GLOBAL SEISMIC FOCUSING AT THE ANTIPODE

R. Brian White

Department of Earth and Planetary Sciences; Washington University; St. Louis, MO, USA

October 25, 2002

ABSTRACT

There is a remarkable focusing of seismic body and surface wave energy at the antipode of seismic sources. This phenomenon is explained and the difficulties with creating synthetic seismograms at the antipode are discussed. Examples of antipodal focusing are then given and compared to synthetics. Applications of antipodal data to studying Earth structure are then discussed with special attention given to the inner core. Finally, implications for the focusing of seismic energy from impacts on the Earth and other planets are discussed.

INTRODUCTION

The antipode is the point at the opposite side of a planet (=180). For our purposes, we are

interested in the antipode of seismic sources such as earthquakes, explosions, or impacts. For a

spherical planet, all great circles paths from a given source pass through the antipode, meaning the

antipode receives seismic energy from all azimuths. The possibility of antipodal focusing of seismic

energy has been realized since the earliest days of seismology [Gutenberg and Richter, 1934], and it was

noted that obscure inner core phases such as PKIIKP could be enhanced at the antipode [Bolt and

O'Neill, 1964]. However, the rarity of antipodal source-receiver geometries has limited observations of

antipodal focusing. Studying seismic phases at antipodal distances thus provides the investigator with a

global view of the Earth within a very compact area.

The compressional wave that passes directly through the Earth is PKIKP. This phase is not

amplified at the antipode because it arrives from only one azimuth. However, other phases that undergo

boundary interactions such as PKP, PP, and PKIIKP are focused at the antipode since they arrive over a

range of azimuths. The contribution of energy from all azimuths adds constructively to give greatly

amplified arrivals. Phases diffracted from the inner/outer core boundary (PKP(BC)) and the core/mantle

boundary (Pdiff) are also focused in this manner. The antipode is also a focusing point for Rayleigh waves.

Thus, we can study the global structure and homogeneity of the Earth by looking at the relative timing,

amplitude, and shape of antipodal waves.

Figure 1: Compressional Wave Phases Focused at the Antipode (Pdiff and others not shown).

Figure 2: Antipodal observations at two WWSSN long-period vertical component stations for an event in New Zealand [Rial and Cormier, 1980].

BODY WAVE FOCUSING

One way to estimate the amplitude of arrivals at the antipode is with Richter’s magnitude formula,

which relates the body wave magnitude mb to amplitude A, period T, and an empirical correction factor for

distance and depth B(,h,).

€

mb = log A T( ) +B Δ,h( ) (1)

For a given earthquake with known location and magnitude, we can solve for the amplitude at any period.

Butler [1986] did the calculation for a magnitude 4.6 shallow focus earthquake recorded at 1Hz. Table 1

summarizes his results. Here, we can see how the P-wave amplitude quickly dies off over the first few

tens of degrees. Note that P is higher in amplitude than PP even at 175 . However, at 180, PP’s

amplitude increases by a factor of 25 compared to only 5 degrees away. We can see that the highest

amplitudes occur at the antipode and that the amplitude of PP is about three times higher than PKP.

Note also the small amplitudes for PKIKP and PKIIKP, even though their amplitudes are also maximized

at the antipode. Not shown here is the Pdiff phase, which is the first arrival at antipodal distances. It has

2

an amplitude considerably smaller than PKIKP. On average, the seismic energy per unit area is about

750 times higher at the antipode than at a distance only 5 away (175), meaning the amplitude is about

€

750 = 27 higher there [Butler, 1986].

Phase Distance()

Amplitude(nm)

P 17 50P 45 8PP 175 5PP 180 125PKP 180 40PKP(BC) 180 1.35PKIKP 180 0.6PKIIKP 180 1

Table 1: Relative Amplitudes of Phases Computed from the Richter magnitude formula [from Butler, 1986].

A more convincing argument for antipodal focusing can be found by looking at synthetic

seismograms. However, the usual form of ray theory equations have a caustic at the antipode, meaning

€

d 2T dΔ2 → ∞. We can easily see this by considering the energy per unit area for a uniform spherical

Earth.

