Exact combination rule for roughness under weighted sums of terrain elements

Derive an exact expression for the roughness r of a terrain generated by the WeightedSum module, z(x, y) = Σ_i w_i z_i(x, y), in terms of the individual roughness values r_i of the terrain elements z_i and the weights w_i. Roughness r is defined as the ratio of the surface area of the original terrain to the surface area of a Gaussian-smoothed version of the same terrain (with a specified smoothing standard deviation).

Background

The library controls terrain characteristics by combining multiple terrain elements via WeightedSum and adjusting weights to meet target metrics. For slope, the resultant gradient combines linearly, allowing exact control. In contrast, roughness—defined as the ratio of the original surface area to the area of a smoothed terrain—does not have a straightforward exact combination rule from component measures.

To make progress, the paper introduces an empirical proxy r_proxy = 1 + [Σ_i (r_i^α − 1) w_i^β]^{1/α}, finding α ≈ 1 and β ≈ 2 to fit data reasonably well. However, an exact formula for the resultant roughness under weighted sums remains unavailable, motivating a formal derivation to enable precise parameter setting without reliance on approximations.