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\title{A Framework for Mitigating the Biases in Barometric Dust Devil Surveys}
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\author[
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]{Brian Jackson}
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\section*{Background}
Dust devils are small-scale (few to many tens of meters) low-pressure vortices rendered visible by lofted dust. They usually occur in arid climates on the Earth and ubiquitously on Mars. Martian dust devils have been studied with orbiting and landed spacecraft and were first identified on Mars using images from the Viking Orbiter \citep{Thomas_1985}. On Mars, dust devils may dominate the supply of atmospheric dust and influence climate \cite{Basu_2004}, pose a hazard for human exploration \citep{Balme_2006}, and they may have lengthened the operational lifetime of Martian rovers \citep{Lorenz_Reiss_2014}. On the Earth, dust devils significantly degrade air quality in arid climates \citep{Gillette_1990} and may pose an aviation hazard \cite{Lorenz_2005}.
The dust-lifting capacity of dust devils seems to depend sensitively on their structures, in particular on the pressure wells at their centers \citep{Neakrase_2006}, so the dust supply from dust devils on both planets may be dominated by the seldom-observed larger devils. Using a martian global climate model, \citet{Basu_2004} showed that observed seasonal variations in Mars' near-surface temperatures could not be reproduced without including the radiative effects of dust and estimated the dust contributes more than 10 K of heating to the heating budget. Thus, elucidating the origin, evolution, and population statistics of dust devils is critical for understanding important terrestrial and Martian atmospheric properties and for in-situ exploration of Mars.
Studies of Martian dust devils have been conducted through direct imaging of the devils and identification of their tracks on Mars' dusty surface \citep[cf.][]{Balme_2006}. Studies with in-situ meteorological instrumentation have also identified dust devils, either via obscuration of the Sun by the dust column \citep{Zorzano_2013} or their pressure signals \citep{Ellehoj_2010}. Studies have also been conducted of terrestrial dust devils and frequently involve in-person monitoring of field sites. Terrestrial dust devils are visually surveyed \citep{Pathare_2010}, directly sampled \citep{Balme_2003}, or recorded using in-situ meteorological equipment \citep{Sinclair_1973,Lorenz_2012}.
As noted in \citet{Lorenz_2009}, in-person visual surveys are likely to be biased toward detection of larger, more easily seen devils. Such surveys would also fail to recover dustless vortices \citep{Lorenz_2015}. Recently, terrestrial surveys similar to Martian dust devil surveys have been conducted using in-situ single barometers \citep{Lorenz_2012,Lorenz_2014,Jackson_2015} and photovoltaic sensors \citep{Lorenz_2015}. These sensor-based terrestrial surveys have the advantage of being directly analogous to Martian surveys and are highly cost-effective compared to the in-person surveys (in a dollars per data point sense).
In single-barometer surveys, a sensor is deployed in-situ and records a pressure time series at a sampling period $\lesssim 1$ s. Since it is a low-pressure convective vortex, a dust devil passing nearby will register as pressure dip discernible against a background ambient (but not necessarily constant) pressure. Figure \ref{fig:conditioning_detection_b_inset} from \citet{Jackson_2015} shows a time-series with a typical dust devil signal.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/conditioning-detection-b-inset1/conditioning-detection-b-inset-thin-lines}
\caption{{\label{fig:conditioning_detection_b_inset}
Example dust devil profile from \protect\citet{Jackson_2015}. The black dots show the pressure $p$ measurements (in hectoPascal hPa) as a function of local time in hours. The blue line shows the best-fit Lorentz profile model.%
}}
\end{center}
\end{figure}
Three spacecraft landed on Mars carried meteorological instrumentation that allows investigators to conduct such surveys, providing a wealth of data on Martian dust devils and other meteorological phenomena, but such surveys suffer from important biases when it comes to studying dust devils. Foremost among these biases is the fact that a pressure dip does not necessarily correspond to a dust-lofting vortex. Indeed, recent studies \citep{2014DPS....4630006S} suggest pressure dips are often unaccompanied by lofted dust, likely because the attendant wind velocities are insufficient to lift dust. The problem of identifying dustless vortices can be mitigated if barometers are deployed alongside solar cells, which can register obscuration by dust, and such sensor pairs have recently been deployed in terrestrial studies \cite{Lorenz_2015}. However, such an arrangement is not fool-proof since a dusty devil can easily pass by the sensors on the anti-sunward side and not register an obscuration signature.
Another key bias arises from the miss distance effect: a fixed barometric sensor is more likely to have a more distant than closer encounter with a dust devil. Since the pressure perturbation associated with a devil falls off with distance from the devil's center, the deepest point in the observed pressure profile will almost always be less than the actual pressure well at the devil's center. The observed shape of the profile will be distorted as well. These miss distance biases are intrinsic to the detection methods, and additional biases can influence the inferred statistical properties. For instance, noise in the pressure time series from a barometer may make more difficult detection of a dust devils with smaller pressure perturbations, depending on the exact detection scheme.
Although it is inherent in single-barometer surveys for dust devils, simple geometric considerations can mitigate the influence of the miss distance effect, allowing single-barometer surveys to be corrected. In particular, the physical parameters for dust devils estimated from the pressure time-series can be corrected for the miss distance effect, at least in a statistical sense. In this study, we consider the geometry of encounters between dust devils and barometers and present a statistical model for correcting the miss distance effect. This study is similar to and motivated by \citet{Lorenz_2014}, but, where that study used a numerical simulation to investigate biases in the recovered population of dust devil properties, we employ an analytic framework that allows wider applicability and provides more insight into the problem. As we discuss, though, our model involves some important assumptions and simplifications.
The plan of this paper is as follow: In Section \ref{sec:formulating_the_signal_distortions_and_recovery_biases}, we discuss the typical encounter geometry for a dust devil detection, how the geometry distorts and biases the recovered parameters, and how to convert from the observed to the underlying distribution of dust devil parameters. In Section \ref{sec:comparison_to_observational_data}, we apply our scheme to recent datasets for martian and terrestrial dust devils. Finally, in Section \ref{sec:discussion_and_conclusions}, we discuss some of the limitations and future improvements for our model.
