Bulletin of the Seismological Society of America, Vol. 97, No. 5, pp. 1570–1586, October 2007, doi: 10.1785/0120060253
Accurate Calculation of Fault-Rupture Models Using the High-Order Discontinuous Galerkin Method on Tetrahedral Meshes by Martin Ka¨ser, P. Martin Mai, and Michael Dumbser
Abstract We present a new method for near-source ground-motion calculations due to earthquake rupture on potentially geometrically complex faults. Following the recently introduced Discontinuous Galerkin approach with local time stepping on tetrahedral meshes, we use piecewise polynomial approximations of the unknown variables inside each element and achieve the same approximation order in time and space due to the new ADER time integration scheme that uses Arbitrary high-order DERivatives. We show how an external source term and its heterogeneous properties in space and time, given by a fine discretization of an extended rupture surface, can be included in much coarser tetrahedral meshes due to the subcell resolution of the high-order polynomial representation. Hereby, the rupture surface is represented very generally as a point cloud of the center locations of individual subfaults at which each polynomial test function is evaluated exactly inside an element and the space– time integration of the source term is accurately computed at each timestep. Besides the incorporation of complex source kinematics we also present the effects of model boundaries that can degrade the accuracy of seismograms due to weak artificial reflections. We propose an extended computational domain of a coarsely meshed buffer region and show that our scheme using the local timestepping completely avoids such boundary problems with only slightly increasing the computational cost. We validate the new approach against different test cases, comparing our results with analytic, quasianalytic, and a series of reference solutions. Our work shows that adding the functionality of accurately treating finite source-rupture models into the general framework of the ADER-Discontinuous Galerkin approach is an important contribution to modeling realistic earthquake scenarios, allowing the efficient inclusion of heterogeneous source kinematics and complex rupture-surface geometries in near-source ground-motion simulations. Introduction The prediction of the intensity and variability of nearsource strong ground motions for future large earthquakes hinges on several key factors: (1) an appropriate earthquake source characterization that captures the spatiotemporal variation of the rupture process; (2) the accurate calculation of wave propagation through a three-dimensional complex Earth structure; (3) the correct treatment of site effects in the shallow near-surface region underneath the observation site. This study presents a new numerical method to accurately treat near-source wave propagation in arbitrarily complex media for heterogeneous source rupture on potentially geometrically complex or segmented faults. The favorable numerical accuracy and the large flexibility of the method make it an ideal approach for large-scale ground-motion simulations if the source geometry and crustal structure exhibit a large degree of spatial complexity. In recent years, several researchers developed numerical
methods for near-fault wave propagation in three-dimensional media using finite differences (Olsen et al., 1995; Graves, 1996; Pitarka, 1999) or finite-element techniques (Bielak et al., 2005). These techniques are mainly used to simulate the ground motions for potential ruptures on specific faults (Olsen and Archuleta, 1996; Graves, 1998; Olsen, 2001), to understand the details of strong-motion generation for past events (Pitarka et al., 1998; Graves and Wald, 2004), or to assess the shaking hazard due to future large earthquakes in a given region (Benites and Olsen, 2005). These studies all have in common that they compute the seismic wave field for a single, or at best a small number of earthquake source models, given as a kinematic characterization of the rupture process. Only a few studies actually consider a dynamically derived rupture model for ground-motion simulation (Aochi and Fukuyama, 2002; Aochi et al., 2002; Olsen et al., 2006), or even calculate the near-source wave
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Accurate Calculation of Fault-Rupture Models Using the High-Order Discontinuous Galerkin Method on Tetrahedral Meshes
field for a larger number of dynamic source models (Aochi and Douglas, 2006). These works have demonstrated the increased need for efficient and flexible methods to compute near-source seismograms for arbitrarily complex fault geometries and crustal structures, incorporating also the spatiotemporal complexity of earthquake sources. Images of past earthquakes (e.g., Beroza and Spudich, 1988; Wald and Heaton, 1994; Salichon et al., 2004) show that slip on the fault planes is heterogeneous at all scales (Mai and Beroza, 2002; Mai, 2004). Moreover, the temporal rupture evolution can also be quite complicated, showing variations in the slip-velocity function, hence the slip duration (rise time), and in rupture speed, which can even increase to supershear rupture velocity on parts of the fault (Archuleta, 1984; Bouchon and Vallee, 2003). This spatiotemporal complexity of the earthquake source contributes strongly to the large variability in waveform characteristics and shaking intensities in near-source ground motions. Wave propagation through complex threedimensional geological structures, wave scattering in inhomogeneous media, and local site effects will add even more variability to the ground motions. The source effects will be even more accentuated if additional geometric complexity in the fault is considered, which strongly affects the dynamic rupture process (Aochi and Fukuyama, 2002; Aochi et al., 2002; Oglesby et al., 2004); hence, the seismic radiation emitted from the fault. In this study we use a recently introduced method in computational seismology for addressing near-source ground-motion simulation due to the heterogeneous rupture process on a fault embedded in a three-dimensional unstructured mesh. The ADER discontinuous Galerkin (ADER-DG) approach (Dumbser and Ka¨ser, 2006; Ka¨ser et al., 2007; de la Puente et al., 2007) achieves an arbitrary high order of accuracy solving seismic-wave propagation problems on tetrahedral meshes accounting for heterogeneous, anisotropic, and viscoelastic material properties. Recently, Dumbser et al. (2007a) introduced an alternative, the ADER-finitevolume method (ADER-FV), which also can handle geometrically complex subsurface models due to the high flexibility of tetrahedral meshes. However, the ADER-DG approach can make particular use of a local timestepping technique, where each element can be updated using its own optimal timestep based on a local stability criterion (Dumbser et al., 2007b). One advantage of this new approach is the accurate consideration of very detailed source models, consisting of numerous small subfaults, each of which is characterized by its own source-rupture properties, hence allowing for spatiotemporally variable rupture process. Note that the discretization of the computational domain has a much larger mesh spacing than the source-model description, but no interpolation or upscaling of the rupture-model information is necessary because the wave generation process can be incorporated by the accurate treatment of external source terms (Ka¨ser and Dumbser, 2006). The high accuracy, especially of the ADER-DG method, is due to the piecewise high-order
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polynomial approximation providing subcell resolution. An enormous advantage is that the mesh does not have to be aligned with the rupture surface. Instead, the rupture surface can have any arbitrary shape or can be divided into different independent segments as the rupture surface is described by a a large number of independent points that represent the centers of individual subfaults. The arbitrary orientation of the rupture surface and the locations of the subfaults with respect to the tetrahedral mesh allow for extremely high flexibility in the process of mesh generation as additional geometrical constraints through the rupture surface are avoided. In the following we summarize the main ideas of the ADER-DG method, commenting briefly also on the ADERFV approach, and how earthquake ruptures are incorporated as external source terms considering both point sources or finite sources. The method is then applied to and validated against well-documented case studies for uniform-slip models on extended faults. We conclude by computing nearsource seismograms for a simulated heterogeneous rupture model, showing the applicability of the ADER-DG technique on ground-motion scenarios.
