domain reduction method for three-dimensional earthquake

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Bulletin of the Seismological Society of America, Vol. 93, No. 2, pp. 825–840, April 2003

Domain Reduction Method for Three-Dimensional Earthquake Modeling

in Localized Regions, Part II: Verification and Applications

by Chiaki Yoshimura,* Jacobo Bielak, Yoshiaki Hisada, and Antonio Fernandez

Abstract Several examples are used to verify the domain reduction method(DRM), a two-step finite-element methodology described in a companion article formodeling earthquake ground motion in highly heterogeneous three-dimensional lo-calized regions. The first set involves a simple, flat-layered system. Verification ofthe DRM for this problem is carried out by comparing the results to those calculateddirectly by the theoretical Green’s function method. Its applicability to more generalproblems is illustrated by two examples: a basin and a hill with and without a weath-ered surface layer and with the same stratigraphy. We use a Green’s function ap-proach for the first step, which for the examples under consideration needs to beperformed only once. For the second step, the domain of computation is reduced ineach case to a small neighborhood of the geological feature at hand. The secondapplication considers the ground motion due to a strike-slip double couple buried14 km below the free surface in an 80-km � 80-km � 30-km region that enclosesentirely the Los Angeles basin. This problem is solved first by the finite-elementmethod using the single-step traditional approach, in which the ground motion iscalculated simultaneously near the seismic source, along the propagation path, andwithin the region of interest with a single model that encompasses the entire geolog-ical structure, from the source to the region of interest. The DRM is then used todetermine anew the ground motion over a much smaller (6-km � 6-km � 0.6-km)region contained within the original domain, and the results of the two methodswithin this region of interest are compared.

These examples serve to demonstrate that in many applications the DRM can besignificantly more efficient than the traditional approach. The DRM can be particu-larly advantageous (1) if the source is far from the local structure and the localstructure is much softer than that of the exterior region, (2) if the localized featureexhibits nonlinear behavior, or (3) if for a prescribed source, one wishes to considera sequence of simulations in which the properties of the local feature, which mightinclude man-made structures, are varied from one simulation to the next.

Introduction

In a companion article (Bielak et al., 2003), hereafterreferred to as Paper I, we have described a two-step, domainreduction method (DRM) for modeling efficiently source,path, and site effects during earthquakes. A particular caseof this method, in which the seismic excitation consisted ofa plane-wave incident wave and the examples were restrictedto two dimensions, was presented earlier (e.g., Loukakis,1988; Loukakis and Bielak, 1995). Alternative two-stepmethods that share the attractive features of the DRM havebeen presented also by other authors (e.g., Aydinoglu, 1993;

*Present address: Technology Center, Taisei Corporation, 344-1 Nase-cho, Totsuka-ku, Yokohama 245-0051, Japan; [email protected].

Zahradnık and Moczo, 1996; Moczo et al., 1997). A moredetailed bibliography is included in Paper I. Here we assessthe DRM by comparing our results with those from estab-lished methods, for two particular three-dimensional prob-lems. We also illustrate the applicability to other problemsand discuss extensions and limitations of the method.

The DRM procedure and definition of regions, bound-aries, and variables described in Paper I are summarized inFigure 1. In step 1 (Fig. 1a), one stores the free-field dis-placement within two boundaries C and Ce for a0 0u and ub e

simpler auxiliary problem without local features (X0 � X�

bounded by C�). These displacements are used for calculat-ing effective seismic forces in step 2 (Fig. 1b).eff effP and Pb e

826 C. Yoshimura, J. Bielak, Y. Hisada, and A. Fernandez

Figure 1. Summary of two-step domain reductionmethod (DRM). (a) Step 1 defines the auxiliary prob-lem over background geological model. Resultingnodal displacements within C, Ce and the region be-tween them are used to evaluate effective seismicforces Peff required for step 2. (b) Step 2, defined overreduced region made up of X and � (a truncatedXportion of X�). The effective seismic forces Peff areapplied within C and Ce. The unknowns are the totaldisplacement fields ui in X and ub on C and the resid-ual displacements we in �.X

Figure 3. Slip function used for seismic source(double couple applied within elastic half-space at1 km beneath the free surface).

Table 1Soil Parameters of Layered System

LayerThickness

(m)Vs

(m/sec)Vp

(m/sec)Density(g/cm3)

1 200 250 500 2.02 400 500 1000 2.03 � 1000 2000 2.0

Figure 2. Flat-layered system used to verify theDRM, with seismic source and region of interest(ROI).

