deletions | additions       diff --git a/section_Formulating_the_Recovery_Biases__.tex b/section_Formulating_the_Recovery_Biases__.tex  index a7e18da..894297c 100644  --- a/section_Formulating_the_Recovery_Biases__.tex  +++ b/section_Formulating_the_Recovery_Biases__.tex 
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\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 profile width in time observed is $\Gamma^\prime_{\rm obs}$.  \item Many dust devil surveys (refs) \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 $\Gamma_{\rm act} = \Gamma_{\rm max}$. \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. 
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b = \left( \dfrac{\Gamma_{\rm act}}{2} \right) \sqrt{ \dfrac{P_{\rm act} - P_{\rm obs}}{P_{\rm obs}}}.  \end{equation}  There is a maximum closest approach distance $b_{\rm max}$ beyond which a devil will produce an undetectably small pressure signal, $P_{\rm obs} < P_{\rm min}$, and $b_{\rm max} = \left( \dfrac{\Gamma_{\rm act}}{2} \right) \sqrt{ \dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm min}}}$. Devils with $b > b_{\rm max}$ will not be detected, which will bias our recovered population of devil parameters in ways that depend on the parameters themselves. It's worth noting that $\Gamma_{\rm act}$ may depend on $P_{\rm act}$, a point we will return to in Section. Again, this recovery bias results from the miss distance effect. Next, we use these equations to formulate the recovery biases and signal distortions resulting from the miss distance effect.