deletions | additions       diff --git a/Figure_ref_fig_n_Pobs_from_uniform_n__.tex b/Figure_ref_fig_n_Pobs_from_uniform_n__.tex  index 996df2a..003cb71 100644  --- a/Figure_ref_fig_n_Pobs_from_uniform_n__.tex  +++ b/Figure_ref_fig_n_Pobs_from_uniform_n__.tex 
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Figure \ref{fig:n-Pobs_from_uniform_n-Pact} shows how significantly \subsection{Converting Between  the miss distance effect modifies Observed and Underlying Distributions}  We can use  the observed distribution of pressure depths as compared encounter geometry  to model  the underlying distribution. Well away from the $P_{\rm max}$, the distribution $n(P_{\rm obs})$ follows a statistical probability for  $P_{\rm obs}^{-2}$ power-law, similar obs}$ and $\Gamma_{\rm obs}$  to what has been observed in several studies. However, the lack fall within a certain range  of devils with $P_{\rm act} > P_{\rm max}$ means that eventually the values, given a  distribution declines toward zero. 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 $b \le b_{\rm max}$.  As Consider  a check distribution  of this result, we also conducted a Monte-Carlo simulation observed values $\rho(P_{\rm obs}, \Gamma_{\rm obs}) = \dfrac{d^2N}{dP_{\rm obs}\ d\Gamma_{\rm obs}}$. The small number  ofdust devil encounters, similar to that in \citet{Lorenz_2014}. We modeled 100,000 dust  devils with a uniform distribution of $P_{\rm $dN = f\ \rho(P_{\rm act}, \Gamma_{\rm act})\ dP_{\rm act}\ d\Gamma_{\rm  act}$ contributing are those that had closest approach distances  between $P_{\rm min}$ $b$  and $P_{\rm $b + db$ 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 assumed a fixed value 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 need to re-cast the upper limit $b_{\rm max}$ in terms of the observed parameters, $b_{\rm max} = \sqrt{\Gamma_{\rm obs}^2 - \Gamma_{\rm min}^2}$. This expression shows, for example, that  for $\Gamma_{\rm act}$. For each devil, obs} = \Gamma_{\rm min}$,  we chose a random closest encounter distance as should only consider  $b = \sqrt{\Phi}\ b_{\rm max}$, where $\Phi$ is 0$, i.e. only  a random variable uniformly distributed between 0 and 1. direct encounter.  The dashed, red line in Figure \ref{fig:n-Pobs_from_uniform_n-Pact} illustrates the result, showing agreement with the prediction within statistical uncertainties. 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 obs}^2 - \Gamma_{\rm min}^2}} f\ \rho(P_{\rm act}(b), \Gamma_{\rm act}(b)) \dfrac{2b\ db}{\Gamma_{\rm obs}^2 - \Gamma_{\rm min}^2}.  \end{equation}