Topographic Effects on the Seismic Response of Trapezoidal Canyons Subjected to Obliquely Incident SV Waves

. Te topography and the incident angle of seismic waves both have considerable efects on the seismic ground motions of canyons in a half-space. In this paper, the theory of wavefeld decomposition and the artifcial boundary is used to develop a method for inputting obliquely incident SV waves. Formulas for the equivalent nodal forces applied to the truncated boundary are derived and implemented in the fnite element method. Te validity of the proposed method is verifed by a test case. A parametric study is then performed to investigate the infuence of canyon geometry and incident angle of SV waves on the seismic response of trapezoidal canyons. Te numerical results indicate that the canyon inclination has a more signifcant efect on the ground motion amplifcation than its height and width. Te amplifcation efects are strongly related to the canyon inclination and the incident angle of SV waves. Additionally, the dominant frequency corresponding to the acceleration of the canyon crests is not sensitive to the incident angle of SV waves.


Introduction
Numerous postearthquake damage surveys [1][2][3][4][5] have shown that surface irregularities have a strong infuence on ground motion during earthquakes, which is known as the topographic efect [6][7][8][9].Canyons are common natural landforms in the mountainous region, and the topographic efect is important for the seismic design of structures located in the canyons, such as bridges across canyons and dams in valleys, where the peak ground accelerations vary signifcantly between the bottom of the canyon and the upper corners [10].Although the consideration of topographic efects is recommended in several seismic codes [11], there are no design principles or regulations applicable to the seismic design of canyons in the current engineering practice.In this regard, it is necessary to further understand the infuence of canyon topography on the topographic efects along the canyon surface.
Over the past decades, the topographic efects on the scattering and difraction of seismic waves induced by the canyon topography have been extensively investigated in the community of earthquake engineering.Te analytical methods and the numerical methods are the two main methods used to study this problem.Te problem of the scattering of antiplane waves (SH waves) in simple and smooth shapes of canyons, including semicircular canyons [12], semielliptical canyons [13], semiparabolic canyons [14], and U-shaped canyons [15], has received more attention in the analytical investigations due to the scalar form of the associated wave equation.Te analytical solutions of the scattering of SH waves by a complicated canyon topography have received less attention due to the rapidly increasing difculty in obtaining an analytical solution for it.In addition, the in-plane incident waves, such as longitudinal waves (P waves) or transverse waves (SV waves), cause mode conversion during the refection of waves at the half-plane surface (Figure 1), so the analytical solution for the in-plane scattering by canyon is also very difcult to obtain.Tis has led to the application of various numerical methods to solve the problem of in-plane scattering induced by canyon topography, such as boundary methods [16], hybrid methods [17], and domain methods [18].Tese numerical methods can not only simulate the diferent incident conditions of P and SV waves but also study the more complex scattering of the in-plane waves by diferent shapes of canyons.
Te seismic response of trapezoidal canyons has also been studied by many researchers.Based on the coupled fnite and infnite element method, Zhao and Valliappan [19] studied the efects of the shape of the trapezoidal canyon on wave scattering due to the vertical incidence of P and SV waves.Tey reported that the canyon topography can have a dramatic efect on both the peak value and frequency content of the ground motion along the canyon surface during an earthquake and that a steeper canyon bank can induce a stronger wave mode conversion efect.Zhang et al. [20] proposed an analytical closed-form solution for the scattering of SH waves induced by a trapezoidal valley during earthquakes.Teir results showed that the dynamic response at the ground surface is highly dependent on the steepness of the canyon and the incident angle of the excitations.Li et al. [21] developed a hybrid method to study a wave feld while considering the efects of layered topography on the spatially variable motions through a symmetrical trapezoidal canyon in a layered half-space.Tey found that the surface motions of the trapezoidal canyon in the layered half-space are signifcantly diferent from those in the uniform case, and the layer conditions play important roles in determining the displacement amplitude along the canyon surface.Although much work has been done to date to study the wave scattering of trapezoidal canyons, the combined efect of trapezoidal canyon geometry and incident angle of SV waves on the topographic amplifcation and frequency-domain ground motions remains to be better understood.On the one hand, the signifcant infuence of the incident angle on ground motion amplifcation has long been recognized [18,22,23].Also, the oblique efect, formed by a combination of the direction of the incident wavefronts and the trapezoid bottom, can lead to apparent diferences in peak ground accelerations at diferent locations on the canyon surface [10].On the other hand, the detailed efects of the trapezoidal canyon geometries (e.g., the height of the canyon, the width of the canyon bottom, and the inclination of the canyon) on the amplifcation pattern of the ground motion require a further understanding, especially in the case of obliquely incident waves.
Te main objective of this paper is to contribute to a better understanding of the efects of canyon topography and incident angle of SV waves on the seismic response of trapezoidal canyons through a parametric investigation.First, the input mechanisms of SV waves combined with an artifcial boundary are introduced.Ten, the numerical method for the incidence of vertical and oblique SV waves is implemented in the fnite element (FE) method and verifed by a test example.Subsequently, a parametric study is conducted to investigate the efects of the following factors on the topographic efect: (a) the normalized height of the canyon; (b) the normalized width of the canyon bottom; (c) the inclination of the canyon; and (d) the incident angle of SV waves.Four incident angles including one vertical incidence and three oblique incidences are considered in each numerical model.In addition, both amplifcation factors and Fourier amplitude spectra along the canyon surface are obtained to reveal the seismic response of the canyon in time and frequency domains.Finally, the complexity of seismic wave interference in the trapezoidal canyon and the differences in numerical results are discussed.

