Fractal Dimensions of Blocks Using a Box-counting Technique for the 2001 Bhuj Earthquake, Gujarat, India

Fractal Dimensions of Blocks Using a Box-counting Technique

for the 2001 Bhuj Earthquake, Gujarat, India

AVADH RAM1 and P. N. S. ROY

Abstract—Several destructive earthquakes have occurred in the Kachchh region of Gujarat during the

past two centuries, among them Allah Bund earthquake (M7.8) in 1819, Anjar earthquake (M6) in 1956

and the recent Bhuj earthquake (M7.6) in 2001. The Anjar earthquake was on KMF (Kachchh Mainland

Fault) and the recent Bhuj events were caused by a hidden fault north of KMF. The present study discusses

the fractal analysis of tectonics governing seismic activity in the region. The region has been divided into

five blocks and the fractal dimension of each block has been calculated using the box-counting technique.

The results show significantly low value of fractal dimension of the Kachchh rift block consisting of the

KMF compared to the other surrounding blocks, which also contain faults and rifts of higher fractal

dimension. This indicates that the cause of earthquakes in this block may be asperities and barriers.

However, the predominance of aftershocks over foreshocks signifies that barriers may be the main cause.

The other results, such as the lower value of dimension of fault clustering show that the Kachchh rift block

has faults which are distributed in a clustered manner. In this context, the seismicity of this block seems to

be high.

Key words: Fractal dimension, faults, rifts, tectonics, asperities, barriers.

Introduction

MANDELBROT (1967) coined fractal from the Latin adjective fractus. The

corresponding Latin verb frangere means ‘‘to break’’ to create irregular fragments.

He used the word ‘‘fractal’’ to describe the characteristics of such objects and events

which have fractional dimensions.

A fractal distribution requires that the number of objects larger than a specified

size has a power-law dependence on the size,

N a 1=r; ð1Þ

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

dimension r (specified size).

Equation (1) can be written as,

1 Department of Geophysics, Faculty of Science, Banaras Hindu University, Varanasi-221005, India.

E-mail: [email protected]; [email protected]

Pure appl. geophys. 162 (2005) 531–5480033 – 4553/05/030531 – 18DOI 10.1007/s00024-004-2620-4

� Birkhauser Verlag, Basel, 2005

Pure and Applied Geophysics

N ¼ crD ; ð2Þ

where c is the constant of proportion and D is the fractal dimension which gives N a

finite value, otherwise with the decrease of size r in equation (1) the value would have

reached infinite.

The empirical applicability of power-law statistics to geological phenomena was

recognized long before the concepts of fractal were conceived. A striking example is

the GUTENBERG-RICHTER (1954) relationship for the frequency-magnitude statistics

of earthquake,

Log N ¼ a� bm; ð3Þ

where a and b are constants, the logarithm is to the base 10 , and N is the number

of earthquakes per unit time with a magnitude greater than m occurring in a

specified area. The proportionality factor in the relationship between the number

of earthquakes and earthquake magnitude is known as the b value. It is now

accepted that the Gutenberg-Richter relationship is equal to a fractal relationship

between the number of earthquakes and the characteristic size of the rupture

(TURCOTTE, 1992), the value of the fractal dimension D is simply twice the b value;

the power-law distribution must be applicable to the scale-invariant phenomena.

Power law can be understood by taking the example of the Richardson

measurement of the British coastline. RICHARDSON (1961) found that the smaller

the length of the measuring rod the longer the length of the coastline as in

equation (1). In other words, the measurement of the coastline length was done in

the order of the minute ruggedness.

The fault system is analyzed with the help of the box-counting method which is a

method of fractal analysis. The method was earlier used by HIRATA (1989) 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 Andrea’s fault

system; by SUKMONO et al. (1996, 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. 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.

In this paper the geology and tectonics of the Gujarat region have been studied

systematically with the help of fractal geometry analysis. Initially, we will study

qualitatively then quantitatively with the aid of fractal geometry.

532 Avadh Ram and P. N. S. Roy Pure appl. geophys.,

The geology and tectonics of the Kachchh region and its surrounding area

suggest that the box-counting method may be used to calculate the surface fractal

dimension of a distribution of the faults. The entire region lying between

14�N� 24�300N and 68�E � 76�200E, has been subdivided into five blocks (I-V).

The surface and volume fractal dimensions for the entire region as well as for blocks

(I-V) are separately calculated. The volume fractal dimension is equal to one plus

surface fractal dimension (TURCOTTE, 1992; MAUS and DIMRI, 1994; SUNMONU and

DIMRI, 2000). Thus, for the entire graph the boxes to be used will be between 1.56 km

to 50 km long.

