IMPLICATION OF FRACTAL DIMENSION ON PROPERTIES OF RIVERS AND RIVER BASINS

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1 INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND TECHNOLOGY (IJCIET) Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14) ISSN (Print) ISSN (Online) Volume 5, Issue 12, December (2014), pp IAEME: Journal Impact Factor (2014): (Calculated by GISI) IJCIET IAEME IMPLICATION OF FRACTAL DIMENSION ON PROPERTIES OF RIVERS AND RIVER BASINS Sowparnika M. 1, Dr. P.G. Jairaj 2 1, 2 (Department of Civil Engineering, College of Engineering, Trivandrum, India) ABSTRACT Fractals are geometric figures that are made up of patterns which repeat themselves at smaller scales. The concept of fractals is made use of in the determination of hydrological parameters of individual streams and stream networks. The present study deals with the estimation of fractal dimension of individual streams of seventeen rivers in Kerala. The scaling exponent of catchment area of three river basins in Kerala was also determined. The scaling behavior of drainage network within the drainage basin was also studied. The runoff from a river catchment depends mainly on catchment characteristics and these are generally estimated from topographical maps of specified scales. Thus, hydrologic modeling computations are scale dependent making fractal dimension useful in hydrologic studies to characterize geomorphologic measures of rivers and river basins. Keywords: Drainage Network, Fractal Dimension, Geographic Information System, River Morphometry, Scaling Behavior. 1. INTRODUCTION Euclidian or classical geometry consists of describing physical objects using lines, circles, ellipses etc. This type of geometry is appropriate for discussing man-made technological objects, and not for describing natural entities. Mandelbrot [1] developed fractal geometry the so called geometry of nature to describe the natural objects found in the nature. A fractal is a rough or fragmented geometric shape that can be subdivided into parts each of which is (at least approximately) a reduced size copy of the whole. Fractals are generally self similar and independent in scale. Fractal geometry is useful for describing irregular and fragmented patterns found in many disciplines. Fractal shapes are characterized by their details, the more the fractals are zoomed more properties of it are revealed. River and river networks are well known for scaling and many studies were conducted to study this scaling nature [2] [3]. The dimension of the fractals characterizes the scaling properties of rivers and indicates how an associated measure changes with the change in map scale. Geomorphologic measures like stream length, drainage density and slope are directly measured from maps, and hence those fractal dimensions have to be incorporated to characterize these measures. The scaling behavior of landscape morphological properties has been investigated using mono fractal and multi fractal analysis [4]. Hence a study on fractal dimensions of river networks and its influence on the characteristics of basin, which in turn affect the runoff computations from a basin, are very essential. The present study focuses mainly to determine the relationship of basin geomorphologic characteristics like mainstream length and drainage area, so as to find the fractal dimension for the watersheds. Moreover the space filling nature of stream networks was studied by computing the scaling exponents of catchment area. The study was conducted on the river basins of Kerala state, India. 155

2 2. METHODOLOGY The fractal dimension of individual stream is a measure of its irregularity; it is a measure of the extent of stream meanderings. The fractal dimension for a stream network is a measure of the ability of a network to fill a plane and it arises from the branching nature of the network and sinuosity of individual streams. If the stream network is truly space filling the fractal dimension of the stream network should be 2.0. But studies have shown that networks are not space filling at some level, the stream network stops and the hill slopes begin. Accordingly it is anticipated that the fractal dimension of a stream network is less than 2.0 and it is further acknowledged that the fractal dimensions may vary from one location to another. 2.1 Main stream The fractal dimension of individual stream is estimated by calculating the main stream length from different scale maps [5]. This main stream length is plotted against the scale ratio in a log- log graph Log L = log K + (1-d) log E (1) L= K E (1-d) (2) Where, L is the measured length, E is the scale ratio and d is the fractal dimension 2.2 Catchment Area The catchment area of river basin was determined from maps of two different scales. The drainage network for the basin was determined using DEM and the total network length was obtained. The scaling properties of catchments were examined using modified form of Hack s law. Here, network length is related to catchment area by the following relation: γ l = ka logl = log k + γ log A Where, l is the Network length, γ is the Scaling exponent and A is the Catchment area. A log scale plot of the network length along the Y axis and catchment area along X axis results in a straight line fitting. The slope of the line gives the scaling exponent for the catchment area. From the scaling exponent, the space filling nature of catchment area was studied. Using the maximal length of basin measured from the outlet and maximal width of basin orthogonal to maximal length, the Hurst exponent was determined using Equation 5. w H = (5) Where, H is the Hurst exponent, l t is the Maximal length of basin measured from the outlet and w is the Maximal width of basin orthogonal tol t. The scaling behavior of catchment area can be determined using the Hurst exponent. If H= 1, then scaling is said to be self similar and if H< 1, then scaling is said to be self affine. 2.3 Scaling of Drainage network within a river basin The implication of scaling on drainage network at sub basin level was studied using the law proposed by Mandelbrot [1]. According to this law, the main river length (L) and total drainage length (l) of drainage network within a sub watershed is related to its sub watershed area (A) as given in Equation 6 and 7. l t (3) (4) α l A γ L A (6) (7) Where, L is the main river length, l is the Drainage network length, A is the Sub basin Area, γ, α are the Scaling exponents. Scaling exponent α was obtained from the slope of log-log plot between stream network length and catchment sub basin area. Scaling exponent γ was obtained from the slope of log-log plot between main river length and catchment sub basin area. Based on the values obtained, the implication of scaling on the drainage network within a river basin in the sub basin level was studied. 156

