Statistical roughness

Statistical roughness is an important descriptor of surface topography (Turcotte 2007). While statisticalRoughness has been developed for terrain analysis in the context of geomorphology, there is a vast array of applications to statistical roughness from microscopic scales (e.g., material science) to studying craters in the Solar Systems.

There are multiple main to derive estimates of the statistical roughness for a given surface (Shepard et al. 2001). In statisticalRoughness, the statistical roughness is estimated from the self-affine scaling relationship between the standard deviation of the elevation in regions of a given lateral dimension and the size of that lateral dimension (Family and Vicsek 1991).

Self-affine scaling of surfaces

The width of a surface \(S\) of vertical dimension \(z\) and lateral dimension \((x,y)\) is defined as:

\[\left\langle w \right\rangle_R = \sqrt{\left\langle \left\langle u^2(x,y) \right\rangle_R \right\rangle_s} \] with \(u_R\) the deviation function: \[u_R(x,y) = z(x,y) - \left\langle z(x,y) \right\rangle_R \]

Notice that \(\left\langle w \right\rangle_R\) is an average over all regions the surface \(S\) with lateral dimensions \(R\) and corresponds to the average over \(S\) of the standard deviations over \(R\).

\[\left\langle w \right\rangle_R \sim R^H\]

\(H\) depends on the size \(L_S\) of the surface \(S\) and the lateral dimensions \(R\). In the following, I drop the subscript and equate \(L_s\) with \(L\).

When dealing with data on a grid, it is obviously convenient if the lateral dimensions \(R\) corresponds to factors of \(L\). A factor \(b\in\mathbb{N}\) of \(a\in\mathbb{N}\) is such that the remainder of \(a/b\) is zero and \(b \notin \left\{1;a\right\}\). In other words, for a given value of \(L\), we are interested in finding all the factors of \(L\), excluding \(1\) and \(L\) itself.


The workflow of statisticalRoughness then is divided into three main components:

  1. Factoring
  2. Tiling the elevation data
  3. Computing the statistical roughness for each tile


Family, Fereydoon, and Tamás Vicsek. 1991. Dynamics of Fractal Surfaces. World Scientific.

Shepard, Michael K., Bruce A. Campbell, Mark H. Bulmer, Tom G. Farr, Lisa R. Gaddis, and Jeffrey J. Plaut. 2001. “The Roughness of Natural Terrain: A Planetary and Remote Sensing Perspective.” Journal of Geophysical Research: Planets 106 (E12): 32777–95.

Turcotte, Donald L. 2007. “Self-Organized Complexity in Geomorphology: Observations and Models.” Geomorphology 91 (3-4): 302–10.