The paper "Multiscale Analysis of Topographic Surface Roughness in the Midland Valley, Scotland" by Grohmann et al., 2011 describes the differences between a six methods of calculating surface roughness measurements from 2D digital topography. His paper was helpful since he provides a quantitative comparison of each method using a single test region at various spatial resolutions and window sizes. At the very end of his paper he states:
Standard deviation of slope remains the single most effective measure
of surface roughness due to the simplicity of calculation, detection
of fine scale/regional relief, and performance at a variety of scales.
He also recommends using vector dispersion and the standard deviation of profile curvature, based on their ability to depict terrain features. He downvotes the method of area ratio since it "fails to distinguish between landforms in areas of low relief." The area ratio method is similar to the Rugosity index from Lundblad et al., but there may be some small difference in calculation (I have not looked at the code for the Rugosity index vs. the area ratio method used in Grohmann et al.).
I chose two methods: standard deviation of slope and vector dispersion; standard deviation of slope for simplicity/accuracy and vector dispersion since it is sensitive to local variations in elevation, which is suitable for my study areas.