Geophysical Journal International Geophys. J. Int. (2015) 202, 102–118
doi: 10.1093/gji/ggv129
GJI Seismology
Elastic wave finite-difference simulation using discontinuous curvilinear grid with non-uniform time step: two-dimensional case Hong Li,1,2,3 Wei Zhang,1,2,3 Zhenguo Zhang1,2,3 and Xiaofei Chen1,2,3 1 School
of Earth and Space Sciences, University of Science and Technology of China, Hefei, China. E-mail:
[email protected] of Seismology and Physics of Earths Interior, University of Sciences and Technology of China, Hefei, China 3 Mengcheng National Geophysical Observatory, Hefei, Anhui 230026, China 2 Laboratory
Accepted 2015 March 13. Received 2015 March 11; in original form 2014 October 23
SUMMARY A discontinuous grid finite-difference (FD) method with non-uniform time step Runge–Kutta scheme on curvilinear collocated-grid is developed for seismic wave simulation. We introduce two transition zones: a spatial transition zone and a temporal transition zone, to exchange wavefield across the spatial and temporal discontinuous interfaces. A Gaussian filter is applied to suppress artificial numerical noise caused by down-sampling the wavefield from the finer grid to the coarser grid. We adapt the non-uniform time step Runge–Kutta scheme to a discontinuous grid FD method for further increasing the computational efficiency without losing the accuracy of time marching through the whole simulation region. When the topography is included in the modelling, we carry out the discontinuous grid method on a curvilinear collocated-grid to obtain a sufficiently accurate free-surface boundary condition implementation. Numerical tests show that the proposed method can sufficiently accurately simulate the seismic wave propagation on such grids and significantly reduce the computational resources consumption with respect to regular grids. Key words: Numerical solutions; Computational seismology; Theoretical seismology; Wave propagation.
1 I N T RO D U C T I O N Numerical simulation of seismic wave propagation in complex Earth models is a fundamental subject in seismology. The finitedifference method (FDM) is one of the most popular numerical methods and has been widely used in seismology studies because of its robustness and easy implementation. The conventional implementations of FDM generally uses uniform grids (i.e. constant grid spacing) to descretize the computational domain and a constant time step in time marching. Because the grid spacing is determined by the lowest velocity according to the dispersion error criterion, and the time step is limited by the highest velocity following the stability condition of explicit schemes, spatial-temporal uniform grids can lead to wasting vast computational resources due to unnecessary over-sampling for models containing complex structures (low-velocity zone, topographic free surface). One effort to reduce the limitation of the uniform grids in the conventional FDM is using non-uniform grids, in which the grid spacing varies continuously along one direction from finer in the low-velocity zone to coarser in the remainder region (e.g. Moczo 1989; Pitarka 1999; Zhang et al. 2012a). But in this way, the finer horizontal grid spacing in the low-velocity zone at shallow depth will extend to greater depth, where velocity is high and the vertical grid spacing could be coarser, due to the fact that the number of
102
C
horizontal grid points is same along vertical direction for such continuous non-uniform grids. The expansion of finer horizontal grid spacing from top to bottom throughout the computational domain limits the time step and the efficiency of the methods. In order to fully explore the capability and efficiency of the FDM, we need discontinuous grids (e.g. Jastram & Behle 1992; Jastram & Tessmer 1994; Aoi & Fujiwara 1999). The computational region is divided into several subregions. Each subregion is discretized by a grid with a spatial spacing determined by the local accuracy condition. Between subregions, the spatial spacings may be different by some integer ratio. The major challenge of implementing a discontinuous grid in the FDM is exchanging of wavefield between the finer grid and coarse grid. To get the values on the missing points around the finer grid from the coarse grid, several methods have been developed: trigonometric interpolation (Jastram & Behle 1992; Jastram & Tessmer 1994; Wang et al. 1996), linear interpolation (Aoi & Fujiwara 1999) or interpolation in the wavenumber domain (Wang et al. 2001; Wang & Takenaka 2001). Moczo et al. (1996) also presented a method based upon using an adjusted FD approximation to avoid the interpolation of the wave field values at the missing grid points. To get the values on the missing points around the coarse grid from the finer grid, one way is to directly use the values on the fine grid points. But as Kristek et al. (2010) pointed out, such treatment can cause numerical noise, and
The Authors 2015. Published by Oxford University Press on behalf of The Royal Astronomical Society.
2D discontinuous grid and nonuniform time step eventually leads to instability for long time simulations. In order to avoid instability, some specific smoothing or filtering techniques should be used (Hayashi et al. 2001; Kristek et al. 2010; Zhang et al. 2013). For the sake of simplicity, the discontinuous grids are mostly implemented as layer-like (Aoi & Fujiwara 1999; Jastram & Behle 1992; Jastram & Tessmer 1994; Wang et al. 2001; Wang & Takenaka 2001). However, a laterally localized discontinuous grid can provide a more efficient and flexible discretization in seismic wave simulations (Hayashi et al. 2001; Kang & Baag 2004a,b; Huang & Dong 2009a,b). Besides the spatial discontinuous grid, non-uniform time stepping can further save computational time. Falk et al. (1998) and Tessmer (2000) introduced a locally adjustable time stepping method to solve the seismic wave equation formulated as the second-order partial differential equations. Kang & Baag (2004a,b) developed a discontinuous grid implementation with a locally variable time step for the staggered-grid FDM based on the first-order velocity-stress equations. In the context of the computational aeroacoustics (CAA) simulations, Liu et al. (2010) developed a non-uniform time step Runge–Kutta (NUTS-RK) method and tested it with the discontinuous Galerkin method. Aforementioned discontinuous grids for seismic wave modelling are most implemented on Cartesian grids. Nevertheless, in the presence of surface topography or complex fault geometry, a curvilinear grid is more appropriate (Zhang & Chen 2006; Appel¨o & Petersson 2009; Zhang et al. 2012b, 2014; de la Puente et al. 2014). Zhang et al. (2013) have developed a layer-like discontinuous grid implementation for collocated-grid seismic wave finite-difference (FD) simulations following Kristek et al. (2010)’s idea of adopting a Gaussian filtering. In this paper, we extend the method of Zhang et al. (2013) to a laterally discontinuous curvilinear grid together with a localized variable time step by adapting the NUTS-RK scheme (Liu et al. 2010) to improve the computational efficiency. Numerical examples are presented to demonstrate the accuracy, stability and efficiency of the implementation.
2 V E L O C I T Y- S T R E S S E Q UAT I O N A N D C O L L O C AT E D - G R I D FINITE-DIFFERENCE METHOD We solve the following first-order hyperbolic velocity-stress formulation of the elastic wave equations, ρvi,t = σi j, j + f i ,
(1)
˙ ij, σi j,t = λδi j vk,k + μ(vi, j + v j,i ) − M
(2)
where i and j represent the x and y coordinate directions, v i means the i-direction component of the particle velocity, σ ij is the stress component, λ and μ are the Lam´e parameters, ρ presents the density, δ ij is the Kronecker tensor, fi is the body force, and Mij is the seismic moment component, over which the dot means a time derivative. When surface topography is present, curvilinear grid can be used to avoid the staircase approximation at the surface (Hestholm & Ruud 1998, 2002; Zhang & Chen 2006; Wang & Liu 2007; Appel¨o & Petersson 2009; Lan & Zhang 2011; Zhang et al. 2012b; de la Puente et al. 2014). The coordinate transformation between the physical space and the computational space is x = x(ξ, η), y = y(ξ, η).
