∼ Mw holds in the sea areas aroundJapan. However, a different formula is necessary in the land areas because surfacewave developed with intermediate periods result in larger displacement amplitudes.

A.2 Seismic Intensity

A.2.1 Characteristics of Seismic Intensity

Seismic intensity is “the intensity of ground motion at a certain place is representedby several grades categorized according to the human sensation, the size of the effecton surrounding objects, structures or nature, etc.” [37]As can be understood from thisdefinition, the seismic intensity can be determined instantaneously with no specialequipment, because it is determined from what observers feel and what they see. Inaddition, because there is little possibility for the seismic intensity to be misreportedor mismeasured and the seismic intensity holds robustness as data, this is publishedearliest among earthquake information, and many disaster prevention organizationsuse this as a standard when taking emergency measures such as personnel allocationand facility inspection [12]. In general, the higher the level of information, the morecomplicated the procedures needed. Therefore, it is more likely that unexpectedvalues or null results occur when information is needed urgently, and so it is difficultto use complex procedures in disaster prevention organizations. For example, wecannot say that automated systems for determining information such as hypocenterlocation and fault mechanism are currently robust and accurate.

However, seismic intensity forcibly expresses the ground motion, combining var-ious information such as amplitude, frequency, and duration, to give a simple integer

Appendix 315

Table A.4 Rossi-Forel intensity scale [5, 6] (rewritten by Richter [28])

I Microseismic shock Recorded by a single seismograph or by seismographs ofthe same model, but not by several seismographs ofdifferent kinds; the shock felt by an experienced observer

II Extremely feeble shock Recorded by several seismographs of different kinds; felt bya small number of persons at rest

III Very feeble shock Felt by several persons at rest; strong enough for thedirection or duration to be appreciable

IV Feeble shock Felt by persons in motion; disturbance of movable objects,doors, windows, cracking of ceilings

V Shock of moderateintensity

Felt generally by everyone; disturbance of furniture, beds,etc., ringing of some bells

VI Fairly strong shock General awakening of those asleep; general ringing of bells;oscillation of chandeliers; stopping of clocks; visibleagitation of trees and shrubs; some startled persons leavingtheir dwellings

VII Strong shock Overthrow of movable objects; fall of plaster; ringing ofchurch bells; general panic, without damage to buildings

VIII Very strong shock Fall of chimneys; cracks in the walls of buildings

IX Extremely strong shock Partial or total destruction of some buildings

X Shock of extremeintensity

Great disaster; ruins; disturbance of the strata, fissures in theground, rock falls from mountains

scale using ambiguous filters such as human sensation, objects, and structures. There-fore, the seismic intensity is not suitable formodern seismology based on quantitativeevaluation and theoretical reproduction. However, the history of seismic observationthat can support modern seismology is only about 100 years, and the accumulation ofdata is insufficient, as this time period is remarkably short compared with the occur-rence rate of strong ground motion at any particular place. This means that thereis still a role for seismic intensity as data. The seismic intensity also has anotherimportance as an interface with the society, including its use in disaster preventionorganizations.

According to the simple measurement of the seismic intensity, the origin of thiscan be traced back to the 19th century, and the seismic intensity scale, which aretables referred to for determining the seismic intensity, were born simultaneouslyworldwide. In Europe, de Rossi [5] in Italy and F. A. Forel in Switzerland startedindependently, and in 1884, the Rossi-Forel intensity scale [6] (Table A.4) wasmade by the cooperation of both. However, this scale has defects such as that it wasoutdated by the later advance of technology, that the area covered by X of the highestgrade was too wide, and that the descriptions depended too much on the situationof Europe [28]. Mercalli [23] proposed to improve the Rossi-Forel intensity scaleby resolving these defects. This was further revised by Cancani [3] and Sieberg [31]and resulted in the Mercalli intensity scale.

316 Appendix

Table A.5 History of the seismic intensity scale in Japan (based on JMA [12]). “(0) and “(−)”indicate “(unfelt)” and “(weaker)”, respectively. Arabic numerals in italic 0, 1, 2, · · · , 6 representChinese numerals. “+” and “-” in the last column indicate “Upper” and “Lower”, respectively

1884 1898 1908 1936 1949 1996

Feeble (0) 0 Unfelt O 0

Feeble Feeble 1 I I 1

Weak Weak (−) 2 II II 2

Weak 3 III III 3

Strong Strong (−) 4 IV IV 4

Strong 5 V V 5−5+

Severe Severe 6 VI VI 6−6+

VII 7

In Japan, K. Sekiya, who was one of the founders of the first seismological soci-ety in the world and later the first professor of seismology in the world, defined theseismic intensity scale in Column 1 of Table A.5 which consists of four grades of“feeble”, “weak”, “strong”, and “severe” in 1884 [12]. The annual reports of theGeographic Bureau of the Ministry of Home Affairs and later the Central Meteo-rological Observatory, which was independent from the Bureau, included seismicintensities based on this. Since 1898, in annual reports of the Central MeteorologicalObservatory, “feeble (unfelt)” was added under “feeble”. “weak” was divided into“weak (weaker)” and just “weak”. “strong”was replacedwith “strong (weaker)”, and“severe” was divided into “strong” and “severe”. As a result, the seismic intensityscale was expanded to seven grades (Column 2 of Table A.5). After 1908, the sevengrades were written using Chinese numerals corresponding to Arabic numerals 0, 1,2, · · · , 6 (Column 3 of Table A.5).

