WHY THE SECOND DISPLAY IS ALWAYS APPROXIMATELY LINEAR

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 $Y \ne constant$ 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 $\lambda/\rho$ plays the role of X and the variable $v_s^2 = \mu/\rho$ plays the role of Y. The plots are approximately linear because this method of display puts the most highly variable combination of constants $\lambda/\rho$ in the role of X, and the least variable combination of constants v_s² in the role of Y. Furthermore, the slope of the observed lines is therefore correlated inversely with the porosity $\phi$because the slope is approximately the average value of v_s² which is well-known to decrease monotonically with increasing porosity.