29

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 (D0) 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 quasi-

periodically 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

30

distribution of fault system is studied with the help of box counting method by

obtaining capacity dimension (D0), 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

31

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 (D0) 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,

( )0

1D

rrN ∝ (3.1)

32

where N(r) is the number of objects (i.e., fragments) with a characteristic linear

dimension r (specified size) and D0, the fractal dimension.

Equation (3.1) can be written as,

( )r

DcrN

0= (3.2)

where c is the constant of proportion and D0 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 box-

counting 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 (D0) 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 – D0 log r (3.3)

where c is the constant of proportion and the slope, D0 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 0s . The unit square 20s was sequentially divided

into small squares of size .,.........8,4,2 0001 ssss = . The number of squares or boxes

( )isN intersected by at least one fault line is counted each time. If the fault system is

a self-similar structure, then following Mandelbrot (1983), ( )isN is given by,

( ) ( ) 00 ~~ 0D

i

D

ii ssssN−

(3.4)

33

where, D0 is interpreted as the fractal capacity dimension of the fault system.

The fractal capacity dimension D0 was determined from the slope of the log ( )isN

versus ( )iss0log line of the data points obtained by counting the number of boxes

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.

34

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 (Do). R2

represents correlation coefficients of the regression line.

y = 0.678x - 2.208R² = 0.950Block A

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

0.4 0.8 1.2 1.6 2

Log

(1

/r)

LogN(r)

y = 0.536x - 2.110R² = 0.842

Block L

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

0.4 0.8 1.2 1.6 2

Log

(1

/r)

Log N(r)

y = 0.619x - 2.123R² = 0.986Block N

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

0.4 0.8 1.2 1.6 2

Log (

1/r

)

Log N(r)

y = 0.550x - 2.097R² = 0.981Block S

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

0.4 0.8 1.2 1.6 2

Log

(1

/r)

Log N(r)

35

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 (D0). R2 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 D0 care has been taken in all

respect. Fig.3.3 shows the utilized structural elements for the box counting

y = 1.043x - 2.352R² = 0.995

Block I

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

0.4 0.8 1.2 1.6 2

Log

(1/r

)

Log N(r)

y = 1.231x - 2.512R² = 0.949

Block P

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

0.4 0.8 1.2 1.6 2

Log

(1/r

)

Log N(r)

y = 0.805x - 2.223R² = 0.993Block K

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

0.4 0.8 1.2 1.6 2

Log

(1

/r)

Log N(r)

y = 0.824x - 2.288R² = 0.961Block D

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

0.4 0.8 1.2 1.6 2

Log

(1

/r)

Log N(r)

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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 D0 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 D0 values where distribution

of faults system is maximum. The block – “L” has the highest capacity dimension and

block-P has the lowest D0 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 D0 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

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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 (Do) 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.

38

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 (D0) (Modified after

Dasgupta et. al., 2000).

39

Fig.3.4 The map of the region shows the capacity dimension value (D0) of each block

of the tectonic features (Modified after Dasgupta et. al., 2000).

40

Table.3.1 Box counting dimension and volume fractal dimension for twenty five

blocks, where R2 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(D0)

R2

(Regression Coefficient)

Volume

Fractal Dimension

=D0+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 - - -

41

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 D0. Moreover, the existence of correlation between capacity dimension value

of some blocks and clustering of earthquakes defined by fractal correlation dimension

(DC) 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).

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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.