The Numerical Method To solve three-dimensional seismic-wave propagation problems, we use the velocity-stress formulation of the seismic-wave equation leading to a linear hyperbolic system, including an external source term (e.g., moments or body forces) of the compact form Qp Qq Qq Qq Ⳮ Apq Ⳮ Bpq Ⳮ Cpq ⳱ Sp , t x y z
(1)
where Q is the vector of the p unknown variables, that is, Q ⳱ (rxx, ryy, rzz, rxy, ryz, rxz, u, v, w)T denoting the stresses and particle velocities. The matrices Apq, Bpq, and Cpq are the space- and material-dependent Jacobian matrices and Sp is the external source term that does not depend on the solution Q. Note the classical tensor notation, which implies summation over each index that appears twice. In the following, we concentrate on the numerical treatment of the source term, as the mathematical derivation of the fully discrete ADER-DG scheme or the ADER-FV scheme would go beyond the scope of this article. However, we point out that the ADER-DG and ADER-FV methods are both one-step explicit schemes that do not require intermediate stages (e.g., as known from Runge–Kutta methods). For a detailed desription of the ADER-DG method or the ADERFV method for seismic-wave propagation the reader is referred to previous work (Ka¨ser and Dumbser, 2006; Dumbser and Ka¨ser, 2006; or Dumbser et al., 2007a). We discretize the computational domain X 僆 R3 by a conforming tetrahedral mesh of elements T(m) uniquely addressed by the index (m). In the ADER-DG approach the numerical solution of equation (1) is approximated inside
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each tetrahedron T(m) by polynomials given by a linear combination of space-dependent but time-independent polynomial basis functions Ul(n, g, f) of degree N (Cockburn et al., 2000) and with only time-dependent degrees of freedom ˆ (m) Q pl (t): ˆ (m) (Q(m) h )p (n, g, f, t) ⳱ Qpl (t)Ul(n, g, f ),
(2)
where n, g, and f are the coordinates in a reference element TE, as shown in detail by Dumbser and Ka¨ser (2006). Qh denotes the numerical approximation, the index p stands for the number of unknowns in the vector Q, and l indicates the lth basis function. Multiplying equation (1) by a test function Uk and integrating over a tetrahedral element T(m) gives
冮
Uk
T (m)
Qp dV Ⳮ t
冮
冢
Qq Qq Ⳮ Bpq x y Qq dV ⳱ (m)Uk Sp dV. z T
Uk Apq
T (m)
Ⳮ Cpq
冣
冮
t nⳭ1
冮 冮 tn
T
Point Sources In many geophysical applications and classical validation test cases with analytical solutions the source term in equation (4) is represented by a point source at a location r x s, that is, a Dirac Delta distribution in space and a given source time function sp(t) such that Sp( rx, t) ⳱ sp(t) • d( rx ⳮ rx s).
(3)
In the ADER-DG approach the number L of degrees of freedom for each variable in Q depends on the desired poly1 nomial degree N and is given through L ⳱ 㛭6 (N Ⳮ 1) (N Ⳮ 2) (N Ⳮ 3), with l ⳱ 0, . . ., L ⳮ 1. However, in the ADER-FV approach there is only one degree of freedom per variable, that is, the cell average value. Here, the high-order polynomial approximation of the unknown solution is achieved by reconstructing a polynomial using the cell average values in a stencil around the element T(m). Furthermore, in the ADER-FV approach equation (1) is integrated directly over the element T(m), without multiplication by a test function Uk. In the preceding ADER-DG framework this is formally given through the multiplication of a constant test function Uk ⳱ 1. Omitting the details of the derivation of the ADER-DG or ADER-FV schemes, in both cases the integration of the right-hand side of equation (3) in time over a timestep Dt ⳱ tnⳭ1 ⳮ tn and considering only the external source term that depends on space rx ⳱ (x, y, z) and time t results in Ip(m) ⳱
source terms in three space dimensions, describing, for example, earthquake rupture on extended faults. Because the treatment of large earthquakes as a single-point source is not appropriate, an extended rupture fault area is used that can be divided into several smaller subfaults for which the pointsource approximation is valid (Pujol, 2003). The effects of the large rupture fault on the seismically radiated wave field is then obtained by summation of the contributions of all individual subfaults.
(5)
In the ADER-DG and ADER-FV frameworks all tetrahedral elements of the physical space with coordinates rx are transformed into the canonical reference element TE with coorr dinates n ⳱ (n, g, f). The source term integral of equation (4) can be then rewritten as t nⳭ1
Ip(m)
⳱
冮 冮 U s (t) • d(n ⳮ n )dV dt. tn
r
TE
k p
r
s
(6)
Because the order of integration can be changed, the source time function sp(t) does not depend on space, and using the properties of the Dirac Delta distribution, we compute the integral of equation (6) as
冦
t nⳭ1
r
Uk(ns) •
Ip(m) ⳱
冮 s (t)dt, p
r
if ns 僆 TE
tn
0,
(7) otherwise.
r
U S ( rx, t)dV dt. (m) k p
(4)
The integral in equation (4) has to be calculated in each timestep to correctly update the unknown variables in Q under the influence of the source term. In the following, we will mainly consider the ADER-DG framework and remark that the ADER-FV case is obtained by simply setting Uk ⳱ 1.
