Download - SPECFEM3D - usuarios.geofisica.unam.mxusuarios.geofisica.unam.mx/vala/cursos/Sismologia_Avanzada_2013_2... · FUNCIONES DE FORMA •Funciones de forma utilizadas para parametrizar

SPECFEM3D(ver Komatitsch y Tromp, 1999, 2002a, 2002b)

1. DIVIDIR EN ELEMENTOS

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FUNCIONES DE FORMA

• Funciones de forma utilizadas para parametrizar los elementos. Mapeo entre “Tierra” y elementos.

• Más puntos en las funciones: Más precisamente podemos seguir las fronteras en el medio.

• En SPECFEM (como muchos métodos de elementos finitos) se utiliza Lagrange polinomios de orden bajo para describir los elementos.

FUNCIONES DE FORMA 2D

Polinomias Lagrange

Función de forma

Polinomias Lagrange

Derivadas de funciones de forma

FUNCIONES DE FORMA 3D

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Derivadas de funciones de forma

Un elemento de volumen

Jacobian

Tiene que ser bien definido!!! Eso pone límites en las formas de los elementos

2. DESCRIBIR CANTIDADES FÍSICAS EN LOS ELEMENTOS

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INTERPOLAR FUNCIONES EN ELEMENTOS

• En los elementos utilizamos funciones Legendre de orden más altos. Escogemos los “Gauss-Lebatto-Legendre” puntos como puntos de control.

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GRANDE DE MATRIZES!!

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'/!' 3&. + -#$'!3$& '/( 23&,+!-("($' )(-'#* !' !++ '/( %*32,#3$'&3$ '/( %+#4!+ "(&/. -+!&&3-!++= *(:(**(2 '# !& '/( !"#$%" ,'!-''(#. .-'',#& #: '/( &=&'("0 F/( #*23$!*= 23G(*($'3!+ (H8!'3#$'/!' %#)(*$& '/( '3"( 2(,($2($-( #: '/( %+#4!+ &=&'(" "!= 4(5*3''($ 3$ '/( :#*"

/ !+!0 "+!1+"2 . IJKL

5/(*( / 2($#'(& '/( %+#4!+ "!&& "!'*3M. 0 '/( %+#4!+!4&#*43$% 4#8$2!*= "!'*3M. 1 '/( %+#4!+ &'3G$(&& "!'*3M!$2 2 '/( &#8*-( '(*"0 @M,+3-3' (M,*(&&3#$& :#* '/( +#-!+ -#$9'*348'3#$& '# '/( &'3G$(&& "!'*3M. '/( &#8*-( !$2 '/( !4&#*43$%4#8$2!*3(& !*( %3)($ 3$ '/( N,,($23M0 ?8*'/(* 2('!3+& #$ '/(

-#$&'*8-'3#$ #: '/( %+#4!+ "!&& !$2 &'3G$(&& "!'*3-(& :*#"'/(3* (+("($'!+ :#*"& "!= 4( :#8$2 3$ O#"!'3'&-/ P Q3+#''(IRSSTL0N /3%/+= 2(&3*!4+( ,*#,(*'= #: ! U@A. 5/3-/ !++#5& :#* !

)(*= &3%$3;-!$' *(28-'3#$ 3$ '/( -#",+(M3'= !$2 -#&' #: '/(!+%#*3'/". 3& '/( :!-' '/!' '/( "!&& "!'*3M / 3& 23!%#$!+ 4=-#$&'*8-'3#$0 F/(*(:#*(. $# -#&'+= +3$(!* &=&'(" *(&#+8'3#$!+%#*3'/" 3& $((2(2 '# "!*-/ '/( &=&'(" 3$ '3"(0 N' '/((+("($'!+ +()(+. '/( "!&& "!'*3M 3& %3)($ 4= '/( ;*&' '(*" 3$'/( 5(!V :#*"8+!'3#$ #: '/( (H8!'3#$& #: "#'3#$ IRKLW!

!'

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5"R#$ 5"6!

