Seismic Focusing of Impact Energy

473 Final Project White, 2001

SEISMIC FOCUSING OF IMPACT ENERGY

R. Brian White

January 2, 2002

INTRODUCTION

Strange pitted and hummocky terrains have been described at the antipodes of the Caloris basin on Mercury [e.g.,

Strom, 1984], the Imbrium and Orientale basins on the Moon [e.g., Schultz, 1974], and perhaps on some of the icy

satellites [Watts et al., 1991]. The terrain on Mercury is characterized by 100-180 m high hills spaced 5-10 km apart

with 2 km wide hummocky plains material covering parts of the hilly terrain [Trask and Dzurisin, 1984]. These

hills and troughs are preferentially oriented along the Mercurian grid, suggesting the impact reactivated ancient

tectonic weaknesses [Melosh and McKinnon, 1988]. The terrain at the antipode of the Imbrium basin on the moon

consists of mounds and radial grooves on crater rims [Steuart-Alexander, 1978]. Similarly, the region at the

antipode of Orientale basin is “furrowed and pitted.” [Moore et al., 1974]. For the Caloris and Imbrium antipodes,

the strange terrains cover areas about 3500 km2 and 2300 km2, respectively. Magnetic anomalies at the antipodes of

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

and magnetize the crust [Hood and Huang, 1991]. The magnetic anomaly at the antipode of the Imbrium basin is

large enough to perturb the solar wind [Lin and al., 1998]. Although no hilly and lineated terrain is seen at the

antipodes of large impacts on Mars, Alba Patera volcanism does occur at the antipode of the Hellas basin, and

convergence of seismic waves could account for deep-seated fractures that allowed volcanism to form there

[Peterson, 1978; Williams and Greeley, 1994].

BACKGROUND

Spalation at the antipodes of micrometeoroid impacts into glass spheres suggests that seismic effects could be

important at the antipodes for large impacts [Gault and Wedekind, 1969]. Schultz and Gault [1975a; 1975b]

proposed that these antipodal terrains on the Moon and Mercury were produced by impact-generated seismic waves.

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They found antipodal ground motions in excess of 10 m for Imbrium and showed that reflected body waves from the

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

code to study the effects of large impacts into either solid or liquid planets. They also found sizable antipodal

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

The waves reflected from the free surface and focused below the antipode surface were found to have a much

greater effect than the direct waves (see Figure 2). In general, the antipodal disturbances were 2-3 times greater for

a liquid planet than for a solid one because a solid is more efficient at dissipating energy. Watts et al. [1991]

extended this work using a Lagrangian Eulerian finite element code to model impact effects in a two-layer planet

(mantle and core), and Williams and Greeley [1994] continued the work of Watts et al. [1991] for impacts on Mars

with a three-layer model (crust, mantle, and core).

Figure 1: (Left) Geologic map of the northeast corner of the Discovery (H11) quadrangle of Mercury [Trask and Dzurisin, 1984]. The large blue region is the hilly and lineated terrain, and the small teal regions within it are hummocky plains material. (Right) Mariner 10 photo (FDS 27370) of part of the hilly and lineated terrain [Robinson, 2001]. The scene is 543 km across [Strom, 1984].

While the work of these authors is noteworthy, there are some fundamental problems with it. Schultz and Gault

[1975a; 1975b] and Hughes et al. [1977] consider only a uniform planet without a core. Hughes et al. [1977]

assumes the transient crater diameter is equal to the observed crater and that the shock propagates throughout the

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whole planet, ignoring the elastic transition due to the shock decay. Although the later work or Watts et al. [1991]

and Williams and Greeley [1994] properly introduces simple layering of a core and crust, they do not report how

they arrived at transient crater dimensions, which are essential for impact energy estimates [Melosh, 1989]. All

previous studies have largely ignored surface waves, attenuation, and any fine-scale layering or lateral

heterogeneities.

Figure 2: A compressive wave (A) generated by an impact generates a train of reflected tensile waves (B, C). At the antipode (D), these waves reflect and turn inward and converge below the antipode surface (E) with opposing tensile stresses (F) [Schultz and Gault, 1975a].

