A 3-D hybrid finite-difference finite-element viscoelastic modelling of seismic wave motion

Transcription

1 The definitive version is avaiabe at Geophys. J. Int. 008) 75, 5 84 doi: 0./j.65-46X x A -D hybrid finite-difference finite-eement viscoeastic modeing of seismic wave motion Martin Gais,, Peter Moczo, and J. Kristek, Facuty of Mathematics, Physics and Informatics, Comenius University, Mynska doina F, Bratisava, Sovak Repubic E-mai: moczo@fmph.uniba.sk Geophysica Institute, Sovak Academy of Sciences, Dubravska cesta 9, Bratisava, Sovak Repubic Accepted 008 May 6. Received 008 Apri 0; in origina form 007 October 4 S U M M A R Y We have deveoped a new hybrid numerica method for -D viscoeastic modeing of seismic wave propagation and earthquake motion in heterogeneous media. The method is based on a combination of the fourth-order veocity stress staggered-grid finite-difference FD) scheme, that covers a major part of a computationa domain, with the second-order finite-eement FE) method which can be appied to one or severa reativey sma subdomains. The FD and FE parts causay communicate at each time eve in the FD FE transition zone consisting of the FE Dirichet boundary, FD FE averaging zone and FD Dirichet zone. The impemented FE formuation makes use of the concept of the goba restoring-force vector which significanty reduces memory requirements compared to the standard formuation based on the goba stiffness matrix. The reaistic attenuation in the whoe medium is incorporated using the rheoogy of the generaized Maxwe body in a definition equivaent to the generaized Zener body. The FE subdomains can comprise extended kinematic or dynamic modes of the earthquake source or the free-surface topography. The kinematic source can be simuated using the body-force term in the equation of motion. The traction-at-spit-node method is impemented in the FE method for simuation of the spontaneous rupture propagation. The hybrid method can be appied to a variety of probems reated to the numerica modeing of earthquake ground motion in structuray compex media and source dynamics. Key words: Numerica soutions; Earthquake dynamics; Site effects; Computationa seismoogy; Wave propagation. GJI Seismoogy I N T RO D U C T I O N Athough many computationa and numerica-modeing methods have been deveoped so far for seismic wave propagation and earthquake motion, no singe method can be considered the most efficient and, at the same time, accurate for a wavefied-medium probem configurations. Each method has its advantages and drawbacks either in computationa efficiency or accuracy. Athough ooking for new and more efficient/accurate methods is desirabe, possibe and, we beieve, necessary, a considerabe improvement of the present-day modeing toos can be achieved by combining individua methods in hybrid approaches. In some cases it is advantageous to sove time dependence of the dispacement using one method and spatia dependence using some other method. For exampe, Aexeev & Mikhaienko 980) combined partia separation of variabes and finite-difference FD) method, Mikhaienko & Korneev 984) combined finite integra Fourier transform and FD method. In some other cases it is reasonabe to spit the computationa domain into two or more parts and sove each part by a proper method. Severa hybrid methods were deveoped, mainy in -D, in an effort to achieve reasonabe computationa efficiency in appications to reativey compex structura modes. They incude methods by Ohtsuki & Harumi 98), Shtiveman 984, 985), Van den Berg 984), Kummer et a. 987), Stead & Hemberger 988), Kawase 988), Gaffet & Bouchon 989), Emmerich 989, 99), Fäh 99), Fäh et a. 99), Rovei et a. 994), Bouchon & Coutant 994), Robertsson 996), Zahradník & Moczo 996), Moczo et a. 997), Lecomte et a. 004) and Ma et a. 004). Here we focus on the -D viscoeastic modeing of earthquake motion in the heterogeneous medium with, optionay, kinematic point or finite) or dynamic earthquake source. We combine the fourth-order veocity stress staggered-grid FD scheme with the second-order finite-eement FE) method. Being we aware of the recenty eaborated powerfu spectra-eement method e.g. Komatitsch et a. 005; Chajub et a. 007; Tromp et a. 008) and ADER-DG method e.g. Käser & Dumbser 006; Käser et a. 007; de a Puente et a. 007) we beieve that our hybrid C 008 The Authors 5

2 54 M. Gais, P. Moczo and J. Kristek approach can be usefu in a variety of probem configurations in wave propagation and source dynamics where it can be computationay efficient and, at the same time, sufficienty accurate. The FE method more easiy incorporates boundary conditions at the free surface and materia interfaces compared to the FD method if the free surface or interna interface coincides with a surface of gridpoints. Therefore, the FE method is better suited for simuation of the traction-free condition and rupture propagation than the FD method. On the other hand, a FD scheme can be computationay more efficient on a reguar space time grid for modeing seismic wave propagation for exampe, if the seismic wave produced by the dynamicay rupturing faut are to be propagated away from the faut. It is therefore, very natura to think of a hybrid combination of the two methods if we want to comprise both the dynamic earthquake source and the wave propagation in the compex heterogeneous medium. The FD and FE methods were combined in the -D modeing. Moczo et a. 997) combined the second-order conventiona FD scheme with the second-order FE method for the -D viscoeastic P-SV modeing of seismic motion in the near-surface sedimentary/topographic structure. Ma et a. 004) combined the fourth-order veocity stress staggered-grid scheme with the second-order FE method for the -D eastic P-SV modeing. In our hybrid formuation we aso combine the second-order FE scheme with the fourth-order FD scheme. In the FD FE transition zone the size of the grid spacing in the FE grid is twice smaer compared to that in the FD grid. Away from the transition zone the size of an eement can in principe vary. The twice smaer FE grid spacing is agorithmicay the most natura option: any other ratio between the FD and FE grid spacings woud make the schemes for the transition zone much more compicated and it is ikey that it woud produce more numerica noise. Intuitivey, in a rough estimate, such a choice seems reasonabe given the second- and fourth-order approximations in the FE and FD schemes, respectivey. Stricty speaking, however, the spatia samping does not simpy scae with the approximation order. For exampe, the increase in the approximation order by two does not mean that as much as twice arger grid spacing can be used. This indicates that an appropriatey sma FE grid spacing may require a FD grid spacing smaer than that usuay used by many modeers in the pure fourth-order staggered-grid modeing. This is additionay and independenty supported by resuts obtained by Kristek & Moczo 006). They indicated by their numerica investigations for the -D probem that the fourth-order staggered-grid scheme requires denser spatia samping than that usuay used by many users who consider it approximatey twice coarser than that in the second-order conventiona schemes. One other aspect of the spatia samping is due to the fact that the numerica modeing of the rupture propagation and free-surface topography requires denser spatia samping compared to that usuay used for the wave propagation. In the paper we first present, in a concise form, the -D viscoeastic fourth-order veocity stress staggered-grid FD formuation. We continue with a more detaied exposition of the FE method formuated using the restoring force. We then briefy characterize the computationa domain. The dominant methodoogica part of the paper is devoted to the FD FE transition zone which is a core of the hybrid approach. Resuts of extensive parametric test simuations are then presented to demonstrate numerica behaviour of the transition zone. In order to iustrate possibe appications of the hybrid method, we incude partia resuts of simuations of two hypothetica earthquakes near the Grenobe vaey in France. E Q UAT I O N O F M O T I O N, C O N S T I T U T I V E L AW, A N D F I N I T E - D I F F E R E N C E S C H E M E Rheoogy of a -D isotropic viscoeastic medium can be assumed as made of rheoogies of two GMB-EK bodies. GMB-EK means generaized Maxwe body as defined by Emmerich & Korn 987): severa cassica Maxwe bodies and one Hooke body connected in parae. One GMB-EK body represents the compex frequency-dependent buk moduus. The other represents the compex frequency-dependent shear moduus. Note that rheoogy of the GMB-EK body is identica to that of the generaized Zener body made of severa cassica Zener bodies connected in parae. The equivaence was shown by Moczo & Kristek 005). The foowing formuations of the equation of motion and the stress strain reation are used Kristek & Moczo 00; Moczo et a. 007a): ρ v i = σ i j, j + f i and σ i j = κ ε kk δ i j + µ ε i j ) ε kk δ i j ξ i j n = [ κ Y κ ξ kk δ i j + µ Y µ ξ i j ) ] ξ kk δ i j, ) + ω ξ i j = ω ε i j ; =,..., n. Here, in a Cartesian coordinate system x, x, x ) or x, y, z), i, j, k {,, }, ρx i ) is density, κx i ) unreaxed eastic) buk moduus, µx i ) unreaxed shear moduus, Y κ and Y µ aneastic coefficients, vx i, t) partice-veocity vector, t time, v i time derivative of the ith component of the partice-veocity vector, f x i, t) body force per unit voume, σ i j x k, t) stress tensor, σ i j time derivative of the stress tensor, σ i j, j spatia derivative of the stress tensor with respect to x j, ε i j x k, t) time derivative of the strain tensor, ξ i j x k, t) materia-independent aneastic functions memory variabes), ω anguar reaxation frequency for the th reaxation mechanism, n number of reaxation mechanisms, and δ i j ) )

