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rates will not capture detailed data. Thus, Method 4-4 is no more restrictive than
Method 4-3. However, the present method is powerful in its own right because
the presence of the wp/wx derivative implies that one can approximate it by more
than two (transducer) values of pressure at different positions using higher-order
finite difference formulas - in operational terms, one can employ multiple
transducers and transducer arrays to achieve higher accuracy. Similarly, the
presence of wp/wt means that one can utilize more than two time levels of
pressure in processing in order to achieve high time accuracy. The required
processing in space and time is inferred from the use of finite difference
formulas in approximating the derivatives shown and numerous such
computational molecules are available in the numerical analysis literature. The
illustrative calculation used below, however, assumes only two transducers and
pressures stored at two levels in time.
4.4.3 Run 1. Validation analysis.
SAS14D presently runs with only Option 3 fully tested and other options
are under development. We explain below some hard-coded assumptions and
interpret computed results. The software model assumes the following upgoing
MWD signal, as noted in output duplicated below.
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)}
This upgoing MWD wave signal is presently hard-coded. H is the Heaviside
step function. At time t = 0, the pressure P(x,0) contains three rectangular
pulses with amplitudes (1) 5 for 150 < x < 400, (2) 10 for 600 < x < 1000, and
(3) 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 one second, this represents 16
bits/sec, approximately. Below we define the noise function, which propagates
in a direction opposite to the upgoing signal. For our upgoing signal we have 16
bits/sec. In our noise model below, we assume a sinusoidal wave (for
convenience, though not a requirement) of 15 Hz, amplitude 20 (which exceeds
the 5, 10, 15 above). These approximately equal frequencies provide a good test
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