deletions | additions       diff --git a/subsection_The_P__rm_act__.tex b/subsection_The_P__rm_act__.tex  index c033aa8..dd986cd 100644  --- a/subsection_The_P__rm_act__.tex  +++ b/subsection_The_P__rm_act__.tex 
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\subsection*{The $P_{\rm act}$ Recovery Bias and Distortion}  \label{sec:the_Pact_recovery_bias_and_distortion} Not surprisingly, the devils with the smallest $P_{\rm act}$-values are the least likely to be detected since relatively close (small $b$) encounters are required for $P_{\rm obs} > P_{\rm min}$ and such encounters are more rare. For example, $b = 0$ would be required for a devil with A dust devil's pressure contour  $P_{\rmact} = P_{\rm  min}$ to be detected, an encounter geometry with carves out  a infinitesimally small probability.     The probability for long, narrow area $A(P_{\rm act}, \Gamma_{\rm act})$ as it travels on the surface of the observational arena. If  a barometer lies within that area, the  devil to will  be detected is proportional to the area detected, in principle.  $A$over its pressure profile for which $P(r) \ge P_{\rm min}$, which  is given by  $A = \pi r^2(P_{\rm act}) b_{\rm max}^2 + \upsilon \tau b_{\rm max}  =\pi  \left( \dfrac{\Gamma_{\rm act}(P_{\rm act})}{2} \right)^2 \Gamma_{\rm act}/2 \right) \sqrt{ \dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm min}} } \left[ \pi  \left( \Gamma_{\rm act}/2 \right) \sqrt{  \dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm min}} \right)$. We're interested in } + \upsilon \tau \right].$  The probability to recover a devil with a given $P_{\rm act}$ and $\Gamma_{\rm act}$ is proportional to this total area. Thus devils with deeper, i.e. larger $P_{\rm act}$, pressure wells are more likely to be recovered, skewing  the way in which recovered population.  We can cast  this bias shifts by ratioing this area to that of  the recovered distribution $n(P_{\rm obs})$ toward larger $P_{\rm obs}$, so we deepest and widest devil expected $A_{\rm max}$. We  will calculate the relative recovery probability by taking the ratio, assuming all devils have the same $\Gamma_{\rm act} = {\rm const.}$ \begin{equation}  f_{\Gamma_{\rm $$f_{\Gamma_{\rm  act}} \equiv A(P_{\rm act})/A(P_{\rm max}) \propto \left( \dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm max} - P_{\rm min}} \right).   \end{equation} \right).$$  As expected, the probability to detect a devil with $P_{\rm act} = P_{\rm min}$ is zero, and the probability for detecting a devil with $P_{\rm act} = P_{\rm max}$ is not actually unity, just the relative probability. Calculating the actual probability would require us to define the total area of the arena over which observations were made, e.g. the area of the playa where the barometer was deployed.