V_s30 Prediction Models Based on Measured Shear-Wave Velocities in Tangshan, China

Abstract

V_s30 (equivalent shear-wave velocity of soil layers within a depth of 30 m underground) is widely used in the field of seismic engineering; however, due to the limitation of funds, time, measuring devices, and other factors, the depth for testing shear-wave velocity in an engineering site rarely reaches 30 m underground. Therefore, it is necessary to predict Vs30 effectively. We analyzed the existing models using 343 boreholes with depths greater than 30 m in Tangshan, China. It shows that the topographic slope method is not suitable for predicting V_s30 in Tangshan. The Boore (2011) model overestimates, while Boore (2004) underestimates V_s30 in Tangshan, while Junju Xie’s (2016) model has ideal prediction results. We propose three new models in this paper, including the bottom constant velocity (BCV) model, linear model, and conditional independent model. We find that the BCV model has limited prediction ability, and the linear model is more suitable when z ≤ 18 m, while the conditional independent model shows good performance under conditions where z > 18 m. We propose that the model can be accurately and effectively applied in Tangshan and other regions with low shear-wave velocity.

1. Introduction

Earthquakes have brought a series of engineering challenges for the design and service of constructions and buildings [1,2]. Soil, as a wave dispersion medium and bearing layer for structures, has dynamic characteristics that have important implications for the seismic design and structural analysis of construction materials under ground motion [3,4,5]. Shear-wave velocity, V_s, is one of the most basic parameters for characterizing the dynamic behaviors of soils. V_s30 has been widely applied in the field of seismic engineering for inground vibration prediction [6,7], site classification [8], site effect evaluation [9], seismic performance assessment, or designing structures on a regional scale [10]; however, due to the limitation of funds, time, and measuring devices, the depth for testing shear-wave velocity generally cannot reach 30 m and sometimes there is no shear-wave velocity available. At present, V_s20 (equivalent shear-wave velocity of soil layers within a depth of 20 m underground) is still used to describe the characteristics of site soil in China. Although some studies have suggested that seismic motion can predict V_s without drilling, due to the scarcity of seismic records and the limited effect of this method itself, this method is not widely used. Therefore, it is very important to predict V_s30 rapidly, accurately, and effectively.

When the project site lacks the measured data of shear-wave velocity, other indicators that can reflect the geological characteristics of a site are usually used to predict V_s30, such as surface geology [11,12,13], topographic classification [14,15], topographic slope [16], category of geotechnical engineering works [17,18], mixed geological features [19], and Quaternary System isopach [20]. These methods of using other indicators to characterize V_s30 can quickly and widely predict V_s30, but the accuracy is low, and the methods are significantly affected by the accuracy of original geological features. While, for an engineering site with shear-wave velocity but the depths are less than 30 m underground, the shear-wave velocities (V_sz, z < 30 m) at known depths can be used to predict V_s30 more accurately, such as the BCV model (a constant recursive model) [21] and a linear model [22,23,24]. Although such models using the known data of shear-wave velocity to predict V_s30 show a high accuracy, there is certain regional applicability in the established V_s30 models due to the limitations of the basic data, especially in areas with low shear-wave velocity, there is still no effective prediction method.

In this paper, the measured data of shear-wave velocities from boreholes in Tangshan were used for V_s30 prediction. Based on the data of shear-wave velocities of 343 boreholes with depths more than 30 m in the city, the topographic slope was first used as an alternative indicator for rapid prediction of V_s30. Then, three new V_s30 prediction models were established based on V_sz, and the goodness of fit of the new models was ranked by using the log-likelihood (LLH) method. The prediction model of V_s30 proposed in this paper can be accurately and effectively applied to areas with low shear-wave velocity, such as Tangshan.

2. Introduction of V_s30 Prediction Models and Sorting Method of the Goodness of Fit

2.1. Prediction Using Topographic Slope Method

Geomorphologic fluctuations indicate topography and lithology. For example, the steep peaks suggest the rock-dominated geological property of the site, while the nearly flat basins indicate soil-dominated geological properties. Hard sites tend to have steep slopes, while thicker sedimentary basins generally show shallower slopes, indicating there is a certain correlation between topographic slope and V_s30. Based on this, Wald et al. [16] proposed a V_s30 prediction model based on the topographic slope in 2007. This study analyzed the corresponding relationship between V_s30 and NHRP different site categories, including a tectonically stable zone and a tectonically active zone, respectively. The United States Geological Survey constructed an empirical model of the relationship between global V_s30 and topographic slope using this model.