The DP value is a measure of fault cluster which will be calculated to derive the

fault distribution. The fractal dimension D in a region is twice the b value

(TURCOTTE, 1992). A similar result was obtained by KING (1983) by means of a

fractal-faulting model in three dimensions. With the help of the above relation, the b

value for the entire region as well for the subdivided blocks has been calculated.

These fractal dimensions thus obtained will be corroborated with the strong motion

and broadband data of the Bhuj earthquake to understand the mechanism of the

earthquake due to barriers. This technique will be beneficial to analyze the potential

of destructive earthquakes along active fault systems satisfying fractal distribution.

Geology and Tectonics of the Region

Basically, the tectonics of this region has three palaeorifts, namely Kachchh, Cambay

and Narmada (Fig. 1), which are tectonically very active as evident from neotectonic

movements and seismicity. Intra-continental rifting takes place generally along

crustal weak zones such as ancient orogenic belts and palaeosutures between

colliding proto-continents. Such zones are prone to reactivation by later tectonic

episodes. Gujarat rifts are located in such weak zones of Precambrian orogenic belts.

The Kachchh rift was first to form, followed sequentially by the Cambay rift in early

Cretaceous and the Narmada rift in the Bombay offshore region mutually displacing

each other (BISWAS, 1987). The eastern fault of the Cambay rift continues to the

south along the west coast as the west coast fault (WCF).

Kachchh rift is the northernmost pericontinental embayed basin present between

the subsurface Nagar Parkar Uplift (NPU) in the north, the Radhanpur – Barmer

arch in the east, and the Kathiwar uplift in the south. The two uplifts are the rift

shoulders respectively along Nagar Parkar Fault (NPF) and North Kathiwar Fault

(NKF) and the Radhanpur–Barmer arch is the western shoulder of the Cambay rift

which orthogonally terminates the Kachchh rift.

The most typical feature of a rift basin is shown in structural style, such as tilted

footwall uplifts and half-grabens form respectively, the highlands and intervening

plainlands of Playas and Salinas. The uplifts appear to be the basement blocks

draped with marginal monoclinal flexures (knee and ankle folds) of sediment which

Vol. 162, 2005 Fractal Dimensions of Blocks 533

cover over the faulted-up edges. In all the uplifts, second-order folds, e.g., anticlines

and domes of varying shapes and sizes, complicate the flexures within a narrow

deformation zone along the master faults in the western part of the basin, the uplifts

are tilted down southerly forming a series of step faults, whereas in the eastern part

the style is different which is described as follows.

The major intra-baisnal faults bounding grabens, half-grabens, and horsts within

the rift system are Island Belt Fault (IBF), Gora Dongar Fault (GDF), Banni fault

(BF), Kachchh Mainland Fault (KMF), Katrol Hill Fault (KHF), South Wagad

Figure 1

Tectonics of the Kutch region in Gujarat, India. This region is divided into five blocks in which Block I is

the region containing the Kutch Mainland Fault. The star indicates the epicenter of the main shock of

January 26, 2001 (after BISWAS AND KHATRI, 2002).


Fault (SWF) and Gedi Fault (GF) as shown in Figure 2. These faults are near

vertical basement faults and follow the mid-Proterozoic orogenic trend of Delhi fold

belt (BISWAS, 1980,1987, 2000). The Great Rann subbasin is a narrow graben

bounded by NPF and IBF between Nagar Parkar Uplift (NPU) and Island Belt

Uplift (IBU) which is segmented into four separate uplifts—Pachham (PU), Khadir

(KU), Bela (BU), and Chorar (CU) uplifts, by cross faults. Southward tilting of IBU

along KMF formed the Banni half-graben. Kachchh Mainland uplift (KMU) is

bounded by KMF and is tilted to the south along KHF. The KHF divides the

Mainland uplift into Mainland half-graben (Bhuj syncline) and Katrol Hill Uplift in

the south, which is again tilted to the south to form the Gulf of Kachchh half-graben

(GOK HG) against the Kathiawar (Saurashtra) block along NKF.

Wagad uplift (WU) is in the eastern part of the rift and is bounded by (SWF)

along its southern margin. SWF consists of a system of curved, converging and

diverging faults with associated tight folds. This part of WU is extensively deformed

with complicated pattern of folds. WU is tilted to the north against BU (of IBU

chain) which is a horst bounded by GF and IBF. The north tilting of WU formed

Rapar half-graben (RHG) against GF. The over-stepping relationship of KMF and

Figure 2

The faults and tectonic settings of the studied region are shown (after BISWAS AND KHATRI, 2002).