These forces are rigorously equivalent within discretizationerror to the fault fource Pe. Since they are distributed onlyin a layer around X, which contains the local feature, thedomain size for step 2 can be reduced to a smaller region ofinterest. In step 2, the total wave field ui, ub and the residualwave field we (we � ue � , where ue is the total displace-0ue

ment in �) for a complex problem with local features (XX� � bounded by �) are calculated using .eff effˆ ˆX C P and Pb e

In this article we consider, as a first particular instance,a flat-layered system for the background structure in step 1(X0 � X�). Free-field displacements are calcu-0 0u and ub e

lated by the Green’s function method (Hisada, 1994, 1995).Verification of the DRM is carried out by taking the materialin X to be identical to that of the background structure X0

and by comparing the results to those calculated directly bythe Green’s function method. Applicability of the DRM isdemonstrated by replacing X in step 2 with local structures,including a basin or a hill. Subsequently, we consider anadditional problem involving the Los Angeles basin to il-lustrate and verify the applicability of the DRM for more

realistic situations, which entail a more complex geometryand highly heterogeneous material properties.

Flat-Layered System

Model Verification

We consider the two-layer system underlain by an elas-tic half-space shown in Figure 2; its properties are listed inTable 1. These can, of course, be readily scaled. No materialdamping is considered, for simplicity. The seismic source isa dip-slip double couple buried at a point 1 km below thefree surface. The strike, dip, and rake are 0�, 90�, and 90�,respectively, the seismic moment M0 � 6 � 1015 N m, andthe slip function is shown in Figure 3. North is aligned with

Domain Reduction Method for Three-Dimensional Earthquake Modeling in Localized Regions 827

Figure 4. Layered system within region of inter-est. (a) Finite-element mesh tailored to shear-wavevelocity of each layer and the half-space; (b) crosssection on vertical plane through AA�. The bolddashed lines show surfaces C and Ce where effectiveforces Peff are applied in step 2.

the y axis. The region of interest is a 1-km � 1-km �1-km cube located 10 km east of the epicenter.

Because of the simplicity of the physical setting and ofthe source, in this example we evaluate the displacements

on C and Ce in step 1 using the three-dimensional0 0u and ub e

Green’s functions in the computer code by Hisada (1994,1995). In step 2 the material in X is taken to be identical tothat of the background material, as shown in Figure 4. Weuse an elastic wave propagation finite-element simulationcode developed for modeling earthquake ground motion inlarge sedimentary basins (Bao et al., 1998). The wave prop-agation code is built on top of Archimedes, an environmentfor solving unstructured-mesh finite-element problems onparallel computers (Bao et al., 1998; Archimedes, 1998).Archimedes includes two- and three-dimensional mesh gen-erators, a mesh partitioner, a parceler, and a parallel codegenerator. We use linear tetrahedral elements. We haveadded the capability of automatically determining the sur-faces C and Ce once a box that defines the region of interesthas been prescribed and have built in the necessary opera-tions for evaluating the effective forces Peff. These calcula-tions are also performed in parallel since the contribution toPeff from each element within the layer adjacent to X can beevaluated independently of the other elements within thatlayer. In addition, in this study we have used a lumped massmatrix approach, in which one-fourth of the mass of eachtetrahedron is assigned to each node. Therefore, the off-diagonal elements of the mass matrix vanish, and conse-quently, the evaluation of the effective seismic forces in-volves only a multiplication of the stiffness matrix by afree-field displacement (see equation 8 in Paper I). We uselumped mass matrices to avoid the need of solving a systemof algebraic equations at each timestep. With this choice, theonly significant algebraic operation at each timestep is amatrix–vector multiplication. In addition, since linear ele-ments have nodes only at the vertices, no displacements needto be stored inside the layer between C and Ce.

The solution from the DRM for points on a fictitiousborehole that passes through points B and B� (Fig. 4b) isshown in Figure 5. Figure 5a depicts the x (east–west) com-ponent of the displacement, and Figure 5b the vertical com-ponent, at various depths. (Displacements in the y [north–south] direction are not shown as they essentially vanish,due to symmetry.) The complete wave field, including bodywaves and surface waves, can be clearly observed in thisfigure. The corresponding results from the Green’s functionsevaluations are also shown in Figure 5, for comparison. Peakvalues, with their signs, are listed on the right columns forboth solutions next to the synthetic seismograms. The agree-ment between the two sets of waveforms is quite good, withmaximum differences in amplitude on the order of 5%. Thisis consistent with the accuracy we can expect from our finite-element approximation, which is tailored to 10 points perwavelength, according to the shear-wave velocity withineach element and a maximum frequency of 1 Hz. The ap-proximate dominant frequency of the surface waves is 0.3

Hz. There are a total of 102,402 elements and 19,143 nodesin the mesh shown in Figure 4. It took 12 min of CPU timeon eight processors of the T3E computer at the PittsburghSupercomputing Center to solve the corresponding equationof motion, using a second-order central difference methodwith a timestep of 0.01 sec.

