Simulation of Spatially Correlated Multipoint Ground Motions in a Saturated Alluvial Valley

Based on Biot’s theory, the boundary element method, and spectral representation method, an eﬀective simulation method for multiple-station spatially correlated ground motions on both bedrock and surface is developed, incorporating the spectral density function, coherence function, and site transfer function that consider both the wave scattering eﬀect and the medium saturation. The accuracy and feasibility of the present method are validated by a typical numerical example. Our results indicate that the local site conditions and the saturation property of the medium signiﬁcantly aﬀect the multipoint spatially correlated earthquake ground motions, especially in the long-period range. It is necessary to use spatially varying ground motions with the rational consideration of local site eﬀects and medium saturation as input during the seismic analysis of large-span structures.


Introduction
Many cities are located on alluvial valleys, and numerous seismic observations and damage surveys have confirmed the significant influence of alluvial valleys on ground motion [1][2][3][4][5]. A typical example is a great deal of large-span bridges and high dams built in saturated alluvial valleys. However, the dynamic response of large-span structures is greatly affected by the seismic spatial variation effect, which may lead to errors in calculation. Accurate and reliable ground motion parameters are needed in the seismic design of important engineering structures. erefore, the influence of multiple input conditions should be considered when studying the seismic response of large-span structures [6]. In a word, the multistation ground motion in a saturated alluvial valley studied in this paper has a profound influence on the theoretical research value and engineering application of future seismic research and development directions. eoretically, the choice of power spectral function and coherence function model is the key to simulate spatially varying seismic motions. Furthermore, many scholars have proposed several coherency function models to simulate spatially varying seismic ground motions, such as those reported by Loh, Feng et al., Hao et al., Harichandran et al., and Qu et al. [7][8][9][10][11]. ey obtained the ground motion data of the dense array by statistical regression using stochastic signal processing techniques. Some scholars [12][13][14] established the model of coherence function from theoretical analysis and then fitted the parameters in the model based on ground motion records. Kiureghian attributed the spatial variation of ground motion to the Wave Passage Effect, Incoherence Effect, Attenuation Effect, and Site Response Effect. Studying the influence of a single factor makes the analysis of ground motion more in line with the actual engineering site.
e key point of constructing the power spectrum function is the calculation of the transfer function. e current research [15][16][17] calculated the transfer function mostly based on the one-dimensional wave propagation theory [18] of layered half-space medium, without considering the scattering effect of seismic waves at local sites and the saturation state of the medium. e scattering effect of local topography on seismic waves cannot be ignored, and the medium in the valley site is often in a saturation state, which is quite different from the usual single-phase elastic model. Many scholars have studied seismic wave scattering [19][20][21][22][23][24][25][26]. It is worth pointing out that the above studies all focus on the seismic response of the alluvial valleys with elastic media. However, it should be noted that saturated soil is a kind of natural soil that is common in the coastal and valley areas, and the foundation soil of many cities is saturated, such as Tianjin, China. Based on Biot's theory [27] of wave propagation in saturated soil, the propagation characteristics of seismic waves in saturated poroelastic medium are quite different from those in the elastic medium model. It is more practical to use the saturated model to study it. Due to the complexity of the problem, there are relatively few studies based on saturated porous media models. Lin et al. [28,29] discussed the dynamic response of plane wave reflection in saturated porous half space. Li et al. [30][31][32][33] gave the analytical solutions of P and SV wave scattering in semicircular saturated depressions and saturated alluvial valleys. e scattering law of Pwave around a canyon was studied by Ba et al. [34] using the wave function expansion method. Liu [35] used the indirect boundary element method (IBEM) to study the scattering of P waves in saturated alluvial valleys. e above papers also illustrate the necessity of considering the wave scattering effect in the multistation seismic motion simulation. He et al. [36,37] considered the seismic wave scattering of complex terrain in elastic medium but did not consider the saturated medium. Based on the above literature, it can be seen that spatially related ground motion simulation considering saturated media is rare and still has room for development.
So far, many scholars [38][39][40][41][42][43][44][45] have studied spatially correlated seismic motions with local field effects in mind. At present, there are mainly two methods of spatially related multipoint ground motion simulation that have a great influence. e first method is based on the assumption of stationary random process and reflects the correlation of all points in space through the ground motion power spectrum matrix, and the representative one is the HOP [9] method. e second method is to use the random field interpolation method to obtain the time history of ground motion at other points. e representative method is Vanmarcke et al.'s [46] method.
In this study, an effective method is proposed to simulate spatially correlated ground motions at multiple stations by considering the scattering effects of seismic waves at local sites, such as saturated alluvial valley sites. Among all kinds of numerical methods, the boundary element method [30] (BEM) chosen in this paper is more suitable for calculating the broadband frequency of two-dimensional transfer function in half space. It has the unique advantage of reducing the dimension of the problem and automatically satisfying the infinite boundary conditions, which is particularly obvious in this paper. erefore, considering the spatial coherence effect, wave scattering effect, and saturated medium comprehensively, the BEM combined with the prototype spectral representation method [9] (PSRM) was adopted to obtain multistation ground motion in the saturated alluvial valley site. e method presented in this paper provides a new idea for the scattering of elastic waves in saturated media, which has important theoretical significance and engineering application, and provides a certain theoretical basis for the seismic design of engineering projects and the scientific basis of urban planning.