2.2. V_s30 Prediction Models Based on V_sz

When the depth for testing shear-wave velocity does not reach 30 m, V_sz can be generally used to predict V_s30. At present, there are two categories of commonly used models for predicting V_s30 based on V_sz: the constant-velocity extrapolation model and the gradient-velocity extrapolation model.

For the constant-velocity extrapolation model, the main assumption of the BCV model is that V_s is constant from z to 30 m [21] as given by:

$$V_{s30}=\frac{30}{\mathsf{∆}{t}_z+\frac{30-z}{V_{sz}}}$$

(1)

where z, Δt_z, and V_s(z) represent the depth of the hole bottom, the travel time of shear waves, and the instantaneous shear-wave velocity at the depth (z m) of the bottom, respectively. For the convenience of description in the following sections, the model is referred to as the BCV model.

The assumption of the constant-velocity extrapolation model is contrary to common sense that shear-wave velocities increase with the burial depth of the soil layers, so the model generally underestimates V_s30. Boore found that there is a strong linear correlation between log V_s30 and log V_sz in 2004 and established a corresponding relationship between V_s30 and V_sz by using Formula (2) based on data collected from 135 boreholes in California [22].

$$\log V_{s30}=a_0+a_1\log (V_{sz})$$

(2)

where a₀ and a₁ are regression coefficients. This study is known as the Boore (2004) model.

Furthermore, based on Kik-net borehole data from Japan, Boore et al. [23] established the following new linear relationship between V_s30 and V_sz.

$$\log V_{s30}=b_0+b_1\log (V_{sz})+b_2{(\log V_{sz})}^2$$

(3)

where b₀ and b₁ are regression coefficients. The regression coefficient of 5 m ≤ z ≤ 29 m is finally determined. The model is referred to as the Boore (2011) model.

Dai et al. [25] built a conditional independent prediction model of V_s30 based on a Markov process. This model supposes that shear-wave velocity from the bottom of boreholes to 30 m underground is unrelated to that from zero to z m underground. Therefore, this model was the first to establish the corresponding relationship (Formula (4)) between V_sz and V_s(z,30) (equivalent shear-wave velocity from z to 30 m underground), and then V_s30 was predicted using Formula (5):

$$\log V_{s(z,30)}=c_0+c_1\log (V_{sz})$$

(4)

$$V_{s30}=\frac{30}{\mathsf{∆}{t}_z+\frac{30-z}{V_{s(z,30)}}}$$

(5)

where c₀ and c₁ are regression coefficients. The model is called the conditional independent model.

2.3. Sorting Method for Goodness of Fit of the Prediction Models Based on the LLH Method

At present, R² (the adjusted coefficient of determination) and similar parameters are used to sort the goodness of fit of prediction models; however, Weimin He et al. [26] found that there is a misjudgment that arises when using the adjusted coefficient of determination to determine the regression effects. Based on this, we adopted a more reliable LLH method [27] to sort goodness of fit data about the prediction models.

The LLH method was proposed based on information theory, allowing the selection of the optimal model based on data [27]. The LLH value of the prediction model is given by:

$$LLH(g,x)=-\frac{1}{N}{\displaystyle {\sum }_{i=1}^N{\log }_2\left[g\left(x_i\right)\right]}$$

(6)

where g(x) and N indicate the probability density function (PDF) of the model with samples from x₁ to x_N and sample size, respectively. The smaller the LLH value of the model, the better the prediction.

Based on the measured data of shear-wave velocities in Iran, Shafiee et al. [28] analyzed and compared hypothesis testing and the likelihood method of optimal fitting of standard residuals. Different sorting results of V_s30 prediction models using the LLH method show that LLH methods can accurately sort the goodness of fit of V_s30 prediction models.

