Teleseismic Shot-profile Migration

J. Shragge1, B. Artman1, and C. Wilson1,2

1Department of Geophysics, Stanford University, Stanford, CA 94305-2215

2 Now at Lamont-Doherty Earth Observatory, Columbia University, Palisades, NY 10964-1000

Received 2005 March 31; revised paper submitted xxx

ABSTRACT

Lithospheric images generated from multi-channel teleseismic data reveal important aspects of crustal

and mantle structure, and offer windows into past and present tectonic processes. However, the imag-

ing techniques used to generate these images can be improved through further adaption of exploration

seismology practices. We introduce the shot-profile representation of wave-equation migration as a

novel way to cast the teleseismic imaging problem. We show how this technique can be tailored to

suit teleseismic acquisition geometry and wavefields, and detail a procedure for performing kinematic

and structural imaging (migration) with all first-order forward- and backscattered phases in teleseis-

mic coda. We apply the shot-profile migration algorithm to the IRIS-PASSCAL CASC-1993 data set

to generate images of the Cascadia subduction zone. We generate a structural image that shows a

fairly continuous crust-mantle reflector extending from the continental interior nearly to the mantle

wedge. We suggest that differences between our results and other images of the Juan de Fuca subduc-

tion zone are attributable to a combination of the different filters applied during the imaging process

and more coherent stacking of events due to migration with improved migration velocity profiles.

1

INTRODUCTION

Deployments of regionally extensive and reasonably spatially sampled seismic arrays in the last

decade have provided earthquake data that yield images of lithospheric structure at scales hitherto

unattainable (Dueker and Sheehan, 1997; Rondenay et al., 2001; Poppeliers and Pavlis, 2003; Wil-

son et al., 2003). These images now play a crucial role in improving our understanding of past and

present processes of tectonic evolution. Owing to the EARTHSCOPE program and the associated

US-ARRAY, the next decade will see a dramatic increase in the volume of three-component broad-

band seismic data available for imaging the North American lithosphere. However, to fully exploit

this opportunity, earthquake seismologists must improve their current lithospheric imaging practices.

Most early teleseismic imaging experiments used data recorded on sparsely distributed three-

component stations. Hence, seismologists interested in the structure beneath a seismic station were

forced to use scattered phases identifiable from a single set of three-component seismograms. For ex-

ample, isolating forward-scattered P-to-S converted waves through receiver-function analysis (Phin-

ney, 1964; Langston, 1977). The introduction of spatially extensive and relatively dense three-

component arrays allowed earthquake seismologists to adapt exploration seismology practices to

the teleseismic imaging problem. Initial work in array-based processing used simple work flows in-

volving linear moveout corrections combined with common conversion point stacking (Dueker and

Sheehan, 1997). More recent efforts involved profiling an Archean continental suture in Southwest-

ern Wyoming (Sheehan et al., 2000), and applying a Kirchoff-like migration/inversion based on a

generalized Radon transform (GRT) to produce images of the Cascadia subduction zone (Rondenay

et al., 2001). Work by Bostock et al.(2001) and Aprea et al. (2002) also expanded the number of scat-

tering phases available for delineation of crustal and mantle structure by demonstrating the imaging

potential of backscattered modes (i.e., free-surface multiples).

2

The move toward applying seismic exploration technology to teleseismic wavefields has fostered

overall improvement in teleseismic imaging practices; however, additional imaging technology re-

mains to be transferred. In this paper, we examine teleseismic wave-equation depth migration, a

method analogous to that commonly applied in seismic exploration. This technique extrapolates

recorded wavefields into the subsurface using a one-way wave-equation, and constructs subsurface

images by evaluating a physical imaging condition (Claerbout, 1971). Importantly, the shot-profile

configuration of wave-equation imaging easily conforms to the geometry of teleseismic experiments,

and affords a robust methodology accommodating imaging of all first-order, forward- and backscat-

tered phases. Moreover, due to the first-order planarity of source functions this method allows for

polarization information easily accounted for in Kirchhoff-based formulations to be approximately

incorporated into the formalism.