3 3. ANALYSIS OF THE PROBLEM The values of the fractal dimension for individual streams were estimated for seventeen rivers in Kerala. The scaling of catchment area of three river basins in South Kerala was also studied by determining the scaling exponent and Hurst exponent. A study on effect of scaling on catchment properties of a river basin in North Kerala was also done. 3.1 Main stream For the determination of fractal dimension of individual streams, the main stream length of seventeen rivers in Kerala was extracted from the digitized maps in two scales (1 in 50,000 and 1 in 2, 50,000). This main stream length of each river computed from two maps scales are used for the computation of fractal dimension of individual streams. The rivers selected for the study are: Vamanapuram, Karamana, Neyyar, Kallada, Achenkovil, Pamba, Manimala, Meenachil, Muvattupuzha, Periyar, Chalakudy, Karuvannor, Bharathapuzha, Chaliyar, Kuttiyadi, Mahe and Kadalundi. For each river the measured length was plotted against scale ratio on a log-log scale. The slope of the straight line thus fitted was calculated and the fractal dimension of individual rivers was obtained using Eqn.2. The measured rivers length of the seventeen rivers of Kerala obtained from the maps of scales 1 in 50,000 and 1 in 250,000 are tabulated as shown in the Table 1. The map showing the seventeen rivers selected is shown in Fig. 1. The influence of map scale on measured length of rivers in Kerala is illustrated in Fig. 2. Table 1: Main stream length of 17 Rivers in Kerala in two map scales Rivers Length (km) 1 in 50,000 1 in 2,50,000 Vamanapuram Meenachil Karuvannur Manimala Neyyar Achankovil Karamana Chalakudy Muvatupuzha Pamba Periyar Kadalundi Kuttiyadi Mahe Bharathapuzha Chaliyar Kallada

4 Fig 1: Map showing the rivers in Kerala selected for the study Fig 2: Influence of map scale on measured length of 17 rivers in Kerala 3.2 Catchment Area To estimate of scaling exponent of catchment area, the catchment area was digitized from SOI toposheets of two different scales (1:50, 000 and 1:2, 50, 000) using ArcGIS 9.1 software. The three river basins used in the study are Vamanapuram, Karamana and Neyyar. The Vamanapuram river basin lies between to N latitudes and to E longitudes. The Karamana basin lies between 8 o N and 8 o N latitudes and 76 o E and 77 o 44 E longitudes. The Neyyar river basin is located between latitudes 8 o 30 N to 8 o 40 N and longitudes 77 o 8 E to 77 o 20 E. Using SRTM data, the DEM of the catchment area was extracted by masking using the digitized boundary in both scales. The stream network for each river basin was determined with Arc Hydro tool in Arc GIS 9.1. After deriving stream network, the total network length was obtained from the attribute table. A visual comparison of the delineated stream network with SOI toposheet was done and it was found that the stream network up 158

5 to third order was accurately determined. The network length was plotted against catchment area in a log scale plot to obtain a straight line. The slope of this graph gives the scaling exponent. A summary of catchment area, total network length in both scales is given in Table 2. The digitized boundary of Vamanapuram river basin and derived stream network is shown in Fig. 3. NAME Table 2: Summary of catchment area, network length for the 3 river basins Catchment Catchment area in area in Network length 1:50,000 1:2,50,000 in 1:50,000 Network length in 1:2,50,000 Vamanapuram Karamana Neyyar (a) (b) Fig. 3: Catchment area of Vamanapuram River in scales (a) 1:50,000 and (b) 1:2, 50,