SWF indicates that these are segments of a strike-slip fault (AYIDIN and NUR, 1985).

Across, the basin meridional high is one of the most unique features of the basin. It is

manifested as a geomorphic high across the Banni half-graben and KMU. This high

is a first-order basement ridge and appears to be the relic of the hinge zone of the

Indus shelf basin prior to Kachchh rifting (BISWAS, 1982, 1987). A number of small-

scale extensional faults occurs in the crustal region of this NNE-SSW striking high,

parallel to its axis as seen in the central part of the Mainland uplift. These faults

extend distantly with throws varying from 1 to 100 m. Basic dykes accompany most

of them, indicating their extensional origin.

Application of Fractals to Fault System

In this method the fault on the map was initially superimposed on a square grid

size s0. The unit square s20 was sequentially divided into small squares of size

s1 ¼ s0=2; s0=4; s0=8; . . . : The number of squares or boxes N sið Þ 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 sið Þ is given by,

N sið Þ � s0=sið ÞD� s�Di ; ð4Þ

where D is interpreted as the fractal dimension of the fault system. The fractal

dimension D was determined from the slope of the log N sið Þ versus log s0=sið Þ line ofthe data points obtained by counting the number of boxes covering the curve and the

reciprocal of the scale of the boxes.

The area of the present study covers the belt between 14�N� 24�300N and

68�E � 76�200E, and it was divided into five blocks (Fig. 1). Surface and volume

fractal dimensions were obtained for the entire region as well as for blocks (I-V)

separately (Figure 3(a) and Table 1. From Figure 3(a) the Block I Dimension

DI ¼ 0.9784 was determined from the plot Log N(S) – Log(1/S). Similarly other

blocks in its neighborhood DII ¼ 1.1005, DIII ¼ 1.0770, DIV ¼ 1.0840, DV ¼ 1.0207

were also determined from Figure 3(a). Also the LANDSAT images of Figures 6(a)

and 6(b) used for the KMF region, i.e., BLOCK I which produced values of

D1 ¼ 0.9227 (Jawaharnagar-Devisar), D1 ¼ 0.9465 (Lodai), which is quite analogous

the above result of low value. Though the images depict traces of active faults, which

are meager, but the images have higher resolution. Hence it is used for a comparative

study of the region containing few traces. The volume fractal dimension is equal to

one plus surface fractal dimension (TURCOTTE, 1992; MAUS and DIMRI, 1994;

SUNMONU and DIMRI, 2000). For the entire graph, the boxes between 1.56 km to 50

km long have been used. Figure 4 shows a three-dimensional map of volume fractal

dimension distribution of the studied area. Figure 4 clearly depicts that BLOCK I

corresponding to Figure 1 has very low volume fractal dimension compared to other

surrounding BLOCKS. Thus the region has less rugged or three-dimensional


Figure 3

(a) Log (N(S) – Log (1/S) plot to determine fractal dimension of Block I-V as well as for the whole region.

Figure 3(b) Log (P(S) – Log (1/S) plot to determine Geometrical probability dimension (Dp) of Block I-V

as well as for the whole region.

Table 1

The surface/capacity, volume fractal dimension and the b value for the entire region as well as for the blocks

(I-V). The Surface fractal dimensions are obtained from the plot of Log (N(S) – Log(1/S) and DP is

obtained from the plot of Log(P(S) – Log(1/S).

BLOCKS Surface fractal

dimension (D1)

b value for c = 1.5

(D1/2)

Fractal dimension DP

for Probability

distribution

Volume fractal

dimension

(D1+1)

Block I 0.9784 0.4892 1.0894 1.9784

Block II 1.1005 0.5503 1.3709 2.1005

Block III 1.0770 0.5385 1.3203 2.0770

Block IV 1.0840 0.5420 1.3114 2.0840

Block V 1.0207 0.51035 1.1184 2.0207

Whole Region 1.0983 0.54915 1.1706 2.0983


heterogeneity, which need to be analyzed an other way as the possible cause of such

stress liberation. These findings have thrown light upon the asperity or barrier as the

possible cause of such large earthquakes, which will be undertaken in later sections.

