deletions | additions       diff --git a/subsection_Converting_Between_the_Observed__.tex b/subsection_Converting_Between_the_Observed__.tex  index ecb61bd..4897b09 100644  --- a/subsection_Converting_Between_the_Observed__.tex  +++ b/subsection_Converting_Between_the_Observed__.tex 
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$P_{\rm act}(b) = P_{\rm obs}\left[ 1 + \left( \dfrac{2b}{\Gamma_{\rm act}} \right)^2 \right]$. To calculate the integral, we also need to re-cast the upper limit $b_{\rm max}$ in terms of the observed parameters, $b_{\rm max} = \frac{1}{2} \sqrt{\Gamma_{\rm obs}^2 - \Gamma_{\rm min}^2}$. This expression shows, for example, that for $\Gamma_{\rm obs} = \Gamma_{\rm min}$, we should only consider $b = 0$, a direct encounter. The integral to convert from $\rho(P_{\rm act}, \Gamma_{\rm act})$ to $\rho(\Gamma_{\rm obs}, P_{\rm obs})$ is therefore  \begin{eqnarray}  \label{eqn:convert_from_actual_to_observed_density}  \rho(\Gamma_{\rm obs}, P_{\rm obs}) = & \int_{b = 0}^{b_{\rm max}} & &  f\ \rho(\Gamma_{\rm act}(b), P_{\rm act}(b))\ act}(b)) &\  \dfrac{2b\ db}{b_{\rm max}^2} \\ = & \int_{b = 0}^{\frac{1}{2} \sqrt{\Gamma_{\rm obs}^2 - \Gamma_{\rm min}^2}} & &  f\ \rho(\Gamma_{\rm act}(b), P_{\rm act}(b))\ act}(b))& \  \left( \Gamma_{\rm obs}^2 - \Gamma_{\rm min}^2 \right)^{-1}\ 4b\ db. \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).