  Spatially-explicit estimates of population density, together with appropriate
estimates of uncertainty, are required in many management contexts. Density
Surface Models (DSMs) are a two-stage approach for estimating spatially-varying
density from distance-sampling data. First, detection probabilities -- perhaps
depending on covariates -- are estimated based on details of individual
encounters; next, local densities are estimated using a GAM, by fitting local
encounter rates to location and/or spatially-varying covariates while allowing
for the estimated detectabilities. One criticism of DSMs has been that
uncertainty from the two stages is not usually propagated correctly into the
final variance estimates. We show how to reformulate a DSM so that the
uncertainty in detection probability from the distance sampling stage
(regardless of its complexity) is captured as an extra random effect in the GAM
stage. In effect, we refit an approximation to the detection function model at
the same time as fitting the spatial model. This allows straightforward
computation of the overall variance via exactly the same software already
needed to fit the GAM. A further extension allows for spatial variation in
group size, which can be an important covariate for detectability as well as
directly affecting abundance. We illustrate these models using point transect
survey data of Island Scrub-Jays on Santa Cruz Island, CA and harbour porpoise
from the SCANS-II line transect survey of European waters.