Wieczorek and Zuber [2001] and Zuber [2001] attempt to explain the generation of strange antipodal terrains using

convergence of ejecta rather than seismic modification. This study was motivated by the observation that this

terrain is slightly offset to the east and north from the true antipode of Imbrium. They could not explain the terrain

at Imbrium’s antipode with Imbrium ejecta convergence, but taking into account the Moon’s prograde rotation, they

did find a good fit to the observations by assuming the terrain was due to an oblique Serenitatis impact rather than

Imbrium. One very important aspect of seismic wave propagation that these two papers overlooked is that Rayleigh

waves are very sensitive to lateral changes in the shallow crust and upper mantle [Zuber, 2001]. In a perfect

spherical planet with homogeneous structure, surface waves will converge exactly at the antipode, but in reality

there will be some bias due to both the path heterogeneities [Zuber, 2001] and obliquity of impact [Dahl and

Schultz, 2001]. The generally eastward offset of the Imbrium antipode terrain is expected given the Moon’s

dichotomy in crustal thickness. Understanding seismic wave propagation through planets could yield a better

understanding of impact-induced antipodal effects and help resolve the relative contributions of other processes on

the formation of anomalous antipodal terrain.

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SEISMIC WAVE FOCUSING

The focusing and defocusing of seismic waves has been noted under a few specific conditions on the Earth. For

example, at the Pahute Mesa nuclear test site in Nevada, a high velocity body beneath the area caused consistent

azimuthal variations in P wave amplitudes [Cormier, 1987; Lynnes and Lay, 1984]. The body was interpreted as a

focusing agent of the P waves. Another type of local seismic wave amplification occurs in sedimentary basins. S

waves from the 1994 Northridge earthquake in California were focused by the Los Angeles basin, causing

anomalously high damage in Santa Monica [Davis and al., 2000; Frankel, 1999; Gau et al., 1996]. Similarly, the

1995 Hyogo-ken Nambu earthquake in Kobe, Japan was associated with amplified damage due to constructive

interference of S-waves and diffracted Rayleigh waves [Kawase, 1996]. Other basin effects on site amplification

have been noted by Chin and Aki [1991], [Olsen et al., 1995], Su and Aki [1995], and O’Connell [1999].

The preceding examples are due to local anomalies in the source region, but global focusing effects have also been

investigated. Chael and Anderson [1982] found that since the antipode is a caustic (i.e., singularity) for the Eikonal

equation of ray theory propagation, Rayleigh wave energy from all over the globe is focused at the antipode. In this

case, the whole Earth behaves as a giant lens, focusing all of the Rayleigh wave energy at the antipode. This

amplification due to focusing has a maximum when the source is an isotropic explosion [Chael and Anderson,

1982], which is a good approximation for a large impact. Lay and Kanamori [1985] studied antipodal focusing of

both Rayleigh and Love waves and the effect of near-source anomalies on the antipodal amplitudes. Rial [1978] and

Rial and Cormier [1980] have studied the problem of whole-earth body wave focusing most extensively. Using a

uniform asymptotic expansion for the high order Legendre functions in terms of spherical Bessel functions, they

eliminate the singularity at the antipode (∆ = 180°) and use ray theory to study the propagation of body waves. They

found an order of magnitude amplification of normally weak high frequency phases such as Pdiff, PKP, PKIIKP, PP,

PPP, PcPPKP, SKSSKS, and SS at the antipode in the region of 178° ≤ ∆ ≤ 182°. In general, they find both core

phases (such as PKP) and surface reflections (such as PP) to be most amplified at the antipode.

An isotropic explosion will radiate compressional energy with uniform phase at all azimuths from the source, and

for a spherically symmetric globe, this results in complete constructive interference for vertical motion and

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cancellation of horizontal motion at the antipode [Chael and Anderson, 1982]. This leads to strong amplification of

the signal-to-noise for surface wave arrivals at the antipode. Because Rayleigh waves are emitted in two wide

phase-symmetric lobes, they are amplified better than Love waves, and even order energy has been found to focus

more completely than odd order energy [Lay and Kanamori, 1985]. If the source is purely compressional, we expect

to observe a circularly symmetric radiation pattern for SV, Love, and Rayleigh waves [Archambeau, 1972], however

non-symmetric Rayleigh wave patterns and strong Love waves are often observed [Lay and Kanamori, 1985].