We begin this paper with a discussion of the limitations on teleseismic imaging methods from

acquisition geometry, and the requirements for imaging each first-order scattering phase contained

within teleseismic wavefields. We then present an overview of the wavefield continuation and imag-

ing condition steps comprising the shot-profile migration approach. Subsequently, we outline how

this technique can be tailored to suit a teleseismic experiment, and discuss the parameters required

to image the various scattering phases. Finally, we present shot-profile migration results for the

subduction-suture synthetic data set presented in Shragge et al. (2001), and for the IRIS-PASSCAL

CASC-93 data set discussed in Rondenay et al. (2001).

TELESEISMIC EXPERIMENTS

Array geometry places a number of limitations and requirements on the class of applicable litho-

spheric imaging methods. In this section, we review the limitations generated by the current ap-

3

proach to instrument deployment, how they affect our measurement of teleseismic wavefields, and

the constraints they place on teleseismic imaging algorithms.

Teleseismic acquisition geometry

The wavefields used in teleseismic imaging are measured with linear deployments of three-component

broadband instruments. As with all imaging experiments, the class of methods available to image

with these wavefields is inextricably linked to acquisition geometry. An ideal array configuration for

3D migration would densely sample the 2D acquisition surface with regularity over a lithospheric tar-

get of interest. At present, though, the limited number of broadband instruments and the ubiquitous

presence of field site obstacles necessitate deployments of 1D arrays characterized by irregular station

spacing and off-axis instrument positioning. Hence, recorded wavefields are greatly under-sampled

in one coordinate, which precludes the use of fully 3D migration methods. Accordingly, a 2.5D

approximation incorporating 3D propagation through a 2D medium is currently the most practical

representation.

Teleseismic wavefields

Teleseismic earthquake sources originate, by definition, at large epicentral distances (30◦ < δ < 90◦)

from deployed arrays. These wavefields exhibit source-time functions longer in duration and more

complex than those found in seismic exploration. Teleseismic wavefields radiate outward from nu-

cleation points, turn in the middle-to-lower mantle, and arrive as quasi-planar wavefronts at the base

of the lithosphere (see Figure 1). These wavefields interact with discontinuous, receiver-side litho-

spheric structure generating forward-scattered P- and S-wave energy. The source energy further inter-

acts with the free-surface, giving rise to downgoing reflected P- and converted S-waves. These effec-

4

tive sources again interact with structure generating additional scattering phases that radiate upward

to the surface. Thus, the summation of the source term, first-order scattering and their higher-order

scattering analogs constitute the teleseismic (P-wave) coda.

We consider a recorded teleseismic wavefield displacement vector, u, to be the superposition of

three terms: source wavefield, usrc, first-order scattered wavefield, u f irst , and higher-order scattered

wavefield, uhigher ,

u = usrc +u f irst +uhigher , (1)

where the boldface letters indicate vector wavefields. Note that each of these wavefield scatter-

ing terms implicitly contain a convolution between the wavefield’s Green’s function and the event-

specific source signature. However, for conciseness we refer to only the recorded displacement wave-

field and postpone to below our discussion of source-signature deconvolution. The first-order scat-

tering term consists of two forward-scattered modes: PP diffractions, and PS conversions. (We use

a convention where the caret, P, and circumflex, P, characters mark upgoing and downgoing wave-

fields, respectively.) The reflected and converted downgoing P- and S-wave energy then gives rise

to five first-order backscattering modes: PP, PS, SP, SSv and SSh . Thus, the first-order scattering

displacement term is the vector superposition of seven phases:

u f irst = PP+ PS+ PP+ PS+ SP+ SSv + SSh . (2)

Here, SSv and SSh phases represent SV → SV and SV → SH scattering, respectively. Although

higher-order scattering phases, uhigher , generally exist in teleseismic wavefields, we neglect these

arrivals and concentrate our imaging efforts on the following approximate teleseismic wavefield coda,

u ≈ usrc + PP+ PS+ PP+ PS+ SP+ SSv + SSh . (3)

