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CHAPTER 6

Kappa () Model for Kachchh and Saurashtra

Regions of Gujarat

6.0 Introduction

The spectrum of earthquake strong ground motions has been found to be

attenuated more rapidly at high frequencies above the corner frequency than predicted

from the Brune (1970) model. The attenuation can be considered as decrease of

amplitude with distance or time. The effective attenuation within the crust is modeled

by the inverse of quality factor (Q). It was recognized that the decay of acceleration

spectra at high frequencies could not be explained by whole path attenuation within

the crust. Hanks (1982) introduced a frequency parameter, fmax, to explain the rapid

decay of acceleration spectra at high frequency. According to this model, the

acceleration spectra are flat above the corner frequency (as per Brune model), and fall

off rapidly above a second corner frequency (fmax) mainly due to propagation path

effects/local site effects. Hanks (1982) called this phenomenon of high-frequency

band limitation of radiated earthquake energy as “the crashing spectrum syndrome”.

Boore (1983) considered this high frequency decay as high cut filter in the

generalization of his stochastic technique of simulating accelerograms. Papageorgiou

and Aki (1983) agreed with Hanks‟ fmax but suggested that its origin was related to the

source rather than site attenuation. Singh et al. (1982) introduced a site attenuation

parameter, t*, as e-ft*

to explain the spectral attenuation of the SH waves. A similar

formulation was proposed by Cormier (1982). The t* has been considered as the

predecessor of the most frequently used factor to model high-frequency spectral

attenuation (Ktenidou et al., 2013). This spectral decay parameter was introduced by

Anderson and Hough (1984) and denoted by „‟ (Kappa). This factor was introduced

on the basis of observation that in log-linear space the decay of the acceleration

spectrum, A(f), can be considered as linear for the frequencies higher than a specific

frequency, fE (Anderson and Hough):

A(f) = A0 exp (-f) (6.1)

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A0 depends on source properties, epicentral distance and other path related factors, fE

is the frequency above which the spectral shape is linear on log-linear scale.

A linear relation between the observed values of and the distance was

suggested where the intercept of the variation with distance (0) corresponds to the

attenuation of S waves and slope of the variation corresponds to the incremental

attenuation (R). The linear relation can be written as:

= 0 + R R (6.2)

The above relation is based on the observation that the intercept varied

systematically according to the different site conditions and distance dependence was

not correlated and therefore assumed to be same for all the sites. There is no

theoretical basis to assume a linear dependence of . The studies have been done to

estimate the dependence of on various factors. These studies revealed the different

shapes of distance dependence of including non-linear (Anderson, 1991), concave

shape dependence (Gentili and Franceschina, 2011), convex shape dependence

(Fernández et al., 2010) and also no significant correlation with distance (Kilb et al.,

2012). Tsai and Cheng (2000) proposed a model where depends primarily on the

source, secondarily on the site and only slightly on distance. The studies have been

done to suggest that depends strongly on source (magnitude and focal mechanism)

effects (Purvance and Anderson, 2003; Papageorgiou and Aki, 1983; Petukhin and

Irikura, 2000; Halldorsson and Papageorgiou, 2005).

These above mentioned studies indicate that debate on origin, physical

meaning and its dependence on various factors continue. In spite of this, is being

used in number of applications like for the computation of site amplification factors

(Boore and Joyner, 1997); prediction of ground motion and calibration of ground

motion predication equations (Toro et al., 1997; Campbell, 2003; Atkinson and

Boore, 2006), for the characterization of sites (e.g. Drouet et al., 2010; Awashti et al.

2010), for its correlation with the engineering parameters such as peak ground

acceleration (pga) and Arias intensity (Mena et al., 2010). The has been found to

be a very useful parameter for modeling the acceleration spectra at high frequencies

and therefore necessary to constrain the attenuation and pga values.

In this chapter, the high frequency spectral amplitude decay of local

earthquakes has been analyzed by estimating for Kachchh and Saurashtra regions of

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Gujarat. The model has been developed by estimating at different sites, different

source-station distances and possible dependence of with earthquake size.