3. Engineering Geology beneath Tangshan City

Tangshan City is located at the border of the North China Plain and the southern slope of the Yanshan Mountains, with eroded low mountains and hills in the northeast that extend to the Tangshan uplift zone in the southwest and belongs to a zone with shallow bedrock. The other part presents flat alluvial plains of the Luanhe River system, sporadically distributed relic mountains of bedrock and striking to the southeast on the whole. Due to the combined effects of rivers and transgression, the coastal region is low and flat, where sand dunes and many shell ridges remain (Figure 1).

The Quaternary strata in this region are extremely well developed, accounting for about four-fifths of the total area. An area of low mountains and hills is found in the northeast, and the central region is dominated by a mixture of slope deposits and diluvial deposits from the late Pleistocene, showing a small thickness therein. Late Pleistocene alluvial and diluvial deposits are developed in the piedmont belt, and the southern region is dominated by alluvial and diluvial strata in the Holocene epoch, interspersed with relatively thick alluvial deposits of modern rivers. The underlying bedrock is generally high in the north and low in the south on the whole, and the thickness of the Quaternary strata gradually increases from north to south, while sediment particles gradually decrease in prevalence from north to south. The surface is covered by Holocene strata.

4. Analysis of the Existing Shear-Wave Velocity Data in Tangshan City

The shear-wave velocities data of 418 boreholes in Tangshan City (Figure 1) were mainly obtained from seismic safety evaluation and seismic micro-zoning in the region. Most of the data were obtained using the excitation method in a single borehole using an XG-I instrument, and the test depth of shear-wave velocity varied from 6 m to 110 m (Figure 2a). The 343 boreholes with depths greater than 30 m accounted for about 82% and formed the basis for the dataset used in this research. The V_s30 of borehole data used in this study ranges from 169 to 325 m s⁻¹ (Figure 2b), and V_s30 > 300 m s⁻¹ was found in four boreholes.

5. Vs₃₀ Prediction Based on Topographic Slope

Wald et al. [16] studied the correlations between two topographic parameters (surface slope and elevation) with V_s30. The results demonstrate that the effects of using surface slope were better than that of using surface slope and elevation. Therefore, here, we directly used topographic slope as a proxy to predict V_s30. As the topographic slope cannot be directly obtained, it needs to be converted from digital elevation model (DEM) data, so the accuracy of DEM data can directly affect the results of the topographic slope. In addition, considering the current DEM data sources in Tangshan City, we separately downloaded DEM data at 30 m and 90 m over this region from the Geospatial Data Cloud website. Moreover, using the “Slope” function in the “Spatial Analyst” module of ArcGIS, geographic information system software, DEM data were converted into topographic slope through the fitting surface method. After obtaining the topographic slope in the city, they were allocated to 343 boreholes deeper than 30 m.

Figure 3a,b show site categories of 343 boreholes deeper than 30 m and a box plot of topographic slopes with 30 m and 90 m in Tangshan City. Using these kinds of figures, Wald et al. [16] analyzed the correlation between site category and topographic slope in California. In the figure, shear-wave velocities correspond to the boundary value of V_s30 in NEHRP site categories, and the site categories are subdivided. V_s30 ranges of site categories E, D1, D2, and D3 are <180, 180 to 240, 240 to 300, and >300 m s⁻¹, respectively. It can be seen from the figure that for the topographic slope with different resolutions, the scopes of site categories are different. However, as the site soil gradually hardens, the corresponding topographic slope shows a slightly upward trend. Figure 4 shows the relationships between V_s30 of the 343 boreholes and topographic slope with resolutions of 30 m and 90 m. The plot relates to V_s30, and the topographic slope was fitted by the least squares method. Although this curve gradually rises with increasing topographic slope, data points in the figure are very discrete, showing no statistically significant corresponding relationship. Moreover, the ascent stage of the curve appears when the slope exceeds 0.02, and a sharp rise appears when the slope exceeds 0.05, but there are few data points; therefore, this trend is not reliable.

Figure 5 compares the research results in this study with those obtained by Wald et al. [16]. In the figure, polygons represent the scope of V_s30 in the tectonically unstable zone determined by Wald and Allen corresponding to the topographic slope. It can be seen from the figure that the topographic slope corresponding to V_s30 in Tangshan City is systematically lower than the results obtained by Wald and Allen in some regions of the United States. This is similar to the research results obtained by Wald in Salt Lake City in the United States in 2007.