Surface wave ray tracing reveals the defocusing of energy at the antipodes can cover a region up to 1000’s of km

wide for R3 arrivals, and the degree of deflection is a function of the amplitude of the phase velocity variations due

to small perturbations along the ray paths [Lay and Kanamori, 1985]. Obliquity of impacts also produces stress

waves asymmetries [Dahl and Schultz, 2001]. Furthermore, at the antipode, Love and Rayleigh wave arrivals are

not perpendicularly polarized because the transverse spheroidal and longitudinal toroidal mode components become

significant and rotate the polarization directions [Chael and Anderson, 1982].

Figure 3: (A) Examples of PKP and PP ray paths that focus at the antipode. Dashed lines represent wavefronts with propagating time in minutes [Rial and Cormier, 1980]. (B) Theoretical PP spectral amplitudes near the antipode for periods 5-65 seconds [Rial, 1978].

IMPACT SCALING AND ENERGIES

Scaling rules relate the geometry and size of observable craters to parameters such as impactor velocity, size,

composition, target composition, and local gravity [Holsapple and Schmidt, 1982]. For the purpose of planetary

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seismic wave propagation, we can consider the impact as a point source since the time and length scales are long

compared to the cratering event. Near-source effects are discussed by [Cooper and Sauer, 1977]. For distances less

than one or two impactor radii, the point source approximation is not valid [Holsapple, 1993]. Holsapple and

Schmidt [1987] developed a scalar “coupling parameter” to measure the coupling of energy and momentum of the

impactor into the planetary surface. For an impactor with radius a, velocity U, and density δ, the energy scale

transferred to the target with density ρ and compressive sound velocity c is given by

33

2 3 UE mv a cc

νµ δρρ

= =

2 (1)

The parameter ν = 1/3 for conditions of interest, and µ depends on the target composition. Porous materials with

high dissipation of energy have low µ values around 0.4, and a perfectly solid nonporous target has µ = 2/3. Most

planetary surfaces of interest have µ ≈ 0.58, but this could be as low as 0.37 if porosity reaches 30%. High values of

µ mean that energy dominates, and low µ values correspond to momentum-dominated impacts.

Equation 1 is only the energy scale, and it ignores the observable crater size parameters. A better way to estimate

the energy associated with an impact is the Schmidt-Holsapple scaling equation [Melosh, 1989]. For a vertically

incident projectile into competent rock, this relation is

4.540.33 0.22

0.11 0.131.8tD g

Ea

ρδ

=

(2)

Here, Dt is the transient crater diameter, a is the projectile diameter, ρ is the projectile density, δ is the target

density, and g is the gravity of the target body. We can see that E is most dependent on Dt. The effect of U for

silicate planets is difficult to distinguish from the effect of g, so the absence of U in Equation 2 should not be

alarming [Pike, 1988]. The observable crater rim diameter D is the result of modification from the original transient

crater size. For large-scale complex craters, the transient crater dimensions are controlled by the impactor

parameters and planetary gravity and not the planetary strength [O'Keefe and Ahrens, 1999]. Although indications

of the transient crater diameter are lacking, both scaling rules and crater data suggest that the transient crater

diameter is approximately half the final crater diameter for large, complex craters dominated by gravity [Croft,

1985; O'Keefe and Ahrens, 1993].

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2tDD = (3)

When the impactor and planetary surface have similar material properties (δ = ρ), about half of the kinetic energy of

the impactor is transferred to the planet [O'Keefe and Ahrens, 1993]. However, only a small portion of the energy

goes into elastic wave production. This so-called “seismic efficiency” is approximately 10-4 [Schultz and Gault,

1975a, b].

(4) 410elasticE −= E

For terrestrial underground explosions, the energy partitioning between compressional and shear waves is a ratio

1:10, meaning anomalously large SH polarized waves (i.e.., Love waves) result, and Rayleigh waves are affected too

[Archambeau, 1972]. This suggests the effect of surface waves could be much greater than that of body waves,

although differences in source mechanism, dimension, spectral content, and rise time make it difficult to compare

explosions to impact sources. Nowroozi [1986] found that, for impacts, the surface wave magnitudes are generally

1-1.5 times less than those generated by an equivalent tectonic source.