6.1 Method Adopted for the Estimation of

A number of different approaches have been suggested to estimate . These

include use of displacement spectra (Biasi and Smith, 2001), fitting of observed

response spectra (Silva and Darragh, 1995; Silva et al., 1997), full inversion of source,

path and site parameters (Margaris and Boore, 1998; Drouet et al., 2010; Oth et al.,

2011) and broad band inversions (Anderson and Humphrey, 1991; Humphrey and

Anderson, 1992). In this study, we follow the classical method of Anderson and

Hough (1984) which is still one of the most widely used method (e.g. Douglas et al.,

2010; Van Houtte et al., 2011; Ktenidou et al., 2013; Ktenidou et al., 2014).

According to this classical method, equation (6.1) can be written as:

log [A(f)] = log (A0) - f . log (e) (6.3)

This implies that can be estimated with a linear least square fit to the log-linear

observed spectra. The spectral decay parameter is given by:

= /.log (e) (6.4)

is the slope of the linear fit. The has been determined in the present study using

the following steps:

(i) Select the S-wave portion of the corrected acceleration time history

(ii) Compute the Fourier amplitude spectrum and plot it on log-linear scale i.e.

with a logarithmic y-axis (amplitude) and a linear x-axis (frequency).

(iii) The two frequencies are selected by visual inspection of Fourier spectrum:

first from the start of the linear downward trend in the acceleration

spectrum (F1) and second (F2), end of the linear downward trend. The F1

and F2 have been selected to the right of the corner frequency.

(iv) Fit a line in a least square sense between F1 and F2 and estimate using

equation (6.4).

The two frequencies F1 and F2 have been selected manually to reduce the

bias estimates of as F1 is found to be varied from seismogram to

seismogram. Also F2 shows variation to different signal to noise ratios.

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6.2 - Model for Kachchh Region

A total of 159 accelerograms of 34 earthquakes with magnitude range

3.3-4.9 have been analyzed to develop model for Kachchh region. Figure 6.1

shows the location of epicenters and recording stations.

Figure 6.1: Location of the epicenters and SMA stations used for the estimation of

in Kachchh region. The lower left panel shows the study area on the map

of Gujarat state.

An example of accelerograms of M 3.5 earthquake which occurred on April 19, 2014

and recorded at Badargarh (BDR) is shown in Figure 6.2.

Figure 6.2: Accelerograms of M 3.5 earthquake which occurred on April 19, 2014

and recorded at Badargarh (BDR).

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The log-linear plots of the acceleration spectra along with the best fitted line

between F1 and F2 are shown in Figure 6.3.

Figure 6.3: Log linear plot of acceleration spectra along with best fitted line for

Kachchh region of Gujarat.

.

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Figure 6.3 contd.

.

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The average values of κ estimated for different sites are given in Table 6.1.

Table 6.1: Average Kappa values at different stations in Kachchh

Sr. Station Code No. of

records.

Average

1 Anjar ANJ 3 0.028

2 Badargarh BDR 9 0.031

3 Bela BEL 8 0.022

4 Bhachau BHU 15 0.019

5 Chobari CHO 6 0.019

6 Desalpar DES 3 0.018

7 Dudhai DUD 4 0.039

8 Fathegarh FAT 2 0.025

9 Kandla KAN 6 0.02

10 Khavda KHA 12 0.026

11 Lakadia LAK 16 0.027

12 Lodrani LOD 7 0.027

13 Nakhatarana NAK 5 0.029

14 Rapar RAP 15 0.016

15 Suvai SUV 16 0.02

16 Vamka VAM 8 0.016

We observe that the value of varies from 0.016 to 0.039 with an average of 0.023.

We note that Narsaiah and Mandal, (2014) have also estimated κ for the Kachchh

region using different data set and inversion technique. They have obtained the values

as 0.02 - 0.03 which are in agreement with those estimated in the present study.

Figure 6.4: Distance dependence of for all stations in Kachchh region

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Douglas et al. (2009) have estimated average value of mainland France as

0.022 (Douglas et al., 2009). The value for Sino-Korean paraplatform has been

found to vary from 0.022 to 0.039 (Table 6.4).