Notice that the agreement between the finite-elementsolutions and the corresponding Green’s functions remainsquite close down to the interface C, at 700 m. Right belowthis point, the finite-element solution almost vanishes. Thesame behavior is observed in Figure 6 for the seismograms


Figure 5. Synthetic seismograms for displace-ments along downhole line BB� (Fig. 4b). The depthfrom free surface and shear-wave velocity of each ma-terial is indicated to left of seismograms. (Other prop-erties are listed in Table 1.) The scale, in centimeters,is shown above the origin of the first seismogram.Peak displacements from finite-element DRM simu-lations and corresponding values from Green’s func-tions calculations are shown to the right of the seis-mograms. (a) x- (east–west) component; (b) z- (up–down) component.

Figure 6. Synthetic seismograms for displace-ments along free-surface (horizontal) line AA� (Fig.4a). The distance x is measured from the origin of thex axis. Other nomenclature is as in Figure 5.

on the free surface along AA� (Fig. 4a). The difference be-tween the results from the DRM approach and the Green’sfunctions does not exceed 5% at these locations, and thedisplacements beyond C also essentially vanish. Recall thatin the outer region �, our formulation yields residual dis-Xplacements; since the material in X is the same as that in thebackground structure for this example, we must vanish. The

fact that the numerical values of these residual displacementsare close to zero provides a useful numerical check. An in-teresting consequence of the vanishing of we for this problemis that, theoretically, the outer boundary � must play noCrole in the solution, regardless of the absorbing boundaryconditions one uses there. For the present application theboundary nodes were left unconstrained, thereby implyingthat the outer boundary is traction free. The fact that residualdisplacements in � are barely visible confirms that for theXvalidation problem the boundary condition on � has anCinsignificant numerical effect. Moreover, since there are nowaves leaving the region of interest X, one could modifythe material in the exterior region beyond a single-element


Figure 7. Homogeneous basin embedded in flat-layered system. (a) Finite-element mesh; (b) crosssection through AA�.

thick layer surrounding the surface Ce, and the results withinX would not change.

Flat-Layered System with Basin and Hill

Idealized Basin. The first example we use to illustrate theapplicability of DRM to more complex situations is one thatinvolves a local structure X with a sedimentary basin em-bedded into the same two-layer stratigraphic system we con-sidered in the previous section. The basin is in the shape ofa spherical cap and has a maximum depth of 100 m and a150-m radius at its intersection with the free surface, asshown in Figure 7. It has a uniform shear-wave velocity of125 m/sec, P-wave velocity of 250 m/sec, and density of2 gm/cm3. The seismic source is identical to that for theunperturbed flat-layered system. Thus, there is no need torecalculate the free-field ground motion on C and0 0u and ub e

Ce, as we can reuse the seismograms obtained previously.On the other hand, in contrast to the flat-layered system forwhich the residual displacement vanished outside the regionof interest, in this problem the basin generates a scatteredwave. Hence, an absorbing boundary must be introduced on

� to limit the occurrence of spurious reflections. We useCa simple dashpot approach (Lysmer and Kuhlemeyer, 1969)for this purpose, which consists of adding viscous dampersat each node on �. This gives rise to a diagonal dampingCmatrix C with nonzero terms associated only with boundarynodes.

The resulting displacement synthetics along the lineBB� (Fig. 7) are shown in Figure 8, together with the cor-responding values for the background (flat-layered) struc-ture. As expected, the basin has the effect of magnifying theamplitude of the free-field ground motion. This amplificationis confined primarily to points within the basin, where itreaches values of about 50% in the x (east–west) direction.The vertical amplification is only of the order of 20%.

Figure 9 shows synthetics for locations along AA� (Fig.7) for the x (east–west) component of the displacement. Thecorresponding results for the vertical component are in Fig-ure 10. The top panel in each figure depicts the free-fieldground motion in the background structure with no valley.There is a discontinuity across the interfaceC because withinthe region of interest X we plot the total displacement, butin the exterior region only the residual displacement isshown. The middle panel shows the corresponding resultswith the basin present, and the bottom panel the differencebetween the previous two. The latter is the residual displace-ment along the line AA� over the entire region of interest.In this case, since the stratigraphy in steps 1 and 2 is thesame, the residual wave field corresponds identically to thescattered wave field. The effect of the basin on the groundmotion can be seen explicitly from the bottom panel or bycomparing the top two panels in each figure. The basin ef-fects in the x (east–west) direction are of the same order inthe basin’s interior as the free-field motion, especially in thedeepest part of the basin. This region is affected most be-

cause the prescribed ground motion excites primarily thebasin’s fundamental mode. Effects for the vertical groundmotion are less pronounced. The residual motion is contin-uous across C, as expected, because this fictitious interfacehas no physical meaning and has been introduced merely forcomputational convenience. It is clear from Figure 9b and cand Figure 10b and c that the wave motion outside the basinis purely outgoing and that no visible spurious reflectionsare being generated at the absorbing boundary. The peakamplitude of the scattered wave field at the edges of thecomputational domain (x � 0, 1000 m) is of the order of10% of the peak amplitude of the background free-field mo-tion with no valley.