(3)
Consequently, by using the chain rule, we can transform the velocity-stress equations (eqs. 1 and 2) in the Cartesian coordinate
103
to the curvilinear coordinate, which can be written in a compact matrix form as ∂U ∂U ∂U ˜ ˜ =A +B + F, ∂t ∂ξ ∂η
(4)
where U is the velocity-stress vector, U = (vx , v y , σx x , σ yy , σx y )T , F is the source vector, T fx f y ˙ yy , − M ˙ xy , ˙ xx , −M F= , , −M ρ ρ
(5)
(6)
where the superscript T represents the transpose operator. The co˜ and B˜ can be found in Appendix A. efficient matrices A The staggered-grid FDM has been widely used in seismic simulation for its simplicity and robustness (Madariaga 1976; Virieux 1986; Levander 1988; Graves 1996; Olsen & Archuleta 1996; Kristek et al. 2002; Moczo et al. 2004). An alternative choice is collocated-grid schemes or non-staggered-grid schemes, in which all variables are located at the same grid position. However, a collocated-grid central difference scheme can lead to the odd–even decoupling mode (grid to grid oscillation). In order to void this problem, we solve the velocity-stress equation (eq. 4) by using the DRP/opt MacCormack scheme (Hixon 1997), which alternately uses the following forward and backward difference operators (with respect to the x-axis as examples) in the fourth-order Runge–Kutta time marching scheme: L xF (U)i =
3 1 an Ui+n , x n=−1
(7)
L xB (U)i =
3 1 −an Ui−n , x n=−1
(8)
where Lx represents the spatial difference with respect to the x -axis, the subscript i is the grid index, the superscript F means a forward operator and the superscript B denotes a backward operator. The coefficients a−1 = −0.30874, a0 = −0.6326, a1 = 1.2330, a2 = −0.3334 and a3 = 0.04168 are obtained by minimizing the dissipation error at eight points or more per wavelength (Hixon 1997). In general, the scheme requires minimal eight points per wavelength for a successful (dispersion-free) propagation. If both directional derivatives in eq. (4) are calculated using the forward and backward operator, the right hand side of eq. (4) is discretized as ˜ xF (U) + BL ˜ yF (U) + F, Lˆ F F = AL
(9)
˜ xB (U) + BL ˜ yB (U) + F. Lˆ B B = AL
(10)
Consequently, in order to update the wave-field from time n t to (n + 1) t, the fourth-order Runge–Kutta scheme, alternatively using the two operators in eqs. (9 and 10), can be written as: k1 = Lˆ F F (Un ) , k2 = Lˆ B B Un + tα2 k1 , k3 = Lˆ F F Un + tα3 k2 , k4 = Lˆ B B Un + tα4 k3 , Un+1 = Un + t
4 i=1
βi ki .
(11)
104
H. Li et al.
y distance
(b)
y distance
(a)
x distance
x distance
Figure 1. (a) Topographic model with low-velocity zone near the surface. (b) Discontinuous curvilinear grid used to discretize the model: the region containing the low-velocity zone (dark region) with a surface topography is discretized by a finer curvilinear grid, and the other region with high velocity (grey region) is discretized by a coarse grid.
where Un is the wavefield at time step n, Un + 1 is the wavefield at time step n + 1. ki is the ith intermediate stage values within one Runge–Kutta step; the coefficients are α 2 = 0.5, α 3 = 0.5, α 4 = 1, β 1 = 1/6, β 2 = 1/3, β 3 = 1/3, and β 4 = 1/6. The detailed implementation and verification could be found in Zhang & Chen (2006) and Zhang et al. (2012b).
3 IMPLEMENT DISCONTINUOUS GRIDS AND NON-UNIFORM TIME STEP The real Earth model generally has a topographic free surface and contains low-velocity zones (e.g. a basin) near the free surface (Fig. 1a), which may significantly affect the propagation of the seismic waves. To accurately simulate topographic effects and seismic wave propagation in the low-velocity zone, a small spatial spacing is required to suppress the dispersion error. A simple uniform grid to discretize the whole region can result in spatial over-sampling in the high-velocity bedrock and temporal over-sampling in the lowvelocity basin, thus wasting vast computational resources. In order to improve the utilization of the computational power, discontinuous grid should be employed. When using discontinuous grids, based on velocity distributions, the computational region is divided into several subregions (Fig. 1b). A fine grid with spacing hf is used in low-velocity regions, and a coarse grid with spacing hc is used in high-velocity regions. The spatial spacing of each subregion is determined by its local dispersion condition, and usually is set to an integer ratio n = hc / hf between two neighbour subregions for easy implementation. The key of a discontinuous grid implementation is the wavefield exchange between two sides of the spatial and temporal discontinuous interfaces that can ensure wavefield continuity across these interfaces. In this paper, in order to construct the discontinuous grid with both spatial and temporal step discontinuities, we introduce two transition zones (Fig. 2): one is the spatial transition zone (along the spatial discontinuous interface), and the other is the temporal transition zone, where the spatial spacings are the same at both sides of the zone but the time step changes across the interface. For curvilinear grids, we use eq. (3) to map the curvilinear grid in the physical space to a uniform Cartesian grid in the computational space. The velocity-stress equation (eqs 1–2) is solved in the computational space in the form of eq. (4). The influence of the curvilinear grid is combined with the medium coefficients in ˜ and B ˜ in eq. (4). We can deem the coeffithe coefficient matrices A cient matrices as effective coefficient matrices in the computational space. With this consideration, we can implement the discontinuous grid on the uniform Cartesian grid in the computational space.
3.1 Localized spatial discontinuous grid For spatial discontinuous grids, the spatial transition zone contains two types of ghost grid points (light grey zone in Fig. 2): the ghost points for the coarse grid (yellow points) on which the wavefield is down-sampling from the fine grid, and the ghost grids for the fine grid (cyan points) on which the wavefield is interpolated from the coarse grid. In order to interpolate the wavefield from the coarse grid to the ghost points for the fine grid, we adopt a bilinear interpolation as in Zhang et al. (2013). To down-sample the wavefield from the fine grid to the ghost coarse grid points, one simple way may just take the values on the fine grid points that coincide with the ghost coarse points. However such simple treatment will lead to unstable long time simulation. To overcome this problem, Hayashi et al. (2001) used some averaging or weighting to suppress the artificial numerical noise generated in the down-sampling procedure. Kristek et al. (2010) obtained the values on the ghost coarse grid points by applying the Lanczos filter to remove the noise. Zhang et al. (2013) used the Gaussian filter in the collocated-grid FD scheme. In this work, we use the Gaussian filter in Zhang et al. (2013), which can be expressed in the curvilinear coordinate system as (along ξ -direction for example), (ξi ) =
2r
wmG (ξi+m ),
(12)
m=−2r
where r is the spatial spacing ratio, and wmG is the Gaussian weighting coefficients given by wmG = A · e−0.663(ξi+m −ξi )
2 /(r ξ )2
,
The factor A is a constant to normalize 2r
wmG = 1.
(13) wmG , (14)
m=−2r
For the localized discontinuous grid, the spatial transition zone may reach the free surface. To get the values of the ghost coarse grid points near the surface, a modified Gauss filter is used to smooth values on the fine grid (along η-direction): pr wlG η j+l , ηj =
(15)
l=−2r
where wlG is also given by eq. (13) but the factor A is changed to normalize wlG as pr l=−2r
wlG = 1,
2D discontinuous grid and nonuniform time step
FSFT Grid
CSCT Grid
Ghost FSFT Grid
Ghost CSFT Grid
105
CSFT Grid
Spatial discontinuous interface
Spatial transition zone
Temporal discontinuous interface
Temporal transition zone
Coincided Grid
Figure 2. Spatial transitional zone and temporal transitional zone in discontinuous grid with the spacing ratio r = 2. The blue dashed line represents the spatial discontinuous interface, and the red dashed line represents the time discontinuous interface. The light grey zone is the spatial transition zone, and the dark grey zone is the temporal transition zone. FSFT Grid: finer spacing hf with finer time step grid tf . CSCT Grid: coarse spacing hc with coarse time step grid tc . CSFT Grid: coarse spacing hc with finer time step tf grid.
where p = 0 for the ghost coarse grid points located at the free surface, and p = 1 for the ghost points of the first coarse grid layer below the free surface.
