I have two surfaces represented as raster images with heights as grayscale values. One is natural landscape elevations; the other is just distance from a line.
I have computed Minkowsky D = 2 - H where, for surface area A and cell size s at aggregation level k:
$$ A_k = s_k^{-H} \quad \mathrm{rewritten} \quad \log A_k = -H \log s_k$$
and H is found by linear regression.
D for the elevation surface is 2.31, but D for the distance surface comes out as 1.81. Does this indicate an error in computation? a wrong choice of dimension formula? Is there some meaning associated with a fairly simple 2D surface having a fractal dimension less than 2?
A locally Lipschitz map can't increase Hausdorff dimension. In particular, affine projections are Lipschitz maps. So if your surface $S$ lies over a nonempty open set $U$ in the $xy$ plane, in that the image of $S$ under the projection $(x,y,z) \to (x,y)$ contains $U$, then the Hausdorff dimension of $S$ must be at least the Hausdorff dimension of $U$, namely $2$.