\section*{Formulating the Signal Distortions and Recovery Biases}
\label{sec:formulating_the_signal_distortions_and_recovery_biases}
To develop a model for the recovery biases and signal distortions induced by the miss distance effect, we will make the following assumptions:
\begin{enumerate}
\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}. The time it takes for a devil to travel a distance equal to its own full-width at half-max $\Gamma_{\rm act}$ is $\tau_{\rm act} = \Gamma_{\rm act}/\upsilon$. By contrast, the pressure signal registered by a devil is observed to have a width in time $\tau_{\rm obs} = \Gamma_{\rm obs}/\upsilon$, where $\Gamma_{\rm obs}$ is the inferred full-width/half-max of the devil's pressure profile.
\item A dust devil appears and disappears instantaneously, traveling a distance $\upsilon L$ over its lifetime $L$. As pointed out by \citet{Lorenz_2013}, $L$ seems to depend on dust devil diameter $D$ as $L = 40\ {\rm s}\ \left( D/{\rm m} \right)^{2/3}$, 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, $P_{\rm th}$ and $P_{\rm max}$, respectively. $P_{\rm th}$ 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 \citep{Renn__1998}. Likewise, the profile widths must fall between $\Gamma_{\rm th}$ and $\Gamma_{\rm max}$, possibly set by the ambient vorticity field in which a devil is embedded \cite{Renn__2001} and/or the depth of the planetary boundary layer \citep{Fenton_2015}. The two sets of limits may not be related, i.e. devils with $P_{\rm max}$ do not 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 \citet{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, we formulate the recovery biases and signal distortions resulting from this miss distance effect.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/encounter-geometry/encounter-geometry}
\caption{{\label{fig:encounter_geometry}
(Left) Mars Global Surveyor Mars Orbiter Camera image of crater showing dust devil tracks. Available at \url{http://www.msss.com/mars\_images/moc/2003/12/02/2003.12.02.R0901707.gif}. (Right) Geometry of dust devil encounter. The blue circles show pressure contours, the y-axis runs along the devil's velocity vector measured relative to the sensor $\mathbf{\upsilon}$, and $b$ is the closest approach distance between the barometer and the devil center. The shaded blue region shows the area of the surface carved out by the traveling $P_{\rm th}$ contour. The inset shows a close-up of the encounter.%
}}
\end{center}
\end{figure}
\subsection{The Signal Distortion}
\label{sec:the_signal_distortion}
The deepest point observed in the pressure profile $P_{\rm obs}$ is given by
\begin{equation}\label{eqn:P_obs}
P_{\rm obs} = \dfrac{P_{\rm act}}{1 + \left( 2 b/\Gamma_{\rm act} \right)^2}.
\end{equation}
Clearly, unless $b = 0$, $P_{\rm obs} < P_{\rm act}$. Likewise, non-central encounters will distort the profile full-width/half-max, giving a full-width/half-max $\Gamma_{\rm obs}$.
The observed pressure signal drops to half its value at a time $t = \pm \frac{1}{2} \tau_{\rm obs} = \pm \frac{1}{2} \Gamma_{\rm obs}/\upsilon$ by definition. At these times, the center of the devil is a radial distance from the barometer $r(t = \pm \tau_{\rm obs}/2) = \sqrt{b^2 + \left( \frac{1}{2} \Gamma_{\rm obs} \right)^2}$ and $P(r) = \frac{1}{2} P_{\rm obs} = \frac{1}{2} \dfrac{P_{\rm act}}{1 + \left( 2 b /\Gamma_{\rm act} \right)^2} = \dfrac{P_{\rm act}}{1 + \left( 2r(\pm \tau_{\rm obs}/2)/\Gamma_{\rm act} \right)^2 }$. Solving for $\Gamma_{\rm obs}$ gives
\begin{equation}
\label{eqn:Gamma_obs}
\Gamma_{\rm obs}^2 = \Gamma_{\rm act}^2 + \left( 2b \right)^2.
\end{equation}
Figure \ref{fig:compare_profiles} shows how a non-central encounter modifies the observed pressure profile, and we call this modification the signal distortion. Combining the Equations \ref{eqn:P_obs} and \ref{eqn:Gamma_obs} gives the following useful expression:
\begin{equation}
\label{eqn:P_obs_Gamma_obs}
P_{\rm obs}\ \Gamma_{\rm obs}^2 = P_{\rm act}\ \Gamma_{\rm act}^2.
\end{equation}
We can solve Equation \ref{eqn:P_obs} for $b$:
\begin{equation}\label{eqn:b}
b = \left( \dfrac{\Gamma_{\rm act}}{2} \right) \sqrt{ \dfrac{P_{\rm act} - P_{\rm obs}}{P_{\rm obs}}}.
\end{equation}
A single barometer at a fixed location can sense a dust devil only over a certain area, spanning a maximum radial distance $b_{\rm max}$, beyond which devils will produce pressure signals smaller than the detection threshold, $P_{\rm obs} < P_{\rm th}$, but the value of $b_{\rm max}$ depends on the context. For example, the very largest radial distance at which a devil could be sensed is $b_{\rm max} = \left( \dfrac{\Gamma_{\rm max}}{2} \right) \sqrt{\dfrac{P_{\rm max} - P_{\rm th}}{P_{\rm th}}}$.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/compare-profiles2/compare-profiles2}
\caption{{\label{fig:compare_profiles}
Dust devil profiles for $b = 0$ and $b = \Gamma_{\rm act}$, i.e. for a closest approach equal to the profile's diameter.%
}}
\end{center}
\end{figure}
We can use the encounter geometry to model the statistical probability for $P_{\rm obs}$ and $\Gamma_{\rm obs}$ to fall within a certain range of values, given a distribution of $P_{\rm act}$- and $\Gamma_{\rm act}$-values. The probability density for passing between $b$ and $b + db$ of a devil is $dp(b) = 2 b\ db / b_{\rm max}^2 $ for $b \le b_{\rm max}$. In this context, we take $b_{\rm max} = \left( \dfrac{\Gamma_{\rm act}}{2} \right) \sqrt{\dfrac{P_{\rm act} - P_{\rm th}}{P_{\rm th}}}.$ This expression allows us to calculate the average miss distance $\langle b/\Gamma_{\rm act} \rangle = \int b/\Gamma_{\rm act}\ dp = 2/3\ b_{\rm max}/\Gamma_{\rm act} \approx 1/3 \sqrt{P_{\rm act}/P_{\rm th}}$, assuming $P_{\rm act} \gg P_{\rm th}$. If, for example, $P_{\rm act} \approx 10\ P_{\rm th}$, $\langle b \rangle \approx \Gamma_{\rm act}$, meaning that, on average, $P_{\rm obs} \approx P_{\rm act}/5$ and $\Gamma_{\rm obs} \approx 5\ \Gamma_{\rm act}$.