3 A -D hybrid finite-difference 55 Tabe. Foca and source time function parameters of the point doube-coupe source used in the numerica tests for the US-, US-8, FS-, FS-8, convergence, and contact of two haf-spaces probem configurations. Source parameters Source time-function parameters Strike S Dip δ Rake λ M 0 f p γ s t s N m 0.5 Hz s Kronecker deta. The equa-index summation convention does not appy to index. Coefficients Y κ and Y µ are obtained from Y κ = α Y α 4 ) / β Y β α 4 ) β 4) and Y µ = Y β, where =,..., n, α = [ κ + 4 µ )/ρ ]/ and β = µ/ρ) / are eastic P- and S-wave veocities. Aneastic coefficients Y α and Y β are obtained from measured/desired quaity factor vaues Q α and Q β using the system of equations n Q ω ω k + ω Q ν ω k ) ν ω k ) = Y ν ω + ω k ; k =,..., n ; ν {α, β}. 6) = System 6) can be soved using the east-square method. For modeing constant or amost constant Q ω) it is reasonabe to sampe the frequency range of interest with at east ω /ω = 0. Frequencies ω k may be chosen as ω = ω, ω n = ω n, and ω k / ω k = 5. If the unreaxed modui κ and µ or, equivaenty, eastic veocities α and β are not known, and, instead, phase veocities αω r ) and βω r ) at certain frequency ω r are known from measurements, the unreaxed quantities can be determined from aneastic coefficients Y α and Y β, and veocities αω r ) and βω r ), assuming the rheoogy of the GMB-EK. For more detais see Moczo et a. 997, 007a). The FD scheme soving eqs ) ) is the -D fourth-order veocity stress staggered-grid FD scheme presented by Moczo et a. 00, 004, 007a,b), Kristek et a. 00), and Kristek & Moczo 00). Here we ony note that the smooth and discontinuous heterogeneity of the viscoeastic medium is accounted for by effective grid materia parameters assigned to the grid positions and evauated as integra voume arithmetic for density) and harmonic for viscoeastic modui) averages. A materia discontinuity can intersect a grid ce. In Fig., we show the spatia distribution of the materia and fied quantities in the staggered grid. Because the coarse spatia distribution is not appied in the FD FE transition zone the figure shows a grid ce with a aneastic functions. The figure shoud hep to understand the FD FE communication in the FD FE transition zone which is described ater. Note that Fig. shows discrete grid quantities for which we use symbos different than for the continuous quantities. 5) T H E - D S E C O N D - O R D E R D I S P L A C E M E N T R E S T O R I N G - F O RC E F I N I T E - E L E M E N T S C H E M E We use the second-order FE scheme for an isotropic viscoeastic medium. Instead of the standard FE scheme with the goba stiffness matrix we use the concept of the restoring-force vector Frazier & Petersen 974; Archueta 976). Because the FE scheme with the restoring-force vector is, as far as we know, not so we known or documented in the seismoogica iterature and because we have to expain the incorporation of the GMB-EK rheoogy in the FE formuation, here we briefy outine the derivation of the FE scheme with the restoring-force vector from the Gaerkin formuation of the discretized variationa form of the equation of motion for one eement e. A detaied expanation of the concept can be found in Moczo et a. 007a). The equation is s k ρ s det J d ü i + s k, j σ i j det J d M M s k f i det J d s k h e M i dŵ = 0; 7) Ŵ N k, {,,..., L}, where s k are the shape functions, indices i and j denote spatia coordinates x, y, z), s k, j partia spatia derivative of the shape function, ρ density of the medium in eement e, σ ij stress tensor, f i body force per unit voume, Ŵ N part of the boundary of the eement with the acting traction h e i, M master eement, and L the number of nodes in eement e. u i is the component of the oca vector of the discretized dispacements at the nodes. The atter vector is u e = u x u y u z = [u x,...,u x L, u y,..., u yl, u z,..., u zl ] T. 8) The integrations in eq. 7) are performed in the master eement M, that is, eement in the oca coordinates. Matrix J is the Jacobian of transformation of the master eement M from the oca coordinates to eement e in the goba coordinates.

