Equation of earthquake magnitude is classified as follows:
A=
C (
M)
r-μ I
A=
C (
M)
r-ν e
-κr, κ=const. II
and log
C (
M)=α
M+β
where
A is the maximum amplitude and
r is the hypocentral distance. However, Type I is not applicable to the large range of
r and Type II is not for the large range of
M. Accor-dingly, the following equation is proposed:
A=
C (
M)
r-ν e-r/r c
(M) III
Type III is applicable to the observed values of the large range of both
r and
M. As examples, the results of observation for 23 large and middle earthquakes and several explosions are shown.
It is reasonable to use the velocity amplitude for
A. Then, the equation of Type I is expressed by
0.80
M-0.80=log
V (kine)+2 log
r (km) (3)
and that of Type III is expressed by the following equations:
α
M+β=log
C (
M)=log (
Vrνer/rc)=log (
Vc
rc
νe1) (4)
where
Vc is the velocity amplitude at
r=
rc. For the data of 23 earthquakes of which the magnitudes are from 4.6 to 8.3, the values of
M determined by Eq. (4) are well consistent with those by Eq. (3) and we get the following relations:
log
C (
M)=0.655
M-1.719 (5)
and log
rc (km) = 0.210
M+ 0.947 (6)
where ν is appropriately assumed to be 1. Then we have the
V-
r curves for various
M. These curves are well consistent with the observed values of explosions and microearthquakes, too.
The original definition of
M is given by the maximum amplitude at a certain distance
r0, but it is approximately available for the small range of
M. Therefore, for the large range of
M,
r0 should be changed into .
rcIf the duration of the wave group of the maximum amplitude is proportional to the hypocentral distance, the total kinetic energy of an earthquake can be related to its magnitude as follows:
log
E (erg)=2 log
C (
M)+log
c (
M)+coast. (15')
=1.52
M+ 11.8 (17)
for the physical constants given commonly in the earth's crust.
Equation of Type I is applicable in the vicinity of
rc, so that the coefficient of log
r is
-μ= (
d log
V/
d log
r)
r=
rc=(-ν -
r/
rc)
r=
rc=- (μ+1) (9)
Therefore, from the equations
M (
V,
r) and M (
M,
r) of Type I and
V=
A2π/
T, we have
log
Tc (sec)=0.257
M-1.32, at
r =
rc (19)
From Eq. (6) and (19), we have approximately
rc∼const.×
Tc (20)
and accordingly, we see that the equation of Type III is due to the specific dissipation function
Q-1.
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