Generating and scaling fractional Brownian motion on finite domains

Power variograms of statistically isotropic or anisotropic fractal fields (common in earth science) are weighted integrals of variograms representing statistically homogeneous fields (modes) having mutually uncorrelated increments. Large- and small-scale cutoffs were previously assumed proportional to length scales of the sampling window and data support. We verify this assumption numerically for two-dimensional isotropic fractional Brownian motion (fBm). It was previously concluded semi-empirically that, for Hurst coefficient H = 0.25, the constant of proportionality is μ = 1/3. We confirm this but find μ to vary with mode type and H. We find that due to lack of ergodicity, sample fBm variograms generated on finite windows exhibit directional dependence and differ sharply between realizations. Many realizations are required to obtain an average sample variogram resembling the theoretical power model, especially for persistent fields. We propose generating fBm on finite windows using truncated power variograms and provide guidance for doing so effectively.