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U3"3+!*+=. 5( -/##&( '(&' :8$-'3#$& #: '/( :#*"

!I#I$. %. &LL""J

7"R#$ 7"6!

*.+.,"X8*+,7 !*I$L!+I%L!,I&L 0 IJZL


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5/(*( "*+,""I#I$*. %+. &,LL. !$2 5/(*( ! 2#' 2($#'(& 23G(*9($'3!'3#$ 53'/ *(&,(-' '# '3"(0 >' 3& 3",#*'!$' '# *(!+3^('/!' '/( 5(!V :#*" #: '/( (H8!'3#$& #: "#'3#$ /#+2& :#*%6* '(&' )(-'#* !0 F/(*(:#*(. '/( %+#4!+ &=&'(" IJKL 3& 483+'4= 3$2(,($2($'+= &(''3$% :!-'#*& #: 8*+,

R . 8*+,K !$2 8*+,

J (H8!+'# ^(*#0 _/!' 3& *("!*V!4+( !4#8' '/( *(&8+' IJ]L 3& '/( :!-''/!' '/( )!+8( #: !--(+(*!'3#$ !' (!-/ ,#3$'. !( *+,7 I3L. 3& &3",+="8+'3,+3(2 4= '/( :!-'#* -*-+-,"*+,4*+,

' 6 '/!' 3&. '/( (+("($'!+"!&& "!'*3M 3& 23!%#$!+0 F/3& ,*#,(*'= !+&# /#+2& '*8( :#*'/( %+#4!+ "!&& "!'*3M !:'(* !&&("4+= #: '/( &=&'("0 >' 3& '/3&2(&3*!4+( ,*#,(*'= '/!' /!& "#'3)!'(2 '/( 8&( #: E!%*!$%(3$'(*,#+!$'& :#* '/( *(,*(&($'!'3#$ #: :8$-'3#$& #$ '/( (+("($'&.3$ -#$`8$-'3#$ 53'/ '/( 8&( #: '/( [!8&&\E#4!''#\E(%($2*(3$'(%*!'3#$ *8+(0 a#'( '/!' 2($&3'= " $((2 $#' 4( -#$&'!$' #)(*!$ (+("($'. 48' "!= )!*= :*#" #$( %*32,#3$' '# !$#'/(*0F/(*(:#*(. '/( U@A 3& !4+( '# /!$2+( :8++= /('(*#%($(#8&"(23!0



TRK <= 1#&%373(>) %6, 4= ?-#&@

Utilizando esto se hace integración en tiempo utilizando un escema de diferencias finitas (solo en tiempo!)

SPECFEM3D_GLOBE

1. DIVIDIR EN ELEMENTOSDividir la tierra en 6 “chunks” más una caja en el centro de la Tierra. Dividir los “chunks” en NxN cajas a

lo largo del superficie. Incrementar el tamaño de los elementos con profundidad para asegurar estabilidad. “Conforming mesh”.

394 D. Komatitsch and J. Tromp

Figure 6. (a) Mesh used for the simulations presented in this study. It honors first-order discontinuities at depths of 24.4 km, 220 km, 400 km, and 670 km,the CMB, and the ICB; it also honours second-order discontinuities at 600 km, 771 km, and at the top of D!!. The mesh is doubled in size once below theMoho, a second time below the 670 km discontinuity, and a third time just above the ICB. Each of the six chunks has 240 " 240 elements at the free surfaceand 30 " 30 elements at the ICB. The triangle indicates the location of the source, situated on the equator and the Greenwich meridian. Rings of receivers witha 2# spacing along the equator and the Greenwich meridian are shown by the dashes. We also show a close-up of the two mesh doublings in the mantle (b) .

3 T H E S P E C T R A L - E L E M E N T M E T H O D

In this article we ignore the effects of self-gravitation and rotationon global wave propagation. Self-gravitation and rotation are onlyrelevant in the context of long-period surface waves and will beconsidered in a subsequent article (Komatitsch & Tromp 2002).

