Anisotropic Multifractal Scaling of Mount Lebanon Topography: Approximate Conditioning

We used multifractals to analyze the Lebanese topography focusing on Mount Lebanon. The elevation data were obtained from NASA STRM Global Digital Elevation of Earth Land, spaced at 80m in the East-West direction, and at 90m in the North-South direction. After transforming the grid to be perpendicular and parallel to the range, we found anisotropic scaling from 500m to 10,000m, and it reflected the fact that the Lebanese topography was more correlated in the direction perpendicular to the mountain range, probably due to occurrence of valleys and ridges in that direction. We estimated the parameters of the Universal Multifractal (UM) model and found α = 1.45 and c1 = 0.05, consistent with values reported for topography. The UM parameter H was found to be 0.72 across the range and 0.57 along the range, the latter value agrees with prior observations. However, the larger value across the range is consistent with the higher spatial correlation in that direction. We introduced a new expression for the 2D power spectral density, and we showed that it can decently capture the anisotropic scaling. We also generated multiple realizations and we showed that the generation of anisotropic scaling did not alter the underlying parameter values α and c1 of the UM model. We also proposed an approximate method for generating conditional simulations, and we showed that through a judicious selection of values, one may reproduce approximately the observed field values at the desired locations. We believe such an approach could be used to generate realistic simulations of fields that are time-invariants, such as topography and soil properties.