3.2 Non-uniform time step In heterogeneous models, the time step of an explicit time integration scheme is generally determined by the high velocity at greater depth if a uniform grid and uniform time step are used. Such regular time step is much smaller than that the stability condition allows at shallow low-velocity regions. As the spatial discontinuous grid employed, the subregion descretized with fine grid also requires a small time step. To further save computational time, we need nonuniform time step schemes to allow different time steps at different regions. In the context of CAA simulations and the discontinuous
Galerkin method, Liu et al. (2010) developed a NUTS-RK method that does not need interpolation or extrapolation at the interface between different time steps. In this paper, we extend the NUTS-RK method to the FDM on curvilinear grids. We introduce a temporal transition zone to handle the temporal resampling (dark grey zone in Fig. 2). The temporal transition zone has a width (in terms of points) same as double the FD half stencil and is divided in two regions: one region has a coarse time step tc (blue grids in the dark grey zone), and the other has a fine time step tf (red grids in the dark grey zone). The temporal discontinuous interface (red dashed line in Fig. 2) is located between two points and does not coincide with any grid layer in our FD scheme. The time step ratio is usually equal to the spatial spacing ratio for the sake of simplicity. Here, we use a time step ratio r = tc / tf = 2 as an example (Fig. 3), which corresponds to the spatial spacing change ratio in Fig. 2. When the wavefield in the coarse time step region advances once step, the wavefield in the fine time step region
106
H. Li et al. Time
tn+1 = tn + 2Δtf
tn+1 = tn + Δtc
Δtf tn+12 = tn + Δtf
Δtc
4b
5b
6b Δtf
tn
tn 1
2
3
Gc
4
5
6
Gf
Figure 3. Time marching scheme with non-uniform time step for the ratio r = 2. Gc is the spatial coarse grid, and Gf is the spatial finer grid. We refer the time tn and tn + 1 as the synchronous time level, and tn + 1/2 as the intermediate time level. At the interface several grid layers (here five layers) to Gc /Gf interface (blue dashed line) in Gc , time step changes from tf to tc to avoid unnecessary small time step in the coarse grid, where we denote as temporal discontinuous interface (red dashed line).
advances twice. As in Liu et al. (2010), we call the step tn and tn + 1 as synchronous time level, and the step tn + 1/2 as the intermediate time level. To simplify the implementation, the temporal discontinuous interface is generally set in the coarse grid with several grid layers apart from the spatial discontinuous interface. The exact number of layers between the spatial discontinuous interface and the temporal discontinuous interface is determined by the FD scheme. It could be equal to or slightly larger than the FD half stencil. In this work, we use nCGFT = 3 in the simulations, while in Figs 2 and 3, nCGFT is 5 for visual clarity. Because the time steps are different in the temporal transition zone, the wavefield in the coarse time step region is missing on synchronous time levels to calculate the spatial difference operator ( Lˆ in eqn. 11) at points in the fine time step region, and vice versa. One solution is to use interpolation/extrapolation to get the missing values from known temporal values, but may reduce the overall accuracy of the scheme. The NUTS-RK method utilizes the connection of the stage values of the RK scheme for different time steps to directly calculate the wavefield on the missing time levels to avoid interpolation and extrapolation and maintain the same high-order accuracy of the original RK scheme. In the following, we denote the intermediate stage values with time step tf as kfi , and stage values with time step tc as kci . The intermediate stage value ki of the fourth-order Runge–Kutta scheme (eq. 11) with step t can be written as a Tylor expansion (Liu et al. 2010), ⎡ 2 ⎤ ∂ U ⎡ 1⎤ k 2 ⎢ ∂t2 ⎥ ⎢∂U⎥ ⎢ k2 ⎥ ⎢ 2 ⎥ ⎢ ⎥ = CP t ⎢ ∂t2 ⎥ , (16) ⎢ 3⎥ ⎢ ∂ U2 ⎥ ⎣k ⎦ ⎣ ∂t ⎦ ∂2U k4 tn →tn + t ∂t 2
where C and P t ⎡ 1 0 ⎢1 c 22 ⎢ C=⎢ ⎣ 1 c32
are
1
c43
c42
0
0
0 ⎥ ⎥ ⎥, 0 ⎦ c44
⎡
1
0
0
⎢0 ⎢ P t = ⎢ ⎣0
t
0
0
t
0
0
0
(17)
0
⎤
0 ⎥ ⎥ ⎥, 0 ⎦
2
(18)
t 3
where the coefficients cij in matrix C are calculated from α i in eq. (11) (see Appendix B). In the following, we shall describe how to utilize eq. (16) to advance points 1–3 (Fig. 3) in the coarse time zone and points 4–6 (Fig. 3) in the fine time zone on the synchronous time level, and point 4b–6b (Fig. 3) on the intermediate time level. (i) On the synchronous time level tn , advancing points 4–6 from tn to tn + tf will require the intermediate stage values kfi at points 1–3 at the same intermediate time level as at points 4–6 to calculate the spatial difference. Note that we can write eq. (16) for different time steps at the same point and have ⎡
kf1
⎤
⎡
⎢ k2 ⎥ −1 ⎢ f ⎥ P−1 tf C ⎢ 3 ⎥ ⎣ kf ⎦ kf4
⎤
kc1
⎢ k2 ⎥ −1 ⎢ c ⎥ = P−1 tc C ⎢ 3 ⎥ ⎣ kc ⎦ kc4
tn →tn + tf
,
(19)
tn →tn + tc
Then we can derive kfi at points 1–3 from kci by the following transformation, ⎡
kf1
⎤
⎡
⎢ k2 ⎥ ⎢ f⎥ ⎢ 3⎥ ⎣ kf ⎦ kf4
⎤
0 c33
t=tn
and
kc1
⎤
⎢ k2 ⎥ −1 ⎢ c ⎥ = CP tf P−1 tc C ⎢ 3 ⎥ ⎣ kc ⎦ tn →tn + tf
kc4
,
(20)
tn →tn + tc
where the subscript f and c correspond to fine time step tf and coarse time step tc , respectively. We can denote the transformation matrix as −1 Tc→ f,synch = CP tf P−1 tc C .
(21)
2D discontinuous grid and nonuniform time step Similarly, advancing points 1–3 from tn to tn + tc will require kci at points 4–6, which can be obtained through ⎡ 1⎤ ⎡ 1⎤ kc kf ⎢ k2 ⎥ ⎢ k2 ⎥ ⎢ c⎥ −1 ⎢ f ⎥ = CP tc P−1 , (22) ⎢ 3⎥ tf C ⎢ 3 ⎥ ⎣ kc ⎦ ⎣ kf ⎦ kc4
kf4
tn →tn + tc
tn →tn + tf
and −1 T f →c,synch = CP tc P−1 tf C .
(23)
As mentioned by Liu et al. (2010), the transformation matrices Tf → c, synch and Tc → f, synch are both lower-triangular matrices. Consequently, when calculating kci , only preceding stage values of knf , n = 1, . . . , i–1 are required, and vice versa. So the loop of transformation procedure would not be locked. Once the stage values are known, we can use the Runge–Kutta scheme to get the missing wavefield in the temporal transition zone. (ii) On the intermediate time level tn + 1/2 , advancing points 4b-6b to tn + 1 also needs the corresponding wavefield available at points 1–3. Using the Taylor series expansion, the analysis form of Runge– Kutta scheme for the intermediate time level can be written as ⎡ (1) ⎤ ⎡ 1⎤ U k ⎢ U(2) ⎥ ⎢ k2 ⎥ ⎢ ⎥ ⎢ ⎥ = CP tf B tf ⎢ (3) ⎥ , (24) ⎢ 3⎥ ⎣U ⎦ ⎣k ⎦ k4
U(4)
tn + tf →tn +2 tf
t=tn
where B tf is the expansion coefficients ⎡ ⎤ 1 b12 tf b13 tf2 b14 tf3 ⎢0 1 b23 tf b24 tf2 ⎥ ⎢ ⎥ B tf = ⎢ ⎥, ⎣0 0 1 b34 tf ⎦ 0
0
0
(25)
1
where bij = 1/(j − i)! Thus we can derive the stage values kif for the intermediate time level at the coarse time step region from kci as ⎡ 1⎤ ⎡ 1⎤ kf kc ⎢ k2 ⎥ ⎢ k2 ⎥ ⎢ f⎥ −1 ⎢ c ⎥ = CP tf B tf P−1 , (26) ⎢ 3⎥ tc C ⎢ 3 ⎥ ⎣ kf ⎦ ⎣ kc ⎦ kf4
tn + tf →tn +2 tf
kc4
tn →tn + tc
and the transformation matrix is −1 Tc→ f,inter = CP tf B tf P−1 tc C .