Figure 8. Synthetic seismograms for displace-ments along downhole line BB� (Fig. 7b). The solidlines show the response with basin present. Thedashed lines correspond to free-field motion (withoutthe valley). The right-hand columns show peak valueswith and without basin. Traces for points within sur-face C represent total displacement; those for pointsoutside this surface show residual displacements withrespect to free-field surface motion of the correspond-ing points for the flat-layered system. (a) East–westcomponent; (b) up–down component.

Figure 9. Synthetic seismograms for the east–west component of displacement along free-surfacehorizontal line AA� (Fig. 7a). Panels (a) and (b) showtotal displacements inside and on C and residual dis-placements outside of it; panel (c) shows residual dis-placement at all locations along AA� and thus depictsdirectly the effect of the basin on surface groundmotion.

To summarize the results from the surface response ofthe basin and its immediate vicinity to the incoming seismicwaves, we have plotted in Figure 11 the distribution of themaximum value of the total surface displacements overthe entire region of interest. Here we have included also theresponse in the y (north–south) direction. Notice the differ-

ent scales used for each component; also, the smallest andlargest values of these maxima after spatial smoothing arereported in each panel. The effect of the basin on the free-surface ground can be clearly seen for the various compo-nents of the wave field. The peak amplification occurs offcenter, especially for the vertical motion. In addition, thereare noticeable backward and forward scattering effects in thevicinity of the basin. The former leads to an increase in theground motion with respect to that of the free-field motion,while the latter has the opposite effect (Fig. 11a,c).

Idealized Hill. To illustrate the applicability of our pro-cedure to the analysis of topographic effects from exposedgeological structures, we consider, as a second example, thecase of a hill supported on the two-layered system, as shownin Figure 12. We model the ground motion for two variationsof the hill problem. In the first instance, the hill is assumed


Figure 10. Same as Figure 9, but for the up–downcomponent of displacement.

to be homogeneous with the same properties as the top layerof the background material; in the second, the hill has aweathered surface layer 25 m thick, with the same propertiesas those of the basin in the previous example. The hill hasa square base 350 m � 350 m, it is 100 m high, and thelateral sides have a slope of 45�. The seismic excitation isthe same as before, as is the numerical procedure in almostevery detail. The one difference here is that the localizedtopographical feature is located above the free surface of thelayered system, and its lateral and top surfaces are tractionfree. It is worth noting that contrary to other methods suchas finite differences, no treatment of any kind is required inthe finite-element method to enforce traction boundary orinterface conditions. These are satisfied automatically as aconsequence of the variational principle that underlies thefinite-element formulation.

Seismograms for several locations along the line AA�on the free surface in Figure 12 are shown in Figure 13 forthe homogeneous hill. The x (east–west) and z (up–down)components of the displacement are depicted, together witha list of their peak values. A comparison of these results with

the corresponding free-field displacements in Figures 9a and10a reveals an amplification on the order of 2.5 at the crestof the hill in the maximum amplitude of the east–west com-ponent of displacement. For the vertical component this am-plification is only about 1.5. This topographic amplificationcan be clearly observed in Figure 14, which shows the re-sidual displacement along the same line. In this case, theresidual motion is the scattered motion from the hill, sincethe layered structure beneath the hill is the same as that ofthe background structure. The residual motion is signifi-cantly greater in the east–west than in the up–down direc-tion. The maximum response of the hill occurs late in theexcitation, and the wave scattering is significantly strongerthan for the basin. The peak value of this scattered motionis of the order of 35% of the free-field motion at the edgesof the computational domain. This means that the hill’s ef-fect on the free-field ground motion is far from negligible.

The distribution of maximum response of the hill’s freesurface and of its neighboring region is shown in Figure 15.From this figure it is seen that the prescribed seismic sourceexcites primarily the fundamental mode of the hill. The peakamplitudes of the x (east–west) and z (up–down) componentsof displacement increase from the base to the top, and themaximum peak values occur on the eastward side of the crestfor the x component of displacement and on the westwardside and uphill plane for the vertical component. The dis-placement in the y (north–south) direction is much smallerthan in the previous two cases, but the peak values occur atthe foot of the hill and outside of it. Interestingly, in contrastto the basin problem, backward scattering here causes deam-plification and forward scattering amplification (Fig. 15a).Even though this example represents an idealized situation,it suggests that interpreting ground motion in the vicinity ofa topographic feature as free-field ground motion must bedone with caution.

Figures 16–18 show the corresponding results for thehill with the weathered layer. The results are qualitativelysimilar to those of the homogeneous hill, except that thesofter layer amplifies significantly the hill response. Com-pared to the free-field amplitude without the hill, the ampli-fication ratio of the layer east–west displacement componentis about 3.6.