Seismic intensity observation in Japan is determined by law to be the business ofthe Central Meteorological Observatory, and after the reorganization in 1956 to bethe responsibility of the JapanMeteorological Agency (JMA). Seven-grade intensityscales have been in use from 1898 to the present in Japan, though, since 1936, Romannumerals have been used instead of Chinese numerals (Column 4 of Table A.5), andArabic numerals are used afterwards. There were also revisions in 1949 and 1996(Columns 5 and 6 of Table A.5), which are explained in Sect. A.2.3.

A.2.2 Sensory Seismic Intensity

The Mercalli intensity scale, whose history is explained in the previous section, hasspread to the United States, and theModifiedMercalli intensity scale (“MM intensity

Appendix 317

scale” for short, 12 grades), which Wood and Neumann [40] improved further, hasbeen widely used worldwide. The MM intensity scale simplified by Richter [28] isshown in Table A.6. Conventional seismic intensities such as represented by theMMintensity scale are here called sensory seismic intensity.

However, in Europe, where the Mercalli intensity scale originated, there was amovement to internationally standardize seismic intensity scale on a basis otherthan the MM intensity scale. In 1964, UNESCO recommended the use of the MSKintensity scale (12 grades; same as theMM intensity scale) byMedvedev, Sponheuer,and Kárník [22]. The MSK intensity scale has been adopted in Europe and hasdeveloped into the European Macroseismic Scale (EMS). However, this approachwas not taken upmuch outside of Europe, because of differences in architectural styleand lifestyle of each country and the use of compilations of prior seismic intensitydata. Although Japan investigated this scale, theMSK intensity scale was not adoptedbecause detailed damage investigation is often necessary and hence the approach isnot suitable for quick reports right after earthquakes [12, 21].

The JMA intensity scale is next explained as a sensory seismic intensity scaleused in Japan. The columns of Table A.5 show its history: Column 5 indicates theaddition of intensity VII as the 7th grade in 1949. This addition was triggered bythe Fukui earthquake (1948, M 7.1, Ms 7.3) in June of the preceding year [37].This earthquake caused 3,769 fatalities and 36,184 collapsed houses [35], and it wasthe most damaging earthquake after the Kanto earthquake (1923) until the Kobeearthquake. In particular, in the Fukui basin just above the source region of the 1948event, a collapse rate of houses over 90% was observed everywhere. At that time,the officers of the Central Meteorological Observatory shared the feeling that thedescription of intensity VI: “Severe shaking. Shaking such that houses are collapsed,landslides and cracks in the ground are caused” was insufficient for these obser-vations. Then, in January 1949, the operational rules for earthquakes and tsunamiswere revised, and intensity VII was added: "Extreme shaking. The collapse rate ofhouses reaches 30% or over. Landslides, cracks and faults in the ground are caused.”(Table A.7). After the addition, intensity VII was not announced for a long time, butwas announced with an assigned area for the first time for the Kobe earthquake.

For historical earthquakes, in the period during which there were no seismo-graphs, the only data available are sensory seismic intensities and arrival times, whichare roughly estimated from descriptions in historical documents. However, buildingsin the historic periods are likely to be more weakly constructed than modern build-ings. Therefore, seismic intensities are determined by comparing the descriptionswith Table A.5 or A.6 and can be adjusted by subtracting certain values. For exam-ple, Usami [34] used 0.5 ∼ 1 as the certain values for historical earthquakes in Japanand seismic intensities in the JMA intensity scale. If a distribution of concentric cir-cles appears by plotting the adjusted intensities on a map and drawing isoseismals,which show the range of each intensity, it can be assumed that the vicinity of thecircle center is the epicenter, and the magnitude of the historical earthquake can alsobe obtained from relationships between epicentral distances and seismic intensities.4

Usami [35] is a compilation of such works in Japan.

4Many relationships are listed in Sect. 3.5 of Utsu [36].

318 Appendix

Table A.6 Modified Mercalli intensity scale [40] with the simplified descriptions [28]

I Not felt. Marginal and long-period effects of large earthquakes

II Felt by persons at rest, on upper floors, or favorably placed

III Felt indoors. Hanging objects swing. Vibration like passing of light trucks. Durationestimated. May not be recognized as an earthquake

IV Hanging objects swing. Vibration like passing of heavy trucks; or sensation of a jolt likea heavy ball striking the walls. Standing motor cars rock. Windows, dishes, doors rattle.Glasses clink. Crockery clashes. In the upper range of IV wooden walls and frame creak

V Felt outdoors; direction estimated. Sleepers wakened. Liquids disturbed, some spilled.Small unstable objects displaced or upset. Doors swing, close, open. Shutters, picturesmove. Pendulum clocks stop, start, change rate

VI Felt by all. Many frightened and run outdoors. Persons walk unsteadily. Windows,dishes, glassware broken. Knickknack, books, etc., off shelves. Pictures off walls.Furniture moved or overturned. Weak plaster and masonry D cracked. Small bells ring(church, school). Trees, bushes shaken (visibly, or heard to rustle)

VII Difficult to stand. Noticed by drivers of motor cars. Hanging objects quiver. Furniturebroken. Damage to masonry D, including cracks. Weak chimneys broken at roof line.Fall of plaster, loose bricks, stones, tiles, cornices (also unbraced parapets andarchitectural ornaments). Some cracks in masonry C. Waves on ponds; water turbid withmud. Small slides and caving in along sand or gravel banks. Large bells ring. Concreteirrigation ditches damaged