(27)
By the transform matrices Tf → c, synch , Tc → f, synch and Tc → f, inter , we can derive the stage values and corresponding wavefield for different time step in the temporal transition zone, thus we can calculate the spatial differences at the points along the time discontinuous interface without any interpolation or extrapolation, and maintains the same high order accuracy as that of the fourth-order Runge–Kutta scheme in the main domain. We should note that these transformation relationship can be easily extended to an arbitrary integer ratio r > 2 case, as shown in Liu et al. (2010). 4 N U M E R I C A L V E R I F I C AT I O N S In order to verify the implementation of the spatial-temporal discontinuous approach on curvilinear grids, we present six numerical
107
tests on: a full homogeneous space model with Cartesian grid, a full homogeneous space model with curvilinear grid, a half-space model with a flat surface, a low-velocity layer over an half-space model, a basin model with flat surface, and a basin model with a topography. We use an explosive point source in all the tests. The source time function is a Ricker wavelet with central frequency at 1.0 Hz (thus the maximum frequency is about 2.5 Hz) and time-delay of 1.0 s. The spatial singular source is approximated by a spatial Gaussian distribution in the curvilinear coordinate system exp(−(ξ − ξ0 )2 /( ξ )2 − (η − η0 )2 /( η)2 ),
(28)
where (ξ 0 , η0 ) is the source location in the curvilinear coordinate. To absorb the out-going waves, ADE CFS-PML (Zhang & Shen 2010) is used. If different spacing grids reach the PML, we ensure that the total thickness of the PML is the same at different spacing grids (a thickness of 12 grid layers in the coarse grid) and the inner boundaries of the PML align. The parameters α 0 and β 0 are calculated by the distance to the PML inner boundary, so the absorbing effect is less affected by different numbers of PML layers in discontinuous grid. We find that the conventional PML (by setting the parameters α 0 = 0 and β 0 = 1) is unstable for long time simulation in discontinuous grid, but ADE CFS-PML can achieve stable simulation results. In this work, we choose α 0 = 1.5π fc and β 0 = Vmax /PPW hfc suggested by Zhang & Shen (2010), where Vmax is the max velocity in the model, PPW is the minimum number of points per wavelength, h is the spatial spacing, and fc is the centre frequency of the source wavelet.
4.1 Homogeneous full-space model with Cartesian grid In the first numerical test, we use a homogeneous full-space model with Cartesian grid to verify the proposed discontinuous grid method with non-uniform time step scheme. By using a homogeneous model, we can isolate the effects of the discontinuous grid and the non-uniform time step scheme from complex structures. If there are any artificial reflections caused by the discontinuous spatial and temporal interfaces, we should be able to observe them in this simulation results. The P-wave velocity VP is 4.5 km s−1 , S-wave velocity VS is 2.6 km s−1 and the density ρ is 2600 kg m−3 . The model is discretized by a fine grid ( hf = 50 m) in the upper part and a coarse grid ( hc = 100 m) in the lower part (Fig. 4). The spatial discontinuous interface (blue dashed line) is located at y = 0 km. Three grid layers below the interface (red dashed line), the time step increases from tf = 0.005 s (in the upper part) to tc = 0.01 s (in the lower part). We then perform two numerical simulations. In one simulation, the source is located in the fine grid (Fig. 4a), and the wave propagates from the fine grid into the coarse grid. In the other simulation, the source is located in the coarse grid (Fig. 4b), and the wave propagates from the coarse grid into the fine grid. The locations of the source (star) and the receivers (triangles) are shown in Fig. 4. We use the results from a uniform fine grid ( h = 50 m and t = 0.005 s) as the references for comparisons. Fig. 5 illustrates the snapshot comparisons between two simulations at time t = 3 s (middle column) from discontinuous grid with the reference solution in the uniform grid (left column). No obvious artificial reflections either from the spatial discontinuous interface (blue dashed lines in Fig. 5) or temporal discontinuous interface
108
H. Li et al.
Figure 4. Homogeneous full-space model with Cartesian grid and the discontinuous grid structures for two cases. (a) The source (star) is located at (0, 5) km in the fine grid, there are three receivers (triangles) located at (−10, −5) km, (0, −5) km and (10, −5) km. (b) The source is located at (0, −5) km in the coarse grid, and there are three receivers located at (−10, 5) km, (0, 5) km and (10, 5) km. The blue dashed line represents the spatial discontinuous interface, while the red dashed line is the temporal discontinuous interface.
Figure 5. Snapshot comparisons of vertical velocity component v y between discontinuous grid and uniform grid for the homogeneous full-space model with Cartesian grid. Left panels show the uniform grid results; middle panels show the discontinuous grid results, and right panels are the differential snapshots between the left and middle panels amplified 100 times. (a) Source in fine grid (Fig. 4a). (b) Source in coarse grid (Fig. 4b). The blue dashed line is the spacial discontinuous interface, and the red dashed line is the temporal discontinuous interface. The maximum amplitude of each snapshot and relative error are indicated.
(red dashed lines in Fig. 5) can be observed from the snapshot comparisons. The right column of Fig. 5 shows the differential snapshots amplified 100 times, which reveals there are indeed very weak artificial reflection, but the amplitude is generally less than 0.2 per cent of the physical waves. The results show that the spatial transition zone and temporal transition zone in our method can exchange the wavefield across the spatial-temporal discontinuous interface as expected. To further check the accuracy of the discontinuous grid implementation, Fig. 6 shows the comparisons of velocity seismograms at three receivers. The seismograms from the discontinuous grid (red dashed lines) compare very well on the ones from the uniform fine grid (black solid lines). There is no apparent
discrepancy between two solutions. The maximum relative error (the ratio of the maximum pointwise difference to the maximum amplitude calculated by the uniform fine grid) is within 0.1 per cent in both cases. We also perform a long time simulation up to 40 000 time steps to check the numerical stability of the scheme. Fig. 7 shows the vertical velocity component v y at receiver 2, which demonstrates the scheme is stable even up to 40 000 time steps. From this simple model test, we can see that the proposed spatialtemporal discontinuous grid method can accurately and stably simulate seismic wave propagation through the discontinuous interfaces without obvious artificial scattering.
2D discontinuous grid and nonuniform time step (a)
(b) Vx
Vy
Vx
UniG DisG Diff 4.7087 4.7082
3
4.7087 4.7087
3
0.0034(0.1%) 0.0000 0.0001
2
0.0022(0.0%) 7.9127 7.9190
2
0.0001(--%)
0.0116(0.1%)
4.7087 4.7082
1
4.7087 4.7087
1
0.0034(0.1%) 0
Vy
UniG DisG Diff
5
10
Time(s)
Receiver Number
Receiver Number
109
4.7048 4.7014
3
0.0038(0.1%) 0.0000 0.0001
2
5
0.0035(0.1%) 7.9055 7.9107
2
0.0001(--%)
0.0053(0.1%)
4.7048 4.7014
1
0.0021(0.0%) 0
4.7048 4.7014
3
4.7048 4.7014
1
0.0038(0.1%)
10
0
5
Time(s)
10
0.0035(0.1%) 0
Time(s)
5
10
Time(s)
Figure 6. Comparisons of seismograms at three receivers in the homogeneous full-space model with Cartesian grid for (a) source in fine grid (Figs 4a) and (b) source in coarse grid (Fig. 4b). Black solid lines are the results of uniform grid (UniG), red dashed lines are the results of discontinuous grid (DisG), and blue solid lines are the difference between UniG and DisG. The maximum amplitude of each seismogram and relative error is indicated.
4.2 Homogeneous full-space model with curvilinear grid
(a)
In this numerical test, we use the same full-space velocity model as in the previous test but with a curvilinear grid to check the performance of the discontinuous grid on curvilinear grid. The model is also discretized into a fine grid ( hf ≈ 50 m) and coarse grid ( hc ≈ 100 m) as shown in Fig. 8. The interface is located along y = A sin (2π x/L), where A = 1 km, and L = 30 km. The time step increases from tf = 0.005 s to tc = 0.01 s at three grid layers below the spatial discontinuous interface. Similar to the previous test, we perform a simulation by locating the source in the fine grid (Fig. 8a), and also perform another simulation by locating the source in the coarse grid (Fig. 8b). The locations of the source (star) and the receivers (triangles) are shown in Fig. 8. We use the results of a uniform fine grid ( h ≈ 50 m and t = 0.005 s) as the reference for comparisons. Fig. 9 shows the snapshot comparison of two cases at time t = 3 s with the reference results. We can see that the wavefield smoothly propagates through the spatial discontinuous interface (blue dashed lines) and the temporal discontinuous interface (red dashed lines). The differential snapshots in right panels of Fig. 9 show that the artificial reflections are negligible. Fig. 10 depicts the detailed comparisons of the velocity seismograms at the three receivers in Fig. 8. We see the fit between the seismograms of the discontinuous grid
Figure 7. Seismogram at receiver 2 (v y component) for long time simulations up to 40 000 time steps (400 s with 0.01 s time step) for Figs 6a (Case 1) and 6b (Case 2).
