Chapter 3 Box counting method for fault system 3.1 Introduction The Himalayan Range with the lofty mass on earth is characterized by very complex structural elements that play a vital role in determining the seismicity of the region in response to the northward movement of the Indian Plate impinging on Asian mainland. The geologically complex and tectonically active central seismic gap of Himalaya is characterized by unevenly distributed seismicity. The highly complex geology with equally intricate structural elements of Himalaya offer an almost insurmountable challenge to estimating seismogenic hazard using conventional methods of Physics. Here we apply integrated unconventional hazard mapping approach of the fractal analysis of box counting fractal dimension of structural elements to quantify the distribution of structural elements of the region in order to understand the seismogenesis of the region properly. Several researchers (Korvin 1992; Preuss 1995; Turcotte 1997; Ram and Roy 2005; Roy et al., 2012b) have applied the box counting method of fractal dimension or capacity dimension (D 0 ) for understanding the distribution of structural elements (faults, thrusts, lineaments etc.) in different parts of the earth. Earth s crust is complex with faults present at all scales and orientations. The faults play an important role in releasing the accumulated energy in the form of Earthquakes. The study of fault systems is necessary for understanding the earthquakes occurrences. Moreover, the active faults are considered to be the source for earthquakes in the seismically active zones of the world. Their identification bears significant importance towards recognizing the seismic hazard of such zones. There is extensive evidence that characteristic earthquakes do occur quasiperiodically on major faults. Many studies have been carried out to quantify the recurrence time statistics of these characteristic earthquakes (Utsu, 1984; Ogata, 1999; Rikitake, 1982). Faults and spatial distribution of earthquakes epicentres, statistically obey a power law distribution which can be quantified by a fractal dimension. The 29
distribution of fault system is studied with the help of box counting method by obtaining capacity dimension (D 0 ), which quantifies in terms of minimum or maximum coverage of structural elements in a region (Gonzato et al., 1998; Ram and Roy 2005; Chauveau et al., 2010; Roy et al 2012b.). Several researchers (Korvin 1992; Turcotte 1997; Sunmonu and Dimri 2000; Thingbaijam et al., 2008) have applied the method for understanding the scaling nature of the fault systems in different parts of the world. Major structural and tectonic elements exposed in and around the study region that are used for analysis are Indus Suture, Karakoram Fault, Main Central Thrust (MCT), Main Boundary Thrust (MBT), Main Frontal Thrust (MFT), Jwala Mukhi Thrust, Drang Thrust, Sundar Nagar Fault, Kaurik Fault system, Alaknanda Fault, Martoli Thrust, Mahendragarh Dehradun Fault, North Ramgarh Thrust, South Ramgarh Thrust, South Almora Thrust, North Almora Thrust, Moradabad Fault, and Great Boundary Fault (Joshi 1999; Dasgupta et al. 2000). The study region has experienced copious earthquakes in past and hence led to the huge destruction. Episodic events occurrence in the study region are well explained geotectonically in previous Chapter 2 (section 2.4 tectonic setting) with substantial evidences. Moreover it has been noticed that one of the events is associated with severe landslides as reported by (Sarkar, et al.2004). The main reasons for these are explained by various workers (Sarkar et al, 2004 and Nawani, 2009). The 1991 Uttarkashi earthquake has shaken many parts of Uttarkashi district. This quake has induced instability nearby specifically the Varunavat Parvat. This south-facing slope of Varunavat Parvat was in an unstable condition due to the Tambakhani slide and the landslide scarp at the hilltop on the right side of the Tambakhani slide. The cracks resent in the uphill slope further forced the instability of the hill slope. The study suggests that the scarp of Tambakhani slide was enlarging easterly and was finally merging with the existing scarp on the eastern side. Anthropogenic activities at the toe of the hill slope have resulted in loosening of the rock mass and increase in slope gradient. Hence these activities have also disturbed the stability of the slopes by increasing the shear stress. All these factors might have shifted the stability of the hill from a marginally stable to an unstable condition. This might have been the cause of severe landslide in Uttarkashi in September 2003. So the 30