The distance dependence of for all the stations is shown in figure 6.4. A

linear fit to this gives the relation:

= 0.00004458 *R + 0.0208 (6.5)

This implies that the dependence of with distance is not significant in

Kachchh region.

Figure 6.5 shows the variation of with distance for two sites of Kachchh

region – LAK (soft soil site) and RAP (hard rock site) where sufficient numbers of

earthquakes have been recorded. The best fit lines are also shown.

Figure 6.5: Distance dependence of for two sites (a) LAK, soft soil site (b)

RAP, hard rock site

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Figure 6.6: Plot between kappa and magnitude for different sites of Kachchh region

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Figure 6.6: Contd.

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Figure 6.6: Contd.

The equations of the best fit lines are:

κ = 0.000062 * R + 0.016 for the hard rock site RAP

and

κ = 0.000222*R + 0.0201 for the soft soil site LAK

We note that the dependence of on distance is similar for these sites and 0

( at R = 0) is lower for the hard rock site (0.016) as compare to for the soft rock site

(0.0201). This indicates the difference in site characteristics for these two sites.

Douglas et al. (2010) developed a regional kappa model for mainland France and

found that kappa depends on both local geology (Soil or Rock) and source to site

distance, and estimated 0 (soil) = 0.0270, 0 (rock) = 0.0207. Similarly 0 (soil) and

0 (rock) have been found to be 0.036 and 0.030 for Southern California (Kilb et al.

2012).Similar observations has been found out by Parolai and Bindi 2004).

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The estimated values of have been plotted versus magnitudes in Figure 6.6

for different sites of Kachchh region in order to investigate the dependency of on

earthquake size. These plots show that there is no significant correlation between

and magnitude at most of the stations for Kachchh region and therefore is not

related to source effect for the region.

6.3 - Model for Saurashtra Region

As has been discussed in chapter 5 of this thesis, the seismic activity in the

Saurashtra region has been confined to two sub-regions – Junagarh and Jamnagar.

Therefore the analysis has been done for these two regions separately. This study is

reporting first time the estimates of κ for Saurashtra region. Figure 6.7 shows the

location of epicenters and recording stations for these two sub-regions of Saurashtra.

The seismograms recorded at BBS stations and converted to accelerograms have been

used for estimating for Saurashtra region. The acceleration spectra have been

obtained by multiplying the velocity spectra with (=2πf) for the purpose.

Figure 6.7: Location of epicenters and seismic station in Junagarh and Jamnagar

region of Saurashtra.

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6.3.1 Estimation of for Junagarh Region

The 65 good quality seismograms of 20 earthquakes recorded at stations TAL,

MAL, JUN, AMR, and UNA (Figure 6.7) have been analyzed to estimate for the

Junagarh region. The two frequencies F1 and F2 have been selected as per procedure

described in the section 6.1 of this chapter. Figure 6.8 shows the log-linear plots of

the acceleration spectra along with the best fitted line between F1 and F2 for Junagarh

region.

Figure 6.8: Log linear plot of acceleration spectra along with best fitted line for

Junagarh region of Saurashtra (Gujarat).

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Figure 6.8: Contd.

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Figure 6.8: Contd.

The average values of for different stations of Junagarh region are given in

Table 6.2. The values vary from 0.014 to 0.026 for different stations with an

average of 0.0206.

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Table 6.2 shows the average kappa for 5 stations of Junagarh

Sr.

No.

Station Code No. of

records

k

Avg.

1 Amreli AMR 8 0.014

2 Junagarh JUN 18 0.022

3 Malia MAL 10 0.021

4 Talala TAL 10 0.026

5 Una UNA 18 0.020

The variation of values with distance is shown in Figure 6.9.

Figure 6.9: Variation of with distance for Junagarh region along with the best fit

line.

The equation of the best fit line is (Figure 6.9):

κ = 0.0177 + 0.0000686 *R (6.6)

This shows that the distance dependence of is not significant but it more than that

obtained for Kachchh region. The value of 0 is 0.0177 which is less than that

obtained for Kachchh region (0.0208). One of the implications of lower values is

that the high frequency strong ground motions may be higher in the regions with low

as compare to those regions with higher values.