VIII Steering of motor cars affected. Damage to masonry C; partial collapse. Some damage tomasonry B; none to masonry A. Fall of stucco and some masonry walls. Twisting, fall ofchimneys, factory stacks, monuments, towers, elevated tanks. Frame houses moved onfoundations if not bolted down; loose panel walls thrown out. Decayed piling broken off.Branches broken from trees. Changes in flow or temperature of springs and wells.Cracks in wet ground and on steep slopes

IX General panic. Masonry D destroyed; masonry C heavily damaged, sometimes withcomplete collapse; masonry B seriously damaged. (General damage to foundations)Frame structures, if not bolted, shifted off foundations, Frames racked. Serious damageto reservoirs. Underground pipes broken. Conspicuous cracks in ground. In alleviatedareas sand and mud ejected, earthquake fountains, sand craters

X Most masonry and frame structures destroyed with their foundations. Some well-builtwooden structures and bridges destroyed. Serious damage to dams, dikes, embankments.Large landslides. Water thrown on banks of canals, rivers, lakes, etc. Sand and mudshifted horizontally on beaches and flat land. Rails bent slightly

XI Rails bent greatly. Underground pipelines completely out of service

XII Damage nearly total. Large rock masses displaced. Lines of sight and level distorted.Objects thrown into the air

Masonry A, B, C, D:

A Good workmanship, mortar, and design; reinforced, especially laterally, and boundtogether by using steel, concrete, etc.; designed to resist lateral forces

B Good workmanship and mortar; reinforced, but not designed in detail to resist lateralforces

C Ordinary workmanship and mortar; no extreme weaknesses like failing to tie in atcorners, but neither reinforced nor designed against horizontal forces

D Weak materials, such as adobe; poor mortar; low standards of workmanship; weakhorizontally

Appendix 319

Table A.7 The JMA intensity scale revised in 1949 [4] (at the time of revision, JMA was calledthe Central Meteorological Observatory). The supplements were added in 1978 [11]

Grade Description Supplement

O Unfelt. The degree to which something isrecorded on a seismograph without beingfelt by a human body

Even if the slight shaking of a hangingobject is seen or a rattling sound is heard, itis unfelt that a human body does not feelthe shaking

I Feeble shaking. Shaking felt only bypeople at rest and especially people withkeen sense of shaking

People at rest slightly feel it but do not feelit for a long time. Most standing people donot feel it

II Light shaking. Shaking felt by manypeople, and the slight movement of asliding door can be seen

People can sense the movement of ahanging object, and people can slightly feelthe shaking when standing, but hardly feelit when moving. Some wake up even whensleeping

III Weak shaking. Shaking such that housesshake, sliding doors rattle, hangingobjects like a light shake considerably, andthe movement of the water surface in avessel is recognized

So startling that people asleep are wakenup but do not rush out of houses, and feelno fear. Many people sense it outdoors, butsome walking do not

IV Moderate shaking. Shaking such thathouses shake severely, unstable vases etc.fell down, and the water in vesselsoverflows. People walking also sense it,and many people rush out of houses

People asleep jump out of bed and feel fear.People can see utility poles and treesshaking. In ordinary houses, even if someroof tiles are shifted, it is not likely to bedamaged yet. People feel dizzy

V Strong shaking. Shaking such that cracksare made in walls, gravestones and stonelanterns fall down, and chimneys, stonewalls, etc. are damaged

It is quite difficult to stand. Minor damageto ordinary houses begins to occur. Softground can crack or collapse. Unstablefurniture falls

VI Severe shaking. The collapse rate of housesis less than 30%. Shaking such thatlandslides and cracks in the ground arecaused, and many people cannot stand

Walking is difficult and crawling isrequired to move

VII Extreme shaking. The collapse rate ofhouses reaches 30% or over. Landslides,cracks and faults in the ground are caused

A.2.3 Instrumental Seismic Intensity

In Japan and Taiwan, seismic intensity observations are performed by governmentsnationwide, and intensity data have been positively utilized for measures follow-ing earthquakes. For the official observations in Japan, observers at JMA officesdetermine seismic intensities by comparing their own sensations and observations ofindoor and outdoor with the descriptions and supplements of the JMA intensity scaleshown in Table A.7 (Sect. A.2.2). This determination leads to the characteristics of

320 Appendix

sensory seismic intensity such as simplicity, immediacy, and robustness, but it worksin reverse causing the following problems [12]:

(a) Because the intensity of ground motion is strongly affected by the ground, thereis no guarantee that the seismic intensity of a JMA office is a representative valueof the surrounding region.

(b) It is inevitable that differences arise depending on the type of building in whichthe observer is located.

(c) There are individual differences among observers.(d) Sensory seismic intensity cannot be observed in an unmanned place.

In addition, after the Kobe earthquake, in which intensity VII was announced forthe first time since its establishment in 1949, the JMA pointed out the followingproblems [12]:

(e) Because the definition of seismic intensity VII is strict as 30% or more of thecollapse rate of houses, several days or more are required for detailed investi-gation (in the Kobe earthquake, it took three days, and several more days for itsaffected area).

(f) For seismic intensities V and VI, the ranges of the corresponding damage aretoo large and it is difficult to perform appropriate measures.

To solve these problems of the sensory seismic intensities, the JMA developedseismic intensity meters, whichmeasures the seismic intensity, and distributedmanyof them nationwide. This solves the problems of individual difference, unmannedobservation, and immediacy of intensityVII (problems (c), (d), and (e)). If the seismicintensity meters are densely distributed, the seismic intensity is observed at pointswith various ground conditions, so that the problem (a) is solved. The problem (b)is also solved, if the seismic intensity meters are located away from buildings.

