deletions | additions       diff --git a/section_Formulating_the_Recovery_Biases__.tex b/section_Formulating_the_Recovery_Biases__.tex  index 9a44ef1..6680d81 100644  --- a/section_Formulating_the_Recovery_Biases__.tex  +++ b/section_Formulating_the_Recovery_Biases__.tex 
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\begin{enumerate}  \item Each dust devil pressure profile has a well-defined, static profile, which follows a Lorentzian: $P(r) = \dfrac{P_{\rm act}}{1 + \left( 2r/\Gamma_{\rm act} \right)^2 }$. Here $r$ is the distance from the devil center, $P_{\rm act}$represents  the actual pressure depth at the devil's center, and $\Gamma_{\rm act}$ the profile full-width at half-max. Alternative profiles have been suggested, including Burgers-Rott or Vastitas profiles that might provide more accurate physical description of \citep{Lorenz_2014}, but using  adust devil profile \citep{Lorenz_2014}. A  different profile would likely not  modify our resultshere but not  substantially. \item A The  dust devil center  is carried past the sensor with by  the ambient wind field at a velocity $\upsilon$. For simplicity, we will assume the wind velocity vector $\upsilon$, which  is constant in magnitude and direction, while in direction. In  reality, the ambient wind field carrying a devil can be complex, even causing multiple encounters between devil and sensor and consequently more complex pressure signals \citep{Lorenz_2013}. The upshot of this assumption is that a A  devil whose center passes directly over the sensor will register a pressure dip with a full-width at half-max in time $\Gamma^\prime_{\rm act} = \Gamma_{\rm act}/\upsilon$. The act}/\upsilon$, so that the observed  profile width in timethat is actually  observed is $\Gamma^\prime_{\rm obs}$. \item Many dust devil surveys \citep[e.g.][]{Ellehoj_2010} impose The deepest point observed for  a devil that is recovered is $P_{\rm obs}$, which must exceed some fixed  minimumpressure threshold  $P_{\rm min}$, below which a putative pressure fluctuation is deemed statistically insignificant.For distant encounters with a devil, the observed pressure will fall below $P_{\rm min}$, and the devil will not be recovered.  At the other end of the scale, basic thermodynamic limitations restrict the maximum pressure depth a devil can have to some finite value, $P_{\rm max}$. Thus, we will assume the pressure signals for detected devils fall between these two limits. Likewise Likewise,  the $\Gamma_{\rm act}$-values fall between $\Gamma_{\rm min}$ and $\Gamma_{\rm max}$. $\Gamma_{\rm min}$ might be set by the sampling rate of the barometric logger, while $\Gamma_{\rm max}$ might be set by the requirement that detected devils are narrow enough to be discernible against long-term (e.g., hourly) pressure variations in the observational time-series. variations.  The two sets of limits aren't necessarily may not be  related, i.e. devils with $P_{\rm act} = P_{\rm max}$ don't necessarily have $\Gamma_{\rm act} = \Gamma_{\rm max}$. \item The distributions of $P_{\rm act}$ and $\Gamma_{\rm act}$, $n(P_{\rm act})$ and $n(\Gamma_{\rm act})$ respectively, are integrable and differentiable. The same is true for the distributions of observed dust devil parameters.  \item The uncertainties on the profile depth and width estimated for a dust devil are negligible.  \item Dust devils all have the same lifetimes. \cite{Lorenz_2013} suggested a dependence of dust devil lifetime on diameter, $D$ as $D^{0.66}$. In principle, this effect probably skews the distribution of observed dust devils toward larger ones. However, for single-barometer surveys, the encounter geometry plays the dominant role in setting the observed distribution, and as a first cut, we neglect the difference in dust devil lifetimes in this study.  \end{enumerate}  With these assumptions, we can relate the geometry of an encounter directly to the observed profile parameters, and Figure \ref{fig:encounter_geometry} illustrates a typical encounter. As a devil passes the barometer, it will have a closest approach distance $b$. If the dust devil passed directly over the sensor, i.e. $b = 0$, the radial distance evolves as $r(t) = \upsilon t$, with time $t$ running from negative to positive values, and the devil passes directly over at $t = 0$. If $b \ne 0$, then $r(t) = \dfrac{b}{\cos\left[ \arctan\left( ^{\upsilon t}/_{b} \right) \right]}$. Figure \ref{fig:compare_profiles} compares profiles for $b = 0$ and $b = \Gamma_{\rm act}$. The deepest point observed in the pressure profile $P_{\rm obs}$ is given by