Effect of Randomness of Parameters on Amplification of Ground Motion in Saturated Sedimentary Valley

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Considering the randomness of saturated media parameters such as porosity and permeability coefficient and the randomness of geometric boundary, based on Biot’s theory and the indirect boundary element method (IBEM), in this paper, the Monte Carlo method is used to analyze the stochastic amplification of ground motion in a saturated sedimentary valley. The stochastic seismic response law of a saturated complex site is obtained, which is useful in engineering.

Abstract

Based on Biot’s theory and the indirect boundary element method (IBEM), the Monte Carlo method is utilized to generate random samples to calculate the displacement response of a saturated sedimentary valley under SV wave incidence. The purpose of this paper is to explore the effects of randomness of porosity, permeability coefficient, and geometric shape on the seismic amplification effect of saturated complex sites. It is shown that the change of media porosity in the saturated site with defined boundary has relatively little influence on the ground motion, and the influence of the permeability coefficient is slightly larger. While in the site with a random boundary, the influence of both the porosity and permeability coefficient are significant, which cannot be ignored. The conclusion plays an important guiding role in earthquake disaster prevention and mitigation, such as seismic risk analysis and earthquake microzonation in saturated sedimentary valleys.

1. Introduction

A large number of large-scale projects, such as bridges and dams, are built in sedimentary valleys. The amplification effect of ground motion in river valley sites has been confirmed by many earthquake damage observations and theoretical studies (Trifunac, M.D. et al. [1], Ashford, S.A. et al. [2], Huang, H.C. et al. [3]). The existing ground motion simulation methods of sediment valleys can be divided into numerical methods and analytical methods. The analytical methods mainly refer to the wave function expansion method (Trifunac M.D. [4,5], Todorovska M.I. et al. [6], Chen J.T. et al. [7] etc.). The numerical methods include the finite element method (Koketsu K. et al. [8], De Lautour O.R. et al. [9], etc.), the finite difference method (Frankel A. et al. [10], etc.), the boundary element method (Luzón F. et al. [11], Liu Z.X.. et al. [12], Ba Z.N. et al. [13], etc.), finite element–boundary element coupling method (Gatmiri B. et al. [14], etc.) and so on. In order to simplify the calculation, the research models mainly focus on geometric models in ideal uniform elastic space (Liang J.W. et al. [15], Sheikhhassani R. et al. [16], Lee V.W. et al. [17]).

However, in fact, soil is not a single-phase ideal elastic medium. Saturated soft soil exists widely in coastal and river valley areas. The propagation characteristics of seismic waves in saturated porous elastic media are quite different from those in elastic media models. For coastal valleys, it is more reasonable to study a saturated soil model. Biot [18,19] established the dynamic equation of a saturated two-phase medium, which laid the theoretical foundation for saturated soft soil. Zienkiewcz O.C. et al. [20] simplified and modified the basic model of Biot to a simpler practical model. Liang J.W. et al. [21] extended Biot’s theory to layered saturated sites by dynamic stiffness matrix. Kumar R. et al. [22] studied the reflection and propagation of plane waves in different fluid-saturated porous half-spaces. Gupta S. et al. [23] studied torsional surface waves in fluid-saturated porous half-spaces under heterogeneous layers. Barak M. S. et al. [24] studied the reflection and propagation of elastic waves on the non-ideal interface between a micropolar elastic solid half-space and a fluid-saturated porous solid half-space. In recent years, the simulation of saturated sites has developed in the direction of complex site, in combination with seismic resistance of underground engineering, and so on. Li W.H. et al. [25,26,27], Liu Z. X. et al. [28], Ba Z.N. et al. [29], and others have provided the analytical solutions of P wave and SV wave scattering by various underground structures in saturated half-spaces by using the wave function expansion method, finite element method, and indirect boundary element method (IBEM), respectively. Fattah M.Y. et al. [30,31,32] studied the response and behavior of dry sand and saturated sand by experiment. Han B. et al. [33] studied the mechanism of compression wave propagation in saturated soil by hydrodynamics coupling analysis and numerical method. It is worth pointing out that most of the current studies are based on the certain saturated site seismic response studies, and there are few studies on uncertain sites.

Furthermore, in view of the difficulty of the accurate determination of the geotechnical parameters and the uncertainty of the actual site characteristics (site shape, geotechnical parameters, anisotropy (Carcione J.M. [34]), etc.), the research on the randomness of the site develops accordingly. In the sedimentary valley, the medium randomness is the randomness of the medium parameters (shear modulus, porosity, permeability coefficient, etc.), and the boundary randomness is the randomness of the boundary shape. In order to reveal the response of ground motion in various conditions, more studies focus on the influence of the change and randomness of the site medium, boundary, and others on ground motion. Djilali Berkane H et al. [35] used the random method to solve the response of underground pipelines under spatially randomly distributed earthquakes. Liu Z.X. et al. [36] studied the influence of the randomness of geotechnical media on the amplification of seismic waves. Ge Z.X. [37] studied the seismic response of a semicircular sedimentary valley with a gradient of shear wave velocity in any direction to the incident SH wave. Dravinski M. et al. [38] studied the steady-state scattering of plane simple harmonic SH waves in multilayer homogeneous isotropic soils of arbitrary shape in half-space by using the direct boundary integral equation method. Li J. et al. [39] proposed an analysis method which considers the influence of random characteristics of geotechnical media on the ground motion coherence function of engineering site. He Y. et al. [40] analyzed the influence of shape randomness on seismic wave scattering when SH waves are incident. Liu Z.X et al. [41] provided a simulation method of stochastic seismic response for river valleys.

