deletions | additions       diff --git a/subsection_Converting_Between_the_Observed__.tex b/subsection_Converting_Between_the_Observed__.tex  index 1378016..8d5786a 100644  --- a/subsection_Converting_Between_the_Observed__.tex  +++ b/subsection_Converting_Between_the_Observed__.tex 
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\subsection{Converting Between the Observed and Underlying Distributions}  Consider a distribution of observed values $\rho(\Gamma_{\rm obs}, P_{\rm obs}) = \dfrac{d^2N}{d\Gamma_{\rm obs}\ dP_{\rm obs}}$. The small number of devils $dN = f\ \rho(\Gamma_{\rm obs}, P_{\rm obs})\ d\Gamma_{\rm act}\ dP_{\rm act}$ contributing are those that had closest approach distances between $b$ and $b + db$ of the detector. Thus, we can convert $f\ \rho(\Gamma_{\rm obs}, P_{\rm obs})$ to $\rho(\Gamma_{\rm obs}, P_{\rm obs})$ by integrating from $b = 0$ to $b_{\rm max}$ and setting $\Gamma_{\rm act}(b) = \sqrt{\Gamma_{\rm obs}^2 - \left( 2b \right)^2}$ and  $P_{\rm act}(b) = P_{\rm obs}\left( \Gamma_{\rm obs}/\Gamma_{\rm act} \right)^2$. 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{equation} \begin{eqnarray}  \label{eqn:convert_from_actual_to_observed_density}  \rho(\Gamma_{\rm obs}, P_{\rm obs}) &=& &\int_{b  = \int_{b 0}^{b(\Gamma_{\rm obs}, P_{\rm obs})} &f\ \rho(\Gamma_{\rm act}(b), P_{\rm act}(b))\ &\dfrac{2b\ db}{b_{\rm max}^2}&\\  &=& &\int_{b  = 0}^{\left( \Gamma_{\rm obs}/2 \right) \sqrt{\left( P_{\rm obs} - P_{\rm min} \right)/P_{\rm min}}} min}}}&  f\ \rho(\Gamma_{\rm act}(b), P_{\rm act}(b))\ \dfrac{2b\ &\dfrac{2b\  db}{b_{\rm max}^2}.  \end{equation} max}^2}&.  \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).