In conclusion, it can be found that there is a weak correlation between V_s30 and topographic slopes with multiple resolutions in Tangshan City; that is, the topographic slope is not suitable for predicting V_s30 in this region. The main reasons are as follows: (1) the prediction capacity of the topographic slope for V_s30 is weaker than other geological features, which has been demonstrated in many studies; (2) Tangshan City is mainly located in the North China plain with a flat topography, where the topographic gradient changes are less obvious than that in hilly and mountainous areas; (3) the slope calculated by DEM in urban areas is usually significantly affected by the canopy effects; (4) most of the boreholes collected in this project are located in plain areas, while there are only a few boreholes in hilly and mountainous areas.

6. V_s30 Prediction Models Based on V_sz

6.1. Establishment of V_s30 Prediction Models

Considering that V_s30 prediction based on V_sz has not been carried out in Tangshan, this paper first analyzed the correlation between V_sz and V_s30 in this region. Figure 6 shows the relationship between V_sz (z = 10, 15, 20, and 25 m) and V_s30 in the city. In the figure, the circle indicates the measured shear-wave velocities, while the dotted line represents V_s30 = V_sz. Moreover, r is Pearson’s correlation coefficient between them. As displayed in the figure, even when z = 10 m, r is still greater than 0.9, indicating that it has a very strong correlation between V_sz and V_s30. Furthermore, r increases with z, indicating that the correlation between V_sz and V_s30 increases with z increasing.

The above analysis shows that V_sz is strongly correlated with V_s30 in the Tangshan area, so three new V_s30 prediction models were established based on the data of shear-wave velocities measured in 343 boreholes across the city and compared with existing prediction models. Finally, the new models in this paper were prioritized.

(1)

BCV model

The simplest model of predicting V_s30 is to assume bottom constant velocity; therefore, we built the BCV prediction model for V_s30 in Tangshan. Figure 7 demonstrates the comparison between V_s30est. (predicted value of V_s30 using the BCV model) and V_s30obv. (measured value of V_s30) when z is 10, 15, 20, and 25 m. The dotted line represents V_s30est. = V_s30obv. As is shown in the figure, when z = 10 m, V_s30est. is lower than V_s30obv. With the constant increase in z, V_s30est. is gradually close to V_s30obv., indicating that the predicted results using the BCV model are gradually close to the measured values. This is because the BCV model assumes that V_s(z,30) (equivalent shear-wave velocity from the bottom of boreholes to 30 m underground) is equal to V_s(z); when z is small, V_s(z,30) is also correspondingly small, resulting in small V_s30est. With the gradual increase in z, V_s(z) rises correspondingly; therefore, the predictive capacity of the model is enhanced. In addition, when the shear-wave velocity in the shallow parts of the boreholes is low or changes rapidly with z, the BCV model underestimates V_s30 more obviously. As most borehole data used in this study are from regions with thick overburden in the North China Plain where shear-wave velocities obtained from boreholes in shallow strata are generally low, the phenomenon whereby the BCV model underestimates V_s30 is significant when z is small.

(2)

Linear prediction model

Generally, in a soil profile, the shear-wave velocity linearly increases with depth. Based on the data of shear-wave velocities in 343 boreholes deeper than 30 m in Tangshan, we built new V_s30 linear prediction models using Formula (2).

Table 1. Regression parameters of the V_s30 linear prediction model (Formula (2)).

Depth (m)	a0	a1	r	σresi	Depth (m)	a0	a1	r	σresi
10	0.288	0.925	0.952	0.022	20	0.207	0.932	0.988	0.011
11	0.281	0.925	0.959	0.021	21	0.185	0.939	0.991	0.010
12	0.272	0.926	0.964	0.019	22	0.157	0.949	0.993	0.009
13	0.262	0.926	0.968	0.018	23	0.134	0.957	0.994	0.008
14	0.232	0.926	0.968	0.018	24	0.114	0.963	0.996	0.007
15	0.263	0.921	0.975	0.016	25	0.097	0.969	0.997	0.005
16	0.255	0.921	0.978	0.015	26	0.080	0.969	0.997	0.005
17	0.247	0.922	0.980	0.014	27	0.059	0.981	0.999	0.003
18	0.236	0.924	0.983	0.013	28	0.039	0.987	0.999	0.002
19	0.224	0.927	0.986	0.012	29	0.020	0.993	1.000	0.001

presents the regression parameters of the prediction models when 10 m ≤ z ≤ 29 m. In the models, a and b are regression coefficients; r represents Pearson’s correlation coefficient between V_s30est. and V_s30obv. at different depths, and σ_resi is the standard deviation of the fitted residuals. It can be observed from Table 1 that when the depth gradually increases from 10 m to 29 m, r rises from 0.952 to 1, while σ_resi decreases from 0.022 to 0.01, indicating that the linear model shows good predictive capacity when z is small and the predicted results gradually approach the measured values with increasing z.