Caloris Imbrium Orientale D (km) 1300 670 320 c = 1 km/s E (J) [Equ. 1] Eelastic (J) [Equ. 4]

4.99×1025 4.99×1021

4.99×1025 4.99×1021

4.99×1025 4.99×1021

c = 4 km/s E (J) [Equ. 1] Eelastic (J) [Equ. 4]

7.45×1025 7.45×1021

7.45×1025 7.45×1021

7.45×1025 7.45×1021

ρ = 2.94 g/cm3, δ = 2.94 g/cm3 E (J) [Equ. 2] Eelastic (J) [Equ. 4]

1.24×1027 1.24×1023

2.57×1025 2.57×1021

8.97×1023 8.97×1019

ρ = 7.85 g/cm3, δ = 2.94 g/cm3 E (J) [Equ. 2] Eelastic (J) [Equ. 4]

5.38×1027 5.38×1023

1.12×1026 1.12×1024

3.91×1024 3.91×1020

Table 1: Impact and seismic energies for impact of gabbroic anorthosite into gabbroic anorthosite (ρ = 2.94 g/cm3, δ = 2.94 g/cm3) and iron into gabbroic anorthosite (ρ = 7.85 g/cm3, δ = 2.94 g/cm3) with a = 50 km and U = 25 km/sec. The results from Equation 1 use ν = 1/3, µ = 0.58, and U = 25 km/s. c = 1 km/s corresponds to the brecciated surface layer, and c = 4 km/s solid crust [Bullen and Bolt, 1987]. Crater diameters are taken from Wood and Head [1976]. Densities are found from impedance matching of Hugoniots [Ahrens and O'Keefe, 1977; O'Keefe and Ahrens, 1975].

Geometrical spreading, attenuation, and dispersion work to decrease the energy of the seismic waves as they travel

across the planet. For a spherical wavefront in a homogeneous medium, the seismic wave amplitude A decreases as

1/r, where r is the radius from the source. Since energy scales as A2, it decreases as 1/r2. For Mercury with r = 2440

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km and the Moon with r = 1738 km, this means that the seismic energy is decreased by a factor of 13 and 12 by the

time it reaches the antipode. Thus, depending of the value of Eelastic chosen from Table 1, Caloris delivered 108-1010

J of energy to its antipode, Imbrium 109-1012 J, and Orientale 108-109 J.

THE SPECTRAL ELEMENT APPROACH

Typically, one uses normal-mode summation to calculate synthetic seismograms for spherically symmetric Earth

models. In order to accurately calculate synthetic seismograms for global wave propagation, three-dimensional

planetary models are a necessity. At the Caltech Seismological Laboratory, Komatitsch and Tromp have developed

a promising new technique called the spectral element method (SEM) that combines aspects of finite difference,

finite element, and discrete wavenumber modeling approaches [Komatitsch and Tromp, 1999]. Initially developed

for use in computational fluid dynamics, the SEM combines the flexibility of the finite element method with the

accuracy of the spectral method. It can accurately model wave propagation on local and regional scales in both 2-D

and 3-D [Komatitsch and Tromp, 2001c]. By including attenuation, anisotropy [Komatitsch and Tromp, 2001c],

lateral velocity and density variations, 3-D crustal model, ellipticity, free-surface topography, bathymetry, oceans,

rotation, and self-gravitation [Komatitsch and Tromp, 2001d], these simulations are the most realistic ever achieved.

Komatitsch and Tromp [1999; 2001c; 2001d] thoroughly review the SEM method. In short, the SEM determines the

displacement field produced by an earthquake in a finite Earth model the equations of motion for seismic waves.

The displacement s produced by a disturbance in a planet is governed by the momentum equation

2

2tρ ∂

= ∇ +∂

s T fi (5)

Here, the density is ρ, and the stress tensor T is linearly related to the displacement gradient ∇s via Hooke’s law,

which for an elastic anisotropic solid is

:= ∇T c s (6)

The fourth-order elastic tensor c has 21 independent components in the general case with full 3-D anisotropy. If we

add the effect of attenuation, Hooke’s law must be modified so the stress is determined by the entire strain history.

