Geology Reference
In-Depth Information
Internal MWD upgoing (psi) signal available as
P(x,t) = + 5.000 {H(x- 150.000-ct) - H(x- 400.000-ct)}
+ 10.000 {H(x- 600.000-ct) - H(x- 1000.000-ct)}
+ 15.000 {H(x- 1400.000-ct) - H(x- 1700.000-ct)}
consisting of three closely spaced and short rectangular pulses (H is the
Heaviside step function). At time t = 0, the pressure P(x,0) contains three
rectangular pulses with amplitudes (a) 5 for 150 < x < 400, (b) 10 for 600 < x <
1000, and (c) 15 for 1400 < x < 1700. Thus, the pulse widths and separations,
going from left to right, are
x
400 - 150 = 250 ft
x
600 - 400 = 200 ft
x
1000 - 600 = 400 ft
x
1400 - 1000 = 400 ft
x
1700 - 1400 = 300 ft
The average spatial width is about 300 ft. If the sound speed is 5,000 ft/sec
(as assumed below) then the time required for this pulse to displace is 300/5,000
or 0.06 sec. Since sixteen of these are found in a single second, this represents
16 bps, approximately. Below we define the noise function, which propagates
in a direction opposite to the upgoing signal. For our upgoing signal we have 16
bps. In our noise model below, we assume a 15 Hz sinusoidal wave (for
convenience, though not a requirement) with an amplitude of 20 (which exceeds
the 5, 10, 15 above). These equal frequencies provide a good test of effective
filtering based on directions only - conventional frequency methods will not
work since both signal and noise frequencies are similar.
For the MWD pulse, the far right position is 1,700 ft. We want to be able
to “watch” all the pulses move by in our graphics, so we enter “1710” ( >1700
below). We also assume a transducer separation of 30 ft. This is about 10% of
the typical pulse width above, and importantly, is the length of the standpipe;
thus, we can place two transducers at the top and bottom of the standpipe.
Recall that Method 4-4 is based on derivatives. The meaning of a derivative
from calculus is “a small distance.” Just how small is small? The results seem
to suggest that 10% of a wavelength is small enough.
Downward propagating noise (psi) assumed as
N(x,t) = Amplitude * cos {2Ŧ f (t + x/c)} ...
o Enter noise freq "f" (hz):
15
o Type noise amplitude (psi):
20
o Enter sound speed c (ft/s):
5000
o Mean transducer x-val (ft):
1710
o Transducer separation (ft):
30
Note that the noise amplitude is not small, but is chosen to be comparable
to the MWD amplitudes, although only large enough so that all the line
drawings fit on the same graphical display. The method actually applies to
much larger amplitudes as we will shortly demonstrate. After SAS14D.exe
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