deletions | additions       diff --git a/section_Formulating_the_Recovery_Biases__.tex b/section_Formulating_the_Recovery_Biases__.tex  index de98af0..a6d1941 100644  --- a/section_Formulating_the_Recovery_Biases__.tex  +++ b/section_Formulating_the_Recovery_Biases__.tex 
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\item The radial pressure profile for dust devils follows a Lorentz function. That is, each dust devil pressure profile has one well-defined center, and the pressure dip (relative to the ambient pressure) observed at a radial distance $r$ from the center of the dust devil is $P(r) = \dfrac{P_{\rm act}}{1 + \left( 2r/\Gamma_{\rm act} \right)^2 }$. Here $P_{\rm act}$ represents the actual pressure depth at the devil's center, and $\Gamma_{\rm act}$ the profile full-width at half-max. This profile is also assumed not to change with time.  \item A dust devil is carried past the sensor with the ambient wind field at a velocity $\upsilon$. For simplicity, we will assume the wind velocity vector is constant in magnitude and direction, while 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 devil whose center passes directly over the sensor will register a pressure dip with a full-width at half-max in time $\Gamma^\prime_{\rm act} = \Gamma_{\rm act}/\upsilon$. For a more distant encounter, the The  profile width in time that is actually  observed is $\Gamma^\prime_{\rm obs}$. \item Many dust devil surveys \citep[e.g.][]{Ellehoj_2010} impose a minimum pressure 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 thermodynamical limits likely 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 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. The two sets of limits aren't necessarily related, i.e. devils with $P_{\rm act} = P_{\rm max}$ don't necessarily have $\Gamma_{\rm act} = \Gamma_{\rm max}$. 
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