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Multifractal Earth topography

机译：多形地球地形

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This paper shows how modern ideas of scaling can be used to model topography with various morphologies and also to accurately characterize topography over wide ranges of scales. Our argument is divided in two parts. We first survey the main topographic models and show that they are based on convolutions of basic structures (singularities) with noises. Focusing on models with large numbers of degrees of freedom (fractional Brownian motion (fBm), fractional Levy motion (fLm), multifractal fractionally integrated flux (FIF) model), we show that they are distinguished by the type of underlying noise. In addition, realistic models require anisotropic singularities; we show how to generalize the basic isotropic (self-similar) models to anisotropic ones. Using numerical simulations, we display the subtle interplay between statistics, singularity structure and resulting topographic morphology. We show how the existence of anisotropic singularities with highly variable statistics can lead to unwarranted conclusions about scale breaking. We then analyze topographic transects from four Digital Elevation Models (DEMs) which collectively span scales from planetary down to 50 cm (4 orders of magnitude larger than in previous studies) and contain more than 2 x 108 pixels (a hundred times more data than in previous studies). We use power spectra and multiscaling analysis tools to study the global properties of topography. We show that the isotropic scaling for moments of order <= 2 holds to within +/-45% down to scales; 40 in. We also show that the multifractal FIF is easily compatible with the data, while the monofractal fBm and fLm are not. We estimate the universal parameters (alpha, C-1) characterizing the underlying FIF noise to be (1.79, 0.12), where alpha is the degree of multifractality (0
机译：本文展示了如何使用现代比例尺概念对各种形态的地形进行建模，以及如何在各种比例尺上准确地表征地形。我们的论点分为两部分。我们首先调查主要的地形模型，并表明它们是基于基本结构（奇异性）与噪声的卷积。着眼于具有大量自由度的模型（分数布朗运动（fBm），分数利维运动（fLm），多重分数分数积分通量（FIF）模型），我们表明它们的区别在于潜在的噪声类型。另外，现实模型需要各向异性奇异性。我们展示了如何将基本的各向同性（自相似）模型推广为各向异性模型。使用数值模拟，我们可以显示统计量，奇异性结构和所产生的地形形态之间的微妙相互作用。我们展示了具有高度可变统计量的各向异性奇点的存在如何导致关于尺度破坏的无端结论。然后，我们分析了四个数字高程模型（DEM）的地形断面，这些模型共同将尺度从行星向下扩展到50厘米（比以前的研究大4个数量级），并且包含超过2 x 108像素（比100倍多的数据）之前的学习）。我们使用功率谱和多尺度分析工具来研究地形的整体属性。我们证明，对于阶数<= 2的矩，各向同性缩放比例在缩放比例下保持在+/- 45％以内; 40 in.。我们还表明，多重分形FIF易于与数据兼容，而单一分形fBm和fLm不兼容。我们估计表征潜在FIF噪声的通用参数（alpha，C-1）为（1.79，0.12），其中alpha是多重分形的程度（0

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