4 56 M. Gais, P. Moczo and J. Kristek Figure. A staggered-grid FD ce with positions of partice-veocity components U, V, Ẇ, stress tensor components T xy, T yz, T zx, T xx, T yy, T zz, aneastic functions ξ xy, ξ yz, ξ zx, ξ xx, ξ yy, ξ zz, effective eastic buk and shear modui κ H, µ H, µ xy H, µh yz, µh zx and aneastic coefficients Y µ, Y κ, Y µ xy, Y µ yz, Y µ zx. Indices A and H indicate integra voume arithmetic and harmonic averages. Figure. Iustration of the computationa domain. The FE region can cover a part of the mode with a free-surface topography eft-hand side) or a dynamicay rupturing surface right-hand side). The rest of the domain is covered with the FD grid. One computationa domain can, in principe, incude severa FE regions. The atter option is avaiabe in the Fortran 95 program D Hybrid FDFE avaiabe at Codes). Eq. 7) is a set of L equations and can be written in a matrix form. By appying the standard FE procedure to the first term on the eft-hand side of the equation we obtain the we known oca mass matrix M e. From the third and fourth terms we obtain a oca oading-force vector f e. The appication of the standard FE procedure to the second term woud ead to the oca stiffness matrix usuay denoted by K e ). Here we proceed in other way in order to obtain a oca restoring-force vector. For i = x we rewrite the second term on the eft-hand side of eq. 7) in the form s k, j σ i j det J d i = x [ ] sk, x σ xx + s k, y σ xy + s k, z σ xz det J d. 9) M M Considering the vector of shape functions s = [s,..., s L ] T we can define a oca vector of the x-components of the restoring force as r x = s, x σ xx + s, y σ xy + s, z σ xz ) det J d. M 0) )

5 A -D hybrid finite-difference 57 Figure. Iustration of an agorithmicay minima transition zone for the causa FD FE communication at each time eve. For simpicity, ony the vertica grid pane with the x- and z-components of the partice veocity, U and Ẇ, norma stress tensor components, T xx, T yy and T zz, and shear stress tensor component T zx is shown compare with Fig. ). h FD is the spatia grid spacing in the FD grid. h FE is the spatia grid spacing in the uniform part of the FE grid in and near the transition zone; the rest of the FE grid can be non-uniform. Note that no specia symbo is used to indicate positions of the dispacement vector in the FE grid. Each intersection of the grid ines gridpoint) in the FE region, is a position of a components of the dispacement vector. Anaogousy for i = y and i = z we obtain r y = s, x σ xy + s, y σ yy + s, z σ yz ) det J d M and r z = s, x σ xz + s, y σ yz + s, z σ zz ) det J d. M Then r e = r x r y ) ) 4) r z is the oca restoring-force vector. A component of the oca restoring-force vector, rik e, represents the i-component of the force acting at node k due to eastic forces acting in the eement. These eastic forces represent reaction to the current state of deformation of the eement. The forces tend to return the eement back to the equiibrium. The stress tensor for the isotropic eastic medium is given by Hooke s aw: σ xx = λ + µ) s, T x u x + λ s, T y u y + λ s, T z u z, σ yy = λ s, T x u x + λ + µ) s, T y u y + λ s, T z u z, σ zz = λ s, T x u x + λ s, T y u y + λ + µ) s, z T u z, σ xy = µ s, T y u x + s, T x u ) y, σ yz = µ s, T z u y + s, T y u ) z, σ xz = µ s, T z u x + s, T x u ) z. Using the oca mass matrix M e, the oca oading-force vector f e, and oca restoring-force vector r e we can rewrite eq. 7) in a matrix form M e ü e = r e + f e. 5) 6)

6 58 M. Gais, P. Moczo and J. Kristek Figure 4. Possibe spatia configurations of the grid position at which interpoations are necessary. Each of the nine stencis indicates a possibe spatia position of the FE-gridpoint empty circe) of the FE Dirichet boundary with respect to the FD-grid partice-veocity positions fu circes) used for interpoation. Number stencis in the eft coumn) refers to the FE-gridpoint ocated directy at a grid ine of the FD-grid vaues that can be used for the interpoation. The ower-case a, b and c then distinguish three possibe configurations aong such a grid ine. Number refers to the FE-gridpoint ocated at a grid pane of the FD-grid vaues that can be used for the interpoation. Number refers to the FE-gridpoint ocated out of a grid panes and ines of the FD-grid vaues that can be used for the interpoation. Foowing the standard FE procedure to assembe the oca systems of equations for a eements in the mesh we obtain a goba system of ordinary differentia equations M ü = r + f, where M is the goba mass matrix, ü goba vector of discretized dispacements at nodes, f goba oading-force vector and r is the goba restoring-force vector. 7)

7 A -D hybrid finite-difference 59 Figure 5. Iustration of the FD FE transition zone used in the hybrid modeing for the causa FD FE communication at each time eve. The difference compared to the agorithmicay minima transition zone consists in a presence of the averaging zone between the FE Dirichet boundary and FD Dirichet zone. For simpicity, ony the vertica grid pane with the x- and z-components of the partice veocity, U and Ẇ, norma stress tensor components, T xx, T yy and T zz, and shear stress tensor component T zx is shown compare with Fig. ). h FD is the spatia grid spacing in the FD grid. h FE is the spatia grid spacing in the uniform part of the FE grid in and near the transition zone; the rest of the FE grid can be non-uniform. Note that no specia symbo is used to indicate positions of the dispacement in the FE grid. Each intersection of the grid ines gridpoint) in the FE region is a position of a components of the dispacement vector. Approximating the second time derivative in eq. 7) by the centra-difference formua eads to an expicit FE scheme for updating dispacements at the time eve m + : u m+) = t) M [ r m + f m ] + u m u m ). Note that the oca vector of the restoring force can be expressed using the oca stiffness matrix K e : r e = K e u e. Substitution of expression 9) into eq. 6) woud eventuay ead to the standard formuation of the FE method with the goba stiffness matrix. This, however, impies that the stabiity and grid dispersion for the FE scheme with the restoring-force vector are exacty the same as those for the standard FE scheme with the goba stiffness matrix. The main reason for the FE scheme with the goba restoring-force vector is reduction of the memory required by the goba stiffness matrix K. The matrix is very sparse. Let N n be the tota number of nodes in the mesh of the hexahedra eements. We use the so-caed HEX8 eements with eight nodes at the corners. Then the tota number of eements in matrix K is N n N n, whereas the number of non-zero eements is, approximatey, N n 8. Because the atter number is consideraby smaer than the former one, storing ony the non-zero eements means reativey considerabe reduction of the memory requirements. At the same time, N n 8 is sti a too arge number if we reaize that even in the modest -D modeing miions to tens of miions of nodes are necessary. In other words, the number of the non-zero eements of the goba stiffness matrix poses a serious probem. The goba restoring-force vector r contains N n vaues. It is 8 times ess than the number of the non-zero eements in the goba stiffness matrix. As a consequence, the FE formuation 7) needs consideraby smaer memory compared to that required by the standard FE formuation with the goba stiffness matrix assuming the same time discretization. On the other hand the stiffness matrix is time-independent and thus it is computed once at the beginning of computation. The restoring force varies with time, and therefore, it has to be updated at each time eve. This means that the restoring force reduces memory requirements but increases computationa time. For competeness note that the stiffness matrix is time-independent ony if materia parameters are time-independent. If we considered time-dependent materia parameters, for exampe, non-inear behaviour of a medium, the stiffness matrix woud vary with time. Such a 8) 9)