Holding $\Gamma_{\rm act}$ fixed, we can also use the probability density expression and Equation \ref{eqn:Gamma_obs} to calculate the probability density for an encounter to give an observed profile width between $\Gamma_{\rm obs}$ and $\Gamma_{\rm obs} + d\Gamma_{\rm obs}$:
\begin{equation}
\label{eqn:dp_dGamma_obs}
\dfrac{dp}{d\Gamma_{\rm obs}} = \dfrac{\Gamma_{\rm obs}}{b_{\rm max}^2} = 4 \Gamma_{\rm act}^{-2} \left( \dfrac{P_{\rm th}}{P_{\rm act} - P_{\rm th}}\right) \Gamma_{\rm obs}.
\end{equation}
We require that $b \le b_{\rm max}$ in order for a devil to be detected, which limits the range of allowable values for $\Gamma_{\rm obs}$, given $P_{\rm act}$ and $\Gamma_{\rm act}$. We can use Equation \ref{eqn:Gamma_obs} to solve for $\Gamma_{\rm obs}/\Gamma_{\rm act}$:
\begin{equation}
\label{eqn:Gamma_obs_limits}
1 \le \dfrac{\Gamma_{\rm obs}}{\Gamma_{\rm act}} \le \sqrt{P_{\rm act}/P_{\rm th}}.
\end{equation}
We can employ an analogous procedure involving Equation \ref{eqn:P_obs} to calculate the probability density for an encounter to give an observed profile depth between $P_{\rm obs}$ and $P_{\rm obs} + dP_{\rm obs}$:
\begin{equation}
\label{eqn:dp_dP_obs}
\dfrac{dp}{dP_{\rm obs}} = \left( \dfrac{P_{\rm act}}{P_{\rm obs}^2} \right) \left(\dfrac{\Gamma_{\rm act}}{2 b_{\rm max}}\right)^2 = \left( \dfrac{P_{\rm act}}{P_{\rm obs}^2} \right) \left( \dfrac{P_{\rm th}}{P_{\rm act} - P_{\rm th}} \right).
\end{equation}
We also require that $P_{\rm obs} \le P_{\rm act}$.
Figure \ref{fig:distortion_probabilities} shows the probability densities for $\Gamma_{\rm obs}$ and $P_{\rm obs}$, as well as the corresponding limits. To interpret the figure, consider $\Gamma_{\rm act} = 10^{1.5}$ m in panel (a). Contours of $\dfrac{dp}{d\Gamma_{\rm obs}}$ increase going up, indicating that the miss distance effect drives $\Gamma_{\rm obs}$ toward larger values (within the limited range allowed). In panel (b), contours of $\dfrac{dp}{dP_{\rm obs}}$ increase going down, showing $P_{\rm obs}$ tends toward smaller values for a fixed $P_{\rm act}$.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/distortion-probabilities/distortion-probabilities}
\caption{{\label{fig:distortion_probabilities}
(a) Contours of the probability density for $\Gamma_{\rm obs}$, $\log_{10} \left( \dfrac{dp}{d\Gamma_{\rm obs}} \right)$, assuming $P_{\rm act} = P_{\rm max}$. (b) Contours of the probability density for $P_{\rm obs}$, $\log_{10} \left( \dfrac{dp}{dP_{\rm obs}} \right)$. In both panels, the grey regions are forbidden by definition.%
}}
\end{center}
\end{figure}
\subsection{The Recovery Bias}
\label{sec:the_recovery_bias}
As it travels on the surface of the observational arena, a dust devil's pressure contour $P_{\rm th}$ carves out a long, narrow area $A(P_{\rm act}, \Gamma_{\rm act})$. If a barometer lies within that area, the devil will be detected, in principle. $A$ is given by
\begin{equation}
\label{eqn:dust_devil_area}
A = \pi b_{\rm max}^2 + \upsilon L b_{\rm max} = \left( \Gamma_{\rm act}/2 \right) \sqrt{ \dfrac{P_{\rm act} - P_{\rm th}}{P_{\rm th}} } \left[ \pi \left( \Gamma_{\rm act}/2 \right) \sqrt{ \dfrac{P_{\rm act} - P_{\rm th}}{P_{\rm th}} } + \upsilon L \right],
\end{equation}
The probability to recover a devil is proportional to this total track area. Thus devils with deeper and wider pressure profiles are more likely to be recovered. Using the lifetime scaling from \citet{Lorenz_2014}, Figure \ref{fig:relative_areas} shows that the second term dominates over the first term for all but the smallest, slowest dust devils, so, for simplicity, we will neglect the first term, giving
\begin{equation}
A \approx \left( \Gamma_{\rm act}/2 \right) \sqrt{ \dfrac{P_{\rm act} - P_{\rm th}}{P_{\rm th}} } \upsilon L.
\end{equation}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/relative-areas/relative-areas}
\caption{{\label{fig:relative_areas}
The two area terms from Equation \ref{eqn:dust_devil_area} for dust devils with a range of diameters $D$ traveling with a range of velocities, 0.1, 1, and 10 m/s.%
}}
\end{center}
\end{figure}
The fact that larger, faster dust devils cover more area means that they are more likely to be recovered. We can quantify the recovery probability $f$ by taking the ratio of track areas for a given dust devil to the largest area for a dust devil, $A_{\rm max}$:
\begin{equation}
\label{eqn:recovery_bias}
f = \dfrac{A(\Gamma_{\rm act}, P_{\rm act})}{A_{\rm max}} = A_{\rm max}^{-1}\ \Gamma_{\rm act} \sqrt{\dfrac{P_{\rm act} - P_{\rm th}}{P_{\rm th}}} \upsilon\ L.
\end{equation}
The devil with the deepest profile need not also have the widest profile or the largest velocity. \citet{Renn__2001} argue that the diameter of a vortex is set, in part, by the local vorticity field, while \citet{Balme_2012}, from their field studies, find no correlation between diameter and velocity from their field work. However, in quantifying the recovery probability $f(\Gamma_{\rm act}, P_{\rm act})$, it is only important that we apply a uniform normalizing factor to the whole population, and the denominator in Equation \ref{eqn:recovery_bias} just provides a convenient expression for that. Any other uniform normalization (e.g., using average parameters) would suffice.