Establishment of the Canyon Model.
Figure 2 shows a simplifed model of a symmetrical trapezoidal canyon.Te height of the canyon is h, the width of the canyon bottom is L, and the inclination of the canyon is i.Te length of the upper fat ground surface behind both crests of the canyon, as well as the depth of the model, is at least three times the shear wavelength λ s .Te incident SV wave is assumed to propagate from the left side of the model at an angle of θ s .Te incident angle θ s is defned as the angle between the direction of propagation of the SV wave and the vertical direction.Points A, D, and B, C are labeled to represent the crest and foot of the canyon, respectively.Te y-axis of the coordinate system lies on the symmetry axis of the model, and the x-axis is parallel to the fat ground.

Governing Equations.
Based on the decomposition of the wavefeld, the total motion of the wave u is composed of two parts: the motion of the scattering feld u S and the freefeld ground motion u F .Terefore, the total wavefeld can be expressed as u � u S + u F .On the truncated boundary, the equations of motion and the total motion of the wave at a given boundary node l can be expressed as follows: where m l is lumped mass of node l, c linj and k linj represent the damping and stifness coefcients, respectively; _ u T nj and u T nj indicate the velocity and displacement of the boundary node n, respectively; and f S li and f F li are the loads in node l induced by the motion of the scattering feld and the free-feld motion, respectively.Te subscripts i and j indicate components of the Cartesian coordinate, and i, j � 1, 2 correspond to x, y in the two-dimensional problem.
Te force of the boundary node corresponding to the motion of the scattering feld is described as a function of the displacement and velocity felds: 2 Shock and Vibration where A l represents the infuence area of all elements around node l in the artifcial boundary, and K li and C li are coefcients of the viscous-spring artifcial boundary that are introduced in Section 2.3.
By substituting equations ( 2) and ( 3) into (1), we obtain the equation of motion at the boundary node l, that is, By combining the coefcients, equation ( 4) can be simplifed as follows: where Te right side of equation (5) shows the equivalent nodal force in artifcial boundary nodes induced by free-feld motion.By replacing the force term with the stress term on the right side of equation ( 5), the equivalent nodal force at l can be given as follows [24]: It is clear from equation ( 6) that the motion of the incident wave can be converted into the equivalent nodal force applied to the corresponding artifcial boundary nodes.

Te Viscous-Spring Artifcial Boundary.
To eliminate the infuence of wave refection from the boundary, a viscous-spring artifcial boundary is established by setting a series of springs and dashpots along the boundary [24], as shown in Figure 3. Te elastic spring coefcient K and the damping coefcient C can be written as follows: where the superscripts N and T denote the normal and the tangential directions, respectively, A and B are the modifed dimensionless coefcients, with suggested values of 0.8 and 1.1, respectively [24], λ is the Lame constant, G is the shear modulus, r represents the distance between the source of the wave and the artifcial boundary, and c p � ��������� (λ + 2G)/ρ  and c s � ��� G/ρ  stand for the velocities of the compression wave and the shear wave in the medium, respectively.

