Indian Journal of Marine Sciences Vol. 36(2), June 2007, pp. 162-166 Fractal dimensions of selected coastal water bodies in Kerala, SW coast of India - A case study *Srikumar Chattopadhyay & S. Suresh Kumar Centre for Earth Science Studies, Trivandrum-695 031, India *(E-mail: radresource@vsnl.com) Received 11 October, 2006, revised 9 March 2007 Fractal dimension (D) of shorelines of four coastal water bodies, namely Paravoor, Ashtamudi, Kayamkulam and Vembanad have been worked out applying three methods. The D values ranged from 1.09 to 1.40. Edge of Ashtamudi, which is genetically different from the rest, is rougher compared to other water bodies as is evident from the higher D values recorded by this water body. The method of log N-log G plots appeared to be the most suitable for this study. Coast perpendicular water body surrounded by laterites like Asthamudi estuary could be well separated from the rest based on fractal dimension. It is suggested that fractal dimensions may be used for classifying coastal water bodies as linked to their genesis. [Key words: Fractal dimension, SW coast of india, coastal water body, Kerala, log N-log G plot] Introduction Terming something fractal implies a distinctive relationship among structures observed in different scales although the fractal is scale invariant. Fractal dimension is a measure of roughness of geographic features or it is the descriptor of the complexity of surfaces. Earth s topography evolved through tectonic activities modified by erosion and sedimentation. It is complex, chaotic and yet there is orderliness as shown by the applicability of fractal statistics 1. Fractal concepts are increasingly being applied in earth science research 1-3. Since 1980s, there were studies demonstrating application of fractals in geomorphology 4. There were studies on fractal dimension of various geographic features like coastline, drainage basin perimeters, sink holes etc 5-7. These studies demonstrated that many relations in geomorphology had a fractal nature. Coastal water bodies in Kerala are either coast parallel or coast perpendicular. This orientation is partly linked to their genesis. The coast parallel water bodies are situated in the coastal alluvium (Recent sediment) belt and coast perpendicular water bodies are foundered river mouths mostly encircled by laterite surfaces. The present study is one of the trial *Corresponding Author: Ph.: + 471 2442451-54 Fax: + 471 2442280 efforts to work out fractal dimension of perimeters/shorelines of four selected coastal water bodies in Kerala. One of these four water bodies is completely surrounded by laterites. The objectives of this paper are (i) to examine the scope of using fractal dimension to classify the coastal water bodies as linked to their genesis and (ii) to select a suitable method for calculating fractal dimension based on the results obtained from this analysis. So far, there had been no such attempt for coastal waters of Kerala (south west coast of India). Materials and Methods Kerala is located in the south western corner of Indian peninsula. More than 60% of Kerala s geographical area is covered by laterites. The narrow coastal plain of Kerala (Fig. 1) is characterized by a series of estuaries and lagoons. These water bodies cover 8 an area of around 2426 km 2. Four water bodies, namely, Vembanad-part (area-160 km 2 ), Kayamkulam (20 km 2 ), Ashtamudi (54 km 2 ), and Paravoor (5 km 2 ), were taken up for the analysis. The Vembanad, running parallel to the coast, is the largest lagoon in the west coast. It s formation is linked to the emergence of barrier bar, an unique feature of Kerala coast, and transgression and regression of sea. Conversely, the Ashtamudi estuary is a foundered river mouth controlled by horst and graben structure.