8 60 M. Gais, P. Moczo and J. Kristek Figure 6. Probem configurations for numerica tests of the numerica behaviour of the FD FE transition zone. A point doube-coupe source indicated by a star) and receiver profie indicated by a thick ine) are ocated in an unbounded homogeneous eastic space. The source is ocated at the centre of the cube FE region. The FE region is surrounded by the FD region. The receiver profie extends from the FE region through the FD FE transition zone into the FD region. Two spatia sampings are considered. grid spacings per minimum waveength are appied in the FE region in the US- configuration eft-hand pane) whereas 8 grid spacings per minimum waveength are appied in the US-8 configuration right-hand pane). Six and nine grid spacings per minimum waveength are appied in the FD regions in the two configurations, respectivey. matrix woud have to be updated at each time eve. In other words, the stiffness-matrix formuation woud oose the advantage of smaer computationa time. In a hybrid combination of the FE and FD methods it is necessary that both methods have the same mode of reaistic attenuation. For the FE scheme we therefore, assume the same GMB-EK rheoogy as described in the previous section. Because we work with the dispacement

9 A -D hybrid finite-difference 6 Figure 7. For each probem configuration shown in Fig. 6 numerica simuations were performed for four different FD FE transition zones. The four zones are iustrated using simpe geometry here. B = 0 corresponds to the agorithmicay minima transition zone with no averaging zone AZ. B =,, correspond to the transition zones with different thicknesses of the averaging zone AZ. formuation, we consider the stress strain reation in the form Kristek & Moczo 00; Moczo et a. 007a) σ i j = κ ε kk δ i j + µ ε i j ) ε kk δ i j and ζ i j n = [ κ Y κ ζ kk δ i j + µ Y µ ζ i j )] ζ kk δ i j + ω ζ i j = ω ε i j ; =,..., n; ) compare with the anaogous reations for the veocity stress formuation, eqs ) and ). Reca that indices i, j and k i, j, k {x, y, z}) are spatia indices, index denotes the th reaxation mechanism, n denotes the number of reaxation mechanisms, and the equa-index summation convention does not appy to. Rewrite the stress strain reation 0) in the form more suitabe for the FE impementation: σ i j = σ E i j n = σ A i j. We spit the stress tensor into the eastic and aneastic parts. The eastic part σ i E j is given by Hooke s aw 5). The aneastic part is determined by a sum of the aneastic terms σ i A j for a reaxation frequencies due to superposition of a reaxation mechanisms. A vector of the aneastic 0) )

10 6 M. Gais, P. Moczo and J. Kristek Figure 8. Gabor signa used as the source time function in the numerica tests of behaviour of the FD FE transition zone eft-hand side), and its ampitude Fourier spectrum right-hand side). terms σ i A j σxx A σyy A σzz A σxy A σyz A σ A xz where Y + ζ i j = can be written as Y + Y Y Y + Y = κ Y κ + 4 µ Y µ m + ) = Y Y Y Y µy µ µy µ µy µ, Y = κ Y κ µ Y µ. ζ xx ζ yy ζ zz ζ xy ζ yz ζ xz, ) For soving eq. ) we use the scheme by Kristek & Moczo 00). Considering eq. ) at the time eve m, an approximation of the time derivative using the second-order centra-difference formua, and ζ i j m) by an arithmetic average of ζ i j m ) and ζ i j m + ) yieds a recurrent formua ω t ε i j m) + ω t) ζ i j m ) Substituting reation 5) into ζ i j m) = [ ζ i j m ) + ζ i j + ω t m + )] we obtain formua for ζ i j m). The strain tensor components can be computed using reations ε xx = s, T x u x, ε yy = s, T y u y, ε zz = s, z T u z, ε xy = s, T y u x + s, T x u y), 4). 5) 6) ε yz = s, T z u y + s, T y u z), 7) ε xz = s, T z u x + s, T x u z).

11 A -D hybrid finite-difference 6 Figure 9. Resuts of the numerica tests comparison of the FD FE hybrid synthetics with the DWN soutions for the probem configuration US-. Left-hand, centra and right-hand coumns resuts for the x-, y- and z-components of the dispacement vector denoted by U, V and W), respectivey. Top pane: FD FE and DWN synthetics aong the receiver profie extending from the FE region through the transition zone into the FD region for the agorithmicay minima transition zone B = 0). Midde pane: The same as in the top pane but for the transition zone with the smaest possibe averaging zone B = ). Bottom pane: Enveope EM) and phase PM) misfits between the FD FE and DWN synthetics at a receiver positions for a four considered transition zones B = 0,,, ). The vertica ine in each coumn indicates position of the FE Dirichet boundary. The oca restoring-force vector was defined by reations ) 4). For the stress strain reation ), the oca restoring-force vector takes form r i = r E i n = r A i ; i {x, y, z}, 8)

12 64 M. Gais, P. Moczo and J. Kristek Figure 0. The same as in Fig. 9 but for the probem configuration US-8. where r E x r E y r E z = = = M s,x σ E xx + s, y σ E xy M s,x σ E xy + s, y σ E yy + s, z σ E yz s,x σ E M xz + s, y σ E yz + s, z σ E zz + s, ) z σ E xz det J d, ) det J d, ) det J d, 9)

13 A -D hybrid finite-difference 65 Figure. Probem configurations for numerica tests of the numerica behaviour of the FD FE transition zone. A point doube-coupe source indicated by a star) is ocated in a homogeneous haf-space. A receiver profie indicated by a thick ine) is ocated at the panar free surface of the haf-space. The source is ocated at the centre of the cube FE region. The FE region is surrounded by the FD region. The receiver profie at the free surface extends from the FE region through the FD FE transition zone into the FD region. Two spatia sampings are considered. grid spacings per minimum waveength are appied in the FE region in the FS- configuration eft-hand pane) whereas 8 grid spacings per minimum waveength are appied in the FS-8 configuration right-hand pane). Six and nine grid spacings per minimum waveength are appied in the FD regions in the two configurations, respectivey. and r A x r A y r A z = s,x σ A M xx = s,x σ A M yx = s,x σ A M xz + s, y σ A xy + s, y σ A yy + s, y σ A yz ) + s, z σ A xz det J d, ) + s, z σ A yz det J d, ) + s, z σ A zz det J d. 0)

14 66 M. Gais, P. Moczo and J. Kristek Figure. The same as in Fig. 9 but for the probem configuration FS-. A numerica quadrature for computation of the restoring force requires, in genera, aneastic functions at a integration points. The update of the aneastic functions at the time eve m + needs vaues at the time eve m that have to be stored in the computer memory. Standard numerica quadratures for the hexahedra eement with eight nodes use eight integration points. The corresponding memory requirements are: 8 integration points 6 components of the aneastic functions n reaxation frequencies the number of eements in the mesh N e = 48 n N e. For modeing constant or amost constant Q we use 4 reaxation frequencies. Then the estimate gives 9 N e. Obviousy, the incorporation of the attenuation dramaticay increases the memory requirements; the number of quantities required by the eastic restoring force itsef is ony N n. Note that the number of nodes N n and the number of eements N e for arge modes are approximatey the same.