Equivalent Node Force for Obliquely Incident SV Waves.
As shown in Figure 4, the obliquely incident SV wave with angle α decomposes into two parts when reaching the surface of the ground: one is the refected SV wave with the Shock and Vibration same angle α, and the other is the refected P wave with angle β.Te refection angle β and the ratio of the amplitudes of the refected wave and the incident wave can be expressed as follows: , (8b) where A 1 represents the ratio of the amplitude of the refected SV wave to that of the incident SV wave, and A 2 represents the ratio of the amplitude of the refected P wave to that of the incident SV wave.Te length and the height of the truncated computational region are L x and L y , respectively (Figure 4).Te total wave feld at a given boundary node is a superposition of the incident SV waves, the refected SV waves, and the refected P waves.For a given boundary node l (x 0 , y 0 ) in the free feld, the displacement and stress on the left boundary are as follows [25]: where the subscripts x and y represent the components of the waves, u 0 (t) and _ u 0 (t) indicate the displacement-time history and velocity time history of the incident SV waves, respectively, A 1 and A 2 are amplitude ratios of the refected waves to the incident waves that are determined by equations (8b) and (8c), respectively, and ∆t is the time lag in the propagation of incident waves from the wavefront at t � 0 to the left boundary.It can be given as follows: FEM mesh Boundary node 4

Shock and Vibration
Te displacement and stress on the bottom boundary are as follows: where the time lags of the propagation of incident waves from the wavefront at t � 0 to the bottom boundary can be written as follows: Te formulas describing the displacement on the right boundary are the same as those on the left boundary, but an additional time L x sinα/c p needs to be added to ∆t 1 , ∆t 2 , and ∆t 3 owing to the additional distance L x traveled by the wave.Te stresses on the right boundary are the same as those on the left boundary but in the opposite direction.

Verifcation.
In this section, a test case is considered to assess the overall accuracy of the presented numerical methodology, which involves the propagation of in-plane SV waves in a homogeneous elastic half-space with an oblique incident angle.Shock and Vibration a semi-infnite space.Te region is assumed to be an elastic homogeneous medium with Young's modulus E � 6 Ga, mass density ρ � 2450 kg/m 3 , and Poisson's ratio ] � 0.3.Te corresponding velocities of the shear wave c s and the compression wave c p are 971 m/s and 1816 m/s, respectively.
Te size of the computational domain is 2000 m × 1000 m, and the incident angle is θ s � 20 °.An impulse wave with an amplitude of 1 m and an acting time of 0.3 s is used as the incident SV wave, as shown in Figure 5(b).Te corresponding defnition of the incident wave is given as follows: where t denotes time, H(τ) is the Heaviside function, P 0 is the amplitude of the impulse, and P 0 � 1.0 m.Herein, T is the acting time of the impulse, and T � 0.3 s. Figure 6 shows the contours of the displacement magnitude at diferent arrival times of the obliquely incident SV waves.Te fgure clearly shows the propagation and refection of the incident wave, which means that the propagation of SV waves in the semi-infnite space has been efectively simulated without wave refection along the artifcial boundary.To verify the accuracy of the input method, two reference points A (1000, 1000) and B (1000, 500), labeled in Figure 5, are selected to monitor the displacement components in the x-and y-directions (denoted by U x and U y , respectively).A comparison of the displacement-time histories between the theoretical solution and the numerical results at points A and B is shown in Figure 7.It is clearly shown that the numerical results are in good agreement with the results of the theoretical solution, indicating that the introduced input method is appropriate for simulating obliquely incident SV waves.

Description of the Parameters
Te variations in the topography of the canyon (h, L, and i) are shown in Figure 8. Te incident angle θ s varies from 0 °to 30 °in increments of 10 °for each numerical model.Te Ricker wavelets are used as the incident SV waves since they are commonly used as idealized input seismograms (Figure 9).Te acceleration time history of the Ricker wave is defned as follows: where f c and t 0 represent the central frequency of the Fourier spectrum and the time when the acceleration reached its peak, respectively.A Ricker wavelet with an amplitude of 1 m/s 2 and a central frequency of 4 Hz is selected as the incident wave.
A homogeneous elastic medium with mass density ρ � 2650 kg/m 3 , elastic modulus E � 20 Ga, and Poisson's ratio ] � 0.25 is used to represent the rock material in the numerical model.We also introduce dimensionless frequency for normalizing the height of the canyon and the width of its bottom; therefore, the geometric parameters h and L are both normalized by the shear wavelength λ s in this study: In summary, this study mainly considered variations in the following parameters: (1) Te angle of incidence: θ s varies from 0 °to 30 °for the incidence of SV waves in increments of 10 Normalized height of the canyon: η h varies from 0.5 to 2.0, in increments of 0.5 (3) Normalized width of the canyon bottom: η L varies from 0.5 to 2.0, in increments of 0.5 (4) Te inclination of the canyon: i varies from 15 °to 60 °, in increments of 15 °4.