3.1 Mantle and Crust

The wave equation for the Earth’s mantle and crust may be writtenin the form

!"2t s = $ · T + f, (5)

where ! denotes the 3-D distribution of density and T the stresstensor which is linearly related to the displacement gradient $s byHooke’s law:

T = c : $s. (6)

In a transversely isotropic earth model, such as PREM, the elastictensor c is determined in terms of the five elastic parameters A, C,L, N, and F (Love 1911).

In an attenuating medium, Hooke’s law (eq. 6) needs to be mod-ified such that the stress is determined by the entire strain history:

T(t) =! t

%&"t c(t % t !) : $s(t !) dt !. (7)

In seismology, the quality factor Q is observed to be constant overa wide range of frequencies. Such an absorption-band solid may

be mimicked by a series of L standard linear solids (Liu et al.1976; Carcione et al. 1988; Moczo et al. 1997). In practice, twoor three linear solids usually suffice to obtain an almost constant Q(Emmerich & Korn 1987). Attenuation in the Earth is mainly con-trolled by the shear quality factor, such that only the time dependenceof the isotropic shear modulus needs to be accommodated (the bulkquality factor is several hundred times larger than the shear qual-ity factor throughout the Earth). In a transversely isotropic earthmodel one keeps track of the time dependence of the effective shearmodulus. The shear modulus of such a standard linear solid may bewritten in the form (Liu et al. 1976)

µ(t) = µR

"

1 %L

#

#=1

$

1 % $ %#%

$ &#&

e%t/$&#

'

H (t). (8)

Here µR denotes the relaxed modulus, H (t) is the Heaviside func-tion and $ &# and $ %# denote the stress and strain relaxation times,respectively, of the #th standard linear solid. Using the absorption-band shear modulus (eq. 8), the constitutive relation (eq. 7) may berewritten in the form

T = cU : $s %L

#

#=1

R#, (9)

where cU is the unrelaxed elastic tensor determined by the unrelaxedshear modulus

µU = µR

"

1 %L

#

#=1

$

1 % $ %#%

$ &#&

'

. (10)

C' 2002 RAS, GJI, 149, 390–412

Spectral-element simulations of global seismic waves 399

predictor–multicorrector format (Park & Felippa 1980; Felippa &Deruntz 1984; Thompson & Pinsky 1996; Komatitsch et al. 2000a),iterating on the coupling conditions (i.e. the surface integrals overthe CMB and the ICB). Such an iterative scheme converges veryrapidly (Komatitsch et al. 2000a), after only two iterations in prac-tice in the cases presented in this study. The iterations have a negli-gible impact on the cost of the method since we only need to iterateon the degrees of freedom that are coupled at the interface (i.e. onlythe layers of elements in contact with the CMB and the ICB, whichrepresent a very small percentage of the total number of elements).

The memory-variable equation, eq. (11), is solved for R! usinga modified second-order Runge–Kutta scheme in time, since suchschemes are known to be efficient for this problem (Carcione 1994).We do not spread the memory variables across the grid.

Two quantities that reflect the quality of the mesh are the numberof grid points per wavelength, i.e. the resolution of the mesh in termsof how well it samples the wavefield,

N = "0(v/#h)min, (47)

and the stability condition

C = #t(v/#h)max, (48)

which illustrates how large the time step of the explicit time inte-gration scheme can be while maintaining a stable simulation. Here"0 denotes the shortest period of the source, (v/#h)min denotes theminimum ratio of wave speed v and grid spacing#h within a givenelement, and (v/#h)max denotes the maximum ratio of wave speedand grid spacing. Fig. 8 illustrates that we maintain a relatively sim-ilar number of grid points per wavelength, N, throughout the mesh

