Page 272 - Well Logging and Formation Evaluation

262 Well Logging and Formation Evaluation

(
(
(
¢
¢
(
P SC¢) = [ P S 2 * P C S 2 )] [ P S 2 * P C S 2 )+ ()* P C S 1 )] (A4.22)
¢
()
()
P S 1
2
()
()
P C S 2 ).
P S 1 *
P S 2 * R [ P S 2 * R+ () ( 1- R)], since R = ( ¢ (A4.23)
Note that it is always true that:
P SC)+ ( 2 1 and P S C¢)+ ( 2 1.
P S C¢) =
(
P S C) =
(
1
1
Note also that only in the special case that:
P S 1 = () = 05
()
.
P S 2
is it true that:
(
P SC) = (
P C S 1 )
1
A4.6 LEAST SQUARES FIT AND CORRELATION
Consider a series of points (x 1, y 2 )...(x n, y n). When plotted these points
may lie approximately on a straight line or a curve or be scattered ran-
domly. Least squares fit is a way to find the coefficients of a function
approximating the behavior of the data.
Consider data which may be approximated by the formula y = a*x + b.
The line yielded by this equation is known as the line of regression of y
on x. Forming the sum of the squares of all the deviations of the given
numerical value of y from the theoretical values:
-
S = ( S y a x b) 2 (A4.24)
- *
S is minimized when ∂S/∂a =∂S/∂b = 0. This occurs when:

y (
-
-
x (
-
S 2 ** y a x b) = 0 and S 2 * * y a x b) = 0. (A4.25)
-
*
*
Solving these equations:
2
2
a = [ * S x y - S x * S y] [ * S x - (S x) ] (A4.26)
n
n
*
2
2
x y] [
2
b = [S y x - S x S * n S x - (S x) ] (A4.27)
*
*
*
where n is the number of samples.