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\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, frequently used for Martian devils and more recently for terrestrial ones, 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. However, the formulation here also serves to highlight the many important uncertainties and degeneracies involved in those relationships.   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. Since a devil's dust-lift capacity seems to depend very sensitively on its depth \cite{Neakrase_2006}, it's very likely the dust flux from devils on Mars and Earth are significantly underestimated since the deepest pressure wells are probably not observed directly.   As discussed in Section \ref{sec:the_gammaact_recovery_bias_and_distortion}, 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 aren't clear, so it's not clear how the recovered properties are skewed by this bias. However, 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.  The advection velocity for devils remains an critical uncertainty for relating physical and statistical properties, and our assumption of a constant velocity is an important limitation of the model. 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 advection velocity 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 can also account for this uncertainty  uncertainty may be circumvented somewhat , by considering a distribution of advection velocities  Predictions from models of distributions and structures needed!  Since number densities observed haven't turned over at large P_obs yet, likely we haven't seen P_max 
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How to determine max/min values?      As noted in \citet{Lorenz_2011}, the choice of histogram binning procedure (bin size, etc.) can shape the histogram in non-trivial ways, and so the resulting functional form used to describe the histogram 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 to they are binned. For example, the histograms in Figure \ref{fig:} used