Damping of a seismic wave
Cand. Real. Knut Sørsdal
To
illustrate damping on a seismic wave we discuss Fourier-components
. For a function g(t) to repeat itself with
period T the only frequencies f that can be used in the summations are integer
multiples of the fundamental frequency f=1/T. Such a Fourier summation using
frequencies 1/T,1/2T,1/3T,… will converge to g(t) as
desired. That this must be so can be seen by examination of Fig.1. At t=0 and
t=T each of the sinusoids starts and ends at the same place so that for each
times t<0 and t>T the summation will repeat itself with period T.
When we
discuss the gradual loss of amplitude of a seismic wave travelling through an
imperfect medium it should be noted that the change in amplitude of every
Fourier component is exponential, so that each cycle bears the same ratio of
amplitude to the preceding one. This ratio is called the logarithmic decrement
δ. We also have a specific dissipation constant that is the ratio 1/Q. One
Fourier-component is shown on fig.1.
Blue lines
is undamped and red dotted line is damped with attenuation b=0,8 that is the logarithmic decrement.
Fig.1. A Fourier-component of a function g(t) with frequency f=1. Undamped – blue line, attenuated
b=0,8 red dotted line.
To illustrate the concept that each cycle bears
the same ratio of amplitude to the preceding one we can plot several cycles in
the same plot with and without damping. Fig.2. Here we have to frequencies (two
multiples of the fundamental frequency) and the same amplitude decrease ratio
for each of them.
Fig.2. Two Fouriercomponents with same logarithmic decrement damping
Waters (1978) has set up some rules
for damping of seismic waves in imperfect medium:
1.
The
specific dissipation constant 1/Q is essentially independent of frequency.
2.
It
appears that 1/Q is substantially independent of amplitude for strain below 10-4.
The strain in rocks due to the passage of seismic waves used for exploration
purposes is usually no more that 10-8, except within a few meters of
the source.
3.
Observations shows that dissipation is less for a
single crystal than for an aggregate of such crystals.
4.
The
rate of dissipation decreases with increased pressure.
5.
The
exsiting data suggest that dissipation is relatively independent of
temperature.
As suggested
above we could measure attenuation as a percentage of stored energy lost per
cycle of a Fourier component , and from (1) deduce
that this percentage is the same independent of the length of the cycle. And
from here we can introduce other measures:
1/Q =
specific dissipation constant. It is related to the rate at which the
mechanical energy of vibration is converted irreversibly into heat energy and
does not depend on the detailed mechanism by which the energy is
dissipated.
δ =
Logarithmic decrement – the natural logarithm of the ratio of amplitudes of two
successive maxima or minima in an exponentially decaying free vibration.
A = damping
amplitude coefficient in the expression for a free vibration:
e-at
sin (2
ft) (1)
Closely related to the damping coefficient is the
attenuation coefficient α as a measure of attenuation for plane wave in an
infinite medium:
e-αx sin (2f(t-x/c)) where c = wave velocity (2)
∆f/f = Relative bandwidth of a resonance curve
between the half-power or 0.707 amplitude points for a solid undergoing forced
vibrations is a measure of the sharpness of the resonance curve.
∆E/E = Fraction of strain energy lost per cycle.
Now we can put up a relation between the different
parameters on basis of a specific damping capacity b. This damping measure b is
related to others by
b
= 2π/Q =
2δ = 2 a/f
= 2c α/f =
2π ∆f/f = ∆E/E (3)
Example of attenuation of a
Fourier-component.
Propagation of a plane wave of a constant frequency is
illustrated on fig.3. Damped wave is in red. As time (or space) increase the reduction of the
original amplitude increase. How far
will a plane wave of frequency 30 Hz travel before it is reduced to one tenth
of its original amplitude when we consider transmission in PierreShale? We
calculate with Q=17.15 in Pierre Shale and velocity 7000 ft/sec. We have the
formula:
α = πf/Qc (4)
and with
the values above we get α=0.000785
Then we can calculate using (2):
e-αx = 1/10
solving
for x will give x=2929 ft. Then with attenuation alone the amplitude drops by a
factor of 104 in travelling to a depth of 5858 ft. when we correct
for two-way traveltime.
Fig.3. One Fourier component attenuated.
This is
very high attenuation. In limestone Q= 200 and c=20.000 ft sec, giving a
smaller damping α=0.00002355 and the distance will be x=90766 ft.
Compared with spherical spreading attenuation is insignificant. In exploration
seismology the presence of layering has a profound effect on the change in
amplitude of a signal as it travels through the earth.
Waters:
Reflection seismology – Wiley-Interscience publication 1978