Accounting for the uncertainties in the estimation of average shear wave velocity using V _S– N correlations

Abstract

Site-specific seismic hazard analysis is crucial for designing earthquake resistance structures, particularly in seismically active regions. Shear wave velocity ( V _S) is a key parameter in such analysis, although the economy and other factors restrict its direct field measurement in many cases. Various V _S–SPT– N correlations are routinely incorporated in seismic hazard analysis to estimate the value of V _S. However, many uncertainties question the reliability of these estimated V _S values. This paper comes up with a statistical approach to take care of such uncertainties involved in V _S calculations. The measured SPT– N values from all the critical boreholes were converted into statistical parameters and passed through various correlations to estimate V _S at different depths. The effect of different soil layers in the boreholes on the Vs estimation was also taken into account. Further, the average shear wave velocity of the top 30 m soil cover ( V _S30) is estimated after accounting for various epistemic and aleatoric uncertainties. The scattering nature of the V _S values estimated using different V _S– N correlations was reduced significantly with the application of the methodology. Study results further clearly demonstrated the potential of the approach to eliminate various uncertainties involved in the estimation of V _S30 using general and soil-specific correlations.

Graphical abstract

Keywords

uncertainties / V _S– N correlations / V _S30 / SPT data / statistical methodology

1 Introduction

Seismic microzonation is a key tool in any city planning, hazard analysis, or hazard mitigation procedure. The incoming seismic waves during seismic activity and their characteristics primarily control the seismic hazard of a particular region. The local ground conditions significantly influence these incoming seismic waves during wave propagation and amplify/deamplify them before reaching the ground surface. The propagation of seismic waves causes ground motions, leading to ground failures like liquefaction, lateral spreading, slope failure, and other structural damages. Numerous studies have been conducted on the seismic microzonation and hazard map development of different cities around the globe. These studies revealed the importance of site characterization on seismic vulnerability and the intensity of predicted earthquakes of the study region. Kanlı et al. [ 1] produced a V _S30 map of the Dinar region in Turkey that exposed the severity of predicted earthquakes in that region. It was clear that the poor soil conditions pushed the damage distribution of the region. Similar studies were also conducted for various other areas of the world [ 2– 7]. In the Indian scenario, more than half of the Indian sub-continent is susceptible to moderate to severe earthquakes. Seismic microzonation conducted in major Indian cities pointed out the necessity of estimating ground amplification through ground response studies for the accurate seismic hazard assessment of these regions [ 8– 10]. All these studies exhibit the influence of site conditions on an earthquake’s destructive nature by amplifying the ground motion. Therefore, to carry out an efficient seismic hazard analysis with site response studies, site characterization, particularly in terms of shear wave velocity ( V _S), is conducted [ 11].

Shear wave velocity is a principal parameter in geotechnical earthquake engineering for its direct influence on the ground motion amplification and ground response. Researchers have developed various tests to evaluate the shear wave velocity of soil in the laboratory and the field in the recent past. In the laboratory, tests like piezometric bender element test, ultrasonic pulse test, and resonant column test serve this purpose. However, these test results are not reliable for the characterization of an existing ground because of the small representative sample size and difficulties in collecting undisturbed samples. Even though there are many in situ tests like Seismic reflection tests, Seismic refraction tests, Spectral analysis of surface wave tests (SASW), seismic downhole to take care of these issues, cross-hole tests and/or Multichannel analysis of surface wave (MASW) test are routinely used. The requirement of right quality equipment and computation facilities with expert knowledge and practice makes the field measurement of shear wave velocity quite challenging and uneconomical for small studies or projects. This urged researchers to develop a new methodology for evaluating the shear wave velocity of soil using any other feasible parameter. Standard penetration resistance (SPT– N) is a widely used geotechnical parameter of the in situ soil, correlated with various index properties, and directly influences the soil's strength behavior [ 12]. Researchers started developing a correlation between the static field test, SPT– N of the soil, and V _S from the past decades. The shear wave velocity is related to SPT– N values based on power law (Eq. (1)).

