deletions | additions       diff --git a/section_Formulating_the_Recovery_Biases__.tex b/section_Formulating_the_Recovery_Biases__.tex  index c75b32f..04dea09 100644  --- a/section_Formulating_the_Recovery_Biases__.tex  +++ b/section_Formulating_the_Recovery_Biases__.tex 
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\item A dust devil appears and disappears instantaneously, traveling a distance $\upsilon \tau$ over its lifetime $\tau$. As pointed out by \citet{Lorenz_2013}, $\tau$ seems to depend on dust devil diameter $D$ as $\tau = 40\ {\rm s}\ \left( D/{\rm m} \right)^{2/3}$, with diameter in meters. We assume $D \approx \Gamma_{\rm act}$ \cite{Vatistas_1991}.   \item The deepest point in a dust devil There are minimum and maximum  pressure profile must exceed depths that can be recovered by a survey, $P_{\rm min}$ and $P_{\rm max}$, respectively. $P_{\rm min}$ may be set by the requirement that a pressure signal exceeds  somefixed  minimum $P_{\rm min}$. At threshold set by  the other end of noise in  the scale, datastream, while  basic thermodynamic limitations likely  restrict the maximum pressure depth a devil can have to some finite value, $P_{\rm max}$. value.  Likewise, the profile widths must fall between $\Gamma_{\rm min}$ and $\Gamma_{\rm max}$. 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, $\rho(P_{\rm obs}, \Gamma_{\rm obs})$.  \item The uncertainties on the profile depth and width estimated for a dust devil are negligible. In \cite{Jackson_2015}, for example, uncertainties on $P_{\rm obs}$ were about an order of magnitude less than the inferred $P_{\rm obs}$-value for a detected devil, with uncertainties on $\Gamma_{\rm obs}$ at least a factor of three smaller. Robust recovery of a devil against noise requires relatively small uncertainties.  \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 $b$, and the radial distance between devil center and sensor $r(t) = \sqrt{b^2 + \left( \upsilon t \right)^2}$, where time $t$ runs from negative to positive values. The fact that $b$ is usually greater than zero biases the devils that are detected and the way in which their pressure signals register. Next, weuse these equations to  formulate the recovery biases and signal distortions resulting from this miss distance effect.