The seismic intensity meter is a device to calculate seismic intensity from strongmotion records, and its specifications are determined so that calculated seismic inten-sity can reproduce sensory seismic intensity. Therefore, a new processing procedureis established by collecting strong motion records at sites where sensory seismicintensities are obtained. Using this procedure, a seismic intensity is calculated sothat the result is close to the sensory seismic intensity. To increase the accuracy byincreasing the data, strong motion records at locations other than the JMA offices areused as far as possible when the questionnaire seismic intensity [26] are available[12].

The instrumental seismic intensity thus measured is a continuous quantity, andis rounded up to the 1st decimal place. The integer value is then obtained by roundingthe instrumental seismic intensity off (6.5 or more and less than 0.5 are regarded as7 and 0, respectively). This has been the official seismic intensity of JMA in Arabicnumeral form since 1996. In addition, to cope with the problem (f), intensities 5and 6 were divided into “Upper” and “Lower”, and finally the JMA intensity scalereached 10 grades (Column 6 of Table A.5). The above is the major revision ofthe JMA intensity scale after the Kobe earthquake. While the descriptions in anintensity scale play a deciding role in the determination of sensory seismic intensity,

Appendix 321

Fig. A.3 95-type seismicintensity meter consisting ofthe measurement unit (left)and processing unit (right)(reprinted from the JMAwebsite)

the processing procedure instead plays this role in the determination of instrumentalseismic intensity. However, the descriptions of the JMA intensity scale survive in theexplanatory table after being revised to be suitable for Japan in 1996 [13].

The seismic intensity meter, which is the key component of the JMA intensityscale revised in 1996, consists of measurement and processing units, as shown inFig. A.3. Because the seismic intensity meter observes strong motions and calcu-lates seismic intensities from the observed records, the measurement unit is actuallya strong motion seismograph (Sect. 4.1.2). The seismic intensity meters deployednationwide by the JMA are called 95-type seismic intensity meter and are designedaccording to the following specifications [12]. Seismic intensity meters used by localgovernments are also designed according to these specifications. For the measure-ment unit,

(1) Three-component acceleration seismograph with servo mechanism, whichcan observe up to ±2048 gal of each component and has an effective resolutionof 8 mgal or less, and flat frequency characteristics for acceleration at 0 ∼ 50Hz.

(2) Trigger judgment is performed every 10 s, and transfer to the processing unitand data storage are performed every 60 s (therefore, the delay time is 10 s atthe longest).

(3) Transfer to theprocessingunit anddata storage are performeddigitally (Sect. 4.2),and the records for about 50 minutes can be stored in an IC memory card.

(4) The communication function is by ground line or satellite line.(5) By receiving 40 kHz standard radio wave JG2AS, continuous time calibration

with accuracy within 0.01 s is possible.(6) With an external battery, everything including the processing unit can be operated

for at least 3 hours even when the power fails.(7) Everything including the processing unit can withstand intensity 7.

The calculation of the instrumental seismic intensity by the processing unit issummarized as follows [12].

(1) The frequency spectrum (Sect. 4.2) is taken for each component of groundacceleration.

(2) The filter (Sect. 4.3.1) in Fig. A.4 is applied to the spectrum of (1).(3) The inverse Fourier transform (Sect. 4.2.2) is applied to the filtered spectrum

of (2) for the filtered acceleration waveforms of the three components.

322 Appendix

Fig. A.4 The filter appliedto ground accelerations forcalculating instrumentalseismic intensities (based onJMA [12], modified fromKoketsu [20] withpermission of KindaiKagaku)

f (Hz)1010.1

0.1

0.01

1

0f

√1/ ffilter

gain

(4) The vector sum of filtered accelerations in the three components is calculated ingal at every t (time) and the maximum a0 is obtained under the condition thatthe total time when the vector sum is equal to or greater than a0 is 0.3 s or more.

(5) I = 2 log a0 + 0.94 is rounded to the second decimal place and the result is thenrounded down to the second decimal place to obtain the instrumental seismicintensity. Intensities 5 and 6 are designated as “Upper” if they are 5.0 and 6.0 orabove, or “Lower” for other cases.

The filter of (2) (Fig. A.4) is the combination of the three filters:

· The filter of (1/ f )1/2

· high-cut filter (1 + 0.694X2 + 0.241X4 + 0.0557X2 + 0.009664X8 +0.00134X10 + 0.000155X12)−1/2, X = f/ fc, f c = 10 Hz

· low-cut filter (1 − exp( f/ f0)3))1/2, f0 = 0.5 Hz

where f is frequency.In this way, the instrumental seismic intensity is not simply calculated from the

maximum acceleration, but is calculated by adding the effects of the frequency andduration of strong motion, which affect human sensation and damage, using the filterof (2) and the total time of a0 of (4). In particular, the fact that the center frequencyf0 of the filter was taken to be 0.5 Hz, which is a considerably low frequency, seemsto be a result of emphasizing the correlation with building damage for large seismicintensity.

The controversy over whether acceleration, which represents the force of strongmotion, or velocity, which represents energy, most affects damage and human sen-sation, has not been settled. It can be understood that the use of the (1/ f )1/2 filter,which is not the use of no filter for acceleration nor the use of the 1/ f filter forvelocity, is at an intermediate position between the two.

