The theory underlying line transect and variable circular plot surveys-distance sampling-begins with an assumed detectability function, giving the probabilities of detecting animals at different distances from the observer's path. The nature of these probabilities is unspecified in the general development, leaving users to question whether the actual probability structure matters. In particular, may one use the methodology in surveys where animals at the same distance have different probabilities of detection? This paper presents three examples where probabilities come from different assumptions: from the random placement of transects; from the uniform distribution of animals over the study region; and from cues randomly detected by the observer. These exemplify situations where detectability may not be a function of distance alone. Horvitz-Thompson estimators are displayed which can be used in each example, but some estimators require measuring features other than distance. A result concerning optimally weighted Horvitz-Thompson estimators shows that all three can be brought under the same umbrella if detection areas are measured instead of detection distances and if animals are uniformly distributed.