Efects of the Normalized Height of the Canyon.
Figure 10 compares the acceleration amplifcation along the canyon surface for vertically and obliquely incident SV waves with η h varying from 0.5 to 2.0.Note that the horizontal coordinates of the ground surface are normalized by the shear wavelength (λ s ) in each subplot, and the amplifcation factor curves are plotted in both the three-dimensional perspective view and the corresponding two-dimensional front view to better illustrate the variation of amplifcation factors.Te vertical solid lines in the subplots represent the location of the left and right canyon crests while the vertical dashed lines represent the location of the foot of the canyon.Te horizontal and vertical amplifcation factors (HAF and VAF) are defned as the ratio of the maximum horizontal and vertical ground accelerations measured at the surface to the maximum horizontal acceleration in the far feld, respectively. 6

Shock and Vibration
As shown in Figure 10(a), the horizontal amplifcation factors (HAF) fuctuate strongly along the canyon surface, alternating between amplifcation (i.e., HAF > 1.0) and deamplifcation (i.e., HAF < 1.0).Te results associated with the HAF in Figure 10(a) reveal the following fndings.(1) Te curves of the HAF are almost symmetrically distributed along the canyon surface due to the symmetrical shape of the trapezoidal canyon when θ s � 0 °, while the curves of the HAF become unsymmetrical when θ s > 0 °.(2) Te horizontal amplifcation of the ground motion is pronounced around the canyon crests in both cases of vertical incidence and oblique incidence.(3) For vertical incidence, the HAF in the vicinity of the canyon crests shows an increasing trend with the increasing normalized canyon height (η h ).(4) It is also observed that the amplifcation patterns are highly dependent on the incident angle of SV waves for oblique incidence.Specifcally, the HAF around the right crest of the canyon is signifcantly larger than that around the left crest when θ s � 10 °, and the peak values of the HAF increase with increasing η h .However, in the case of θ s ≥ 20 °, the pronounced amplifcation appears on the left side of the canyon, and the right side of the canyon is nearly deamplifed, which is opposite to the amplifcation pattern at θ s � 10 °.Tis indicates that the interference of the wave felds on the right side of the canyon is signifcantly afected by a larger θ s (such as 30 °), which in turn leads to a change in the amplifcation pattern.
Te results associated with the VAF in Figure 10(b) reveal that the overall vertical amplifcation under oblique incidence (θ s > 0 °) is obviously greater than that under vertical incidence (θ s � 10 °).In the case of oblique incidence, the vertical amplifcation on the right side of the canyon is predominant compared to that on the left side of the canyon, which means that the refected P waves and generated By applying the fast Fourier transform (FFT) to the recorded ground accelerations, the Fourier amplitude spectra of surface ground motions are obtained.Te magnitude of the ground motion in a given frequency component can be represented by the corresponding Fourier amplitude.Contours of the Fourier amplitude spectrum for horizontal and vertical accelerations along the canyon surface at diferent values of η h and θ s are given in Figure 11.Te white solid lines and dashed lines in each subplot represent the crest and the foot of the canyon, respectively.It is clearly shown that the Fourier amplitude is mainly distributed in the range of 1-8 Hz, which corresponds to the width of the frequency band of the incident Ricker wave.As expected, the pronounced ground response is concentrated in the frequency range of 3-5 Hz, where most of the energy is carried by the incident waves.In the horizontal direction, the Fourier amplitudes near and behind the crests of a deeper canyon (η h � 2.0) are greater than those of a shallower canyon (η h � 0.5) for vertically incident waves (θ s � 0 °), while the Fourier amplitudes behind the right crest of a shallower canyon are greater than that of a deeper canyon for obliquely incident waves (θ s � 30 °).In the vertical direction, the Fourier amplitudes under obliquely incident waves are signifcantly larger than those under vertically incident waves, indicating a strong oblique efect of the incident waves.In addition, the vertical Fourier amplitudes behind the right crest of a shallower canyon are also larger than those of a deeper canyon under oblique incidence, which is similar to that in the horizontal direction.
Figure 12 clearly shows the distribution of the Fourier amplitude at the left and right canyon crests in the frequency domain for diferent η h and θ s .Te vertical dashed lines in each subplot indicate the central frequency (4 Hz) of the incident Ricker wavelet.In the horizontal direction (Figure 12(a)), the results reveal the following fndings.(1) Te frequency corresponding to the peak Fourier amplitude (PFA) at the left and right canyon crests is close to the central frequency of the incident wave.( 2) Te PFA at the left canyon crest does not change signifcantly with the increase of the incident angle θ s , regardless of η h � 0.5 or η h � 2.0, while the PFA at the right canyon crest decreases rapidly with the increase of the incident angle θ s .In the vertical direction (Figure 12(b)), the results reveal the following fndings.(1) Te frequency corresponding to the PFA at the canyon crests varies between 3 Hz and 5 Hz.(2) For a shallower canyon (η h � 0.5), the PFA increases with increasing incident angle θ s , especially at the right canyon crest.Te diference between the PFA at the left and right canyon crests is obvious at θ s � 30 °. (3) For a deeper canyon (η h � 2.0), the diference between the PFA on the left and right canyon crests is apparent at θ s � 10 °− 20 °.