An attractive feature of the DRM methodology exhibitedby the preceding examples is the relative efficiency of theassociated absorbing boundary conditions. We mentionedearlier that by choosing the residual displacements as theunknown field in the exterior region that surrounds the localgeological features, the residual ground motion in the exte-rior region is strictly outgoing and corresponds to the devi-ation of the actual structure from the background structure.It appears that this perturbation can be small even if theproperties of the local feature differ significantly from thoseof the background structure. In that case, the absorbingboundary is required to dissipate only a small amount ofenergy compared to that of the free-field motion. This effectis illustrated in Figure 19, which shows snapshots at various


Figure 11. Spatial distribution of maximum absolute value of total displacementcomponents of free-surface ground motion within the basin and its vicinity. The pointsource is located 10 km west (x � �10 km). (a) East–west component; (b) north–south component; (c) up–down component. Notice the different scales in each panel.

times taken from an animation of the ground motion for thethree cases considered thus far: the background structure(left-hand column), the basin (middle column), and the ho-mogeneous hill (right-hand column) under the prescribeddouble-couple excitation. The displaced configuration on thevertical plane of symmetry through the line AA� (see Figs.4a, 7a, and 12a) is superposed on top of the initial configu-ration for each system. Visible scattered waves emanatefrom the two structures and reach the absorbing boundaries.These scattered waves, however, are smaller than the free-field ground motion of the background structure. The reasonfor this is that some of the input energy is trapped within thestructure and is released only gradually. This implies that theamount of energy that the exterior boundary needs to absorbat a given instant when the residual wave field is chosen asthe basic unknown in the outer region can be significantlysmaller than that which would need to be absorbed if thetotal displacements were regarded as the unknowns. Thus,even if the percentage of error is the same for a particularchoice of absorbing boundary condition, its performance canbe expected to be superior for the DRM.

Figure 19 also serves to illustrate the relative responseof the background structures, basin, hill, and their immediatevicinity. It is clear that at any given instant the response ofthe modified systems differs significantly from that of thebackground region. Both the basin and the hill exhibitmarked spatial variation over short distances compared tothat of the free-field ground motion. Even though it is stifferthan the basin in this example, the hill responds morestrongly because the basin is confined within the backgroundstructure, whereas the hill vibrates freely above the freesurface.

Los Angeles Basin

Model Verification

In the preceding examples, because the backgroundstructure consisted of a set of horizontal layers overlying anelastic half-space, we were able to use a theoretical Green’sfunction approach to evaluate the free-field motion in step 1of the DRM. If the geometry is complex or the material prop-erties are highly heterogeneous, it becomes necessary to usea purely numerical procedure, such as finite differences orfinite elements. To test the DRM in a more realistic situationagainst the traditional approach, in which the source and theregion of interest are incorporated into a single model, inthis section we consider an 80-km � 80-km � 30-km re-gion that encloses entirely the Los Angeles basin and usethe southern California reference three-dimensional seismicvelocity model, version 2 (Magistrale et al., 2000), devel-oped at the Southern California Earthquake Center (SCEC),to characterize this region. We use the displacement for-mulation of the finite-element method both for the traditionalapproach and for the DRM to determine the ground motionwithin a small subdomain (region of interest) of the originalmodel, using as seismic source a buried double couple,which is located well outside the region of interest.

The SCEC velocity model consists of detailed rule-basedrepresentations of the major southern California basins, em-bedded in a three-dimensional crust over a variable-depthMoho. Outside of the basins, the model crust is based onregional tomographic results. Figure 20 presents a plan viewof the shear-wave velocity distribution at the free surface ofthe region considered, together with a vertical cross sectionalong AA� of the top 15 km of the 30-km deep computa-


Figure 12. Hill on flat-layered system. (a) Finite-element mesh; (b) cross section through line AA�,which traverses the free surface of the flat-layeredsystem and that of the hill. Two cases of hill are con-sidered in the simulations: one for a homogeneoushill, in which its properties are the same as those ofthe top surficial layer, and the second in which thehill has a weathered layer with the same properties asthose of the basin in Figure 7.

Figure 13. Synthetic seismograms for displace-ments of uniform hill along line AA� (Fig. 12a). (a)East–west component; (b) up–down component. Dis-placements inside C are total; outside they are resid-ual. This is the reason that the seismograms exhibit adiscontinuity across Ce.

tional domain across the epicenter of the seismic source. Thescale in the plan view has been capped at 350 m/sec to high-light the large degree of heterogeneity of the model. Thisvelocity is shown to take values beyond 4000 m/sec in thecross-sectional view.