Note that ns 僆 TE is only fulfilled if rx s 僆 T (m) such that only a physical point-source location rx s inside the tetrahedral element T(m) contributes to the integral value Ip(m). The remaining time integral in equation (7) can be computed using one-dimensional Gaussian integration of the appropriate order leading to r
Ip(m) ⳱ Uk(ns) •
nG
兺 xj • sp(sj),
(8)
j⳱1
Source Terms The details of the discretization of continuous external source terms in the framework of ADER-DG methods are outlined in previous work by Ka¨ser and Dumbser (2006) for two space dimensions. Here we focus on the treatment of
where nG is the number of Gaussian integration points, xj is the weight and sj 僆 [tn, tnⳭ1] are the corresponding point locations of the Gaussian quadrature rule, which is exact up to polynomial degree 2nG — 1. In our implementation the source time function sp(t) for each subfault can be specified
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Accurate Calculation of Fault-Rupture Models Using the High-Order Discontinuous Galerkin Method on Tetrahedral Meshes
as an analytic function or as a discrete time series. Note that in the ADER-DG approach the integral in equation (8) can be computed up to a desired order determined by the degree of the polynomial basis function Uk, which is evaluated exactly in space at the source location rx s at any point of the computational domain. In the ADER-FV approach, where Uk ⳱ 1, the source term is approximated by its average inside a tetrahedral element providing no subcell resolution. However, we emphasize that neither the point-source locations nor a receiver locations in both approaches have to coincide with a vertex of the tetrahedral mesh, as the numerical solution is represented by a polynomial within each element.
Extended Source Models Extended source models characterize the earthquake rupture occurring on a fault surface, where the material on one side of the surface is moving relative to the other. As shown by Aki and Richards (2002) the effect of this rupture process on a fault plane can be described in the far field as a point-source double couple (DC), mathematically expressed by the source-moment tensor (Ben-Menahem and Singh, 1981), that is, a symmetric 3 ⳯ 3 matrix MDC pq . The strength of an earthquake is determined by the scalar seismic moment, M0 ⳱ l • A • d, where l is the rigidity of the surrounding rock, A is the area of the fault plane, and d is the average displacement on the fault. The time-dependent seismic-moment rate m(t) incorporates the source time function of the slip rate s(t) and reads as m(t) ⳱ l • A(t) • s(t). Assuming a rupture plane of area A, described by a single DC at a point location rx s in the center of the rupture plane, the time-dependent symmetric seismic-moment rate tensor is then given as MDC pq • l • A • s(t). To account for an arbitrary orientation of double couple, that is, the rupture plane, in three-dimensional space, the symmetric source-moment tensor MDC pq can be rotated according to the fault parameters strike , dip d, and rake k. Because the rotation depends on the choice of the initial double couple MDC pq , the definition of the angles , d, and k and the selected coordinate system (Ben-Menahem and Singh, 1981; Pujol, 2003), the following conventions are imporatant: the x axis is pointing positive south, the y axis is pointing positive east, and the z axis is pointing positive up. Furthermore, the strike 0 ⱕ ⱕ 2p is the azimuth of the intersection of the fault plane with the horizontal xy plane and is measured clockwise from north. The dip 0 ⱕ d ⱕ p/2 is the angle between the horizontal plane and the fault plane and is measured down from the horizontal plane in a vertical plane perpendicular to the strike direction. The rake 0 ⱕ k ⱕ 2p is the angle between the strike direction and the slip direction and is measured in the fault plane counterclockwise from the strike direction. The rake determines the direction of movement of the hanging wall relative to the foot wall on the rupture plane. To avoid ambiguities in a vertical fault plane, that is, d ⳱ p/2,
the hanging wall is defined as the one on the right when looking in strike direction. For our particular choice of the double couple in the y plane MDC pq ⳱
冢
冣 冢
Mxx Mxy Mxz Mxy Myy Myz Mxz Myz Mzz
冣
0 0 1 0 0 0 1 0 0
⳱
,
(9)
we get the final time-dependent seismic-moment tensor for slip in arbitrary orientation as T DC ˜ pq (t) ⳱ Rpr M Mrs Rsq • l • A • s(t) M1(t) M4(t) M6(t) ⳱ M4(t) M2(t) M5(t) M6(t) M5(t) M3(t)
冢
冣
,
(10)
with the rotation matrix Rpq ⳱
冢
冣
R11 R12 R13 R21 R22 R23 R31 R32 R33
,
(11)
and its transposed RTpq, where the entries are given as R11 R12 R13 R21 R22 R23 R31 R32 R33
⳱ ⳱ ⳱ ⳱ ⳱ ⳱ ⳱ ⳱ ⳱
cos(k) cos() Ⳮ sin(k) cos(d) sin(), ⳮcos(k) sin() Ⳮ sin(k) cos(d) cos(), sin(k) sin(d), ⳮsin(k) cos() Ⳮ cos(k) cos(d) sin(), sin(k) sin() Ⳮ cos(k) cos(d) cos(), ⳮcos(k) sin(d), sin(d) sin(), sin(d) cos(), cos(d).
