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Geotechnical Earthquake

Engineeringby

Dr. Deepankar ChoudhuryHumbo ldt Fel low, JSPS Fel low, BOYSCAST Fel lowProfessor

Department of Civil Engineering

IIT Bombay, Powai, Mumbai 400 076, India.Email: [email protected]

URL: http://www.civil.iitb.ac.in/~dc/

Lecture30

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IIT Bombay, DC 2

Module 7

Seismic Hazard Analysis

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IIT Bombay, DC 3

ExampleCase Study

on Gujarat, India

Ref: Ph.D. Thesis o f Jaykumar Shukla (2013),

IIT Bombay, Mumbai, Ind ia.

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Seismic Zones of Gujarat, IS: 1893-Part I (2002)

Location of urban areas selected

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Earthquake Moment Magnitude (Mw)

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

LogN

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

Entire Gujarat

Saurashtra

Mainland Gujarat

Kachchh

Regional Seismicity parameters

7/11/2013

GutenbergRichter recurrencerelations are derived using Least

Square Fit method using prepared

earthquake catalogue for Mw 4.

RegionPast seismicity

used (Year)a-value b-value G-R relation R2

Saurashtra 135 4.03 0.64 Log N=4.030.64 Mw 0.9783

Mainland 175 4.02 0.62 Log N=4.020.62 Mw 0.9370

Kachchh 189 3.41 0.41 Log N=3.410.41 Mw 0.9821

Gujarat 189 4.13 0.51 Log N=4.130.51 Mw 0.9899

Choudhury and Shukla (2011) in Disaster Advances, 4(2), 47-59.

Rastogi et al. (2013)

recommended b value

= 0.67for Saurashtra

region.

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b-value using ML method

Another popular method for estimation of b-

value is by using Maximum Likelihood (ML)method (Aki, 1965; Utsu, 1965)

7/11/2013

min

1

ln(10)( )b

u m

u is the sampling average of the magnitudes

Region b-value using ML estimate

Kachchh 0.526

Saurashtra 0.572

Mainland Gujarat 0.642

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Various Probability DistributionsNo Year Month Date Date

(Years)Latitude

(0N)Longitude

(0E) MwRecurrenc

e Time

(YearsLocation

1 1819 6 16 1819.5 24 69 7.8 Kachchh2 1845 4 19 1845.333 23.8 68.9 6.3 25.833 Lakhpat3 1848 4 26 1848.333 24.4 72.7 6 3 Mount Abu4 1856 12 25 1857 20 73 5.7 8.667 Surat5 1864 4 29 1864.333 22.3 72.8 5.7 7.333 Ahmedabad 6 1871 1 31 1871.083 21.2 72.9 5 6.75 Surat7 1872 4 14 1872.372 21.75 72.15 5 1.289 Bhavnagar8 1882 6 10 1882.5 23.2 71.38 5 10.128 Bhachau9 1903 1 14 1903.083 24 70 5.6 20.583 Kachchh

10

1919

4

21

1919.391

22

72

5.7

16.308

Bhavnagar

11 1921 10 26 1921.833 25 68 5.5 2.442 Kachchh12 1935 7 20 1935.583 21 72.4 5.7 13.75 Surat13 1938 3 14 1938.25 21.6 75 6 2.667 Satpura14 1950 6 14 1950.5 24 71.2 5.3 12.25 Kachchh15 1956 7 21 1956.583 23.3 70 6 6.083 Kachchh16 1963 7 13 1963.583 24.9 70.3 5.3 7 Pakistan17 1965 3 26 1965.25 24.4 70 5.1 1.667 Kachchh18 1966 5 27 1966.417 24.46 68.69 5 1.167 Pakistan19 1970 2 13 1970.167 24.6 68.61 5.2 3.75 Kachchh20 1976 6 4 1976.5 24.51 68.45 5.1 6.333 Allah Band21 1985 4 7 1985.333 24.36 69.74 5 8.833 Kachchh22 1993 8 24 1993.732 20.6 71.4 5 8.399 Rajula23 2001 1 26 2001.083 23.44 70.31 7.7 7.351 Kachchh24 2006 3 7 2006.25 23.79 70.73 5.7 5.167 Gedi,25 2007 11 6 2007.933 21.16 70.54 5 1.683 Junagadh