Figure 8. (a) We maintain a relatively similar number of grid points per wavelength, N defined in eq. (47) for a 25 s reference period, throughout the meshshown in Fig. 6. The color scale indicates the average number of points per wavelength from 4 (dark blue) to 12 or more (red). Note that the doubling regionright below the Moho oversamples the wavefield because the size of the elements in the doubling layer is too small relative to the wave speeds. Note also thatthe number of grid points per wavelength for the inner-core shear wave (!4) is slightly too small since the SEM needs roughly 4.5 points per wavelength to beaccurate (Seriani & Priolo 1994). This is acceptable in practice because the inner-core shear wave is a very small phase. (b) Illustration of the stability condition,C defined by eq. (48), throughout the mesh. The stability value goes from 0.10 (dark blue) to 0.46 (red). Note that the size of the time step is controlled byelements in the doubling region just above the ICB, where the stability value has a maximum.

shown in Fig. 6, and also shows the stability condition, C, obtainedwith the time step used in the numerical simulations. This underlinesthat the mesh coarsening of Fig. 6 used in this article is a simple andefficient solution for meshing the entire Earth.

It is worth mentioning that the two time schemes used in this studyare only second-order accurate, contrary to the high-order spatial ac-curacy provided by the spectral-element discretization. Therefore itmight be of interest in the future to switch to higher-order timeschemes, as proposed for instance by Tarnow & Simo (1994). How-ever, in the current implementation this problem is not critical sincethe stability condition of the explicit time scheme imposes a reason-ably small time step, which provides an accurate evolution in time,even with a simple second-order scheme.

4 P A R A L L E L I M P L E M E N T A T I O N

The mesh designed for the Earth in Fig. 6 is too large to fit in memoryon a single computer. We therefore implement the method on a clus-ter of PCs using a message-passing programming methodology. Re-search on how to use large PC clusters for scientific purposes startedin 1994 with NASA’s Beowulf project, named after the famousOld English poem narrating the adventures of the Scandinavianprince Beowulf (e.g. Heaney 2000), later followed by the Hyglacproject at Caltech and the Loki project at Los Alamos (Taubes 1996;Sterling et al. 1999). The name of the initial project is now used as ageneric name for this type of architecture: these PC cluster comput-ers are referred to as ‘Beowulf ’ machines. Clusters are now beingused in many fields in academia and industry. Hans-Peter Bungefrom Princeton University was among the first to use such clustersto address geophysical problems. Their main advantage is that they

C" 2002 RAS, GJI, 149, 390–412


Figure 1. Mantle model S20RTS (Ritsema et al. 1999) is superimposed onthe mesh. A 3-D density model is obtained by scaling the shear-wave velocityvariations by a factor of 0.4, in accordance with mineral physics estimates.The figure shows lateral variations in shear velocity projected on to the foursides of the six chunks that constitute the cubed sphere mesh (see Figs I.1–6for details). Blue colours denote faster than average shear-wave velocitiesand red colours denote slower than average shear-wave velocities.

we will use model S20RTS of Ritsema et al. (1999) (Fig. 1). Lateralvariations determined by this model are superimposed on PREM(Dziewonski & Anderson 1981). Variations in density are obtainedby scaling the shear-wave velocity variations by a factor of 0.4, inaccordance with mineral physics estimates.

Every phase observed in a seismogram is affected by the Earth’scrust, so it is important to incorporate a detailed crustal model in themesh. We use Crust 2.0 (Bassin et al. 2000), which is a global 2! "2! crustal model (Fig. 2). This model is a significantly improvedversion of the 5! " 5! model Crust 5.1 (Mooney et al. 1998). We donot incorporate the ice layer that is present in some regions of Crust2.0, but we do include the sedimentary layers. The Moho depthin Crust 2.0 varies between 6.65 km (oceanic crust) and 75 km(underneath the Himalaya). The compressional-wave velocity at thesurface of Crust 2.0, excluding the sedimentary layers, varies be-tween 5.0 and 6.2 km s#1, the shear-wave velocity varies between 2.5and 3.2 km s#1, and density varies between 2600 and 2800 kg m#3.For comparison, PREM has an upper-crustal P velocity of5.8 km s#1, an S velocity of 3.2 km s#1 and a density of 2600 kg m#3.We do not honour the shape of the Moho in the mesh, since it is tooshallow in many locations (e.g. the oceanic crust) to squeeze spec-tral elements between the Moho and the surface without creatingstability problems in the time-integration scheme. Instead, we as-sign Crust 2.0 velocities and density to the pre-existing mesh. Wesmooth Crust 2.0 to suppress its sharp transitions between 2! " 2!

blocks. The grid spacing along the surface is roughly 10 km, as inPaper I (4 " 240 spectral elements along a great circle, with fivegrid points in each lateral direction of an element).

