Let $X_n$ be a 1D discrete fBm. Then, its 1st order difference, $W_n=X_n-X_{n-1}$ is fractional Gaussian noise (fGn). This case is simple.
But what happens in 2D? Let $Y(m,n)$ be a 2D fBm, then we have several ways to get "1st order" increments: $$W_x(m,n) = Y(m,n)-Y(m-1,n)$$ $$W_y(m,n) = Y(m,n)-Y(m,n-1)$$ $$W_{xy}(m,n) = \frac{1}{2}(W_x(m,n)+W_y(m,n))$$ etc. Any of these functions will have a different behaviour and spectrum, but usually, considering the 2D fGn, it has a well-known, isotropic hyperbolic-decaying spectrum: $S_{fGn}(r)=\frac{1}{|r|^d}$ where $r=\sqrt{m^2+n^2}$ and $d$ is some function of the fractal dimension.
I can't seem to be able to get a meaningful connection between the spatial definition of fGn and the spectral definition. Any of the above fGns are not really isotropic.
Appreciate any assistance or references for this matter.
Thanks, Yoki.