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ABSTRACT

 Earthquakes accompanied by faults and their source dimensions were investigated.The magnitudes of these earthquakes are related to the length of faults and their slip displacements.The dimensions of earthquake source volume are estimated from the observational dimensions of earthquake fault.The greater the earthquake magnitude,the larger is the fault length at the earth’s surface and its upper limit exists for each magnitude.Observed upper limit is somewhat long in comparison with the length expected from the crustal strength.The ratio of the thickness of the earth’s crust to the focal depth of an earthquake is related to the fault length.The geometrical situation of an earthquake source dimension with respect to the earth’s crust and the probability of fault appearance are investigated.The shallow earthquakes having magnitude greater than 7.0 are accompanied by a surface fault when the crustal thickness is forty kilometers.

INTRODUCTION

 A system of fractures of land surface has been generally observed in the epicentral region related to the occurrence of a great earthquake with its epicenter on land.Faulting is a typical event occurring simultaneously with an earthquake and it is believed to have sometimes brought about the release of a large amount of strain energy stored in the earth’s crust.Further,the length and direction of a fault are generally known to be related to the magnitude and source mechanism of the pertaining earthquake.It is,therefore,important to see the nature of faulting.The relationship between the fault length and the earthquake magnitude was investigated in a previous paper [8].After that,a number of earthquakes accompanied by fault were known and fault data were extensively corrected.In the present paper,the relationship between the magnitude of earthquakes and dimensions of faults,and source dimensions based on faulting are discussed by using world data.

DIMENSIONS OF EARTHQUAKE FAULTS

 A number of destructively large earthquakes are known to have taken place in Japan,but these have not always accompanied by fault.For instance,of about 140 destructive large earthquakes with epicenters on land since 1840,only twelve shocks are known to have been accompanied by surface faults.The earthquake fault data collected [14] for the whole world are given in Table1 which includes more additional and corrected data than those in a previous paper [8] From Table1 the number of faults accompanying land earthquakes totaled sixty four during the period from 1811 to 1964;twenty faults in North America,seven in New Zealand,six in Turkey,three in India,four in Mongolia and some in other countries.The world data show that the observed relative slip displacement varies from a few centimeters,in a very small earthquake,up to 14.3meters,in the largest observed on the dip−slip fault of the Yakutat earthquake of September 1899 in Alaska.The strike−slip displacement also varies with the size of the earthquake,from tens of centimeters in the smallest up to about 10meters in the largest observed on the fault of the outer Mongolia earthquake of April 1957.The fault where the slip is horizontal is designated a strike slip or transcurrent fault and the fault where the slip is vertical or perpendicular to the line formed by the intersection of the fault with the horizontal surface is known as a dip−slip fault.Combinations of dip−slip and strike−slip faulting also occur,but some of faults are predominant in the former and some are predominant in the latter.
 The horizontal extent of the fault varies with the magnitude of the earthquake,from a few meters in the smallest up to an order of 1000kilometers in the largest.The longest directive observed break was that on the fault of Mongolia earthquake of 1905,in which the trace was visible on the ground for a distance of about 700 kilometers.
  A large number of the great earthquakes of the world occur along the margins of the circum−Pacific continents.Some of these earthquakes are accompanied by faults which intersect the surface offshore,and consequently the extents of the faults cannot be observed directly.Fault extents can be estimated indirectly from the lengths of the distribution patterns of after−shocks parallel to the fault and by a method based upon the spectra of surface waves such as Rayleigh and Love waves [4,5,13].In the great Chilean earthquake of 20 May 1960,a length of fault break of about 1000kilometers was estimated by both methods [13].

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TABLE1.EARTHQUAKE MAGNITUDE AND EARTHQUAKE FAULT DATA

ESTIMATION OF SPEED OF SURFACE FAULT FORMATION

 The speed of surface fault formation will be estimated as follows after H.Benioff [3]:The horizontal speed of rupture propagation is found to be about 3 to 4kilometers per second for strike−slip faulting and somewhat less for dip slip fault formation [4,5].Strike−slip faulting proceeds faster because it is propagated from point to point by compressional stresses,while dip−slip faulting is maintained by the smaller shearing stresses.
 The duration of slip at any one point is not accurately known but is estimated to vary from a fraction of a second,in the smallest earthquakes,to about 10 seconds in the largest.The duration of the earthquake at the source is totally equal to the length of rupture divided by the speed of rupture propagation.In the largest earthquakes,with 1000kilometers of surface fault,the duration of the source is thus about 280 seconds,approximately.In the case of the Kern County earthquake of 1952 the time required for the faulting to reach the surface from the focus was also estimated by Benioff [2,3] and was greater than about 3 seconds.The time [2] required for the faulting to progress horizontally from the focus to the end point in the vicinity of Caliente,California was greater than 16 seconds.
 The accumulation rate of elastic strain is approximately estimated by Whitten and Claire [19] for the San Andreas fault by data from triangulation surveys of the strained zone.They found that the two opposed fault blocks are moving relative to each other at a rate of about 5 centimeters per year.If this rate is constant and if each break of the fault occurs at the same value of strain,earthquakes on any given segment should repeat at intervals of about 125years [3].However,there are many unknown factors to estimate the repetition of earthquakes with fair accuracy.Especially the actual vertical extent of the slip is not known even approximately for any earthquake,though it is generally assumed that the break in shallow earthquakes does not extend below the Moho discontinuity,but is restricted to near the surface [7].

