The length‐scaling properties of topography
The length‐scaling properties of topography
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地形的长度尺度特性
DOI：
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发表时间：
1994
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通讯作者：
A. Malinverno
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作者：
J. Weissel;L. Pratson;A. Malinverno
通过分析合成地形表面和地形的数字高程模型（DEM）的“结构函数”，即，绝对高程差的q阶幂：Δhq（l）= E{|h（x + l）- h（x）|q}。我们发现Δ h_1（l）Δ H_（clH）关系很好地描述了以3角秒为网格的DEM所代表的自然地形表面的标度行为。标度指数H的平均值在0.5和0.7之间，表征了埃塞俄比亚、沙特阿拉伯和索马里的DEM在长度尺度l（0.1-150公里）的3个数量级范围内的特征。这三个区域的表观地形粗糙度的差异很可能反映了振幅因子c的差异。在x和y坐标方向上的缩放属性的单独确定允许我们评估缩放指数是否是方位角相关的（各向异性），或者它们是否是各向同性的，而表面本身在有限的长度尺度范围内是各向异性的。我们探索的方法来确定地形表面的特点是简单的或多尺度属性。DEM的标度指数Δh1（l）和Δh2（l）之间的差异很小，但为正，并且结构函数的标度指数的这种发散与多标度行为一致。分数布朗表面的超越和周长集未能产生平凡的分形维数预期的已知单分形集，这表明在分析中使用有限分辨率的数据集所产生的实际限制。通过比较“真实的”地形（表示为DEM）与分数布朗表面的高度，我们表明，基于高斯统计的合成表面是有限的自然地形模型。测高曲线可能反映了构造和侵蚀过程在地形塑造中的相对重要性，它清楚地表明，高于第二个统计矩的统计矩在描述地形表面时很重要。尺度分析是一个有价值的工具，用于评估地球地形的DEM表示的质量和准确性。
The scaling properties of synthetic topographic surfaces and digital elevation models (DEMs) of topography are examined by analyzing their “structure functions,” i.e., the qth order powers of the absolute elevation differences: Δhq(l) = E{|h(x + l) - h(x)|q}. We find that the relation Δh1(l) ≈clH describes well the scaling behavior of natural topographic surfaces, as represented by DEMs gridded at 3 arc sec. Average values of the scaling exponent H between ∼0.5 and 0.7 characterize DEMs from Ethiopia, Saudi Arabia, and Somalia over 3 orders of magnitude range in length scale l (∼0.1–150 km). Differences in apparent topographic roughness among the three areas most likely reflect differences in the amplitude factor c. Separate determination of scaling properties in the x and y coordinate directions allows us to assess whether scaling exponents are azimuthally dependent (anisotropic) or whether they are isotropic while the surface itself is anisotropic over a restricted range of length scale. We explore ways to determine whether topographic surfaces are characterized by simple or multiscaling properties. The difference between scaling exponents of Δh1(l) and Δh2(l) for the DEMs is small, but positive, and such divergence in the scaling exponents of the structure functions is consistent with multiscaling behavior. Exceedance and perimeter sets of fractional Brownian surfaces fail to yield the trivial fractal dimensions expected of sets of known monofractals, suggesting a practical limitation arising from the use of finite resolution data sets in the analysis. By comparing the hypsometry of “real” topography (represented as DEMs) with that of fractional Brownian surfaces, we show that synthetic surfaces based on Gaussian statistics are limited as models for natural topography. Hypsometric curves, which probably reflect the relative importance of tectonic and erosional processes in shaping topography, clearly show that statistical moments higher than the second are important in describing topographic surfaces. Scaling analysis is a valuable tool for assessing the quality and accuracy of DEM representations of the Earth's topography.