Once the mantle and crustal models have been added, we makethe Earth elliptical in shape (Fig. 3). The ellipticity as a functionof depth is determined by solving Clairaut’s equation (Dahlen &Tromp 1998), and the mesh is stretched or squashed accordingly.

Figure 2. The 2! " 2! crustal model Crust 2.0 (Bassin et al. 2000) is super-imposed on the mesh. Because the model consists of blocks with constantproperties (top), we smooth it by averaging over spherical caps with a 2!

radius (bottom). The figure shows Moho depth (which varies between 6.65and 75 km in the model). Red represents thicker than average crust and bluethinner than average crust.

Free-surface topography and bathymetry are also incorporated in themesh (Fig. 4). We use the global 5 " 5 min2 ETOPO5 bathymetryand topography model (NOAA 1988). The bathymetry map is alsoused to define the thickness of the oceans at the surface of the meshin order to take into account the effects of the oceans on global wavepropagation. As will be explained in Sections 3 and 4, the oceansare incorporated in the SEM by introducing an equivalent load atthe ocean floor, without having to explicitly mesh the water layer.

C$ 2002 RAS, GJI, 150, 303–318


Figure 1. Mantle model S20RTS (Ritsema et al. 1999) is superimposed onthe mesh. A 3-D density model is obtained by scaling the shear-wave velocityvariations by a factor of 0.4, in accordance with mineral physics estimates.The figure shows lateral variations in shear velocity projected on to the foursides of the six chunks that constitute the cubed sphere mesh (see Figs I.1–6for details). Blue colours denote faster than average shear-wave velocitiesand red colours denote slower than average shear-wave velocities.

we will use model S20RTS of Ritsema et al. (1999) (Fig. 1). Lateralvariations determined by this model are superimposed on PREM(Dziewonski & Anderson 1981). Variations in density are obtainedby scaling the shear-wave velocity variations by a factor of 0.4, inaccordance with mineral physics estimates.

Every phase observed in a seismogram is affected by the Earth’scrust, so it is important to incorporate a detailed crustal model in themesh. We use Crust 2.0 (Bassin et al. 2000), which is a global 2! "2! crustal model (Fig. 2). This model is a significantly improvedversion of the 5! " 5! model Crust 5.1 (Mooney et al. 1998). We donot incorporate the ice layer that is present in some regions of Crust2.0, but we do include the sedimentary layers. The Moho depthin Crust 2.0 varies between 6.65 km (oceanic crust) and 75 km(underneath the Himalaya). The compressional-wave velocity at thesurface of Crust 2.0, excluding the sedimentary layers, varies be-tween 5.0 and 6.2 km s#1, the shear-wave velocity varies between 2.5and 3.2 km s#1, and density varies between 2600 and 2800 kg m#3.For comparison, PREM has an upper-crustal P velocity of5.8 km s#1, an S velocity of 3.2 km s#1 and a density of 2600 kg m#3.We do not honour the shape of the Moho in the mesh, since it is tooshallow in many locations (e.g. the oceanic crust) to squeeze spec-tral elements between the Moho and the surface without creatingstability problems in the time-integration scheme. Instead, we as-sign Crust 2.0 velocities and density to the pre-existing mesh. Wesmooth Crust 2.0 to suppress its sharp transitions between 2! " 2!

blocks. The grid spacing along the surface is roughly 10 km, as inPaper I (4 " 240 spectral elements along a great circle, with fivegrid points in each lateral direction of an element).

Once the mantle and crustal models have been added, we makethe Earth elliptical in shape (Fig. 3). The ellipticity as a functionof depth is determined by solving Clairaut’s equation (Dahlen &Tromp 1998), and the mesh is stretched or squashed accordingly.