(b) 10
10
5
5 y distance (km)
y distance (km)
(a)
0
−5
−5
−10 −15
0
−10 −10
−5
0 5 x distance (km)
10
15
−15
−10
−5
0 5 x distance (km)
10
15
Figure 8. Homogeneous full-space model with curvilinear grid and the discontinuous grid structures for two cases. (a) The source (star) is located at (0, 5) km in the fine grid; there are three receivers (triangles) located at (−10, −5) km, (0, −5) km and (10, −5) km. (b) The source is located at (0, −5) km in the coarse grid, and there are three receivers located at (−10, 5) km, (0, 5) km and (10, 5) km. Note the grid is shown by every four points for clarity.
110
H. Li et al.
Figure 9. Snapshot comparisons of vertical velocity component v y between discontinuous grid and uniform grid for the homogeneous full-space model with curvilinear grid. Left panels show the uniform grid results, middle panels show the discontinuous grid results, and right panels are the differential snapshots between the left and middle panels amplified 100 times. (a) Source in fine grid (Fig. 8a). (b) Source in coarse grid (Fig. 8b). The blue dashed line is the spacial discontinuous interface, and the red dashed line is the temporal discontinuous interface. The maximum amplitude of each snapshot and relative error are indicated.
(a)
(b) Vx
Vx
Vy
4.7340 4.7365
3
0.0037(0.1%) 0.0037 0.0037
2
0.0026(0.1%) 7.9045 7.9106
2
0.0008(--%)
0.0114(0.1%)
4.6946 4.6959
1
4.6786 4.6803
1
0.0027(0.1%) 5
Time(s)
10
Receiver Number
Receiver Number
4.7180 4.7200
3
0
Vy
UniG DisG Diff
UniG DisG Diff
4.6921 4.6878
3
0.0045(0.1%) 0.0037 0.0041
2
5
Time(s)
7.8971 7.9014
2
0.0045(0.1%)
4.7162 4.7142
1
10
0.0041(0.1%)
0.0004(--%)
0.0017(0.0%) 0
4.6762 4.6721
3
4.7322 4.7301
1
0.0028(0.1%) 0
5
Time(s)
10
0.0023(0.0%) 0
5
10
Time(s)
Figure 10. Comparisons of seismograms at three receivers in the homogeneous full-space model with curvilinear grid for (a) source in fine grid (Fig. 8a) and (b) source in coarse grid (Fig. 8b). Black solid lines are the results of uniform fine grid (UniG), red dashed lines are the results of discontinuous grid (DisG), and blue solid lines are the difference between UniG and DisG. The maximum amplitude of each seismogram and relative error are indicated.
(red dashed lines) and of the uniform fine grid (black solid lines) is as good as in the Cartesian grid (see Fig. 6). The maximum relative error is 0.1 per cent. This numerical test reveals that the spatial-temporal discontinuous grid approach also works very well on curvilinear grids.
4.3 Homogeneous half-space model with a flat surface In this numerical test, we include the flat free surface in the homogeneous half-space velocity model. We test a two-level hierarchically nested discontinuous grid (Fig. 11a, referred as DisG_1) and also
a three-level hierarchically nested discontinuous grid (Fig. 11b, referred as DisG_2), both with non-uniform time steps. The base coarse grid has a grid spacing of h1 = 100 m and the fine grid is refined by a factor r = 2. For the DisG_1, the two level grid spacing has a ratio h1 : h2 = 2 : 1, while for the DisG_2, the three level has a ratio h1 : h2 : h3 = 4 : 2 : 1. The base coarse grids advance with a time step tc = 0.01 s. The time step will change with the same refine ratio as the grid spacing across the temporal discontinuous interface (three points distance to the spatial discontinuous interface in the coarse grid). The source is located at (−12.6,−5) km. Three receivers are all located on the free surface at (−12.6,0) km, (0,0) km and (12.6,0) km. The locations of the source (star) and
2D discontinuous grid and nonuniform time step (a)
(b) 0
0
Level 2
Level 1
−5
−15 −15
−10
−5
0
5
Level 3
−1.5 −2.5
y distance (km)
−1.5
y distance (km)
111
10
x distance (km)
15
Level 2 Level 1
−5
−15 −15
−10
−5
0
5
10
15
x distance (km)
Figure 11. Homogeneous half-space model with two kinds of multi-level hierarchically nested discontinuous grid structures: (a) two-level and (b) three-level. The source (star) is located at (−10, −5) km; there are three receivers (triangles) located at (−10, 0) km, (0, 0) km and (10, 0) km.
the receivers (triangles) are shown in Fig. 11. The half-space has the same elastic parameters as in the two previous example, that is P-wave velocity VP = 4.5 km s−1 , S-wave velocity VS = 2.6 km s−1 and the density ρ = 2600 kg m−3 . To evaluate the accuracy of the discontinuous grid method, we compare the simulation results with solutions obtained by the uniform grid finite-difference method (UniGFD) with spatial spacing h = 50 m and time step t = 0.005 s. Figs 12(b) and (c) show v y snapshots at t = 4 s of DisG_1 and DisG_2, respectively. Comparing to the reference v y snapshot (Fig. 12a) of UniGFD, the agreement among the snapshots of different grid structures is excellent, which demonstrates the spatial-temporal discontinuous grid method can sufficiently accurately simulate seismic wave propagation with the free-surface boundary condition. Especially, the corner of the localized discontinuous grid interface (Figs 12b and c) does not cause scatterings or distort the wavefront when the waves pass through it, which indicates we can employ this flexible localized discontinuous grid to improve the performance of seismic wave numerical simulation. Fig. 13 shows the comparisons of the velocity seismograms. The seismograms from both discontinuous grids (green dashed lines for DisG_1 and red dashed lines for DisG_2) agree well to the reference solution (black solid lines for UniGFD). The maximum relative errors are 2.5 per cent for DisG_1 and DisG_2, respectively. The comparisons demonstrate the accuracy of the laterally localized discontinuous grid implementation with the NUTS-RK scheme even when the discontinuous interfaces intersect with the free surface.
4.4 A low-velocity layer over a half-space model In this numerical test, we perform a numerical simulation in a model consisting of a low-velocity layer over a half-space model (Fig. 14) to investigate the capability of the discontinuous collocated-grid FD method with the NUTS-RK scheme on models with high-velocity contrast. The first layer is 1.0 km thick, and has the properties VP = 2.4 km s−1 , VS = 1.0 km s−1 , and ρ = 1800 kg m−3 . The underlying half-space medium has VP = 4.5 km s−1 , VS = 2.6 km s−1 , and ρ = 2600 kg m−3 . The locations of the source (star) and the receivers (triangles) are shown in Fig. 14. Since the velocity of the whole first layer is low, we employ a layer-like discontinuous grid with non-uniform time step (Fig. 14). The fine grid ( hf = 50 m) covers all the low-velocity layer and also includes 600 m high-velocity region. The rest model is discretized by a coarse grid of hc = 100 m. So the spatial discontinuous interface of two grids is located at 1.6 km below the free surface (0.6 km below the layer interface). The temporal discontinuous interface of the non-uniform time step is located three grid layers below the
spatial discontinuous interface. Specifically, the time step increases from tf = 0.005 s (for points above the temporal interface) to tf = 0.01 s (rest points). To assess the accuracy and efficiency of the discontinuous grid approach, we compare the simulation results with those derived from a uniform fine grid (UniGFD) with h = 50 and t = 0.005 s. Fig. 15 shows the comparisons of velocity seismograms along a receiver line on the free surface by the discontinuous grid and the uniform grid. The waveform is much complex due to multiple reflections in the low-velocity layer. We can see that even for this complex waveform, the results by the discontinuous grid show an excellent agreement with the results by the uniform grid. Fig. 16 shows the detailed comparisons of the seismograms at the three receivers in Fig. 14. The two solutions are almost same; the maximum difference is within 3.1 per cent of results from UniGFD, which indicates the accuracy of the discontinuous grid approach. For this low-velocity layer model, if the discontinuous grid approach was not used, we would need to use the fine grid over all the region, which wastes computational resources and causes unnecessary long time simulation. In contrast, by using the spatialtemporal discontinuous grid, we reduce the computational memory to 33.0 per cent and the computational time to 35.3 per cent (Table 1). If the velocity contrast in the model is even larger, we can use a larger discontinuous grid ratio with a larger time step ratio and save the computational effort even more. 4.5 Basin model with flat free surface In this numerical test, there is an elliptic basin embedded in a halfspace model with flat free surface (Fig. 17). The lower boundary of the basin is defined by a half-ellipsoid function
(x − x0 )2 , (29) y = y0 − c 1 − a2 where (x0 , y0 ) is the centre coordinates of the ellipsoid, x0 = 2.5 km and y0 = 0 km. The lengths of the axes of the ellipsoid are a = 5.5 km and b = 1.5 km. The basin has a low velocity of VP = 2.4 km s−1 , VS = 1.0 km s−1 , and ρ = 1800 kg m−3 . The bedrock has the properties of VP = 4.5 km s−1 , VS = 2.6 km s−1 , and ρ = 2600 kg m−3 . The locations of the source (star) and the receivers (triangles) are shown in Fig. 17. The three receivers are placed vertically along the centre of the basin at (2.5, 0.0) km, (2.5, –1.0) km and (2.5, –2.0) km. Among these three receivers, one is located at the free surface of the basin, one is inside the basin, and the last one is in the bedrock below the basin. We use a laterally localized discontinuous grid (DisGFD) in this basin model as shown in Fig. (17) to attain maximum efficiency. In
112
H. Li et al. Vy
Vx DisG_1
Diff_1 Diff_2
DisG_2 UniGFD
31.9269 31.9606 32.4378
x distance (km)
10
12.5309 12.5435 12.6096
10
0.1461(1.2%) 0.1012(0.8%)
0.4771(1.5%) 0.5108(1.6%) 49.0983 49.1543 50.1519
0
23.6106 23.6932 24.1712
0
0.9798(2.0%) 1.0624(2.2%) 0.1843 0.0989 0.0649
-10
0.4718(2.0%) 0.5454(2.3%) 88.8171 88.7865 90.4957
-10
1.6243(1.8%) 1.5965(1.8%)
0.1418(--%) 0.1840(--%) 0
5
0
10
5
10
Time(s)
Time(s)
Figure 13. Comparisons of seismograms at three receivers in the homogeneous half-space model with flat surface. Black solid lines are the results of uniform fine grid (UniGFD), green dashed lines are the results of twolevel discontinuous grid (DisG_1), and red dashed lines are the results of three-level discontinuous grid (DisG_2), blue solid lines are the difference between UniGFD and DisG_1, and cyan solid lines are the difference between UniGFD and DisG_2. The maximum amplitude of each seismogram and relative error are indicated. 0
y distance (km)
−1.6
−5
−15 −15
−10
−5
0
5
10
15
x distance (km)
Figure 14. The low-velocity layer over half-space model and the discontinuous grid structure. The source is located at (−10, −0.5) km in the low-velocity layer; there are three receivers located at (−10, 0) km, (0, 0) km and (10, 0) km. The grid is shown by every four points for clarity.