Kawasumi [19] obtained I ′ = 2 log a + 1.2 where I’ is the boundary between theintensities I and I − 1 and a is the maximum acceleration observed by Ishimoto-type accelerograph (Sect. 4.1.2). Utsu [37] converted this into I = 2 log a + 0.7 bysubtracting 0.5. I = 2 log a0 + 0.94 in (5) is similar to this equation; however, theconstant changes from 0.7 to 0.94. This difference should be due to the difference

Appendix 323

in seismographs and the detailed investigation for the instrumental seismic intensity(Sect. A.2.3) [12].

Outside of Japan, although no special instrument such as the seismic intensitymeter is prepared, many strong motion seismographs have been deployed, and theinstrumental seismic intensity is calculated using their observational records. How-ever, as ordinary equipment without processing units is used, complicated process-ing such as the JMA requests, is not possible; thus, simple indexes such as the peakground acceleration (PGA, Sect. 4.1.2) and peak ground velocity (PGV,maximumamplitude of a velocity record) are used. For example, in California, the intensity Iin the MM intensity scale can be calculated from

I = 3.66 log PGA − 1.66 , V ≤ I ≤ VIII ,

I = 3.47 log PGV + 2.35 , V ≤ I ≤ IX (A.14)

as shown by Wald et al. [39], and (A.14) is widely used not only in California butalso in other regions and countries.

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World Conference On Earthquake Engineering, 1, A-2, 143–153.23. Mercalli, G. (1902). Sulle modificazioni proposte alla scala sismica De Rossi-Forel. Bollettino

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Index

AABIC, 104, 302Acceleration feedback, 272Acceleration seismograph, 267, 321Accelerograph, 267Accelerometer, 267Active fault, 35A/D conversion, 274A/D converter, 274Addition rules, 144Adjoint equation, 255Adjoint method, 253Adjugate matrix, 128Aftershock distribution, 84, 104AIC, 302Airy phase, 169, 177Aki-Larner method, 192, 225Aliasing, 160, 279Alternating tensor, 244Amplification factor, 141Amplification factor matrix, 141Amplitude characteristics, 266Amplitude spectrum, 95, 276Analog, 274Anelasticity, 18, 241Angle of incidence, 75Anisotropy, 9Anti-aliasing filter, 279ART, 251Artificial reflection, 215, 223Asperity, 107Asperity model, 107Attenuation, 2, 18Attenuation relations, 310Auxiliary plane, 83Azimuth, 74Azimuthal angle, 179, 204

BBackward difference, 220Band-pass filter, 285, 290Barrier model, 108Bayes’ theorem, 299Beach ball solution, 84Beat, 165Beltrami’s theorem, 45Bending method, 208Benioff seismographs, 313Bessel equation, 17, 58Bessel function, 58, 121, 123Bi-directional, 95Bilateral faulting, 95, 99Bisection method, 207Body force, 6, 9, 12Body force equivalent, 41, 44, 94Body wave, 15, 53, 74, , 308Body wave magnitude, 309Boundary, 15Boundary method, 192, 224Boundary wave, 15Bracketing, 207Branch cut, 153Branch point, 152, 158Broadband, 271Broadband seismograph, 88, 272Brune model, 98Bulk modulus, 9Butterworth filter, 287

CCagniard path, 153Calculus of variations, 238Calibration function, 308Canonical equation, 78, 195

© Springer Nature Singapore Pte Ltd. 2021K. Koketsu, Ground Motion Seismology, Advances in Geological Science,https://doi.org/10.1007/978-981-15-8570-8

325

326 Index

Cartesian coordinate system, 35Cauchy’s integral theorem, 155Causality, 18, 21, 277Caustic, 203Centering, 77, 251Central difference, 214Central Meteorological Observatory, 316Centroid, 92Characteristic equation, 163Characteristic oscillation, 20, 162Characterization, 105Characterized source model, 105Chebyshev filter, 292Chi-Chi earthquake, 114Chronograph, 269Clock, 269CLVD, 85, 86CMT inversion, 44, 92, 100, 177, 293Coda wave, 18Cofactor expansion, 182Collocation method, 193, 226Compensated linear vector dipole, 86Composite model, 108Computational Galerkin method, 217Conical wave, 58Conjugate, 83Conjugate gradient method, 251Conrad discontinuity, 253Constitutive law, 7Constraint, 81, 102, 299Constructive interference, 112Continental crust, 253Continuity condition, 27, 120Continuum, 2Convolution, 24, 97, 155, 276Corner frequency, 96Cosine-tapered window, 285Couple of forces, 33, 37, 85Coupling, 120, 127Courant condition, 223Crack, 107Crack model, 106Cross correlation, 241, 248Crust, 18, 34, 184, 253Crustal deformation, 31, 103, 185Crustal earthquake, 34, 314CT, 249Cut-off frequency, 165, 176Cylindrical coordinate system, 11, 16, 58,

121Cylindrical harmonics, 121Cylindrical wave, 17, 58, 121Cylindrical wave expansion, 58, 147, 158

DD/A conversion, 274D93-type seismographs, 311Damped oscillation, 20, 184, 265Damping coefficient, 20Damping constant, 20, 265DB, 275Decimation filter, 275Delay unit, 269Delta function, 27, 40, 44, 277Delta function sequence, 278Delta-sigma modulation type, 275Density, 10Deviatoric component, 86Deviatoric strain, 9Deviatoric stress, 9Difference quotients, 212Diffracted wave, 203Digital, 274Dip angle, 35, 62, 69, 93Dip slip fault, 35, 66, 69, 72, 113Dipole, 85Direct methods, 251Direct wave, 75Direction cosine, 51, 79Directivity effect, 110Discontinuity vector, 90, 100, 125, 129Discrete Fourier transform, 281Discrete inverse Fourier transform, 281Discrete wavenumber method, 160Dislocation, 35Dispersion, 164, 231Dispersion curve, 165, 176Displacement, 3Displacement detection type, 271Displacement feedback, 272Displacement potential, 15, 58Displacement seismograph, 267Dissipation filter, 185Divergence, 12Divergence theorem of Gauss, 25, 45Domain method, 192, 217Double couple, 37, 43, 50, 86, 94Double difference method, 80Driver, 2721-D velocity structure, 18, 73, 74, 90, 1192-D velocity structure, 1913-D velocity structure, 80, 191, 249Dynamic range, 273, 275Dynamic ray tracing, 211