MAGNITUDES OF EARTHQUAKES AND DIMENSIONS OF FAULTS

Array The magnitude of earthquakes which have been accompanied by fault is generally larger than 5.6 which is also the smallest magnitude of earthquake accompanied by crustal deformation or tsunami,as shown in Figs.1 and 2.Earthquakes with a magnitude greater than 7.3 are always accompanied by fault in Japan,and shallow submarine earthquakes with a magnitude greater than 7.3 are also always accompanied by tsunami,as already pointed out [9].
 Since the maximum strain of the earth’s crust is of the order of 10^−4 [18],the elastic strain energy stored in the crust just before an earthquake may be derived from the earthquake fault data such as maximum length at the earth’s surface and maximum displacement at the fault.The relationship between the length l of surface fault and the earthquake magnitude M is shown in Fig.3.This relationship may be determined by the least squares method such as
   M=(0.76±0.08)log l+(6.07 ±0.13),(1)
where l is measured in kilometers.This relationship shows generally that the greater the earthquake,the larger is the fault length at the surface.It is found that the exponent of l is about 0.8.Considering the scattering of the data in Fig.3,we may take the exponent of l as unity for the range of earthquake magnitude from 5.6 to 8.5.(1)is the revised relationship obtained from more additional data of land earthquakes than those in a previous paper [8].
 The relationship between the fault length 1 and the maximum displacement D is shown in Fig.4,which may be determined by the least squared method such as
   logl=(0.93±0.22)logD+(1.25±0.12),(2)
   logD=(0.55±0.07)M−(3.71±0.50),(3)
where D is the largest value of either the maximum dip−slip D_v or maximum strike−slip D_h,in meters.Fig.5 shows the relationship between the earthquake magnitude and the largest displacement D.Fig.6 shows the relationship between the maximum dip−slip D_v,and the maximum strike slip D_h.Combining 
(1)or(3)with Gutenberg−Richter’s formula of seismic wave energy E_s:
   logE_s=11.8+1.5 M,(4)
we obtain the following relation respectively:
   E_s=7.8 × 10^20 × l^1.2,(5)
   E_s=6.6×10^21×D^2.7. (6)
Here,the significance of the exponent of l in(5)or(6)is considered to be related to the spatial shape of rock which releases the elastic strain energy.If only a certain range of length of ground breakage contributed to the release of strain energy,the exponent of l should be close to unity.Seeing that the exponent of l in(5)is nearly equal to unity,the fault length is considered to contribute to the wave energy of the pertaining earthquake.As seen in(5),the wave energy of an earthquake released would be approximately proportional to the horizontal extent of surface break for large earthquakes which had foci in the crust as reported in a previous paper [8].(6)shows that the released earthquake energy would be approximately proportional to the cube of the slip displacement.
 The extent perpendicular to the earthquake fault of the zone of strained rock is considered to be proportional to the displacements of the ground at the fault as a first approximation.Assuming that the elasticity of rocks near the earth’s surface is the same in both the horizontal and vertical directions,the area at the earth’s surface of the zone of strained rock is proportional to the product √(D^2_v,+ √D^2_hl)in which D_v and D_h represent dip−slip and strike−slip,respectively.The relationship between M and √D^2_v+ D^2_hl,(=Rl)or M and D_vl or D_hl for each shock in Table1 can also be determined by the method of least squares to the linear relation,as shown in Figs.7 and 8.
 We get the relations as
  M=(6.16±0.30)+(0.58±0.15)log Rl,(7)
  M=(6 44±0.34)+(0.53±0.20)log Dl,(8)
in which l and R are measured in meters.
 Combining(4)and(7),or(4)and(8)to eliminate M as before,we get
  log E_s=21.0+0.9 log Rl,
  log E_s=21.4+0.8 log Dl (9)
or   E_s=1.0 × 10^21 ×(Rl)^0.9,
    E_s=2.8 × 10^21 ×(Dl)^0.8.(10)
Once the displacements and the length of earthquake faults are determined,the maximum released energy of earthquakes can be estimated from(8)as before.However,from these equations the exponent of R1 or D1 is close to unity.It may be,therefore,concluded that the energy released in the larger shallow earthquakes is approximately proportional to the area at the earth’s surface of the zone of strained rock,as already pointed out [1,8,16].