(1) V_{\mathrm{S}} = A \cdot N^B ,

where A and B are coefficients depending on the type and properties of the soil [ 13]. The power law has been modified widely by various researchers, and several correlations were developed considering different factors influencing the shear wave velocity of the soil [ 14– 30]. Many studies have also been developed considering the Indian scenarios and site-specific properties of Indian cities [ 23, 31– 35]. Estimating V _S from SPT– N is widely adopted these days because of the ready availability of SPT– N data from the geotechnical site reports. Even though the estimation of shear wave velocity at a site using a particular correlation is a simple procedure, selecting appropriate correlation and borehole data are challenging and confusing for practicing engineers. Many correlations are available between SPT– N values and shear wave velocity for a particular location or similar site conditions. Moreover, SPT measurements change from borehole to borehole because of soil heterogeneity and uncertainties involved in the testing. Some correlations were developed for the specific soil type in the available correlations, while some other relationships can be applied for all soil types. All these ambiguities are involved during the hazard analysis questions the capability and reliability of the typically estimated shear wave velocity values used in seismic hazard analysis.

In brief, for a set of borehole data and V _S– N correlations, there is a need for a practical estimation procedure for shear wave velocity to perform a reliable hazard analysis. This paper discusses an approach for estimating shear wave velocity ( V _S) and thereby average shear wave velocity up to 30 m depth ( V _S30), addressing the uncertainties mentioned above, i.e., 1) borehole data uncertainty and 2) correlation uncertainty. Borehole uncertainty can be induced due to various reasons such as soil heterogeneity, and SPT measurement errors. This type of systematic uncertainties is known as epistemic uncertainties. In some cases, even in the same borehole, the SPT varies due to some unknown factors. This leads to developing different correlations for the same soil as correlation uncertainty, known as aleatoric uncertainty. Sensitivity analysis quantifies the influence of each factor causing these uncertainties. This measures the degree of reliability of any methodologies or models developed to eliminate uncertainties [ 36]. Various researchers have developed and demonstrated stochastic models to perform sensitivity analysis considering different input parameters numerically and mathematically in the past [ 36– 40].

The statistical approach adopted here helps eliminate the uncertainties involved in V _S calculations to develop the V _S profile and estimate V _S30. The detailed procedure for estimating V _S and V _S30, considering the measured SPT– N values from different boreholes and V _S– N correlations is outlined in the following sections.

2 Methodology

Developing V _S profile from SPT– N values using different V _S– N correlations is commonly used in seismic hazard analysis. However, various uncertainties are involved in this procedure. Therefore, a simple statistical method is applied here to eliminate these possible uncertainties in the evaluation of V _S and V _S30. This section describes the appropriate selection of boreholes from a set of data and eliminating different uncertainties.

2.1 Selection of appropriate borehole

Figure 1 shows a sample plan for any proposed project site with borehole locations, where nine boreholes (borehole No. 1, 2, 3, 4, 5, 6, 8, 9, and 10) drilled under the plan area of the proposed structure. To generate a uniform V_S profile for the entire site, boreholes placed in the site’s critical location (i.e., the boreholes under the proposed structure) have been considered for the analysis. Hence, data from these nine boreholes are considered while evaluating the entire site’s representative V_S profile. This eliminates the ambiguity in using dummy borehole data to calculate V_S below the proposed project structure.

2.2 Elimination of borehole data uncertainty

For a set of SPT– N values of different boreholes at a particular depth, the SPT– N value varies from borehole to borehole. Thus, to carry out a reliable Vs estimation, N values are converted into a single set of values having three statistical elements, mean, mean + Standard deviation ( SD), and mean – SD. According to the 68–95–99.7 rule of statistics, 68% of the entire set’s values lie in this range. This helps represent the whole set collectively in a range of mean ± SD with the least variations [ 41]. In this particular problem, the set's mean value suggests the central tendency of the discrete set of values, which is influenced by all the values in the set. The mean value, µ is calculated using the relation

(2) \mu = \frac{1}{n} \sum _{i=1}^n x_i ,

where n is the number of elements in the set and x _i is the value of ith element of the set. The SD represents the dispersion of a particular set of values. It indicates how much the set of values can stretch from the mean values. A high SD shows the tendency of the values to remain far from the mean value of the set. The SD of a set of values has calculated using Eq. (3).