EEarthquake, 1

Index 327

Earthquake Research Institute, 268Earthquake source, 2Eikonal equations, 195El Centro seismograms, 268Elastic body, 2, 3Elastic constant, 3, 8Elastic displacement, 49Elasticity, 3, 18Elastic rebound theory, 2, 31, 105Elastodynamics, 2, 3, 192Electromagnetic seismograph, 270Elementary operations, 296Empirical Bayes estimation, 104, 301Energy integrals, 236Engineering seismology, 268Ensemble average, 247Epicenter, 2, 58Epicentral distance, 2, 58, 75, 169, 180, 307Equal-area projection, 114Equation of motion, 10Error propagation, 298European Macroseismic Scale, 317Evidence approximation, 301Explicit method, 209, 220

FFalse position method, 207Far-field term, 52, 81, 95, , 178Fault area, 94Fault displacement, 35, 49Fault parameters, 35, 84Fault plane, 2Fault plane model, 92Fault plane solution, 84Fault rupture, 2, 32, 37, 92Fault-normal direction, 113Faulting, 32, 307Feedback circuit, 272Fermat’s principle, 78, 197, 210FFT, 160, 217, 229, 282Filter, 280, 286Filtering, 285Final slip, 103Finite difference method, 208, 212Finite element method, 192, 217Finite fault, 92, 100Finite strain, 5First-motion approximation, 156fmax, 99F-net, 92Focal mechanism solution, 84Focal sphere, 82, 183

Foot wall, 35Force balance type, 272Forward direction, 112Forward problem, 92, 253Fourier–Hankel transform, 122, 124, 128Fourier series, 281Fourier spectrum, 276Fourier transform, 19, 57, 89, 121, 226, 276Four-quadrant type, 83, 112Free oscillation, 92, 162Free surface, 126, 219Frenet’s formulas, 202Frequency characteristics, 266Frequency response, 266Frequency spectrum, 276, 321Fukui earthquake, 268, 317Full waveform inversion, 253Full wave theory, 157, 161Fundamental mode, 164, 174

GGalerkin method, 193, 217Gaussian beam method, 204Gaussian elimination, 210, 295Gauss–Newton method, 295Generalized ray, 148Generalized ray theory, 148, 161, 163Genetic algorithm, 239Geometrical optics, 193Geometrical spreading, 17, 18, 53, 169, 183Gibbs’ phenomenon, 280, 285Global CMT Project, 89, 92GNSS, 185GPS, 185Gradient method, 239Gravimeters, 314Great earthquake, 310Great earthquakes, 232Green’s function, 27, 91, 101Ground acceleration, 321Ground motion, 1, 3, 264Ground motion simulation, 217Ground motion simulations, 232Ground surface, 15, 74Group velocity, 167, 177, 236

HHalfspace, 122Hamilton’s principle, 197, 235Hamiltonian, 195Hanging wall, 35, 105

328 Index

Hankel function, 17Hankel functions, 123Hankel transform, 122Hanning window, 285Hanshin-Awaji earthquake disaster, 110Haskell matrix, 132, 184Haskell model, 93, 98Head wave, 155Head waves, 148Helmholtz equation, 121, 245Helmholtz potential, 57Helmholtz potentials, 53Helmholtz’s theorem, 12, 44, 48Heuristic search method, 239High-cut filter, 285, 322High-pass filter, 285, 290High-sensitivity seismograph, 268, 272Higher mode, 164Higher modes, 174Hilbert transform, 277Historical earthquakes, 317Homogeneous, 15, 121Homogeneous boundary condition, 26, 27Hooke’s law, 3, 7, 105Horizontal component seismograph, 264Horizontal radiation pattern, 73Horizontal radiation patterns, 122Horizontally layered structure, 58, 119Huygens’ principle, 204H/V spectral ratio, 233Hyperbolic type, 27Hyperparameter, 301Hypocenter, 2, 74, 92, 249Hypocenter determination, 74, 76, 92, 249Hypocentral distance, 2, 53

IIASPEI, 312Ichibai strong motion seismograph, 268Ichibai strong motion seismographs, 310Imperial Valley earthquake, 111, 230, 268Implicit method, 209Impulse, 28Infinitesimal strain, 4, 5Inhomogeneous, 13, 46, 242Initial motion, 76, 92Inner fault parameters, 105InSAR, 185Instrumental seismic intensity, 320, 323Integration by parts, 19, 25, 42, 78, 217Interface, 15Interfaces, 119

Intermediate-field term, 52Intermediate-period seismograph, 267, 308International Seismological Centre, 34Intrinsic attenuation, 18, 159, 184, 242Inverse Fourier transform, 276, 321Inverse Fourier transforms, 19Inverse problem, 92, 253Inversion, 92Inversions, 300IRIS, 177Irregularly layered structure, 224, 253ISC, 312Ishimoto-type accelerograph, 268, 322Isoseismals, 317Isotropic component, 85Isotropy, 9Iterative methods, 251Iterative refinement, 250, 295, 298