consequence of an event may lead to destruction after many years. The application of fractal analysis for fault system in the region is done for understanding the future earthquakes occurrence. The fault system is analyzed with the help of the box-counting method which is a popular tool of fractal analysis. This method was earlier used by Hirata (1989b) for fault systems in Japan; by Idziak and Teper (1996) to study fractal dimensions of fault networks in the upper Silesian coal basin, Poland; Angulo-Brown et al. (1998) studied the distribution of faults, fractures and lineaments over a region on the western coast of the Guemero state in southern Mexico. A similar technique was used by Okubo and Aki (1987) to study the fractal geometry of the San Andreas Fault system; by Sukmono et al. (1994, 1997) to study the fractal geometry of the Sumatra fault system. Sunmonu and Dimri (2000) studied the fractal geometry and seismicity of Koyna-Warna, India by using the same technique. Ram and Roy (2005) used the technique for the Bhuj Earthquake analysis. Thingbaijam et al. (2008) applied the method for determination of capacity dimension (D 0 ) for the fault system. Gonzato et al. (1998) and Gonzato (1998) have developed computer programs to evaluate fractal dimension through the box-counting method, which potentially solves the problem of the process of counting. The main disadvantage of the program is that it deals directly with the images. Joshi and Rai (2003a) used fractal geometry for appraisal of inverted Himalayan metamorphism and Joshi and Rai (2003b) demonstrated fractal dimension as a new measure of neotectonics activity. The technique has also been widely used in various fields for characterization of specific mineral occurrence in nature. Chauveau et al. (2010) used capacity dimension and correlation dimension for the study of colour images. Recently, researchers (Ford and Blenkinsop 2008; Raines 2008; and Carranza 2008 & 2009) applied the same method in order to characterize spatial pattern of occurrences of mineral deposits. 3.2 Fractals in Geological Structural system A fractal distribution requires that the number of objects larger than a specified size has a power-law dependence on the size, ( r) 1 D 0 N (3.1) r 31
where N(r) is the number of objects (i.e., fragments) with a characteristic linear dimension r (specified size) and D 0, the fractal dimension. Equation (3.1) can be written as, ( r) = c (3.2) N r D 0 where c is the constant of proportion and D 0 is the fractal dimension which gives N(r) a finite value, otherwise with the decrease of size r in equation (3.1) the value would have reached infinite. In case of box counting method, N(r) represents the number of occupied boxes and r is the length of the square box side. Hirata 1989b used box counting (e.g. Turcotte 1989, 1997) to cover the fault traces. The fractal dimension of the fault system can be determined by the boxcounting method (Fig. 3.1). The procedure is repeated for different values of r and the results are plotted in a log log graph. As depicted in Fig. 3.2(a) & Fig. 3.2(b) the slope values so obtained are inverted to get the capacity fractal dimension (D 0 ) as mentioned in equation (3.3) and listed in Table 3.1 (Turcotte, 1997).If the fault system under investigation is a fractal, the plots of N(r) versus r can be described by a power law function (Mandelbrot 1985; Feder 1988) as in equation (3.2).The relation in Eq. (3.2) can be represented as a linear function in a log log graph: Log N(r) = log c D 0 log r (3.3) where c is the constant of proportion and the slope, D 0 of the linear log log plots of N(r) versus r represents a measure of the fractal dimension of the system. In this method the faults and other structural elements on the map were initially superimposed on a square grid size s 0. The unit square 2 s 0 was sequentially divided into small squares of size s = s, s 4, 8,..... The number of squares or boxes 1 0 2 0 s0 N ( s i ) intersected by at least one fault line is counted each time. If the fault system is a self-similar structure, then following Mandelbrot (1983), N ( s i ) is given by, N D 0 ( ) ~ ( ) 0 D s s 0 s ~ s i i i (3.4) 32