Figure \ref{fig:recovery_bias} shows contours of $f(\Gamma_{\rm act}, P_{\rm act})$, assuming $\upsilon = \upsilon_{\rm max}$. Not surprisingly, the recovery probability increases toward the upper-right corner, indicating that the deepest, widest dust devils are the most likely to be recovered. Taking the distribution of observed devils as $\rho(\Gamma_{\rm act}, P_{\rm act})$, the product $f(\Gamma_{\rm act}, P_{\rm act}) \times \rho(\Gamma_{\rm act}, P_{\rm act})$ would represent the population of devils that are detected but not how the recovered population would actually look.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/recovery-bias/recovery-bias}
\caption{{\label{fig:recovery_bias}
The recovery bias $f(\Gamma_{\rm act}, P_{\rm act})$, assuming the same velocity for all dust devils.%
}}
\end{center}
\end{figure}
\subsection{Converting Between the Observed and Actual Parameter Distributions}
Consider a distribution of observed values $\rho(\Gamma_{\rm obs}, P_{\rm obs}) = \dfrac{d^2N}{d\Gamma_{\rm obs}\ dP_{\rm obs}}$. The small number of devils $dN = f\ \rho(\Gamma_{\rm obs}, P_{\rm obs})\ d\Gamma_{\rm act}\ dP_{\rm act}$ contributing are those that had closest approach distances between $b$ and $b + db$ of the detector. Thus, we can convert $\rho(\Gamma_{\rm act}, P_{\rm act})$ to $\rho(\Gamma_{\rm obs}, P_{\rm obs})$ by integrating the former density over $b$ and accounting for the bias and distortion effects. To calculate the integral, we also need to re-cast the upper limit to express the maximum possible radial distance, i.e. the distance at which $P_{\rm act} = P_{\rm max}$. Using Equation \ref{eqn:b} and making the replacements $P_{\rm act} = P_{\rm max}$ and $\Gamma_{\rm act} = \left( P_{\rm obs}/P_{\rm max} \right)^{1/2} \Gamma_{\rm obs}$ from Equation \ref{eqn:P_obs_Gamma_obs} gives $b(\Gamma_{\rm obs}, P_{\rm obs}) = \left(\Gamma_{\rm obs}/2\right) \left[ \left(P_{\rm max} - P_{\rm obs}\right)/P_{\rm max} \right]^{1/2}$. The integral to convert from $\rho({\rm act}) \equiv \rho(P_{\rm act}, \Gamma_{\rm act})$ to $\rho({\rm obs}) \equiv \rho(\Gamma_{\rm obs}, P_{\rm obs})$ is then
\begin{equation}
\rho({\rm obs}) = \int_{b^\prime = 0}^{b({\rm obs})} f\ \rho({\rm act}(b^\prime))\ \dfrac{2b^\prime\ db^\prime}{b_{\rm max}^2} \\
= 2\ A_{\rm max}^{-1}\ \upsilon\ \kappa\ b_{\rm max}^{-2} P_{\rm th}^{-1/2}\ \int_{b^\prime = 0}^{b({\rm obs})} \left( \Gamma_{\rm act}(b^\prime)/{\rm m} \right)^{5/3} \left( P_{\rm act}(b^\prime) - P_{\rm th} \right)^{1/2} \ \rho({\rm act}(b^\prime))\ b^\prime\ db^\prime \label{eqn:convert_from_actual_to_observed_density},
\end{equation}
where $\kappa = 40\ {\rm s}$ and $\Gamma_{\rm act}$ is measured in meters, m. Figure \ref{fig:uniform_actual_distribution_to_observed_distribution} shows the result for a uniform distribution of actual values, $\rho({\rm act}) = \left( P_{\rm max} - P_{\rm th} \right)^{-1}\ \left( \Gamma_{\rm max} - \Gamma_{\rm th} \right)^{-1}$ and compares it to the simulated results of an observational survey (blue circles). (For the uniform distribution, the integral has a closed form expression that is unwieldy, so we opt to perform the integration numerically.)\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/uniform-actual-distribution-to-observed-distribution/uniform-actual-distribution-to-observed-distribution}
\caption{{\label{fig:uniform_actual_distribution_to_observed_distribution}
(a) Contours of number density for observed dust devil parameters, $\log_{10}\ \rho(P_{\rm obs}, \Gamma_{\rm obs})$, assuming a uniform distribution of underlying values $\rho(P_{\rm act}, \Gamma_{\rm act}) = \left( P_{\rm max} - P_{\rm th} \right)^{-1}\ \left( \Gamma_{\rm max} - \Gamma_{\rm th} \right)^{-1}$. The blue dots show a simulated dust devil survey. (b) Density marginalized over $P_{\rm obs}$. (c) Density marginalized over $\Gamma_{\rm obs}$.%
}}
\end{center}
\end{figure}
In panel (b), the $\Gamma_{\rm obs}$ density monotonically increases with $\Gamma_{\rm obs}$ since devils with larger $\Gamma_{\rm act}$-values are more likely to be observed (Equation \ref{eqn:recovery_bias}), and they're likely to be observed with yet wider profiles. By contrast, in panel (c), the density peaks at a moderate value of $P_{\rm obs}$, falling off to either side. Few devils are observed with $P_{\rm obs} \approx P_{\rm max}$ because nearly central ($b \approx 0$) would be required, which are unlikely. Looking at the other side of the peak from small $P_{\rm obs}$, devils with larger $P_{\rm act}$ are more likely to be detected. Since we require $\Gamma_{\rm obs} \le \Gamma_{\rm max}$, those devils are confined to a limited range of $b$ and therefore a minimum value of $P_{\rm obs} \ge P_{\rm act} \left( \Gamma_{\rm act}/\Gamma_{\rm max} \right)^2$, often greater than $P_{\rm th}$.
Figure \ref{fig:uniform_actual_distribution_to_observed_distribution} clearly contradicts results from real dust devil surveys, such as \citet{Ellehoj_2010}, who found a $\Gamma_{\rm obs}$ density that dropped with increasing $\Gamma_{\rm obs}$ and a $P_{\rm obs}$ density that increased with decreasing $P_{\rm obs}$. The obvious conclusion is that the distribution of underlying actual values for that survey was not uniform in $\Gamma_{\rm act}$ and $P_{\rm act}$.
