We can understand both the linearity and the apparent dependence of the
data correlation on porosity in the second plotting method
shown in Figure 3 by understanding
some simple facts about such displays. Consider a random variable
*X*. If we display data on a plot of either *X* vs. *X* or 1/*X* vs. 1/*X*,
the result will always be a perfect straight line.
In both cases the slope of the straight line is exactly unity and the
intercept of the line is the origin of the plot (, ).
Now, if we have another variable *Y* and plot *Y*/*X* vs. 1/*X*, then we need
to consider two pertinent cases: (1) If *Y* = *constant*, then the plot
of *Y*/*X* vs. *X* will again be a straight line and the intercept will
again be the origin, but the slope will be *Y*, rather than unity.
(2) If but is a variable with small overall variation
(small dynamic range), then the plot of *Y*/*X* vs. 1/*X* will not generally be
exactly a straight line. The slope will be given approximately by the
average value of *Y* and the intercept will be near the origin, but its
precise value will depend on the correlation (if any) of *Y* and *X*.
In our second method of plotting, the variable plays
the role of *X* and the variable plays the role of
*Y*. The plots are approximately
linear because this method of display puts the
most highly variable combination of constants in the
role of *X*, and the least variable combination of constants
*v*_{s}^{2} in the role of *Y*. Furthermore, the slope of the observed lines
is therefore correlated inversely with the porosity because the slope is approximately the average value of *v*_{s}^{2}
which is well-known to decrease monotonically with increasing
porosity.

10/25/1999