Geometrical Probability of Fault System to Study Fault Distribution

A phenomenon ‘‘A’’ (here fault’s distribution) which appears on the two-

dimensional surface, the geometrical probability for that phenomenon, is defined as

(IDZIAK and TEPER, 1996):

P ðAÞ ¼ SumðAÞ=SumðtotalÞ; ð5Þ

where the summation area of squares traced by faults (the surface covered by squares

with side length si) is:

SumðAÞ ¼ N sið Þ � s2i : ð6Þ

The total area of squares covering the surface is,

SumðtotalÞ ¼ Ni � s2i ; ð7Þ

Ni is the total number of boxes, and N sið Þ is the number of traced boxes. The

probability that a square of size si will include the trace of faults which can be

estimated as follows,

P sið Þ ¼ N sið Þ=Ni: ð8Þ

Figure 4

A three-dimensional map of the volume fractal dimension distribution of the studied area.


If the geometrical probability is self-similar, it should have the fractal distribution

(TURCOTTE, 1992),

P sið Þ ¼ si=s0ð Þ2�DP ð9Þ

where DP is the fractal dimension of the probability distribution P sið Þ. The DP value

is a measure of fault cluster, varying from 0 to 2. The values near zero signify that

faults are either extremely concentrated in the small limited area or sparsely

distributed, whereas the values equals to 2 mean that faults are densely spread in the

whole area.

The results of the fractal cluster analysis made for the entire area, as well as for

the subdivided blocks’ are shown in Figure 3 (b). Figure 3(b) shows the plot Log

P(s) – Log (1/s) for the whole region as well as for each block. The DP value obtained

with the relationship shown above by equation (9) from Figure 3(b). Here the value

obtained for DP shown in Table 1 clearly shows the BLOCK I which has DP 1.0894,

whereas other BLOCKS are higher than BLOCK I. Since DP for BLOCK I of lesser

value compared to other blocks indicates that faults are concentrated in small region

in BLOCK I. Figure 5 depicts aftershocks for the 150 events which are concentrated

Figure 5

The three-dimensional distribution of aftershocks of one year after the main shock of January 26, 2001.


in the small volume space. Hence, such a highly concentrated fault system can

support the cause of such a catastrophic event within the intraplate region.

Fractal Dimension of Faults and b Value

The b value represents the frequency relationship among earthquakes with

different magnitudes. It is related to the stress state and fracture strength of the

crustal medium in the region (LEE et al., 1997). The b value is determined by the

GUTENBERG-RICHTER relationship:

NT ¼ am�b; ð10Þ

where m is magnitude, a and b are regression coefficients. The b value is of greater

importance because by using the concepts of geometrical self-similarity, it has been

shown that the ‘‘b value’’ in the GUTERNBERG-RICHTER relation can be directly

related to the fractal dimension of the active fault system involved in the seismic

activity (AKI, 1981; KING, 1983, TURCOTTE, 1986) by:

D ¼ 3b=c ; ð11Þ

where c is constant and if c ¼ 1.5 (KANAMORI and ANDERSON, 1975; AKI, 1981),

then,

D ¼ 2b : ð12Þ

The fractal dimension D, in a region, is twice the b value (TURCOTTE, 1992). A

similar result was obtained by KING (1983) by means of a fractal-faulting model in

three dimensions. With the help of equation (11), the b value for the entire study

region as well for the subdivided blocks has been obtained as shown in Table 1. For

crystalline rocks, D is equal to b since c is 3.0. Here the value of c is taken as 1.5 and

the corresponding b value obtained from the above relation of equation (12). The

BLOCK I has a b value of 0.48922, whereas other surrounding blocks have a b value

higher than the BLOCK I. This is a clear indication that the chance of occurrence for

a large event is very high. The possibility of asperity and barriers in the BLOCK I is

supported from this finding.

Capacity Dimension and Seismotectonics of the Region

Earlier seismic behavior of a fault has been correlated with the fault geometrical

distribution (SUNMONU and DIMRI, 2000; SUKMONO et al., 1997, TURCOTTE, 1989). In

order to understand the geometry of the fault distribution of the Kachchh region and

its surrounding area, the capacity dimension (D0) or surface dimension has been

used. The surface dimension quantifies the geometrical distribution of faults, which


especially advances fractal analysis of tectonics governing seismic activity in the

region. The region has been divided into five blocks and the fractal dimension of each

block has been calculated using the box-counting technique. The results show

significantly low value of fractal dimension of the Kachchh rift block, consisting of

the KMF compared to the other surrounding blocks which also have faults and rifts

such as the Cambay and Narmada rifts of higher fractal dimension. This indicates

that the cause of earthquakes in this block may be the asperities and barriers

(KHATRI, 1995). To understand the evolution of the fault system, KING (1983) used a

geometric description. Earthquakes initiate and terminate in regions where fault

systems bend, because the bends becomes zones subject to multiscale faulting.