7/11/2013

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

CumulativeProbability

Earthquake (Mw>= 5) in Gujarat region

Pareto Distribution

Rayleigh Distribution

Weibull Distribution

Exponential Distribution

Earthquakes


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Recurrence Estimation

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Probability

Distribution

Model

Recurrence

interval(years)

Predicted

Last Eventoccurred on

Next

Earthquake

Expected on

Study Date

Considered(Nov 10th

2009)

Year Left

from Present

Date

Next

earthquakeexpected

before

Exponential 7.853 2007.933 2015.786 2009.85 5.936 Oct 2015*

Rayleigh 16.173 2007.933 2024.106 2009.85 14.256 Feb 2024

Pareto 3.135 2007.933 2011.068 2009.85 1.218 Jan 2011

Weibull 7.011 2007.933 2014.944 2009.85 5.094 Dec 2014*

*Note: This research output published in Journal Disaster Advances in Aug.

2011 was validated by actual occurrence of earthquake of September 2011.


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b-value proposed & those by other researchers

Study

Number

Application area b- value Reference Periods for

study taken

1 Kachchh 0.417 Based on least square fit, Present Study (1820-2008)

2 Saurashtra 0.64 (1872-2008)

3 Mainland 0.62 (1872-2008)

4 Entire 0.51 (1820-2008)

5 Kachchh 0.526 Based on ML estimate, Present Study (1820-2009)

6 Saurashtra 0.572 (1872-2009)

7 Mainland 0.642 (1872-2009)

8 Saurashtra 0.67 Rastogi et al. (2013) (1970-2010)

9 Gujarat 0.87

( 0.06)

WCE NDMA (2010) (*1800-2009)

10 Gujarat 0.72 Tripathi et al., (2005) -

11 Kachchh 0.43 Ashara et al., (2006) -

12 Kachchh 0.71

0.03

Jaiswal (2006) (1842-2002)

13 Gujarat 0.7 to 0.9

0.07

Raghukanth (2010) (1250-2008)

14 Gujarat 0.4 to 0.6 Kolathayar et al. (2011)

for Clustered catalogue

(250 B.C. -2010)

15 Gujarat 0.4 to 0.8 Kolathayar et al. (2011)

for declustered catalogue

(250 B.C. -2010)

16 Peninsular 0.92 Jaiswal and Sinha (2007) (1842-2002)

17 Gujarat region 0.55 Bhatia et al. (1999) -

18 Gujarat 0.89 Thaker et al. (2012) 1818-2008

Shukla and Choudhury (2012) in NHESS, 12, 2019-2037.

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Entire Gujarat is divided into three regions

Kachchh

Saurashtra

Mainland Gujarat

Earthquake catalogue is divided as per these three regions

Only fault sources are used as seismic sources

Poisson distribution for earthquake occurrence

All the faults are Normal faults, depth ranging 10 to 15km from

ground surface.

DSHA - Some starting points

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1 Describes the potential for dangerous, earthquake-related natural

phenomena i.e. Maximum Considered Earthquake (MCE)

2

The earthquake hazard for the site is a peak ground accelerationof 0.57g resulting from an earthquake of magnitude 5.7 on theNarmada Son Fault at a distance of 11.42 km from the site.

3 Sometimes called Deterministic Scenario in Magnitude,

Distance pair i.e. (5.7, 11.42)

DSHA

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67 68 69 70 71 72 73 74 75

67 68 69 70 71 72 73 74 75

19

20

21

22

23

24

25

26

19

20

21

22

23

24

25

26

F18

F17

F14 F15

F13

F12 F25A F5

F2

F1

F4

F3F6

F7

F8

F10

F9

F33

F35

F34

F37

F38 F42

F41

F43

F45 F46

F31

F32

F24

F49F23

F48

F27

F26

F21

F28F29

F30

Legend :

n thFault, Fn

Fault Map of study region

7/11/2013

Total 40 major faults

are considered.

Length derived from

referred literature and

maps.