It should be pointed out that there are few studies on the random seismic responses of saturated complex sites in the above literature. This paper, considering the randomness of saturated media parameters, such as porosity and the permeability coefficient and the randomness of geometric boundary, and based on Biot’s theory, and indirect boundary element method, the Monte Carlo method is used to analyze the stochastic amplification of ground motion in a saturated sedimentary valley. The stochastic seismic response law of saturated complex sites is obtained, which is useful in engineering.

2. Calculation Model and Method

2.1. Calculation Model

The calculation model used in this article is shown in Figure 1. It is assumed that there is a sedimentary valley site in the saturated half-space, and the excitation is incident at an angle $\theta$ at infinity from the bedrock half-space. $L_1$ and $L_2$ are the boundary.

2.2. Calculation Method

2.2.1. Biot’s Theory

According to Biot’s theory [18], the constitutive equation of homogeneous saturated media can be expressed as follows:

$${\sigma }_{ij}=\lambda e{\delta }_{ij}+2\mu {\epsilon }_{ij}-{\delta }_{ij}\alpha P(i,j=x,y)$$

(1)

$$P=-\alpha \mathrm{M}{u}_{i,i}-M{w}_{i,i}$$

(2)

$$\mu u_{i,jj}+(\lambda +{\alpha }^{2}M+\mu )u_{j,ji+}\alpha M{w}_{j,ji}=\rho {\ddot{u}}_i+{\rho }_f{\ddot{u}}_i$$

(3)

$$\alpha M{u}_{j,ji}+M{w}_{j,ji}={\rho }_f{\ddot{u}}_i+m{\ddot{w}}_i+b{\dot{w}}_i$$

(4)

where ${\sigma }_{ij}$ is the total stress tensor of soil, P is pore water pressure. ${\epsilon }_{ij}$ is the average strain tensor of soil, $e$ is the volumetric strain of the soil skeleton. $\mu ,\lambda$ is the lame constant of the soil skeleton, ${\delta }_{ij}$ is the Kronecker function. $u_i$ and $w_i$ represent the displacement vector of the soil skeleton and the displacement of water with respect to the soil skeleton, respectively. $\alpha$ and $M$ are parameters to characterize the compressibility of soil particles and pore fluids. $\rho$ is the total density of the soil, ${\rho }_f$ is the mass density of the fluid, $\mathrm{m}$ is a parameter, which is the same as the physical meaning of the mass.

$b={\rho }_{f}g/k$ is the dissipation coefficient, which reflects the soil viscous coupling degree. $g$ is the gravity acceleration, k is the permeability coefficient. The unit of the permeability coefficient is the same as the unit of flow velocity. When the permeability coefficient $\mathrm{k}=\infty$, that is $b=0$, the fluid can flow freely in the pores. The internal friction is not taken into account. When the permeability coefficient $\mathrm{k}=0$, that is $\mathrm{b}\to \infty$, the viscous coupling effect is significant, and the soil is almost impermeable.

2.2.2. IBEM Method

According to IBEM and elastic wave theory, the total displacement $u^{\left(t\right)}$ of any point in the plane is obtained as follows:

$$u^{\left(\mathrm{t}\right)}=u^{\left(\mathrm{f}\right)}+u^{\left(\mathrm{s}\right)}$$

(5)

where $u^{\left(\mathrm{f}\right)}$ represents the free field displacement at any point, and $u^{\left(\mathrm{s}\right)}$ is the scattered wave field displacement constructed by a virtual wave source.

The free field displacement is calculated by using the wave reflection principle. When the SV wave is obliquely incident, three kinds of reflected waves (${\mathrm{P}}_1$ wave, ${\mathrm{P}}_2$ wave, and SV wave) are generated on the ground reflector in the saturated half-space. Then the potential function expressions of the above three waves in a Cartesian coordinate system are as follows:

$${\varphi }_1^{\mathrm{r}}=a_1{\mathrm{e}}^{-\mathrm{i}{k}_{\alpha 1}\left(xsin{\theta }_{\alpha 1}+ycos{\theta }_{\alpha 1}\right)}$$

(6)

$${\varphi }_2^{\mathrm{r}}=a_2{\mathrm{e}}^{-\mathrm{i}{k}_{\alpha 2}\left(xsin{\theta }_{\alpha 2}+ycos{\theta }_{\alpha 2}\right)}$$

(7)

$${\psi }^{\mathrm{r}}=b_1{\mathrm{e}}^{-\mathrm{i}{k}_{\beta }\left(xsin{\theta }_{\beta }+ycos{\theta }_{\beta }\right)}$$

