بررسي نظريه فركتال در رودخانه زرينه‌رود با استفاده از روش شمارش جعبه‌اي

Fractal Analysis in Zarrineroud River Using Box-Counting Method

هندسه فركتال كاربردهاي بسياري در قلمرو علوم در دهه هاي اخير پيدا كرده است. استفاده از مدل هاي فركتالي در بررسي پديده­ هاي ژئومورفولوژيكي طي سال­ هاي اخير گسترش يافته است. يكي از پركاربردترين روش‌ها در مطالعه اين پديده ­ها، بعد فركتالي است. در اين مطالعه، تجزيه و تحليل فركتال براي حوضه رودخانه زرينه‌رود واقع در شمال غرب ايران با استفاده از روش شمارش جعبه ­اي انجام گرفت. براي دست­يابي به اين هدف، مطابق روش رودريگز و رينالدو، با استفاده از تصاوير ماهواره‌اي و نقشه‌هاي توپوگرافي، تعداد سلول‌هاي پوشش‌دهنده تمامي طول رودخانه زرينه‌رود به همراه انشعابات فرعي آن در هفت مقياس مختلف(25، 50، 100، 200، 400، 800 و 1600 متر) محاسبه و شبكه­ هاي رودخانه‌اي به كمك نرم افزار Arc GIS استخراج گرديد. بعد فركتالي محاسبه شده (1/04) به مقدار اقليدسي نزديك است. اين بدان معني است كه بعد فركتالي مشخصه الگوي پيچ‌وخم اين رودخانه را نشان مي‌دهد. در نتيجه، بعد فركتالي پايين رودخانه زرينه‌رود مي‌تواند وجود فرآيندهاي كنترلي تكتونيك روي تكامل الگوي زهكشي مورد مطالعه را نشان دهد. بنابراين دلايل مي‌توان نتيجه گرفت رودخانه مذكور از نوع خود الحاقي هست و شكل رودخانه به‌طور يكسان در تمام جهات منتشر نشده است و در سيماي كنوني شبكه زهكشي، شاخه‌هاي جانبي مراتب كمتر، مستقيماً به شاخه اصلي با بالاترين مرتبه ريخته مي‌شوند.

Introduction At first glance, river morphology seems complex, because two phenomena (in this case rivers) are not similar. According to Chaos theory, the river network geometry can be regulated using mathematical techniques. This pattern is applied for every structure that shows chaotic behavior. The study of nonlinear dynamics on earth surface processes and landscapes has been dominated by the application of analytical techniques derived largely from mathematics, statistics, physics, computational science, and other fields characterized by experimental laboratory techniques and numerical models. While much of this has been quite fruitful, geomorphology is dominantly (and appropriately) a field-based discipline where the ground truth is paramount. One powerful method is calculating the fractal dimension relative to landscape. Fractals are defined as geometric objects that are self-similar under a change of scale, i.e. their shape remains the same under any magnification or reduction. The fractal dimension characterizes the extent to which the fractal 'fills up' the embedding space and, in this example, will attain a value between 1 and 2. Generally, meanders are described in terms of radius of curvature, wavelength, and other parameters applied on simple geometric configurations. Numerical analysis principles of any drainage basin characteristics, is connected with the order concept. So the first step in the study of system nonlinear properties, is the combined analysis of channels, and consider them as the lines, are an equal level. In fractals, dimension is very important and measurement scales don't play a role as the effective factor. two kinds of fractals could be found: (a) self-affine fractals: the object changes when the scale varies whereas its fractal dimension does not change, and (b) self-similar fractal: the object is statistically identical to any scale. This property causes the application of fractal in different sciences. Determination of fractal dimension has a great importance in behavior recognition and the prediction of change in river trajectory. Furthermore, fractal dimension can provide information about branch1S length and sources of river that are not accessible. Matherials and Methods Zarrineroud River is located in the northwest of Iran, and it is one of the longest rivers in this region. It is 302 km long, arising from the mountains of Kurdistan Province, south of Saqqez, where it is also known as the Jaqtoo (Jaqalu) River. The river continues north and slightly west to the cities of ShahinDezh, Kashavar and Miandoab and pours into Lake Urmia. The structure observed in the demonstrations of chaotic systems does not seem to be a space filling nor a simple curve (a line). This complex geometry can be characterized by a non-integral dimension, and the structure is then called a fractal. The capacity or box counting dimension is a simple way of defining a dimension. It is related to the Hausdorff dimension, and is usually equal to this (and often assumed to be so in the context of dynamical systems), although there are counter examples. The construction is as follows. Suppose we have a set in an m-dimensional space. Imagine covering the space with equal size m-cubes of side; and count how many mcubes contain points in the set, say N. The capacity is defined as Fractal dimension is a useful concept in describing natural objects, which gives their degree of complexity. There are various closely related notions of fractional dimension. From the theoretical point of view, the most important are the Hausdorff dimension, the packing dimension. However, the box counting dimension is widely used in practice, which may be due to the ease of implementation. Satellite imagery and topographic maps, at 1:50000-scale, are used for calculations. The number of covering cells of Zarrineroud River are calculated at seven different scales (25, 50, 100, 200, 400, 800 and 1600 meters) and river systems are extracted by using Arc GIS program. Then the input of calculating two-dimensional fractal number was inserted to the software through correlation function. Regarding the fact that in analyzing fractal dimension through reducing scale more accurate detail can be achieved, for coastal lines, rivers or any two-dimensional phenomenon L=NS equation is used to generally estimate length (L). N, the number of cells, and S, the side-length, are required to measure that phenomenon. Therefore, the apparent length of a line, like a river, increases in nonlinear manner; however, the scale decreases in proportion to the measured pattern; that is as N increases, based on the rule of proportion, S decreases. Discussion First, according to Rodriguez-lturbe and Rinaldo's method in seven various scales, we count the covering cells by river, then from equation 1 acquire the box-counting dimension. In each step taken, the value of the dimension decreases. Thus, in order for the dimension value to reach values of an open interval (1, 2) more steps are required than for the other mentioned fractal dimensions. Since the fractal dime'nsion has a limitation of this logarithm value when the grid size is too small, we