I typed that last one too quickly– too many typos, but my wife says I’m not supposed to revise blogs, but move on… .
So, for clarity, let’s talk a little more about McNab indices. Field-derived McNab indices are a measure of average angle from the observer to the horizon (mesoscale landform index), or from the observer to another field person a set distance away, e.g. 10m or 18m, or whatever the plot size or settled upon (minor landform index). Typically these are done in the field at a discreet number of directions, e.g. 4 cardinal directions, or 4 cardinal plus 4 ordinal (NE,NW
SE, SW, SE) directions. The landform image in this post and the last is a calculation of mesoscale landform, which is harder to do in a classic GIS than minor landform (I’ll have a follow-up post on minor landform, probably using ArcGIS).
To calculate the values for mesoscale landform computationally, we require calculating the angle to the horizon for a certain number of directions for each point in the landscape. This has the potential to be computationally intensive and complicated. If, however, we conceive of the problem differently, as a 3D calculation we can perform in PovRay, arriving at the answer is simplified by the coding already done by the PovRay programmers.
Essentially, McNab mesoscale indices are a field proxy for the steradian of a site.
What does that mean? Well, essentially, a map of steradians is a proxy for the shading of a white uneven surface on a cloudy day– or how much diffuse light is available to a given site. Used in combination with site aspect, this is enough information to determine most of of the light conditions of a site, which is why McNab indices in combination with other factors are a good predictor of site productivity, and correlated with different plant communities across the landscape.
How does an uneven white surface shade on a cloudy day? Steeper areas with more of the sky occluded are darker while wide open spaces, like the bottom of river valleys and the tops of ridges and plateaus are brighter. If you want to witness this effect, look to snow on the ground on a cloudy day (and what a great winter to do it). (The only difference with snow is subsurface scattering which evens out the light quite a bit.) You’ll notice the divots in the snow to be darker than the peaks, and the edges of the divots to be darkest of all.
The question then, is how to we compute this within PovRay? We could use radiance as a global illumination model, but the calculation of inter-reflectivity that is at the core of radiance, while an important real factor in the landscape, would fail to replicate the original mesoscale landform index. Instead, we set up a series of lights in a dome to illuminate the whole sky sphere, blur them a bit, and call it a day, a technique developed for HDRI rendering by Ben Weston, whose code I borrowed heavily. The more lights the element of the landscape is exposed to the brighter, and vice versa, essentially making the brightness of an element proportional to the exposure to the sky sphere. Unlike Ben, I used a simple white sphere to get even lighting.
Rendered at 16-bit resolution, we have a possible range of values from 0-65535. Let’s assume that a linear transformation of these values will result in values representing the proportion of the sky sphere. From there, transformation to steradians and then to solid angles in degrees is trivial. Once it’s solid angles in degrees, it represents the same kind of value as a mesoscale landform index would give us (more later…).