deletions | additions       diff --git a/section_Formulating_the_Recovery_Biases__.tex b/section_Formulating_the_Recovery_Biases__.tex  index 4dc0b1d..42ef1b7 100644  --- a/section_Formulating_the_Recovery_Biases__.tex  +++ b/section_Formulating_the_Recovery_Biases__.tex 
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\begin{enumerate}  \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 However, alternative profiles have been suggested. \cite{Lorenz_2014} suggested either a Burgers-Rott or Vastitas profile might provide more accurate physical description of a dust devil profile. Using a different profile would likely change the results here but not substantially. Given that the Lorentz profile is more commonly used in the field, we opt to use it here. The  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 $\Gamma^\prime_{\rm act} = \Gamma_{\rm act}/\upsilon$. The profile width in time that is actually observed is $\Gamma^\prime_{\rm obs}$. 
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\end{enumerate}  It is important to note that we have assumed dust devil pressure profiles follow a Lorentz function. However, alternative assumptions have been suggested. \cite{Lorenz_2014} suggested either a Burgers-Rott or Vastitas profile might provide more accurate physical description of a dust devil profile. Using a different profile would likely change the results here but not substantially. Given that the Lorentz profile is more commonly used in the field, we opt to use it here.  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 $-\infty$ to $+\infty$, 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   \begin{equation}\label{eqn:Pobs_from_Lorentz_profile}  P_{\rm obs} = \dfrac{P_{\rm act}}{1 + \left( \dfrac{2 b}{\Gamma_{\rm act}} \right)^2}. 
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