deletions | additions       diff --git a/section_Formulating_the_Signal_Distortions__.tex b/section_Formulating_the_Signal_Distortions__.tex  index ed2086c..a35f5b5 100644  --- a/section_Formulating_the_Signal_Distortions__.tex  +++ b/section_Formulating_the_Signal_Distortions__.tex 
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\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, $P_{\rm act}$ 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 Vatistas profiles that might provide more accurate physical description \citep{Lorenz_2014}, but using a different profile would not modify our results substantially.   \item The dust devil center travels at a velocity $\upsilon$, which is constant in magnitude and direction. In reality, a devil's trajectory can be complex, even encountering a sensor multiple times and consequently producing complex pressure signals \citep{Lorenz_2013}. 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 $\tau_{\rm  act} = \Gamma_{\rm act}/\upsilon$, so that the profile width observed in time is $\Gamma^\prime_{\rm $\tau_{\rm  obs}$. \item A dust devil appears and disappears instantaneously, traveling a distance $\upsilon \tau$ L$  over its lifetime $\tau$. $L$.  As pointed out by \citet{Lorenz_2013}, $\tau$ $L$  seems to depend on dust devil diameter $D$ as $\tau $L  = 40\ {\rm s}\ \left( D/{\rm m} \right)^{2/3}$, with diameter in meters. We assume $D = \Gamma_{\rm act}$ \cite{Vatistas_1991}. \item There are minimum and maximum pressure profile depths that can be recovered by a survey, $P_{\rm min}$ th}$  and $P_{\rm max}$, respectively. $P_{\rm min}$ th}$  may be set by the requirement that a pressure signal exceeds some minimum threshold set by the noise in the datastream, while basic thermodynamic limitations likely restrict the maximum pressure depth for a devil. Likewise, the profile widths must fall between $\Gamma_{\rm min}$ th}$  and $\Gamma_{\rm max}$, possibly set by the ambient vorticity field in which a devil is embedded \cite{Renn__2001}. The two sets of limits may not be related, i.e. devils with $P_{\rm max}$ don't necessarily have widths $\Gamma_{\rm max}$. As it turns out, our results are not sensitive to the precise values for each of these limits. \item The two-dimensional distribution of $P_{\rm act}$ and $\Gamma_{\rm act}$, $\rho(P_{\rm act}, \Gamma_{\rm act})$, is integrable and differentiable. The same is true for the distributions of observed dust devil parameters, $\rho(P_{\rm obs}, \Gamma_{\rm obs})$.