Figure 12. Snapshot comparisons of vertical velocity component v y between two discontinuous grids and uniform fine grid for the homogeneous half-space model with flat surface. (a) The uniform fine grid result. (b) Twolevel discontinuous grid result. (c) Three-level discontinuous grid result. The blue dashed lines are the spacial discontinuous interfaces, and the red dashed lines are the temporal discontinuous interfaces.
the laterally localized discontinuous grid, an embedded rectangular box surrounds the basin, which is discretized by a fine grid with hf = 50 m, while the rest region is discretized by a coarse grid with hc = 100 m. The fine time step is tf = 0.005 s, which is used in the fine grid and also three-layer points in the coarse grid surrounding the fine grid. All the rest points use a time step of tc = 0.01 s.
We use the simulation results derived from a uniform fine grid (UniGFD, h = 50 and t = 0.005 s) as the reference for comparisons. Fig. 18 shows the comparisons of velocity seismograms along the receiver line on the free surface by the discontinuous grid and the uniform fine grid. The basin traps the energy and causes long time shaking in the basin. We can see the discontinuous grid can accurately calculate the ground motion in this basin model, even though the waveforms are very complex. Fig. 19 shows the detailed comparison of seismograms at the three receivers. The seismograms from the discontinuous grid and the uniform fine grid are almost same over the whole time window. The maximum relative error is only 0.7 per cent. The localized spatial-temporal discontinuous grid can efficiently reduce the number of points and save the needed computational memory. In this basin model, the discontinuous grid only requires 30 per cent computational memory comparing to the uniform grid approach. The localized discontinuous grid also reduces the length of transition zone between the fine/coarse grids, which in turn
2D discontinuous grid and nonuniform time step
113
(a) Vy 15
10
10
x distance (km)
x distance (km)
Vx 15
5
5
0
0
0
5
15
10
10
5
0
25
20
15
20
25
15
20
25
Time(s)
Time(s)
(b) Vy 15
10
10
x distance (km)
x distance (km)
Vx 15
5
0
5
0
0
5
10
15
20
25
0
5
10
Time(s)
Time(s)
Figure 15. Comparisons of synthetic seismograms of the low-velocity layer over half-space model. Left panel shows the horizontal velocity component v x , and right panel shows the vertical velocity component v y . (a) Results of discontinuous grid (DisGFD). (b) Results of uniform fine grid (UniGFD).
(a)
(b) Vx
Vy
UniGFD DisGFD Diff
UniGFD DisGFD Diff
3.4449 3.4275
10
1.6216 1.6172
10 0.0483(3.0%)
3.8263 3.8215
x distance (km)
x distance (km)
0.1054(3.1%)
5 0.0657(1.7%) 5.0554 4.9909 0
1.8849 1.8472 5 0.0487(2.6%) 3.3026 3.2764 0
0.0664(1.3%)
0.0342(1.0%)
0.0154 0.0095
16.0516 16.0560
-5
-5 0.0249(--%)
0
5
10
15
20
0.0185(0.1%)
25
0
5
Time(s)
10
15
20
25
Time(s)
Figure 16. Detailed comparisons of the seismograms at three receivers in the low-velocity layer over half-space model. (a) The horizontal velocity component v x . (b) The vertical velocity component v y . Black solid lines are the results of uniform fine grid (UniGFD), red dashed lines are the results of discontinuous grid (DisGFD), and blue solid lines are the difference between UniGFD and DisGFD. The maximum amplitude of each seismogram and relative error are indicated.
reduces the overhead of the wavefield exchange between the transition zones. Combined with the non-uniform time step, the computation time is reduced to 23.5 per cent comparing to the run time of the uniform grid (Table 1).
shape is defined by function
4.6 Basin model with a hill-canyon topography
where (x0 , y0 ) is the centre coordinates of the ellipsoid: x0 = 2.5 km and y0 = 0 km. The lengths of two axes of the ellipsoid are a = 5.5 km and b = 1.5 km. The surface topography is defined
A more realistic model containing a basin and a hill-canyon shape topography (Fig. 20) is studied in this numerical test. The basin
y = y0 − c 1 −
(x − x0 )2 , a2
(30)
114
H. Li et al. The remainder region is discretized by a rectangular coarse grid of hx = 100 m and hy = 100 m. The fine time step is tf = 0.005 s, which is used in the fine grid and also three-layer points in the coarse grid surrounding the fine grid. All the rest points use a time step of tc = 0.01 s. We use the results from a fine continuous curvilinear grid (ConGFD) with hx = 50 m, hy = 25 ∼ 100 m and time step tf = 0.005 s as the reference for comparison. Fig. 21 shows the comparison of velocity seismograms along the receiver line on the free surface with the discontinuous grid and the fine continuous curvilinear grid. The major features of the waveforms are similar to Fig. 18 since the basins in these two models have the same bottom shape. The topography causes scatterings and also modifies the waveform to some extent, which can be observed in Fig. 21. Despite the effects of the basin and topography, the results by the discontinuous grid (top row in Fig. 21) show an excellent agreement with those by the fine continuous curvilinear grid (bottom row in Fig. 21), which demonstrates the spatial-temporal discontinuous curvilinear grid method can be used to simulate seismic wave propagation in complex models with enough accuracy. This excellent agreement can be further confirmed from the detailed comparisons of the seismograms (Fig. 22) of the five receivers at the free surface. The seismograms from the discontinuous grid and the fine continuous curvilinear grid are very similar in the whole time window. The maximum relative error is about 3.0 per cent. By using a localized discontinuous curvilinear grid, the computation memory requirement is reduced to 33.8 per cent and computational time to 25.7 per cent compared to the fine continuous curvilinear grid. This numerical test verifies that the localized
0
y distance (km)
−2 −3
−15 −15
15
10
5
0
−5
−10
x distance (km)
Figure 17. Basin model with flat surface and the discontinuous grid structure. The source located at (−10, −3) km; there are three receivers located at (2.5, 0) km, (2.5, −1) km and (2.5, −2) km. The grid is shown by every four points for clarity.