Figure 8 displays the comparisons of predicted results of V_s30 using the Boore (2004) model [22], Boore (2011) model [23], Junju Xie (2016) model [20], and the new linear model built in this study when z = 10, 15, 20, and 25 m. As shown in Figure 8, at any depth, the Boore (2011) model overestimates V_s30 in Tangshan. By comparing different V_s30 prediction models based on shear-wave velocities measured in Greece, Stewart et al. [29] obtained similar conclusions because the measured V_s30 in basic shear-wave velocity data from the boreholes used by Boore (2011) is generally large. However, the Boore (2004) model underestimates V_s30 in Tangshan to some extent. These results demonstrate that shear-wave velocities in the region change faster with depth compared to those in California while changing slower than those in Japan. In addition, the boreholes selected in this study and by the Junju Xie (2016) model are located in the North China Plain, and the measured values of V_s30 are relatively close, so the prediction results are also similar.

(3)

Conditional independent model

This model supposes that shear-wave velocity from the bottom of boreholes to 30 m underground is unrelated to that from zero to z m underground. In this paper, the corresponding relationship between V_s(z) and V_s(z,30) at 10 m ≤ z ≤ 29 m is first analyzed using Formula (4).

Table 2. Regression parameters of the linear prediction model of V_s(z,30).

Depth (m)	c0	c1	r	σresi	Depth (m)	c0	c1	r	σresi
10	0.938	0.643	0.830	0.043	20	0.919	0.642	0.804	0.042
11	0.947	0.637	0.838	0.042	21	0.798	0.690	0.818	0.041
12	0.937	0.641	0.827	0.043	22	0.751	0.708	0.812	0.042
13	0.959	0.631	0.813	0.044	23	0.772	0.699	0.821	0.042
14	1.134	0.558	0.794	0.045	24	0.843	0.670	0.815	0.041
15	1.124	0.561	0.777	0.046	25	0.827	0.677	0.820	0.040
16	1.106	0.569	0.766	0.047	26	0.767	0.701	0.803	0.042
17	1.090	0.575	0.762	0.047	27	0.698	0.728	0.815	0.040
18	0.951	0.631	0.800	0.043	28	0.466	0.818	0.839	0.039
19	0.927	0.639	0.801	0.043	29	0.267	0.896	0.886	0.036

shows the regression coefficients of V_s(z) and V_s(z,30). Figure 9 presents the relationship between V_s(z) and V_s(z,30) when z is 10, 15, 20, and 25 m, and the dotted line represents V_s(z,30) = V_s(z). As is shown in Figure 9, (1) the convergence of data points of the measured shear-wave velocities at any depth is basically consistent, and (2) observing the fitting parameters of the prediction models, it can be found that the standard deviation of residuals of the prediction models at different depths is approximately 0.04, indicating that the influence of z on the linear relationship between V_s(z) and V_s(z,30) is limited, and (3) some measured shear-wave velocities fall below the dotted line, that is, V_s(z) > V_s(z,30), which is more prominent when z > 10 m. In general, shear-wave velocities of soil layers gradually will increase with the increase in burial depth, so V_s(z) will be less than V_s(z,30). However, due to the existence of soft soil, the shear-wave velocity of many boreholes in Tangshan decreases with burial depth. Figure 10 shows the schematic diagrams of shear-wave velocities as they change with depths shallower than 30 m in the No. 76 and No. 112 boreholes. It can be seen that shear-wave velocities decrease monotonically when the depths of the two boreholes increase from 25 m to 30 m, resulting in V_s(25) exceeding V_s(25,30).