(8)

where $a_1,a_2,b_1$ represent the amplitude coefficient, while $k_{\alpha 1},k_{\alpha 2},k_{\beta }$ represent the wave number. According to the relationship between the potential function and displacement, free field soil skeleton displacement $u_i^{\left(\mathrm{f}\right)}$, fluid relative displacement $w_i^{\left(\mathrm{f}\right)}$ and pore water pressure $p^{\left(\mathrm{f}\right)}$ can be expressed as follows:

$$u_x^{\left(\mathrm{f}\right)}=-\mathrm{i}{k}_{\alpha 1}\left({\varphi }_1^{\left(\mathrm{r}\right)}+{\varphi }_2^{\left(\mathrm{r}\right)}\right)sin{\theta }_{\alpha 1}+i{k}_{\beta }\left({\psi }^{\left(\mathrm{i}\right)}-{\psi }^{\left(\mathrm{r}\right)}\right)cos{\theta }_{\beta }$$

(9)

$$u_y^{\left(\mathrm{f}\right)}=\mathrm{i}{k}_{\beta }\left({\psi }^{\left(\mathrm{i}\right)}+{\psi }^{\left(\mathrm{r}\right)}\right)sin{\theta }_{\beta }-i{k}_{\alpha 1}{\varphi }_1^{\left(\mathrm{r}\right)}cos{\theta }_{\alpha 1}-\mathrm{i}{k}_{\alpha 2}{\varphi }_2^{\left(\mathrm{r}\right)}cos{\theta }_{\alpha 2}$$

(10)

$$w_x^{\left(\mathrm{f}\right)}=-\mathrm{i}{k}_{\alpha 1}\left[{\chi }_1{\varphi }_1^{\left(\mathrm{r}\right)}+{\chi }_2{\varphi }_2^{\left(\mathrm{r}\right)}\right]sin{\theta }_{\alpha 1}+i{k}_{\beta }{\chi }_3\left({\psi }^{\left(\mathrm{i}\right)}-{\psi }^{\left(\mathrm{r}\right)}\right)cos{\theta }_{\beta }$$

(11)

$$w_y^{\left(\mathrm{f}\right)}=\mathrm{i}{k}_{\beta }\left[{\chi }_3\left({\psi }^{\left(\mathrm{i}\right)}+{\psi }^{\left(\mathrm{r}\right)}\right)\right]sin{\theta }_{\beta }-\mathrm{i}{k}_{\alpha 1}{\chi }_1{\varphi }_1^{\left(\mathrm{r}\right)}cos{\theta }_{\alpha 1}-\mathrm{i}{k}_{\alpha 2}{\chi }_2{\varphi }_2^{\left(\mathrm{r}\right)}cos{\theta }_{\alpha 2}$$

(12)

$$p^{\left(\mathrm{f}\right)}=\left(\alpha +{\chi }_1\right)M{k}_{\alpha 1}^2{\varphi }_1+\left(\alpha +{\chi }_2\right)M{k}_{\alpha 2}^2{\varphi }_2$$

(13)

When the sedimentary valley exists, the scattered waves will be generated in the half-space and inside the deposit. The displacement, stress of soil skeleton and pore water pressure can be expressed by the following formula.

$$u_i^{\left(\mathrm{s}\right)}\left(x\right)=b_{n^{\prime }}{G}_{i,1}^{\left(\mathrm{s}\right)}\left(x_n,x_{n^{\prime }}\right)+c_{n^{\prime }}{G}_{i,2}^{\left(\mathrm{s}\right)}\left(x_n,x_{n^{\prime }}\right)+d_{n^{\prime }}{G}_{i,3}^{\left(\mathrm{s}\right)}\left(x_n,x_{n^{\prime }}\right)$$

(14)

$$w_i^{\left(\mathrm{s}\right)}\left(x\right)=b_{n^{\prime }}{G}_{wi,1}^{\left(\mathrm{s}\right)}\left(x_n,x_{n^{\prime }}\right)+c_{n^{\prime }}{G}_{wi,2}^{\left(\mathrm{s}\right)}\left(x_n,x_{n^{\prime }}\right)+d_{n^{\prime }}{G}_{wi,3}^{\left(\mathrm{s}\right)}\left(x_n,x_{n^{\prime }}\right)$$

(15)

$${\sigma }_{ij}^{\left(\mathrm{s}\right)}\left(x\right)=b_{n^{\prime }}{T}_{ij,1}^{\left(\mathrm{s}\right)}\left(x_n,x_{n^{\prime }}\right)+c_{n^{\prime }}{T}_{ij,2}^{\left(\mathrm{s}\right)}\left(x_n,x_{n^{\prime }}\right)+d_{n^{\prime }}{T}_{ij,3}^{\left(\mathrm{s}\right)}\left(x_n,x_{n^{\prime }}\right)$$

(16)

$$p^{\left(\mathrm{s}\right)}\left(x_n\right)=b_{n^{\prime }}{T}_{p,1}^{\left(\mathrm{s}\right)}\left(x_n,x_{n^{\prime }}\right)+c_{n^{\prime }}{T}_{p,2}^{\left(\mathrm{s}\right)}\left(x_n,x_{n^{\prime }}\right)+d_{n^{\prime }}{T}_{p,3}^{\left(\mathrm{s}\right)}\left(x_n,x_{n^{\prime }}\right)$$

