Transient Constant Area Surface and Downhole Wave Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 119

4.3.7 Run 5. Non-periodic background noise.
In Runs 1-4, we assumed a periodic wave to model (a Fourier component
of) the mudpump noise. However, we can easily model non-sinusoidal
distortions caused by the desurger (Chapter 6 provides more realistic models).
Numerous non-sinusoidal functions have been tested successfully. To
demonstrate the general nature allowed for the XNOISE function, we assume a
straight line time function to locally represent a large scale slowly varying noise
function. This can be an idealization of a signal distorted by the desurger.
FUNCTION SIGNAL(T)
C MWD upward wave signal function
C Train of pulses, 0.5 sec width, 0.5 sec separation
C CASE 2. NARROW PULSE WIDTH (Considered in single xducer
method)
C Clearly see interference between upgoing and reflected pulses
A = 10.0
R = 100.0
SIGNAL = A*(TANH(R*(T-0.100))-TANH(R*(T-0.101)))/2.
RETURN
END
C
FUNCTION XNOISE(T)
C Mud pump noise function may also include reflected MWD signal,
C but it is not necessary to add the wave reflection to the
C total noise to demonstrate directional filtering.
C FRQPMP = Hertz freq of pump noise, propagates downward
PI = 3.14159
FRQPMP = 15.
C AMP = 0.25
AMP = 5.
C XNOISE = AMP*SIN(2.*PI*FRQPMP*T) + 0.0*SIGNAL(T)
XNOISE = AMP*T
RETURN
END
The time delay between transducers is 0.01 sec or 10 ms. In Figure 4.3f,
the black curve is the upgoing pulse, the red is a linearly growing non-periodic
noise function in time that may be representative of other more general forms of
noise, and the green is the superposition of the upgoing MWD signal and the
noise function. The blue curve shows how the black signal is successfully
recovered from the green input data using Method 4-3.
The reader may again ask, “Why are we so successful at recovering signals
even with signal-to-noise ratios less than 10%, when the signal processing
literature is much less optimistic?” The reason: conventional signal processing
deals typically with random noise. Our noise is not random, but a propagating
wave with wave-like properties - noise that a “smart algorithm” such as ours
can remove. If our noise also contains random or other types of noise, then that
noise must be separately removed before or after application of our filters.