Page 264 - Well Logging and Formation Evaluation

254 Well Logging and Formation Evaluation

example function y = a*x + b evaluated between x 1 and x 2, the integral
becomes:

x 2 2 2 2
+
Ú ( ax b dx = [05 a x + bx c] = 05 a*( 2 x )+ b*( x - x )
x -
. *
+ )
*
. *
*
1
2
1
x 1
In many real engineering problems, data are presented as sampled at
discrete intervals (e.g., 0.5-ft sampling increment for logs) and cannot be
described by simple mathematical functions. For these data, a numerical
differentiation or integration may also be performed without resort to
calculus.
Say, for instance, one wanted to make a differential of a GR log with
respect to depth. The procedure would be to simply take the difference
between successive data values at each increment and divide by the depth
increment. Taking the integral would involve just adding successive data
values multiplied by the depth increment.
A4.2 SPECTRAL (FOURIER) ANALYSIS
For any wireline log sampled in depth, it is possible to think of the log
as being composed of a complex mixture of cosine waves that, when
added together in the right proportions, yield the log. The cosine func-
tions will have the form:

(
p
i f
y i = A i *cos * *x l i + )
2
where
A i = the amplitude of the component i
(1/l i), or k i, = the wavenumber of the component i
f i = the phase of the component i.
If L(x) is the complete log, we can say:

Lx) =S yi.
(

Spectral analysis is the mathematical determination of the set of A i and
f i as a function of k i. The determination of the spectra is performed using
computer algorithms, which will not be discussed here.
The range of k i that needs to be used is 0 (corresponding to a cosine
wave of infinite wavelength) to 1/sample increment (since variations at a
smaller scale than the sampling increment cannot be detected anyway).