After obtaining the corresponding relationship between V_s(z) and V_s(z,30) at different depths, this paper predicted V_s30 based on Formula (5). Figure 11 displays the comparison between V_s30eat. from the new conditional independent model and V_s30obv. when z values are 10, 15, 20, and 25 m. It can be seen from the figure that data points are uniformly distributed on both sides of the straight line, denoting V_s30est. = V_s30obv. when z = 10 m. With the increase in z, the data points become more convergent, indicating that the prediction result of V_s30 is better.

6.2. Sorting of Goodness of Fit of the New Vs30 Prediction Models

As described above, three new V_s30 prediction models, the BCV model, the linear model, and the conditional independent model, are established in this study, based on the data of shear-wave velocities of 343 boreholes deeper than 30 m in Tangshan.

Then, the LLH values were calculated at different depths using the three new V_s30 prediction models, i.e., the BCV model, the linear model, and the conditional independent model (CIP model) (Figure 12).

(1)

It can be observed from the figure that when the depth z of the boreholes changes from 10 m to 25 m, the LLH value of the BCV model is greater than that of the other models. When z is not less than 26 m, the LLH value of the BCV model gradually decreases; that is, when z ≥ 26 m, the predicted value of the BCV model approaches the measured value. This is because the BCV model assumes that shear-wave velocities from the bottom of the boreholes to 30 m underground are equal to that at the bottom. As z approaches 30 m, the better the prediction effect of this model is.

(2)

The LLH value of the linear model is smaller than that of the other two models when z < 19 m, but it gradually increases and even becomes the largest when z > 19 m. This is mainly because the model can only perform simple linear estimation when predicting shear-wave velocities from z to 30 m underground, which cannot reflect the complex correspondence between the shear-wave velocity and the burial depth, especially in Tangshan where the thick Quaternary layer makes shear-wave velocities decrease with increasing burial depth.

(3)

When 10 m ≤ z ≤ 18 m, the LLH value of the conditional independent model is greater than that of the linear model, but when z > 18 m, its LLH value is the minimum of the three models. It suggests that Formula (4) based on a Markov process can better predict V_s(z,30) when z is large. In other words, when z is large, the difference between V_s(z) and V_sz may be large, and the correlation between them is stronger.

In conclusion, it is suggested to use the linear model and the conditional independent model separately as V_s30 prediction models in Tangshan when z is not larger than 18 m and exceeds 18 m.

7. Conclusions

When V_s measurements are not available in an area, proxy-based relationships can be used. By collecting the data from 343 boreholes with a depth greater than 30 m in Tangshan, this paper predicted V_s30 first using a topographic slope with resolutions of 30 m and 90 m as an alternative indicator. Then, the three new V_s30 prediction models were established based on the measured V_sz in the region. Finally, the goodness of fit of the new models was sorted. The main conclusions are as follows:

(1)

Topographic slope is not suitable for the prediction of V_s30 in Tangshan, mainly because of the limitations of the topographic slope method itself. Additionally, the variation of the topographic slopes in this region is smaller than that in hilly and mountainous areas. The slope calculated by DEM in urban regions is generally significantly affected by canopy effects. In addition, the relatively low V_s30 of the borehole data collected in this study is one of the reasons.

(2)

The Boore (2011) model established based on Japanese Kik-net data overestimates V_s30 in Tangshan, while the Boore (2004) model based on data from California underestimates V_s30 to some extent. This indicates that the gradient change of shear-wave velocities with depth in Tangshan is smaller than that in Japan and larger than that in California. In addition, the boreholes selected in this study and by the Junju Xie (2016) model are located in the North China Plain, and the measured values of V_s30 are relatively close, so the prediction results are also similar. The prediction results between this study and the Junju Xie (2016) model are also similar because the boreholes selected are both located in the North China Plain. It also shows that the V_s30 prediction models are regionally applicable.

(3)

Three new prediction models (the BCV model, the linear model, and the conditional independent model) established in this study based on shear-wave velocity data from Tangshan all have a certain prediction ability. The BCV model has limited prediction ability at all depths. The linear model is suitable when z ≤ 18 m, while the conditional independent model shows good predictive capacity when z > 18 m. Therefore, it is suggested to use the linear model and the conditional independent model separately as V_s30 prediction models. A possible limitation of this study is that our models should be used with care because of the relatively small range of V_s values used.