Movement of many faults in these regions distributes the stress concentration of a

propagating rupture front and terminates motion. The multiple faults create offsets

in the next fault to move. These offsets are the asperities that must break before a

new earthquake occurs. The low fractal dimension also signifies that the region has

thrust rather than the tensional structure.

The data of the active faults used in this study were taken from the tectonic map

of BISWAS and kHATRI (2002). The whole region’s surface fractal dimension DW has

been calculated as 1.0983. In spite of faults and rifts in the Cambay rift (Block II) and

the Narmada rift (Block III), the Kachchh rift block (Block I) is the active one in the

region. The Block I has Dimension DI= 0.9784, unlike other blocks in its vicinity

DII= 1.1005, DIII= 1.0770, DIV = 1.0840, DV = 1.0207. Also the LANDSAT

images of Figures 6(a) and 6(b) used for the KMF region, i.e., BLOCK I which gave

values of D1 ¼ 0.9227 (Jawaharnagar-Devisar), D1 ¼ 0.9465 (Lodai) which is quite

analogous to the above result of low value. This lower surface fractal dimension of

Block I suggests the presence of asperities and barriers.

The production of a fracture surface with a high fractal dimension requires more

external energy (NII et al., 1985). Here the fracture surface is made under minimum

energy, as the volume fractal dimension is around 2 (Table 1 and Figure 4). DAVIDGE

and GREEN (1968) has shown that the crack paths in the materials including the

obstacle depend on the relative strength, size and distribution of obstacles. Probably

the asperity or barriers in the rock control the fractal dimension (HIRATA, 1989).

The variability of the fractal dimension in different zones may be related to

geological heterogeneity (AVILES et al., 1987). A value of D close to 3 implies that

earthquake fractures are filling up a volume of the crust (Fig. 4). A value close to 2

suggests that it is a plane that is being filled up, and a value close to 1 means that line

sources are predominant (AKI, 1981). The volume fractal dimension is also very small

for Block I as shown in the three-dimensional map (Fig. 4) of volume fractal

distribution of the entire region. This signifies that the region is less rugged or

crenulated. Consequently the possible cause of high energy liberation seems to be due

to barriers. The stress field is rougher, as the aftershock predominance justifies the

presence of barriers qualitatively (AKI, 1984). Another significant feature is the low

DP= 1.0894 of Block I, i.e., the DP value is a measure of fault cluster, varying from 0


to 2. Here the value which is near to 0 for Block I compared to other blocks

(Table 1), and signifies that faults are extremely concentrated in the small limited

area (clustered) in Block I.

Discussion and Conclusions

In the early earthquake cycle, the correlation length is short, reflecting a stress

field, which is very rough on the regional scale. This implies that the distribution of

stress on the regional scale is highly heterogeneous and that the physical dimensions

of regions capable of producing a rupture are smaller than the physical dimensions of

the individual faults which are capable of producing a rupture. Thus, when an

earthquake nucleates it will reach the edge of the highly stressed patch before it can

grow to become a large event. As these events occur, they smooth the stress

field at long length scales by redistributing stress to neighboring regions, while

roughening the stress field at short length scales through aftershocks. The process of

smoothing the stress field at long wavelengths while roughening it at short

wavelengths increases the correlation length of the regional stress field without

shutting off small events. As the process continues, the growing correlation of the

stress field allows ruptures in the smoothed stress field to grow to greater lengths,

smoothing the stress field at even longer length scales. Thus, earthquakes which

nucleate later in the cycle, and therefore in a more correlated stress field, are able to

rupture barriers which would have halted the earthquake in an earlier, less correlated

stress field. Only when criticality is reached is the stress field correlated on all scale

lengths up to and including the largest possible event for the given fault network. The

study of fault geometry indicates that earthquakes do not occur on a single surface

but on fractal structure of many closely correlated faults. The DP value is a measure

of fault cluster varying from 0 to 2. TOSI (1998) illustrated that possible values of

fractal dimension are bound to range between 0 and 2, which is dependent on the

dimension of the embedding space. Interpretation of such limit values is that a set

with D fi 0 has all events clustered into one point, and at the other end of the scale,

D fi 2 indicates that the events are randomly or homogeneously distributed over a

two-dimensional embedding space. Here the value is close to 0 for Block I compared

to the other blocks, which signifies that faults are extremely concentrated in the small

limited area (clustered) in Block I. The total number of these small faults can be very

large, practically infinite. On the other hand, critical behavior is fundamentally a

Figure 6

(a) The LANDSAT image detailed distribution of active faults around Jawaharnagar-Devisar along the

Kachchh Mainland Fault Zone. The white lines indicate the Active Fault traces (after MALIK et al.,2001).

