Investigating multi-scale and multi-fractal topographic properties of Iran

  Geomorphology is de_ned as the science of Earth's diverse physical landforms with an emphasis on their origin and distribution across the landscape as well as the dynamic processes that shape the topographic features . Enhancing the uptake of geomorphic understanding and its underlying processes play a key role in understanding of physical geography, as one of the major research challenges in geography Fractal and multifractal analysis of topography has long been a very useful method to obtain synthetic topography in geology and geography which has led to a variety of di_erent results. The concept of fractals was rst introduced by Mandelbrot [1967] as a measure of the Earth's topography, the length of a rocky coast line [Mandelbrot,1982] The fractal dimension is a measure of global property of the system in question and in many cases is independent of various details and so it is used to classify di_erent systems and models in terms of the value of the fractal dimension besides other statistical measures.
However, in various studies in the past (see e.g., Lovejoy and Schertzer (1990); Lavallee et _ al. (1993)), it has been noti_ed that in some cases the topography is not far than simple to be explained with a single scaling exponent such as the fractal dimension, it is rather more appropriate to study topography as a scale invariant quantity that generally requires multifractal measures and exponent functions. This gives rise to an in_nity of fractal dimensions for di_erent statistical moments of a variable are needed to completely characterize the scaling properties.
There exist a few multifractal studies of topography showing the multiscale properties of the height pro_le over various ranges in length scale (see for example Lovejoy and Schertzer (1990); Lavallee et al. _ (1993); Weissel et al. (1994); Lovejoy et al. (1995); Pecknold et al. (1997); Tchiguirinskaia et al. (2000); Gagnon et al. (2003), J.-S. Gagnon et al. (2006). A similar mono vs multifractal study exists for the arti_cial growth surface models (e.g., Morel et al., 2000). The growth models are studied in both context of monofractality and multifractality (Bouchaud et al., 1993; Schmittbuhl et al., 1995).
In this Section we use the box-counting method to estimate the fractal dimension of the height pro_le in Iran. In this method we consider the Iran's topography shown in Fig. 1. We used the Iran's topography grid data with high resolution on a square lattice of size 1500 _ 1320.
The heights fhi;jg are known at each grid node (i; j) distributed between the minimum height hmin = -4398 m and the maximum height hmax = 5149 m. In the box-counting method, we consider a cube of size 500 1320 9547 which covers all the height topography.
The mesh sizes are scaled to unity. At each grid point (i; j), we consider a height column of size 1 1 hi;j with hmin _ hi;j _ hmax. We cover the entire cube with boxes of size 1 1 b, for di_erent length scales b, and then count how many boxes of the grid i.e, Nb, are covering part of the height columns. Then we do the same thing by using a _ner grid with smaller boxes (smaller b). By shrinking the size of the grid repeatedly, we end up more accurately capturing the structure of the pattern of the topography.
If the the topography has a fractal structure, then one would expect the following power-law relation Using the box counting method, the fractal dimension Df of the height pro_le is thus given by the slope of the line when we plot the value of log(Nb) on the y-axis against the value of log(b) on the x-axis i.e., (6)As shown in FIG. 2, we _nd that the fractal dimension of the Iran's topography is Df = 2:20(1).
Result and discuion:We find that _(2) = 0:99(0) which gives the roughness exponent _ = _(2)=2 = 0:495(5) in accord with our previous estimates obtained from di_erent methods like the power spectrum analysis and two methods mentioned in Subsections IVA and IV B. We also _nd the higher moment exponents _(3) = 1:39(1) and _(4) = 1:53(1). As mentioned above, for self-a_ne surface one should _nd _(q) = q_ or equivalently, _(q)=q = _. If we test this criteria for Iran's topography, we _nd that _(3)=3 = 0:46(1) and _(4)=4 = 0:38(1) which di_er from our estimate for the roughness exponent _ = 0:495(5). This completes our conclusion that the height pro_le of Iran's topography has a multifractal statistics.