Inferences about the spatial density or phase-space structure of stellar
populations in the Milky Way require a precise determination of the effective
survey volume. The volume observed by surveys such as Gaia or near-infrared
spectroscopic surveys, which have good coverage of the Galactic mid-plane
region, is highly complex because of the abundant small-scale structure in the
three-dimensional interstellar dust extinction. We introduce a novel framework
for analyzing the importance of small-scale structure in the extinction. This
formalism demonstrates that the spatially-complex effect of extinction on the
selection function of a pencil-beam or contiguous sky survey is equivalent to a
low-pass filtering of the extinction-affected selection function with the
smooth density field. We find that the angular resolution of current 3D
extinction maps is sufficient for analyzing Gaia sub-samples of millions of
stars. However, the current distance resolution is inadequate and needs to be
improved by an order of magnitude, especially in the inner Galaxy. We also
present a practical and efficient method for properly taking the effect of
extinction into account in analyses of Galactic structure through an effective
selection function. We illustrate its use with the selection function of
red-clump stars in APOGEE using and comparing a variety of current 3D
extinction maps.Comment: Code available at https://github.com/jobovy/mwdust and at
https://github.com/jobovy/apogee-map
