deletions | additions       diff --git a/section_Formulating_the_Recovery_Biases__.tex b/section_Formulating_the_Recovery_Biases__.tex  index b1952a6..17f68b8 100644  --- a/section_Formulating_the_Recovery_Biases__.tex  +++ b/section_Formulating_the_Recovery_Biases__.tex 
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\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 $b$. 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 $b = \Gamma$. \Gamma_{\rm act}$.  The deepest point observed in the pressure profile $P_{\rm obs}$ is given by \begin{equation}  P_{\rm obs} = \dfrac{P_{\rm act}}{1 - \left( \dfrac{2 b}{\Gamma_{\rm act}} \right)^2}.  \label{eqn:Pobs} 
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\label{eqn:b}  \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.