Ultimately, we are interested in converting the density of observed parameters to the density of actual parameters, and Equation \ref{eqn:convert_from_actual_to_observed_density} provides a way to do that. Figure \ref{fig:integration_path} illustrates the integration track involved in Equation \ref{fig:integration_path}, and we define the end points of the integration track as $\Gamma_0 \equiv \Gamma_{\rm obs} \left( P_{\rm obs}/P_{\rm th} \right)^{1/2}$ and $\Gamma_1 \equiv \Gamma_{\rm obs} \left( P_{\rm obs}/P_{\rm max} \right)^{1/2}$. The equation involves an integral over $b$, which, given $\Gamma_{\rm obs}$ and $P_{\rm obs}$, represents a fixed curve in $\Gamma_{\rm act}-P_{\rm act}$. In other words, $\Gamma_{\rm obs}$ and $P_{\rm obs}$ define a locus of points for $\Gamma_{\rm act}$ and $P_{\rm act}$, and the integral over $b$ involves traveling along the locus from the point $(\Gamma_{\rm act}, P_{\rm act}) = (\Gamma_{\rm obs}, P_{\rm obs})$ up to $(\Gamma_1, P_{\rm max})$. In fact, any points $(\Gamma_{\rm obs}^\prime, P_{\rm obs}^\prime)$ satisfying Equation \ref{eqn:P_obs_Gamma_obs}, $P_{\rm obs}^\prime\ \Gamma_{\rm obs}^{\prime 2} = P_{\rm obs}\ \Gamma_{\rm obs}^2$, lie on this locus. Consequently, the only difference between $\rho(\Gamma_{\rm obs}^\prime, P_{\rm obs}^\prime)$ and $\rho(\Gamma_{\rm obs}, P_{\rm obs})$ is where on the track the integral starts -- the integrals for both end at the same point.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/integration-path/integration-path}
\caption{{\label{fig:integration_path}
Example of an integration track for Equation \ref{eqn:convert_from_actual_to_observed_density}. The shading illustrates an example $\rho(\Gamma_{\rm act}, P_{\rm act})$ distribution, discussed below -- brighter shades represent smaller densities. The blue track represents the locus of points passing through given $\Gamma_{\rm obs}$- and $P_{\rm obs}$-values. For example, $\rho(\Gamma_{\rm obs}, P_{\rm obs})$ involves integrating from $b = 0$, where $(\Gamma_{\rm act}, P_{\rm act}) = (\Gamma_{\rm obs}, P_{\rm obs})$ up to a maximum $b$-value, at $(\Gamma_{\rm act}, P_{\rm act}) = (\Gamma_1, P_{\rm max})$.%
}}
\end{center}
\end{figure}
These considerations suggest the following:
\begin{eqnarray}
\label{eqn:difference_between_observed_density_points}
\rho(\Gamma_{\rm obs}, P_{\rm obs}) - \rho(\Gamma_0, P_{\rm th}) &=& &\int_{(\Gamma_{\rm obs}, P_{\rm obs})}^{(\Gamma_1, P_{\rm max})} \cdots db^\prime& - &\int_{(\Gamma_0, P_{\rm th})}^{(\Gamma_1, P_{\rm max})} \cdots db^\prime& \\ &=& &\int_{(\Gamma_0, P_{\rm th})}^{(\Gamma_{\rm obs}, P_{\rm obs})} \cdots db^\prime& = &\int_{b^\prime = 0}^{b} \cdots db^\prime &,
\end{eqnarray}
where we have suppressed the integrands for clarity. We can then differentiate both sides with respect to $b = \left( \Gamma_{\rm obs}/2\right) \left[ \left( P_{\rm obs} - P_{\rm th} \right)/P_{\rm th} \right]^{1/2}$, but, for the left-hand side, we will convert the $b$-derivative:
\begin{equation}
\label{eqn:b_derivative_into_P_obs_derivative}
\dfrac{d}{db} = 2 \left( \dfrac{2}{\Gamma_{\rm obs}} \right) \left( \dfrac{P_{\rm obs} - P_{\rm th}}{P_{\rm th}} \right)^{1/2} \left( P_{\rm obs}\ \dfrac{\partial}{\partial P_{\rm obs}} - \left( \dfrac{\Gamma_{\rm obs}}{2} \right) \dfrac{\partial}{\partial \Gamma_{\rm obs}} \right).
\end{equation}
Thus,
\begin{equation}
\label{eqn:convert_from_observed_to_actual_density}
\dfrac{d}{db} \bigg( \rho(\Gamma_{\rm obs}, P_{\rm obs}) - \rho(\Gamma_0, P_{\rm th}) \bigg) = \dfrac{d}{db} \left( \int_{b^\prime = 0}^{b} f\ \rho({\rm act}) \dfrac{2b^\prime\ db^\prime}{b_{\rm max}^2} \right) \\
\left( \dfrac{2}{\Gamma_{\rm obs}} \right) \left( \dfrac{P_{\rm obs} - P_{\rm th}}{P_{\rm th}} \right)^{1/2} P_{\rm obs}\ \left( \dfrac{\partial \rho({\rm obs})}{\partial P_{\rm obs}} - \left( \dfrac{\Gamma_{\rm obs}}{2 P_{\rm obs}} \right) \dfrac{\partial \rho({\rm obs})}{\partial \Gamma_{\rm obs}} \right) = b_{\rm max}^{-2}\ f(b)\ \rho({\rm act}(b))\ b \\
\Rightarrow \rho({\rm act}) = \left[ 4 b_{\rm max}^2\ \Gamma_{\rm obs}^{-2}\ f(b)^{-1}\ P_{\rm obs} \left( \dfrac{\partial \rho({\rm obs})}{\partial P_{\rm obs}} - \left( \dfrac{\Gamma_{\rm obs}}{2 P_{\rm obs}} \right) \dfrac{\partial \rho({\rm obs})}{\partial \Gamma_{\rm obs}} \right) \right]_{\rm obs \rightarrow act} \\
\boxed{ \rho(\Gamma_{\rm act}, P_{\rm act}) = k\ \Gamma_{\rm act}^{-11/3} \left( P_{\rm act} - P_{\rm th} \right)^{-1/2} \left[ P_{\rm obs} \dfrac{\partial \rho(\Gamma_{\rm obs}, P_{\rm obs})}{\partial P_{\rm obs}} - \left( \dfrac{\Gamma_{\rm obs}}{2} \right) \dfrac{\partial \rho(\Gamma_{\rm obs}, P_{\rm obs})}{\partial \Gamma_{\rm obs}} \right]_{\rm obs \rightarrow act} },
\end{equation}
where $k = 4 b_{\rm max}^2 A_{\rm max} \kappa^{-1}\ \upsilon^{-1}\ P_{\rm th}^{1/2}$, $d \rho(\Gamma_0, P_{\rm th})/db = 0$, and ${\rm obs \rightarrow act}$ indicates that ${\rm obs}$ quantities should be replaced with ${\rm act}$ quantities after the derivatives are taken.