Maximum earthquake

magnitude calculated

from relationships

recommended by few

researchersconsidering one third

length as rupture

surface.Shukla and Choudhury (2012) in NHESS, 12, 2019-2037.

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GMPEs selected

7/11/2013

GMPE Applicability Remark

Abrahamson andSilva (1997) Worldwide shallow crustalearthquake

Boore et al.

(1997)

Shallow crustal earthquake

of Western north America

(Rock site definition is in accordance

with NEHRP seismic code)

Campbell (1997) Worldwide shallow crustal

earthquake

(for Mw > 5 and sites with distance to

seismogenic rupture 60 km in active

tectonic region)Sadig (1997) Shallow crustal earthquake

of California

(Moment magnitude Mw = 4 to 8 and

distance up to 100 km).

Toro et al. (1997) Crustal earthquake of

Intraplate region in Eastern

and Central North America

(For spectral period less than 0.2 sec,

values limited to 1.5 g and periods less

than 1 sec are limited to 3 g.)

Frankel et al.

(1996)

Intraplate region of Central

and Eastern North America

Raghukanth and

Iyengar (2007)

Peninsular India (For sites with shear wave velocity Vs

3.6 km/sec.)

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Various GMPEs

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0 20 40 60 80 100 120 140 160 180 200

1E-3

0.01

0.1

Spectralaccele

rations(g)

Distance from Hypocenter (km)

Abra.-Silva (1997)

Boore-Joyner-Fumal (1997)

Campbell (1997)Frankel (1996)

Sadigh (1997) Rock

Toro (1997)

Raghu Kanth & Iyengar (2007)

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DSHA ResultsName of

City/Urban Area

Location Deterministic Seismic Scenarios (Controlling Fault/Magnitude-Distance pairs)

Short Period (0.2 sec)

(two stored building)

Long Period (2 sec)

(highrise buidlings)

N0

E0

Fault Mw Distance (km) Fault Mw Distance (km)Ahmedabad 23.030 72.580 F24 5.5 14.290 F17 7.0 171.240

Amreli 21.602 71.218 F6 5.5 15.500 F7 6.0 25.330

Anjar 23.112 70.023 F14 7.0 13.320 F14 7.0 13.320

Baroach 21.715 72.977 F33 5.5 7.190 F33 5.5 7.190

Bhavnagar 21.770 72.143 F30 5.5 8.000 F30 5.5 8.000

Bhuj 23.252 69.662 F14 7.0 14.500 F14 7.0 14.500

Dholavira 23.883 70.215 F17 7.0 5.000 F17 7.0 5.000

Dholera 22.248 72.195 F31 5.5 20.330 F28 6.0 33.030

Dwarka 22.241 68.966 F2 5.5 12.000 F13 7.0 89.240Gandhidham 23.071 70.135 F14 7.0 15.960 F14 7.0 15.960

Gandhinagar 23.229 72.651 F26 5.5 31.570 F17 7.0 174.240

Jamnagar 22.466 70.066 F13 7.0 34.480 F13 7.0 34.480

Junagadh 21.515 70.456 F7 6.0 14.370 F7 6.0 14.370

Mandavi 22.833 69.346 F13 7.0 34.720 F13 7.0 34.720

Mehsana 23.598 72.380 F24 5.5 15.010 F17 7.0 143.610

Morvi 22.814 70.829 F14 7.0 59.090 F14 7.0 59.090

Palanpur 21.171 72.438 F24 5.5 10.250 F17 7.0 152.810

Patan 23.850 72.114 F49 5.5 15.600 F17 7.0 115.390

Porbundar 21.643 69.611 F2 5.5 11.640 F43 7.0 110.190

Rajkot 22.283 70.800 F13 7.0 83.280 F13 7.0 83.280

Surat 21.194 72.819 F34 5.5 17.760 F13 5.5 17.760

Surendranagar 22.718 71.637 F28 6.0 30.570 F17 7.0 105.020

Vadodara 22.306 73.187 F24 5.5 22.190 F24 5.5 22.190

Valsad 20.610 72.925 F37 5.5 19.000 F37 5.5 19.000

Veraval 20.912 70.353 F2 5.5 12.000 F2 5.5 12.000


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Typical scenarios for Ahmedabad City

7/11/2013

0 1 2 3 4

0.01

0.1

F24 (Mw 5.5,14.29 km)

F17 (Mw 7, 171.24 km)

SpectralAccelera

tions(g)

Spectral Period (sec)

Ahmedabad city

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Typical Deterministic SpectraCities representing

Kachchh region hasvery high ground

motions and

Mainland Gujarat

has lowest ground

motions. For

Saurashtra mixed

results are obtained.