Figure 6(b) The LANDSAT image detailed distribution of active faults around Lodai along the

Kachchh Mainland Fault Zone.The white lines indicate the active fault traces (after MALIK et al., 2001).

c



cooperative phenomenon, resulting from the repeated interactions between ‘‘micro-

scopic’’ elements, which progressively ‘‘phase up’’ and construct a ‘‘macroscopic’’

self-similar state. The word ‘‘critical’’ describes a system at the boundary between

order and disorder, and is characterized by both extreme susceptibility to external

factors and strong correlation between different parts of the system.

The surface fractal dimension D1 for Block I is smaller and the corresponding b

value is also low, which signifies that the occurrence of large events is very high.

Similar low fractal dimension value was found for strike-slip faults, such as the

Xianshuihe Fault zone, the Longshoushan Fault zone, the Qilianshan Fault zones,

the Changma Fault zone, and continuing (DIAO and CHAO, 1994). The fractal

dimension obtained for the Vilarica strike-slip fault, NE Portugal ranged 1.0062 to

1.0171 (ANTONIO RIBERIO et al., 1991). SHIMAZAKI and NAGAHAMA (1995)

demonstrated that active fault systems in Japan possess self-similarity with fractal

dimensions of 0.5 to 1.6. The fractal dimension characterizes the degree to which the

fractures fill the surrounding space. We can predict the fracture characteristic by

ascertaining the value of D.

Fifty-one aftershocks of M ‡ 4.0 recorded during 26 January to 30 June, 2001

at Hyderabad observatory, located about 1000 km from the epicentral zone, have

Figure 7

(a) Log (N(S) – Log (1/S) plot to determine fractal dimension of LANDSAT image of the Jawaharnagar –

Devisar region. Figure 7(b) Log (N(S) – Log (1/S) plot to determine fractal dimension of LANDSAT

image of Lodai region.


been used to estimate the generalized fractal dimension which is found to be 0.68

(KELKAR, 2002). Moreover, the volume fractal dimension is also very small for the

Block I shown in the three-dimensional map of the volume fractal distribution of

Figure 8

The stress condition due to aftershocks of the studied region at a depth of 20–35 km for a period of one

year from the main shock of January 26, 2001.


the entire region. It signifies that the region is less rugged or crenulated, hence the

possibility of lesser energy liberation. Then the question may arise as to what is

the cause of such high-energy liberation? It is due to barriers—the stress

roughener— as the aftershock predominance justifies qualitatively. Still some

quantitative justifications are required, which are shown in Figure 8. Figure 8

clearly justifies that there are heterogeneities which have produced such a rapid

change of stress drop from the aftershocks for one year near the main shock

region at the depth range 20–35 km. The possibility of barrier is very high after

analyzing the plot of stress condition (stress drop) of the region for one year from

the main shock. This has been obtained with the support of SEISAN software on

broadband data using spectral analysis for surface wave of approximately 100

aftershocks for a period of about one year. The seismic reflection and strong

motion data studies in that region are still required to provide an exact

understanding quantitatively. The presence of asperities and barriers are also

supported by other findings. The tomography study suggests (MANDAL et al.,

2003,) the presence of an ultra-mafic body characterized by high V and low r at

10–40 km depths beneath the epicentral zone resting on the base of the lower

crust. The body might have intruded during the process of rifting. The study has

also shown a fluid-filled weak asperity body along the NWF at 20–30 km depths

within 1/3rd area of the high velocity body in the focus of the earthquake. MISHRA

et al. (2002), carried out a tomographic study which includes P-, S- and Poisson

ratio crack density (fracture density) and saturation rate to estimate in situ

material heterogeneities at the main shock hypocenter. They also inferred that the

presence of a fluid-filled fracture rock matrix at the main shock hypocenter may

be a driving force to trigger the 2001 Bhuj earthquake in the continental interiors,

which is quite analogous to the above cause of the 2001 Bhuj earthquake due to

asperities and barriers. Thus, these are the possible causes of the Bhuj earthquake

of 2001. In conclusion, the region with active faults which are fractally distributed

can be analyzed quantitatively and the results might yield such relevant

information for the cause of earthquake. Consequently this information, may

help to mitigate earthquake hazard in the region.

Acknowledgement

One of us (P.N.S. Roy) is thankful to the Council of Scientific and Industrial

Research (CSIR), New Delhi, for an award of the Senior Research Fellowship

(SRF) under the direct scheme. India Meteorological Department, Government of

India is greatly acknowledged for providing the broadband data. The anonymous

reviewers’ comments and suggestions have considerably improved the text of the

manuscript.


