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.

中文翻译：

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黎巴嫩山地形的各向异性多分形缩放：近似条件

我们使用多重分形分析以黎巴嫩山为重点的黎巴嫩地形。高程数据来自 NASA STRM Global Digital Elevation of Earth Land，间隔为 80m 在东西方向，在 90m 在南北方向。在将网格转换为垂直并平行于范围后，我们发现从 500 开始的各向异性缩放米到 10,000m，它反映了黎巴嫩地形在垂直于山脉的方向上更相关的事实，可能是由于在该方向上出现了山谷和山脊。我们估计了通用多重分形（UM）模型的参数并发现α = 1.45和C1 = 0.05，与报告的地形值一致。UM 参数H发现在整个范围内为 0.72，在整个范围内为 0.57，后一个值与先前的观察结果一致。但是，该范围内的较大值与该方向上较高的空间相关性一致。我们为 2D 功率谱密度引入了一个新表达式，并且我们证明它可以很好地捕捉各向异性缩放。我们还生成了多个实现，并且我们证明了各向异性缩放的生成并没有改变基础参数值α和C1UM 模型。我们还提出了一种用于生成条件模拟的近似方法，并且我们表明，通过明智地选择值，可以在所需位置近似地再现观察到的场值。我们相信这种方法可用于生成时不变场的真实模拟，例如地形和土壤特性。