Height Fluctuations and Surface Gradients in Topographic Measurements.

Topographic maps are composed of pixels associated with coordinates (, , ) on a surface. Each pixel location (, ) is linked with fluctuations in a measured height sample (). Fluctuations here are uncertainties in heights estimated from multiple topographic measurements at the same position. Height samples () are measured at individual locations (, ) in topographic measurements and compared with gradients on topographies. Here, gradients are slopes on a surface calculated at the scale of the sampling interval from inclination angles of vectors that are normal to triangular facets formed by adjacent height samples ( = (, )). Similarities between maps of gradients logs and height fluctuations are apparent. This shows that the fluctuations are exponentially dependent on local surface gradients. The highest fluctuations correspond to tool/material interactions for turned surfaces and to regions of maximum plastic deformation for sandblasted surfaces. Finally, for abraded, heterogeneous, multilayer surfaces, fluctuations are dependent on both abrasion and light/sub-layer interactions. It appears that the natures of irregular surface topographies govern fluctuation regimes, and that regions which are indicative of surface functionality, or integrity, can have the highest fluctuations.