€

E

area=

kc

2πr 3

tan i

cos i sinΔ

d 2T

dΔ2 (2)

Here, E is energy, k is a wavenumber, r is radial distance from the source, i is the takeoff angle, T is travel

time, and is epicentral distance. Thus, the solution is undefined at the antipode because sin(180)=0.

This singularity occurs because high-order asymptotic expansions for the Legendre functions fail to be

valid at the antipode [e.g. Rial, 1978]. We can see this by looking at the expression for the zeroth-order

asymptotic approximation to the normalized Legendre functions [Romanowicz and Roult, 1988].

€

Plm Δ( ) =

1

π sin Δcos l +

1

2

⎛

⎝ ⎜

⎞

⎠ ⎟Δ−

π

4+

mπ

2

⎡

⎣ ⎢

⎤

⎦ ⎥ (3)

Rial [1978] and Rial and Cormier [1980] were the first researchers to successfully deal with this

problem by using uniform asymptotic expansions for the Legendre functions of large order in terms of

Bessel functions. In order to obtain finite amplitudes at the antipode, they used Legendre functions or

order l (Pl) away from the antipode and zero-order Bessel functions (J0) near the antipode.

€

Pl cos π − Δ( )[ ] for 150o ≤ Δ ≤ 178o

J0 l π − Δ( )[ ] for 178o ≤ Δ ≤ 180o(4)

By integrating over suitable contours on the complex ray parameter plane, they obtained the frequency

domain response of a realistic Earth model to a point source. They used high-frequency approximations

3

to the solutions of the Helmholz equation, which included the effects of frequency-dependent transmission

and reflection coefficients [Rial and Cormier, 1980]. Figure 3 shows theoretical spectral amplitudes for a

PP wave in the vicinity of the antipode. We can see that amplitude is proportional to the square root of

the frequency.

Figure 3: Theoretical PP spectral amplitudes near the antipode for periods between 5 and 65 s [Rial, 1978].

By taking the inverse Fourier transform of this frequency domain response, time domain Green’s

functions could be generated for the phases of interest. Synthetics were then computed by convolving the

source time function with Green’s function response. Hence, they obtained full wave theory solutions in

terms of displacement potentials for Pdiff, PKIKP, PKIIKP, PKP(BC), PKP, and PP in a spherical Earth

using realistic structural models. Figure 4 illustrates the Green’s functions and associated synthetics for

arrivals between 175 -180. With the Green’s functions, it is easy to see that the arrivals coming together

from opposite directions add constructively to produce an order of magnitude amplification at the

antipode. PKIKP is not amplified because it is not a sum of multiple arrivals. PKP and PP undergo

opposite /4 phase shifts when they cross the axial caustic at the antipode [Rial and Cormier, 1980].

4

Figure 4: (A) Green’s functions and (B) synthetic seismograms near the antipode [Rial and Cormier, 1980].

SURFACE WAVE FOCUSING

The whole earth acts as a lens for surface waves, which spread out over the spherical surface of

the earth and converge at the antipode. For a spherically-symmetric Earth, Rayleigh waves constructively

interfere to give amplified arrivals (Figure 3), but Love waves destructively interfere to give no net Love

wave motion at the antipode. Like body waves, the approximations generally used for Rayleigh wave

propagation break down close to the 180 (see Equation 3) because at the antipode wave trains arrive

simultaneously from all azimuths and do not allow for the usual high-frequency great circle path

representation [Romanowicz and Roult, 1988]. This occurs for both ray theory and normal mode

summation surface wave synthetics. To avoid this problem, various alternate asymptotic approximations

to the Legendre functions [Dahlen, 1980] or first-order perturbation theory [Romanowicz and Roult, 1988]

have been used. More recently, the spectral element approach has been shown to faithfully generate

antipodal synthetic seismograms [Komatitsch and Tromp, 1999,2002a,b].