Calculation Model and Method.
e model used in this paper is shown in Figure 1. In the saturated half space, there is an alluvial valley with the shape of a semicircle. jand kare any two points on the overburden, and j' and k' are the corresponding points on the hypothetical bedrock surface. e excitation is assumed to be incident at an angle αfrom the bedrock half space.

Multipoint Ground Motion Calculation Method.
Because the distance from the seismic source to the surface is much larger than the size of the alluvial valley, it can be assumed that the ground motion at the bedrock is a zeromean stationary random process with the same intensity and frequency components. In this paper, the Clough-Penzien model [47] is used to simulate the power spectral density function. e spatially related auto and cross power spectrum can be expressed as where H j (iω) and H k (iω)are the transfer functions at locations jand k, which reflect the influence of local soil on wave propagation. e symbol * represents the complex conjugate. |c j′k′ (iω)|denotes the lagged coherency, which reveals the influence of valley topography on the spatial correlation of ground motion. e Sobczyk model (Sobczky, 1991) is used in this study. erefore, the power spectrum density functions of spatial ground motion at the bedrock can be expressed as a positive definite Hermite matrix S(iω), which can be simplified by Cholesky decomposition.

Shock and Vibration
According to PSRM, the stationary time histories can be directly expressed as where are the amplitudes and phase angles of the simulated time histories.
To obtain a nonstationary time histories that is consistent with the actual ground motion, an envelope function f(t)is applied to q j (t).

Biot's eory.
According to Biot's saturated poroelastic medium theory [48], the constitutive equation of saturated medium is as follows: where σ ij and Prepresent the total stress component and pore pressure of soil, respectively; e and ε ij , respectively, represent the volumetric strain and strain component of soil skeleton; λ and μare the Lame constant of soil skeleton; δ ij is the Kronecker function, with a value of 0 or 1; and α and M are parameters used to indicate the compressibility of soil particles and pore fluids (0 ≤ α ≤ 1, 0 ≤ M ≤ ∞). e solid frame displacement and fluid displacement relative to solid frame are denoted by u i and w i , and the motion equations in saturated poroelastic medium can be expressed as follows [49]: where ρis the total density of soil, ρ f is the mass density of fluid, m is a parameter corresponding to the physical meaning of mass, and b is the coefficient used to reflect the degree of viscosity coupling of soil. If internal friction is not considered, then b � 0.