5

Imaging restrictions and requirements

As the last two sections show, a number of characteristics of teleseismic acquisition and wavefields

restrict the class of applicable teleseismic imaging algorithms. Specifically, a teleseismic migration

algorithm should handle irregular array geometry and arbitrary source distribution in both epicen-

tral distance and backazimuth, be amenable to 2.5D (and eventually 3D) geometry, and incorporate

all scattering modes listed in equation 3 into the imaging formalism. We demonstrate below that,

with limited data preprocessing, each of these requirements can be realized with a shot-profile wave-

equation migration approach. Finally, although the extension of the theory below to 3D is straight-

forward, we elect to treat the case of 2.5D experimental geometry in deference to the acquisition

geometry of most existing teleseismic data sets.

EXPLORATION SHOT-PROFILE MIGRATION

Seismic exploration surveys acquire large data volumes consisting of seismic traces indexed by source

and receiver position. One strategy for imaging the subsurface with such a data set is to sort the

data volume into profiles indexed by the source coordinate, and then migrate each shot-profile in-

dependently. To illustrate how this shot-profile migration approach works, we present an idealized

experiment of a point scatterer in a homogeneous 2D half-space in Figure 2.

A shot-profile is generated when a scalar source wavefield, S(x , z = 0, t |s), is excited at shot

point s = s(x0, z = 0) (Figure 2a), forming a downgoing one-way wavefield. (Hereafter, we assume

that all media are 2D and invariant in the y-direction.) At depth level z = z1 the source wavefield,

S (x , z = z1, t |s) (Figure 2b), interacts with discontinuous structure, generating an upgoing one-way

wavefield, R (x , z = z1, t |s) (Figure 2c), that propagates to the surface and is recorded as receiver

wavefield, R (x , z = 0, t |s) (Figure 2d). This process is then repeated at every source point to generate

6

the complete data volume.

To formalize the kinematic/structural imaging experiment, we are required to perform at least

the adjoint process (i.e., an approximate inverse of the propagation and backscattering described

above). Shot-profile wave-equation migration is one such adjoint imaging technique. This procedure

approximately reverses the experiment by extrapolating source and receiver wavefields, S and R,

from the acquisition surface to depth using a one-way wave-equation, and estimating the scattering

coefficient through a physical imaging condition. These two procedures are described below.

Wavefield Extrapolation

The first step of shot-profile migration is to propagate the source and receiver wavefields into the

subsurface. We use the Helmholtz equation to model the propagation of scalar acoustic wavefield

Ws ,

[

∇2 + s2ω2]Ws = 0, (4)

where ∇2 is the Laplacian operator, s the propagation slowness (i.e., reciprocal of velocity), and ω

angular frequency. One-way wavefield extrapolation operators can be derived from the frequency-

wavenumber dispersion relation representation of equation 4,

kz =

√

s2ω2 − k2x , (5)

where kz and kx are the vertical and horizontal wavenumbers, respectively. Wavenumber kz forms

the basis of the operators that propagate the source and backscattered receiver wavefields into the

subsurface through,

S(x , z +1z,ω | s) = S(x , z,ω | s) e−ikz 1z, (6)

R(x , z +1z,ω | s) = R(x , z,ω | s) eikz 1z .

7

where 1z is the depth of the propagation step. The source wavefield propagates downward from the

surface advancing forward in time; however, the operator applied to propagate the receiver wavefield

has the opposite sign to preserve causality (i.e., the scattering generating the receiver wavefield oc-

curred prior its measurement). The two wavefields are extrapolated to all depths within the model

through a recursive application of equation 6.