JJacobian, 200, 295Jacobian matrix, 238, 295JMA, 84, 111, 268, 272, 310, 316JMA intensity scale, 317JMA magnitude, 311, 314JMA-87 type, 272Joint hypocenter determination, 80Joint inversion, 103Jordan’s lemma, 155, 168

KK-NET, 110, 232, 272K-NET95, 272Kanto earthquake, 32Kanto earthquake disaster, 33Kirchhoff’s integral, 204Kobe earthquake, 104, 110, 312, 317Kocaeli earthquake, 112Kramers–Kronig relation, 21Kumamoto earthquake, 84, 114, 232

LLagrange interpolation, 222Lagrange’s equations, 78Lagrangian, 78, 197, 235Lamé’s constants, 8Lamé’s theorem, 13Landers earthquake, 111Landslide, 32Laplace transform, 150Large earthquake, 310

Index 329

Least-squares method, 103, 250, 293Left-lateral strike slip fault, 36, 37L’Hôpital’s rule, 186Likelihood, 294, 300Linear least-squares method, 92, 102, 104,

294Linearity, 23, 243Line source, 96Long Beach earthquake, 268Long-period ground motions, 230Long-period seismograph, 267, 308Love wave, 163, 233Low-cut filter, 285, 322Low-pass filter, 275, 285, 288Lumped mass, 220

MMagnitude, 1, 44, 307Mainshock, 84, 104Mantle, 18, 34, 184, 253MAP estimation, 300Marginal likelihood, 301Marginal probability, 300Maslov seismogram, 204Mass-proportional damping, 21, 218Master event method, 80Maximum likelihood estimate, 294Maximum likelihood estimation, 294, 300Maximum posterior estimate, 300Mean stress, 9Mechanical seismograph, 269Medium, 2Mercalli intensity scale, 315Method of steepest descent, 168Method of weighted residuals, 191, 217, 224Michoacan earthquake, 231Microearthquake, 310Microtremor exploration, 232, 238Microtremors, 233, 240Mixed method, 192Moderate earthquake, 310Modified Bessel function, 150Modified Gram–Schmidt algorithm, 298Modified Mercalli intensity scale, 316Moho discontinuity, 253Moment magnitude, 44, 313Moment rate function, 44, 52, 95Moment tensor, 85Moment time function, 44, 52, 95, 100Momentum, 195Motion-stress vector, 124, 133Motion-stress vectors, 128

Moving coil type, 270Multi-time window, 101, 294Muramatsu-type seismograph, 271

NNatural period, 264Near-field term, 52Negative feedback mechanism, 273NIED, 110, 272Niigata earthquake, 231Nodal plane, 83Nodes, 219Nonlinear least-squares method, 238, 294Non-negative condition, 81, 102Normal distribution, 293Normal equation, 295Normal fault, 36Normal mode, 177Normal modes, 162Normal oscillations, 162Normal strain, 4Normal stress, 7Northridge earthquake, 113Nyquist sampling rate, 279

OOblique slip fault, 36Observation equation, 76, 250, 293Observational error, 92, 293Observational errors, 104Observed deformation, 103Observed seismogram, 91, 102, 253ω-cube model, 98ω-square model, 98Omori-type strongmotion seismograph, 268Optical seismograph, 269Origin time, 76, 249Outer fault parameters, 95, 100Over-damping, 272Overflow, 130, 143, 229Oversampling, 275

PParameterized shooting method, 209Partial differential equation, 27P axis, 83, 86Peak ground acceleration, 268, 323Peak ground velocity, 323Pendulum, 263Perfectly elastic body, 7Period, 2, 20

330 Index

Phase angle, 276Phase characteristics, 266Phase shift, 292Phase spectrum, 276, 292Phase velocity, 21, 129, 163, 233Pivoting, 296, 298Plane wave, 16Plane wave expansion, 152Plate, 33Plate boundary, 34Plate boundary earthquake, 34Plate motion, 33Plate tectonics, 33Point force, 44, 53Point source, 18, 37, 50Poisson equation, 12Poisson’s ratio, 9, 171Pole, 165Poles, 159Potential vector, 124, 133Potential vectors, 128Power spectrum, 247Press-Ewing seismograph, 312Principle of superposition, 23, 43, 73, 91,

102Propagation invariant, 229Propagator matrix, 90, 101, 132Pseudo-spectral method, 217Pull, 84Push, 84P wave, 13, 52

QQ, 20, 159, 184, 221QR-factorization, 297Quality factor, 20Questionnaire seismic intensities, 320

RRadiation boundaries, 215Radiation boundary, 192Radiation boundary condition, 126, 192, 234Radiation boundary conditions, 130Radiation pattern, 33, 82Rake angle, 35Ramp function, 44, 95, 277Ray, 16, 74, 148Ray equations, 197Ray Jacobian, 182, 200, 203, 211Rayleigh ansatz, 229Rayleigh damping, 219Rayleigh wave, 159, 170, 233

Rayleigh wave velocity, 160, 171, 176Ray parameter, 180Ray theory, 78, 178, 194Ray tracing, 75, 178, 204Reciprocity relation, 28, 240Reciprocity theorem, 26, 28, 39, 107, 184,

242Record section, 167Rectangular function, 96, 280, 285Recursive equation, 286Recursive filter, 286Reflected wave, 136Reflection/transmission coefficients, 137,

