\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.