REFERENCES

AKI, K., A probabilistic synthesis of precursor phenomena. In Earthquake Prediction (eds. Simpson, D.W.

and Richards, P.G.)(American Geophysical Union, Washington, D.C.). (1981)

AKI, K. (1984), Asperities, Barriers, Characteristic Earthqukes and Strong Motion Prediction, J. Geophys.

Res. 89, 5867–5872.

ALLEGRE, CLAUDE J., JEAN LOUIS LE MOUEL, HA DUYEN CHAU CLEMENT NARTEAU (1995), Scaling

Organization of Fracture Tectonics (SOFT) and Earthquake Mechanism, Phys. Earth and Planet. Inter.

92, 215–233.

ANGULO-BROWN, F., RAMIREZ-GUZMAN, A.H., YEPEZ, E., RUDOIF-NAVARRO, A., and PAVIA- MILLER,

C.G. (1998), Fractal Geometry and Seismicity in the Mexican Subduction Zone, Geofisica Internat. 37,

29–33.

ANTONIO RIBERIO, ANTONIO MATEUS, MARIA C. MOREIRA, MARIA F. COUTINHO, The Fractal Geometry of

an Active Fault (Vilarcia Srike-Slip Fault, NE Portugal) and its Implications on Earthquake Generation

(Fractals in the Fundamental and Applied Sciences H.O. Peitgen, (eds. J.M. Henriques and L.F.

Penedo) (Elsevier Science Publishers B.V., North-Holland 1991).

AVILES, C. A., SCHOLZ, C. H. and BOATWRIGHT, J. (1987), Fractal Analysis Applied to. Characteristic

Segments of the San Andreas Fault, J. Geophys. Res. 92, 331–344.

AYIDIN and NUR (1985), The types of stopovers in strike-slip tectonics. In (K.T. Biddle and N. Christie-Blick,

eds.), Strike-slip Deformation, Basin Formation, and Sedimentation, Soc. Eco., Pal. and Min. Spec. Publ.

37, 35–44.

BARNSLEY M., Fractals Everywhere, (Academic Press, Inc. 1998).

BISWAS, S.K. (1980), Structure of Kutch – Kathiawar region, Western India, Proc. 3rd Indian Geological

Congress, Poona, pp. 255–272.

BISWAS, S.K. (1982), Rift Basin in Western Margin of India with Special Reference to Hydrocarbon

Prospects, Bull. Am. Assoc. Petrol. Geol. 66(10), 1497–1513.

BISWAS, S.K. (1987), Regional Tectonic Framework, Structure, and Evolution of the Western Margin Basins

of India, Tectonophysics 135, 307–327.

BISWAS, S.K. (2000), Kutch Basin: Geology and Evolution. National Symp. On Recent Advances in Geology

and Resource Potential of the Kachchh Basin, December 21–23, 2000, B.H.U., Varanasi.

BISWAS, S.K. and KHATRI, K.N. (2002), A Geological Study of Earthquakes in Kutch, Gujarat, India,

J. Geol. Soc. of India 60, 131–142.

DAVIDGE, R.W. and GREEN, T.J. (1968), The Strength of Two-phase Ceramic/Glass Material, J. Mater. Sci.

3, 629–634.

DIAO, SHOUZHONG and CHAO, HONGTAI, Main Topics of Fractal Research into Earthquakes in China, A

Review, Fractals and Dynamic Systems in Geoscience (ed. Kruhl, John H.) (Springer-Verlag, New York,

1994), 197–211.

GONZATO, G., MULARGIA, F., and MARZOCCHI, W. (1998), Practical Application of Fractal Analysis:

Problems and Solutions, Geophys. J. Int. 132, 275–282.

GONZATO, G. (1998), A practical Implementation of the Box-counting Algorithm, Comp. Geosci. 24, 95–100.

GUTENBERG and RICHTER, Seismicity of the Earth and Associated Phenomenon, 2nd. Ed. (Princeton

University Press, Princeton. 1954).

HIRATA, T. (1989), Fractal Dimension of Fault Sytems in Japan: Fractal Structure in Rock Fracture

Geometry at Various Scales, Pure Appl. Geophys. 131, 157–170.

IDZIAK, A. and TEPER, L. (1996), Fractal Dimension of Faults Network in the Upper Silesin Coal Basin

(Poland): Preliminary Studies, Pure Appl. Geophys. 147, 239–247.

KANAMORI, H. and ANDERSON, D. (1975), Theoretical Basis for Some Empirical Relations in Seismology,

Bull. Seismol. Soc. Am. 65, 1073–1095.