Figure 2. The 2! " 2! crustal model Crust 2.0 (Bassin et al. 2000) is super-imposed on the mesh. Because the model consists of blocks with constantproperties (top), we smooth it by averaging over spherical caps with a 2!

radius (bottom). The figure shows Moho depth (which varies between 6.65and 75 km in the model). Red represents thicker than average crust and bluethinner than average crust.

Free-surface topography and bathymetry are also incorporated in themesh (Fig. 4). We use the global 5 " 5 min2 ETOPO5 bathymetryand topography model (NOAA 1988). The bathymetry map is alsoused to define the thickness of the oceans at the surface of the meshin order to take into account the effects of the oceans on global wavepropagation. As will be explained in Sections 3 and 4, the oceansare incorporated in the SEM by introducing an equivalent load atthe ocean floor, without having to explicitly mesh the water layer.

C$ 2002 RAS, GJI, 150, 303–318

SE simulations of global seismic wave propagation—II 305

Figure 3. The ellipticity of the Earth is incorporated in the mesh. As aresult of its rotation, the Earth is slightly flattened at the poles (blue colours)and elongated at the equator (red colours). The ellipticity at the surface issmall (!! 1/300).

At this stage we therefore simply store a map of the thickness of theoceans (Fig. 5).

It is important to mention that, as in Paper I, the stability of thetime-integration scheme, i.e. the value of the time step"t , remainscontrolled by the size of the mesh near the inner-core boundary. Thestability condition is therefore not affected by the introduction of3-D heterogeneity.

Figure 4. Topography and bathymetry of the Earth, obtained from the ETOPO5 model (NOAA 1988), is added to the mesh. Left: surface elements of theactual mesh used in this paper. The colour scale represents elevation with respect to the reference ellipsoid. One can see how accurately the mesh honourstopography. Right: close-up of Mexico and the Southern United States showing the spectral elements in the mesh at the surface (grey squares). In each spectralelement we use a polynomial degree N = 4 (see Paper I for details), therefore each surface mesh element contains (N + 1)2 = 25 grid points, which translatesinto an average grid spacing of approximately 10 km at the surface.

Figure 5. Map of the thickness of the oceans and large lakes at the surfaceof the mesh. In Section 3 we use this map to represent the effects of theoceans on global wave propagation based upon an equivalent load, withouthaving to explicitly mesh the oceans. The oceans represent about 75 per centof the surface of the Earth (the colour scale indicates ocean depth) and thecontinents (yellow) about 25 per cent. The bathymetry map is taken frommodel ETOPO5 (NOAA 1988).

3 T H E S P E C T R A L - E L E M E N T M E T H O D

In this paper we incorporate the oceans, which are mostly relevantfor free-surface reflected phases, such as PP, SS and SP, and forthe dispersion of Rayleigh waves. We also include the effects of

C" 2002 RAS, GJI, 150, 303–318


Figure 19. Vertical component of displacement convolved with the instrument response and low-pass filtered at a corner frequency of 40 s at TriNet stationPAS in Pasadena, California, for the PREM normal-mode solution (left, dotted line), and a fully 3-D SEM simulation (right, dotted line), compared with realdata (solid line). The event is located in Vanuatu and the path is therefore mostly oceanic. The epicentral distance is 86!. The Rayleigh wave arrives 85 s earlierthan in PREM. Note that the 3-D SEM synthetic matches this early arrival very nicely and tracks the phase of the Rayleigh wave quite well.