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Figure 6.10 shows the variation of with magnitudes for different sites of

Junagarh region. We note that there is no significant correlation between the and

magnitude of earthquakes. This shows that is not related to source effect in

Junagarh region. The similar observation has been found for Kachchh region.

6.3.2 Estimation of for Jamnagar Region

The 54 good quality seismograms from 20 earthquakes recorded at four

stations have been analyzed to estimate the for Jamnagar region. Figure 6.11 shows

the log-linear plots of the acceleration spectra along with the best fitted line between

F1 and F2 for Jamnagar region.

Table 6.3 gives the average values of at different stations of the region. The

values vary from 0.033 to 0.036 with an average of 0.034.

Figure 6.10: Variation of with magnitude at different sites of Junagarh.

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Figure 6.11: Log linear plot of acceleration spectra for Junagarh region along with

the best fitted line between F1 and F2.

Table 6.3: Average kappa for 4 stations of Jamnagar region

Sr.

No

STN No. of

records

κ

avg.

1 MOR 5 0.033

2 SUR 10 0.036

3 RAJ 13 0.033

4 LAL 7 0.035

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The variation of with distance for Jamnagar region is shown in the Figure 6.12. The

best fitted line is also shown.

Figure 6.12: Variation of with distance for Jamnagar region along with the best

fitted line

The equation of the best fitted line is (figure 6.12):

κ= 0.00004214 * R + 0.029 (6.6)

The distance dependence of is not significant for Jamnagar region.

Figure 6.13 is shows the variation of with magnitude for 4 sites of Jamnagar region.

We note from figure 6.13 that there is no significant correlation between the

and magnitude for the earthquakes recorded at the sations RAJ and LAL. However,

for the stations SUR and MOR, the has an increasing trend with the increase in

magnitude. This implies that values are affected by source effect for these two

stations. It needs further investigation with more number of waveforms.

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Figure 6.13: Variation of Kappa with magnitude for the stations of Jamnagar

region

6.4 Conclusion

The average values of are found to be 0.023, 0.0206 and 0.034 respectively

for the Kachchh, Junagarh and Jamnagar regions of Gujarat. The distance

dependence of is found to be similar for these three regions while 0 differ for the

three regions. These are 0.0208, 0.0177 and 0.029 respectively for the Kachchh,

Junagarh and Jamnagar regions. This implies the average site characteristics for these

three regions. The values of estimated in the present study have been compared

with those of other regions of the world in Table 6.4. We note that the values are

comparable. It has been found that the is not affected by the source effect in

Kachchh and Junagarh region. For the two stations of Jamnagar region – SUR and

MOR- the values show some source effect also. The values estimated here are

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Table 6.4: Kappa values from different regions of the world

(Modified after Chandler et al., 2006) Sr.

No.

Region Vs30 Kappa

(km/sec)

1 Central and eastern north America 2.8 0.003

2 Western north America 0.65-0.7 0.066-0.073

3 Sino-Korean paraplatform 1.2 0.022-0.039

4 South China fold 1.5 0.014-0.028

5 Australia 0.65-2.4 0.011-0.045

6 Southern Iberia 0.48 0.069

7 NE Japan 0.53 0.067

8 Taiwan 0.48 0.081

9 Generic rock 0.85 0.035-0.040

10 Italy 0.62 0.045-0.07

11 Southern California 0.7 0.056

12 Iceland 0.65 0.04

13 NEHRP C class 0.7 0.04-0.05

14 Eastern North America 2.8 0.006

15 NE Sonora, Mexico - 0.04

16 Mainland France - κ_soil = 0.0270

κ_rock=0.0207

17 M5-7 0.62 0.0412

0.8 0.0407

1 0.0394

18 Southern California ANZA - κ_ soil = 0.036

κ _rock = 0.030

19 Shillong, India - κ_ soft rock =

0.0735

κ _firm rock =

0.00.0637

20 Kachchh ,Gujarat, India.

(This Study)

- 0.023

21 Junagarh, Gujarat, India.

(This study)

- 0.0206

22 Jamnagar, Gujarat, India.

(This study)

- 0.034

useful for the simulation of earthquake strong ground motions in the Kachchh and

Saurashtra regions and therefore important for the evaluation of seismic hazard of the

regions.