15 A -D hybrid finite-difference 67 Figure. The same as in Fig. 9 but for the probem configuration FS-8. The coarse spatia distribution of the aneastic functions Day 998; Graves & Day 00; Kristek & Moczo 00) can significanty reduce the additiona memory requirements. The techniques are particuary suitabe for structured meshes. It is important to consider the FE mesh unstructured. Therefore, we simpify the agorithm and evauate the aneastic functions ony at the centre of an eement. This means that for cacuation of the aneastic functions we consider constant strain within an eement, the strain being ocated at the eement s centre. Then the number of quantities to store is reduced to 4N e. We have impemented the traction-at-spit-node TSN) method for a numerica modeing of spontaneous rupture propagation in order to incorporate dynamic source modes. The TSN was independenty deveoped by Andrews 97, 976a,b, 999) and Day 977, 98). There are some differences in the formuations by Andrews and Day. We cosey foowed Day s formuation. A detaied exposition of this TSN formuation and impementation can be found in the monograph by Moczo et a. 007a).

16 68 M. Gais, P. Moczo and J. Kristek Figure 4. Probem configuration and 7 discretizations for the convergence test. The configuration is very simiar to that of the US- test shown in Fig. 6. A point doube-coupe source indicated by a star) and receiver positions trianges) are ocated in an unbounded homogeneous eastic space. The numbers of eements and FD grid spacings per λ min, grid spacings in FE and FD, and the tota number of eements in the FE region for each discretization are summarized in the tabe. The bottom part indicates the sizes of eements, FD FE transition zones, and receiver positions for a considered discretizations.

17 A -D hybrid finite-difference 69 Figure 5. Numerica resuts for the convergence test enveope misfit EM) as a function of the number of eements per λ min for the x-, y- and z-components of the dispacement vector denoted by U, V and W) at three receiver positions. 4 C O M P U TAT I O NA L D O M A I N As aready pointed out, the main idea of the hybrid combination of the two methods is to enabe efficient numerica simuations considering probem configurations that are as reaistic as possibe. Such configurations may incude a free-surface topography, materia heterogeneity as we as a dynamicay rupturing faut. Correspondingy, a computationa domain of the hybrid FD FE method may incude one or more FE regions that woud cover those parts of the mode where the free-surface topography or faut have to be considered. As expained in

18 70 M. Gais, P. Moczo and J. Kristek Figure 6. Probem configuration in case when a materia interface intersects the FD FE transition zone. The cube FE region is centred around a square of the panar interface between two homogeneous eastic haf-spaces. A point doube-coupe source indicated by a star) is ocated in a stiffer haf-space. A receiver profie is ocated in the softer haf-space two grid spacings h FD ) from the materia interface. The receiver profie extends from the FE region through the FD FE transition zone into the FD region. The receiver profie is shifted from the vertica pane with the source in the x-direction by 600 m. the introduction, the FE regions shoud be as sma as possibe compared to the FD region which shoud cover a major part of the whoe computationa domain. The computationa domain is schematicay iustrated in Fig.. Dispacement components in the FE regions), and partice-veocity and stress tensor components in the FD region are first updated independenty by the FE and FD schemes, respectivey. Then the FE regions have to causay communicate at each time eve with the FD region. Given the structures of the both schemes, the schemes cannot communicate at a singe grid surface. A particuar FD FE transition zone at the contact of the FD and FE regions is necessary for a sufficienty accurate and stabe communication. The structure of the transition zone and the agorithm of the FD FE communication is expained in detai in the next section. 5 T H E F D F E T R A N S I T I O N Z O N E 5. Principe of the FD FE communication an agorithmicay minima transition zone The FD and FE schemes can communicate with each other at each time eve ony in the region, where the FD and FE grids overap in the transition zone. The shape and size of the transition zone are basicay determined by the FD schemes for updating partice-veocity and stress tensor components. The transition zone is iustrated in Fig.. The figure shows an exampe of a vertica cross-section of a particuar transition zone. The zone consists of the FE Dirichet boundary and FD Dirichet zone. It is cear that the FE Dirichet boundary for the second-order dispacement FE scheme consists of a singe staircase grid surface that has to go through the gridpoints of the FD staggered grid. At the same time, a finite-thickness Dirichet zone is necessary for the fourth-order veocity stress staggered-grid FD scheme. The oca thickness and staircase shape of the FD Dirichet zone are determined by the requirement that the partice veocity at the FD gridpoints ocated at the FE Dirichet boundary be cacuated using the fourth-order veocity stress staggered-grid FD scheme for an interior gridpoint. This is possibe if reevant stress tensor components are avaiabe. The FD Dirichet zone has to incude those stress tensor components. Moreover, those stress tensor components have to be updated by the FD scheme from the partice veocities, because it woud be very difficut to cacuate the stress tensor components by the FE schemes at eement nodes. For the agorithmic reasons it is reasonabe that the reevant partice-veocity components be part of the FD grid. The FE grid near the transition zone has to be uniform because the staggered grid is uniform. The agorithm of the causa FD FE communication in the agorithmicay minima transition zone can be summarized in the foowing steps. For brevity we use U for any of the dispacement component, U for any partice-veocity component, and T for any stress tensor component. Subscripts FD and FE refer to the corresponding grids. Lower-case m denotes a time eve. ) Dispacements U FE m + ) are updated at the gridpoints of the interior FE region the FE gridpoints except the FE Dirichet boundary). ) Stress tensor components Tm) are updated at the gridpoints of the FD region incuding the stress tensor grid positions inside the FD Dirichet zone).