Efects of Normalized Width of the Canyon Bottom.
HAF and VAF curves for diferent values of η L from 0.5 to 2.0 are shown in Figure 13.For a given θ s , the horizontal amplifcation curves for diferent η L have similar shapes.In all cases except η L � 1.0, the maximum values of HAF near the canyon crests are almost the same for a given incident angle θ s despite the increasing η L .In addition, the vertical amplifcation curves for diferent η L also have similar shapes for a given θ s .Te vertical amplifcation generally increases with increasing θ s for a given η L .For both horizontal and vertical amplifcations, the shapes of the amplifcation factor curves at diferent η L are similar for the same incident angle θ s , and no signifcant diferences in the maximum amplifcation are observed for a given θ s , implying that the width of the canyon bottom has little efect on the seismic response of the canyon.
Figure 14 shows plots of the contours of the Fourier amplitude in the case of horizontal and vertical ground accelerations at diferent values of η L and θ s .For a given incident angle θ s , the variation of η L has no signifcant efect on the horizontal and vertical Fourier amplitude spectra.Terefore, similar conclusions can be drawn as in the previous section.

Efects of Inclination of the Canyon. Te variation of horizontal and vertical amplifcations versus the inclination of the canyon for diferent incident angles of SV waves is plotted in
Figure       (1) For a given θ s , the maximum VAF increases with increasing inclination of the canyon.(2) As the θ s increases, the difference in VAF between diferent inclinations of the canyon gradually decreases.Figures 16(a) and 16(b) show the variation of Fourier amplitude spectra with diferent i and θ s in the horizontal and vertical directions, respectively.In the case of vertical incidence (θ s � 0 °), the magnitude of the horizontal ground motion was almost uniformly distributed over the entire canyon surface for i � 15 °.As the canyon inclination increases, the pronounced horizontal ground motions are concentrated near or behind the canyon crest.It is also clear that the vertical ground motions are signifcantly enhanced with increasing canyon inclination, and the pronounced surface motions gradually converge at the crests.In the case of obliquely incident waves (θ s � 30 °), the intensity of ground motions behind the right crest decreases in both horizontal and vertical directions with increasing canyon inclination.While the horizontal and vertical ground motions on the left side of the canyon are enhanced with increasing inclination.Te distribution of the Fourier spectrum becomes more complex with increasing inclination, indicating that the inclination of the canyon has a signifcant efect on the seismic ground motions of the canyon.
Te horizontal and vertical Fourier amplitudes at the left and right canyon crests for diferent i and θ s are shown in Figure 17.In the horizontal direction, for a gentler canyon (e.g., i � 15 °), the diference in PFA between the left and right canyon crests becomes apparent when the incident angle of SV waves is larger (e.g., θ s � 30 °), while the difference in PFA between the left and right canyon crests for a steeper canyon (e.g., i � 60 °) is not obvious under diferent θ s .In the vertical direction, the PFA at the left and right canyon crests increases with increasing incident angle θ s for a gentler canyon (e.g., i � 15 °).However, for a steeper canyon (e.g., i � 60 °), the PFA at the left crest decreases and then increases with increasing incident angle θ s , and the PFA at the right crest increases and then decreases with increasing incident angle θ s .It is also observed that for oblique incidence, in some cases, the frequencies corresponding to the PFA at the left and right crests of a steeper canyon are distributed on either side of the incident wave frequency (i.e., 4 Hz).For example, in the case of i � 60 °and θ s � 20 °for the horizontal direction, and the cases of i � 60 °and θ s � 30 °for the vertical direction.In summary, the frequency corresponding to the PFA in the horizontal and vertical directions is in the range of 3-5 Hz, which is close to the incident wave frequency.

