Deterministic investigation of effect of Stress drop on Seismic Site Response Analysis of Allahabad City

Due to the high stress of Faizabad ridge close to Allahabad city and the absence of strong-motion records for any engineering studies, it is essential to use a stochastic model to study the deterministic earthquake scenario of Allahabad city. The work investigates the effect of stress drop for an earthquake on 30 sites (83 boreholes) located across the city using 1-D seismic site response analysis. The ground motion has been simulated for Allahabad fault using stochastic finite fault model for stress drop ranges from ~70 bar to ~200 bars. Simulation results show the Peak Ground Acceleration (PGA) value of 0.026 g and 0.085 g at 70 and 200 bars stress drops, respectively. Site response results reveal that Indian Standard IS: 1893-2002 underestimates the PGA at higher stress drop compared to the estimated spectral acceleration values. Further, the lower stress drop can give a higher mean spectral acceleration at a long-period. Contour plot of surface-level PGA, low and high period spectral acceleration with response spectra for Allahabad city shows the variation with stress drop.


Introduction
Researchers in the early 1960's understood that the effect of an earthquake is not alike on every type of soil. Medvedev (Medvedev, 1962) has correlated intensity increment with the surface geology for the events that happen in the Asian region. Many other similar relationships were developed for other cities California (Evernden and Thomson, 1988), Japan (Shabestari et al., 2004), Turkey (Picozzi et al., 2009), Darfield (Bradley, 2012) and Xanthi (Stamati, Klimis and Lazaridis, 2016) are used to know the response of an earthquake on local surface geology. Regional soil site characteristics play a significant role in determining the damage potential of arriving seismic waves. Destruction of Mexico City after Michoacan earthquake (1985) is an example of the effects of local surface geology on-site response (Avilés and Pérez-Rocha, 1998). The impact of amplified ground motion is having a destructive effect on structures when periods matches with the site periods. In the Loma-Prieta earthquake (1989), soft soil sites were severely affected in the San Francisco, while adjacent hard rock sites were amplified by 2 to 4 times (Housner and Thiel, 1990;Segall, Bürgmann and Matthews, 2000). Substantial damages have been reported at Ahmedabad due to Bhuj earthquake (2001) despite a large epicentral distance (160 miles) because of site amplification of thick alluvial deposit under Ahmedabad (Sitharam and Govindaraju, 2004). Many studies have performed on the site-specific response by considering local soil properties of sub-surface layers of SPT-N values. As per authors knowledge, none of the studies reported on the effect of stress drop on-site response analysis. Parameters such as thickness, type of soil, density, plasticity index, groundwater table and shear wave velocity (calculated by the existing relationship between shear wave velocity and SPT-N values) are required. Pitilakis (2004) stated that to estimate the site classification based on SPT-N or shear wave velocity along with site response couple with response spectra is the most efficient way to account for site effects. Further, no attempt was made to determine the site classification and site response of Allahabad city, which lies in the eastern section of Indo-Gangetic plains (IGP).

About Prayag city
Allahabad formerly known as Prayag (place of offerings) is one of the most important cities of Uttar Pradesh. It is situated in the north-central section of the country in the southeastern section of the state. Allahabad city lies between latitude 25°30 ′ . 04.1" N to 25°29 ′ 58.3" N with its centre coordinates as 25.45° N & 82.5° E. and situated at the intersection of river Ganga and Yamuna. The city hosts the world's largest religious gathering (about 100 million people) known as 'MahaKumbh' held after every 12 years and 'ArdhKumbh' after every six years. Further, the one-month carnival takes place every year, known as 'Magh Mela'. All Saints Cathedral, KhusroBagh, Allahabad Highcourt, Alfred Park, Anand Bhavan and New Yamuna Bridge are important heritage structures. Allahabad is 7 th most crowded city of the state and 36 th most populous in the country with an approximate population of 1.11 million (Census, 2011).
Allahabad is strategically vital for politics, education, economy and tourism. Headquarters of North-Central Railway and four National Highways (NH), viz., NH 2, NH 27, NH 76 and NH 96. NH 2 is also known as Grand Trunk road which connects Delhi to Kolkata via Allahabad. NH 2 is a part of Asian Highway AH 1. the Waterway 1, the longest in India connects Allahabad and Haldia (Kolkata) (Figure 1). There are seven bridges out of six are in use for connectivity across the rivers. Allahabad (now Prayag) is well known for its academics. It is an education hub of the state and establishment of headquarters of the world's largest examining body viz., 'Board of High School and Intermediate Education'. The city has renowned universities, Research Institutions, and Technical and Medical Institutions which attract students from across the country. Many Infrastructural projects for residence, industrial projects, roads, metro rail, flyovers and hospital buildings, etc., have been planned. As a result, the seismic vulnerability of the city needs proper attention. Many of the existing structures were designed using IS codes recommendation but didn't consider the local soil site conditions and restitution characteristics of the earthquake.

Geology of Allahabad District
Geologically, the city is situated at IGP, a vast fore-deep region of Himalayas. As per Dasgupta. (Dasgupta, 2000), the study area has characterised into Ganga alluvial plain, Yamuna alluvial plain and Vindhyan plateau. The geomorphic features of the area as follows: (1) Active flood plains, confined in the vicinity of the river system; (2) Older alluvial plains, defined by depositional and erosional terraces, generally existing in patches along the active plain; and (3) Rocky surface (Denudational hills), mainly of quartzite in nature and prominently found in the Trans-Yamuna area. The surface lithological behaviour is found to be quite different in Trans Ganga and Trans Yamuna area, with some hard rock strata in the northern part of Trans-Yamuna area (Pandey, 2008). This is the reason that the borehole data is only up to 10m. The Ganga basin, which dominates the geological setup of the region is a part of IGP, with most massive modern alluvial sedimentary terrain having a length of about 1000 km and width varying from 200 -450 km, being more extensive in the western part and narrower in the east (Singh, 1996) (Bagchi and Raghukanth, 2017) (Singh 1996;). The IGP, a down-warp of the Himalayan foreland converted into alluvial plains of variable sediment depth.   Table 1 shows the density of stones found at the bed level. In this research work, the average rock density of 2533 kg/m 3 is used for site response analysis.

Seismotectonic Setup
The IGP is the most significant contemporary alluvial sedimentary plain that inhibits a population of more than 200 million people. These plains exhibit several faults having three prominent trends, viz, North North East (NNE)-South South West (SSW) to North East (NE)-South West (SW), North North West (NNW)-South South East (SSE) to North East (NE)-South East (SE) and East (E)-West (W). Out of these, the E-W trending fault set in Azamgarh-Gorakpur area defines the MirganjGraben (Szulc et al., 2006). The E-W elongated IGP shaped in response to the collision of the Indian plate, and Eurasian Plate that caused the uplift of Himalaya initiated in the Palaeogene is an active foreland basin (DEWEY JF and BIRD JM, 1970), (Singh, 2013)(Acharyya and Saha, 2018). The region integrates numerous covered faults and ridges in its basement (Valdiya, 1970) (Acharyya and Saha, 2018). In the IGP, groups of significant ridges added due to seismotectonic movements of the basements. The Faizabad ridge and Munger ridge, close to Allahabad, are bounded by various faults. They are a prolongation of Bundelkhand massif and Satpura massif (Singh, 2015). Quittmeyer and Jacob (Quittmeyer and Jacob, 1979) compared the seismicity of the IGP with Himalaya and considered it as moderately seismic. (Sinha, R; Tandon, SK; Gibling, MR; Bhattacharjee, PS; Dasgupta, 2005) studied all these ridges bounded by faults and its tectonic extension from the Indian shield.  Kayal (2008). In this study, only faults having the length of more than 50 km are considered and are shown in Figure 2. Figure 2 shows the presence of many potential seismic sources of Home Affairs has shown that Allahabad City lies near Faizabad Ridge, which is inactive over around thirty decades. The ridge is highly stressed due to a large seismic gap and can, therefore, cause a high magnitude earthquake shortly (Mishra, D. and Uniyal, 2010). Further, it also concluded that subduction of Indian plate under the Asian plate by 5.25 m could generate a Great earthquake (Anbazhagan et al., 2017). Due to the possible seismic gaps and tectonic set up of IGP, it is vital to carry out seismic site characterisation and the site-specific seismic ground response of the city like Allahabad along with the effect of stress drop on ground motion response.

Local Soil Condition
Soil data obtained from various Government and private geotechnical agencies for 30 sites at Allahabad city( Fig. 2). Essential properties like a type of soil, density of soil, plasticity index, depth of groundwater table and SPT-N value extracted from the dataset. Shear wave velocity has calculated from the correlation established between SPT-N value and shear wave velocity proposed by (Anbazhagan, Kumar and Sitharam, 2013) for IGP for all type of soils. = 68.96 ( ) 0.51 (1) The shallow sub-surface soil characteristics play an important role in site-specific ground response analysis and also contribute to the amplification potential of a site. So it is necessary to account near-surface soil properties to determine surficial ground motion. The shallow depth (30 m) shear wave velocity profiles are unavailable for Allahabad city. However, soil characteristics and SPT-N values are available from bore log information of 83 boreholes drilled at 30 sites covering the main area of Allahabad city. The depth ranges from 10 m to 30 m in depth. The category of the sites is estimated as per its shear wave velocity using SPT-N value (International Buiding Code, 2009). Boore (Boore, 1983) methodology used to estimate shear wave velocity at 30 m from shallow depth boreholes. For 30 -250 m soil column, a linear variation of shear wave velocity is assumed.

Simulation of synthetic ground motion
Engineers are primarily interested in the strong ground motion, which can affect the people and their environment. In general, the earthquake strength is directly proportional to the amplitude values if they last for sufficient duration. Much information inferred from the amplitude content, but a dynamic response of a structure is very sensitive to the frequency at which it loaded. In the present research work, the finite fault seismological model of Motazedian and Atkinson (Motazedian and Atkinson, 2005) has used in the simulation of rock level time history. It is a modified form of point source stochastic seismological model (Boore, 1983). The faults divided into N number of sub-faults, and every single sub-fault symbolises as the point source. Acceleration time histories of ground motion have calculated for the point source by using a stochastic seismological model. The simulated synthetic ground motions are added up by using time lags. Equation 2 shows the Fourier amplitude spectrum of ground motion provided by the point source seismological model due to i th sub-fault.
where, = i th sub-fault's moment magnitude; G = geometric attenuation ; V S = Shear wave velocity (3.6 km/sec); = site to sub-fault distance; Q = regional quality factor; = scaling factor to convert sub-faults energy of high-frequency spectral level; 0 ( ) = dynamic corner frequency; ( )= site amplification in comparison to the source; the average shear-wave velocity may be less than 1500 m/sec. − 0 = high cut filter (rapid spectral decay at high frequencies) (Boore, 2003) (Anderson and Hough, 1984). 〈 ∅ 〉 =Average Radiation coefficient; = density of the crust at the focal depth. The coefficient √2, is the product of the free surface amplification and partitioning of energy in orthogonal directions. The Moment of the i th sub-fault from the slip distribution expressed as: where, 0 = total seismic moment of all the sub-faults; = average final slip acting on the i th sub-fault. The dynamic corner frequency 0 ( ), the seismic moment 0 moreover, the stress drop ∆ relationship is shown in equation 4. For simulating rock level ground motion at Allahabad, the kappa factor for soft rock site is obtained as 0.057 using Eq. (6). In the present study, Table 3 shows the parameters used to simulate the synthetic ground motion. Further, the maximum magnitude of the earthquake for all faults (Fig. 1) has been calculated from Wells and Coppersmith (1994) relation using fault rupture length as 1 3 ⁄ of total fault length (Wells and Coppersmith, 1994 al., 2015). The estimated Peak Ground Acceleration (PGA) is 0.248g, as shown in Fig. 4 whereas recorded PGA at KTP station is 0.260g Mainshock of the earthquake (also known as Gorkha earthquake),  Table 4 provides the primary source parameters of Nepal earthquake.

Simulated ground motion for Allahabad City
The ground motion at Allahabad city have simulated for the stress drops ranges from 70 bars to 200 bars at Allahabad fault(~70 km from city). Acceleration time history has been plotted using MATLAB, as shown in the Figs. 5 (a)-(f). The PGA ranges from 0.026g to 0.085g. The simulated time history is used in the scrutiny of stress drops on-site response analysis at a different time period.

Site response analysis
Local soil site conditions have proven a significant effect on ground surface parameters. The one-dimensional equivalent linear approach is used to carry out site response analysis for all 83 boreholes of the Allahabad city using 1-D analysis. The hyperbolic model is selected and calibrated for 250 m thick soil column using EPRI (Inst:, 1993) curves data (Park and Hashash, 2005). Reference strain calculated for at the middle of each 100 m thickness. As the profiles are not available for Allahabad city, so they have been computed using SPT-N values from the correlation, = 68.96 ( ) 0.51 (Anbazhagan, Kumar and Sitharam, 2013) which is based on IGP. To account for nonlinear behaviour of soil, damping ratio versus shear strain curves and shear modulus versus shear strain curves proposed by Vucetic and Dobry (1991) for clay and sand-based on plasticity properties. Synthetic ground motion developed for bedrock level is applied to carry out site-specific response analysis.

Results and Discussions
A comprehensive set of analyses is conducted to understand the effect of stress drop on surfacelevel ground motion and response spectra of the sites. The simulated ground motion has used as an input motion at bedrock level to known the response of all 83 boreholes at 30 sites. The ground motions due to fault 1 (Allahabad fault) for different stress drops (70, 100, 125, 150, 175 and 200 bars) with 0.06 kappa factor has used at the level of 250 m thick soil columns (below the 30 m depth). Simulated acceleration time history was used as input ground motion, applied at bedrock level to carry out seismic site response analysis for Allahabad city. From the results of site response analysis, surface-level acceleration time history has extracted for all the sites at different stress drops. Input ground motion, output ground motion and mean of all the 30 sites acceleration time history for Allahabad city was plotted and has shown in Figures. 6 (a) Table 5 shows the Maximum input, maximum and minimum output PGA values with their occurrences at similar sites. Government Press (Civil Lines) which lies in the centre of the city shows maximum output PGA for all the stress drops except 125 bar whereas minimum PGA value has observed at Kilaghat which is near to river having shallow bedrock depth. Maximum average PGA was found to be 0.12 g at Government Press (Civil Lines) while a minimum of 0.05 g at Kilaghat.
Results demonstrated that the Civil lines area is more susceptible to ground motion amplification. Table 6 shows the amplification of PGA for all stress drops. Response spectrum for Allahabad city has been plotted at 5 % damping for different stress drops and compared with suggested response spectrum by Indian Standard code IS: 1893-2002 and are shown in Figures. 7 (a)-(f). All response spectrum is discussed below in terms of amplification/de-amplification for different time period ranges.    In addition to the above-detailed response spectra of the city, the average response spectra of all stress drops are also plotted and compared with response spectra proposed by IS: 1893 -2002 (Fig. 8). The figure shows that average output is amplified almost 2 to 4 times for the entire range of period. At zero period (PGA), the value of mean spectral acceleration has estimated as 0.12 g, which is more than the recommended value of 0.1 g for the Allahabad city. Maximum mean spectral acceleration is found to be 0.14 g, 0.28 g, 0.   Figure. 9 (b), maximum PGA value of 0.35 g occurs at District court, while minimum PGA value of 0.12 g is at Kilaghat and Baluaghat. These contour maps show that the area like Civil Lines are more prone to seismic amplification, whereas Kilaghat and Baluaghat are earthquake amplification. The surface level PGA contour has been plotted only for 200 bar stress drop, which gives maximum PGA for all the sites (Fig. 10). Maximum and minimum estimated PGA value was found to be 0.24 g and 0.06 g, respectively.

Conclusions
The study on the effect of stress drop on-site response analysis has been performed out for Allahabad city using six synthetic acceleration-time histories generated for Allahabad fault (~70km from Allahabad city). Thirty sites across the city are classified as per IBC-2009, out of which 12 are of C type, 15 are of D type, and 3 falls in E category. The response of Allahabad city for the simulated ground motion are presented in terms of response spectra, and its variation is shown in the form of contour plots (PGA contour maps low (0.3 s) and high (1.0 s) natural time periods). For a short period, the maximum Spectral acceleration values observed for stress drop of 200 bar is 0.29g. It is observed that the civil lines area have significant amplification than the Kilaghat area. For period range (0.67-1.0 sec), mean spectral acceleration response at 100 bar is more than 125 bar whereas 150 bar response is more than 175 bar. Similarly, for the period range of 1.0 sec-3.0 sec, mean spectral acceleration response at 175 bar is higher than 200 bar stress drop. Hence, it is not always that the higher stress drop produces significant amplification on the site; even the lower stress drop at long period can produce higher amplification. The site response analysis shows that almost all the sites in Allahabad are well above the recommended PGA values for Allahabad city. Further, the civil lines area has 2.8 times higher amplification compared to the other parts of the city. The plotted response spectra may be used for designing structures in and around the Allahabad city. The contour plots may also give an idea for the selection of sites and redesigning/retrofitting of old structures.