1. The stereoscopic approach: Fundamental assumptions: Lambertian reflection from the surface The difference between measured radiances at two view- angles can be used as a proxy for relative surface roughness

2. Fundamentals:  For a given pixel, surface-reflected solar irradiance L (Wm -2 sr -1 ) at a given view angle  can be approximated as: (1)  The ratio between L at  1 and L at  2 is then: (2) I sol – incident solar irradiation (Wm -2 ) R e - surface reflectivity S  - down-welling sky irradiance (Wm -2 ) f sh - effective shade fraction (3)

3. Atmospheric effects: I sol – incident solar irradiation (Wm -2 ) R e - surface reflectivity S  - path radiance (Wm -2 sr -1 ) S  - down-welling sky irradiance (Wm -2 ) f sh - effective shade fraction  (  ) – atmospheric transmissivity  Per pixel, the ratio between at-sensor surface-reflected solar irradiance values L  (Wm -2 sr -1 ) at view angles  1 and  2 can be approximated as: (5) (4) Can be removed using ‘dark object subtract’ Becomes a multiplicative scaling actor Assuming a laterally ~homogeneous atmosphere at the image scale   (  1) /  (  2) can be regarded as constant for the whole image.

4. Atmospheric effects: I sol – incident solar irradiation (Wm -2 ) R e - surface reflectivity S  - path radiance (Wm -2 sr -1 ) S  - down-welling sky irradiance (Wm -2 ) f sh - effective shade fraction  (  ) – atmospheric transmissivity - can be regarded as a proxy for relative surface roughness between similarly sloping pixels within a single image. incorporates roughness variations at all sub-pixel scales is independent of surface composition fairly insensitive to atmospheric effects surface atmosphere 30° Atmospheric transmissivity is a function of path length