KELKAR, R.R. (Sept. 2002), Bhuj Earthquake of January 26, 2001 (Monograph on Seismology No. 3/2002,

India Meteorological Department, New Delhi).

KHATRI, K.N. (1995), Fractal Description of Seimicity of India and Inference Regarding Earthquake

Hazard, Current Sci. 69, 361–366.


KING, G. (1983), The Accommodation of Large Strains in the Upper Lithosphere of the Earth and Other

Solids by Self-similar Fault System: The Geometrical Origin of b value, Pure Appl. Geophys. 121, 761–

815.

LEE, C. F., YE, HONG, and ZHOU, QING (1997), On the Potential Seismic Hazard in Hong Kong, Episodes

20, 89–94.

MALIK, J. N., NAKATA, T., SATO, H., IMAIZUMI, T., YOSHIOKA, T., PHILIP, G., MAHAJAN, A. K., and

KARANTH, R. V. (2001), January 26, 2001, The Republic Day (Bhuj) Earthquake of Kachchh and Active

Faults, Gujarat, Western India, J. Active Fault Res. Japan 20, 112–126.

MANDAL, P., RASTOGI, B.K., SATYANARAYANA, H.V.S., and KOUSALYA, M. ( 2003), Presence of Large

Crustal Strain around an Asperity/Heterogeneous Body in the Lower Crust beneath Kachchh: A Possible

Explanation for the Occurrence of 2001 Bhuj Earthquake Sequence, International Workshop on Earth

System Processes Related to Gujarat Earthquake Using Space Technology, January 27–29, 2003, I.I.T.,

Kanpur.

MANDELBROT, B.B. (1967), How Long is the Coast of Britain? Statistical Self-similarity and Fractional

Dimension, Science 165, 636–638.

MANDELBROT, B.B., The Fractal Geometry of Nature (Freeman, New York 1983).

MAUS, S. and DIMRI, V.P. (1994), Scaling Properties of Potential Fields due to Scaling Sources, Geophys.

Res. Lett. 21, 891–894.

MISHRA, O.P.,ZHAO DUPENG, KAYAL J.R., DE REENA, and SINGH, O.P. (2002), Tomography of the Source

Area of the 2001 Bhuj Earthquake: Evidence for Fluids at the Hypocenter, Geoph. Rs. Lett. (In press).

OKUBO, P.G. and AKI, K. (1987), Fractal Geometry in the San Andreas Fault System, J. Geophys. Res. 92,

345–355.

RICHARDSON, L.F. (1961), The problem of Contiguity. An Appendix of Statistics of Deadly Quarrels,

General Systems Yearbook 6, 139–187.

SHIMAZAKI, K. and NAGAHAMA, H. (1995), Does earthquake breaks out irregularly? Collective and

individual characters of earthquakes, Kagaku (Sciences) Iwanami Shoten. Tokyo, Vol. 65, No. 4, 241–

256.

SUKMONO, S., ZEN, M.T., KADIR, W.G.A., HENDRAJJYA, L., SANTOSO, D., and DUBOIS, J. (1994), Fractal

Pattern of the Sumatra Active Fault System and its Geodynamical Implications, J. Geodyn. 22, 1–9.

SUKMONO, S., ZEN, M.T., HENDRAJJYA, L., KADIR, W.G.A., SANTOSO, D., and DUBOIS, J. (1997), Fractal

Pattern of the Sumatra Fault Seismicity and its Application to Earthquake Prediction, Bull. Seismol. Soc.

Am. 87, 1685–1690.

SUNMONU, L.A. and DIMRI, V.P. (2000), Fractal Geometry and Seismicity of Koyna- Warna, India, Pure

Appl. Geophys. 157, 1393–1405.

TOSI, P. (1998), Seismogenic Structure Behavior Revealed by Spatial Clustering of Seismicity in Umbria-

Marche Region (Central Italy), Annali di Geofisica 41, 215–224.

TURCOTTE, D.L. (1986), A Fractal Model of Crustal Deformation, Tectonophysics, 132, 261–269.

TURCOTTE, D.L. (1989), Fractals in Geology and Geophysics, Pure Appl. Geophys. 134, 171–196.

TURCOTTE, D.L.,Fractals and Chaos in Geological and Geophysics, (Cambridge Univ. Press, New York.

1992).

(Received May 21, 2003, Accepted April 27, 2004)

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Fractal Dimensions of Blocks Using a Box-counting Technique for the 2001 Bhuj Earthquake, Gujarat, India

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