6 The log scale plot of total network length and catchment area for Vamanapuram river basin is shown in Fig. 4. The maximal length of basin from the outlet and the maximal width perpendicular to the length measured is given in Table 3. The slope of the straight line obtained gives the scaling exponent for the catchment area of river basins. Using the maximal length of basin measured from outlet and maximal width of basin orthogonal to maximal length, the Hurst exponent is determined. Fig. 4: Log-log plot of catchment area and network length for Vamanapuram River Table 3: Summary of maximal length and maximal width for three river basins Name Maximal length(km) Maximal width (km) Vamanapuram Karamana Neyyar Scaling of Drainage network within a river basin For studying the scaling of drainage network within the catchment area, the stream network was delineated from digital elevation model (DEM). The river basin selected for this study is Chaliyar river basin located in latitude 10 o 05 N to 10 o 35 N and longitude 76 o 15 E to 76 o 55 E. The stream network is extracted using Arc Hydro tool in Arc GIS 9.1. The drainage network extracted using Arc Hydro tools is shown in Fig. 5. A visual comparison of the delineated stream network with that of toposheet was done and it was found that the stream network up to third order has been accurately determined. After obtaining the drainage network, the sub watersheds were delineated up to third order as shown in Fig. 6 in Arc GIS 9.1. Fig. 5: Drainage network of Chaliyar river basin Fig. 6: Sub watersheds up to third order for Chaliyar river basin 160

7 The drainage network within each sub-watershed is extracted using intersect tool in Arc GIS 9.1. This is done for sub-watersheds of all orders from sixth to third order. The network length is obtained from the attribute table. Next, the mainstream length within each sub-watershed was determined, i.e. the third order stream length within third order sub-watershed, fourth order stream length within fourth order sub-watershed etc. The maximal length of basin measured from outlet and maximal width of basin orthogonal to maximal length was also determined. From this, the Hurst exponent was also determined to study the scaling nature of Chaliyar river basin. After determining the network lengths and mainstream lengths, a log scale graph was plotted. The graph between sub-watershed network length and sub-watershed area is shown in Fig.7. The graph between sub-watershed mainstream length and sub-watershed area is shown in Fig. 8. Scaling exponent α was obtained from the slope of log-log plot between stream network length and catchment sub basin area. Scaling exponent γ was obtained from the slope of log-log plot between main river length and catchment sub basin area. Fig. 7: Log-log plot of sub watershed area and network length Fig. 8: Log-log plot of sub watershed area and sub watershed main river length 161

8 4. RESULTS AND DISCUSSION The estimation of fractal dimension for the main channel stream length and for the river networks of Kerala was done. Similarly, the scaling behavior of catchment area was also studied. The implication of scaling on drainage network of a river basin at sub-watershed level was also studied. 4.1 Main streams For each main stream the estimated stream length obtained from the maps of scale 1 in 50,000 and 1 in 2,50,000 were plotted. The slope of the resulting straight lines fitted for each main stream, is found out and the fractal dimension of main channel length of each river is derived as in Table 4. Majority of the rivers in Kerala is found to exhibit fractal nature. From the table it can be seen that fractal dimension of 17 rivers in Kerala is in the range of 1 to 1.24 with an average value of The variations in the computed values maybe due to the fact that fractal dimension vary from one location to another. Table 4: Estimated fractal dimensions of rivers in Kerala Rivers Fractal Dimension (d) Vamanapuram 1.07 Karamana Neyyar 1.09 Achenkovil 1.08 Bharathapuzha 1.1 Chalakudy Chaliyar 1 Mahe 1.07 Karuvannor 1.09 Pamba Periyar Kallada Meenachil 1.1 Manimala 1.01 Muvattupuzha Kadalundi 1.1 Kuttiyadi 1.1 Also, lower fractal dimension indicates that the object is less complex. A higher value for fractal dimension indicates that the object is more complex or irregular. From this study, Mandelbrot s suggestion that rivers are fractals is also substantiated. Chaliyar River in North Kerala has a fractal dimension one, which indicates that it does not have a fractal nature. 4.2 Catchment Area The scaling effect on catchment area can be determined from scaling exponent and Hurst exponent determined. A summary of scaling exponents and Hurst exponents for the three river basins is given in Table 5. From the scaling exponents determined for the catchment area, the space filling nature of stream network is determined and from the Hurst 162

