Page 324 - Rock Mechanics For Underground Mining

ENERGY, MINE STABILITY, MINE SEISMICITY AND ROCKBURSTS

equations of the form of equation 10.95 are required for a solution. Therefore, at least
four sensors, in a well-conditioned, non-planar array, are required. A least-squares
method is used to invert the seismic travel time data to obtain the three coordinate
components and the time of occurrence of the event. More than four sensors are
required for more accurate solution through the use of redundant data.

10.10.2 Seismic moment, M 0
The seismic moment, M 0 , is a measure of the strength of a seismic event in terms
of parameters described by the double couple, shear dislocation model of a seismic
source. According to Aki and Richards (1980), seismic moment can be expressed as

M 0 = Gu s A (10.96)
where
G is the shear modulus at the source
u s is the average displacement across the discontinuity
A is the slip area of the discontinuity

Calculation of seismic moment from this expression is not possible in a mine set-
ting, because u s and A cannot be determined readily. In practice, as described by
McGarr (1984), seismic moment can be estimated from various spectral parameters
derived from seismic records. These are calculated from the displacement spectrum
of a waveform, which is obtained from the Fourier transformation of the seismic
waveform from the time domain into the frequency domain. The particular spectral
parameters of interest are the low frequency far-field displacement level, (0), and
the corner frequency, f 0 , both of which are identified on the spectral density plot in
Figure 10.27.
Seismic moment M 0 is then estimated from the expression due to Hanks and Wyss
(1972):

3
M 0 = 4 0 C R (0)/F c R c S c (10.97)
0
where 0 is the mass density of the source medium
C 0 is the P- or S-wave velocity of the medium
R is the distance between source and receiver
(0) is the low frequency plateau of the far-field displacement spectrum
of the P-wave or S-wave
F c is a factor to account for the radiation pattern
R c accounts for free-surface amplification of either P-wave or S-waves
S c is a site correction factor

10.10.3 Seismic energy
The radiated seismic energy represents the total elastic energy radiated by a seis-
mic event, and is a relatively small proportion of the total energy released. One
method of calculating seismic energy transmitted is given by Boatwright and Fletcher
(1984):

2
E c = 4 0 C 0 F (R/R c F c )J c (10.98)
c
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