(12)
˜ (t) of equation The rotated seismic-moment tensor M (10) is then applied as an external source term Sp in the hyperbolic system of the velocity-stress formulation in equation (1). Because the seismic-moment tensor source only acts on the stresses in the first six equations of the system in equation (1), the source time function sp(t) in equation (5) is given as sp(t) ⳱ Mp(t),
for p ⳱ 1,K, 6;
(13)
where Mp(t) are the entries of the seismic-moment tensor as defined in equation (10) and sp(t) ⳱ 0 for p ⱖ 7. The preceding discussion is strictly valid for a point source, but it can be extended to finite-source rupture models by dividing the fault plane into a number of subfaults at locations rx s(i), each of which is characterized by its own seismic-moment tensor. Each subfault is then representative ˜ (i)(t) with for an area A(i) and has a seismic-moment tensor M (i) (i) (i) its local fault parameters , d , and k and its local
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seismic-moment rate m(i)(t) and source time function s(i)(t). The superposition of the contributions of the subfaults finally accounts for the effect of the large, extended, and possibly heterogeneous slip or slip-rate distribution across the entire fault plane. In the ADER-DG approach, we numerically accomplish this superposition by the summation of all subfault r (i) contributions s(i) p (t), whose locations x s are inside a tetrar (i) (m) hedral element T , that is, if ns 僆 TE is fulfilled. Therefore, using equation (13) the total source term integral in equation (7) is then given by ns
Ip(m)
⳱
s(n, g, t) ⳱
t nⳭ1
兺 Uk(ns ) 冮 M(i)p (t) dt, r (i)
with its origin at the top western corner of the fault plane, the hypocenter is located at (nH, gH) ⳱ (1, 2) km. The fault plane is divided into 80 ⳯ 40 subfaults, each representing a DC source. The shape and amplitude of the slip-rate function s(n, g, t) is constant over the fault surface, and the onset time s is proportional to the distance from the hypocenter, with the rupture front propagating at a constant speed velocity vrup ⳱ 3000 m/sec. The explicit form of the slip-rate function is given as:
(14)
i⳱1
tn
where ns is the number of subfaults located in the tetrahedral element T(m).
Numerical Examples and Validation LOH.2
To validate the results of the proposed ADER-DG and ADER-FV methods and to analyze their accuracy and computational cost we present the solutions obtained for a welldefined three-dimensional test problem proposed by the Pacific Earthquake Engineering Research Center (PEER) (Day et al., 2001). This study tests and validates several methods for numerical modeling of earthquake ground motion and calibrates the results to analytical solutions. We therefore compare our numerical results with the ground motions computed with these well-established codes. The different methods are denoted by four-character abbreviations indicating the respective institutions: UCBL (D. Dreger and S. Larsen, University of California, Berkeley/Lawrence Livermore National Laboratory); UCSB (K. Olsen, University of California, Santa Barbara); WCC1 (R. Graves, URS Corporation); WCC2 (A. Pitarka, URS Corporation); CMUN (J. Bielak, Carnegie-Mellon University). The first four codes use a finite-differences method on uniform, structured grids with staggered locations of the velocity-stress components and fourth-order accuracy in space. The CMUN code uses piecewise linear interpolation on a unstructured tetrahedral mesh in a finite-element method. The setup of the test problem LOH.2 (layer over halfspace) is shown in Figure 1a, displaying for clarity only one of four symmetrical quarters of the computational domain X ⳱ [ⳮ15, 15] ⳯ [ⳮ15, 15] ⳯ [ⳮ17, 0] km. The material parameters for the two-layer model (top layer with thickness 1 km over half-space) are given in Table 1. The seismic source is a finite-fault rupture in the vertical plane x ⳱ 0, representing a right-lateral strike-slip earthquake with strike ⳱ 90, dip d ⳱ 90, and rake k ⳱ 180. The fault size measures 8 ⳯ 4 km, the hypocenter is located at (0, 1, ⳮ4) km as displayed in Figure 1a. In the local fault-plane coordinate system (n, g) aligned with strike and dip direction and
冢 冣
s s H(s), 2 exp ⳮ T T
(15)
where H is the Heaviside step function and T ⳱ 0.1 sec is a smoothing factor controlling the frequency content and the amplitude of the source time function. The total seismic moment of the rupture source is given by M0 ⳱ 1018 N m. The ground-motion signals are computed at 10 receivers on the free surface as indicated in Figure 1a for a simulation time of 9 sec. The receiver locations are (xi, yi, zi) ⳱ (i • 600, i • 800, 0) m, for i ⳱ 1, . . ., 10. Here, we discretize the extended computational domain Xe ⳱ [ⳮ30, 30] ⳯ [ⳮ30, 30] ⳯ [ⳮ30, 0] km, which is much larger than the suggested domain X to completely avoid boundary effects. The unstructured, tetrahedral mesh shown in Figure 2 consists of 5.8 ⳯ 105 elements and is generated in a problemadapted manner: in the zone of interest the waves traveling from the fault to the receivers pass through tetrahedral elements with an average edge length of 350 m, whereas in other regions the mesh is coarser, with average edge lengths of 5000 m to reduce the number of elements and hence computational cost. Note that in the ADER-DG approach, neither the subfault locations nor the receiver locations have to coincide with nodes of the tetrahedral mesh. However, the mesh respects the material interface between medium 1 and medium 2 as the faces of the tetrahedral elements are aligned with the interface (Figs. 1a and 2). Analogous to the PEER study (Day et al., 2001), we show the radial, transversal, and vertical components of the seismic velocity field computed at receiver 10 at (x10, y10, z10) ⳱ (6, 8, 0) km. Note that for all seismograms shown, the raw seismograms are converted by convolution with a Gaussian moment rate function of spread 0.06 sec and time shift 0.24 sec as described by Day et al. (2001). Additionally, each plot gives the relative seismogram misfit nt
E⳱
nt
兺 (sj ⳮ saj )2冫j⳱1 兺 (sja)2, j⳱1
(16)
where nt is number of time samples of the seismogram, sj is the numerical value of the particular seismogram at sample j, and sja is the corresponding analytical value. As shown by Kristekova´ et al. (2006) this misfit criterion only quantifies the misfit between two signals in the case of a pure amplitude modification of the whole signal. However, we use this sim-
Accurate Calculation of Fault-Rupture Models Using the High-Order Discontinuous Galerkin Method on Tetrahedral Meshes
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Figure 1. (a) One of four symmetric quarters for the LOH.2 test case, where the rupture plane of 4 ⳯ 8 km is aligned with the y axis with the hypocenter at 4 km depth. (b) One of four symmetric quarters for the LOH.4 test case, where the rupture plane of 6 ⳯ 6 km is oblique to all coordinate axes with the hypocenter at 6 km depth. In both cases a layer of 1 km (medium 1) is lying on top of another layer (medium 2).