Discussion
Tis study investigates the efects of the canyon geometry and incident angle of SV waves on the seismic response of trapezoidal canyons.Te incident angle of SV waves and the inclination of the canyon play important roles in the seismic ground motions of the canyons.For a given incident angle θ s , the amplifcation factor curves for similar topographic geometries (e.g., the dimensionless height h/λ s or dimensionless width L/λ s ) have similar shapes, which means that they result from similar patterns of the waveform [26].However, the inclination of the canyon shows a more signifcant infuence on the seismic amplifcation of the canyon than the other two canyon geometry parameters.Figure 18 shows the seismogram synthetics of the horizontal and vertical ground motion components, which include the following waveforms: (1) direct SV waves (denoted by SV), (2) refected P waves (denoted by P), and (3) difracted Rayleigh waves generated by sharp corners (denoted by R 1 and R 2 ).It is clearly shown that the sharper corners signifcantly distort the seismic waves.Tis may be attributed to the sharper corners that a steeper canyon has compared to a fatter canyon, leading to strong scattering waves.Meanwhile, a stronger efect of mode conversion of the waves is induced by a larger incident angle θ s , indicating that the intensity of seismic ground motions may be underestimated if the oblique incidence of the seismic wave is ignored.In addition, it is found that the canyon surface is subjected to stronger horizontal ground motion than its vertical motion.
Te amplifcation patterns and seismic ground motions under diferent incident conditions can be explained by further analysis of the canyon wavefelds.Te snapshots of the acceleration wave feld and the contours of the absolute acceleration amplitude for diferent canyon confgurations are shown in Figures 19 and 20.As shown in Figure 19, two canyon models with diferent normalized heights (η h � 0.5 in Figures 19(a) and η h � 2 in 19(b)) are subjected to the obliquely incident wave (θ s � 20 °).It can be seen that the interference between the refected waves from the horizontal ground surface and the scattering waves from the canyon surface is mainly concentrated on the left part of the canyon, i.e., the side of the incident wave.Hence, the wavefelds on the left side are more complicated than those on the right side, which can be explained by the "canyon-decay efect."Te total internal wavefelds are similar for both shallower and deeper canyons despite their diferent heights.However, a deeper canyon may show a more pronounced "canyondecay efect" compared to a shallower canyon since the complicated wavefelds after interference are blocked by the left side of the deeper canyon, and thus the incident waves traveling to the right side of the canyon are less afected.Although the incident SV waves mainly infuence the horizontal ground motions of the canyon surface, they also cause ground motions in the vertical direction (see Figure 18(b)).Te vertical ground motions are mainly induced by the refected P waves and Rayleigh waves which are less afected on the right side of the canyon, hence the vertical ground motions on the right side of the canyon are stronger than those on the left side of the canyon.On the other hand, the refected P waves on the left side of the canyon interfere with the scattered waves, and the resulting vertical ground motions will be further reduced.For canyons with diferent inclinations (shown in Figure 20), the degree of complexity of the wavefelds becomes diferent.A more complex interference between the refected waves and the scattering waves is observed for a canyon with a steeper      , indicating that the sharper corners of the canyon can signifcantly distort the incident waves and refected waves and result in stronger scattering waves.Terefore, the inclination of the canyon has a great infuence on the seismic ground motions of the canyon which should be paid more attention to when performing seismic design for a canyon.