(17)

where $n^{\prime }$ is the point on the virtual source plane, and $b_{n^{\prime }},c_{n^{\prime }},d_{n^{\prime }}$ are the source densities of ${\mathrm{P}}_1$ waves, ${\mathrm{P}}_2$ waves, and SV waves at the $n^{\prime }$th discrete point on the virtual source plane, respectively. $G_{i,l}^{\left(\mathrm{s}\right)}\left(x_n,x_{n^{\prime }}\right),G_{i,l}^{\left(\mathrm{s}\right)}\left(x_n,x_{n^{\prime }}\right),T_{ij,l}^{\left(\mathrm{s}\right)}\left(x_n,x_{n^{\prime }}\right),T_l^{\left(\mathrm{s}\right)}\left(x_n,x_{n^{\prime }}\right)$ represent the Green’s functions of solid phase displacement, fluid relative displacement, total stress and pore water pressure in the saturated half-space, respectively (angle mark 1, 2, 3, corresponding ${\mathrm{P}}_1$ wave, ${\mathrm{P}}_2$ wave and SV wave source, respectively).

The surface free boundary conditions are automatically satisfied by Green’s function. The continuity conditions on the interface between saturated deposition and half-space are as follows:

$$\left\{\begin{array}{l}{u}_x^{\mathrm{I}}=u_x^{\mathrm{II}},u_y^{\mathrm{I}}=u_y^{\mathrm{II}},w_n^{\mathrm{I}}=w_n^{\mathrm{II}}\\ {\sigma }_{nn}^{\mathrm{I}}={\sigma }_{nn}^{\mathrm{II}},{\sigma }_{nt}^{\mathrm{I}}={\sigma }_{nt}^{\mathrm{II}}\\ p^{\mathrm{I}}-p^{\mathrm{II}}=0\end{array}\right.$$

(18)

where superscript I and II represent the half-space and sedimentary valley, respectively.

2.2.3. Application of Monte Carlo Method in Site Random Analysis

The random algorithm of Monte Carlo (Shinozuka, M. et al. [42]), also known as the statistical simulation method, is a method of dealing with mathematical problems. It uses a group of random numbers to approximately solve the problem, finds out the similarity of probability and statistics, and then obtains the approximate solution by experimental sampling. It can be used to solve nonlinear, parametric excitation, and other random problems, and is widely used(Caserta A. et al. [43], Hacıefendioğlu, K. et al. [44]).

For the medium randomness, the Monte Carlo method is adopted to randomly generate multiple groups of media samples with different porosity and permeability coefficients for saturated sedimentary valleys. The seismic response of complex sites under different sample conditions is calculated based on the IBEM. The results are compared for statistical analysis to reveal the influence of media randomness on seismic response.

For the boundary randomness, the previous studies are mostly based on regular boundary models, with which it is difficult to reflect the effect of irregular terrain on ground motion. In this paper, multiple groups of boundary samples are randomly generated by using the Monte Carlo method. The IBEM is used to calculate the seismic response of the samples for simulating the influence of boundary randomness on ground motion. A general conclusion is given to solve various problems of random boundary seismic response in engineering.

3. Numerical Examples

3.1. Media Randomness

3.1.1. Calculation Model

The saturated sedimentary valley model is shown in Figure 2, with a depth of 200 m, a width of 400 m. The width of the cover layer is 200 m, and the thickness is 26.8 m. The incident angles are 0° and 30°. The excitation is incident at an angle $\theta$ from the infinity of the bedrock half-space. It is necessary to consider the influence of the randomness of parameters such as the porosity and permeability coefficient on the amplification effect of valley ground motion.

3.1.2. Effect of Porosity on Saturated Sedimentary Valleys

One hundred groups of random sedimentary porosity samples are generated by the Monte Carlo method, which obeys the normal distribution with mean value of 0.3 and coefficient of variation of 0.03. The dimensionless frequency $\eta$ is defined as the ratio of the width of sedimentary valley 2a to the incident wavelength, $\eta =2a/\lambda =\omega a/c\pi$.

By changing the sediment porosity, the displacement response of the random samples is solved, and the mean value $\mu$, standard deviation $\sigma$, and variation coefficient $\sigma /\mu$ of the surface displacement response amplitude of the saturated sediment valley under the incident plane SV wave are calculated. The confidence interval $\mu \pm \sigma$ and variation coefficient are used to analyze the impact of porosity change on the valley site response.

The surface response displacement amplitudes $\mu$ and $\mu \pm \sigma$ of the samples are shown in Figure 3, when the porosity is random while the permeability coefficient is constant $(k=\infty$). The solid line and the dash-dot lines represent $\mu$ and $\mu \pm \sigma$, respectively. The coefficients of variation of horizontal and vertical displacement amplitudes of the random samples are presented in Figure 4. Abscissa $x/a$ represents the horizontal distance between the observation point of the surface and the center of the sedimentary surface, in unit of 1. The ordinate represents the ratio of the amplitude of the surface displacement to the amplitude of the incident wave, in unit of 1.