19 A -D hybrid finite-difference 7 Figure 7. Resuts of the numerica tests comparison of the FD FE hybrid synthetics with the DWN soutions for the probem configuration shown in Fig. 6. Left-hand, centra and right-hand coumns resuts for the x-, y- and z-components of the dispacement vector denoted by U, V and W), respectivey. Top pane: FD FE and DWN synthetics aong the receiver profie extending from the FE region through the transition zone into the FD region. Bottom pane: Enveope EM) and phase PM) misfits between the FD FE and DWN synthetics at a receiver positions. The vertica ine in each coumn indicates position of the FE Dirichet boundary. Figure 8. Iustration of a position of the faut and the FD FE transition zone. Distance between the faut pane and FD FE transition zone is denoted by δ. Simuations were performed for six different distances δ {, 5, 9,, 7, } h FE. ) Partice veocities U FD m + ) are updated at the gridpoints of the interior FD region. 4) Partice veocities within the FD Dirichet zone at the gridpoints indicated by the doube squares and circes in Fig. ) are updated using the FE dispacement vaues at the same gridpoints: U FD m + ) = U FEm + ) U FE m). t )

20 7 M. Gais, P. Moczo and J. Kristek Tabe. Materia and computationa parameters used in the numerica simuations of the rupture propagation. λ = µ v P v S ρ h FD h FE t 4. GPa 6050 m s 500 m s 800 kg m 50 m 5 m s Notes: λ and µ are Lamé eastic parameters, v P and v S are P- and S-wave veocities, ρ is density, h FD is spatia grid spacing in the FD grid, h FE is spatia grid spacing in the FE grid and t is time step. Tabe. Constitutive parameters of the inear sip-weakening friction aw used in the numerica simuations of the rupture propagation. σ 0 τ 0 µ u µ f d 0 S r nuc τ nuc GPa 0.0 GPa m m 8.4 GPa Notes: σ 0 is initia norma traction, τ 0 is initia shear traction, µ u is static coefficient of friction, µ f is kinematic coefficient of friction, d 0 is characteristic sip-weakening distance, S is strength parameter, r nuc is radius of a circuar nuceation zone and τ nuc 0 is initia shear traction in the nuceation zone. 5) Dispacements U FE m + ) are updated at the FE Dirichet boundary using the FE dispacements and FD partice veocities at the same gridpoints: U FE m + ) = U FE m) + t U FD m + ). ) Reca that the FE Dirichet boundary consists of a singe staircase grid surface that goes through the gridpoints of the FD staggered grid. A grid position of the FD staggered grid is a position of either of just one partice-veocity component or one shear stress tensor component or three norma stress tensor components or none component. At the same time the symboic eq. ) requires a partice-veocity components at a given grid position. Consequenty, an interpoation of the missing partice-veocity components is necessary. A possibe spatia configurations of the grid positions at which interpoations are necessary are symboicay shown in Fig. 4. Three aong-grid-ine configurations, abeed a, b and c in Fig. 4, are possibe. Let p denote a true FD grid index aong a grid ine in any of the three Cartesian coordinate directions. Then the fourth-order interpoation formuas for the three configurations are, respectivey, f p = 9 6 [ ] f p + f p + 6 [ f p ], ) f p = 5 6 f p f p f p f p + 5, 4) f p = 5 6 f p f p f p f p ) Three in-grid-pane configurations abeed a, b and c in Fig. 4 require more compicated interpoation formuas. Let p, q denote true FD-grid indices of a grid position in any of grid panes parae with either of the three Cartesian coordinate panes. Then the fourth-order interpoation formuas for the three configurations are, respectivey, [ ] f pq = 5 f 6 p, q, q + f p, q + + f p +, q + [ f p, q + f p, q +, q + f p +, q +, 6) ], q, q +, q, q + [ ] [ ] f pq = f 6 p, q, q + f p, q + + f p +, q + [ ] [ ] 5 f p, q +, q + + f p, q + 5, q + 5, 7) [ ] f p, q, q +, q, q + [ ] f pq = 9 f 6 p, q +, q + [ ] 9 f 8 p, q + + f p +, q + [ ] + f p, q + 5, q + 5 [ ] f p, q, q [ ] f p, q +, q + [ ] f p, q + + f p +, q + [ ]. 8) f p, q + 7, q + 7

21 A -D hybrid finite-difference 7 Figure 9. Receiver positions on the faut pane and contours of the rupture front. R is chosen to be the in-pane-mode and R antipane-mode receiver. R denotes receiver position near the ine of the rupture front spitting. The numbers indicate times in seconds for which the rupture front is shown. Finay, the most compicated but sti pausibe interpoation formuas are required for three out-of-grid-pane configurations abeed a, b and c in Fig. 4. Let p, q, r denote true FD-grid indices of a desired FE-grid position. Then the fourth-order interpoation formuas for the three configurations are, respectivey, f pqr = 64 [ f p, q, r, q, r +, q, r, q, r +, q +, r, q +, r +, q +, r, q +, r + ] 64 [ f p, q, r, q, r +, q +, r, q +, r +, q, r, q, r, q, r +, q, r +, q +, r, q, r, q +, r +, q, r +,, q +, r, q +, r +, q +, r, q +, r +, q, r, q, r +, q +, r, q +, r +, q, r, q, r +, q +, r, q +, r + ] 9) f pqr = 7 64 [ f p, q, r [ f p, q, r [ f p, q, r +, q, r, q, r +, q, r +, q +, r, q +, r +, q +, r +, q +, r ], q +, r + ], q +, r + ] + 64 [ f p, q, r [ f p, q, r, q, r + 5, q, r +, q +, r + 5, q +, r, q +, r + 5 ], 40), q +, r +, q, r, q, r +, q +, r, q +, r +, q, r, q +, r, q, r +, q +, r +, q, r, q +, r, q, r +, q +, r + ]

22 74 M. Gais, P. Moczo and J. Kristek f pqr = 4 64 [ f p, q, r [ f p, q, r [ f p, q, r [ f p, q, r + 7, q, r +, q, r +, q, r + 5, q, r + 7, q +, r +, q +, r +, q +, r + 5, q +, r + 7, q +, r + ], q +, r + ], q +, r + 5 ], q +, r + 7 ]. 4) 64 [ f p, q, r +, q +, r +, q, r +, q +, r +, q, r [ f p, q, r +, q +, r +, q +, r +, q, r +, q, r +, q +, r + ], q +, r +, q, r +, q +, r +, q, r +, q +, r + ] We performed an extensive series of numerica tests for the agorithmicay minima FD FE transition zone iustrated in Fig.. Some of the resuts are shown in the next subsection where they are compared with numerica behaviour of the improved transition zone. The numerica behaviour of the agorithmicay minima FD FE transition zone is not bad but ceary some sight numerica noise is present in a simuations. The presence of the noise ed us to modify the transition zone. The modified zone is presented in the next subsection. 5. Smooth transition zone with FD FE averaging The modification of the agorithmicay minima FD FE transition zone shown in Fig. ) consists in the insertion of the averaging zone between the FE Dirichet boundary and FD Dirichet zone. The modified transition zone thus consists of three distinct parts the FE Dirichet boundary, FD FE averaging zone, and FD Dirichet zone. The modified FD FE transition zone is iustrated in Fig. 5. The modification aso means that the oca thickness and staircase shape of the FD Dirichet zone is determined by requirement that the partice-veocity components at the grid interface between the averaging zone and FD Dirichet zone be cacuated using the fourth-order veocity stress staggered-grid FD scheme for an interior gridpoint. Note that in the agorithmicay minima zone the same requirement was reated to the FE Dirichet boundary. Ceary, the question is how thick the averaging zone shoud be. This can be estimated using numerica tests. Before we present resuts of the numerica tests we wi describe the agorithm of the causa FD FE communication at each time eve in the modified transition zone. The agorithm of the FD FE hybrid method can be summarized in the foowing steps: ) Dispacements U FE m + ) are updated at the gridpoints of the interior FE region the FE gridpoints except the FE Dirichet boundary). ) Stress tensor components Tm) are updated at the gridpoints of the FD region incuding the stress tensor grid positions inside the FD Dirichet zone. ) Partice veocities U FD m + ) are updated at the gridpoints of the interior FD region incuding the dashed ine see Fig. 5) between the averaging zone and FD Dirichet zone. 4) Partice veocities U FD m + ) within the FD Dirichet zone at the gridpoints indicated by doube squares and circes in Fig. 5) are updated using the FE dispacement vaues at the same gridpoints: U FD m + ) = U FEm + ) U FE m). 4) t 5) Partice veocities U FD m + ) in the averaging zone, incuding the dashed ine between the averaging and FD Dirichet zones, are repaced by vaues obtained by weighted averaging of the FE partice veocities and U FD m + ): U w FD m + ) = w U FEm + ) U FE m) t + w) U FD m + ), 4) where w = at the dashed ine between the averaging and FD Dirichet zones, and w = 0 at the FE Dirichet boundary. The weighting coefficient ineary changes between the two vaues over the averaging zone. 6) FE dispacements U FE m + ) in the averaging zone, incuding the dashed ine between the averaging and FD Dirichet zones, are repaced by averaged vaues U w FEm + ): U w FE m + ) = w U FE m + ) + w) U FD m + ), 44) U w FE m + ) = U FEm) + t U w FE m + ). 45) A grid position of the FD staggered grid is a position of either of just one partice-veocity component or one shear stress tensor component or three norma stress tensor components or none component. Eq. 44) requires a partice-veocity components at a given grid position. Consequenty, an interpoation of the missing partice-veocity components is necessary. The interpoation is the same as that described for the agorithmicay minima transition zone, see eqs ) 4).

