Geology Reference
In-Depth Information
The well known two-transducer wave separation method developed in
Foster and Patton (1973), for example, applies to “continuous wave telemetry.”
Since the signal, taken as F, is a periodic function with recurring maxima and
minima, a robust analogue filter taking advantage of that fact is suggested, and
(x
b
- x
a
) is required to take on certain fractional multiples of a wavelength.
Incidentally, in communications theory, the “probability of bit error” depends on
the nondimensional “signal-to-noise (S/N) ratio.” We might note that pump
disturbances do
not
represent noise in the usual sense; they are coherent waves
that can be detected and cancelled by signal processing algorithms such as the
one given here. Noise, in general, refers to random disturbances, and “wave
noise” is not noise - even if it originates from “noisy” mud pumps.
Differential technique.
In this next example, we examine waves with
dissipation. In fact, we consider the superposition of a wave F(t + x/c
1
) having a
dissipation factor e
-
t
, and a wave G(t + x/c
2
) having the dissipation e
-
t
, where
the attenuation constants satisfy > 0. Thus, we take
u(x,t) = e
- t
F( t + x / c
1
) + e
- t
G( t + x / c
2
) (4.2.151)
We next multiply Equation 4.2.151 through by e
+ t
, differentiate the resulting
expression with respect to t, and then x, to successively obtain
ue
+ t
= e
t
F( t + x / c
1
) + G( t + x / c
2
) (4.2.152)
(ue
+ t
)/ t = {e
t
F( t + x / c
1
)}/ t + G' (4.2.153)
(ue
+ t
)/ x = {e
t
F( t + x / c
1
)}/ x + G'/c
2
(4.2.154)
c
2
(ue
+ t
)/ x = c
2
{e
t
F( t + x / c
1
)}/ x + G' (4.2.155)
If we now subtract Equation 4.2.153 from Equation 4.2.155, we obtain
c
2
{e
t
F( t + x / c
1
)}/ x - {e
t
F( t + x / c
1
)}/ t
= c
2
(ue
+ t
)/ x - (ue
+ t
)/ t
(4.2.156)
and
c
2
e
t
F'(t + x/c
1
)/c
1
- e
t
F'(t + x/c
1
) - e
t
F( t + x / c
1
)
= c
2
e
+ t
u/ x - e
+ t
u/ t - e
+ t
u (4.2.157)
The e
+ t
terms cancel throughout, and multiplication by e
t
gives the required
first-order “differential” (not “difference”) equation for F, namely,
(c
2
/c
1
-1)F'(t+x/c
1
) - F( t + x / c
1
) = (c
2
u/ x - u/ t - u) e
t
(4.2.158)
This equation is the sought result, providing a means to determine F from
sampled values of u(x,t). Suppose that = is small, with c
1
= - c and c
2
= c .
This limit models wave propagation problems where upgoing and downgoing
waves have like speeds and dissipation rates. Then, the signal is simply
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