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( ) ( )

'( ) : '

t tt

t

∞

−∞

∂ −= ∇

∂∫c

T ' (7) t dts

Rather than using these equations of motion and their appropriate boundary conditions directly, the SEM uses an

integrated form of the problem, which naturally satisfies the necessary stress-free boundary condition. This means

surface waves can be more accurately simulated because incorporation of free-surface topography is relatively

straightforward. The integrated or “weak” formulation of the problem is achieved by dotting the momentum

equation (1) with an arbitrary test vector w and integrating by parts over the model volume Ω. The stress-free

boundary condition ( is the unit outward normal to all boundaries.) n

ˆ 0=T ni (8)

can be incorporated into the expression to give

( ) ( )2 3

32 3

ˆ: :CMB

d S t pt tΩ Ω

∂ ∂= − ∇ + ∇ +

∂ ∂∫ ∫ ∫ss rw w T r Ω w r n w ri iρ 2d (9)

Here, S(t) is the source-time function, rs is location of the point source. In this case, n is the unit normal to the

core-mantle boundary (CMB), where the additional boundary condition that the normal components of velocity and

traction (p) are continuous is imposed. The formulization in Equation (5) is completely general and valid for an

anelastic, anisotropic material, and it honors continuity across the fluid outer core and solid lower mantle. A similar

condition must be imposed on the inner core boundary (ICB) and the outer core. For a discussion of the equations

of motion in the outer core and their relation to the problem, see Komatitsch and Tromp [2001c; 2001d].

ˆ

To implement the SEM code, a parallel computer is needed because the size of the mesh requires too many memory

variables per gridpoint for a workstation to handle. The memory must be spread out over neighboring gridpoints

using a multiple-processor computer. At Caltech, they use a Beowulf cluster of 312 Intel Pentium-III processors

(nodes) working at 733 MHz with 156 GB of shared memory [Komatitsch and Tromp, 2001a, b]. The Center for

Scientific and Parallel Computing at Washington University currently has a 68-node computer with 64 R10000 SGI

processors at 195 MHz and 4 R12000 SGI processors at 270 MHz [2001]. The total memory is 64 GB for the

R10000 cluster and 512 MB for each R12000 computer. The capability of this system is about a factor of 3-5

slower than the Caltech setup, making implementation of the SEM code somewhat more difficult and time-

consuming here. However, a recent NSF grant will be supplying Washington University with a new 128-node

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cluster with Intel IA64 processors and 64 GB of memory. This system, expected in early 2002, will make

implementation of SEM code more feasible.

An unfortunate advantage of applying the SEM approach to seismic wave propagation through planets other than

Earth is that very little three-dimensional structural information is known for other planets. We can take into

account the varying crustal thickness on the Moon and Mars, which means surface waves can be accurately

modeled. However, no such constraint exists for Mercury yet. This means a spherically symmetric model is

sufficient for Mercury, which greatly reduces the computational cost for the SEM, or a normal mode technique

could be used instead. One caveat with using the SEM is that it deals exclusively with seismic waves, so shock

waves cannot be modeled without serious modification of the code [Tromp, 2001]. This means that rarefactions

generated by the impact and its subsequent shock decay cannot be considered.

DISCUSSION

We have seen that the seismic effects from impacts are probably significant and focusing of seismic energy at the

antipode could cause anomalous deformation there. Strange terrains at the antipodes of large basins on Mercury

(Caloris) and the Moon (Imbrium and Orientale) have been described and attributed to such seismic modification.

Some authors have tried to ascribe these terrains to ejecta convergence, but they have ignored the strong effects of

lateral velocity changes on surface waves, which can account for systematic bias of the focusing away from the

antipode. The estimates for energies of impacts are shown in Table 1. These are generally in agreement with past

calculations [Watts et al., 1991]. This energy is further partitioned between surface and body waves, and it is

attenuated as it passes through the planet. A three-dimensional spectral element approach is proposed to study the

focusing of seismic waves at the antipode. It faithfully reproduces body and surface waves for a very complex Earth

model. Modifying the model for other planets should reduce the cost of the computations since less structure is

known for them. Thus, computations should be feasible on the new Washington University parallel computer

expected in the coming months. In the meantime, normal mode summation that avoids the caustic at the antipode

could provide useful insight into the problem, especially for Mercury.

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Seismic Focusing of Impact Energy

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