Although equation 6 is exact only for vertically stratified media, s(z), techniques exist to extend

its applicability to laterally varying profiles, s(x , z). For the results presented in this paper, we em-

ployed the split-step Fourier approach (Stoffa et al., 1990) that approximates wavenumber kz with a

Taylor series expansion about a reference slowness, s0(z),

kz ≈ ω (s − s0)+

√

s20ω

2 − k2x . (7)

The first mixed-domain (ω, x) term in equation 7 acts as a local correction to the second phase-shift

(ω,kx ) term that handles the bulk of the propagation. Increased accuracy is achieved through the

use of a multi-reference split-step Fourier method (Stoffa et al., 1990) designed to minimize quantity

s(x , z)− s0(z) .

The imaging condition

The second step of shot-profile migration forms a subsurface image by extracting appropriate infor-

mation from the extrapolated source and receiver wavefields. Claerbout’s imaging principle (Claer-

bout, 1971) asserts that energy in the receiver wavefield, R, spatially collocated with energy in the

source wavefield, S, at time t = 0 originated from a reflector at that model point. This can be ac-

complished by extracting the zero-lag of the cross-correlation of the two wavefields (equivalent to a

time-domain multiplication of the two wavefields in Figure 2b and c). In practice, we generate the

image, I (x , z), by multiplying S and R∗ together at all subsurface model points for each frequency

8

and source location, and summing the resulting values (Claerbout, 1985),

I (x , z) =∑

s

∑

ω

S(x , z,ω | s) R∗(x , z,ω | s), (8)

where R∗ denotes the complex conjugate of the receiver wavefield. Note that the summation over

sources is outside the frequency summation and is equivalent to stacking partial subsurface images.

TELESEISMIC SHOT-PROFILE MIGRATION

A teleseismic experiment is analogous to a seismic exploration survey, save for a source function

that is an approximately planar wavefront defined by constant ray-parameter, p. Figure 3 presents

the teleseismic equivalent of the backscattering scenario illustrated in Figure 2. Here, an upgo-

ing plane wave has already generated a free-surface-reflected, downgoing effective planar source,

S(x , z = 0, t |p) (Figure 3a), that propagates into the subsurface. At depth level z = z1 source wave-

field, S(x , z = z1, t |p) (Figure 3b), interacts with the discontinuous structure to generate receiver

wavefield, R(x , z = z1, t |p) (Figure 3c), which propagates to the surface and is recorded as receiver

wavefield, R(x , z = 0, t |p) (Figure 3d). Performing the adjoint of this process through a shot-profile

migration approach is again directly analogous to the exploration scenario described above, save for

a slight modification to a plane-wave source wavefield.

In this section we recast the shot-profile migration method to suit teleseismic wavefields. We be-

gin by justifying how, under certain conditions, elastic wavefields can be approximately propagated

by acoustic one-way wavefield extrapolation. Subsequently, we specify the constituents of the tele-

seismic source and receiver wavefields, and present the parameters required to image each first-order

scattering phase in equation 3.

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Acoustic propagation of teleseismic wavefields

The equations governing the propagation of vector displacement wavefield, u, through a homoge-

neous isotropic elastic solid can be derived from the force-free, spectral Navier-Stokes equation

(Ben-Menahem and Singh, 1981),

α2∇∇ ·u−β2∇ ×∇ ×u+ω2u = 0, (9)

where α and β are the P- and S-wave velocities, and ∇ is the gradient operator. By Helmholtz’s

theorem, vector field, u, separates into irrotational, uirr , and solenoidal, usol , components so that u =

uirr +usol. Applying vector identities, ∇ ×uirr = 0 and ∇ ·usol = 0, allows us to rewrite equation 9

as,

α2

[

∇2 +ω2

α2

]

uirr +β2

[

∇2 +ω2

β2

]

usol = 0. (10)

Equation 10 is satisfied identically if uirr and usol respectively satisfy,

[

∇2 +ω2

α2

]

uirri = 0;

[

∇2 +ω2

β2

]

usoli = 0; (11)

where subscript, i = x , y, z, denotes the three vector components. These two formulae represent inde-

pendent vector Helmholtz equations for the irrotational P-wave and solenoidal S-wave components

of the total displacement wavefield. Note that if displacement vector, u i , can be rotated to a reference

frame that localizes all P-wave energy on a single component, vector equations 10 will reduce to two

scalar acoustic wave-equations appropriate for independently propagating P- and S-wavefields,

[

∇2 +ω2

α2

]

WP = 0,

[

∇2 +ω2

β2

]

WS = 0. (12)

Extrapolation operators for both teleseismic P- and S-wavefields can be subsequently developed

from equations 12 in a manner identical to equation 7. Teleseismic source and receiver wavefields

are then extrapolated according to equations 6.