Comparison of

present study withIS: 1893Part I

(2002) spectra

7/11/2013

IS 1893 (2002)- Zone III

0.5 Fractile0.84 Fractile

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Spectral

Acceleration(g)

Spectral Period (s)

Deterministic Spectra Ahmedabad


Note: 0.5 Fractile = 50th percentile (representing

MCE) and 0.84 Fractile = 84th percentile

(representing SSE), with 5% damping

D i i i i

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Deterministic scenarios

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Name of City/Urban

Area

PGA (g) IS: 1893 Part1 (2002) PGA in (g)

Median

(0.5

percentile)

Zone assigned PGA (MCE) PGA (DBE)

Ahmedabad 0.125 III 0.16 0.08Amreli 0.116 III 0.16 0.08

Anjar 0.530 V 0.34 0.17

Baroach 0.220 III 0.16 0.08

Bhavnagar 0.230 III 0.16 0.08

Bhuj 0.620 V 0.34 0.17

Dholavira 0.670 V 0.34 0.17

Dholera 0.160 III 0.16 0.08

Dwarka 0.089 IV 0.24 0.12

Gandhidham 0.490 V 0.34 0.17Gandhinagar 0.053 III 0.16 0.08

Jamnagar 0.200 IV 0.24 0.12

Junagadh 0.176 III 0.16 0.08

Mandavi 0.154 V 0.34 0.17

Mehsana 0.125 IV 0.24 0.12

Morvi 0.085 IV 0.24 0.12

Palanpur 0.160 IV 0.24 0.12

Patan 0.138 IV 0.24 0.12

Porbundar 0.160 III 0.16 0.08

Rajkot 0.060 III 0.16 0.08

Surat 0.094 III 0.16 0.08

Surendranagar 0.084 III 0.16 0.08

Vadodara 0.073 III 0.16 0.08

Valsad 0.091 III 0.16 0.08

Veraval 0.188 III 0.16 0.08


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IIT Bombay, DC 19

Probabilistic Seismic

Hazard Analysis (PSHA)for Gujarat, India

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Four Seismicity models used

7/11/2013

1E-5

1E-4

1E-3

0.01

0.1

4 5 6 7 8

AnnualR

ateofEvents>M

w=4

Earthquake Moment Magnitude (Mw)

F14 (Exponential, b-value LSF estimate) F14 (Exponential,b-value as per ML estimation)F15 (Exponential, b-value LSF estimate) F15 (Exponential,b-value as per ML estimation)

F17 (Exponential, b-value LSF estimate) F17 (Exponential,b-value as per ML estimation)

F18 (Exponential, b-value LSF estimate) F18 (Exponential,b-value as per ML estimation)

F14 (Exponential, b-value=0.92, Jaiswal and Sinha (2007)) F14 (Characteristic Model)




L i T i l ti

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Logic Tree simulations

7/11/2013

Logic Tree used in seismic hazard computations (Figures in the bracket show the weightage assigned in

seismic hazard computations) (Abbreviations: Raghu Kanth and Iyengar (2007)RI07, Abrahamson and

Silva (1997)AS97, Frankel (1996)F96, Toro et al., (1997)T97, Boore-Joyner and Fumel (1997)

BJF97, Campbell (1997)C97, Sadigh et al. (1997)- S97).

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PSHA Calculation overview

7/11/2013

Number of Faults = 40

Number of Magnitude-Recurrence relations = 4

Number of GMPEs used = 7

For each city computations = 40 x 4 x 7 = 1120

For PSHA map, 8430 grid points means = 8430 x 4 x 7 =233520 computations carried out in present study.

This can be handled using computer software like SEISRISKIII , CRISIS 2007 , EZ-FRISK88 etc. and MS Excel.

