Given a distribution obeying the power-law (fractal) relation, such as the cumulative distribution function $L_{cf}(> X) = CR^{-D}$, if $X$ is given, how does one find the constant $C$ from a given data set?
For example, given this table:
[Source: C. Li et al. / Geomorphology 130 (2011)]
how does one determine how the values for $L_{cf}$ were obtained? It appears that they used the method of least squares fitting to obtain the values for $D$, but I don't seem to understand where the values for $C$ are coming from. Any help would be appreciated.
If the relationship between $R$ and $L_{cf}$ is $$ L_{cf}=C R^{-D}\tag1, $$ then logging both sides of (1) gives you $$ \log(L_{cf})=\log C -D\log R.\tag2 $$ Read equation (2) this way: If you plot $\log(L_{cf})$ on the $y$-axis against $\log R$ on the $x$-axis (equivalently, if you plot $L_{cf}$ against $R$ on log-log paper), you should see a straight line. Moreover, the slope of that line will be $-D$, and the $y$-intercept will be $\log C$. From these two bits of information you can deduce the values of $D$ and $C$.
(The authors found two piecewise linear segments, each with its own parameters that can be deduced in this way. Interesting paper, btw. The data seem to fit, ahem, remarkably well.)