Transient Constant Area Surface and Downhole Wave Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 100

Also note that the upcoming wave, which is closely correlated to 'p(t) only
at low frequencies, does not typically represent the train of desired logging 0's
and 1's since it contains “ghost reflections” generated downhole. Again,
downhole near the MWD signal source, waves are created that travel uphole; in
addition, waves are created which travel downhole and then reflect upward.
Thus, the net signal traveling uphole from the MWD tool is not the siren
position-encoded 'p(t) but the superposition of an intended signal and its
unwanted ghost reflections. Surface signal processing only removes undesired
surface effects. From the signal obtained from surface processing, the models
derived in Methods 4-5 and 4-6 must be used next to extract the intended signal,
that is, 'p(t), which contains the encoded 0's and 1's containing logging
information. All of these filters must be supplemented by additional ones,
introduced in Chapter 6, for other noise mechanisms. Taken together, the
complete system of filters provides the basic foundation of a signal processor
applicable to very high data rates.

4.1.2 Theory.
Let u(x,t) denote the Lagrangian fluid displacement variable for the
acoustic field. Then, the function
u(x,t) = F(t - x/c) - F(t + x/c - 2L/c) (4.1a)

represents the superposition of an upgoing wave F(t - x/c) and a downgoing
wave - F(t + x/c - 2L/c), with

u(L,t) = F(t - L/c) - F(t + L/c - 2L/c) = 0 at x = L (4.1b)
Note that we have assumed u = 0 at the pump piston face x = L (the pressure is
measured at the standpipe location x = 0). The piston speed is very small
compared to the sound speed in the fluid and can be ignored. If we had assumed
a centrifugal pump, the boundary condition at x = L would have been wu/wx = 0.
If p denotes the acoustic pressure and B is the bulk modulus, then

p(x,t) = -B wu/wx = + (B/c) [F'(t - x/c) + F'(t + x/c - 2L/c)] (4.1c)

At the transducer x = 0, p(0,t) = (B/c) [F'(t) + F'(t - 2L/c)] states that the total
measured pressure p(0,t) is the sum of an upward contribution + (B/c)F'(t) and
the delayed function + (B/c)F'(t - h) having the same sign because a solid
reflector has been assumed, where h = 2L/c is the known roundtrip time delay
from the transducer to the solid reflector. In more physically meaningful
nomenclature, we can rewrite Equation 4.1c as

P upgoing (t) + P upgoing (t-h) = P measured (t) (4.1d)
The deconvolution problem is stated as follows. When the total pressure
P measured (t) is available at a single transducer as a discrete array of values at
different instances in time, and P upgoing (t) vanishes at t < 0, find the function