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QuasiCartesian finitedifference computation of seismic wave propagation for a threedimensional subglobal model
Earth, Planets and Space volume 69, Article number: 67 (2017)
Abstract
A simple and efficient finitedifference scheme is developed to calculate seismic wave propagation in a partial spherical shell model of a threedimensionally (3D) heterogeneous global Earth structure for modeling on regional or subglobal scales where the effects of the Earth’s spherical geometry cannot be ignored. This scheme solves the elastodynamic equation in the quasiCartesian coordinate form similar to the local Cartesian one, instead of the spherical polar coordinate form, with a staggeredgrid finitedifference method in time domain (FDTD) that is one of the most popular numerical methods in seismicmotion simulations for localscale models. The proposed scheme may be a localfriendly approach for modeling on a subglobal scale to link regionalscale and localscale simulations. It can be easily implemented using an available 3D Cartesian FDTD localscale modeling code by changing a very small part of the code. We implement the scheme in an existing Cartesian FDTD code and demonstrate the accuracy and validity of the present scheme and the feasibility to apply it to real large simulations through numerical examples.
Introduction
In recent years, there have been remarkable developments in numerical modeling techniques of seismic wave propagation, associated with progress in computer architecture. The numerical simulation has become a dominant tool for understanding seismic events in both earthquake seismology and exploration seismology. There are several numerical methods used for such purposes, such as the finiteelement, spectralelement, and finitedifference methods. Among these the finitedifference method (FDM) is one of the most popular ones. The FDM can be applied in either the time or frequency domain. The finitedifference method in time domain (FDTD) is popular because it is relatively simple and easy to program. For a review on various schemes of the FDTD, see, e.g., Moczo et al. (2014). Simplicity of the FDTD implementation motivates seismologists to apply this method.
The FDTD has been widely utilized for threedimensional (3D) seismic wave simulations on local scales (e.g., Graves 1996; Hayashida et al. 1999; Pitarka 1999). The FDTD has also been successfully applied to globalscale modeling (e.g., Thomas et al. 2000; Toyokuni et al. 2005; Toyokuni and Takenaka 2006). The globalscale modeling usually solves the elastodynamic equation in spherical polar coordinates, while the localscale modeling solves the equation in Cartesian coordinates. We often deal with a problem of an intermediate scale, so we may search for a simple but accurate or stable scheme. For modeling on regional scales, we may have to consider the spherical geometry of the Earth. Historically, Earthflattening transformation has been used in waveform modelings for laterally homogeneous (i.e., spherically symmetric) Earth models (see, e.g., Box 9.2 of Aki and Richards 2002) with, for example, the reflectivity method. It can exactly transform a SHwave propagation problem posed for a medium with spherical symmetry into a problem posed for a planestratified medium and give a useful approximation to a PSV problem in a spherically symmetric medium. However, for laterally or threedimensionally heterogeneous Earth models the validation of the use of Earthflattening approximation should be eventually checked by comparing the solutions with those computed by numerical methods such as the FDTD without the approximation for each of the same models. The FDTD has actually been applied to modeling wave propagation even for media with random properties (e.g., Igel and Gudmundsson 1997). We thus prefer to use a method without Earthflattening approximation. The FDTD has been exploited for intermediatescale, regional or continentalscale modeling as well as localscale modeling. Igel et al. (2002) presented an excellent scheme for a partial spherical shell (spherical section) model of a 3D heterogeneous global earth structure, which is a rather globalfriendly method to link regionalscale and globalscale modelings because it solves the elastodynamic equation in spherical polar coordinates.
In this paper, we propose a localfriendly approach for modeling on subglobal scales to link regionalscale and localscale modelings. This solves the elastodynamic equation in the quasiCartesian coordinate form similar to local Cartesian one, instead of in the original spherical polar coordinate form, with the FDTD. The proposed scheme can be easily implemented in an available 3D Cartesian FDTD code of localscale modeling such as strongmotion simulation by changing a very small part of the code. It is one of the most important merits of use of the quasiCartesian formulations instead of the original spherical polar ones.