Table 1 Elastic Material Parameters for the LOH.2 and LOH.4 Test Cases
Medium 1 Medium 2
cp (m/sec)
cs (m/sec)
q (kg/m3)
4000 6000
2000 3464
2600 2700
k (GPa)
l (GPa)
20.8 32.4
10.4 32.4
ple criterion given by equation (16) because it is not crucial in this case to assess the accuracy of the proposed method. First we present our results obtained by ADER-DG schemes of order O2, O3, and O4 in the extended domain Xe applying the local timestepping (LTS) algorithm introduced in previous work (Dumbser et al., 2007b). As shown in Figure 3a, the ADER-DG O2 solution does not reproduce the analytical solution with satisfactory accuracy. Increasing the spatial and temporal order the ADER-DG O3 in Figure 3b fits the analytical solution much better and leads to a much smaller relative seismogram misfit E. The seismograms of the ADER-DG O4 scheme in Figure 3c coincide with the analytical solution almost perfectly. Considering the accuracy of the method with respect to the highest frequency resolved, we remark that the necessary investigations on the base of convergence tests have been carried out in previous work (Dumbser and Ka¨ser, 2006; Ka¨ser et al., 2007). In general, it is difficult if not impossible to give a rule of thumb of how many elements per shortest wavelength have to be used, as the accuracy depends on several factors, such as the approximation order, the particular measure of the error, the acceptable error level, the
propagation distance, and so on. However, as a very rough estimate for the presented LOH.2 test case with a maximumfrequency contents of 3 Hz, we state that the ADER-DG O4 scheme provides satisfying results as shown in Figure 3c by using less than two elements per shortest wavelength and where the waves have propagated about 15 wavelengths from the source to the receiver. Furthermore, in Figure 3d we present the solution obtained by the ADER-DG O4 scheme on a reduced mesh, which is identical to the original one used for the results in Figure 3c, but with the domain extension of the mesh removed and therefore only discretizing X instead of Xe. In this case, the numerical solution begins to deviate from the analytical one after 5.5 sec due to boundary effects that cause low-amplitude reflected waves. Note that the LTS method allows us to update each element with its own optimal timestep length given from a local stability condition. In contrast to typically used global timestepping (GTS) schemes, where each element has to use the smallest timestep length appearing in the entire mesh, the LTS scheme is computationally much more efficient. In Figure 4a and b we compare the results of the ADERDG O4 scheme in the extended domain Xe with the ADERFV O4 scheme, which uses GTS by default due to the necessary polynomial reconstruction in each timestep. First, we observe that the ADER-DG scheme with GTS in Figure 4a provides the same results as with LTS in Figure 3c, however, at a much higher computational cost. Furthermore, we clearly see that the FV approach is less accurate even though the mesh used is much finer as it contains 1.8 ⳯ 105
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Figure 2. Cut into the discretization of the LOH.2 and LOH.4 models to visualize the problem-adapted tetrahedral mesh, which is refined in the zone of interest from the source to the receiver line to a depth of 6 km. Note the extended boundaries of the mesh to 30 km to avoid boundary effects.
elements with average tetrahedral edge lengths of 100 m in the zone of interest up to 5000 m at the model boundaries. This might be because (1) the subcell resolution in the wavepropagation algorithm of ADER-FV schemes is less than for an ADER-DG scheme of equal order and (2) the spatial approximation of an external source term is less accurate in the ADER-FV approach as shown above. In Figure 5 we plot again our result of the LTS ADERDG O4 scheme in the extended domain Xe in comparison with the results of the reference codes UCBL, UCSB, WCC1, WCC2, and CMUN. The reference solutions shown in Figure 5b–f reproduce the analytical solution with various discrepancies. However, similar to previous comparisons with analytical solutions for similar test cases (LOH.1, LOH.3) with point sources (Dumbser and Ka¨ser, 2006; Ka¨ser et al., 2007), we can clearly show that the ADER-DG O4 scheme produces more accurate results than all reference codes. As mentioned previously we do not use any absorbing boundaries, but simply numerical fluxes to mimic absorbing boundaries as explained in detail in previous work (Ka¨ser and Dumbser, 2006; Dumbser and Ka¨ser, 2006). Comparison of the result of the CMUN code in Figure 5d with ours in Figure 3d suggests that their approach uses a similar treatment of the boundary effects. Trying to compare the CPU times turned out to be a difficult task because all reference codes were run on dif-
ferent machines with different levels of parallelization and no CPU-time data were available in the final PEER report (Day et al., 2001). However, to give a rough indication of the computational cost for our ADER-DG and ADER-FV simulations, we compare the total number of mesh elements Ne, the total number of degrees of freedom Nd to be kept in storage, and the required CPU time in seconds for all methods in Table 2. Our computations were performed on the SGI Altix supercomputer of the Leibniz Rechenzentrum in Mu¨nchen, Germany, using 128 Intel Itanium2 Madison 9M processors with 4 GB of RAM each. Concerning the computational storage requirements, the finite-difference reference codes were using a regular grid of mesh width 100 m leading to 1.5 • 107 grid points to discretize the computational domain X. The unstructured tetrahedral meshes in the ADER-DG can be chosen much coarser, in that even the smallest tetrahedral elements of average edge lengths 350 m include a number of subfaults whose spacing is only 100 ⳯ 100 m to discretize the finitefault plane. To completely avoid boundary effects using the extended domain Xe the storage requirements have to be increased by 23%, whereas the computational time only increases by 16% (Table 2), even though the volume of the computational domain Xe is more than seven times larger than that of X.
Accurate Calculation of Fault-Rupture Models Using the High-Order Discontinuous Galerkin Method on Tetrahedral Meshes
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Figure 3.
Comparison of the radial, transverse, and vertical velocity components for the LOH.2 test case on receiver 10. The numerical solution of the ADER-DG O2 (a), ADER-DG O3 (b), and ADER-DG O4 (c) scheme on the extended mesh (thin solid) with LTS is plotted against the analytical solution (thick dashed). (d) The solution of an ADER-DG O4 scheme (thin solid) on the reduced mesh, without the boundary layer, is plotted against the analytical solution (thick dashed) to demonstrate the boundary effects. The relative seismogram misfit E is given for each trace.