(3) S D = \sqrt{\frac{{\displaystyle \sum _{i=1}^n{\left(x_i-\mu \right)}^2}}{n-1}} .

The use of statistical parameters, mean, and SD helps represent all the boreholes’ behavior to a simpler form. Moreover, the elements, mean – SD and mean + SD, show the range of V _S values. The methodology converges the set of values into a single point and makes sure that the converged values lie in an acceptable range.

2.3 Elimination of correlation uncertainty

Further, there are many correlations available to estimate shear wave velocity considering the general property of the soil and the specific behavior of soil. The selection of correlation is confusing for practicing engineers. Several available correlations considered eliminating this ambiguity, and a convergence technique is applied. The new set of SPT– N values now passed through various available correlations and calculated the shear wave velocity. Each element of the SPT set passing through a particular correlation generates a set of shear wave velocities with three parts. The procedure followed for other correlations also, such that it generated X number of sets, having three elements for each set, where X is the number of correlations. The value of V_S estimated from mean SPT– N from each set is further converged into a single set of statistical elements, mean, mean – SD, mean + SD. To reduce the width of the band of values, only the mean value of the Vs is selected in the case of shear wave velocity calculated using mean – SD and mean + SD. The V_S values and depth are used to calculate the average shear wave velocity at 30 m. A detailed representation of the flow of the entire procedure is shown in Fig. 2(a). The correlations used here were developed for all types of soil. That is, the variation in the properties of the soil is not considered. Soil-specific correlations are also adopted to take care of these variations in properties.

All the borehole data are primarily passed through the different correlations for soil-specific correlations, thereby obtaining mean, mean + SD, mean – SD shear wave velocities. Here, the ambiguity at each borehole is taken care of by different correlations and converging the entire soil-specific correlations into a single set of values. This helps in reflecting the behavior of a particular soil layer more precisely in Vs estimation. Figure 2(b) shows the procedure flow for the assessment of V _S30 using specific soil correlations. The entire approach can be adopted for both measured and corrected N values. However, since the number of correlations available for corrected N values is less, the current study explores the usage of measured N values.

3 Correlations used in this study

Various researchers developed several correlations in the form of Eq. (1). Some of them are based on measured N values, and some are based on corrected N values. Correlations using measured N values are used in this particular method of evaluation. The correlations developed considering different soil types were also selected. Standard relationships used in this study are given in Table 1.

3.1 Average shear wave velocity, V _S30

The surface material of the earth plays a vital role in wave propagation and the release of seismic energy. In addition, the waves propagated through the earth’s surface are highly influenced by the elastic properties of the surface material. Thus, a shallow depth shear wave velocity is used to perform seismic hazard analysis. The average shear wave velocity, V _H up to a depth of H m from the surface layer, is expressed as:

(4) V_{\mathrm{H}} = \frac{{\displaystyle \sum d_i}}{{\displaystyle \sum \frac{d_i}{v_i}}} ,

where v _i is the V _S of the ith soil section having thickness d _i. Σ d _i corresponds to the cumulative sum of the thickness of all the soil sections, H. The average shear wave velocity V _S30 of the soil section up to 30 m depth is given by:

(5) V_{\mathrm{S}30} = \frac{30}{{\displaystyle \sum _{i=1}^N\frac{d_i}{v_i}}} ,

where N is the number of soil sections up to 30 m depth [ 1, 34].