In the next section, we apply Equation \ref{eqn:convert_from_observed_to_actual_density} to datasets from real surveys, but as an example, consider the simple observed distribution $\rho(\Gamma_{\rm obs}, P_{\rm obs}) = \alpha\ P_{\rm obs}^{-2}$. Applying Equation \ref{eqn:convert_from_observed_to_actual_density} gives a distribution of actual parameters as follows:
\begin{equation*}
\nonumber \rho(\Gamma_{\rm act}, P_{\rm act}) = k^\prime \Gamma_{\rm act}^{-11/3} \left( \dfrac{P_{\rm th}}{P_{\rm act} - P_{\rm th}} \right)^{1/2} P_{\rm act}^{-2}.
\end{equation*}
The shaded contour plot in Figure \ref{fig:integration_path} illustrates this $\rho(\Gamma_{\rm act}, P_{\rm act})$ distribution. Note that, for this example, $\partial \rho({\rm obs})/\partial P_{\rm obs} < 0$. In such a case, the signs on the partial derivatives should be flipped since the limits on the integral for Equation \ref{eqn:convert_from_actual_to_observed_density} would be flipped.
The expression blows up as $P_{\rm act} \rightarrow P_{\rm th}$ because such shallow dips are only observed for statistically impossible central encounters ($b = 0$). If we assume $P_{\rm th} \ll P_{\rm act}$ for any observed values, then we have (collecting $P_{\rm th}^{1/2}$ with the other constants at the beginning of Equation \ref{eqn:convert_from_observed_to_actual_density}):
\begin{equation*}
\nonumber \rho(\Gamma_{\rm act}, P_{\rm act}) \approx k^\prime \Gamma_{\rm act}^{-11/3} P_{\rm act}^{-5/2}.
\end{equation*}
Using a direct numerical simulation of a dust devil barometric survey, \citet{Lorenz_2014} found that an observed distribution $\rho(P_{\rm obs}) \propto P_{\rm obs}^{-2}$ required an actual distribution approximately $\rho(P_{\rm act}) \propto P_{\rm act}^{-2.8}$, in line with our results here. Our results show that, for observed distributions following power-laws with a negative indices, a slightly steeper slope is required for the distribution of actual profile depths and considerably steeper for the distribution of widths.
\section*{Comparison to Observational Data}
\label{sec:comparison_to_observational_data}
In this section, we apply our statistical formulation to results from previous single-barometer surveys. A very important assumption we make here is that dust devils are advected past the barometer with some characteristic velocity $\upsilon$, which allows us to treat the width of a profile in time $\tau_{\rm act}$ as a proxy for the width in space $\Gamma_{\rm act}$. As we discuss below, this assumption complicates our analysis of the distribution of observed profile widths but is unavoidable without wind speed data for each observed devil, usually lacking for martian studies.
\citet{Ellehoj_2010} analyzed 151 sols worth of time-series data, including pressures, temperatures, wind speeds, and images, from instruments on-board the Phoenix Lander \citep{Smith_2008}. To identify dust devil passages in the barometric data, they compared the average pressure in a 20-s window to the average pressure in 20-s windows to either side of the former window. Average pressures in the middle window different by more than 0.1 Pa from the average on either side were identified as possible dust devil passages. Then for every pressure event found, they analyzed the surrounding pressure and temperature values, and non-significant and false events, e.g., from data transfer gaps, were removed by hand (the precise criteria used to exclude an event are not given). In this way, \citet{Ellehoj_2010} identified 197 vortices with a pressure drops larger than 0.5 Pa = $10^{-0.3}$ Pa, which we will take as $P_{\rm th}$ for this dataset.
Figure \ref{fig:Ellehoj_data_obs_to_act_dist} shows a scatter plot of their reported detections. The colored contours are calculated using a Gaussian kernel density estimator and a bandwidth of 0.75, analogous to histogram bin widths. The narrower bandwidth, 0.5, provided by Scott's rule \citep{SCOTT_1979} results in apparently spurious structure in the density contours. The marginalized and normalized densities are shown along each axis.
To derive the density of actual parameters, we can apply Equation \ref{eqn:convert_from_observed_to_actual_density} to the density of observed parameters in Figure \ref{fig:Ellehoj_data_obs_to_act_dist}. The white contours within the bottom-left panel illustrate the density estimate, while the top- and rightmost curves show the marginalized and normalized densities for $\Gamma_{\rm act}$ and $P_{\rm act}$. Several features stand out.
Foremost is the fact that the contour lines terminate near a value $\Gamma_{\rm act}^\prime =$ 6 s, indicating that the observed distribution implies very few dust devils with actual profile widths longer than that and the dust devils with apparently wider profiles were actually observed with relatively large miss distances. Assuming all the devils traveled past the sensor with $\upsilon \approx$ 3 m/s, the cutoff in temporal width at 6 s translates to a cutoff in dust devil diameter of about 18 m, consistent with the typical devil width of 10-20 m reported by \citet{Greeley_2006}. The sharp drop-off feature also arises, in part, because our model for the miss distance effects indicates the observed distribution should be skewed toward wider profiles, but the distribution observed by \citet{Ellehoj_2010} actually drops off longward of about 6 s. This decline in density manifests as a very steep drop in the inferred density for $\Gamma_{\rm act}^\prime$.
What does this result mean? Primarily, it implies that converting from the observed temporal widths to the spatial widths is challenging since even a moderate variability in ambient windspeed (from 1 to 20 m/s) could easily contribute an order of magnitude variation in observed width for a single devil. It also highlights the important role played by the detection scheme employed to sift devils out of the pressure time series. \citet{Ellehoj_2010} employed windows 20 s wide, which would have necessarily biased their detections toward devils with profiles less wide than that -- a devil with profile as wide as or wider than 20-s would have been filtered out. This detection bias probably contributes some to the decline in density toward larger $\Gamma_{\rm act}^\prime$. As discussed in Section \ref{sec:discussion_and_conclusions}, a completeness assessment for the detection scheme could help mitigate this bias.