Free
Field. e free field displacement is calculated by using the wave reflection principle; when the SV wave is incident at an angle θ β , three kinds of reflected waves (P I wave, P II wave, and SV wave) are generated on the reflective surface of the ground in the saturated half space.
en, the expression of the potential functions of the above three waves in the Cartesian coordinate system is as follows: where a 1 , a 2 , and b 1 represent the reflection coefficients and k α1 , k α2 , and k β represent the wave number. According to the relationship between the potential function and the displacement, solid frame displacement u i , and pore pressure p (π) in the free field can be, respectively, expressed as follows: PSV waves Shock and Vibration 2.3.3. Scattering Field. Scattered waves will be generated in half space and inside the sedimentary due to the presence of alluvial valleys. According to the single-layer potential theory, the scattering wave can be formed by the superposition of the expansion wave sources and shear wave sources on the virtual wave source surface inside and outside the deposition.
where n ′ is the point on the virtual source plane; b n ′ , c n ′ , and d n ′ are the source densities of P I wave, P II wave, and SV wave at the n ′ discrete point on the virtual source plane, respec- l (x n , x n′ ), respectively, represent Green's functions of solid frame displacement, fluid relative displacement, total stress, and pore pressure in saturated half space (subscript l � 1, 2, 3corresponding to P I wave, P II wave, and SV wave sources, respectively).
According to IBEM and elastic wave theory, the total displacement of any point in the half space can be obtained as follows: where u (s) is the scattered wave field constructed by the virtual wave source and u (f) represents the free field displacement at any point.
Assuming that there is a vertical transfer relationship exists between the ground motion of the alluvial valley points and the imaginary bedrock surface points, then the transfer function can be expressed as

Boundary Condition.
Since the dynamic Green's functions in saturated half space automatically satisfy the free surface boundary condition, only the continuity conditions of the interface between saturated deposition and half space are considered. In the case of boundary permeability, the boundary conditions are as follows: where the superscripts sand vrepresent the half space and alluvial valley, respectively.

Correctness Verification
To verify the correctness of the method in this paper, a comparison between the surface displacement amplitude calculated in this paper and the results in the literature is given. In consideration of degradation, the calculated parameters are as follows: porosity is 0.001, dimensionless frequency η � ωa/c β π, Poisson's ratio ] � 1/3, damping ratio ζ � 0.001, and incident angle θ β � 0°. e results presented in this paper are in good agreement with those of the corresponding elastic media in Figure 2(Sanchez-Sesma and Campillo [21]) and Figure 3(Dravinski and Mossessian [50]), thus verifying the correctness of the method presented in this paper.

Numerical Examples
e site model of a semicircular alluvial valley with permeable boundary is shown in Figure 4. Taking the SV wave incident at 30 degrees from the bedrock half space as an example, three points 1, 2, and 3 were selected along the horizontal ground. e spacing is shown in the figure. Points 1′, 2′, and 3′ are, respectively, the corresponding points of the assumed horizontal field directly above the surface of the alluvial valley. Table 1shows the corresponding physical parameters of the soil layer, the fluid density is 1000 kg/m 3 , and the selection of horizontal ground motion parameters as follows:

Shock and Vibration
To verify the simulation method, the auto/cross-PSDs (power spectrum densities) are compared with the corresponding theoretical values, which are obtained by equation (2). e auto/cross-PSD estimates of the simulated ground motions are obtained by averaging the estimates from 30 generated samples. Figure 5shows the comparisons of the auto/cross-PSDs of the generated acceleration time histories with the corresponding target ones of the alluvial valley. Not only the auto-PSDs of the simulated motions match well with those of the target spectrum but also the cross-PSDs of the simulated motions match well with those of the target spectrum.
erefore, the effectiveness of this method is illustrated. Figure 6shows five sets of acceleration history samples of the saturated alluvial valley. For comparison, Figure 7shows the corresponding equivalent elasticity. e response spectra obtained from samples of the two cases are given in Figure 8. Table 2shows five sets of peak ground acceleration (PGA) samples and the peak response spectrum (PRS) of the two cases. PGA1 and PRS1 mean the PGA and PRS calculated by the method in this paper, respectively, while PGA2 and PRS2  Shock and Vibration mean the PGA and PRS calculated by the elastic method, respectively. According to the numerical comparison in Table 2, it can be clearly seen that the heterogeneity of seismic spatial distribution in the alluvial valley is different from whether the saturation state of the alluvial valley is considered. From the perspective of spatial relationship, compared with 1 point, the acceleration amplitude of 2 point and 3 point is larger. Taking the mean value of PGA1 as an example, the acceleration amplitude of 2 point is 1.7 times more than that of 1 point, and the acceleration amplitude of 3 point is 1.46 times more than that of 1 point. is is mainly because point 1 is located outside the sedimentary overburden, namely, on the bedrock, which reflects the vibration characteristics of the input seismic wave more. e remaining two points are located in the alluvial valley, and the energy focusing and filtering and amplification of the seismic wave occur in the process of propagation. erefore, the spatial effect of ground motion can not be ignored. For example, for a long and large structure across a saturated alluvial valley, the support points located in the inner part of the alluvial valley should be input more severe earthquake motion than those on the outer bedrock instead of the same motion. is is also the significance of this study, which provides a reasonable basis for the spatial correlation multipoint seismic input of local sites. e saturation model and elastic model have great differences in peak acceleration, peak value of response spectrum, and predominant period of the response spectrum. According to the calculation results of the saturated model, the PGA at 1 point on the bedrock decreases by 44.7% compared with that of the elastic model and the PGA  is is because the pore water dissipates the vibration energy greatly in the saturation medium, and the seismic response is reduced to some extent. is is also reflected in the peak of the reaction spectrum. It is worth noting that the predominant period corresponding to the peak value of the response spectrum is also different in the two cases: the 1point predominant period at bedrock has little change; but in the alluvial valley, the high frequency response in saturated condition is much less than that in elastic condition, and the predominant period extends from 0.09 s to 0.18 s at 2 point   To further illustrate the difference between the saturated state and the elastic state, the transfer functions for both are shown in Figure 9. It can be seen that the high frequency response of the elastic model is more than that of the saturated model, and the high frequency response of the elastic model is overestimated. However, in the low-frequency region, especially, the seismic response of the long-period structures is underestimated, and the seismic disaster of the long-period structures in the alluvial valley is more serious.
is is exactly what needs to be noted.

Conclusions
In this study, we proposed an efficient method for simulating spatially varying ground motions in a saturated alluvial valley based on Biot's theory, boundary element method, and spectral representation method. e feasibility of the simulation method was verified. A comparison among the results of the saturated model and elastic model revealed the following conclusions. When the proposed multipoint ground motion simulation method is applied to the saturated deposition site, there is a good agreement between the target and theoretical values of the auto/cross-PSDs of the nonstationary random motion. e error is controlled within 5%, which is less than the engineering standard error. It shows that the method has good applicability and strong feasibility in complex fields. e comparison between the results of the saturation model and elastic model shows that the saturation property of saturated alluvial valley cannot be ignored. e characteristics of wave propagation in saturated two-phase media are essentially different from those in single-phase media, and there are complex fluid-solid coupling effects in the process of wave propagation. e fluid-saturated porous media model can more accurately reflect the actual formation information and can better meet the needs of the actual site response simulation in coastal and valley areas. For saturated soils with high porosity below the groundwater level or in coastal or valley areas, the single-phase model is difficult to describe the variation of pore pressure and flow rate, which will lead to large simulation errors. Especially, in the long-period range which is unfavorable to the long-span structure, the seismic response is underestimated.
For the simulation of spatially correlated multipoint ground motion over a saturated alluvial valley, it is necessary to consider the influence of saturated medium, local site scattering, and spatial coherence effect. e method presented in this paper can provide a new method for spatially related multipoint ground motion simulation of saturated complex site and provide a scientific basis for seismic fortification and seismic safety evaluation of actual complex valley site engineering, which has certain scientific significance and social benefits.

Data Availability
e basic data used to support the findings of this study are included within the article and available from the corresponding author upon request. Shock and Vibration 9