4 Site study and geotechnical investigation

The capital city of India, Delhi, is a severe earthquake-prone region in the country. According to India’s seismic zoning map, Delhi NCR belongs to seismic zone IV, with an earthquake magnitude of 6.5 to 7.0 is expected. The borehole data of the selected region of study is shown in Fig. 3. A Geotechnical investigation program with Standard Penetration Test has been undertaken at a site in Delhi NCR, particularly the Noida region, and obtained the geotechnical properties of the soil based on boreholes samples. However, no specific tests were conducted to determine the V_S of the soil at the site. Due to the lack of any such testing data, a reliable statistical estimation of the V_S30 based on measured N values is developed to carry out the seismic hazard analysis.

SPT has been conducted in 18 boreholes of 100 mm diameter at different locations of the site. A detailed investigation was conducted on the site and observed different soil layers in the boreholes at different depths. The various strata of soil observed in the boreholes are as follows:

a) Stratum I: filled-up soil with stone pieces;

b) Stratum II: sand with traces of silt;

c) Stratum III: clayey sandy silt;

d) Stratum IV: silty sand with occasional gravel.

The depth of the borehole and corresponding strata variation is also presented in Fig. 3.

5 Statistical evaluation

A statistical evaluation of the data has been conducted to account for the variations in the SPT measurements, as discussed earlier. It is also seen that the ground surface of borehole locations are not the same. Moreover, the measured SPT values of different boreholes are available at different depths. To account for these differences, all the depths corresponding to the SPT measurements are expressed in terms of depth elevations, which are calculated based on the ground surface evaluation of each borehole with reference to the adjacent road level. The calculated actual depth elevations and corresponding SPT– N values were arranged according to a standard depth elevation profile. The actual depth elevation (ADE) and SPT– N value lie near the closest standard elevation. The measured N values along with mean ( µ) and standard deviation ( σ) of SPT data at different depth elevations are listed in Figs. 4(a) and 4(b). Figure 4(c) represents the statistical distribution of mean measured N values at 0, 10.5, 21, 30 m depths. The distribution of normalized N values lies in line with the 68–95–99.7 rule of statistics. Hence, to avoid further stretching of the width of the band of values, the statistical evaluation is restricted to µ + σ and µ – σ to proceed further.

6 Results and discussions

6.1 Evaluation of shear wave velocity using all type soil correlations

The shear wave velocity profile of the soil section is estimated based on the corresponding mean, µ + σ and µ – σ of SPT values at each depth using various available V_S– N correlations. Figure 5(a) shows the variations in the V_S profiles obtained using different relationships. The fluctuations observed in the V_S profile for each correlation depict the disparity of estimated shear wave velocity throughout. At 30 m depth of the soil profile, the value of V_S ranges from 292 to 634 m/s. This reflects the ambiguity involved in the selection of a proper V_S− N correlation. The band of values in which the shear wave velocity can exist is also represented in the plot. A statistical evaluation is further conducted on the above data. The estimated V_S using various V_S– N correlations is converged into a set of mean, µ + σ and µ – σ matrix. Each parameter is evaluated for all three statistical elements. However, in the case of µ + σ and µ – σ of µ + σ and µ – σ of SPT– N, the width of the band of values is too large, such that the consistency of the results is questioned. To obtain a reliable conclusion, second-order µ + σ and µ – σ are neglected. With the statistical approach, the V_S lies in the range of 334 to 470 m/s. The average shear wave velocity at 30 m is calculated for all the obtained shear wave velocity profiles.

6.2 Evaluation of shear wave velocity using soil specific correlations

Estimating V _S at different boreholes using various correlations is also carried out along with statistical SPT values. When V _S is estimated at a particular borehole, it is possible to use V _S– N correlations developed for specific soil types. Some relationships use observed N values, while others use corrected N values. The V _S profile of borehole 1 using different correlations is represented in Fig. 5(b).

The variation of mean, µ + σ, and µ – σ of V _S using mean V _S profiles of individual boreholes are shown in Fig. 5(b). In addition, the mean variation in V _S profiles estimated based on µ + σ and µ – σ shear wave velocity profiles are also represented in the same figure. V _S profiles have been generated using various V _S– N correlations and SPT data. The statistical evaluation of both the procedures eliminates the uncertainty induced in the measurement of SPT data, variation of SPT values at different boreholes at different depths, and the variation in factors affecting the development of correlations.