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Designation Chance of

Exceedance

Return

period

(Years)

Earthquake Designation

Level 1 50 % 72 Operational BasisEarthquake (OBE)

Level 2 10 % 475Contingency Level

Earthquake (CLE)

Level 3 2 % 2475 Max. CredibleEarthquake (MCE)

Which means each 1120/233520 computations are repeated for each

level of ground motion computations.

Performance Levels of Ground motions considered

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1E-4 1E-3 0.01 0.1 1

1E-11

1E-9

1E-7

1E-5

1E-3

0.1

2475 years return period


Annualfrequencyofexceedence

Peak ground acceleration, (g)

Ahmedabad


Typical Seismic Hazard Curve


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Comparison of Seismic Hazards

7/11/2013

1E-4 1E-3 0.01 0.1 1

1E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1



Annualfrequencyo

fexceedence

Peak ground acceleration, (g)

Ahmedabad

Amreli

AnjarDholavira

Dholera

Dwarka

Gandhinagar

Mandavi

Mehsana

Morvi

Palanpur

Patan

Porbundar

Surendranagar

Valsad

Veraval

Baroach

Bhuj

Gandhidham

Rajkot

Jamnagar

JunagadhBhavnagar

Vadodara

Surat


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How to use these Hazard Curves..

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How to use these Hazard Curves..

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Retrun Period 2475 years



IS 1893 (2002)- Zone III

SpectralAcceleration(g)

Spectral Period (s)

Ahmedabad

Development of Uniform Hazard Spectra

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Deaggreagation analysisJamnagar city

7/11/2013

0.000

0.025

0.050

0.075

0.100

3 1. 2 5

3 3. 7 5

4 3. 7 5

73.75

81. 25

101 .2

5

108.7

5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

Distancefromsitetosource(km)

ProbabilityDensity

Earthqu

akeMo

mentM

agnitude(M

w)

S iti it A l i U b Citi

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Sensitivity Analysis among Urban Cities

7/11/2013

Ahmedabad

Amreli

Anjar

Baroach

Bhuj

Bhavnagar

Dholavira

Dholera

Dwarka

Gandhidham

Gandhinagar

Jamnagar

Junagadh

Mandavi

Mehsana

Morvi

Palanpur

Patan

Porbundar

Rajkot

Surat

Surendranagar

Vadodara

Valsad

Veraval

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50


PGA(g)

Cities

Exponential model, b-value estimated using LSF method

Characterastic Earthquake Model

Exponential model, b=0.92, as per Jaiswal and Sinha (2007)

Exponential model, b-value estimated using ML method

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Summary The seismicity within the Gujarat is very complex and migrating in nature

and region wise not common. Seismic hazard assessment using singleseismicity parameter for entire Gujarat may not correctly represent to the

actual seismicity distribution.

It is therefore important to carryout the seismic hazard analysis for

Gujarat region using regional seismicity parameterswhich are consistentwith present state of seismicity in the Gujarat.

It is observed that in the prepared earthquake catalogue most of the events

are from historic records i.e before 1960 and seismicity rate was constant

up to 1962. After 1962, the seismicity rate within the Gujarat is observedto be increased. However for larger earthquake magnitude threshold i.e.

Mw 4 catalogue can be considered to be complete for its use in seismic

hazard assessment.7/11/2013

Summary

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The seismicity parameter (b-value) is calculated by two different

approaches namely Least Square Fit method and Maximum Likelihood

method. These parameters constitute the basic framework required forseismic hazard assessment for Gujarat region.

The b-values obtained using Least Square Fit are 0.41, 0.64 and 0.62 for

Kachchh, Saurashtra and Mainland Gujarat, respectively.

The b-values estimated using the Maximum Likelihood estimations (Aki,

1965) are observed to be 0.53, 0.57 and 0.64 for Kachchh, Saurashtra and

Mainland Gujarat, respectively.

For Kachchh region, the recurrence interval for the earthquake magnitudeMw= 6 is less than the 25 yearsand Mw7.5 it approaches 120 to 150

year. For both Mainland and Saurashtra region The recurrence interval

for earthquake magnitude Mw= 5 is greater than 25 years and for Mw= 6

it is more than 125.

