# Earthquake Magnitude and Surface Fault Formation

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1 JOURNAL OF PHYSICS OF THE EARTH, VOL. 12, No. 1, Earthquake Magnitude and Surface Fault Formation By Michio OTSUKA Department of Physics, Faculty of Science, Kumamoto University

2 20 Michio OTSUKA or Fig. 2. Relation between earthquake magnitude M and length of earthquake fault L. Dashed and chain lines show the upper and lower boundaries of fault length respectively. Inset shows the geometrical situation of focal region with respect to the earth's crust.

3 Earthquake Magnitude and Surface Fault Formation 21 If the condition the diversity of the dimension and depth of earthquake source volume in relation to the thickness of the earth's crust. For simplicity, the following two assumptions are made: (A) the earthquake source volume is spherical in shape with radius a and (B) the occurrence of fracture is restricted within the earth's crust of thickness H. From (B), it follows that the depth d of the centre of source sphere must satisfy the following condition: The energy content in a source volume is expressed by d<h-a. (6) is satisfied, then fault will appear on the earth's surface. If earthquakes occur evenly through all depths within the earth's crust satisfying condition (6), the probability p'(a) that a given earthquake having the source volume of radius a is accompanied by surface fault appearance is expressed simply by (8) where N'(a) and Nf'(a) are the total number of earthquakes and the number of earthquakes accompanied by faults respectively, all having the focal radius a. a is related to the earthquake magnitude M, so that a in eq. (8) can be replaced by M as follows: (9) where e denotes the elastic constant appropriate for the stress to which the medium within the source volume is subjected and x represents the ultimate strain of the medium. V is the volume of the source volume and is

4 22 Michio OTSUKA Combining eqs. (10) with (2), the following relation is obtained for connecting a and M: loga(m)= m. (11) The probability that an earthquake of magnitude M is accompanied by surface fault appearance calculated by means of eqs. (9) and (11) is given in Table I, where H is taken to be 40km. Table I Probability of fault appearance representing extraordinarily long faults were not used in deriving eq. (4). If M exceeds 7.4, the source volume can no longer be spherical, because the diameter of the sphere is then larger than the thickness of the earth's crust. The volume must be depressed or flattened between the earth's surface and the bottom of the earth's crust in order to be big enough to store the energy needed for an earthquake of that magnitude, necessarily resulting in some fracture or fracture zone on the earth's surface. This will explain why the minimum fault length exists for earthquakes greater than 7.4 in magnitude, and at the same time, why extraoridinarily long faults sometimes appear accompanying these large earthquakes. That the spread in fault length suddenly decreases as M=7.4 is reached (Faults of zero length exist for M less than 7.4, although they could not be shown in Fig. 2) also suggests the appropriateness of the above consideration. K. HOSHINO (1956) studied the relation between the earthquake magnitude and surface fault appearance in Japan and found that the percentage of the number Nf(M) of earthquakes with faul appearance against the total number N(M) of earthquakes of that magnitude is 60% for 7.0<M<7.4 and 100% for 7.4<M. IIDA also pointed out from world data that shallow earthquakes having magnitude greater than 7.3 are always accompanied by surface fault. Agreement of the calculated probability with the observed is satisfactory. Referring to Fig. 2 again, it is seen that, for earthquakes having M greater than 7.4, a lower limit in fault length begins to appear as shown by the chain line. The six points The observational relation between the fault length and the earthquake magnitude has been explained by the geometrical relation between the dimension of earthquake source volume with respect to the thickness of the earth's crust as schematically shown in the inset of Fig. 2. The observational fact has also been explained that the percentage of the number of earthquakes of a given magnitude accompanied by fault appearance against the total number of earthquakes of that magnitude increases from 60% for 7.0<M<7.4 to 100% for 7.4<M. Thus the idea of voluminal storage of earthquake energy has been found to be useful here also. In calculating the probability of fault appearance for an earthquake of a given magnitude, it has been assumed that earthquakes occur evenly through all depths in the earth's crust. This however, is, not actually the case and the vertical distribution of earthquake origin is known to show a characteristic pattern, especially in Japan. If this is taken into account, the calculated probability