Because the antipode is a caustic for Rayleigh wave propagation, energy arrives at all azimuths,

and constructive interference provides enhanced signal to noise. Dispersion smears out this focusing, so

the focusing is less intense for each successive surface wave arrival at the antipode. In fact, significant

spreading of thousands of kilometers can be seen for antipodal R3 arrivals [Lay and Kanamori, 1985].

5

This degradation of focusing is caused by lateral heterogeneities in the earth’s structure, which cause the

rays to deviate from great circle paths. Since Rayleigh waves are sensitive to the upper mantle, they are

a good way to measure attenuation there. Chael and Anderson [1982] used antipodal observations of

Rayleigh waves to obtain global attenuation estimates and determined that the upper mantle is more

attenuative than is generally accepted. By examining the quality of the focusing near the antipode, they

were able to constrain the effects of lateral heterogeneities in the Earth’s upper mantle.

Figure 3: Radial component Rayleigh wave accelograms near the antipode [Dahlen and Tromp, 1998].

USING ANTIPODAL DATA TO STUDY EARTH STRUCTURE

Since antipodal waves arrive from all azimuths, observations at a single station allow us to

sample a large volume of the Earth. By looking at the relative timing and amplitudes of the PKP

branches, one can study properties of the inner and outer cores and their boundaries. For example,

PKP(AB-DF) can be used to study the core-mantle boundary (CMB) region. Poupinet, et al. [1993] found

that PKP(AB) (which is a diffracted wave at the antipode) has a very similar shape to PKIKP (which is

PKP(DF)), implying that the CMB does not exhibit strong topography or heterogeneities at short

wavelengths. However, Luo, et al. [Luo et al., 2001] studied AB-DF travel time residuals at shorter

distances (approximately 150-170) and found strong lateral variations in the D’’ region from multipathed

PKP(AB). PKP(AB-DF) differential travel times can also be used to study inner core anisotropy. Sun and

Song [2002] found that the AB-DF residuals for polar paths are consistently larger than those for

equatorial paths, implying inner core anisotropy of about 2.5%. However, one should be cautioned that

D’’ heterogeneities can bias inner core anisotropy measurements [e.g. Bréger et al., 2000]. PKP(BC-DF)

residual patterns have been widely used to study inner core anisotropy [e.g. Song, 1997; Bréger et al.,

1999; Sun and Song, 2002], and they also indicate aspherical structure within the Earth’s core [Poupinet

et al., 1983; Shearer and Toy, 1991; Tanaka and Hamaguchi, 1997]. PKIKP can also be used to study

6

inner core attenuation [Cormier et al., 1998] and rotation [Song, 1996; Souriau and Poupinet, 2000;

Collier and Helffrich, 2001]. For good reviews, see Song [1997; 2000] and Tromp [2001].

Other antipodal phases can be studied too. Rial [1978] showed that a rare phase (PKIIKP)

between PKIKP and PKP can be observed at the antipode and suggested that PKIKP-PKIIKP and

PKP(BC)-PKIIKP could be important probes for inner/outer core boundary structure. Since the signal-to-

noise ratio is also enhanced ten-fold at the antipode, other rare phases such as PKJKP can also

potentially be seen only at antipodal distances [Rial, 1978]. Butler [1998] found a new phase between

PKIIKP and PKP(AB), which suggests a low velocity zone at the inner-outer core boundary. Song and

Helmberger [1998] found also explored inner core structure and found a triplication that implied an

isotropic upper inner core over an anisotropic lower inner core. This triplication was due to a velocity

increase. Since PP in the vicinity of the antipode samples vertically and laterally a large region of the

base of the mantle, it is an excellent way to study the D’’ region [Rial and Cormier, 1980]. PP is the

phase most affected by ellipticity and lateral heterogeneity in the Earth, so it is a sensitive probe for these

too [Butler, 1986]. Pdiff is another phase seen at the antipode that can give bulk properties of the CMB

because it samples such a large portion of it [Rial and Cormier, 1980].