23 A -D hybrid finite-difference 75 Figure 0. Sip-rate time histories obtained in the simuations of the rupture propagation. The vaue of δ in mutipes of the FE spatia grid spacing indicates the distance between the rupturing faut pane and FD FE transition zone. Left-hand, centra and right-hand coumns: sip-rate histories at receivers R, R and R, respectivey. At each row one of the five soutions, δ {, 5, 9,, 7} h FE, is potted together with the soution for δ = h FE. 7) Dispacements U FE m + ) are updated at the FE Dirichet boundary using the FE dispacements and FD partice veocities at the same gridpoints: U FE m + ) = U FE m) + t U FD m + ). 46) Aso eq. 46) requires a partice-veocity components at a given FE-grid position. Consequenty, an interpoation of the missing particeveocity components is necessary. Here, however, the weighted-averaged FD partice veocities in the averaging zone are used for the interpoations.

24 76 M. Gais, P. Moczo and J. Kristek Figure. Geometrica configuration of the Grenobe vaey and dynamicay rupturing thrust horizonta) faut: interface between sediments and bedrock with indication of the sediment thickness, the FE region dark grey box) covering the ruptured faut area ight grey area inside the dark grey box), projections of the FE region and ruptured faut area onto the fat free surface. The faut is at depth of 5 km. Figure. Geometrica configuration of the Grenobe vaey and dynamicay rupturing strike-sip vertica) faut: interface between sediments and bedrock with indication of the sediment thickness, the FE region dark grey box) covering the ruptured faut area ight grey area inside the dark grey box), projections of the FE region and ruptured faut area onto the fat free surface. The ruptured faut area reaches depth of.5 km. Note that we introduced the weighted averaging of the FD partice veocities and FE dispacements in the averaging zone because we assumed that such a smoothing might improve numerica behaviour of the FD FE contact. 6 N U M E R I C A L T E S T S O F T H E A L G O R I T H M I C A L LY M I N I M A L A N D S M O O T H T R A N S I T I O N Z O N E S We performed extensive numerica tests of the behaviour of the FD FE transition zone. Ceary, the first question necessary to answer was whether the agorithmicay minima transition zone Fig. ) yieds stabe and sufficienty accurate resuts. We have not found any indication of

25 A -D hybrid finite-difference 77 Figure. Materia parameters in the computationa mode of the Grenobe vaey. instabiity in the considered probem configurations in the practicay sufficient ong time windows. However, as aready noted in the section on the agorithmicay minima zone, some sight but evident numerica noise appeared in a simuations. Therefore, we tried to modify the transition zone by adding an averaging zone Fig. 5) in which pure FD vaues and pure FE vaues are repaced by weighted-averages of the FD and FE vaues. In this sense the averaging zone acts as a smoothing zone. A series of numerica tests was performed for severa canonica probem configurations: ) Unbounded homogeneous space, US, Fig. 6. ) Homogeneous haf-space with a panar free surface, FS, Fig.. ) Convergence test, Fig. 4. 4) Panar contact of two homogeneous haf-spaces, Fig. 6. 5) Dynamicay rupturing panar faut near the FD FE transition zone, Fig Unbounded homogeneous space Fig. 6 shows two configurations for tests in an unbounded homogeneous eastic space. The US- and US-8 configurations differ in the numbers of grid spacings appied to sampe the minimum waveength. grid spacings per minimum waveength are appied in the FE region in the US- configuration whereas 8 grid spacings per minimum waveength are appied in the US-8 configuration. 6 and 9 grid spacings per minimum waveength are appied in the FD regions in the two configurations, respectivey. In both configurations a point doube-coupe source was considered in the FE region. Reguary spaced h FD ) receiver positions were chosen aong a profie extending from the FE region through the FD FE transition zone into the FD region.

26 78 M. Gais, P. Moczo and J. Kristek Figure 4. Sequence of the wavefied snapshots for the simuated thrust earthquake beneath the Grenobe vaey. The grey scae indicates the absoute vaue of the horizonta component of the partice veocity at the free surface. For both probem configurations in Fig. 6 we performed numerica simuations for four different FD FE transition zones, see Fig. 7. The first transition zone, indicated by B = 0 in Fig. 7, corresponds to the agorithmicay minima transition zone with no averaging zone AZ. Three other transition zones, indicated by B =,, in Fig. 7, differ from each other in the thickness of the averaging zone AZ. P-wave veocity in the eastic medium is 596 m s, S-wave veocity 000 m s and density is 700 kg m. The source time function used is Gabor signa, see Fig. 8, st) = exp { [ ] } ω p t t s ) /γ s cos [ ω p t t s ) + ], 47) with ω p = π f p, t [0, t s ] and t s = 0.45γ s / f p. Here, γ s contros the width of the signa and is a phase shift. For certain vaues of γ s and, f p can be a dominant frequency. The signa and foca parameters of the doube-coupe point source are given in Tabe. Resuts of the numerica tests are summarized in Figs 9 and 0, where the FD FE hybrid synthetics are compared with those obtained by the discrete-wavenumber method DWN; Bouchon 98; Coutant 989). The synthetics obtained for the agorithmicay minima transition zone are shown together with the DWN synthetics in the top panes. Sight differences are evident behind the main wave group at amost a receiver positions. The synthetics obtained for the B = transition zone are shown together with the DWN synthetics in the midde panes. The FD FE and DWN synthetics practicay coincide within the thickness of the ine. A four configurations B = 0,,, ) are summarized in the bottom panes where the enveope and phase misfits of the FD FE soutions reative to the DWN soutions are dispayed for a receiver positions. The enveope and phase misfits were cacuated according to Kristekova et a. 006). The enveope and phase misfits ceary show that the accuracy of the agorithmicay minima transition FD FE zone consideraby differs from the accuracy of the three other tested transition zones. At the same time, the enveope and phase misfits for the transition zones with the averaging B =,, ) are comparabe and smaer than 0.5 per cent. This ead us to concusion that the smaest possibe thickness of the averaging zone B = ) yieds sufficienty