The quasiCartesian approach could be easily applied to a multiscale hybrid method or domain decomposition method that divides the computational domain into multiple domains in which wave propagation is calculated separately by changing grid size and type (e.g., uniform or nonuniform), often with different methods (e.g., Moczo et al. 1997; Wen and Helmberger 1998; Robertsson and Chapman 2000; Yoshimura et al. 2003; Opršal et al. 2009; Monteiller et al. 2013). If we consider a local domain embedded in a subglobal domain (Fig. 1a, c), the wave propagation in the subglobal domain is computed with the quasiCartesian FDTD as a background model. In the local domain, the propagation is then calculated with the Cartesian FDTD. In this process, we may be able to couple the computations between the domains through the hybrid of the Cartesian and quasiCartesian formulations without combining different methods. For coupling the wavefields in the two domains, for instance, socalled a finitedifference injection method could be exploited, which allows us to calculate synthetic seismograms efficiently after model alterations in the local domain (Robertsson and Chapman 2000; Borisov et al. 2015). The hybrid may also be applicable to a source inversion that estimates a spatiotemporal slip distribution on a fault plane from local (strongmotion), regional, and teleseismic records (Fig. 1b). In this case, the target fault may be set in the local domain where Cartesian coordinates are used. These issues on the hybrid method might be nearfuture subjects.
Methods
For simplicity, we here consider formulations for elastic waves without anelastic attenuation to explain the scheme. In a spherical polar coordinate system (r, θ, ϕ) (Fig. 2), the velocitystress form of the 3D isotropic linear elastodynamic equation may be written as (e.g., Igel et al. 2002; Toyokuni et al. 2012):
(equation of motion)
and (constitutive equation)
where ρ is the density, v _{ r }, v _{ θ }, and v _{ ϕ } are the particle velocities, σ _{ rr }, σ _{ θθ }, σ _{ ϕϕ }, σ _{ rθ }, σ _{ θϕ }, and σ _{ rϕ } are the stress components, f _{ r }, f _{ θ }, and f _{ ϕ } are bodyforce components, λ and μ are Lamé constants, and \(\mathop{\dot{M}_{rr}}, \mathop{\dot{M}_{\theta \theta}}, \mathop{\dot{M}_{\phi \phi}}, \mathop{\dot{M}_{r\theta}}, \mathop{\dot{M}_{\theta \phi}}\), and \(\mathop{\dot{M}_{r\phi}}\) are the time derivatives of momenttensor components.
A differential position or coordinate vector dr may be written in the spherical polar coordinate system as
where (\(\hat{\varvec{r}},\varvec{ \hat{\theta }},\varvec{ \hat{\phi }}\)) are the spherical polar coordinate unit base vectors (Fig. 2), and ds _{ r }, ds _{ θ }, and ds _{ ϕ } are line elements of the arc lengths s _{ r }, s _{ θ }, and s _{ ϕ }, respectively.
Changing the coordinate variables as
where z is depth, R _{0} is the Earth’s radius, and θ′ is latitude, the differential coordinate vector may be represented as
Relabeling (\(\varvec{\hat{z},\hat{\theta }}^{{\prime }} ,\hat{\varvec{\phi }}\)), ds _{ z }, \({\text{d}}s_{{\theta^{{\prime }} }}\), and ds _{ ϕ } as (e _{ z }, \({\mathbf{e}}_{{x^{{\prime }} }}\), \({\mathbf{e}}_{{y^{{\prime }} }}\)), dz, dx′, and dy′, respectively,
Note that the unit vectors e _{ z }, \({\mathbf{e}}_{{x^{{\prime }} }}\), and \({\mathbf{e}}_{{y^{{\prime }} }}\) are pointing downward, north, and east, respectively, in a local Cartesian coordinate system, which vary in direction as the angles θ′ and ϕ. In Fig. 2, x′ and y′curves are identical to the latitude and longitude lines passing a position (r, θ, ϕ), respectively. We here call the coordinates (z, x′, y′) quasiCartesian coordinates. They all have dimension of length. Equation (7) shows that for lateral derivatives \(\partial \bullet /r\partial \theta^{\prime } = \partial \bullet /\partial x^{\prime }\) and \(\partial \bullet /r\cos \theta^{\prime } \partial \phi = \partial \bullet /\partial y^{\prime }\).
