Visibility analyses, used in many disciplines, rely on viewshed algorithms that map locations visible to an observer based on a given surface model. Mapping continuous visibility over broad extents is uncommon due to extreme computational expense. This study introduces a novel method for spatially-exhaustive visibility mapping using airborne lidar and random forests that requires only a sparse sample of viewsheds. In 24 topographically and vegetatively diverse landscapes across the contiguous US, 1000 random point viewsheds were generated at four different observation radii (125 m, 250 m, 500 m, 1000 m), using a 1 m resolution lidar-derived digital surface model. Visibility index – the proportion of visible area to total area – was used as the target variable for site-scale and national-scale modeling, which used a diverse set of 146 terrain- and vegetation-based 10 m resolution metrics as predictors. Variables based on vegetation, especially those based on local neighborhoods, were more important than those based on terrain. Visibility at shorter distances was more accurately estimated. National-scale models trained on a wider range of vegetation and terrain conditions resulted in improved R2, although at some sites error increased compared to site-scale models. Results from an independent test site demonstrate potential for application of this methodology to diverse landscapes.