of $z =
0.308$–$0.545$, the assumed physical scale together with $(1 + z)^4$ cosmological dimming conspire to turn distant high-$z$ galaxies into objects consistent with UDGs. In order to remove these from the background correction count, our XDF catalog was matched to the UVUDF photo-$z$ catalog of [@rafelski2015]. The photo-$z$ distribution of XDF sources that pass the UDG cuts in each cluster are shown as the black histograms in Figure \[fig:photo-z\].

Image simulations {#sec:sims}
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![ \[fig:completeness\] Recovery fraction as a function of size and magnitude for $n=1$ Sérsic profiles. The lines are contours of mean surface brightness within $R_e$ ($\langle\mu\rangle_{e,F814W}$) in units of $\mathrm{mag}~\mathrm{arcsec}^{-2}$; 26.9 $\mathrm{mag}~\mathrm{arcsec}^{-2}$ is the 50% completeness limit. ](figure4.pdf){width="50.00000%"}

Two sets of image simulations were performed. The first was to determine and inform the selection criteria used to select UDGs, and the second was to determine geometrical completeness distributions of objects that pass our selection criteria in the six clusters.

We began by injecting 2000 simulated objects into each of the cluster and parallel image pairs at random positions in the region with overlapping ACS and WFC3 coverage. The <span style="font-variant:small-caps;">SExtractor</span> segmentation map was used to ensure a chosen location was empty. To prevent crowding, this 