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\subsection{Converting Between the Observed and Underlying Distributions}  We can use the encounter geometry to model the statistical probability for $P_{\rm obs}$ and $\Gamma_{\rm obs}$ to fall within a certain range of values, given a distribution of $P_{\rm act}$- and $\Gamma_{\rm act}$-values. The probability density for passing between $b$ and $b + db$ of a devil is $dp(b) = 2 b\ db / b_{\rm max}^2 $ for $b \le b_{\rm max}$. Consider a distribution of observed values $\rho(P_{\rm obs}, \Gamma_{\rm obs}) = \dfrac{d^2N}{dP_{\rm obs}\ d\Gamma_{\rm obs}}$. The small number of devils $dN = f\ \rho(P_{\rm act}, \Gamma_{\rm act})\ dP_{\rm act}\ d\Gamma_{\rm act}$ contributing are those that had closest approach distances between $b$ and $b + db$ of the detector. Thus, we can convert $f\ \rho(P_{\rm act}, \Gamma_{\rm act})$ to $\rho(P_{\rm obs}, \Gamma_{\rm obs})$ by integrating from $b = 0$ to $b_{\rm max}$ and setting $\Gamma_{\rm act}(b) = \sqrt{\Gamma_{\rm obs} - \left( 2b \right)^2}$ and  $P_{\rm act}(b) = P_{\rm obs}\left[ 1 + \left( \dfrac{2b}{\Gamma_{\rm act}} \right)^2 \right]$. To perform 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 $b = 0$, i.e. only a direct encounter. The integral to convert from $\rho(P_{\rm act}, \Gamma_{\rm act})$ to $\rho(P_{\rm obs}, \Gamma_{\rm obs})$ is following: \begin{equation}  \label{eqn:convert_from_actual_to_observed_density}  \rho(P_{\rm obs}, \Gamma_{\rm obs}) = \int_{b = 0}^{\sqrt{\Gamma_{\rm 0}^{\frac{1}{2} \sqrt{\Gamma_{\rm  obs}^2 - \Gamma_{\rm min}^2}} f\ \rho(P_{\rm act}(b), \Gamma_{\rm act}(b)) \dfrac{2b\ \dfrac{4b\  db}{\Gamma_{\rm obs}^2 - \Gamma_{\rm min}^2}. \end{equation}