5 Earthquake Magnitude and Surface Fault Formation of fault appearance based on the even distribution naturally undergoes change, although it is doubtful if the origin of an earthquake as determined by seismological observations should coincide with the centre of earthquake source volume. IIDA (1963) pointed out that shallow submarine earthquakes having greater magnitude than 7.3 are always accompanied by Tsunamis. As already stated, he also pointed out that shallow earthquakes having greater magnitude than 7.3 are always accompanied by surface faults. The coincidence of these magnitude values must have an important bearing to the storage capacity of energy of the earth's crust material. K. KASAHARA (1957, 1958), in his study of the relation between geodetic and seismic aspects of earthquake phenomena, found that, in case of the Tango earthquake, Japan, of 1927, the characteristic period and push-pull distribution of seismic waves are satisfactorily explained by a spherical source model (Fig. 3). KASAHARA adds to state that "stress change taking place on such a spherical surface can hardly be accepted from our common experience of fracture production in a brittle substance, and furthermore, the model of a spherical origin is less suitable for understanding the most outstanding fact of remarkable shearing along a nodal line of push-pull distribution". But the present author is interested in the fact that the length of Gomura Fault accompanying the Tango earthquake happens to be approximately equal to the diameter of the circle defined as the intersection of the source sphere with the earth's surface. KASAHARA studied the two dimensional problem of production of fault plane and elastic wave radiation from it and showed that the spectral intensity of P wave is the maximum for the component wave of which the wave length seismic wave which gives the maximum spectral intensity and 2c is the length of fracture. Now, if c in eq. (12) is identified with the radius of the source sphere, and if the velocity of P wave in the medium is taken to be 5km/sec, the following relation can be obtained: circle (broken lines) refers to the spherical model origin. (after K. KASAHARA (1958)). (13) where T* is the characteristic period of P wave. Taking the logarithm of both sides of eq. (13) and taking eq. (11) into account, the following relation is obtained for connecting T* and M: Eq. (14) agrees with the relation: logt= m (15) given by KASAHARA by spectral analysis of seismic P waves. The coincidence of eq. (14) and eq. (15) is another evidence of appropriateness of the assumption that the fault length represents the diameter of the focal sphere. The author

6 24 Michio OTSUKA does not, however, insist that this must always be the case. In fact, if the fault must always extend up to the maximum dimension of outcrop area of the source volume, the plots in Fig. 2 must be more heavily concentrated around Lm than is the case in Fig. 2. The most important assumption made in the above discussion is that the mantle of the earth does not participate in fracturing. No essential difference between the mechanism at the origin of crustal and subcrustal earthquakes has however been known and there is no reason why earthquake generating stress should not be accumulated in the earth's mantle. It must be admitted that the assumptions on which the preceding discussion has been developed are not more than simplifying working hypotheses for the time being. Reference HOSHINO, K.: 1956 On Magnitude of Earthquakes Accompanied by Faults. Zisin (J. Seism. Soc. Japan), Ser. 2, 8, IIDA, K.: 1959 Earthquake energy and earthquake fault. J. Earth Sci., Nagoya Univ., 7, IIDA, K.: 1963 Magnitude of Tsunamigenic Earthquake, Aftershock Area, and Area of Tsunami Origin. Geophysical Papers Dedicated to Prof. Kenzo Sassa, KASAHARA, K.: 1957 The Nature of Seismic Origins as Inferred from Seismological and Geodetic Observations (1). Bull. Earthq. Res. Inst. 35, KASAHARA, K.: 1958 The Nature of Seismic Origins as Inferred from Seismological and Geodetic Observations (2). Bull. Earthq. Res. Inst. 36, TOCHER, D.: 1958 Earthquake Energy and Ground Breakage. Bull. Seism. Soc. Amer. 48, TSUBOI, C.: 1956 Earthquake Energy, Earthquake Volume, Aftershock Area, and Strength of the Earth's Crust. J. Phys. Earth, 4, TSUBOI, C. and OTSUKA, M.: 1963 Duration of Energy Radiation from an Earthquake Source Volume. Proc. Japan Acad., 39,

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