The response characteristics of surface displacement in saturated sedimentary valleys show multifactorial correlations with porosity, incident frequency, angle, incident position, and other factors. As shown in Figure 3, the confidence interval of the ground displacement amplitude in the outer half-space of the deposit is narrow, while that in the interior of the deposit is wide, indicating that the change of porosity in the sedimentary domain has a great influence on the surface response inside the deposit. It is consistent with the result of the significant increase in the coefficient of variation of the sedimentary section in Figure 4. The coefficient of variation reflects the discrete degree of the amplitude of surface displacement.

As can be seen in

Table 1. Average variation coefficient of horizontal and vertical displacement amplitude (random porosity).

$\mathit{\eta }$	Horizontal		Vertical
	0°	30°	0°	30°
0.5	0.0050	0.0076	0.0069	0.0096
1	0.0076	0.0191	0.0312	0.0467
2	0.0885	0.0785	0.2475	0.1088

, with the dimensionless frequency of the incident wave from 0.5 to 2, the coefficient of variation increases significantly, and the mean value increases by 35.87 times at most (vertical). The results show that the randomness of porosity has a large effect on the displacement at high frequency, which accords with the fluctuation law. In this paper, the horizontal direction is denoted by (H), and the vertical direction is indicated by (V). When the incident angle is different, the spatial variation of the variation coefficient of surface displacement amplitude is significantly different. On the whole, the influence on the interior of the sediment is more obvious, indicating that the influence of incident angle on seismic wave scattering cannot be ignored.

3.1.3. Effect of Permeability Coefficient on Saturated Sedimentary Valleys

The formation permeability coefficient determines the coupling strength between pore water and soil skeleton, which is one of the key parameters of formation–structure dynamic interaction. One hundred groups of sedimentary permeability coefficient samples are randomly generated by the Monte Carlo method, which obeys the normal distribution with a mean value 0.3 m/s and coefficient of variation 0.1, and other data are consistent with the previous section. The ground motion of the saturated sedimentary valley under the change of permeability coefficient (finite value) is calculated, and its influence law on displacement amplitude is obtained, so as to study the influence mechanism of the permeability coefficient. The surface response displacement amplitudes $\mathsf{\mu }$ and $\mu \pm \sigma$ of 100 random samples when the permeability coefficient is random and the porosity is constant $\left(n=0.1,n_c=0.3\right)$ is shown in Figure 5. $n$ and $n_c$ represent the porosity of bedrock and sediment, respectively. The coefficient of variation of the amplitude of the ground displacement response of the sample is shown in Figure 6.

Different from the effect of porosity randomness, the influence of the randomness of permeability coefficient on the displacement amplitude of the saturated sedimentary valley increases significantly. Comparing Figure 4 and Figure 6, it can be seen that the influence of the permeability coefficient randomness on seismic wave scattering in sedimentary valley is greater than that of the porosity randomness. It can be seen from Figure 5 that the $\pm \sigma$ confidence interval width of the surface displacement amplitude in the sedimentary valley is slightly larger than that in the outer half-space, which is consistent with the larger coefficient of variation of the surface displacement in the sediment in Figure 6. It indicates that the randomness of permeability coefficient has a more significant effect on sediment, while it has less effect on the half-space response.

The incident angle still affects the spatial variation of the amplitude variation coefficient of surface displacement. When the seismic wave is obliquely incident, the variation coefficient of the left ($x/a<0$) surface displacement amplitude is obviously higher than that of the vertical incidence, and the change is more obvious in the half-space as a whole, and especially the difference of the vertical displacement variation coefficient distribution under the incident SV wave of $\eta =2$ is more significant.

As can be seen in

Table 2. Mean variation coefficient of horizontal and vertical displacement amplitude (random permeability coefficient).

$\mathit{\eta }$	Horizontal		Vertical
	0°	30°	0°	30°
0.5	0.11	0.14	0.15	0.16
1	0.09	0.45	0.21	0.61
2	0.14	0.48	0.42	0.64

, compared with 0°, the average coefficient of variation of displacement amplitude of 30° is enlarged by 1.3–4.8 times (H) and 1.1–3.0 times (V), respectively, indicating that the influence of the incident angle on seismic wave scattering cannot be ignored. From the perspective of the position of the incident wave, the coefficient of variation of the left surface $(x/a<0)$ is higher than that of the right surface $(x/a>0)$ as a whole. That is, the incident position has a certain influence on the coefficient of variation. Under the incident of the high frequency wave, the influence of the randomness of permeability coefficient on the seismic response at the edge of the sedimentary valley is also very prominent, and the coefficient of variation can reach 8.23 ($\eta =2,\theta =0°$,V). This phenomenon becomes more significant with the increase in the dimensionless frequency.

3.2. Boundary Randomness

Many measurements have proved the influence of the local site on the seismic wave. There are few regular boundaries in the real site, while most are uncertain boundaries. In this section, the Monte Carlo method is used to generate random boundary samples, and the action mechanism of ground motion in a saturated medium in a random site is explored by discussing the influence of saturated soil porosity and permeability coefficient on the amplitude of random boundary surface displacement.