Rial and Cormier [1980] also suggested that source parameters of large earthquakes can be

constrained by antipodal observations since near-receiver effects are minimized at zero ray parameter,

and very large first arrivals are smallest at the antipode. They suggested that modeling large earthquakes

could be greatly simplified and less ambiguous using antipodal data than anywhere else. The antipode is

probably also the best place to detect small events at teleseismic distances, which has important

implications for nuclear test monitoring [Butler, 1986]. Antipodal monitoring of seismic waves provides a

global view of the Earth’s structure within a very concentrated region. Thus, placing an array of

seismometers antipodal to an active source region (such as Tonga – north Africa) can provide a much

more direct and economical view of the Earth than other distributions of seismometers. Butler, et al.

[1986] proposed placing a seismic array antipodal to the Nevada nuclear test site in gain a high-resolution

picture of the core, since the location of the events would be known exactly. By setting off explosions at

the antipode and recording them back in Nevada, the reverse profile experiment can be carried out to

further improve the results [Butler, 1986]. This same principle could be applied to investigating the

interiors of other planets by installing seismic arrays antipodal to artificial sources such as impacting the

planet at the antipode of the array [Rial and Cormier, 1980]. Synthetics computed and compared to data

gathered from such an array could be used to solve for the best possible models of the planet’s interior.

Currently, antipodal configurations are not being considered for the upcoming NetLander mission on Mars

[Mocquet, 1999; Lognonné et al., 2000], but it was being considered for the INTERMARSNET mission

[Angelis and Chicarro, 1996].

IMPACTS AND ANTIPODAL FOCUSING

7

Strange pitted and hummocky terrains have been observed at the antipodes of the Caloris basin

on Mercury [e.g. Strom, 1997], the Imbrium and Orientale basins on the Moon [e.g. Schultz and Gault,

1975b], and some of the icy satellites [Watts et al., 1991]. Magnetic anomalies at the antipodes of lunar

basins are thought to be due to impact-generated ionized vapor clouds that converged at the antipodes

and magnetized the crust [Hood and Huang, 1991], and a high abundance of thorium antipodal to the

Serenitatis basin could be due to the convergence of impact ejecta there [Wieczorek and Zuber, 2001].

Although no hilly and lineated terrain is seen at the antipodes of large impacts on Mars, Alba Patera

volcanism does occur a the antipode of the Hellas basin, and it is thought that the convergence of seismic

waves could account for deep-seated fractures that allowed volcanism to form there [Peterson, 1978;

Williams and Greeley, 1994]. Furthermore, the lack of such hummocky terrain on Callisto could be

evidence of a subsurface ocean there [Williams et al., 2001]. On Earth, energetic impacts have been

linked to many geophysical processes including continental flood basalt volcanism, mantle plumes,

continental rifting, mass extinctions, and geomagnetic pole reversals [Seyfert and Sirkin, 1979; Alvarez et

al., 1980; Burek and Wänke, 1988; Rampino and Caldeira, 1992; Morgan et al., 2002], although the

possibility of impact-induced volcanism is strongly questioned [e.g. Melosh, 2001; Ivanov and Melosh,

2002].

Spalation at the antipodes of micrometeoroid impacts into glass sphere suggested that seismic

effects could be important at the antipodes for large impacts [Gault and Wedekind, 1969]. Schultz and

Gault [1975a; 1975b] proposed that the antipodal terrains on the Moon and Mercury were produced by

impact-generated seismic waves. Using a simple model of a homogeneous planet, they found antipodal

ground motions in excess of 10 m for Imbrium’s antipode and showed that reflected body waves from the

antipode region will actually converge at a point below the antipode. Hughes et al. [1977] used a finite

difference code to study the effects of large impacts into either homogeneous solid or liquid planets.

Ignoring the shock wave decay into the elastic regime, they also found sizeable antipodal displacements

(approximately 1 km), with velocities of tens of m/s, and accelerations approaching lunar gravity. The

waves reflected from the free surface also focused at a point beneath the antipode, and the antipodal

disturbances were 2-3 greater for a liquid planet than a solid one because the solid is more efficient at

dissipating energy. These early works provided order-of-magnitude estimates of the expected antipodal

seismic effects of large impacts, but the over-simplified models and poor treatment of elastic waves led to

questionable results.