CHATTOPADHYAY & SURESH KUMAR: FRACTAL DIMENSION 163 Fig. 1 Study area and outline of four coastal water bodies
164 INDIAN J. MAR. SCI., VOL. 36, NO. 2, JUNE 2007 Kayamkulam is similar to the Vembanad. Paravoor is characteristically closer to the Ashtamudi, however this water body has a coast parallel northern arm also. The divider method based on a technique of measuring length with a fixed step length (G) and counting the number of steps (N) taken to cover the entire length is widely used in geomorphology 9, 10. The measured length L (G) is equal to N * G. The fractal dimension D of a line can be derived from the equation 5 : L(G) = MG 1-D (1) where L(G)-the estimated length of line measured with a unit length of G (step length), M-positive constant, and D = fractal dimension. D must exceed 1 for a line to be fractal. This indicates that as step length G decreases the measured length L(G) increases as a power function of G. The complexity of the line increases with larger value of D and when D value approaches 2 the line becomes so complex that it fills the plane 10. With only two step lengths D can be estimated using the equation: D = log (N 2 / N 1 ) / log (G 1 /G 2 ) (2) where N1 and N2 are the number of steps of lengths G1 and G2 needed to trail the line, and G 1 > G 2. When the step lengths are more than two, D is estimated by a commonly used method known as Richardson plot in which L(G) is plotted against G in a log-log paper and a best fit line is drawn through the scatter of points in case of linear distribution. D = 1 - slope of the regression line. Another method to estimate D, when more than two step lengths are used, is to plot log N against log G. If the plot appears linear, a best fit line is drawn through the scatter. In this case D is equal to the absolute value of the slope of the regression line. At the present instance, we followed all three methods discussed above and worked out the fractal dimensions of four brackish water bodies in the coastal belt of Kerala. The rationale behind applying all three methods is to compare the results and suggest the appropriate one for undertaking similar studies. Perimeters of all four water bodies considered for this study were drawn from the survey of India topographical maps in 1:50,000 scale (Fig. 1). The step lengths used were 250 m, 500 m, 750 m and 1000 m. Results and Discussion Fractal dimension (D) of perimeters of all four water bodies calculated based on two-step length method showed that the D values were 1.01, 1.03, 1.17 and 1.37 for the Vembanad, Paravoor, Kayamkulam and Ashtamudi respectively. The D values exceeded 1 in all four cases indicating fractal nature 10. However, in case of the Vembanad and Paravoor the values are only marginally above 1 and therefore calling them fractal may not be very appropriate. In view of this observation we applied other two methods to work out D values for examining fractal nature of the perimeters. Following Andrle 10 both log N-log G and Richardson plots [log L(G)-log G] have been constructed for each of the four water bodies. All four log N-log G plots appeared linear (Fig. 2). Therefore, least square regression analyses were performed (Table 1). The high R 2 (coefficient of determination) values (0.99) of the regressions seemed to indicate that the log N-log G relations were indeed linear and hence the lines were statistically self-similar. Table 1 Regression equations for all four water bodies Waterbody Regression equation R 2 D value Paravoor log N = 4.69-1.15G 0.996 1.15 log L(G) = 4.63-0.14G 0.851 1.14 Ashtamudi log N = 6.00-1.40G 0.991 1.40 log L(G) = 6.00-0.40 G 0.891 1.40 Kayamkulam log N = 5.18-1.19G 0.999 1.19 log L(G) = 5.14-0.18G 0.972 1.18 Vembanad log N = 5.32-1.10G 0.998 1.10 log L (G) = 5.19-0.09G 0.701 1.09 Fig. 2 Regression plot of Log N against Log G: A): Y = -1.1549 X + 4.6878, R 2 = 0.9964, B): Y = -1.1858X + 5.1774, R 2 = 0.9999 C): Y = -1.3987 X + 5.9969, R 2 = 0.9911, D): Y = -1.1013x + 5.3231, R 2 = 0.9976
CHATTOPADHYAY & SURESH KUMAR: FRACTAL DIMENSION 165 Table 2 Perimeters of the water bodies corresponding to different step lengths (Values in metres) Step length Paravoor Ashtamudi Kayamkulam Vembanad 250 20,250 110,250 53,000 121,250 500 17,500 87,000 48,000 106,500 750 16,500 63,000 42,750 105,750 1000 17,000 68,000 42,000 108,000 Fig. 3 Regression plot of Log L(G) against Log G. A): Y = -0.0906 X + 5.2905, R 2 = 0.7007, B): Y = -0.1755 X + 5.1442, R 2 = 0.9716, C): Y = -0.3993 X + 5.9983, R 2 = 0.8912, D): Y = -0.1394 X + 4.6315, R 2 = 0.8513 In case of Richardson plot, the trend was also linear for all four cases with some variations (Fig. 3). The R 2 values here are lower than those under the log N-log G plot for all the water bodies and it is lowest (0.7) for the Vembanad. It is observed that the Vembanad and the