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FIG.1.Relationship between magnitude M and focal depth H of earthquake occurred in land during the period from 1923 to 1963.Chain line(M=4.8+0.017H)shows the boundary of appreciable damage.
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FIG.2.Relationship between magnitude M and focal depth H of submarine earthquakes during the period from 1923 to 1963.  Tsunamis having height over 4−6m are located on the right side of line B and tsu
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FIG.3.Relationship between earthquake magnitude M and length of earthquake fault,l.
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FIG.4.Relationship between length of earthquake fault,1 and maximum slip displacement D.
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FIG.5.Relationship between maximum slip displacement D and earthquake magnitude M.
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FIG.6.Relationship between maximum dip−slip D_v and maximum Strike Slip D_h.Chain line is the limiting value of fault displacement.Each point corresponds to one fault.
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FIG.7.Relationship between earthquake magnitude M and the product of fault length and maximum displacement √(D_v^2+D_h^2l).
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FIG.8.Relationship between earthquake magnitude M and the product of fault length and maximum displacement DI.● D_el,○ D_id.

AFTERSHOCK VOLUME,EARTHQUAKE MAGNITUDE AND FRACTION OF ENERGY RELEASED AS SEISMIC WAVES

Array In connection with the energy of earthquakes,aftershock volume was investigated by using the aftershock area and the epicentral depth of aftershocks including main shock,as shown in Fig.9.The data [10] in this figure may be expressed by the least squares method as
   logV=(1.06±0.10)M−(2.78±0.74),(11)
where V is the volume of aftershock activity in km^3 and M is from 6 to 8.5.If the aftershock volume is assumed to be an earthquake volume V where the strain energy is stored,the energy of the released seismic waves,E_s is
   E_s=1/2efx^2−V,(12)
in which ex^2 V/2 is the potential energy of a volume V of the crustal material having an average elastic constant e strained an ultimate amount x immediately before an earthquake,and f the fraction of the energy E_s released as seismic waves.
 Assuming reasonable values of e=5×10^11 C.G.S.and x=10^(−4)for an understanding of the nature of the problem and combining(4),(11)and(12)the following relationship may be approximately derived as [10]
   0.5M = logf+4.5.(13)
Thus,the fraction of the energy released as seismic waves may depend upon the magnitude M of an earthquake.We may calculate the traction of conversion of elastic strain energy to wave energy from(13).For instance,
  f■1 for the earthquake magnitude M=9,
  f■1/3 for M=8,
  f■1/10 for M=7,
  f■1/30 for M=6.
 It is,therefore,concluded that the greater the magnitude of an earthquake,the more often will the efficiency of conversion of earthquake strain energy to seismic wave energy will approach unity.
 In connection with the fraction of conversion of strain energy to wave energy,Bullen [6] obtained the earthquake energy E_s based on elastic theory as
   E_s=S^2V/12qμ’ (14)
where V is the volume of the heavily strained zone in an earthquake,S the Mises strength,μ the rigidity,q the ratio of distorsional strain energy to seismic energy.In this case q is corresponding to 1/f in(12).He gave relevant values of S=10^9 dyne/cm^2,μ=0.4×10^12 dyne/cm^2,q=2 for the greatest earthquake.
From q=2,we get f=1/2 which is equal to f for the case of M=8.4.

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FIG.9.Relationship between aftershock volume V and earthquake magnitude M.