The first study to consider the effect of the core was Watts et al. [1991], who used a finite element

code to calculate the antipodal pressures generated by large impacts in two-layer planets. They found

surface accelerations and pressures that exceed the strength of rock. Focusing is enhanced if the core

size is small or large compared to the planet, but focusing is minimized for a core size approximately 1/3

the planet’s size. Williams and Greeley [1994] extended this work for impacts on Mars and included a

third layer (the crust) in their calculations. They also found very high surface pressures at the antipodes

of Mars’ large basins, but only the antipode of Hellas indicates that these could have been important

8

because of the volcanism associated with Alba Patera. One problem with these two studies is that they

do not report how they arrived at the transient crater dimensions, which are essential for impact energy

estimates [Melosh, 1989].

The most realistic treatment of the impact-generated seismic waves problem was done by

Boslough, et al. [1995]. Because of the wide range of spatial and temporal scales of the problem, they

used a hydrodynamics code to simulate the high stresses and strains in the near-source shock regime

and normal mode synthetics for the seismic wave propagation. They used a 10-km diameter asteroid

impacting vertically at 20 km/s to model the Chixulub impact on Earth. When the pressure at the shock

front dropped below the elastic limit, the results of the near-field shock physics simulations were used to

generate source functions for the subsequent seismological synthetic calculations. For the normal mode

synthetic calculations, they used the spherically-symmetric Earth model of Gilbert and Dziewonski

[Gilbert and Dziewonski, 1975] and the attenuation profile from PREM [Dziewonski and Anderson, 1981].

The work of Boslough, et al. [1995] demonstrated that displacement and strain amplitudes near

the antipode are orders of magnitude higher than over the rest of the Earth’s surface (except the source

region) and that the seismic energy remains sharply focused along the axial line connecting the source

and antipode (Figure 5). Like the simpler studies before them, they also found that focusing occurs

beneath the antipode with the high amplitudes mostly confined to the upper mantle but significant down

the core-mantle boundary. The peak displacement at the antipode approaches tens of meters for the

frequencies considered (up to 0.022 Hz). Some improvements for their work consist of (1) including

higher order modes to better characterize the body wave phases, (2) using a modern Earth model with

lateral heterogeneity and ellipticity, (3) estimating the coupling of energy and momentum of the impactor

into the surface [Melosh, 1989; Holsapple, 1993], (4) adding a heat dissipation model to determine extent

of any resulting thermal anomaly in the Earth’s interior, (5) exploring the effect of an oblique impact on the

resultant seismic waves, since oblique impacts are known to create significant shear waves [Dahl and

Schultz, 2001], and (6) extending this work to impacts on other planets.

9

Figure 5: (A) Peak strain as a function of radial distance from the source; (B) Peak radial displacement, (C) peak strain, and (D) peak stresses as a function of depth along the radius beneath the antipode down to the CMB [Boslough et al., 1995].

CONCLUSIONS

We have seen how many seismic body and surface waves are greatly amplified at the antipode

after sampling a large part of the deep Earth structure. The signal-to-noise is similarly enhanced, making

it easier to see rare phases such as PKIIKP and PKJKP. An important factor in the amount of antipodal

focusing is the nature source. Ideally, one wants the nodal planes of the source to be far from the center

of the focal sphere, meaning the ideal earthquake would have a vertical axis of compression or tension

[Rial and Cormier, 1980]. More complicated sources make the integration paths on the complex ray

10

parameter plane more complicated. An isotropic explosion, for example, radiates energy with uniform

energy and phase at all azimuths, giving complete constructive interference for vertical motion and

cancellation of horizontal motion at the antipode of a spherically symmetric planet [Chael and Anderson,

1982]. Note that if the source is a vertical impact, no toroidal modes are generated—only spherical

modes [Boslough et al., 1995]. It should be noted that the actual focusing amplitude depends upon the

lateral heterogeneity of the Earth. Focusing decreases with increasing heterogeneity. Thus, with

ellipticity taken into account, the measured focusing at the antipode can provide a good measure of the

lateral heterogeneity affecting the seismic waves observed [Butler, 1986].