3.2.1. Calculation Model

One hundred groups of random boundary samples are randomly generated by the Monte Carlo method, which obeys the normal distribution with the mean of radius of 1 times and the coefficient of variation of 0.1 times. The model of the saturated sedimentary valley under the random boundary is shown in Figure 7, with a depth of 220 m, a width of 440 m, a cover width of 200 m and a thickness of 46.8 m. The incident angles $\theta$ are 0° and 30°.

3.2.2. Effect of Porosity of Sedimentary Domain on Saturated Sedimentary Valley with Random Boundary

Porosity is a factor affecting the scattering of elastic waves in saturated sedimentary valleys. Based on Biot’s theory and the IBEM, this section solves the displacement response of random boundary samples by changing the porosity of the half-space. Comparing the results of elastic sedimentary valleys with those of saturated sedimentary valleys, the necessity of considering the saturated sedimentary valleys and its influence on the amplitude of random boundary surface displacement is discussed. Then, by changing the sedimentary porosity, the mean value $\mu$, standard deviation $\sigma$ and coefficient of variation $\sigma /\mu$ of the surface displacement response amplitude of the saturated sedimentary valley under the incident plane SV wave are calculated, and the confidence interval $\mu \pm \sigma$ and coefficient of variation are used to analyze the influence of the change of porosity on the valley site response under the random boundary.

Five dimensionless frequencies of $\eta =0.25,0.5,1,1.5,2$ are selected to calculate the surface displacement amplitude and coefficient of variation of the random boundary when the bedrock half-space porosity $n$ is 0.001 and the sedimentary porosity $n_{\mathrm{c}}$ is 0.1,0.2,0.3, respectively. The physical parameters of saturated soils with different porosity are shown in

Table 3. Calculation parameters of saturated sedimentary valley.

	Damping Ratio	Poisson Ratio	Porosity Ratio	Critical Porosity	SV Wave Velocity (m/s)
Bedrock	0.02	0.25	0.001	0.36	792.6
Overburden layer	0.05	0.25	0.1	0.36	706.3
			0.2	0.36	599.8
			0.3	0.36	444.5

:

The images of the amplitude and coefficient of variation of the ground surface at the time of $\eta =0.5,1,2$, $n_{\mathrm{c}}=0.1,0.2,0.3$ are shown in Figure 8 and Figure 9. In the figure, HDA represents the horizontal surface displacement amplitude. The horizontal surface response displacement amplitudes in the case of vertical incidence is shown in Figure 8. In order to facilitate the comparison, the calculation results of the average radius are also given. The average value of the sample amplitude is consistent with the result of the average radius, which shows that the calculation is reasonable. The coefficients of variation of horizontal displacement amplitudes of the random samples are shown in Figure 9.

When the SV wave is incident, the spatial distribution characteristics of displacement are significantly different at different frequencies. As shown in Figure 8, when the dimensionless frequency increases, the change of the amplitude of surface displacement is more complex. When the dimensionless frequency $\eta$ is from 0.5 to 2, the amplitude of sedimentary displacement is magnified by 4.67 times ($n_{\mathrm{c}}=0.1$), 5.85 times ($n_{\mathrm{c}}=0.2$), and 12.52 times ($n_{\mathrm{c}}=0.3$), respectively. The results show that the change of sedimentary porosity has little effect on the amplitude at low frequency. At high frequency, with the increase in sedimentary porosity, the amplitude of surface displacement increases, and the amplitude of sedimentary displacement oscillates and increases. When the porosity of the sedimentary medium is close to the critical state, the variation of surface displacement is more complex. When porosity $n_{\mathrm{c}}=0.2\mathrm{or}$0.3, the magnification also increases gradually. The results show that with the increase in frequency and porosity, the confidence interval of displacement amplitude becomes wider and the discreteness becomes larger.

As can be seen in Figure 9, the porosity of the sedimentary domain has a great influence on the interior of the sedimentary valley, and the coefficient of variation of the sedimentary part is larger than that of the half-space. Under vertical incidence conditions, the interior of the sedimentary valley is magnified 3.60 times ($n_{\mathrm{c}}=0.1$), 3.97 times ($n_{\mathrm{c}}=0.2$), and 4.27 times ($n_{\mathrm{c}}=0.3$). With the increase in porosity $n_{\mathrm{c}}$, the amplification effect is more obvious, and the influence of boundary randomness on the outer half-space increases gradually. Compared with the case of vertical incidence, the spatial variation of the coefficient of variation of surface displacement amplitude is significantly different at a wave incidence of 30°, especially at $\eta =2$, and the distribution of coefficient of variation of vertical displacement is more discrete. The coefficient of variation of the incident proximal end (the left side of the surface) is generally larger than that of the incident distal end (the right side of the surface), which is consistent with the above conclusion in the valley with certain boundary.

The mean values of variation coefficients of horizontal and vertical displacement amplitudes with sediment porosity are listed in

Table 4. Mean value of variation coefficient of horizontal displacement amplitude (with sediment porosity).