There are simple relations (onetoone correspondence) among the components of the particle velocity vector and the stress tensor in the original spherical polar coordinate and the quasiCartesian coordinate systems. Using these relations and the differential coordinates in Eqs. (6) and (7), we can get the form of the elastodynamic equation in quasiCartesian coordinates:
In derivation of these equations, we have carried out simple variable changes such as Eq. (4) and relabeling some coordinaterelated quantities but have not introduced any approximation. The quasiCartesian form, a set of Eqs. (8), (9) and (7), is equivalent to the original spherical polar form and is exact. We found that each of quasiCartesian Eqs. (8) and (9) is formally identical to the corresponding 3D Cartesian coordinate equation except rdependence of d′, r and θ′dependence of dy x′, and the terms with 1/r which are enclosed by bluedashed lines and called “additional terms” hereafter. This fact means that we could implement a set of Eqs. (8), (9) and (7) by adding the additional terms in an available computer code for 3D local Cartesian form and using the lateral grid intervals Δx and Δy depending on the radius r (or depth z) and the latitude θ′ as
The finitedifference stencil (a set of weights) for each of the x′derivatives in the modified code then depends on the depth coordinate at the evaluation point, while the stencil for each of the y′derivatives varies with both coordinates of the depth and latitude. Note that the grid points for the FDM of the quasiCartesian form are identical to those of the spherical polar form.
In the staggeredgrid FDMs such as the FDTD, the derivatives of every field quantity are naturally defined halfway between the grid points where the field quantity is defined. Thus, terms on the righthand side of the elastodynamic equation [Eqs. (8) and (9)], including spatial derivatives, are consistently evaluated at the same grid position where the field quantity on the lefthand side is defined. However, this is not the case for terms that do not include spatial derivatives, i.e., the additional terms in Eqs. (8) and (9). In our FDTD scheme, these terms are evaluated using Lagrange interpolation of the same accuracy order as the corresponding finite difference (e.g., Fornberg 1988).
This scheme can be easily implemented using an available 3D Cartesian FDTD code of localscale modeling. We have implemented the scheme in an existing Cartesian FDTD code (Nakamura et al. 2012) which can calculate local seismic wave propagation in a 3D land–ocean unified model with sea layer, topography, and anelastic attenuation by the FDTD of fourthorder accuracy in space and secondorder accuracy in time. This code utilizes a unified scheme for fluid–solid boundary to correctly model land and seafloor topography (Takenaka et al. 2009) and employs perfectly matched layer (PML) absorbing boundaries to suppress the artificial reflections from the sides and bottom of the computational domain.
Modeling target for the quasiCartesian approach may be a limited area of the globe that is a section of a spherical shell. When the simulation area is at higher latitude, the grid spacing of the y′direction changes more rapidly. From a point view of the accuracy and the stability of the finitedifference approximation, it is better to keep the variation of spatial grid spacing smaller in the computational domain. A simple approach to do so is to move the target area to around the equator of the computational spherical coordinate system using geometrical rotations of the coordinate axes (e.g., Igel et al. 2002). We exploit the transformation from equatorial coordinates to ecliptic coordinates as one of the methods for this approach, which is widely used in astronomy.
We now describe how to apply the equatorialtoecliptic coordinate transformation to our aim. Before conducting the equatorialtoecliptic coordinate transformation, we choose a central position of the target area (“Original” in Fig. 3) as the reference point with latitude of ε and rotate the computational domain about the north pole so that the longitude of the reference point is shifted into 90°E (“Step 1” in Fig. 3). The equatorial coordinates of the reference point are then (longitude, latitude) = (90°, ε). We set the ecliptic through the reference point and employ the equatorialtoecliptic coordinate transformation which transforms the coordinates of the reference point to (90°, 0°) (“Step 2” in Fig. 3). Let us consider an arbitrary point on the computational area with equatorial coordinates (α, δ), which is indicated by a yellow circle called target point in Fig. 3. Applying the equatorialtoecliptic coordinate transformation to the target point gives its ecliptic coordinates (λ, β) as
The angle between the longitude lines through the target point in the equatorial and ecliptic coordinate systems, Δϕ, is then
This gives the angle change of azimuths in the equatorialtoecliptic coordinate transformation.