9 exponent, the scaling behavior of catchment area is determined. From the computed values of scaling exponents, it can be seen that the stream networks are not space filling. From the estimated values of Hurst exponent, the scaling behavior of the river basins is identified as self affine i.e. there is a change in shape of catchment with scale. Table 5: Estimated values of scaling exponents and Hurst exponents River Scaling exponent Hurst exponent Vamanapuram Karamana Neyyar Scaling of Drainage network within a river basin From the log scale plot of network length and sub-watershed area, the scaling exponent α is determined. The equation of the straight line fitted is given in eqn. 8 L=1.57A (8) The scaling exponent obtained is which approaches unity. From the plot of total network length against catchment area, the expected value of α is near unity (Nikora et al., 1996). Good correlation is also shown between the network length and sub-watershed area with R 2 value From the log scale plot of main river length and subwatershed area, the scaling exponent γ is determined. The equation of the straight line fitted is given in eqn. 9. L=1.12A (9) The scaling exponent obtained is Correlation between main stream length and sub-watershed area is low with R 2 value of The low correlation maybe attributed to the geomorphologic characteristics of the river basin as it plays a major role in the evolution of drainage network. The scaling of main river length with sub basin area needs to be investigated further around this area to characterize such a behavior. 5. CONCLUSION The concept of fractals and fractal dimensions are used in hydrology, which characterizes the geomorphologic measures of rivers and stream network. A study of fractal nature of rivers in Kerala was conducted and the fractal dimension of mainstream length of rivers is determined. The scaling behavior of catchment area was also studied. The scaling of drainage network at sub watershed level was also studied for a river basin. From this study, it was found that most of the rivers in Kerala exhibit fractal nature. The fractal dimension of mainstream length of rivers in Kerala was determined and the values vary from 1 to 1.1 with an average value of Central Kerala rivers like Kadalundi, Kuttiyadi, Bharathapuzha and Meenachil are found to be fractally similar. South Kerala rivers like Vamanapuram, Achenkovil, Neyyar are also found to be fractally similar.central Kerala rivers show lesser fractal nature as they have lower fractal dimension. The scaling exponent and Hurst exponent of catchment area of three river basins in Kerala was determined and scaling behavior studied. All the river basins were found to be of self affine nature i.e. it has directional scaling. The stream network is also not found to be space filling. The scaling of drainage network within a river basin up to third order was also studied. The value of scaling exponent obtained on plotting network length and sub-watershed area is near unity and the result shows good correlation. Thus, the network length and catchment area vary linearly and shows scaling nature. The scaling exponent obtained on plotting main river length and sub-watershed area is and does not show good correlation. Thus, the fractal nature of rivers and river basins in Kerala was studied. Many hydrologic modeling computations are scale dependent. In hydrologic modeling the catchment response is a function of geomorphologic 163

10 measures like stream length, slope, drainage density etc. which are directly measured from the maps, as maps are well known for their scaling properties fractal dimension of individual steam and stream network can be used to characterize this scaling properties. Hence, a study on scaling behavior is useful in hydrological studies to characterize geomorphologic measures of rivers and river basins. REFERENCES [1] Beniot B. Mandelbrot, Fractals and the geometry of nature (Yearbook of science and the future, Encyclopedia Britannica, Inc., Chicago, 1981) [2] Tanzhuo Liu, Fractal structure and properties of stream networks, Water Resources Research, 27(12), 1992, [3] Tarboton, D.G., Bras, R.L., and Rodriquez-Iturbe, I., The fractal nature of river networks, Water Resources Research, 24(8), 1988, [4] Ude Shankar, Charles P. Pearson, Vladimir I. Nikora, Richard P. Ibbitt, Heterogeneity In catchment properties: a case study of Grey and Buller catchments, Hydrology and Earth System Sciences, 6(2), 2002, [5] Hjelmfeit, A.T., Fractals and the river-length catchment-area ratio, Water Resour. Bull., 24(2), 1988, [6] Dr. Ghassanal-Adeem Al-Dulaimi, Evaluation of Runoff Depth for Al-Adeem River Basin by using Remote Sensing Technique and GIS Integration, International Journal of Civil Engineering & Technology (IJCIET), Volume 4, Issue 6, 2013, pp , ISSN Print: , ISSN Online: