Evaluation of variability of shear wave velocity in India

The shear wave velocity is a significant component in seismic hazard studies. This involves standard practice in seismic hazard analysis through ground motion prediction equations (GMPEs), ground response analyses (GRAs), or the computation of amplification factors. In practice, the shear wave velocity may be obtained through invasive methods, noninvasive methods, transformation models, or correlation-based methods. Currently, the uncertainty component of shear wave velocity has either been overlooked or oversimplified. The aim was to quantify the range of variations in shear wave velocity across India. The database included data of reported shear wave velocities in 20 states collected from various project reports and publications. A brief review of the methods used to obtain the data is given, along with a systematic highlight of the range of uncertainties encountered and their statistics. Base-case profiles for response analyses are recommended, and their statistics are examined. These observations aid computation of amplification factors. The transformation uncertainties of the shear wave velocity and the standard penetration test (SPT) were examined. The outcomes of the research will guide reliable seismic hazard studies across countries. Furthermore, this study paves the way for future development toward addressing uncertainties in shear wave velocity and facilitates the use of reliability-based design procedures.


Introduction
The shear wave velocity plays a central role in earthquake engineering; it is required for site classification and to evaluate potential modifications to the soil that may undergo during earthquakes.Further, it is used to assess the liquefaction hazards and other local site effects [1]- [4].In practice, it can be obtained through invasive, noninvasive, transformation models, or correlation-based methods.These field methods are based on the equipment and the required number of boreholes to enable measurements.Some common seismic field methods include cross-hole (XH), down-hole (DH), and suspension P-S velocity logging methods (suspension).These methods are invasive because they require physical drilling for measurements.In contrast, noninvasive methods are much more useful and are more frequently utilised.Common noninvasive seismic field methods include multichannel analysis of surface waves (MASW) and spectral analyses of surface waves (MASWs and SASW, respectively).Table 1 shows a comparison of different field seismic methods used to estimate the shear wave velocity from the literature [5].The purpose of this study was to quantify the range of variations in shear wave velocity across India.As outlined by Phoone et al. [6], the variability of geotechnical properties is a complex attribute that stems from many sources of uncertainty.The three primary sources of geotechnical uncertainty are the inherent variability, measurement errors, and transformation uncertainty.Inherent variability is modelled as a random field, which can be described concisely by the coefficient of variation (COV) and the scale of fluctuation.The measurement error is extracted from field measurements using a simple additive probabilistic model, or it is determined directly from comparative laboratory testing programs.This study highlights the significance of regional variability in shear wave velocity in India through on an extensive literature review.To cope with the length of the proceedings and for brevity, base-case profiles that should be considered for ground response analyses are recommended, and their statistics are examined.Given that the correlation of the shear wave velocity with the standard penetration test (SPT) is considered the most widely adopted in India, the transformation uncertainties of these models were examined.

Shear wave velocity to 30 m depth from shallow sites in India
This database represents a countrywide database of reported shear wave velocities from 20 states across India.The data were collected from studies by Thaker and Rao [7] for Gujarat City and Anbazhagan et al. [8].for Lucknow city, Naik et al. [9] for Kanpur, Mhaske and Choudhury [10] for Mumbai city, Bajaj and Anbazhagan [11] for the Indo-Gangetic Basin, Anbazhagan and Sitharam [12] for the Bangalore region, Uma Maheswari et al. [13] for Chennai soils; Chatterjee and Choudhury [14] for Kolkata; Hanumantharao and Ramana [15] for the Delhi region; Rao and Choudhury [16] for the region in Haryana state.Singh et al. [17] for Varanasi City, Thokchom et al. [18] from Western India; Shukla and Solanki [19] for Indore city; Kirar et al.[20] for Roorkee Region; Maheshwari et al. [21] for Northwest Himalaya; Ataee et al. [22] for Mashhad city; Sil and Sitharam [23] for Tripura State; Kumar et al. [24] for Guwahati area; and Anbazhagan et al. [25] for Coimbatore and Vizag, respectively.The collected database contained 2125 profiles.
Figure (1-a) shows the distribution of different methods used to obtain the shear wave velocity.As evident from the figure, the MASW is frequently used in India.The classification of the dataset with respect to depth is important for determining how the depths are distributed.Fortunately, 25% of the data possessed depths less than 20, whereas 46 % of the depths were found to be greater than 30 m, as shown in Figure (1-b).This suggests the possibility of value correlation in the dataset.The site classification of the data showed that most Indian sites were classified as Classes C and D.
This site classification conforms to the system used in the Western US, even though the regional geotechnical characteristics may be different.Given that the shallow sites of India are relatively stiff compared to those in the Western US, the seismic responses should be different from those in the US.A comparison of the distribution of VS30 for categories C and D as representative of the main site classes in India highlights the differences in geotechnical characteristics between the Western US and other sites.The geotechnical characteristics of VS30 can exist only as values greater than zero; therefore, the probability density distribution is assumed to be a lognormal distribution [26].For comparison, we propose the average shear wave velocities of 30 m (VS30) in India for the same site classes in Korea, the Western US (WUS), and Pakistan shown for the dominant site classes C and D, as indicated in Figure (2-a).The data from the Western US were taken from the file "Resolution of Site Response Issues from the Northridge Earthquake" (ROSRINE), whereas the data from Pakistan and Korea was taken from the papers by Adeel et al. [27] and Sun [26], respectively.To highlight the differences more clearly, the distribution of site classes for each period is shown, which clearly indicates the difference.Figure

Best-case shear-wave velocity profile
The above discussion clearly highlights the differences in the reference site conditions resulting from site-specific measurements.Moreover, site periods differed substantially under the same site conditions.This encourages practitioners to adopt site-specific measurements, rather than generic values.In this study, the shear-wave velocity values at a depth of 30 m were obtained through extrapolation with a depth-dependent function.This function depends on the mean profile extracted from a dataset.The mean profile was extracted based on the assumption that the average shear wave velocity followed a lognormal distribution function [28].Thus, the value of the coefficient of variation (CV) helps understand the dispersion of the data from its mean profile.This CV was calculated by subtracting the square root of the natural log transformation of the standard deviation of the data subtracted from one.Figure 3 shows the base-case profile mean, estimated median profile, and CV.This profile was used to extract depth-dependent functions.A different functional form was used to determine a suitable form that represented the dataset.The results are summarised in Table 2.As evident from Table 2, the use of the cubic function provides a better picture of the nature of the possible correlations.By using the mean profile, a representative curve with depth can be derived to map the depth-function interpolation.The quadratic functional also provided comparable results; however, the cubic function was more effective and yielded higher correlations.These results are in agreement with those reported by Boore [29] and Lodi et al. [30].

Transformation uncertainties of the Shear wave velocity with geotechnical properties:
This section presents the results of the effects of the transformation uncertainties of the shear wave velocity on the geotechnical properties.This effect may be attributed to the intermediate correlations of the shear wave velocity with other geotechnical properties that can be used to estimate shear wave velocity values.Although there are many geotechnical properties to which the shear wave velocity can be correlated, the SPT N-value is most frequently used in India.This may be because the SPT is the oldest test in India, and a majority of consulting companies adopt this test.Moreover, the test is more suitable for residential and intermediate infrastructure construction, which is rapid in India.
Mathematically, the relationship between SPT-N and shear wave velocity is as follows: = where c and d are coefficients determined by regression.Hence, the variability between the shear wave velocity and SPT-N values can be interpreted using the statistics of these coefficients.This study follows the methodology outlined by Xiao et al. [31], using the hierarchical Bayesian formwork with MCMC simulations.

Statistical Characterisation using the Hierarchical Bayesian method:
To apply Bayesian characterisation, the equation was first linearised using a logarithmic transformation.Thus, the relationship that relates the SPT-N value with shear wave velocity is controlled by parameter a, which is the intercept of the linear relationship, whereas parameter b is the slope of the linear model, written as: In the above relationship, represents the error term that is assumed to be normally distributed with mean of zero and standard deviation of .We also considered the ith location in our dataset (20 locations in this case).The parameters for the ith location depend on the volume of data and the model in that region.Further to consider the uncertainty associated with the relationship form, suppose that for each relationship in the ith location, we may adopt a model error term that reasonably follows a normal distribution with a mean of zero and a standard deviation of (σ) as: To model the similarity among them in different relationships at different locations, suppose that for different locations, the samples are based on the following probability function, as given by Xiao et al. [31]: ) Here φ is the probability density function (PDF) of the standard normal distribution and D denote the observed data, then a chance to observe different data points is statistically independent.In this case, the probability of observing D is computed as follows: To characterise the posterior we will use the θ as the random variable which is θ = {, b , , b, , σ ε , , b } and for the prior PDF f(, For Bayesian calculations prior distributions on , b , , b, and σ ε , should be interdicted .A class of prior PDF that have a negligible impact on the data called noninformative priors were recommended [31], [32] from normal and inverse gamma distribution as; µ ~ 0 , 100 2 2 ~ 0.001 , 0.001 µ ~ 0 , 100 2 2 ~ 0.001 , 0.001 ε 2 ~ 0.001 , 0.001 In this study, the Bayesian model was used to generate samples based on Markov chain Monte Carlo sampling with data from 20 regions.The data were used to draw samples based on simulations of 100,000 samples for the mean and standard deviation of the model parameters.The samples were then used to represent the model for each region and interpret the variability.During the simulations, the first 10,000 samples were drawn as burn-ins for the models and were not used for the calibration.