LOH.4
In this section we present the results obtained for a similar three-dimensional benchmark, published in a PEER report (Day et al., 2003). The setup of this test problem LOH.4 is shown in Figure 1b and is identical to the LOH.2 case, with the only difference that the orientation of the rupture plane is oblique with respect to all coordinate axes. In this model, the extended source is a thrust fault with strike ⳱ 115, dip d ⳱ 40, and rake k ⳱ 70, the fault size is 6 ⳯ 6 km and the hypocenter is located at the center of the bottom line of the fault at position (0, 0, ⳮ6) km (see Fig. 1b). Using
the local fault-plane coordinate system (f, g) aligned with strike and dip direction and with its origin at the top northwestern corner of the fault, the hypocenter is located at (fH, gH) ⳱ (3, 6) km. The fault plane is divided into 50 ⳯ 50 subfaults of area A ⳱ 1.44 • 104 m2 each. The source time function for the slip rate is again given by equation (15) and the rupture velocity is constant vrup ⳱ 3000 m/sec. Because no analytical solution is provided for the LOH.4 case, we compare our LTS ADER-DG O4 simulations, obtained on the same mesh as used for Xe in the LOH.2 case, with those of the four reference codes UCBL, UCSB, WCC2,
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Figure 4.
Comparison of the radial, transverse, and vertical velocity components for the LOH.2 test case on receiver 10. The analytical solution (thick dashed) is plotted against the numerical one obtained by the ADER-DG O4 scheme (thin solid) with global time stepping (GTS) on the extended mesh (a) and the ADER-FV O4 scheme (thin solid) on the FV fine mesh (b). The relative seismogram misfit E is given for each trace.
and CMUN. As shown in Figure 6a–d our results are close to the reference solutions, which validates the correct treatment of finite-source models in an arbitrary orientation in three-dimensional space by the proposed ADER-DG approach. However, the reference code UCSB seems to produce a phase shift in comparison with our solution, which was not observed for the LOH.2 case. We assume that this could be due to their interpolation technique to treat a fault plane with an oblique orientation with respect to their staggered regular FD grid. Heterogeneous Slip Distribution in a Layered Medium (HSLM) In the previous test cases the source kinematics was described by a circular crack propagating over a rectangular fault with a constant final displacement over the entire fault plane. We now consider a scenario earthquake that resembles in size, fault-plane geometry, and strong-motion station distribution the Tottori earthquake in Japan in 2000 (Mw 6.8). Although a recent study indicates that this earthquake has ruptured several small segments (Fukuyama et al., 2003), source inversions based on a single-fault approximation provide reasonable fits to the observed waveforms (Semmane et al., 2005). To validate the treatment of extended sources in the ADER-DG method, we restrict our calculations to a planar fault embedded in an unstructured tetrahedral mesh. The planar-fault assumption allows us to compare the ADER-DG synthetics with full-wave-field seismograms calculated with a finite-element/discrete-wavenumber technique (Spudich and Xu, 2002) that is well tested and widely used for near-source ground-motion computation
for finite source-rupture models. Figure 7 shows the sourcereceiver geometry we adopt in our simulation. The finite source is a left-lateral strike-slip fault that strikes at ⳱ 150 and dips at an angle of d ⳱ 90 with a rake k ⳱ 0. The fault plane extends from z ⳱ ⳮ2.5 km to a depth of z ⳱ ⳮ17.5 km and has a length of L ⳱ 35 km. These source dimensions were computed for an Mw 6.7 strike-slip earthquake using the scaling relations by Mai and Beroza (2000). We select nineteen well-distributed sites in the epicentraldistance range of 10 to 50 km located in correspondence to strong-motion stations that recorded the 2000 Tottori earthquake. Table 3 lists the linearly depth-dependent velocitydensity model. To generate an extended-source rupture model with spatially varying slip, we use the spatial random-field model for earthquake slip (Mai and Beroza, 2002). This approach is based on an analysis of a large collection of published finitesource rupture models (Mai and Beroza, 2002), obtained through inversion of seismic and/or geodetic data, showing earthquake slip complexity at all spatial scales (e.g., Beroza and Spudich, 1988; Hartzell and Heaton, 1983; Wald and Heaton, 1994; Liu and Archuleta, 2004). Mai and Beroza (2002) demonstrate that a van Karman autocorrelation function (ACF) with magnitude-dependent correlation lengths ax, az and a scale-independent Hurst exponent H ( 0.7) best matches the power-spectral properties the inferred slip distributions. The power-spectral density P(k) of the van Karman ACF is given as P(k) ⳱
4 •p•H a x • az KH (0) (1 Ⳮ k2)HⳭ1
(17)
Accurate Calculation of Fault-Rupture Models Using the High-Order Discontinuous Galerkin Method on Tetrahedral Meshes
Figure 5. Comparison of the radial, transverse, and vertical velocity components for the LOH.2 test case on receiver 10. The analytical solution (thick dashed) is plotted against the numerical one (thin solid) obtained by ADER-DG O4 on the extended mesh (a), UCBL (b), UCSB (c), CMUN (d),WCC1 (e), and WCC2 (f). The relative seismogram misfit E is given for each trace.
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Table 2 Computational Effort of the Different Simulations with ADER-DG and ADER-FV for the Presented Test Cases Problem
Method
Ne
Nd
LOH.2 LOH.2 LOH.2 LOH.2 LOH.2 LOH.2 LOH.2 LOH.2 HSLM HSLM
ADER-DG O2 (LTS,extended) ADER-DG O3 (LTS,extended) ADER-DG O4 (LTS,extended) ADER-DG O4 (LTS,reduced) ADER-DG O4 (GTS,extended) ADER-FV O4 UCBL, UCSB, WCC1, WCC2 CMUN ADER-DG O4 (LTS,reduced) ADER-DG O4 (LTS,extended)
578,118 578,118 578,118 468,795 578,118 1,823,339 15,300,000 — 857,943 1,128,664
2,312,472 5,781,180 11,562,360 9,375,900 11,562,360 1,823,339 15,300,000 — 17,158,860 22,573,280
Figure 6.