where, D 0 is interpreted as the fractal capacity dimension of the fault system. The fractal capacity dimension D 0 was determined from the slope of the log N ( s i ) versus ( ) log line of the data points obtained by counting the number of boxes s 0 s i covering the curve and the reciprocal of the scale of the boxes. Fig.3.1 A schematic diagram of the box-counting method for determining the capacity fractal dimension of the fault system. The r is measure of side of a square box and N(r) is number of boxes containing at least one or any part of fault system. The area of the present study covers the belt between Latitude 28 N-33 N and Longitude 76 E-81 E and it was divided into twenty five blocks (Fig. 3.3). Surface and volume fractal dimensions were obtained for blocks (A-Y) separately shown in Table 3.1. Box counting dimension for each block marked by capital letters (A-Y) is given in Table.3.1. The volume fractal dimension is also calculated which is equal to one plus surface fractal dimension (Turcotte 1997; Maus and Dimri 1994; Sunmonu and Dimri 2000). Usually the surface exposed faults are the two dimensional expression of embedded faults in the three dimensional volume mass. Hence volume fractal dimension will give the perspective of understanding the three dimensional heterogeneity of the complex Himalayan structure. The geological elements have been taken from Dasgupta., et al. (2000). Each block is of the order of 1 1 on the maps. 33
Log (1/r) y = 0.678x - 2.208 R² = 0.950 Block A LogN(r) Log (1/r) y = 0.536x - 2.110 R² = 0.842 Block L Log N(r) Log (1/r) y = 0.619x - 2.123 R² = 0.986 Block N Log N(r) Log (1/r) y = 0.550x - 2.097 R² = 0.981 Block S Log N(r) Fig.3.2.(a) The log (1/r) versus log N(r) is shown for determination of capacity dimension of the block A, N, L and S. The reciprocal of the value for the slope of the line assigns the value of capacity dimension (D o ). R 2 represents correlation coefficients of the regression line. 34
Log (1/r) y = 1.043x - 2.352 R² = 0.995 Block I Log N(r) Log (1/r) y = 1.231x - 2.512 R² = 0.949 Block P Log N(r) Log (1/r) y = 0.805x - 2.223 R² = 0.993 Block K Log N(r) Log (1/r) y = 0.824x - 2.288 R² = 0.961 Block D Log N(r) Fig.3.2. (b) The log (1/r) versus log N(r) is shown for determination of capacity dimension for some of the blocks having lesser coverage of structural elements. The reciprocal of the value for the slope of the line assigns the value of capacity dimension (D 0 ). R 2 represents correlation coefficients of the regression line. 3.3 Data used In the present analysis of fault system using box counting fractal dimension tool, the area covers from Latitude 28 N-33 N and Longitude 76 E-81 E. The tectonic features of the region have been utilized from Dasgupta et al. (2000) for the determination of capacity dimension. In determination of D 0 care has been taken in all respect. Fig.3.3 shows the utilized structural elements for the box counting 35
dimension. Fig.3.3 shows the distribution of structural elements in the entire study area that has been divided into twenty five blocks as marked by capital letters (A to Y). The corresponding box counting fractal dimensional value is depicted in Table.3.1. The variation of capacity dimension of each block may be related to the physical clustering and distribution of the fault system. 3.4 Results and discussion The surface fractal dimension quantifies the geometrical distribution of fault system, which is the quantitative fractal analysis of complex tectonics governing seismic activity in the region. The determination of fractal capacity dimension value of different blocks depict that the region (in different blocks) is having high and low value depending upon the distribution of faults system. The determined D 0 is found to vary from 0.812 to 1.866.The lower value depicts the minimum coverage of faults system and higher value shows the maximum coverage of faults system in a particular block. The results show some boxes are having highest D 0 values where distribution of faults system is maximum. The block L has the highest capacity dimension and block-p has the lowest D 0 value (Table.3.1). The block L has a system of faults well distributed and covers the maximum area of the block in the region. On the other hand block P has only one fault line and covers very less area of the block. Similarly the other different values of different blocks suggest that higher values are associated with maximum coverage and lower values are associated with minimum coverage of different blocks. So the physical significance of low and high D 0 value may be understood well with capacity dimension. The six blocks G, L, M, S T and U of highest capacity dimension value of 1.742, 1.866, 1.739, 1.818, 1.754 & 1.773 respectively are identified with dense and maximum coverage of tectonic elements in a particular block. The two lowest capacity dimension value blocks are I and P with value of 0.959 and 0.812 respectively are observed to have straight line like features of the faults line in each block. The importance of low and high value may be noted from such distribution of fault system. Similarly the other blocks are having different capacity dimension value associated with the existence of fault system is listed in Table. 3.1. Block- E does not have any structural element and fault line in the block- Y is not fractal. A map 36
