Conceptual overview of the four-step modeling framework for quantifying geographic barriers formed by environmental gradients. A: Our generalized model is parameterized for both marine and terrestrial environments using elevation. B: We then compute slope, or the spatial rate of change in elevation, which mechanistically is designed to capture environmental transitions that impose either direct or indirect barriers to species distribution. C: Considering all possible candidate barriers that emerge under different rates of change, we compute a series of biologically informative summary statistics that follow directly from predictions of geographical ecology and speciation and allow us to quantitatively characterize the topographic traits known to be associated with ring species formation (Table 1). D: Finally, we use the summary statistics for all candidate barriers in a multivariate analysis to explicitly compare among known reference barriers and identify similar candidate barriers that may be promoting similar evolutionary processes. Data for illustration purposes (A-C) provided by R Development Core Team .