SOURCE DIMENSIONS AND PROBABILITY OF FAULT APPEARANCE

Array The probability of fault appearance and source dimensions are investigated.For simplicity,we take the same consideration as that of Otsuka [11] and assume that the earthquake source is spherical in shape with radius a and that the occurrence of fracture is restricted within the earth’s crust of thickness H.Then if the following conditions are satisfied,the fault will appear at the earth’s surface:
   d<H−a,(15)
   d<a,(16)
where d is the depth of the center of the source.If the occurrences of earthquakes satisfy conditions(15)and(16),the probability P(M)of an earthquake,having the spherical source of radius a and the magnitude M,accompanied by surface fault is expressed by
   【数式】 (17)
where n(M)is the total number of earthquakes,n’(M)the number of earthquakes accompanied by faults.The elastic strain energy contained in a spherical source volume is expressed as
   E=2/3πex^2a^3.(18)
If the values of e and x are taken as mentioned in Section 5,then E in ergs
becomes
   E=10^4×a^3(a in cm).(19)
Combining(19),(4),(12)and(13),the following relation can be obtained as
   loga(M)=0.33M−1.0(a in km).(20)
From(20)the probability that an earthquake of magnitude M is accompanied by surface fault can be obtained.If H is assumed to be 50km,the calculation shows that the percentage of number n’(M)of earthquakes with fault appearance against the total number n(M)of earthquakes of that magnitude is 100 per cent for M=7.3.The calculated probability for the case of H=40km is 100 per cent for M=7.0,as given in Table2.These calculations are close to the observed data [8,16] that shallow earthquakes having magnitude greater than 6.5 or 7.3 are almost accompanied by surface faults.Further,the size of the spherical source calculated from(20)is also given in Table2.
From the conditions(15)and(16),we can derive the following relation:
   1<2a. (21)
Then,if the value of a is given,the longest fault length l_m,for certain earthquake magnitude is determined by
   l_m=2a.(22)
The number of earthquakes of which the observed fault length l is larger than l_m=2a is given in Table3 where the number n of earthquakes,of which the observed fault length l is smaller than l_m,is also given.
 These results are considered to depend on the certain limit of fault length at the earth’s surface.The greater the magnitude of the earthquake,the larger becomes the difference between the value observed and that calculated by(22).This suggests that the source volume can no longer be spherical when the earthquake magnitude exceeds a certain value which is different from the thickness of the earth’s crust.

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【数式】(17)
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TABLE2.SOURCE DIMENSIONS AND PROBABILITY OF SURFACE FAULT APPEARANCE
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TABLE3.MAXIMUM FAULT LENGTH FOR EARTHQUAKE MAGNITUDE

LIMIT OF FAULT LENGTH AND SOURCE DIMENSIONS

Array The fault length of each earthquake depends upon the degree of shocks,the size of earthquake source,focal depth,source mechanism,and crustal strength.If the conditions of the earthquake occurrences would be the same,an upper limit of fault length would exist for each earthquake magnitude because of the average energy density being almost the same.
 The relationship between the maximum fault length l_max or minimum fault length l_mja and the earthquake magnitude M may be obtained by the chain and dotted lines drawn in Fig.10 where the ratio of the thickness of the earth’s crust to the focal depth of an earthquake is given.As seen in Fig.10,the large values of this ratio are distributed near the upper limit line and the maximum fault length l_max for earthquakes of magnitude M increases with the increase in M.The lines for l_max and l_min are approximately expressed by
   M=2logl_max+3.5,(23)
and  M=2logl_min+7.0,(24)
respectively.l in(23)or(24)is measured in kilometers.
 Eliminating M from(23),(24)and(4),the seismic wave energy E_s can be obtained as
   logE_s=2.1+3 log l_max,(25)
or   E_s=10^2 × l^3_max,
and   logE_s=7.3 + 3logl_min,(26)
or   E_s=2×10^7×l^3_min
in which l is measured in centimeters.
 Combining(22),(23),(4)and(12),we get the approximate value of the fraction of energy conversion,as f=1/10 which corresponds to the average value of f derived from(13)for 6<M< 8.At any rate,we get the maximum value of the source dimension of an earthquake from(23)and(22)as follows:
【表】
As already mentioned in the former section,if the earthquake magnitude exceeds a certain value,source volume may be not spherical,because the dimension of the sphere is larger than the crustal thickness.Thus,the volume must be flattened between the earth’s surface and the bottom of the earth’s crust.This will explain why extraordinary long faults sometimes are observed in the large earthquakes and also why the minimum length of the fault exists for earthquakes greater than M=7.3.

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FIG.10.Upper and lower limit of fault length.−−−− upper limit,‥‥ lower limit.The numeral outside of the circle is the ratio of the thickness of the earth’s crust to the focal depth of an earthquake.
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【表】

CONCLUDING REMARKS

 The relationship between the observational fault dimensions and the magnitude of earthquakes is investigated.The observed relative slip displacement varies from a few centimeters up to about 14meters.Some of faults are predominant in strike slip displacement and some are predominant in dip−slip displacement.The horizontal extent of the fault varies with the magnitude of an earthquake from a few meters up to an order of 1000 kilometers.The minimum magnitude of the earthquake accompanied by visible fault was 5.6.Earthquakes with a magnitude greater than 7 are almost accompanied by a fault.This may well be explained by the calculation of the probability of fault appearances based on the assumption that the mantle of the earth does not concern with fracturing.Although this may not actually be the case,this assumption seems a reasonable and simple working hypothesis.
 The source dimension of an earthquake is estimated by using the fault length.It’s upper limit is identified.with the maximum linear dimension of the earthquake source volume.The greater the magnitude of an earthquake,the larger becomes the difference between the observed value and the calculated maximum length.Observed length is comparatively long in comparison with the length expected from the mechanical strength of the earth’s crust.Further investigation will be necessary for fracturing of the earth’s crust and the mantle.

REFERENCES

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