with all the tectonic elements and the capacity dimension value of each block is presented in Fig.3.4. A value close to 2 suggests that it is a plane that is filled up by structural elements, while a value close to 1 implies that line sources are predominant (Aki 1981). In general capacity fractal dimension (D o ) should lie between 1 and 2. The observation from the analysis of capacity dimension of fault system reflects the actual pattern properties of structural elements in different blocks. So the result of practical and theoretical values matches well. 37
Fig.3.3 The map of the region shows the tectonic features of the entire study area used for the determination of capacity fractal dimension (D 0 ) (Modified after Dasgupta et. al., 2000). 38
Fig.3.4 The map of the region shows the capacity dimension value (D 0 ) of each block of the tectonic features (Modified after Dasgupta et. al., 2000). 39
Table.3.1 Box counting dimension and volume fractal dimension for twenty five blocks, where R 2 represent regression coefficient for straight line fit to obtain Box counting fractal dimension. Block No. Latitude Longitude of the sub area in 1 1 Box Counting Fractal Dimension(D 0 ) R 2 (Regression Coefficient) Volume Fractal Dimension =D 0 +1 Error A 32 N - 33 N, 76 E - 77 E 1.475 0.950 2.475 ±0.05 B 32 N - 33 N, 77 E - 78 E 1.590 0.982 2.590 ±0.018 C 32 N - 33 N, 78 E - 79 E 1.261 0.990 2.261 ±0.01 D 32 N - 33 N, 79 E - 80 E 1.214 0.960 2.214 ±0.04 E 32 N - 33 N, 80 E - 81 E No structural - - - element exists F 31 N - 32 N, 76 E - 77 E 1.616 0.974 2.616 ±0.026 G 31 N - 32 N, 77 E - 78 E 1.742 0.978 2.742 ±0.022 H 31 N - 32 N, 78 E - 79 E 1.387 0.996 2.387 ±0.004 I 31 N - 32 N, 79 E - 80 E 0.959 0.995 1.959 ±0.005 J 31 N - 32 N, 80 E - 81 E 1.261 0.980 2.261 ±0.02 K 30 N - 31 N, 76 E - 77 E 1.242 0.993 2.242 ±0.007 L 30 N - 31 N, 77 E - 78 E 1.866 0.842 2.866 ±0.158 M 30 N - 31 N, 78 E - 79 E 1.739 0.991 2.739 ±0.009 N 30 N - 31 N, 79 E - 80 E 1.616 0.986 2.616 ±0.014 O 30 N - 31 N, 80 E - 81 E 1.645 0.979 2.645 ±0.021 P 29 N - 30 N, 76 E - 77 E 0.812 0.949 1.812 ±0.051 Q 29 N - 30 N, 77 E - 78 E 1.289 0.958 2.289 ±0.042 R 29 N - 30 N, 78 E - 79 E 1.534 0.968 2.534 ±0.032 S 29 N - 30 N, 79 E - 80 E 1.818 0.981 2.818 ±0.019 T 29 N - 30 N, 80 E - 81 E 1.754 0.980 2.754 ±0.02 U 28 N - 29 N, 76 E - 77 E 1.773 0.984 2.773 ±0.016 V 28 N - 29 N, 77 E - 78 E 1.484 0.986 2.484 ±0.014 W 28 N - 29 N, 78 E - 79 E 1.152 0.976 2.152 ±0.024 X 28 N - 29 N, 79 E - 80 E 1.464 0.990 2.464 ±0.01 Y 28 N - 29 N, 80 E - 81 E Not Fractal - - - 40
In our study region, none of the blocks have a value of 1 or 2. Fig.3.4 clearly indicates about the dense and maximum coverage of fault system within a block with the higher value of D 0. Moreover, the existence of correlation between capacity dimension value of some blocks and clustering of earthquakes defined by fractal correlation dimension (D C ) shall be well studied and will be presented in the next chapter-4. The study of volume fractal dimension is done to find out the three dimensional heterogeneity of the complex Himalayan structure. The volume fractal dimension is achieved from the capacity fractal dimension. The three-dimensional map of volume fractal dimension distribution of the entire study area is depicted in Fig.3.5 as calculated and shown in Table 3.1. Higher volume fractal dimension distribution is observed along MCT. On the other hand there exists a gradual increase of volume fractal distribution through MBT toward Main Central Thrust (MCT).Lower volume fractal dimension distribution is along Indus Suture and Karakoram Fault (Fig.3.5). Hence higher volume fractal dimension gives the view of understanding the three dimensional heterogeneity of the complex Himalayan structure which is predominant along the MCT Fig. 3.5. Here the volume fractal dimension of structural elements present in each block was calculated using the box counting technique (Carlson 1991; Hodkiewicz et al. 2005; Ram and Roy 2005; Ford and Blenkinsop 2008; Raines 2008; Carranza 2008, 2009). 41
Fig.3.5 The figure shows the volume fractal dimension distribution of the studied area. The four black curves which denote Indus Suture, Karakoram Fault, MCT and MBT from north to south are the rough sketch (not to scale) made to show the volume fractal dimension distribution in different parts of the region. 42