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 \(\rho(\Gamma_{\rm act}, P_{\rm act})\) to \(\rho(\Gamma_{\rm obs}, P_{\rm obs})\) by integrating the former density over \(b\) and accounting for the bias and distortion effects. To calculate the integral, we also need to re-cast the upper limit to express the maximum possible radial distance, i.e. the distance at which \(P_{\rm act} = P_{\rm max}\). Using Equation \ref{eqn:b} and making the replacements \(P_{\rm act} = P_{\rm max}\) and \(\Gamma_{\rm act} = \left( P_{\rm obs}/P_{\rm max} \right)^{1/2} \Gamma_{\rm obs}\) from Equation \ref{eqn:P_obs_Gamma_obs} gives \(b(\Gamma_{\rm obs}, P_{\rm obs}) = \left(\Gamma_{\rm obs}/2\right) \left[ \left(P_{\rm max} - P_{\rm obs}\right)/P_{\rm max} \right]^{1/2}\). The integral to convert from \(\rho({\rm act}) \equiv \rho(P_{\rm act}, \Gamma_{\rm act})\) to \(\rho({\rm obs}) \equiv \rho(\Gamma_{\rm obs}, P_{\rm obs})\) is then
\[\rho({\rm obs}) = \int_{b^\prime = 0}^{b({\rm obs})} f\ \rho({\rm act}(b^\prime))\ \dfrac{2b^\prime\ db^\prime}{b_{\rm max}^2} \\ = 2\ A_{\rm max}^{-1}\ \upsilon\ \kappa\ b_{\rm max}^{-2} P_{\rm th}^{-1/2}\ \int_{b^\prime = 0}^{b({\rm obs})} \left( \Gamma_{\rm act}(b^\prime)/{\rm m} \right)^{5/3} \left( P_{\rm act}(b^\prime) - P_{\rm th} \right)^{1/2} \ \rho({\rm act}(b^\prime))\ b^\prime\ db^\prime \label{eqn:convert_from_actual_to_observed_density},\]
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 of actual values, \(\rho({\rm act}) = \left( P_{\rm max} - P_{\rm th} \right)^{-1}\ \left( \Gamma_{\rm max} - \Gamma_{\rm th} \right)^{-1}\) and compares it to the simulated results of an observational 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.)