138Reflection/transmission matrices, 139, 140Reflection/transmission matrix, 143, 229Reflectivity method, 158Relaxation function, 19Relaxation method, 208Representation theorem, 29, 37Residual, 294Residues, 165Retarded potential, 45Return period, 34Reverberations, 143, 157Reverse fault, 36Richter scale, 307Right-lateral strike slip fault, 36Rigidity, 8, 94Ripple, 280, 285Rise time, 93, 95, 109Robustness, 314Rossi-Forel intensity scale, 315Rotation, 12Rotation matrix, 86Rounding error, 296Runge–Kutta method, 205Rupture duration, 97Rupture initiation point, 92, 101Rupture propagation, 92, 95, 112Rupture velocity, 93, 101, 112

SSaddle point, 154, 168Sampling, 91, 102, 274, 277San Andreas fault, 31San Francisco earthquake, 31Scalar potential, 12, 45, 243Scaling, 296, 298Scaling law, 99, 108, 313Scattering, 18Scattering attenuation, 18

Index 331

SCEC, 232Schwarz reflection principle, 151Secant method, 207Seismic basement, 253Seismic intensity, 1, 314Seismic intensity meters, 312, 320Seismic intensity scales, 315Seismic interferometry, 239, 248Seismic moment, 37, 51, 93, 313Seismic pulses, 110, 232Seismic tomography, 248Seismic wave, 13Seismograph, 263Seismology, 1Seismometer, 263, 270Sensory seismic intensities, 317Servo mechanism, 271, 321SH wave, 14, 82, 183Shadow, 203Shape function, 219Shear strain, 4, 221Shear stress, 7Shooting method, 204, 235Short-period seismograph, 267Short-period seismographs, 313Simple harmonic oscillation, 15, 264Simulated annealing, 239Sine wave, 167, 266, 276Sine waves, 162Single force, 44Slip, 35, 43, 93Slip amount, 93Slip angle, 35, 73, 93, 101Slip rate function, 44, 108Slip time function, 44, 100Slip weakening, 109Slowness, 148Slowness integration, 149Slowness methods, 149SMA-1, 269SMAC type, 268Small earthquake, 310Smirnov’s lemma, 200Snell’s law, 74, 180S/N ratio, 233, 275Sommerfeld integral, 18, 58, 158Source-controlled fmax, 109Source fault, 2, 35Source inversion, 102, 177, 294Source potential, 73, 90Source process, 100Source region, 2Source spectrum, 95

Source time function, 18, 44, 53Sparse, 251Spectral element method, 220Spectral methods, 150Spectrum, 95, 167, 276Spherical coordinate system, 17, 89, 178Spherical wave, 18, 54, 58Spherical waves, 121Stacking, 248Staggered grid, 216Static magnification, 265Station correction, 80Steepest descent path, 154Step function, 94, 150, 277Stiffness-proportional damping, 219Strain, 5, 32Strain energy function, 8, 235Strain vector, 7Stress, 3, 7Stress drop, 105Stress tensor, 7Stress vector, 6, 38Stress-free condition, 27, 126, 215, 223, 234Stress-free conditions, 130Strike, 35, 93Strike slip fault, 35, 58, 65, 113Strong ground motion, 1Strong motion, 1Strong Motion Accelerometer Committee,

268Strong motion observation, 273Strong motion seismograph, 267, 321Strong motion seismographs, 110Strong motion seismology, 2Strong motions, 267Subdeterminant, 131, 189Subdeterminants, 138Subduction, 33Subduction zone, 34Subfault, 37, 100Successive approximation type, 274Summation convention, 24, 131, 239Superposition, 121Surface wave, 15Surface wave magnitude, 308Surface wave pole, 165, 177Surface wave poles, 159Surface waves, 162, 308, 314SV wave, 15, 82, 183S wave, 13, 52Symmetric tensor, 7Synthetic deformation, 103Synthetic seismogram, 91, 102, 253

332 Index

TT*, 184Takeoff angle, 179, 204T axis, 83, 86Teleseismic body wave, 177Teleseismic earthquake, 272Teleseismic earthquakes, 177, 308Tensor, 7Tensor Green’s function, 28, 239Three-component seismograph, 265Time history, 276Tokachi-oki earthquake, 230Torsion, 202Total reflection, 141Tottori earthquake, 312Traction, 6Tradeoff, 103, 185, 301Transducer, 270Transmitted wave, 136Transport equation, 200, 203Transverse isotropy, 8Trapezoidal function, 97Trapezoidal rule, 160Travel time, 75, 198Trial function, 191Triangular function, 44, 277Trigger mechanism, 268Tsuboi’s formula, 31095-type seismic intensity meters, 321

UUnderdetermined system, 251Unilateral faulting, 95, 96USCGS standard type, 268USGS, 232, 268, 312

VVariance matrix, 253Variations, 236Vector dipole, 85Vector potential, 12, 243Velocity detection type, 271Velocity feedback, 272Velocity seismograph, 267Velocity structure, 2Velocity structure inversion, 238Vertical component seismograph, 265Viscoelasticity, 18Viscosity, 18Volumetric strain, 4, 9, 23

WWave equation, 13, 46, 121Wave equations, 156Wavefront, 16, 18, 198Wavenumber, 57, 58, 121, 276Wavenumber integration, 147Wavenumber spectrum, 276Weak formulation, 217Weighting functions, 192Weyl integral, 18Wiechert seismographs, 310Window, 285WKBJ approximation, 156, 194WKBJ seismogram, 157Wood–Anderson seismograph, 269Wood-Anderson seismograph, 307W phase, 53WWSSN, 312