Figure 20. Vertical component of displacement convolved with the instru-ment response and low-pass filtered at a corner frequency of 40 s at stationPAS in Pasadena, California, for an SEM simulation in PREM (top, dottedline) and a fully 3-D SEM simulation (bottom, dotted line) compared withreal data (solid line). The event is located in Bolivia and the path is thereforemostly continental. The epicentral distance is 68!. The 3-D SEM simula-tion improves the fit to the data significantly, in particular the large SKS,sSKS and SS phases between 1400 and 1700 s, as well as the Rayleigh wavearound 2000 s. The improvement is less spectacular than in Figs 18 and 19because the fit obtained based upon PREM was already very good for thisstation.

complexity of realistic 3-D Earth models. Specifically, we incor-porate lateral variations in P-, S-wave velocity and density in themantle, as well as a 3-D crustal model, and we show how to in-troduce the ellipticity, topography and bathymetry of the Earth. Wealso show how to take into account the oceans, rotation and self-gravitation. The effect of the oceans on global wave propagation isefficiently introduced based upon an equivalent surface load integralthat does not require an explicit meshing of the oceans, thus greatlysimplifying the method and reducing the CPU time. We validatethe implementations of self-gravitation and the oceans based upon

comparisons with PREM normal-mode synthetics at periods greaterthan 20 s for self-gravitation and 25 s for the oceans. For long-periodmultiorbit surface waves we accurately reproduce the effect of self-gravitation up to R4. Contrary to what is often assumed, we show thatfor some source–receiver configurations the effects of the oceans,self-gravitation and rotation can be significant. Both self-gravitationand the oceans have the effect of slowing down the Rayleigh wave.

As a first fully 3-D application we consider data from two earth-quakes: a shallow 1997 Mw = 7.4 event in Vanuatu and the great1994 Mw = 8.2 deep Bolivia event. For the Vanuatu event we showthat Rayleigh waves on trans-Pacific paths can arrive more than 85 searlier than in PREM, and that Love waves are much shorter in dura-tion than in PREM. For the Bolivia event we demonstrate that witha fully 3-D simulation the fit to the data is improved compared withthe same calculation in PREM.

We believe that the SEM is the method of choice for the simulationof global seismic wave propagation in fully 3-D Earth models. Thusfar, no other technique is capable of accurately incorporating all thecomplexities associated with this problem. The main current draw-back of the SEM lies in the computational cost of the large-scale3-D simulations. The calculations presented in this paper require151 processors on a parallel computer, several tens of Gigabytes ofdistributed memory, and use tens of hours of CPU time, depend-ing on the desired length of the time-series. Such requirements mayseem prohibitive, but several factors play in favour of the SEM.First, very efficient and relatively cheap parallel computers such asPC clusters (also known as ‘Beowulfs’) are now available to individ-ual researchers (Sterling et al. 1999; Komatitsch & Tromp 2001).On such machines the SEM offers superior accuracy compared withother techniques, such as finite-difference or pseudospectral meth-ods, for a comparable or even lower cost. Secondly, extremely pow-erful computers are now under development that will revolutionizeparallel computing over the next few years. The world’s fastest com-puter as of 2002, the Earth Simulator at JAMSTEC, has a capac-ity of 40 Tera floating point operations per second (1 Teraflops =1012 flops). On such a computer we estimate that all the calcula-tions presented in this paper would run in 30 min or less. Fig. 21,adapted from Thomas Sterling’s Supercomputing 2000 presenta-tion (Sterling 2000), shows an extrapolation from 1993 to 2010of the speed of the fastest computer in the world, based upon theTop500 list of supercomputers (Meuer et al. 2001). The curve showsthat we may reach a computer capable of 1 Petaflop = 1 millionGigaflops = 1015 flops around the year 2010. On such a machinethe calculations presented in this paper would very likely run in lessthan 30 s. Fig. 21 also shows a tentative estimate of the expectedevolution of processor technology over the next ten years. One can

C" 2002 RAS, GJI, 150, 303–318

er, they may be of limited validity when wewish to recover velocity variations on lengthscales smaller than approximately 1000 km(11, 12), close to the resolution provided bythe most recent 3D models. To assess thequality of a 3D model, i.e., to evaluate themisfit between the data and the syntheticseismograms, the same approximate methodsthat were used to construct the model areemployed. This means that there is a real riskthat errors in the theory are mapped back intothe model.