Antipodal studies can tell use about departures from lateral homogeneity and sphericity of the

core and mantle to better-constrain models. Inner core anisotropy, rotation, and structure are particularly

well-suited to antipodal observation. Modeling impacts as seismic sources can provide important

information on antipodal seismic focusing for the Earth and other planets. This could have exciting

implications for studying impact-induced volcanism, plume formation, continental rifting, and geomagnetic

pole reversals. Since antipodal observations of seismic phases provide so much information in a

compact area, we should be utilizing this geometry more for seismic arrays on Earth and should seriously

consider as the most efficient initial arrays on other planets.

REFERENCES

Alvarez, L.W., W. Alvarez, F. Asaro, and H.V. Michel, Extraterrestrial Cause of the Cretaceous/Tertiary Extinction, Science, 208, 1095-1108, 1980.

Angelis, G.D., and A. Chicarro, A Catalogue of Potential Landing Sites for the INTERMARSNET Mission, Planet. Space Sci., 44 (11), 1325-1346, 1996.

Bolt, B.A., and M.E. O'Neill, Times and Amplitudes of the Phases PKiKP and PKIIKP, Geophys. J. R. astr. Soc., 9, 223-231, 1964.Boslough, M.B., E.P. Chael, T.G. Trucano, and D.A. Crawford, Axial Focusing of Energy from a Hypervelocity Impact on Earth, Int.

J. Impact Engng., 17, 99-108, 1995.Bréger, L., B. Romanowicz, and H. Tkalcic, PKP(BC-DF) Travel Time Residuals and Short Scale Heterogeneity in the Deep Earth,

Geophys. Res. Lett., 26 (20), 3169-3172, 1999.Bréger, L., H. Tkalcic, and B. Romanowicz, The Effect of D'' PKP(AB-DF) Travel Time Residuals and Possible Implications for Inner

Core Structure, Earth Planet. Sci. Lett., 175, 133-143, 2000.Burek, P.J., and H. Wänke, Impacts and Glacio-eustasy, Plate-tectonic Episodes, Geomagnetic Reversals: A Concept to Facilitate

Detection of Impact Events, Phys. Earth Planet. Int., 50, 183-194, 1988.Butler, R., Amplitudes at the Antipode, Bull. Seism. Soc. Am., 76 (5), 1355-1365, 1986.Butler, R., Antipodal Observations of the Inner-outer Core Boundary and Diffracted P, EOS Trans. AGU, 79 (??), ???, 1998.Butler, R., T.M. Brocher, and J.A. Rial, Inner Core Experiments: Teleseismic Exploration of the Antipode, EOS Trans. AGU, 67 (8),

89, 93-94, 1986.Chael, E.P., and D.L. Anderson, Global Q Estimates from Antipodal Rayleigh Waves, J. Geophys. Res., 87 (B4), 2840-2850, 1982.Collier, J.D., and G. Helffrich, Estimate of Inner Core Rotation Rate from United Kingdom Regional Seismic Network Data and

Consequences for Inner Core Dynamical Behaviour, Earth Planet. Sci. Lett., 193, 523-537, 2001.Cormier, V.F., L. Xu, and G.L. Choy, Seismic Attenuation of the Inner Core: Viscoelastic or Stratigraphic?, Geophys. Res. Lett., 25

(21), 4019-4022, 1998.Dahl, J.M., and P.H. Schultz, Measurement of Oblique Impact-generated Shear Waves, Proc. Lunar Planet. Sci. Conf., 32, 2001.Dahlen, F.A., A Uniformly Valid Asymptotic Representation of Normal Mode Multiplet Spectra on a Laterally Heterogeneous Earth,