Ashtamudi recorded the lowest and the highest D value respectively under all three methods. The origin of Ashtamudi is distinctively different from the other three water bodies. Paravoor, also a foundered mouth has a northern arm developed by barrier bar formation. This water body is of mixed origin. The Vembanad, Kayamkulam and northern arm of Paravoor are confined to recent sediments dominated by sand and clay material and therefore their perimeters/ shorelines are not so complex like Ashtamudi, which is surrounded by laterite cliffs, a result of block faulting. In case of Paravoor, D value increases from 1.03 to 1.13 when northern arm confined to the sandy belt is excluded from calculation in the two step method. It may be inferred that coast perpendicular water bodies have a tendency to register high D values. These water bodies are surrounded by laterites. Therefore an apparent relationship between roughness and material composition of shorelines of water bodies is evident. These observations illustrate the importance of studying fractal dimension of coastal water bodies. Perimeters corresponding to different step lengths provided in Table 2 indicate that in general, perimeter increases with decrease in step lengths as is usual for a feature to be fractal. However except Kayamkulam in all other three cases perimeters decreased when step lengths were reduced from 1000 m to 750 m. Perimeters increased again for the step lengths of 500 m and 250 m. Lower R 2 values corresponding to three water bodies (Paravoor, Ashtamudi and Vembanad) in case of Richardson s plot may be attributed to this distribution. Conclusion Perimeters/shorelines of coastal water bodies, irrespective of their size, appear to be of fractal nature. The D value ranges from 1.09 to 1.40. The Ashtamudi estuary has the rougher edges compared to other three as indicated by higher D value. Geologic factors might have contributed in this context. Although data set is smaller, there is a clear trend. It is suggested that the possibility of utilizing fractal dimension in differentiating coastal water bodies according to their genesis may be explored. There is a need to use larger data set, large scale map and image. The base map used for the present analysis is 1:50,000 scale topographical maps. This scale might have influenced the results. Analyses of these features from large scale maps (say 1:25,000/1:10,000) can possibly improve some of the observations. The study can be extended to other parts of west coast covering more number of water bodies. The numerical value of fractal dimension (D) may be the most important single parameter of an irregular cartographic feature like shoreline of a water body. These values can be used to characterize water bodies irrespective of their size and classify them according to the roughness of their perimeters/shorelines and finally link these observations to their geomorphic evolution. This enhances the scope of using fractal dimension in studying coastal geomorphology. Regarding methodology, the analyses based on log N-log G plot method and Richardson plot method yielded better results compared to that obtained by using two step method. However, in view of the R 2 value the log N-log G plot method appears to be more appropriate for the present study.
166 INDIAN J. MAR. SCI., VOL. 36, NO. 2, JUNE 2007 Acknowledgement Thanks are due to Dr. M. Baba, Director, CESS and Dr. K. Soman, Head, RAD for extending facilities, and Lakshmy, C. S., Neetha Bhubanan and Sachin helped in data processing in computer and drawing maps. References 1 Turcotte D L & Huang J, Fractal distributions in geology, scale invariance and deterministic chaos, In Fractals in the earth sciences, edited by Christopher C Barton and Paul R La Pointe (Plenum Press, New York) 1995, pp. 1-40. 2 Dimri V P (ed), Application of fractals in earth sciences, (A A Balkema, USA/Oxford and IBH Publishing Co., New Delhi), 2000, pp. 242 3 Vedanti N & Dimri V P, Fractal behavior of electrical properties in oceanic and continental crust, Indian J Mar Sci, 32 (2003) 273-278. 4 Snow R S & Mayer L, Introduction, Special Issue: Fractals in geomorphology, Geomorphology, 5 (1992) 1-4. 5 Mandelbrot B B, How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science, 156 (1967) 636-638. 6 Breyer S P & Snow R S, Drainage basin parameters: a fractal significance, Geomorphology, 5 (1992) 143-158. 7 Reams M W, Fractal dimensions of sinkholes, Geomorphology, 5 (1992) 159-166. 8 Kerala State Landuse Board, Land Resources of Kerala State, (Government of Kerala, Thiruvananthapuram) 1995, pp. 248-249 9 Goodchild M F & Mark D M, The fractal nature of geographic phenomena, Annals Asso Am Geog, 77 (1987) 265-278. 10 Andrele R, Estimating fractal dimension with divider method in geomorphology, Geomorphology, 5 (1992) 31-142.