Geology Reference
In-Depth Information
4.4.2
Theory.
Our derivation at first follows the author's United States Patent No.
5,969,638 entitled “Multiple Transducer MWD Surface Signal Processing.” But
the present method, which includes a robust integrator to handle sharp pressure
pulses, substantially changes the earlier work, which is incomplete; the full
formulation is developed here. In Method 4-3, we used time delayed signals for
which there was no restriction on time delay size. For Method 4-4, we invoke
time and space derivatives; thus sampling times should be small compared to a
period and transducer separations should be small compared to a wavelength.
As in Method 4-3, the representation of pressure as an upgoing and
downgoing wave is still very general, and all waves in the downward direction
are removed with no information required at all about the mudpump, the
desurger or the rotary hose. Note that “c” is the mud sound speed at the surface
and should be measured separately. In our derivation, expressions for time and
space derivatives of p(x,t) are formed, from which the downgoing wave “g” is
explicitly eliminated, leaving the desired upgoing “f.” The steps shown are
straightforward and need not be explained.
p(x,t) = f(t - x/c) + g(t + x/c)
(4.4a)
p t = f' + g'
(4.4b)
p x = - c -1 f' + c -1 g'
(4.4c)
cp x = - f' + g'
(4.4d)
p t - cp x = 2f'
(4.4e)
f ' = ½ (p t - cp x ) (4.4f)
Equation 4.4f for “f,” which is completely independent of the downgoing
wave “g,” however, applies to the time derivative of the upgoing signal f(t - x/c)
and not “f” itself. Thus, if a square wave were traveling uphole, the derivative
of the signal would consist of two noisy spikes having opposite signs. This
function must be integrated in order to recover the original square wave, and at
the time the patent was awarded, a robust integration method was not available.
The required integration is not discussed in the original patent, where it was
simply noted that both original and derivative signals in principle contain the
same information. In our recent SAS14D software, a special integration
algorithm is given to augment the numerical representation in Equation 4.4f.
The success of the new method is demonstrated below in Figure 4.4b.
At first glance, the two-transducer delay approach in Method 4-3 seems to
be more powerful because it does not require time integration, and since it does
not involve derivatives, there are no formal requirements for sampling times to
be small and transducer separations to be close. However, in any practical high-
data-rate application, the latter will be the case anyway, e.g., slow sampling
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