Geophys. J. R. astr. Soc., 62, 225-247, 1980.Dahlen, F.A., and J. Tromp, Theoretical Global Seismology, Princeton University Press, Princeton, NJ, 1998.Dziewonski, A.M., and D.L. Anderson, Preliminary Reference Earth Model, Phys. Earth Planet. Int., 25, 297-356, 1981.Gault, D.E., and J.A. Wedekind, The Destruction of Tektites by Micrometeoroid Impact, J. Geophys. Res., 74 (27), 6780-6794, 1969.Gilbert, F., and A.M. Dziewonski, An Application of Normal Mode Theory to the Retrieval of Structural Parameters and Source

Mechanisms from Seismic Spectra, Phil. Trans. Roy. Soc. Lond., A 278, 187-269, 1975.Gutenberg, B., and C. Richter, On Seismic Waves, Gerlands Beitr. Geophys., 43, 56-133, 1934.Holsapple, K.A., The Scaling of Impact Processes in Planetary Sciences, Annu. Rev. Earth Planet. Sci., 21, 333-373, 1993.

11

Hood, L.L., and Z. Huang, Formation of Magnetic Anomalies Antipodal to Lunar Impact Basins: Two-dimensional Model Calculations, J. Geophys. Res., 96 (B6), 9837-9846, 1991.

Hughes, H.G., F.N. App, and T.R. McGetchin, Global Seismic Effects of Basin-forming Impacts, Phys. Earth Planet. Int., 15, 251-263, 1977.

Ivanov, B.A., and H.J. Melosh, Impacts do NOT Initiate Volcanic Eruptions!, Lunar and Planetary Laboratory Conference, 2002.Komatitsch, D., and J. Tromp, Introduction to the Spectral Element Method for Three-dimensional Seismic Wave Propagation,

Geophys. J. Int., 139, 806-822, 1999.Komatitsch, D., and J. Tromp, Spectral-element Simulations of Global Seismic Wave Propagation - I. Validation, Geophys. J. Int.,

149 (2), 391-???, 2002a.Komatitsch, D., and J. Tromp, Spectral-element Simulations of Global Seismic Wave Propagation - II. Three-dimensional Models,

Oceans, Rotation, and Self-gravitation, Geophys. J. Int., 150 (1), 303-???, 2002b.Lay, T., and H. Kanamori, Geometric Effects of Global Lateral Heterogeneity on Long-period Surface Wave Propagation, J.

Geophys. Res., 90 (B1), 605-621, 1985.Lognonné, P., D. Giardini, B. Banerdt, J. Gagnepain-Beyneix, A. Mocquet, T. Spohn, J.F. Karczewski, P. Schibler, S. Cacho, W.T.

Pike, C. Cavoit, A. Desautez, M. Favède, T. Gabsi, L. Simoulin, N. Striebig, M. Campillo, A. Deschamp, J. Hinderer, J.J. Lévéque, J.P. Montagner, L. Rivéra, W. Benz, D. Breuer, P. Defraigne, V. Dehant, A. Fijimura, H. Mizutani, and J. Oberst, The NetLander Very Broad Band Seismometer, Planet. Space Sci., 48, 1289-1302, 2000.

Luo, S.-N., S. Ni, and D.V. Helmberger, Evidence for a Sharp Lateral Variation of Velocity at the Core-mantle Boundary from Multipathed PKPab, Earth Planet. Sci. Lett., 189, 155-164, 2001.

Melosh, H.J., Impact Cratering: A Geologic Process, Oxford University Press, New York, 1989.Melosh, H.J., Impact-induced Volcanism: A Geologic Myth, GSA Annual Meeting, 2001.Mocquet, A., A Search for the Minimum Number of Stations Needed for Seismic Networking on Mars, Planet. Space Sci., 47, 397-

409, 1999.Morgan, J.P., T.J. Reston, and C.R. Ranero, Mass Extinctions, Continental Flood Basalts, and 'Impact Signals': Are Mantle Plume-

induced 'Verneshots' the Causal Link?, Nature, in press, 2002.Peterson, J.E., Antipodal Effects of Major Basin-forming Impacts on Mars, Proc. Lunar Planet. Sci. Conf., 9, 885-886, 1978.Poupinet, G., R. Pillet, and A. Souriau, Possible Heterogeneity of the Earth's Core Deduced from PKIKP Travel Times, Nature, 305,