CPU (sec)
3,960 9,300 21,540 18,420 43,860 50,400 — — 28,920 35,700
Comparison of the radial, transverse, and vertical velocity components for the LOH.4 test case on receiver 10. The ADER-DG O4 solution on the extended mesh (thin solid) is plotted against the reference solutions (thick dashed) obtained by UCBL (a), UCSB (b), CMUN (c), and WCC2 (d).
Accurate Calculation of Fault-Rupture Models Using the High-Order Discontinuous Galerkin Method on Tetrahedral Meshes
Figure 7. Fault geometry and station distribution for the extended-source simulations, adopted from the 2000 Tottori earthquake. The projection of the vertical fault, striking at U ⳱ 150 is shown by the heavy gray line; the star denotes the epicenter.
Table 3 Depth-Dependent, Linearly Varying Elastic Material Parameters for the HSLM Test Case Depth z (m)
cp (m/sec)
cs (m/sec)
q (kg/m3)
k (GPa)
l (GPa)
0 ⳮ2,000 ⳮ16,000 ⳮ38,000 ⳮ260,000
5500 6050 6600 8030 8200
3180 3500 3810 4620 4680
2600 2700 2800 3100 3300
26.066 32.677 40.678 67.556 77.336
26.292 33.075 40.645 66.168 72.278
where k ⳱ 冪a2x k2x Ⳮ a2z k2z and kx, kz are the horizontal and vertical wavenumbers, respectively, and KH(0) is the modified Bessel function of the first kind (order H). The correlation lengths are found to obey approximately the following simple scaling relations: ax 2.0 Ⳮ 0.33 • Leff, az 1.0 Ⳮ 0.33 • Weff, where Leff, Weff are the effective fault dimensions as defined by Mai and Beroza (2000), or alternatively log10(ax) ⳮ2.5 Ⳮ 0.5 • Mw, log10(az) ⳮ1.5 Ⳮ 0.33 • Mw. Using these scaling relations we simulate heterogeneous slip maps by computing the power spectrum, assuming a random phase and carrying out the two-dimensional Fourier transform obeying Hermitian symmetry to ensure a purely real valued slip function. Such slip models will have, statistically seen, the same properties as slip distributions of past earthquakes, and hence can serve as a starting point for ground-motion simulations for scenario earthquakes. It remains to add conditions for the temporal rupture evolution,
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that is, specifying the rupture velocity and rise time on the fault, potentially reflecting first-order principles of rupture dynamics (see Guatteri et al., 2003, 2004). Following the proceding discussions, we calculate ax az 10 km, and we select a Hurst exponent of H ⳱ 0.8. The resulting slip distribution, generated at 70 ⳯ 30 subfaults with an area of A ⳱ 2.5 • 105 m2 is shown in Figure 8. Note the major large-slip asperity northwest of the hypocenter with maximal slip of 2.5 m and several minor slip patches scattered over the fault plane. The hypocenter location is consistent with the findings of Mai et al. (2005) that ruptures tend to start close to zones of large slip or stress-drop, in agreement with energy-budget considerations of rupture dynamics. The temporal rupture evolution is given by a rupture front that spreads across the fault plane at a constant speed of vrup ⳱ 2700 m/sec, the dislocation rise time is fixed at sr ⳱ 0.8 sec for all points on the fault, and we use an isosceles triangle to generate the slip-rate function s(t). We therefore define a kinematic rupture model for which the slip and slip rate is spatially variable; however, nothing would prevent us in our method from using more complex, dynamically constrained rupture models. This rupture model is then inserted into the discrete frequency-wavenumber integration code COMPSYN by Spudich and Xu (2002) to compute full wave-field near-source synthetic seismograms in the frequency range [0.01–3.0] Hz. Note that for these calculations a maximum frequency limit has to be specified and was chosen according to the frequency content of the waves generated by the source model. These seismograms serve as the reference solution against which we then compare the ADER-DG seismograms at various stations. The computational domain X ⳱ [ⳮ60, 60] ⳯ [ⳮ60, 60] ⳯ [ⳮ60, 0] km (Fig. 7) consists of a mesh of 8.5 ⳯ 106 tetrahedral elements. To visualize the rupture plane with the heterogeneous slip distribution and its orientation inside the unstructured tetrahedral mesh used for the computations, Figure 9 shows only part of the mesh, including the free surface. Note that the tetrahedral elements are not aligned with the rupture plane and in general contain several subfaults indicated by the regular mesh on the fault surface. Figure 10a–f depicts the comparison of the three velocity components obtained by the LTS ADER-DG O4 scheme with the reference solution obtained by the COMPSYN code. Stations 3, 5, 11, 12, 16, and 17 are selected to cover different azimuths and distances from the fault. Stations 3 and 5 in Figure 10a and b display almost perfect agreement of the two solutions as the stations are located in the center of the computational domain. Stations 16 and 17 in Figure 10e and f are showing slight deviations after 20 sec, whereas for stations 11 and 12 in Figure 10c and d we observe clear discrepancies in the later part of the seismograms. Computing the numerical seismograms with the same LTS ADER-DG O4 scheme on an extended domain Xe ⳱ [ⳮ120, 120] ⳯ [ⳮ120, 120] ⳯ [ⳮ100, 0] km discretized by 1.1 ⳯ 106 elements, which inside the domain X are identical with the previous reduced mesh, provides the re-
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Figure 8. Heterogeneous slip distribution on the 35 ⳯ 15 km rupture plane; the star denotes the hypocenter located at depth of 14 km. Slip contours are shown every 25 cm, the maximum displacement on the fault is 2.5 m; the thin gray contours extending radially from the hypocenter indicate the rupture front that propagates at a constant rupture speed of vrup ⳱ 2.7 km/sec.
Figure 9.