27 A -D hybrid finite-difference 79 Figure 5. Sequence of the wavefied snapshots for the simuated strike-sip earthquake near the Grenobe vaey. The grey scae indicates the absoute vaue of the horizonta component of the partice veocity at the free surface. accurate resuts with both eves of spatia grid samping. As expected, the denser spatia grid samping in the US-8 simuations yieds sighty smaer misfits compared to those in the US- simuations. Let us note that a sma spatia gap visibe in the top and midde panes between synthetics for the U and V components is due to the fact that the U and V grid positions in the staggered FD grid are shifted by h FD / reative to the ine of the receiver profie extending from the conventiona FE grid positions. Because the DWN synthetics were cacuated for the exact FD grid positions, the comparison of the FD FE and DWN soutions is not affected. The enveope and phase misfits are argest for the V component. This is because the misfits are scaed with respect to the maximum dispacement-component ampitude. In the considered probem configuration the maximum ampitude is in the V component. 6. Homogeneous haf-space with a panar free surface Fig. shows two probem configurations, FS- and FS-8. They are simiar to the US- and US-8 configurations. The difference between the US and FS configurations is that the top side of the FE region as we as the receiver profies are ocated directy at the free surface in the FS configurations. The purpose of this choice was to verify behaviour of the FD FE transition zone in interaction with the panar free surface. The configurations with the free surface can be considered to be more stringent tests for the behaviour of the FD FE transition zones compared to those in the unbounded homogeneous space. The reasons for this are both the physica effect of the free surface on the

28 80 M. Gais, P. Moczo and J. Kristek Figure 6. Spatia distribution of PVA at the free surface for the vertica- and dipping-faut events. Soid ine is used for the PVA isoines for the dipping-faut event, dashed ined for the vertica-faut event. wavefied and appication of the spatiay asymmetric interpoation formuas for situations abeed b and c in Fig. 4. As in the case of the US configurations, we investigated four different FD FE transition zones, see Fig. 7. Resuts of the numerica tests are summarized in Figs and, where the FD FE hybrid synthetics are compared with those obtained by the DWN method. The structure of Figs and is the same as that of Figs 9 and 0. It is cear from Figs and that the addition of the averaging zone consideraby improves the eve of accuracy. At the same time, the enveope and phase misfits for the transition zones with the FD FE averaging B =,, ) are comparabe and smaer than.0 per cent. This ead us to concusion that the smaest possibe thickness of the averaging zone B = ) yieds sufficienty accurate resuts with both eves of spatia grid samping. Again, as expected, the denser spatia grid samping in the FS-8 simuations yieds sighty smaer misfits compared to those in the FS- simuations. 6. Convergence test The probem configuration is shown in Fig. 4 and is very simiar to that of the US- for the unbounded homogeneous test Fig. 6). The purpose of this series of simuations is to check the convergence of the FD FE transition zone. We consider a fixed physica size of the cube FE region in a homogeneous unbounded medium. A point doube-coupe source is ocated at the centre of the FE region. The source time function and the foca parameters are the same as in the previous simuations. Three receiver positions are at fixed physica positions with respect to the FE region. One, R, is ocated in-between the source pane and FE Dirichet boundary, the second, R, is exacty at the FE Dirichet boundary, and the third, R, is in the FD region. Seven discretizations were considered starting with six eements per minimum waveength and ending with 4 eements per minimum waveength. Correspondingy, grid spacings were appied in the FD grid. A physica size of the FD FE transition zone as we as the eement size for each of the seven discretization is iustrated in Fig. 4. Fig. 5 compares resuts obtained with a the used discretizations in terms of the enveope misfits cacuated reative to the exact soutions for three dispacement components at the considered receiver positions. Enveope misfits are shown as functions of the number of

29 A -D hybrid finite-difference 8 Figure 7. Spatia distribution of CAV at the free surface for the vertica- and dipping-faut events. Soid ine is used for the CAV isoines for the dipping-faut event, dashed ined for the vertica-faut event. eements per minimum waveength. It is cear from the figure that the enveope misfits are sufficienty sma ess than.0 per cent for a discretizations except the coarsest one. The rate of convergence is approximatey. Note that reativey sma enveope misfit even for the coarsest discretization shoud not be surprising given the broad spectrum of the source time function Fig. 8) and the fact that six eements sampe the minimum waveength. The broad spectrum aso expains why we do not see effect of the grid dispersion a three receivers are reativey cose to the source and the source discretization effect can dominate the grid dispersion at sma distances. We shoud not pace receivers farther because the effect of arge distances woud mask the behaviour of the FD FE transition zone. Let us note that the phase misfits between the FD FE and exact soutions for a discretizations are smaer than the phase misfits between two identica signas shifted in time by one time step. Consequenty, the phase misfits cannot be used to anayse the convergence. 6.4 Panar contact of two homogeneous haf-spaces It is aso important to check the behaviour of the FD FE transition zone in a configuration with a materia interface. Fig. 6 shows such a configuration. A panar materia interface between two homogeneous eastic haf-spaces intersects the FE region and thus aso the FD FE transition zone. The cube FE region is centred around a square of the panar interface. As in the previous simuations, the wavefied is due to a point doube-coupe source that is ocated in a stiffer haf-space and inside the FE region. Receiver positions are aong a profie in the softer haf-space, two grid spacings h FD ) away from the materia interface. The receiver profie extends from the FE region through the FD FE transition zone into the FD region. The source time function and foca parameters are the same as in the previous tests. The simuation was performed ony for the B = transition zone. The FD FE hybrid synthetics for the considered probem configuration are compared with the DWN synthetics in Fig. 7. The three dispacement components of the synthetics are shown for a reguary spaced 5 h FD ) positions aong the receiver profie. The bottom pane of the figure shows the enveope and phase misfits between the FD FE and DWN synthetics. It is cear that the eve of agreement