The small red square shown in Figure 20 represents theregion of interest selected for this demonstration example.It is 6 km � 6 km in plan and 0.5 km deep. Due to thecomplexity, heterogeneity, and refinement of the finite-

element mesh, we do not sketch the exact location of thetwo bounding surfaces C and Ce on which the effectiveforces are to be applied in step 2 (see Fig. 1 for notation).The lateral limits of this localized region along the crosssection AA� are denoted by vertical yellow lines in Figure20b. Close-up views of the computational domain to be usedin step 2 of the DRM are shown in Figure 21. As in ourprevious models, this domain extends beyond the region ofinterest. The displayed shear-wave velocity exhibits varia-tions between 184 and 274 m/sec at the free surface. Otherregions, outside the limits shown in Figure 21, present shear-wave velocities as low as 60 m/sec at some locations.

The source is defined as a strike-slip double couple lo-cated at (40 km, 56 km, �14 km), as shown in Figure 20b,with a strike, slip, and rake of 0�, 90�, and 0�, respectively.Its seismic moment M0(t) is prescribed as

�t /T0M (t) � M [1 � (1 � t/T )e ],0 0 0

with M0 � 1 � 1018 N m, and T0 � 0.2 sec. The meshgenerated for the simulations is tailored for a maximum fre-quency of 0.5 Hz; thus, the resulting synthetic ground mo-tions are filtered accordingly.

The verification procedure follows the two steps of theDRM method: (1) large-scale simulation and calculation of


Figure 14. Synthetic seismograms for residualdisplacements along free-surface line AA� in Figure12. (a) East–west component; the peak value of thisresidual displacement is about twice that of the cor-responding value for the flat-layered system. (b) Up–down component; the peak value is about 70% thatof the corresponding value for the flat-layered system.

the effective forces at the boundaries of the region of interest,and (2) simulation within the reduced domain. To comparethe results of the DRM with those of the traditional proce-dure, which for this particular problem corresponds to step1 of the DRM, the velocity response in step 1 is recordedalong two lines of free-surface observation points within theregion of interest, as shown in Figure 21a. The cross sectionshown in Figure 21b is taken across the southwest–northeastdiagonal along observation points P. Note that the modelused in step 2 is only 600 m deep. The corresponding syn-thetics from the traditional approach (step 1) will be com-pared with those obtained from step 2, in which only thelocal region is used in the analysis.

For the first step of the calculations, our elastic wavepropagation finite-element simulation code (Bao et al.,1998) takes as input the original mesh for the entire 80-km� 80-km � 30-km model, the DRM limiting surfaces de-noted by red lines in the previous figures, the geological andgeometric characteristics of original computational domain,and the seismic source. Although no material damping isconsidered in this model, a simple viscous damping ap-proach is used as in our previous examples to limit the oc-currence of spurious reflections at the outer boundaries.

The mesh is partitioned, and a communication graph isdeveloped to distribute the computational load among allavailable parallel processors. The finite elements that inter-sect the limits of the DRM box are tagged as DRM elementsbefore the beginning of the simulation, and each processorstores the number and location of its own tagged DRM ele-ments and nodes. The simulation proceeds as a typical wavepropagation analysis, except that in addition to recordingresponses at locations of interest, the displacement field isrecorded at the tagged DRM nodes. Parallel synchronization

Figure 15. Spatial distribution of maximum value of displacement components ofground motion on free surface of uniform hill and its vicinity. (a) East–west component;(b) north–south component; (c) up–down component.


Figure 16. Same as Figure 13, but for weathered hill.

is essential for the sequential output procedures since eachprocessor outputs its DRM information to a single outputbuffer.

For step 2, the elements that belong to the outer regionand the seismic source are discarded, as shown in Figure 21.The calculation proceeds with the reduced mesh and theDRM tagged nodes as multiple seismic sources representedby the effective forces calculated from the displacement fieldin step 1. This analysis requires much smaller computing

and storage capabilities; it may, therefore, be performed ei-ther sequentially or in parallel. In either case, the recordedDRM node displacements are assigned to the new meshnodes with an interpolation scheme. We have developed anautomated procedure for which the meshes for step 1 andstep 2 need not be identical. Likewise, the simulation time-step may also differ. This represents a clear advantage forcode interaction purposes, as it is common to use differentnumerical calculation procedures, meshes, and softwaretools for the large-scale ground-motion simulation (step 1)and the small-scale ground-motion simulation, soil–structureinteraction, and building response (step 2). In this particularexample, the first and second meshes coincide, and such ascheme is not required. Numerous and repeated numericalsimulations, such as those required by nonlinear analyses orparametric studies, may now be performed with just a frac-tion of the original computational resources.

The resulting mesh statistics for the present backgroundand local simulations are shown in Table 2. The reductionin required number of mesh nodes and elements is substan-tial. This fact translates into considerable computing effi-ciency for further analysis within the local region. For ex-ample, step 1 required 3 hr on 128 parallel processors of theCray T3E machine at the Pittsburgh Supercomputing Centerfor 20,000 timesteps of 0.002 sec. In addition, approximately6 Gb of storage space are used to store mesh and data files.By contrast, the second step localized calculation may beperformed in a single-processor personal computer with lessthan 2% of the original data storage capabilities. As shownon Table 2, the second step uses only about 1% of the origi-nal mesh.

