deletions | additions       diff --git a/subsection_Converting_Between_the_Observed__.tex b/subsection_Converting_Between_the_Observed__.tex  index 5f663fc..6d03f19 100644  --- a/subsection_Converting_Between_the_Observed__.tex  +++ b/subsection_Converting_Between_the_Observed__.tex 
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\begin{equation}  \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\ db}{b_{\rm max}^2}\\  = 2\ A_{\rm max}^{-1}\ \upsilon\ \kappa\ b_{\rm max}^{-2} P_{\rm min}^{-1/2}\ \int_{b = 0}^{b(\Gamma_{\rm obs}, P_{\rm obs})} \left( \Gamma_{\rm act}(b)/{\rm m} \right)^{5/3} \left( P_{\rm act}(b) - P_{\rm min}\right)^{1/2} min} \right)^{1/2}  \ \rho( \Gamma_{\rm act}(b), P_{\rm act}(b))\ b\ db, \end{equation}  where $\kappa = 40\ {\rm s}$ and $\Gamma_{\rm act}$ is measured in meters, m. 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). (For the uniform distribution, the integral has a closed form expression that is unwieldy, so we opt to perform the integration numerically.)