$\mathit{\eta }$	0$°$			30 $°$
	${\mathit{n}}_{\mathit{c}}=0.1$	${\mathit{n}}_{\mathit{c}}=0.2$	${\mathit{n}}_{\mathit{c}}=0.3$	${\mathit{n}}_{\mathit{c}}=0.1$	${\mathit{n}}_{\mathit{c}}=0.2$	${\mathit{n}}_{\mathit{c}}=0.3$
0.25	0.06	0.09	0.12	0.07	0.12	0.14
0.5	0.15	0.22	0.21	0.08	0.16	0.20
1	0.39	0.37	0.30	0.64	0.52	0.61
1.5	0.80	0.78	0.62	0.75	0.61	0.71
2	0.52	0.41	0.41	0.34	0.30	0.27

and

Table 5. Mean value of variation coefficient of vertical displacement amplitude (with sediment porosity).

$\mathit{\eta }$	0$°$			30 $°$
	${\mathit{n}}_{\mathit{c}}=0.1$	${\mathit{n}}_{\mathit{c}}=0.2$	${\mathit{n}}_{\mathit{c}}=0.3$	${\mathit{n}}_{\mathit{c}}=0.1$	${\mathit{n}}_{\mathit{c}}=0.2$	${\mathit{n}}_{\mathit{c}}=0.3$
0.25	0.33	0.35	0.34	0.05	0.06	0.07
0.5	0.23	0.30	0.27	0.20	0.18	0.18
1	0.41	0.37	0.37	0.32	0.27	0.26
1.5	1.32	1.42	1.26	1.42	1.05	1.39
2	1.10	0.99	1.46	0.74	0.78	0.69

, respectively. It can be seen that the average value of the coefficient of variation of displacement increases for both directions with the increase in dimensionless frequency, which is 1.3–4 times larger than that of $\eta =$ 0.25, and almost reaches the maximum at $\eta =1.5$. The average value of the coefficient of variation is small when dimensionless frequency $\eta =2$, because at some frequencies (such as $\eta =2$), the amplification is close to 0, resulting in a standing wave phenomenon. The change of incident angle also has an influence on the mean value of the coefficient of variation. On the whole, the variation of the coefficient of variation is more complex due to the influence of the scattering effect, waveform conversion, and other factors when the seismic wave is obliquely incident.

3.2.3. Effect of Permeability Coefficient on Saturated Sedimentary Valley with Random Boundary

In this section, the ground motion of characteristic points of a saturated sedimentary valley under the change of permeability coefficient (finite value and infinity case) is calculated, and its influence on the displacement amplitude of the bedrock point and sedimentary point is obtained.

For the permeability coefficient, the case of $k=0.1,0.5,1,5,\infty$ is calculated in this paper (unit: m/s). Several typical cases of permeability coefficient $k=0.1,0.5,\infty ,\eta =0.5,1,2$ are given in Figure 10 and Figure 11. The horizontal and vertical surface displacement amplitudes of the random boundary under vertical incidence is shown in Figure 10. The horizontal and vertical coefficients of variation of random boundaries at different incident angles is shown in Figure 11.

As shown in Figure 10, under the incidence of the SV wave, the surface displacement response of a saturated sedimentary valley demonstrates a correlation with the permeability coefficient, incident frequency, incident angle, and other factors. When the permeability coefficient $k>1$, the displacement amplitude is close to the pervious case, with the increase in the permeability coefficient, the amplitude slightly changes. Compared with other permeability coefficients, the confidence interval of the surface displacement amplitude of the surface point with $k=0.1$ is wider, that is, the permeable boundary has a greater influence on the surface displacement amplitude of the random boundary. Moreover, the position of the larger confidence interval of the displacement amplitude basically corresponds to the peak position of the amplitude mean value. Within the sedimentary valley, there is the same rule. On the whole, whether in the bedrock half-space or in the sedimentary valley, the influence of the impermeable case on the displacement amplitude of the random boundary is greater than that in the permeable case.

As shown in Figure 11, the coefficient of variation at the boundary point of the sedimentary valley $\left(\frac{x}{a}=\pm 1\right)$ obviously increases to form a wave peak, and this phenomenon becomes more significant with the increase in dimensionless frequency, which is consistent with the law of significant amplification at the corner of the high-frequency sedimentation valley, indicating that under the incidence of high-frequency waves, the influence of boundary randomness on the seismic response at the edge of the sedimentary valley is also very prominent.

The influence of the change of permeability coefficient on the location of sedimentary domain is greater than that of bedrock. The reason is that there is not only the focusing of seismic waves, but also the conversion of wave patterns in sediments. The critical angle corresponding to the permeability coefficient at each frequency is about 32.2°. Because the oblique incident angle (30°) is close to the critical angle, the surface response becomes complex.

The average values of the variation coefficients of horizontal and vertical displacement amplitudes with permeability coefficients are listed in

Table 6. Mean value of variation coefficient of horizontal displacement amplitude (with permeability coefficients).