Figure 22 compares the displacement synthetics at theobservation points for the traditional approach (step 1 forthis problem) and the two-step DRM. Clearly, the resultingsynthetics consist both of body and surface waves. The two

Figure 17. Same as Figure 15, but for weathered hill.


sets of results are essentially identical. Notice that eventhough for this example the material properties within thelocalized region are almost uniform in the lateral direction,the spatial variability of the surface ground motion is quitestrong. The DRM captures accurately the complex groundmotion that is generated as the seismic waves travel fromthe source through the deep and shallow parts of the geo-logical structure within the extended region.

In dealing with the finite-element method, there is oneadditional point that should be mentioned here. It is wellknown that the displacement formulation of the finite-element method fails for incompressible materials (Poisson’sratio, � � 0.5) and leads to inaccurate results as � ap-proaches this limiting value, since the corresponding stiff-ness matrix becomes nearly singular. For the seismic veloc-ity model of the Los Angeles basin we considered here, thereare many locations where � takes values between 0.4 and0.44. Independent comparisons of our results with thosefrom finite difference calculations that use an algorithmbased on stress-velocity formulation that is insensitive toPoisson’s ratio have confirmed the accuracy of our imple-mentation of the traditional finite-element methodology(Day, 2002).

Discussion

The initial motivation for developing the DRM camefrom the desire to deal with problems for which the causative

fault is at some distance from the region of interest. By re-moving temporarily the local geological feature in step 1,we showed via the examples in the previous section that itis possible to greatly simplify the original problem, espe-cially for problems in which some portions of the domainhave very low shear-wave velocity compared to that of thebackground geology. This allows one to use coarser meshesfor the background system than would be needed with asingle-step procedure and, therefore, an increased number oftimesteps if the spatially discretized equations of motion aresolved in step 1 by an explicit step-by-step time integration.Only for step 2 is a finer mesh, and thus smaller timesteps,required in order to represent accurately the ground motionin the presence of a highly contrasting localized geologicalfeature.

There are other problems for which the DRM may beadvantageous even if the fault is not far from the region ofinterest, for example, for situations in which, because of un-certainty of the geometric and material properties of the localfeature, it may be desirable to repeat the calculations fordifferent combinations of the system parameters, such as inseismic inversion. In that case, step 1 need only be appliedonce for a prescribed source. Nonlinear soil behavior ex-tending over a limited region falls in the same category, asthe solution must be determined iteratively. Confining thenonlinearity to step 2 would then be greatly advantageous.With these applications in mind, we have developed an au-tomated interpolating procedure for which the meshes withinthe common domains need not be identical for the two steps.Similarly, the timesteps in the two simulations may also dif-fer. This allows one to use different codes for the large-scalewave propagation simulation in step 1 and for the small-scale ground-motion simulation in step 2, for which the cor-responding code may include provisions for nonlinear ma-terial behavior.

For the DRM procedure to be rigorously valid, as weindicated earlier, the material exterior to X (Fig. 1b) mustbe identical to that of the original problem. However, fromthe numerical results in the preceding section we saw thatfor the basin and hill, the residual wave field in the exteriorregion is only a fraction of the complete wave field withinthe region of interest. This suggests that one might be ableto simplify considerably step 2 for a general case, yet main-tain an acceptable approximation. Nonetheless, we should

Figure 18. Same as Figure 14, but for weatheredhill. Notice the significant increase of peak responsedue to weathering.

Table 2Mesh Statistics for DRM Simulations

ConceptStep 1:

Background CalculationStep 2:

Localized Calculation

Mesh elements �69,000,000 �660,000Mesh nodes 12,783,986 119,905Interior boundary nodes 5,026 5,026Exterior boundary nodes 5,440 5,440Interior domain nodes 109,439 109,439Exterior domain nodes 12,664,081 —DRM elements — 29,427


point out that if in selecting the region � one leaves outXgeological features such as a deep layer or a heterogeneitythat is present in the original lithology, the DRM will not berigorously equivalent to a single-step procedure that modelsthe entire region all at once, since any reflections from theheterogeneity or the deep layer due to the residual wave fieldwill be ignored in step 2. However, provided the backgroundmodel used to determine the free-field motion is0 0u and ub e

identical to the original one in the domain X�, then theequivalent seismic forces Peff will be exact to within dis-cretization error. The only approximation will be due to thesecondary reflections generated by the residual wave field,and these in many cases will be insignificant.