Results and discussion
Figure 4 shows a histogram of 100,000 samples from the MCMC simulations used to estimate the mean and standard deviation of the parameter.The means of a and b are not zero, as shown in this illustration, indicating that there is inter-region variability in the model for various regions.Additionally, the interregional variability caused by the model slopes was less than that linked to the model intercept.
The histograms of samples a and b drawn from the MCMC are shown in Figure 5.The posterior statistics of a and b (I = 1, 2, up to 20) were determined based on the samples obtained using MCMC.The correlation coefficients between various variables can be estimated using the samples obtained from the MCMC.The correlation coefficients between b (i = 2, up to 20) and the error term, and between (i = 2, up to 20) and the error term were both zero.For each region, the correlation coefficients between a and b are not zero.A model was constructed for each region using posterior statistics.These results were used to develop the model, and the calculated statistical values are presented in Table 3. Figure 6 shows the results obtained for all the Indian regions using the calibrated model.Evidently, the obtained results of the 20 regions are within the 95% confidence interval of the relationship.However, in some cases where the volume of data used is less, the results have a wide range of the 95% confidence interval.Nonetheless, the obtained results were data-driven with reduced uncertainties.Furthermore, Figure 7 shows a comparison between the values of the slope and intercept obtained in the literature and the posterior statistics calculated in this study.It is evident from the plot that most of the simulation results are in good agreement with the values reported in the literature, with no significant uncertainty.However, the results of regions ( 4) and ( 12), which represent the Delhi and Lucknow urban centres, are higher than those reported in the literature.Although the volume of data used for the Delhi region was sufficient and similar to that of other regions in the dataset, the reported value in this area was above the mean reported value.This may be attributed to the nature of the data collected in this region or its geological structure.The same issue was observed in the Lucknow urban centre, which had the largest reported values in the dataset.
To clarify the effect of the data used for each model on the results obtained, data from the Bangalore region were used as an example.For the first time, ten randomly selected data points were used to calibrate the MCMC for the Bangalore region.Following the same method, 100,000 simulation samples were drawn from the Bayesian model and the statistics for each parameter were obtained.These values were used to plot the results, as shown in Figure 8.As can be observed from the plot, the obtained model has a wider confidence interval, which reflects a larger uncertainty in the results.In the same plot, the data used are coloured black, whereas the full data for this region are plotted in grey.Second, 10 additional data points were added to the data used for the first time and used for model calibration.This process is repeated for 30,50,70,90,110,130, and 140 data points.The 95% confidence interval was drastically reduced in each case as additional data have been added to the data points considered.This suggests that accurate correlations must be developed using data specific to an area.If region-specific data are unavailable, then the degree of uncertainty in the established model may be high.

Conclusions
This study evaluated the shear wave velocity variability in India using field data obtained from the literature.The primary aim of this study is to develop a base-case shear wave velocity profile and investigate the transformation uncertainties associated with the shear wave velocity obtained through a SPT.The shear wave velocity profiles were grouped based on their maximum depth, and statistical parameters, including the mean, median, and CV, were calculated along the depth.By using the mean profile data from the study sites, a representative curve covering a depth range of 0-30 m was generated using various functional forms.The cubic function exhibited the greatest R-square value of 0.7621, indicating its efficacy.This approach provides a means of extrapolating shallow profile data to estimate the shear wave velocity at greater depths, which is useful for reliable computations in ground response analyses.Furthermore, by using the hierarchical Bayesian approach, the transformation uncertainties of the shear wave velocity with the SPT were examined, emphasising inter-site variability.The transformation uncertainties of shear wave velocity and SPT results were examined for 20 regions in India.The usefulness of these results was demonstrated using a database collected from 20 regions.If region-specific data are not available or are limited, the model borrows information from another region.However, when no regional datasets were available, the results indicated a wide range of uncertainties.Sensitivity analysis, as verified in the Bangalore region, shows that more data points improve accuracy, with a 95% confidence interval achieved with 50 data points or 30% of the entire dataset for that region.

Figure 1 :Figure ( 2
Figure 1: Distribution of different methods used to obtain the shear wave velocity, and (b) distribution of different profile depths in the dataset.

Figure 4 :
Figure 4: Histograms of samples of the mean and the standard deviation drawn from MCMC simulation.

9 Figure 5 :
Figure 5: Histogram distribution of (a) and (b) variables in different region in India.

Figure 6 :
Figure 6: Obtained results for the 20 Indian regions with the calibrated model.

Figure 7 :
Figure 7: Comparison between the values in the literature and the posterior statistics.

Figure 8 :
Figure 8: Obtained results for Bangalore region with different scenarios.

Figure 9
Figure9shows a comparison of the different relationships for Bangalore constructed by varying the number of data points.With 50 data points, the model achieved a higher level of acceptance,

Figure 9 .
Figure 9.Comparison of different relationships for Bangalore region constructed by varying numbers of region-specific data point.

Table 2 :
Summary of different functional forms and obtained coefficient of determination.

Table 3 :
Posterior statistics of model parameters for different regions in the database.