$\mathit{\eta }$	0$°$					30 $°$
	$\mathit{k}$= 0.1	$\mathit{k}$= 0.5	$\mathit{k}$= 1	$\mathit{k}$= 5	$\mathit{k}$ = $\mathit{\infty }$	$\mathit{k}$= 0.1	$\mathit{k}$= 0.5	$\mathit{k}$= 1	$\mathit{k}$= 5	$\mathit{k}$ = $\mathit{\infty }$
0.5	0.28	0.76	0.52	0.20	0.18	0.32	0.38	0.51	0.24	0.19
1	0.47	0.47	0.30	0.26	0.26	0.41	0.47	0.47	0.48	0.50
2	1.94	0.75	0.66	0.65	0.65	1.80	0.86	0.69	0.77	0.90

and

Table 7. Mean value of variation coefficient of vertical displacement amplitude (with permeability coefficients).

$\mathit{\eta }$	0$°$					30 $°$
	$\mathit{k}$= 0.1	$\mathit{k}$= 0.5	$\mathit{k}$= 1	$\mathit{k}$= 5	$\mathit{k}$ = $\mathit{\infty }$	$\mathit{k}$= 0.1	$\mathit{k}$= 0.5	$\mathit{k}$= 1	$\mathit{k}$= 5	$\mathit{k}$= $\mathit{\infty }$
0.5	0.80	0.54	0.32	0.31	0.32	0.78	0.25	0.23	0.22	0.21
1	1.08	0.41	0.27	0.25	0.24	0.65	0.37	0.42	0.47	0.48
2	2.72	1.01	0.94	0.92	0.89	1.81	0.90	0.89	0.91	0.96

, respectively. It can be seen from the tables that the coefficient of displacement variation in both directions increases with the increase in dimensionless frequency. At $k=0.1$, the magnification of high frequency ($\eta =$2.0) is 6.88 times (H, $\theta =0°$), and 3.42 times (V, $\theta =0°$) compared with low frequency ($\eta =0.5,\theta =0°$). The average magnification of $k=0.5~\infty$ coefficient of variation is 2.27 times (H), and 2.63 times (V). From the magnification, it can be seen that compared with the permeable case, the impermeable case is more obviously magnified by frequency. The change of incident angle also has a certain influence on the mean value of the coefficient of variation. In vertical incidence, with the increase in permeability coefficient, the average value of the coefficient of variation tends to decrease, while 30° is more complex, which may be affected by th escattering effect, waveform conversion, and other factors. Oblique incidence will magnify the coefficient of variation as a whole. Compared with the random medium under the certain boundary, under the random boundary condition, the change of porosity or permeability coefficient has a more significant influence on the coefficient of variation, and a greater influence on the response of ground motion.

4. Discussion and Conclusions

In this paper, based on Biot’s theory and IBEM, the Monte Carlo method is used to generate random sedimentary valley samples. The displacement amplitude response under the SV wave incidence is calculated to explore the influence of the randomness of sedimentary media (porosity and permeability coefficients) and boundary randomness on the site response. The following conclusions are drawn:

(1)

The change of porosity has little influence on the seismic response under the certain boundary condition, but it has a greater influence under the random boundary condition, and the coefficient of variation is magnified by 1.5–3 times. The influence cannot be ignored. The coefficient of variation of the surface response is positively correlated with sedimentary porosity and increases continuously with the increase in porosity. Therefore, the change of porosity needs to be considered in the uncertain boundary in practical engineering.

(2)

Compared with the soil porosity, the boundary permeability coefficient has a greater influence on the ground motion amplification effect of the sedimentary valley. Under the certain boundary condition, compared with the permeable case, the coefficient of variation of the impermeable case is magnified by an average of 2–3 times, and the displacement amplitude data are more discrete. The effect is more significant under uncertain boundary conditions, with an average magnification of 5–6 times. The change of media permeability coefficient in the sedimentary valley also leads to a great change in the width of the confidence interval, so it is more necessary to consider the influence of the permeability coefficient in the sedimentary valley.

(3)

The influence of boundary shape change is related to many factors, such as incident frequency, incident angle, incident wave position, and so on. The variability of the displacement amplitude of the sedimentary part is significant at low frequency, but on the contrary, at high frequency, the significant variability is at the edge of the sedimentary valley. Compared with the vertical incidence, the average magnification of the coefficient of variation at 30° is 2–3 times, indicating that the angle also has a great influence. The coefficient of variation of the front wave surface is higher than that of the back wave surface, which proves that the position of the incident wave also has a certain influence on the coefficient of variation.

(4)

The results show that the spatial variation of ground motion is very large. The amplitude of ground motion at the top of the valley slope is much greater than that at the bottom. The amplitude at the front wave surface is greater than that at the back wave surface. These conclusions are consistent with the monitoring conclusions of the Feitsui Canyon(Gao Y.F. et al. [45]). More parameter analysis and verification is needed, in combination with the measured ground motion records.

To sum up, both the randomness of saturated medium and boundary shape have a certain influence on the response of the ground motion of a sediment valley. Compared with porosity, the displacement amplitude is more affected by the randomness of permeability coefficient. In the actual engineering site, selecting the appropriate medium parameters in the confidence interval can greatly simplify the calculation. The critical parameters, such as the permeability coefficient, should be analyzed in detail. In engineering design, special attention should be paid to the spatial variation effect of ground motion in the sedimentary valley site. The conclusion plays an important guiding role in earthquake disaster prevention and mitigation, such as seismic risk analysis, and earthquake microzonation in saturated sedimentary valleys.