Signal Processor Design and Additional Noise Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 172

balloons.” For elastic wave applications, the above analogies can be used to
develop models for complicated bottomhole assemblies with multiple changes in
bottomhole assembly and borehole geometry and acoustic impedance along the
path of signal propagation - models analogous to the six-segment waveguide
formulation in Chapter 2 and the two-part waveguide approaches in Chapters 3
and 5 are easily developed for elastic systems. This is obvious because the only
formulation differences are changes required at acoustic impedance
discontinuities located at drillpipe and drill collar junctions.
For the fluid flows considered earlier, we assumed that acoustic pressure
and volume velocity were continuous, that is, (wu/wx) collar = (wu/wx) drillpipe and
A collar (wu/wt) collar = A drillpipe (wu/wt) drillpipe hold. These would be replaced by
continuity of displacement and net force, that is, (wu/wt) collar = (wu/wt) drillpipe and
A collar (wu/wx) collar = A drillpipe (wu/wx) drillpipe . In terms of Ix,t), we would have
(wI/wx) collar = (wI/wx) drillpipe and A collar (wI/wt) collar = A drillpipe (wI/wt) drillpipe . In other
words, the models and numerical solutions developed in Chapters 2-5 can be
used without modification provided the dependent variables are interpreted
differently ! Our discussion, for simplicity, assumes that the moduli of elasticity
for drillpipe and drill collar are identical, but extensions to handle differences
are easily constructed. The wave equation transforms used here to develop our
physical analogies were originally introduced by the author in Chin (1994). It is
important to note that, while “drillpipe acoustic” methods work in vertical wells,
they perform poorly in deviated and horizontal wells where rubbing of the
drillstring with the formation is commonplace.
6.6 LMS Adaptive and Savitzky-Golay Smoothing Filters
(software reference, all of the filters in Sections 6 and 7 are found
in C:\MWD-06)
As explained in our chapter objective, our aim is not an exhaustive
treatment of standard signal processing, but rather, a concise one which directs
readers to more detailed publications, e.g., Stearns and David (1993) and Press
al et (2007). Exceptions are Sections 1-5 above, which explain relevant
downhole concepts in detail. Of the more complicated methods available, LMS
(least mean squares) adaptive filters provide some degree of flexibility in
applications with slowly varying properties. Figure 6.6a shows a raw
unprocessed wave signal with random noise and a propagating wave, while the
LMS processed waveform remarkably appears in Figure 6.6b. Smoothing filters
may need to be applied to data such as that in Figure 6.6b, or to noisy datasets
such as the one in Figure 6.6c. The Savitzky-Golay smoothing filter, for
instance, removes high-frequency noise in the top curve to produce the lower
frequency red line shown at the bottom.