The issue of covariates differs between conventional distance sampling (of which I spoke during the introductory workshop) and distance sampling when g(0)<1 (which uses the mrds approach).
Because the mrds (or double observer) approach uses mark-recapture analysis (the "mr" in "mrds"), more extensive modelling of the "mr" portion of the double observer analysis is required. This is because mark-recapture modelling does not possess the pooling
robustness property, possessed by conventional distance sampling. In the absence of the pooling robustness property, sources of heterogeneity must be modelled to prevent bias.
This paragraph is from a paper by Borchers et al. (1998):
Using similar arguments to those of Patil et al. (1993), one can show that, when
detection on the trackline is
not certain, the p.d.f. of the observed data is not the same when mean detection
functions, averaged over z, are used in place of detection functions that depend on z, even when
the true forms of the mean detection functions are known. Inferences will in general be biased when
sources of heterogeneity
in detection probabilities are not modelled. This is a phenomenon that has
long been recognized in MR studies. As a consequence, it is not in general possible to have pooling
robust estimators (Burnham, Anderson, and Laake, 1980) in the presence of
uncertain detection of animals
on the trackline. (An estimator is pooling robust if it yields approximately unbiased
estimates whether or not data are pooled over z.)
Borchers, D. L., Zucchini, W., & Fewster, R. (1998). Mark-recapture models for line transect surveys.