Numerical examples
In this section, we show two numerical examples of the quasiCartesian FDTD application. The first numerical example demonstrates seismic wave propagation in a Moon section model. Since the Moon is smaller than the Earth, the effects of the spherical geometry on wave propagation could be stronger than those of the Earth. We compare waveforms (synthetic seismograms) for a spherical Moon model and the corresponding flat model. The synthetic seismograms for the spherical Moon model are calculated by solving a set of Eqs. (8), (9) and (7) (hereafter called “quasiCartesian FDTD”), while those for the flat model are obtained by using the equations without the additional terms and assuming constant Δx and Δy (hereafter called “Cartesian FDTD”). In the first numerical example, we check the accuracy and validity of the quasiCartesian scheme by comparing the synthetic seismograms with those obtained by the spherical FDTD (Toyokuni and Takenaka 2012) which treats an axisymmetric global model including the center of the sphere in spherical polar coordinates. The second numerical example demonstrates a longrange (~1000 km) simulation of seismic wave propagation around Japan for a virtual subduction event. The structural model for this simulation is threedimensionally heterogeneous and has land and ocean topographies and seawater. In the second numerical example, we also compare waveforms for a 3D heterogeneous spherical Earth model with those for the corresponding flat model.
Simulation for a Moon section
Figure 4 shows the model setting. The computational domain is a spherical cubic section of 45° × 45° × 450 km, which is discretized with cells of 0.06172° × 0.06172° × 1 km for the quasiCartesian FDTD. The spacing of 0.06172° corresponds to 1.87 km at the Moon surface. The time step is 0.0125 s. The source (point source) and receivers are located at a depth of 100 km. The datum (acquisition surface) is indicated by light blue surface in Fig. 4. Note that subsurface receivers could be more affected by the spherical geometry than surface receivers because of the shorter radius. The employed seismic structure is spherically symmetric, i.e., 1D, which is shown in Fig. 5. We constructed this structure based on Garcia et al. (2011). This model is perfectly elastic without anelastic attenuation. We put an axisymmetric source of M _{ zz } = 10^{18} Nm whose source time function is a bellshaped pulse with width of 15 s.
Figure 6 shows the vertical and radial components of calculated waveforms at seven receivers on AA′ line shown in Fig. 4. The transverse components are not displayed as they are all zero because of the axisymmetric source. The waveforms are bandpassfiltered in the period range 25–100 s. The traces of the quasiCartesian and the spherical FDTDs (spherical models) are almost identical, which illustrates the accuracy and validity of the present quasiCartesian scheme. The traces of the Cartesian FDTD (flat model) are different from those of the other FDTDs (spherical models). The discrepancy becomes larger as the epicentral distance increases. Figure 7 shows snapshots of vertical particle velocity field over the vertical cross section including the source and the receivers shown in Fig. 6. It is found that the PML works well for the sides and the bottom of the computational domain.
Simulation for an Earth section around Japan
We show the target area (around Japan) for the next simulation in Fig. 8. This simulation area is much larger than that for usual local strongmotion simulations which can often be carried out with Cartesian FDTD. The right panel of Fig. 8 displays the computational area moved around the equator by the equatorialtoecliptic coordinate transformation. The computational domain is discretized with cells of 0.05° × 0.05° × 0.5 km. The total grid size is then 1601 × 2401 × 401. The time step is 0.025 s. The structural model is the 3D model that Nakamura et al. (2015) used for a strongmotion simulation of a real moderate earthquake, one of aftershocks of the 1995 Kobe earthquake, for land and oceanbottom seismic networks. This model incorporates realistic 3D heterogeneous velocity, density, and anelastic attenuation structures including a seawater layer, the seafloor and land surface topography, sediment layers and crust and upper mantle for the continental and the oceanic plates. Anelastic attenuation could be implemented by using a viscoelastic formulation (e.g., Blanch et al. 1995; JafarGandomi and Takenaka 2013) instead of the elastic one, and it is straightforward to apply the quasiCartesian scheme to the viscoelastic formulation. We here demonstrate a simulation of seismic wave propagation using a 3D structural model with anelastic attenuation.
