deletions | additions       diff --git a/The_fact_that_larger_faster__.tex b/The_fact_that_larger_faster__.tex  index 9107099..20ed122 100644  --- a/The_fact_that_larger_faster__.tex  +++ b/The_fact_that_larger_faster__.tex 
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The fact that larger, faster dust devils cover more area means that they are more likely to be recovered by fixed station surveys. We can quantify the recovery probability $f$ by taking the ratio of track areas for a given dust devil to the largest area for a dust devil, $A_{\rm max}$:  \begin{equation}  \label{eqn:recovery_bias}  f = \dfrac{A(P_{\rm \dfrac{A(\Gamma_{\rm  act}, \Gamma_{\rm P_{\rm  act})}{A_{\rm max}} = A_{\rm max}^{-1}\ \Gamma_{\rm act} \sqrt{\dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm min}}} \upsilon\ \tau. \end{equation}  The devil with the deepest profile need not also have the widest profile or the largest velocity. \cite{Renn__2001} argue that the diameter of a vortex is set, in part, by the local vorticity field, while \cite{Balme_2012}, from their field studies, find no correlation between diameter and velocity from their field work. However, in quantifying the recovery probability $f(P_{\rm $f(\Gamma_{\rm  act}, \Gamma_{\rm P_{\rm  act})$, it's only important that we apply a uniform normalizing factor to the whole population, and the denominator in Equation \ref{eqn:recovery_bias} just provides a convenient expression for that. Any other uniform normalization (e.g., using average parameters) would suffice. Figure \ref{fig:recovery_bias} shows contours of $f(P_{\rm $f(\Gamma_{\rm  act}, \Gamma_{\rm P_{\rm  act})$, assuming $\upsilon = \upsilon_{\rm max}$. Not surprisingly, the recovery probability increases toward the upper-right corner, indicating that the deepest, widest dust devils are the most likely to be recovered. The product $f(P_{\rm $f(\Gamma_{\rm  act}, \Gamma_{\rm P_{\rm  act}) \times \rho(P_{\rm \rho(\Gamma_{\rm  act}, \Gamma_{\rm P_{\rm  act})$ represents the population of devils that are detected but not what the recovered population actually looks like.