To overcome this dilemma, seismolo-gists have begun to use numerical methods.Unfortunately, most of the current tech-niques come with severe restrictions, andthey are frequently limited to two-dimen-sional (2D) axi-symmetric models to re-duce the computational burden. Because ofits simplicity and ease of implementation,the finite-difference technique has been in-troduced to simulate global seismic wavepropagation (13, 14 ). In this differential or“strong” formulation of the wave equation,displacement derivatives are approximatedusing differences between adjacent gridpoints, which makes the implementation ofaccurate boundary conditions difficult. Thepseudospectral technique, in which thewave field is expanded in global basis func-tions (typically sines, cosines, or Cheby-shev polynomials) (15, 16 ), has similarlimitations because it is also based on astrong formulation. Nevertheless, thismethod has been used to simulate wavepropagation in a portion of the mantle (17 ).For both finite-difference and pseudospec-tral techniques, gridding the entire globeremains an outstanding challenge (18). Al-ternative approaches, such as the direct-solution method (19, 20) and the coupled-mode method (21, 22), which are based onan integral or “weak” formulation of theequation of motion, are numerically expen-sive because of the wide coupling band-width required in the presence of stronglateral heterogeneities. Also, handling vari-ations in crustal thickness is difficult be-cause effects due to boundary undulationsare linearized. Because direct-solution andcoupled-mode methods involve the manip-ulation of large matrices, they are restrictedto modeling long-period seismograms (typ-ically periods greater than 80 s).

To be practical, a method for the simula-tion of global seismic wave propagationshould accurately incorporate effects due tovelocity and density heterogeneity, anisotro-py, anelasticity, sharp velocity and densitycontrasts, crustal thickness variations, topog-raphy, ellipticity, rotation, self-gravitation,and the oceans without intrinsic restrictionson the level of heterogeneity or the applicablefrequency range (23). The spectral-elementmethod (SEM) is such a method.

The MethodThe SEM was developed more than 15 yearsago in computational fluid dynamics (24). Itcombines the flexibility of the finite-elementmethod with the accuracy of the pseudospec-tral method. In a classical finite-elementmethod, the points that are used to define thegeometry of an element are also used for theinterpolation of the wave field. In a SEM, thewave field is expressed in terms of higher-degree Lagrange polynomials on Gauss-Lo-batto-Legendre interpolation points. This en-sures minimal numerical grid dispersion andanisotropy. The most important property ofthe SEM is that the mass matrix is exactlydiagonal by construction, which drasticallysimplifies the implementation and reducesthe computational cost because one can usean explicit time integration scheme withouthaving to invert a linear system. The SEMwas first introduced in local and regional

seismology in the 1990s (25, 26) and onlyrecently in global seismology (27–29). Whenpart of the model is 1D, the method can becoupled with a normal-mode solution in thespherically symmetric region to reduce thenumerical cost and to facilitate the analysis ofhigher frequency signals (30).

The spectral-element mesh for Earth isbased on the so-called “quasi-uniform gno-monic projection” or “cubed-sphere” (31),which is an analytical mapping from the cubeto the sphere (Fig. 2). The mesh honors thefirst- and second-order discontinuities inmodel PREM (4) and accommodates crustalthickness variations, surface topography, andellipticity. A small cube at Earth’s centermatches perfectly with the cubed-spheremesh for the surrounding globe, therebyeliminating a mesh singularity (27).

For calculations on a parallel computer,the six building blocks that constitute the

Fig. 4. Lowpass filtered (period ! 50 s) recordings of (top) the 19 February 1995 off-shoreCalifornia earthquake (Mw 6.6, event A) at stations MBO (Senegal), LZH (China), LPAZ (Bolivia),KIEV (Ukraine), and EUAT (Tonga) and (bottom) the 14 August 1995, New Britain earthquake (Mw6.7, event B) at stations PET (Russia), NIL (Pakistan), COL (Alaska), PFO (California), and RER (LaReunion island). Superimposed on the recordings (black) are 1D PREM synthetic seismograms (blue)and fully 3D SEM synthetic seismograms (red). The maps plotted above the seismograms indicatestation locations (orange triangles), earthquake epicenters (red stars), and great-circle pathsbetween epicenters and stations. Plate boundaries are drawn with thin black lines.

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