Arbitrary orientation of the rupture plane with heterogeneous slip distribution with respect to the tetrahedral mesh. Subfault centers do not have to coincide with mesh vertices, which does not constrain mesh generation and avoids interpolation procedures.
sults in Figure 11a–f. Now, all seismograms, in particular, at station 11, which previously was located very close to the model boundary, show a perfect match with the seismograms obtained by the COMPSYN code as boundary effects are completely avoided. The computational effort is given in Table 2 and shows that the additional storage requirements increase by 31% whereas the computational time increases only by 23% even though the volume of the extended mesh is again almost seven times larger. As in the previous test cases LOH.2 and LOH.4 this is due to the use of the LTS
approach as the added tetrahedral elements in the extended mesh are very coarse and therefore do not increase the total amount of elements dramatically and have to be updated less frequently because of their much larger size and the corresponding larger local timestep. The preceeding example clearly validates the ADER-DG approach as a viable tool for computing near-source seismograms for finite-rupture models with arbitrary orientation and heterogeneous slip distributions. Hence, our tests confirm the performance of the proposed scheme and its ability
Accurate Calculation of Fault-Rupture Models Using the High-Order Discontinuous Galerkin Method on Tetrahedral Meshes
Figure 10. Comparison of the ADER-DG O4 solutions for the reduced mesh (thin solid) for the velocity components u, v, and w in the north–south, east–west, and vertical directions, respectively, are shown for the stations 3 (a), 5 (b), 11 (c), 12 (d), 16 (e), and 17 (f) together with the reference solution (thick dashed) obtained by the discrete frequency-wavenumber code COMPSYN.
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Figure 11. Comparison of the ADER-DG O4 solutions for the extended mesh (thin solid) for the velocity components u, v, and w in the north–south, east–west, and vertical directions, respectively, are shown for the stations 3 (a), 5 (b), 11 (c), 12 (d), 16 (e), and 17 (f) together with the reference solution (thick dashed) obtained by the discrete frequency-wavenumber code COMPSYN.
Accurate Calculation of Fault-Rupture Models Using the High-Order Discontinuous Galerkin Method on Tetrahedral Meshes
to treat external source terms representing extended rupture models with high accuracy. Though the rupture model here was chosen to be rather simple, with constant rupture speed, constant rise time, and a simple slip-velocity function to validate the method, the ADER-DG scheme can also handle more complex source models, potentially derived from dynamic rupture modeling. Recall that the mesh generation is not affected by the inclusion of the extended source as the mesh does not have to be aligned with the rupture surface. This property of the ADER-DG method offers great flexibility for computing ground motions for geometrically complex fault systems because the geometry of the faults is fully described by numerous points representing individual subfaults with local fault parameters and slip-rate functions.
Conclusion We have presented a new simulation method to accurately treat finite-source rupture models using tetrahedral meshes. In particular, because of the subcell resolution of the high-order polynomial approximation inside a tetrahedral element, the ADER-DG method allows for very coarse meshes compared with the fine discretization of an extended rupture surface via a large number of subfaults. As the polynomial test functions can be evaluated exactly at each location of a subfault center and each subfault can provide its local fault parameters and slip-rate function, the ADER-DG approach allows for a highly accurate treatment of any earthquake source characterization. Although we restricted ourselves in this study for validation purposes to planar rupture planes and simple kinematic rupture models, with either constant or heterogeneous slip but constant rise time, constant rupture velocity and a simple slip-velocity function, nothing in the ADER-DG method prevents the use of more complex source-rupture models. In particular, it is straightforward to apply the ADER-DG approach to ground-motion simulations for geometrically complex fault systems that involve rupture on two or more fault segments at any arbitrary relative position to each other. Moreover, the method can also be applied to more physically motivated source models whose slip distribution and spatiotemporal rupture evolution are obtained from dynamic rupture modeling. Additionally, we demonstrate an approach to avoid boundary effects due to the size of the computational domain that may cause weak, artificial wave reflections in the synthetic seismograms. By embedding the region under study into an enlarged computational domain the absorbing boundaries, which do not perform optimally for incident waves that are not normal to the boundary, are moved further away from the zone of interest. However, the additional computational storage requirements and the increase in run time is relatively small despite the increased mesh. The extended mesh covering the larger domain typically consists of very large and therefore relatively few additional tetrahedral elements. Furthermore, due to the use of the local timestepping of the ADER-DG method, these additional elements have to
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be updated less frequently because of their larger size and hence larger local timestep. The extension of the ADER-DG method to near-fault seismic-wave propagation therefore presents a powerful tool to model scenario earthquakes with space–time variable source kinematics, potentially with geometrically complex rupture surfaces, considering also heterogeneous, anisotropic, and viscoelastic media. To that end, future seismichazard studies that involve large-scale simulation-based ground-motion prediction will profit from the accuracy and flexibility of the ADER-DG method. Moreover, the method offers an attractive property by allowing quick low-order calculations that can be carried out very efficiently to provide rapid initial results and performance checks. Once a particular set of simulations has been tested and verified at low order, the same calculations can be run at the desired accuracy without remeshing the computational domain.
Acknowledgments We thank the DFG (Deutsche Forschungsgemeinschaft), as the work was supported through the Emmy Noether-program (KA 2281/2-1) and the DFG Forschungsstipendium (DU 1107/1-1). This work was also funded by the EC Marie-Curie Training Network SPICE–Seismic Wave Propagation and Imaging in Complex Media: A European Network. We are especially grateful to the Leibniz Rechenzentrum in Mu¨nchen for providing their supercomputing facilities. We thank S. Day for discussions and for providing the PEER results. Dudley Joe Andrews and an anonymous reviewer provided helpful comments. This is contribution number 1479 of the Institute of Geophysics, ETH Zurich.
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[email protected] (M.K.) Institut fu¨r Geophysik Eidgeno¨ssische Technische Hochschule Zu¨rich Ho¨nggerberg, HPP G 1.2 8093 Zu¨rich, Switzerland
[email protected] (P.M.M.) Laboratory of Applied Mathematics Department of Civil and Environmental Engineering University of Trento, Via Mesiano 77 38050 Trento, Italy
[email protected] (M.D.) Manuscript received 8 December 2006.
