deletions | additions       diff --git a/subsection_Converting_Between_the_Observed__.tex b/subsection_Converting_Between_the_Observed__.tex  index 811455c..728f572 100644  --- a/subsection_Converting_Between_the_Observed__.tex  +++ b/subsection_Converting_Between_the_Observed__.tex 
...
$P_{\rm act}(b) = P_{\rm obs}\left( \Gamma_{\rm obs}/\Gamma_{\rm act} \right)^2$. To calculate the integral, we also need to re-cast the upper limit $b_{\rm max}$ to express the maximum possible radial distance, $b_{\rm max} = \left( \dfrac{\Gamma_{\rm max}}{2} \right) \sqrt{\dfrac{P_{\rm max} - P_{\rm min}}{P_{\rm min}}}$. The integral to convert from $\rho(P_{\rm act}, \Gamma_{\rm act})$ to $\rho(\Gamma_{\rm obs}, P_{\rm obs})$ extends from $b = 0$ to $\left( \Gamma_{\rm obs}/2 \right) \sqrt{\left( P_{\rm obs} - P_{\rm min} \right)/P_{\rm min}}$ and is given by  \begin{eqnarray}  \label{eqn:convert_from_actual_to_observed_density}  \rho(\Gamma_{\rm obs}, P_{\rm obs}) &=& &\int_{b = 0}^{b(\Gamma_{\rm obs}, P_{\rm obs})} & &f&\ \rho(\Gamma_{\rm act}(b), P_{\rm act}(b))\ &\dfrac{2b\ \dfrac{2b\  db}{b_{\rm max}^2}&\\ max}^2}\\  &=& &\int_{b = 0}^{\left( \Gamma_{\rm obs}/2 \right) \sqrt{\left( P_{\rm obs} - P_{\rm min} \right)/P_{\rm min}}}& &A_{\rm max}^{-1}\ \Gamma_{\rm act} \sqrt{\dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm min}}} \upsilon\ \kappa\ \left( \Gamma_{\rm act}/{\rm m} \right)^{2/3}& \rho(\sqrt{\Gamma_{\rm \\   && && &\rho(\sqrt{\Gamma_{\rm  obs}^2 - \left( 2b \right)^2}, P_{\rm obs}\left( \Gamma_{\rm obs}/\Gamma_{\rm act} \right)^2)\ &\dfrac{2b\ \right)^2)&\ \dfrac{2b\  db}{\left( \dfrac{\Gamma_{\rm max}}{2} \right)^2 \left( \dfrac{P_{\rm max} - P_{\rm min}}{P_{\rm min}} \right)}&. \right)}.  \end{eqnarray}  Figure \ref{fig:uniform_actual_distribution_to_observed_distribution} shows the result for a uniform distribution for underlying values, $\rho(\Gamma_{\rm act}, P_{\rm act}) = \left( P_{\rm max} - P_{\rm min} \right)^{-1}\ \left( \Gamma_{\rm max} - \Gamma_{\rm min} \right)^{-1}$ and compares it to a simulated dust devil survey (blue circles).