6.3 Evaluation of average shear wave velocity V _S30

Further, V_S30 values estimated from different statistical analyses are tabulated in Table 2. The application of both general correlation and soil-specific correlations is reported. A narrow difference in values of V_S30 is observed in both cases. This is due to the collective effect of a set of correlations rather than depending on a single correlation. Moreover, the entire mean values of both the cases lie in the range of mean ± SD range, even though there is a negligible crossing of limits.

7 Summary and conclusions

The study presents a practical approach for evaluating V _S30 of a particular site. The SPT– N values measured from different boreholes at the site show different values. The heterogeneity in the soil and errors caused in the SPT measurement are the primary reasons for this epistemic uncertainty (borehole data uncertainty). An aleatoric uncertainty, named correlation uncertainty, is also involved during the estimation of V _S from SPT– N values. The presence of these uncertainties affects the reliability of typical V _S estimation procedures. The paper discusses using a statistical methodology to overcome these uncertainties to obtain a reliable value of V _S30.

To eliminate the different uncertainties, a convergence technique with the help of a statistical method is used. The large set of borehole data from nine boreholes is converged into a single set of mean values of the data. Also, two additional sets of mean ± SD are developed to fix the range of possible values. This set of values fully reflects the behavior of the entire nine borehole data. A similar approach was applied in the case of different correlations also. Thus, a single parent set of V _S along with mean ± SD is developed, accounting for the various uncertainties to calculate V _S30 of the profile. The approach can be adopted for estimating the shear wave velocity at sites where direct measurement of V _S is not conducted, and site-specific correlations are unavailable. The following significant conclusions are drawn from the study.

1) The SPT data at different depths from different boreholes exhibited significant scatter in values throughout the site. Besides, the application of different V _S– N correlations result in notable variation in the V _S values for the same SPT value. Therefore, the scatter in SPT– N values and corresponding differences in V _S values show borehole data uncertainty and correlation uncertainty.

2) The entire borehole data matrix converged into a three-column matrix with statistical parameters representing the whole site eliminating the borehole uncertainty involved in measuring SPT– N values. The mean SPT– N values vary from 7 to 60 throughout the site, along with the depth.

3) A very high degree of scattering is found in shear wave velocity values estimated using different correlations. For example, at 30 m depth of the soil profile, the value of V _S ranges from 292 to 634 m/s (variation of 342 m/s) using different general soil correlations. The statistical approach eliminates 60% of the combined borehole and correlation uncertainties in the calculated V _S values by cutting down the variations to 136 m/s.

4) Different correlations induce V _S values from 344 to 725 m/s (at borehole 1) in the case of application of specific soil correlations. The range of values is almost similar to that of general soil correlations. The application of second-order statistical approach narrows down 51% of variations in the V _S values. This shows that the use of general soil correlations works more effectively with this procedure.

5) The use of statistical parameters eliminates the influence of epistemic and aleatoric (borehole and correlation) uncertainties involved in the calculations and narrows down the range of V _S values.

6) Based on the different statistical variations observed in the value of V _S30, three V _S30 values are recommended to consider in further seismic hazard analysis to account for uncertainties involved in the estimation of V _S. The V _S30 values are 320, 285, and 240 m/s. The site belongs to class D of NEHRP soil type classification [ 42].

Various ambiguities involved in calculating V _S and V _S30 from SPT– N values are outlined in this study. The calculated V _S values with a high range of variations are converted into narrow sets of values to make the estimation procedure reliable and convincing. Though the study comprises measured N values, the methodology's scope can also be extended to corrected N values and more number of correlations. Thus, the methodology reduces V _S and V _S30 estimation complexity using SPT– N values in field problems and makes the field test data more reliable for researchers and practicing engineers, particularly for small projects where direct V _S measurements are not taken.

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