Conclusions
In this study, an input method for obliquely incident SV waves is introduced and implemented in the fnite element method.Te amplifcation factors and Fourier amplitude spectra along the trapezoidal canyon surface are then investigated.Te main conclusions of this study are as follows: (1) Te incident angle of SV waves has a signifcant infuence on the seismic response of the trapezoidal canyon.Te amplifcation efect may be signifcantly underestimated if only the vertical incidence is considered.(2) Te horizontal amplifcation is greater than the vertical amplifcation.Te horizontal and vertical amplifcation patterns are highly correlated with the incident angle of SV waves and the inclination of the canyon.(3) Te efect of the inclination of the canyon on ground motion amplifcation is more pronounced compared to that of the canyon height and the width of the canyon bottom.(4) Te distribution of the Fourier amplitude spectra along the canyon surface is mainly infuenced by the inclination of the canyon and the incident angle of SV waves, but the dominant frequency corresponding to the acceleration at the canyon crests is not sensitive to the incident angle.

Figure 1 :
Figure 1: Te mode conversion during the refection of SV waves at the half-plane surface.

Figure 2 :
Figure 2: Numerical model of the trapezoidal canyon subjected to oblique incidence of SV waves.

Figure 3 :
Figure 3: A sketch of the viscous-spring artifcial boundary on the FE model.

Figure 4 :
Figure 4: Te incidence of SV wave in half-space and refection on the ground surface.

Figure 5 :Figure 6 :Figure 7 :Figure 8 :
Figure 5: (a) A sketch of obliquely incident SV wave in half-space and (b) the input seismic motion.

Figure 15 .Figure 9 :
Figure 9: Te acceleration time history of the incident Ricker wavelet (f c � 4 Hz and t 0 � 0.5 s) and the corresponding Fourier amplitude spectrum.

Figure 10 :
Figure 10: Efects of normalized canyon height η h on the (a) horizontal and (b) vertical acceleration amplifcations with η L � 1.0 and i � 30 °under incident angles θ s of 0 °, 10 °, 20 °, and 30 °. Te vertical solid lines in the subplots represent the location of the left and right canyon crests while the vertical dashed lines indicate the positions of the canyon feet.

Figure 11 :Frequency
Figure 11: Efect of normalized canyon height η h on the Fourier amplitude spectrum of (a) horizontal and (b) vertical ground accelerations for confgurations with η L � 1.0 and i � 30 °under vertically and obliquely incident SV waves.Te white solid lines and dashed lines represent the locations of the crests and feet of the canyon model, respectively.

Figure 12 :
Figure 12: Te (a) horizontal and (b) vertical Fourier amplitudes at the canyon crests for diferent heights of the canyon and incident angles of SV waves.

12
Shock and Vibration angle θ s .(3) Te incident angle of SV waves has a signifcant infuence on the horizontal amplifcation, especially for a steeper canyon (e.g., i � 45 °or 60 °).For example, the HAF at the left canyon crest is about 1.0 for i � 60 °and θ s � 0 °, while the HAF increases signifcantly to 1.6 for i � 60 °and θ s � 30 °. (4) Te peak values of HAF at the canyon bottom are almost the same for a given θ s , which is not related to the inclination of the canyon.Te results associated with the

Figure 13 :
Figure 13: Efects of normalized width of the canyon bottom η L on the (a) horizontal and (b) vertical acceleration amplifcations with η h � 1.0 and i � 30 °under incident angles θ s of 0 °, 10 °, 20 °, and 30 °. Te vertical solid lines in the subplots represent the location of the left and right canyon crests while the vertical dashed lines indicate the positions of the canyon feet.

Figure 14 :
Figure 14: Efect of normalized canyon height η L on the Fourier amplitude spectrum of (a) horizontal and (b) vertical ground accelerations for confgurations with η h � 1.0 and i � 30 °under vertically and obliquely incident SV waves.Te white solid lines and dashed lines represent the locations of the crests and feet of the canyon model, respectively.

Figure 15 :xFigure 16 :Figure 17
Figure 15: Efects of canyon inclination i on the (a) horizontal and (b) vertical acceleration amplifcations with η h � 1.0 and η L � 1.0 under incident angles θ s of 0 °, 10 °, 20 °, and 30 °. Te vertical solid lines in the subplots represent the location of the left and right canyon crests while the vertical dashed lines indicate the positions of the canyon feet.

Figure 17 :
Figure 17: Te (a) horizontal and (b) vertical Fourier amplitudes at the canyon crests for diferent inclinations of the canyon and incident angles of SV waves.