by function y(x) = a exp(−(x − x1 )2 /a 2 ) + b exp(−(x − x2 )2 /a 2 ),
(31)
where x1 = −5.0 km, x1 = 5.0 km, a = 1.0 km and b = −1.0 km. The locations of the source (star) and the receivers (triangles) are also shown in Fig. 20. The basin and bedrock have the same parameters as in the previous test. In the basin, we have VP = 2.4 km s−1 , VS = 1.0 km s−1 , and ρ = 1800 kg m−3 . And in the bedrock, we have VP = 4.5 km s−1 , VP = 2.6 km s−1 , and ρ = 2600 kg m−3 . We use a fine curvilinear grid ( hx = 50 m, hy = 25 ∼ 100 m) to cover the topographic region and the basin as shown in Fig. 20. (a)
Vy 15
10
10
x distance (km)
x distance (km)
Vx 15
5
0
-5
0
5
0
-5
5
10
15
20
25
0
5
10
15
20
25
15
20
25
Time(s)
Time(s)
(b) Vy 15
10
10
x distance (km)
x distance (km)
Vx 15
5
0
0
-5
-5
0
5
5
10
15
Time(s)
20
25
0
5
10
Time(s)
Figure 18. Comparisons of synthetic seismograms of basin model with flat surface. Left panel shows the horizontal velocity component v x , right panel shows the vertical velocity component v y . (a) Results of discontinuous grid (DisGFD). (b) Results of uniform fine grid (UniGFD).
2D discontinuous grid and nonuniform time step (a)
(b) Vx
Vy
UniGFD DisGFD Diff
UniGFD DisGFD Diff
0.2398 0.2385
0.3137 0.3129
-1
0.0021(0.7%) 0.5891 0.5871
-2
0.1642 0.1633
0
0.0012(0.5%)
y distance (km)
0
y distance (km)
115
0.0008(0.5%) 0.7111 0.7111
-1
0.0011(0.2%) 0.7826 0.7824
-2
0.0026(0.4%)
0
5
10
15
20
25
30
Time(s)
0.0030(0.4%)
0
5
10
15
20
25
30
Time(s)
Figure 19. Detailed comparisons of the seismograms at three receivers in the basin model with flat surface. (a) The horizontal velocity component v x . (b) The vertical velocity component v y . Black solid lines are the results of uniform fine grid (UniGFD), red dashed lines are the results of discontinuous grid (DisGFD), and blue solid lines are the difference between UniGFD and DisGFD. The maximum amplitude of each seismogram and relative error are indicated.
Table 1. The computational resource consumed by the discontinuous grid comparing with the uniform fine grid. Model
Memory (per cent)
CPU Time (per cent)
LVL Basin B-Topo
33.0 30.0 33.8
35.3 23.5 25.7
LVL: a low-velocity layer over a half-space model. Basin: basin model. B-Topo: basin model with a hillcanyon topography.
0
y distance (km)
−2.5 −5
−15 −15
−10
−5
0
5
10
15
x distance (km)
Figure 20. Basin model with hill-canyon topography and the discontinuous grid structure. The source is located at (−12, −0.5) km, there are five receivers located at the surface. The horizontal coordinates of the five receivers are −10 km, 5 km, 0 km, 5 km and 10 km. The grid is shown by every four points for clarity.
discontinuous curvilinear grid with NUTS-RK scheme can simulate seismic wave propagation in complex structure model with sufficient accuracy and efficiency.
5 C O N C LU S I O N S In this study, we developed a new flexible localized discontinuous grid FD method with NUTS-RK scheme on curvilinear grids for seismic wave simulation. The discontinuous grid consists of fine and coarse grids with different spatial spacings satisfying the dispersion error criterion based on local velocity variations. In order to maximize the power of the
discontinuous grid for seismic wave simulation, we allow the grid to be discontinuous in a box region surrounding the low-velocity zone. Such laterally localized discontinuous grid can attain more computational efficiency improvement than a layer-like discontinuous grid. In the spatial transition zone between fine and coarse grids, a bilinear interpolation is used to get the values from the coarse grid, and a Gaussian filter is used to suppress the artificial numerical noise that can cause instability for long time simulation when retrieving the values from the fine grid to update wave equations in the coarse grid. To further improve the efficiency of the discontinuous grid, a NUTS-RK scheme is adapted to avoid the temporal over-sampling in the coarse grid. We introduced a time transition zone in the coarse grid alongside of the spatial discontinuous interface. In this transition zone, we applied the NUTS-RK scheme to implement the wavefield exchange between two regions with different time steps without interpolation or extrapolation. There is no loss of accuracy due to non-uniform step and the same high-order accuracy is maintained through the whole simulation domain. When topography is present, the boundary-conforming grid is employed to avoid the stair-case approximation. The localized discontinuous grid with non-uniform time step scheme can work well with such curvilinear grid by applying wavefield exchange in the regular computational space. The computational efficiency improvement by employing the discontinuous grid depends on several factors, including how may volumes be discretized by coarse grid according to the velocity distribution, the ratio of the grid spacing and time step between the fine and coarse grids. The implementation of the transition zone will also consume a certain amount of computational resources, but can be compensated by using coarse grids (Table 1 case: LVL). If a laterally localized discontinuous grid is employed, the total length of the transition zone can be reduced, and the efficiency can be improved notably (Table 1 case: Basin and B-Topo). Numerical tests show that the implementation of discontinuous grid method, with NUTS-RK scheme, on curvilinear collocatedgrid, can simulate the wave propagation accurately, while greatly save the computational resources in terms of both memory and computational time. The flexible discontinuous grid scheme described in this paper can be easily extended to 3D case, which will be investigated in the
116
H. Li et al. (a) Vy 15
10
10
x distance (km)
x distance (km)
Vx 15
5
0
-5
0
5
0
-5
5
15
10
20
25
0
5
10
15
20
25
15
20
25
Time(s)
Time(s)
(b) Vy 15
10
10
x distance (km)
x distance (km)
Vx 15
5
0
0
-5
-5
0
5
5
10
15
20
25
0
5
10
Time(s)
Time(s)
Figure 21. Comparisons of synthetic seismograms of basin model with hill-canyon topography. Left panel shows the horizontal velocity component v x ; right panel shows the vertical velocity component v y . (a) Results of discontinuous grid (DisGFD). (b) Results of fine continuous curvilinear grid (ConGFD).
Figure 22. Detailed comparisons of the seismograms at five receivers in basin model with a hill-canyon topography. (a) The horizontal velocity component v x . (b) The vertical velocity component v y . Black solid lines are the results of fine continuous curvilinear grid (ConGFD), red dashed lines are the results of discontinuous grid (DisGFD), and blue solid lines are the difference between ConGFD and DisGFD. The maximum amplitude of each seismogram and relative error are indicated.
following work. The NUTS-RK scheme is not restricted to a specific spatial FD scheme, and can be implemented in other schemes.
Science Foundation of China (grants No. 41090293, 41374056) and partially by the Fundamental Research Funds for the Central Universities (WK2080000053).
AC K N OW L E D G E M E N T S
REFERENCES
The authors thank Dr. Josep de la Puente and Dr. Jozef Kristek for their constructive comments. This work is supported by National
Aoi, S. & Fujiwara, H., 1999. 3D finite-difference method using discontinuous grids, Bull. seism. Soc. Am., 89(4), 918–930.
