Transient Constant Area Surface and Downhole Wave Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 129

h'(W) = - {c/(2B)} 'p(W- L/c) (4.5c)
Next we subtract Equation 4.5b from Equation 4.5a, but this time introduce the
dummy variable W = t - L/c, to obtain

f '(W) = h'(W) + {c/(2B)} 'p(W + L/c) (4.5d)

If we eliminate h'(W) between Equations 4.5c and 4.5d, we obtain 'p(W + L/c) -
'p(W - L/c) = (2B/c) f '(W), which we can express in the more meaningful
physical form

½ ['p(W + L/c) - 'p(W - L/c)] = (B/c) f '(W) (4.5e)
We now recast the above equation in a more useful form, taking the dummy
variable W = t - L/c. Since it is clear that (B/c) f '(t - L/c) is just the upgoing
pressure p 2 (L,t), we can write
½>'p(t) - 'p(t - 2L/c)] = (B/c) f '(t - L/c) = p 2 L,t (4.5f)

where p 2 L,t is the result of using Methods 4-1, 4-2, 4-3 or 4-4 and 'p is to be
recovered. Equation 4.5f can be easily interpreted physically. Suppose that a
pulser creates a 'p signal. A signal 'p/2 goes uphole. It simultaneously sends a
signal - 'p/2 going downhole, so that the net pressure differential is 'p. The
signal - 'p/2 travels downward to the bit, which is assumed as a solid reflector,
and the signal reflects with an unchanged pressure sign, that is, the contribution
- ½ 'p(t - 2L/c), now including the roundtrip time delay 2L/c. This derivation
applies to dipole pulsers where the created MWD pressure is antisymmetric with
respect to the source position. Obvious changes will apply to monopole
(negative pressure) pulsers which, again, are not considered in this topic.
Note that the two terms on the left side of the Equation 4.5f refer to
incident and ghost signals, while p 2 L,t is obtained from measured surface data.
This is the “p pipe ” of Chapter 2. It is obtained from surface signal processing
results of using Methods 4-1, 4-2, 4-3 or 4-4 applied at the standpipe. It is not
necessary to know the amount of energy loss incurred from downhole to surface
to apply Methods 4-5 and 4-6 since the same attenuation applies to all parts of
the signal. Any convenient electronic gain suffices. If we denote H = 2L/c as
the roundtrip delay time between the pulser and the solid bit reflector, we can
write Equation 4.5f as
'p(t) - 'p(t - H) = 2 p 2 L,t (4.5g)
Detection problems are associated with Equation 4.5g. Consider the use of
phase-shift-keying (PSK). When H is “not small,” a phase shift propagates
uphole while one simultaneously travels downward and reflects upward. Two
phase shifts travel up the drillpipe, one real and the other apparent. Solution of
Equation 4.5g, which is possible exactly and analytically, is required to
determine 'p(t) which alone contains the input sequence of 0's and 1's.