Turning to the distribution of $P_{\rm act}$, there is a decline moving from moderate to small $P_{\rm act}$-values. However, we also might expect the peak in the density for $P_{\rm obs}$-values to originate from a peak at a larger $P_{\rm act}$-value due to the miss distance effects. In Figure \ref{fig:Ellehoj_data_obs_to_act_dist}, the observed density peaks at $P_{\rm obs} \approx 10^{-0.15}\ {\rm Pa} = 0.7\ {\rm Pa}$, while the inferred actual density peaks at $P_{\rm act} \approx 10^{0.2}\ {\rm Pa} = 1.6\ {\rm Pa}$. We can compare these peaks to our expected average miss distance from Section \ref{sec:the_signal_distortion}: $\langle b/\Gamma_{\rm act} \rangle \approx \frac{1}{3} \sqrt{P_{\rm act}/P_{\rm min}} = \frac{1}{3} \sqrt{1.6/0.5} \approx 0.6$. With that average miss distance, we'd expect a dust devil with $P_{\rm act}$, on average, to observed with $P_{\rm obs} = P_{\rm act}/\left( 1 + \left( 2\times 0.6 \right)^2 \right) \approx 0.4\ P_{\rm act}$, not too different from where the peaks do occur for two distributions. Deciding whether the observed peak is due primarily to difficulties detecting small $P_{\rm obs}$ signals or the miss distance effect also requires a completeness assessment, however.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Ellehoj-data-to-actual-dist/Ellehoj-data-obs-to-act-dist}
\caption{{\label{fig:Ellehoj_data_obs_to_act_dist}
The bottom left panel shows a scatter plot (blue dots) of Martian dust devil encounters observed by the meteorological instruments on the Pheonix Lander and reported in Table 1 from \protect\citet{Ellehoj_2010}. Copper-to-black shaded contours show the corresponding density estimated using a Gaussian kernel density estimator, with darker colors indicating higher densities. The blue lines in the top and right panels show the marginalized, normalized distributions of $\tau_{\rm obs}$ and $P_{\rm obs}$, respectively. The bottom left panel also shows white contours representing the density of $\tau_{\rm act}$ and $P_{\rm act}$, while the white lines in the top and right panels show the corresponding normalized, marginalized densities.%
}}
\end{center}
\end{figure}
As another recent example, \citet{Jackson_2015} conducted a terrestrial survey at El Dorado Playa in Nevada, deploying several pressure loggers that collected data over the course of two years. This dataset involves the same sampling rate (2 Hz) similar to that of the martian dataset (0.5 Hz) but extends over a much longer baseline -- the martian dataset spanned 151 sols. With more than 250 million time-series data points collected, the latter study applied an automated detection scheme that involved smoothing the data stream over 1,000-s windows and searching for statistically significant outlier points, recovering more than 1,600 putative dust devil pressure dips. Figure \ref{fig:Jackson_data_obs_to_act_dist} shows a scatter plot of their reported detections (note the change in units from Pa to hPa for the pressures). As in \citet{Jackson_2015}, we masked out detections with large $\tau_{\rm obs}$ and small $P_{\rm obs}$ (called ``unexpectedly shallow'' in \citet{Jackson_2015}), giving rise to the absence of points in the lower right-hand corner of the bottom left panel. As for the previous dataset, we employed a Gaussian kernel density estimate with a bandwidth of 1 to suppress spurious structure (somewhat smaller bandwidths do not qualitatively change the results).
Similar features appear for this dataset as for the previous dataset for Mars, for the densities of both observed and actual parameters, although the locations of features are shifted. Again, a sharp decline in the distribution of $\tau_{\rm act}$ coincides with a peak in the $\tau_{\rm obs}$ density, but this time around $\tau_{\rm act} = 10^{1.6}$ s = 4 s. With a typical speed of 5 m/s, $\tau_{\rm obs}$ = 40 s corresponds to a physical width of 200 m, much larger than the typical dust devil width observed on El Dorado Playa \citep{Balme_2012}. This result either indicates the detection of unusually large/slow-moving dust devils or (more likely) some of the widest signals reported in \citet{Jackson_2015} are pressure dips unrelated to dust devils, an issue discussed at length in that study. In any case, if many of the detections can be attributed to dust devil passages, the observed $\Gamma_{\rm obs}$-distribution points to underlying population of much more narrow dust devils.
Again, as for the previous dataset, the peak near $P_{\rm obs}$ is likely due, at least in part, to recovery biases for shallow devils. However, we would also expect the miss distance effect to shift the peak in the $P_{\rm act}$ distribution near $10^0$ hPa = 1 hPa to $10^{-0.6}$ hPa = 0.25 hPa in the $P_{\rm obs}$ distribution: $P_{\rm th}$ = 0.1 hPa, so $\langle b/\Gamma_{\rm act} \rangle = \frac{1}{3} \sqrt{1\ {\rm hPa}/0.1\ {\rm hPa}} = 1$. Thus, a peak at $P_{\rm act}$ = 1 hPa should be shifted to $P_{\rm obs} = 1\ {\rm hPa}/\left( 1 + (2 \times 1)^2 \right)$ = 0.2 hPa.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Jackson-data/Jackson-data-obs-to-act-dist}
\caption{{\label{fig:Jackson_data_obs_to_act_dist}
Similar to Figure \ref{fig:Ellehoj_data_obs_to_act_dist}, except using data for Earth from \protect\citet{Jackson_2015} and instead of white lines for the densities of actual parameters, this figure uses black, dashed lines in all the panels.%
}}
\end{center}
\end{figure}
\section*{Discussion and Conclusions}
\label{sec:discussion_and_conclusions}
Our formulation here provides a starting place for relating the population statistics of dust devils as recovered by single-barometer surveys to their physical structures. Understanding these relationships is critical for understanding the atmospheric influence of devils on both planets since it depends so sensitively on both the devils' statistical and physical properties. As noted in \citet{Jackson_2015} and \citet{Lorenz_2014}, in estimating the total flux of dust injected into the martian atmosphere, it is important to consider the population-weighted flux and not the flux from the average dust devil. Of course, knowing the population is critical to calculating that population-weighted flux. Moreover, lab work reported in \citet{Neakrase_2006} suggested an exponential dependence of dust flux on a dust devil's pressure depth, and so even small shifts in the distribution of dust devil pressure depths can result in large shifts in the dust flux. For instance, using the exponential dependence indicated in Figure 4 of \citet{Neakrase_2006}, we find that the dust flux given by the distribution of $P_{\rm act}$ in Figure \ref{fig:Ellehoj_data_obs_to_act_dist} would be more than 30\% larger than that given by the $P_{\rm obs}$-distribution.