Summary

S

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For earthquake magnitude Mw5 in Gujarat region, it is observed that

Exponential Model and Weibull Model are appropriate probabilitydistributions for Gujarat region.

Estimated seismic ground motions using Deterministic approach

advocates that across Gujarat region, the seismic activities thereby

seismic ground motions are dominated by the faults in the Kachchhregion.

It is again interesting to note that two major faults in Kachchh region

namely F17 (Iceland Belt Fault) and F14 (Kachchh Mainland Fault) are

the major contributors to the seismicity in the entire Gujarat.It is alsonotable that the present seismicity in Kachchh region is concentrated

near F14 (KMF)and the location of Bhuj earthquake of 2001 is also

close to this fault.

Summary

Summary

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Though the peak ground acceleration (PGA) values are not in close

agreement with those recommended by IS:1893 Part 1 (2002) for some

cities but overallseismic ground motion distribution across the Gujaratregion is more or less in agreement with regional distribution

recommended by IS: 1893 Part 1 (2002).

It is observed that the seismic hazards for few cities are found to be

affectedby near sources at lower spectral periods and by distant

sources at higher spectral periods. In presently study it is observed that

many cities of Gujarat like Ahmedabad, Gandhinagar etc. are

influenced by distance source seismicity at higher spectral periods.

For all cities in the Mainland Gujarat, the present PSHA study results

in close agreement with the seismic ground motions recommended by

IS: 1893 Part 1 (2002).

Summary

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IIT Bombay, DC 35

Some Earlier Seismic Hazard Analysis for

India: Case Studies

Studies Carried out in India

Jabalpur city

Sikkim HimalayaDelhi

Dehradun

Guwahati

Bangalore

Kolkata

J b l it

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36

Jabalpur city

Final hazard map of Jabalpur (PCRSMJUA, 2005)

Sikkim Himalaya

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IIT Bombay, DC 37

Sikkim Himalaya

Nath, 2007

Seismic Hazard analysis was

carried out deterministically byconsidering the seismotectonic

parameters and presented

maximum credible earthquake

for Sikkim by Nath et al. (2006).

Further developed new

attenuation relation for Sikkim

Himalaya, a finally he

developed seismicmicrozonation map using

geographical information

system (GIS)

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IIT Bombay, DC 38

Delhi Site-specific Microzonation Study in Delhi Metropolitan City by

2-D Modelling of SH and P-SV Waves by Parvez et al. (2004),Microzonation of earthquake hazard in Greater Delhi area by

Iyengar and Ghosh (2004), and Seismic Microzonation Studies

for Delhi Region by Rao and Neelima Satyam (2005) and First

Order Seismic Microzonation of Delhi, India Using Geographic

Information System (GIS) by Mohanty et al. (2006).

Mohanty et al. (2006) prepared a first order seismic

microzonation map of Delhi using five thematic, layers viz., Peak

Ground Acceleration (PGA) contour, different soil types at 6 mdepth, geology, groundwater fluctuation and bedrock depth,

integrated on GIS platform.

Delhi

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IIT Bombay, DC 39

Delhi

Final hazard

map of Delhi

(Bansal and

Vandana,2007)

D h d

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40

Dehradun

Spectral acceleration map of Dehradun (Ranjan, 2005)

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Bangalore

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IIT Bombay, DC42

Bangalore

Peak ground

acceleration map

of Bangalore

using DSHA.

(Sitharam, 2008)

Based on 950 bore

hole data with SPT

and MASW at 56

locations

Bangalore

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IIT Bombay, DC43

Bangalore

PGA contour at

rock level with

10% probability of

exceedance in 50

years forBangalore city

(Sitharam, 2008)

Kolkata

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IIT Bombay, DC44

Kolkata

the major part of

Kolkata City can

experience PGA value

above 0.25g, whichsuggests that Kolkata

is in seismic zone IV.

(Mohanty 2008)

In the seismic zonation

map, Kolkata lies on theboundary of zone III and

IV (which suggest the

PGA value of 0.2 and

0.25g, respectively)

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IIT Bombay, DC 45

End of

Module 7

Documents

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