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Brian Jackson edited subsection_Combining_the_P__rm__.tex
almost 9 years ago
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\subsection*{Combining the $P_{\rm act}/\Gamma_{\rm act}$ Biases and Distortions}
Now we investigate these same biases and distortions when both $P_{\rm act}$ and $\Gamma_{\rm act}$ are allowed to vary. In this case, the recovery bias $f = f(P_{\rm act}, \Gamma_{\rm act})$ and represents a bias in two dimensions. We can cast it as before, by comparing the area occupied by the measurable portion of a dust devil with $P_{\rm act}$ and $\Gamma_{\rm act}$ to that largest measurable area for a devil. In principle, though, the latter area does not necessarily correspond to a devil with $P_{\rm act} = P_{\rm max}$ and $\Gamma_{\rm act} = \Gamma_{\rm max}$ since the pressure and profile width can be physically coupled in a way that does not allow that combination to occur for a devil. Thus, we will express $f$ using an unspecified maximum area:
\begin{equation}\label{eqn:full_bias_uncoupled}
f(P_{\rm act}, \Gamma_{\rm act}) = \dfrac{
\pi \Gamma_{\rm act}^2 \left( \dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm min}} \right) }{A_{\rm max}},
\end{equation}
where