The model for the miss distance effect developed here serves to highlight the many important uncertainties and degeneracies involved in single-barometer dust devil surveys. In particular, these results show that it is difficult to disentangle the geometry of an encounter between a devil and a detector from the devil's structure. The pressure profile observed for a devil will almost always be wider and less deep than the devil's actual profile.
As discussed in Section \ref{sec:the_recovery_bias}, the miss distance effect biases the recovered population toward the physically widest devils. Because the dynamical processes that form and maintain devils are not well-understood, the relationship between the width of a devil and its other physical properties are not clear. \citet{Fenton_2015} argue that dust devil height is related to the boundary layer depth, while the physical model outlined in \citet{Renn__1998} indicates the profile depth should also scale with boundary layer depth. In any case, the bias definitely plays a role in estimates of the areal density for dust devil occurrence. For example, by assuming a devil profile width of 100 m, \citet{Ellehoj_2010} combine the number of devils recovered from pressure time-series and wind speed data to estimate a local occurrence rate of 1 event per sol per 10 km$^2$. Although useful, that occurrence rate estimate involves an implicit marginalization over the dust devil population and the efficiency function for their detection scheme. The occurrence rate for small dust devils (those with narrow profiles) could be considerably larger since they are less likely to be detected. Likewise, the rate for large devils (wider profiles) could also be larger since the detection scheme probably filters out devils with profiles much wider than 20-s.
An improved understanding of the biases involved in a detection scheme is critical for relating the observed to the underlying population, and a simple way to assess a scheme's detection efficiency is to inject synthetic devil signals (with known parameters) into the real data streams. Then the detection scheme can be applied to recover the synthetic devils and the efficiency of detection assessed across a swath of devil parameters. Such an approach is common in exoplanet transit searches \citep[e.g.][]{Sanchis_Ojeda_2014}, where dips in photometric time series from planetary shadows closely resemble dust devil pressure signals. By injecting synthetic devils into real data, the often complex noise structure in the data is retained and simplifying assumptions (such as stationary white noise) are not required.
Among important limitations of our model, the advection velocity $\upsilon$ for devils remains an critical uncertainty for relating physical and statistical properties. This limitation points to the need for wind velocity measurements made simultaneously with pressure measurements in order to accurately estimate dust devil widths. In particular, correlations between $\upsilon$ and dust devil properties will skew the recovered parameters in ways not captured here. For example, the devils with the deepest pressure profiles seem to occur preferentially around mid-day local time both on Mars \cite{Ellehoj_2010} and the Earth \cite{Jackson_2015}. If winds at that time of day are preferentially fast or slow, then the profile widths recovered for the deepest devils will be skewed toward smaller or larger values. In addition, some field observations suggest devils with larger diameters may be advected more slowly than their smaller counterparts \cite{Greeley_2010}, which would tend to make their profiles look wider.
The formulation described here could, in principle, account for this uncertainty by incorporating a distribution of $\upsilon$ determined observationally, $\rho(\upsilon)$. The distribution $\rho(P_{\rm obs}, \tau_{\rm obs})$ can be converted to $\rho(P_{\rm obs}, \Gamma_{\rm obs})$ using $\rho(\upsilon)$ via the following integral:
Then the physical width of a devil profile could be represented using a probability density $\dfrac{dp}{d\Gamma_{\rm act}} \propto n(\upsilon)\ d\upsilon$.
As highlighted in Section \ref{sec:comparison_to_observational_data} and discussed in \citet{Lorenz_2011}, the choice of the binning procedure (bin size, etc.) in constructing the distributions of physical properties shapes the result in non-trivial ways, and the approach used to describe the distributions will also depend on the procedure. Fortunately, the field of data science has provided several statistically robust and objective procedures for binning data that frequently use the data themselves to determine how they are binned \citep[e.g.][]{Feigelson_2009}. One simple way to ascertain the optimal binning procedure would be to generate synthetic populations according to prescribed distribution functions (power-laws, exponential, etc.) and then investigate which binning procedure allowed the most accurate recovery of the assumed distribution. As an alternative, \citet{Lorenz_2012} suggests plotting cumulative distributions to circumvent the ambiguities involved in binning choices altogether.
Clear predictions of the distributions of physical parameters for dust devils from high resolution meteorological models would be especially helpful for constraining and directing this work, and some progress in this area has been made. For example, \citet{2005QJRMS.131.1271K} applied a large-eddy simulation of a planetary convective boundary layer to study vortical structures and the influence of ambient conditions on their formation. For the handful of vortices formed in the simulations, there was good qualitative agreement with observation. \citet{2010BoLMe.137..223G} also studied vortex formation on Earth and Mars and noted the role of the boundary layer's depth on vortex scale. Given the stochastic nature of boundary layer dynamics, detailed statistical predictions from such models are needed for comparison to observation. However, the computational expense of such high-resolution models makes that prohibitive.
Likely the best way to study dust devil formation and dynamics in the field is not statistically, but directly via deployment of sensor networks that produce a variety of data streams with high spatial and time resolution. Field work with in-situ sensors has a long history but usually involving single-site deployments \citep[e.g.][]{Sinclair_1973}. In the decades since that study, technological developments in miniaturization and data storage now provide a wealth of robust and inexpensive instrumentation, ideally suited for the long-term field deployment required to study dust devils, without the need for direct human involvement. Recently, \citet{Lorenz_2015} deployed an array of ten miniature pressure- and sunlight-logging stations at La Jornada Experimental Range in New Mexico, providing a census of vortex and dust-devil activity at this site. The simultaneous measurements resolved horizontal pressure structures for several dust devils, giving entirely independent estimates of vortex size and intensity.
The rich and growing databases of high-time-resolution meteorological data, both for the Earth and Mars, combined with the wide availability and affordability of robust instrumentation, point to bright future for dust devil studies. The data streaming in from the Mars Science Laboratory Rover Environmental Monitoring Station (REMS) \citep{G_mez_Elvira_2012} may provide new insight into Martian dust devils, although preliminary studies \citep[e.g.][]{2015DPS....4741907M} have found very few dust devils in Gale Crater. The formulation presented here provides a simple but robust scheme for relating the dust devils' statistical and physical properties, and though it has some limitations, it represents an important next step in improving our knowledge of these dynamic and ethereal phenomena.
\section*{Acknowledgements}
The authors acknowledge useful input from Paul Simmonds.
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