Concluding Remarks

The two-step DRM, described in Paper I and illustratedby several three-dimensional examples in this article, pro-vides an efficient and reasonably accurate methodology formodeling earthquake ground motion in complex localizedregions with large contrasts in material properties with re-spect to the background geology. By separating local fea-tures with possibly short wavelengths from the backgroundstructure, this methodology can make it possible to modelearthquake ground motion at higher frequencies and withgreater fidelity than has been practical up to now. While thismethod was originally conceived for cases in which the

Figure 19. Snapshots of ground displace-ment on vertical cross section across AA�(plane of symmetry) at various instants, for thebackground flat-layered system, the homoge-neous basin embedded in the flat-layered sys-tem, and the homogeneous hill atop the flat-layered system. Time is measured from theonset of the excitation at the seismic source.The scale is at the top left of figure. Displace-ments in the interior region to C are total; thosein the exterior are relative to those correspond-ing to the background layered system (residualfield). (a) S-wave arrival; (b) multireflection ofS waves and fundamental Rayleigh mode;(c) fundamental Rayleigh mode; (d) first higherRayleigh mode. Notice the radiated residualwave field from basin and hill, which is con-centrated primarily on surface layers. Observe,also, the deformation along the boundary of theregion of interest. Full animation, as well asFigures 9, 10, 13, 14, 16, and 17, shows thenegligible effect of absorbing boundary con-ditions on ground motion.


Figure 20. Shear-wave velocity model ofthe Los Angeles basin. (a) Free-surface shear-wave velocity distribution, showing a 6-km �6-km region in which the DRM analysis is per-formed; (b) cross section AA� shows the shear-wave velocity distribution down to 15 km. Theblue zones represent softer soils. Notice thatthe localized region includes the southeasternportion of the San Fernando Valley. The lati-tude and longitude of the lower left-handcorner of panel (a) are 33.7275�N and118.9080�W.

source is far from the local structure, it can be especiallyuseful for performing repeated analyses in which the sourceis kept fixed, but the properties of the local feature are variedfrom one simulation to the next. This methodology is equallyappropriate if the localized feature exhibits nonlinear behav-ior or there are engineered structures present within the re-gion of interest. Additional computational savings can begained if the region of interest is restricted to the local geo-logical feature and its vicinity and excludes heterogeneitiesor deep layers some distance away. However, errors due tosecondary reflections generated by the residual wave fieldwill occur if the background region contains heterogeneitiesthat are not included in the reduced region. While our nu-merical results indicate that the outgoing waves are smalldue to the impedance contrast between the material withinthe region of interest and the exterior region, the issue ofsecondary reflections deserves further investigation. We be-

lieve that the DRM provides a useful tool toward the assess-ment of seismic hazard and seismic risk reduction in urbanareas within basins with complex topography.

Acknowledgments

This work was supported by the National Science Foundation, underGrant CMS-9980063 of the KDI program, and under Grant EEC-0121989of the Engineering Research Centers. The cognizant program directors areClifford J. Astill and Joy M. Pauschke, respectively. Computing serviceswere provided by the Pittsburgh Supercomputing Center on the Cray T3E.This work was also partly supported by a special project of U.S.-JapanCooperative Research for Urban Earthquake Disaster Mitigation (No.11209201), Grant-in-Aid for Scientific Research on Priority Area (CategoryB), Ministry of Education, Culture, Sports, Science, and Technology ofJapan (PI: Tomotaka Iwata), and by Taisei Corporation. We are gratefulfor this support. We thank Peter Moczo and an anonymous reviewer, whosereviews helped us improve the manuscript. The work was done while oneof the authors (C.Y.) was visiting Carnegie Mellon University as a researchscholar on leave from Taisei Corporation.


Figure 21. Close-up of the DRM region of anal-ysis with the location of the observation points. (a)Plan view with longitudinal (L) and perpendicular (P)cross sections; (b) 600-m deep cross section along Preceiver line. The red line represents the limits of thelocal region. Note that panel (b) shows only the top600 m of the 30-km deep original model. The dis-played values of the shear-wave velocity present var-iations between 190 and 316 m/sec at the free surface.However, the model used in the simulations includesvelocities as low as 60 m/sec.

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Department of Civil and Environmental EngineeringCarnegie Mellon UniversityPittsburgh, Pennsylvania [email protected], [email protected]

(C.Y., J.B., A.F.)

Department of ArchitectureKogakuin University1-24-2 Nishi-shinjuku, Shinjuku-kuTokyo 163-8677, [email protected]

(Y.H.)

Manuscript received 21 September 2001.


Figure 22. Displacement synthetics (in centimeters). The figure compares the re-sponse at the observation points for the traditional single-step and two-step DRM cal-culations. The results are identical since the effective force nodes of the first- andsecond-step analyses are those from the original mesh; that is, there are no additionalspatial discretization errors between steps 1 and 2. The readings along line L (longi-tudinal) show more pronounced phase differences than those along line P, consistentwith the location of the hypocenter, the magnitude of the shear-wave velocity, and the

distance between the observation points.

domain reduction method for three-dimensional earthquake

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