2D discontinuous grid and nonuniform time step Appel¨o, D. & Petersson, N.A., 2009. A stable finite difference method for the elastic wave equation on complex geometries with free surfaces, Commun. Comput. Phys., 5(1), 84–107. de la Puente, J., Ferrer, M., Hanzich, M., Castillo, J.E. & Cela, J.M., 2014. Mimetic seismic wave modeling including topography on deformed staggered grids, Geophysics, 79(3), T125–T141. Falk, J., Tessmer, E. & Gajewski, D., 1998. Efficient finite-difference modelling of seismic waves using locally adjustable time steps, Geophys. Prospect., 46(6), 603–616. Graves, R.W., 1996. Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences, Bull. seism. Soc. Am., 86(4), 1091–1106. Hayashi, K., Burns, D.R. & Toks¨oz, M.N., 2001. Discontinuous-grid finitedifference seismic modeling including surface topography, Bull. seism. Soc. Am., 91(6), 1750–1764. Hestholm, S. & Ruud, B., 1998. 3D finite-difference elastic wave modeling including surface topography, Geophysics, 63(2), 613–622. Hestholm, S. & Ruud, B., 2002. 3D free-boundary conditions for coordinatetransform finite-difference seismic modelling, Geophys. Prospect., 50(5), 463–474. Hixon, R., 1997. On increasing the accuracy of maccormack schemes for aeroacoustic applications, in AIAA/CEAS Aeroacoustics Conference 3 rd, Atlanta, GA, pp. 29–39. Huang, C. & Dong, L.-G., 2009a. High-order finite-difference method in seismic wave simulation with variable grids and local time-steps, Chinese J. Geophys.-Chinese Edition, 52(1), 176–186. Huang, C. & Dong, L.-G., 2009b. Staggered-grid high-order finitedifference method in elastic wave simulation with variable grids and local time-steps, Chinese J. Geophys., 52(6), 1324–1333. Jastram, C. & Behle, A., 1992. Acoustic modelling on a grid of vertically varying spacing, Geophys. Prospect., 40(2), 157–169. Jastram, C. & Tessmer, E., 1994. Elastic modelling on a grid with vertically varying spacing, Geophys. Prospect., 42(4), 357–370. Kang, T.-S. & Baag, C.-E., 2004a. An efficient finite-difference method for simulating 3D seismic response of localized basin structures, Bull. seism. Soc. Am., 94(5), 1690–1705. Kang, T.-S. & Baag, C.-E., 2004b. Finite-difference seismic simulation combining discontinuous grids with locally variable timesteps, Bull. seism. Soc. Am., 94(1), 207–219. Kristek, J., Moczo, P. & Archuleta, R.J., 2002. Efficient methods to simulate planar free surface in the 3D 4th-order staggered-grid finite-difference schemes, Studia Geophysica et Geodaetica, 46(2), 355–381. Kristek, J., Moczo, P. & Galis, M., 2010. Stable discontinuous staggered grid in the finite-difference modelling of seismic motion, Geophys. J. Int., 183(3), 1401–1407. Lan, H. & Zhang, Z., 2011. Three-dimensional wave-field simulation in heterogeneous transversely isotropic medium with irregular free surface, Bull. seism. Soc. Am., 101(3), 1354–1370. Levander, A.R., 1988. Fourth-order finite-difference P-SV seismograms, Geophysics, 53(11), 1425–1436. Liu, L., Li, X. & Hu, F.Q., 2010. Nonuniform time-step Runge–Kutta discontinuous galerkin method for computational aeroacoustics, J. Comput. Phys., 229(19), 6874–6897. Madariaga, R., 1976. Dynamics of an expanding circular fault, Bull. seism. Soc. Am., 66(3), 639–666. Moczo, P., 1989. Finite-difference technique for SH-waves in 2-D media using irregular grids application to the seismic response problem, Geophys. J. Int., 99(2), 321–329. Moczo, P., Lab´ak, P., Kristek, J. & Hron, F., 1996. Amplification and differential motion due to an antiplane 2D resonance in the sediment valleys embedded in a layer over the half-space, Bull. seism. Soc. Am., 86(5), 1434–1446. Moczo, P., Kristek, J. & G´alis, M., 2004. Simulation of the planar free surface with near-surface lateral discontinuities in the finite-difference modeling of seismic motion, Bull. seism. Soc. Am., 94(2), 760–768. Olsen, K.B. & Archuleta, R.J., 1996. Three-dimensional simulation of earthquakes on the los angeles fault system, Bull. seism. Soc. Am., 86(3), 575–596.
117
Pitarka, A., 1999. 3D elastic finite-difference modeling of seismic motion using staggered grids with nonuniform spacing, Bull. seism. Soc. Am., 89(1), 54–68. Tessmer, E., 2000. Seismic finite-difference modeling with spatially varying time steps, Geophysics, 65(4), 1290–1293. Virieux, J., 1986. P-SV wave propagation in heterogeneous media: velocitystress finite-difference method, Geophysics, 51(4), 889–901. Wang, X. & Liu, X., 2007. 3-D acoustic wave equation forward modeling with topography, Appl. Geophys., 4(1), 8–15. Wang, Y. & Takenaka, H., 2001. A multidomain approach of the fourier pseudospectral method using discontinuous grid for elastic wave modeling, Earth Planet. Space, 53(3), 149–158. Wang, Y. & Schuster, G., 1996. Finite-difference variable grid scheme for acoustic and elastic wave equation modeling, in 1996 SEG Annual Meeting, Society of Exploration Geophysicists. Wang, Y., Xu, J. & Schuster, G., 2001. Viscoelastic wave simulation in basins by a variable-grid finite-difference method, Bull. seism. Soc. Am., 91(6), 1741–1749. Zhang, W. & Chen, X., 2006. Traction image method for irregular free surface boundaries in finite difference seismic wave simulation, Geophys. J. Int., 167(1), 337–353. Zhang, W. & Shen, Y., 2010. Unsplit complex frequency-shifted PML implementation using auxiliary differential equations for seismic wave modeling, Geophysics, 75(4), T141–T154. Zhang, W., Shen, Y. & Zhao, L., 2012a. Three-dimensional anisotropic seismic wave modelling in spherical coordinates by a collocated-grid finite-difference method, Geophys. J. Int., 188(3), 1359–1381. Zhang, W., Zhang, Z. & Chen, X., 2012b. Three-dimensional elastic wave numerical modelling in the presence of surface topography by a collocated-grid finite-difference method on curvilinear grids, Geophys. J. Int., 190(1), 358–378. Zhang, Z., Zhang, W., Li, H. & Chen, X., 2013. Stable discontinuous grid implementation for collocated-grid finite-difference seismic wave modelling, Geophys. J. Int., 192(3), 1179–1188. Zhang, Z., Zhang, W. & Chen, X., 2014. Three-dimensional curved grid finite-difference modelling for non-planar rupture dynamics, Geophys. J. Int., 199(2), 860–879.
A P P E N D I X A : C O E F F I C I E N T M AT R I C E S As the mapping from curvilinear to Cartesian coordinates is x = x(ξ, η),
y = y(ξ, η),
(A1)
then, by using the chain rules, the coefficient matrices in the compact matrix form of the velocity-stress equations (eq. 4) in the computational space can be written as ⎞ ⎛ 0 0 ξ,x /ρ 0 ξ,y /ρ ⎟ ⎜ 0 0 ξ,y /ρ ξ,x /ρ ⎟ ⎜0 ⎟ ⎜ ⎟ ˜ =⎜ 0 0 0 A ⎟ ⎜ (λ + 2μ)ξ,x λξ,z ⎟ ⎜ ⎟ ⎜ (λ + 2μ)ξ,y 0 0 0 ⎠ ⎝ λξ,x μξ,y
μξ,x
0
0
0 (A2)
⎛
0
0
η,x /ρ
0
η,y /ρ
⎞
⎜ ⎜0 ⎜ ˜ =⎜ B ⎜ (λ + 2μ)η,x ⎜ ⎜ ⎝ λη,x
0
0
η,y /ρ
λη,y
0
0
(λ + 2μ)η,y
0
0
⎟ η,x /ρ ⎟ ⎟ ⎟ 0 ⎟. ⎟ ⎟ 0 ⎠
μη,y
μη,x
0
0
0 (A3)
118
H. Li et al. Then the coefficients cij in matrix C (eq. 17) can be expressed as
APPENDIX B: NON-UNIFORM TIME S T E P RU N G E – K U T TA S C H E M E T R A N S F O R M AT I O N C O E F F I C I E N T S
c22 = a21 ,
The general explicit four-stage Runge–Kutta time integration scheme with step t can be written as
c42 = a41 + a42 + a43 ,
c32 = a31 + a32 ,
ˆ n ), k1 = L(U
ˆ n + a31 tk1 + a32 tk2 ), k3 = L(U
Un+1 = Un + t
1
4 i=1
βi ki .
c44 = a43 a32 a21 .
(B2)
a21 = α2
ˆ + a41 tk + a42 tk + a4 tk ), k = L(U n
c43 = a43 a32 + a43 a31 + a42 a21 ,
As the fourth-order Runge–Kutta scheme (eq. 16) used in present work, then we have
ˆ n + a21 tk1 ), k2 = L(U
4
c33 = a32 a21 ,
2
3
(B1)
a31 = 0,
a32 = α3 ,
a41 = 0,
a42 = 0,
a43 = α4 .
where α 2 = 0.5, α 3 = 0.5, and α 4 = 1.
(B3)