204-206, 1983.Poupinet, G., A. Souriau, and L. Jenatton, A Test on the Earth's Core-mantle Boundary Structure with Antipodal Data: Example of

Fiji-Tonga Earthquakes Recorded in Tamanrasset, Algeria, Geophys. J. Int., 113, 684-692, 1993.Rampino, M.R., and K. Caldeira, Antipodal Hotspot Pairs on the Earth, Geophys. Res. Lett., 19 (20), 2011-2014, 1992.Rial, J.A., On the Focusing of Seismic Body Waves at the Epicentre's Antipode, Geophys. J. R. astr. Soc., 55, 737-743, 1978.Rial, J.A., and V.F. Cormier, Seismic Waves at the Epicenter's Antipode, J. Geophys. Res., 85 (B5), 2661-2668, 1980.Romanowicz, B., and G. Roult, Asymptotic Approximations for Normal Modes and Surface Waves in the Vicinity of the Antipode:

Constraints on Global Earth Models, J. Geophys. Res., 93 (B7), 7885-7896, 1988.Schultz, P.H., and D.E. Gault, Seismic Effects from Major Basin Formations on the Moon and Mercury, The Moon, 12, 159-177,

1975a.Schultz, P.H., and D.E. Gault, Seismically Induced Modification of Lunar Surface Features, Proc. Lunar Sci. Conf., 6, 2845-2862,

1975b.Seyfert, C.K., and L.A. Sirkin, Earth History and Plate Tectonics: An Introduction to Historical Geology, Harper & Row, New York,

1979.Shearer, P.M., and K.M. Toy, PKP(BC) versus PKP(DF) Differential Travel Times and Aspherical Structure in the Earth's Inner Core,

J. Geophys. Res., 96 (B2), 2233-2247, 1991.Song, X., Joint Inversion for Inner Core Rotation, Inner Core Anisotropy, and Mantle Heterogeneity, J. Geophys. Res., 105 (B4),

7931-7943, 2000.Song, X., and D.V. Helmberger, Seismic Evidence for an Inner Core Transition Zone, Science, 282, 924-927, 1998.Song, X.D., Anisotropy in Central Part of Inner Core, J. Geophys. Res., 101, 16089-16097, 1996.Song, X.D., Anisotropy of the Earth's Inner Core, Rev. Geophys., 35 (297-313), 1997.Souriau, A., and G. Poupinet, Inner Core Rotation: A Test at the Worldwide Scale, Phys. Earth Planet. Int., 118, 13-27, 2000.Strom, R.G., Mercury: An Overview, Adv. Space Res., 19 (10), 1471-1485, 1997.Sun, X., and X. Song, PKP Travel Times at Near Antipodal Distances: Implications for Inner Core Anisotropy and Lowermost Mantle

Structure, Earth Planet. Sci. Lett., 199, 429-445, 2002.Tanaka, S., and H. Hamaguchi, Degree One Heterogeneity and Hemispherical Variation of Anisotropy in the Inner Core from

PKP(BC)-PKP(DF) Times, J. Geophys. Res., 102 (B2), 2925-2938, 1997.Tromp, J., Inner-core Anistropy and Rotation, Annu. Rev. Earth Planet. Sci., 29, 47-69, 2001.Watts, A.W., R. Greeley, and H.J. Melosh, The Formation of Terrains Antipodal to Major Impacts, Icarus, 93, 159-168, 1991.Wieczorek, M.A., and M.T. Zuber, A Serenitatis Origin for the Imbrian Grooves and South Pole-Aitken Thorium Anomaly, J.

Geophys. Res., 106, 825-867, 2001.Williams, D.A., and R. Greeley, Assessment of Antipodal-impact Terrains on Mars, Icarus, 110, 196-202, 1994.Williams, D.A., J.E. Klemaszweski, F.C. Chuang, and R. Greeley, Galileo Imaging Observations of the Valhalla Antipode: Support

for a Subsurface Ocean on Callisto?, DPS meeting, 2001.

12