We set a virtual point source in the Hyuganada area in the southwestern Japan for the simulation. We assume the source depth of 21 km, which is in the oceanic crust of the Philippine Sea slab (Fig. 9), and a thrusttype event (strike = N22.4°E, dip = 79°, rake = 76°) with seismic moment of 5.33 × 10^{17} Nm (M _{W} 5.8) whose source time function is a bellshaped pulse with width of 2.5 s. We output the seismograms at Fnet stations within the simulation area, which are operated by the National Research Institute for Earth Science and Disaster Resilience (Okada et al. 2004).
Figure 10 shows the synthetic seismograms at three stations (TKD, ABU, and TSK) marked in the right panel of Fig. 8, for the spherical Earth (quasiCartesian FDTD) and the corresponding flat Earth (Cartesian FDTD). The epicentral distances of the three stations are 103.5, 438.7, and 859.3 km for TKD, ABU, and TSK, respectively. The waveforms have been bandpassfiltered in the period range 10–20 s. At station TKD nearest to the epicenter, the traces of all components from the quasiCartesian FDTD are identical to those from the Cartesian FDTD within the line thickness, which means the contribution of the additional terms in Eqs. (8) and (9) is tiny at this distance. At stations ABU and TSK, the discrepancy between the traces for the 3D spherical and the 3D flat Earth models is visible, and at TSK in particular the difference between them looks clear around 200 s (S phase). Figure 11 displays the synthetic seismograms from all stations shown in Fig. 8 along to the epicentral distances. The difference between waveforms obtained by the two FDTDs is clear beyond about 500 km in this case.
Conclusions
We have described a simple and efficient finitedifference scheme, called the quasiCartesian FDTD to calculate seismic wave propagation for a subglobal (regional or larger) scale model represented by a partial spherical shell of a 3D heterogeneous global Earth structure. This scheme solves the elastodynamic equation in the quasiCartesian coordinate system similar to a local Cartesian system. We have demonstrated accuracy and validity of the present scheme and the feasibility to apply it to real large simulations via two numerical examples. The present scheme can be easily implemented using a 3D Cartesian FDTD code of localscale modeling by changing a very small part of the code. The quasiCartesian approach may be able to open a window for multiscale modeling ranging from global to local scales.
Abbreviations
 3D:

threedimensional
 FDM:

finitedifference method
 FDTD:

(staggeredgrid) finitedifference method in time domain
 PML:

perfectly matched layer(s)
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Authors’ contributions
HT developed the quasiCartesian finitedifference scheme for subglobal seismic wave propagation modeling. HT, MK, GT, and TN participated in developing the simulation codes and carrying out the numerical tests. TO participated in the study design. HT and MK drafted the manuscript. All authors read and approved the final manuscript.
Acknowledgements
We are grateful to the editor, Prof. Kimiyuki Asano, and the two anonymous reviewers who provided us with constructive comments and suggestions that have improved this paper. We used the Generic Mapping Tools (Wessel, P. & Smith, W. H. F. EOS Trans. AGU 79, 579, 1998) for drawing some figures. This study is partially supported by KAKENHI (26282105) and JHPCN (jh160029NAH).
Competing interests
The authors declare that they have no competing interests.
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Takenaka, H., Komatsu, M., Toyokuni, G. et al. QuasiCartesian finitedifference computation of seismic wave propagation for a threedimensional subglobal model. Earth Planets Space 69, 67 (2017). https://doi.org/10.1186/s4062301706511
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Keywords
 Finitedifference method
 FDTD
 Seismic wave propagation
 Subglobal model
 Regional scale
 QuasiCartesian coordinates