10

Teleseismic Receiver Wavefields - RP , RSV , and RS H

If equations 12 are to be applicable for downward continuing vector teleseismic wavefields, we must

separate the irrotational P-waves from the solenoidal S-waves. One approach is to exploit the pla-

narity of teleseismic wavefronts and their approximately constant polarization vectors, p =[

px , py]

.

The polarization vector for each event can be calculated using standard radially symmetric Earth

models (e.g., IASPEI-91), or estimated from hodogram analysis applied to the direct teleseismic ar-

rivals. Polarization vector, p, also defines a rotation, R, from the generalized ground orientation of

the instrument, [ux ,u y ,uz], to an event-specific displacement frame, [u R,uT ,uV ], through,

u j = Rj iui , (13)

where i = x , y, z and j = R, T , V .

Estimates of the source polarization vector and near-surface velocities also are used to specify a

free-surface transfer matrix transformation, F, (Kennett, 1991). We apply this matrix to recorded

teleseismic wavefields to rotate and scale the measurements from a ground displacement frame

[u R,uT ,uV ] to a wave-vector frame, [P , SH , SV ], through,

uk ≈ Fkj u j (14)

where j = R, T , V and k = P , SH , SV .

Here we assume that the source function is a direct P-wave arrival; hence, after substituting

Psrck for usrc

j and replacing uk with the expression in equation 3, the composite application of the

free-surface transformation matrix and rotation operators generates the following response,

uk ≈ δpk[

Psrc + P P + P P + S P]

+ δvk[

P S + P S + S Sv

]

+ δhk S Sh ,

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where δkp, δkv, and δkh are Kroneker delta functions signifying symbolically that P-, SV -, and SH -

wave contributions to the teleseismic wavefield are steered to orthogonal vector components (e.g.,

δpv = 0). We emphasize that this approximation strictly holds only for specularly reflected and con-

verted waves. However, the dot product between the estimated and true polarization vectors is nearly

unity for a wide angular band around the estimated value, which effectively steers the majority of

scattered P- and S-wave energy to the correct wavefield component.

Assuming the validity of the above approximations, we derive scalar receiver wavefields from the

vector teleseismic displacement wavefield,

RP ≡ δpk Fkj Rj iui ≈[

Psrc + P P + P P + S P]

,

RSV ≡ δvk Fkj Rj iui ≈[

P S + P S + S Sv

]

, (15)

RS H ≡ δhk Fkj Rj iui ≈ S Sh .

Note that all modes still implicitly contain the source signature of each event. Hence, improved

imaging requires estimating the source signature and deconvolving it from the source and receiver

wavefields to recover the wavefield Green’s function. Fortunately, source term P src is contained

within the receiver wavefield, RP , which provides us with a direct way to measure both the arrival

times of the source wavefield and its event-specific source-signature.

Teleseismic Source Wavefields - S

Shot-profile migration allows us to specify source wavefields of almost arbitrary character. Due to

this flexibility, we can choose our source wavefield representation in a number of ways. We generate

source wavefields by extracting estimates of the direct-wave arrival times, t d .a.(x), and the source

signature from the RP wavefield component. First, we calculate a set of optimal delay times, t d .a.,

using the multi-channel cross-correlation approach of VanDecar and Crosson (1990). The source

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wavefield Green’s function is then generated by allocating a delta-function spike at the direct arrival

time on each wavefield trace (i.e., S(x , t ,p) = δ(t d .a.(x)− t(x ,p)). The resulting source wavefield, S,

is then band-pass filtered to the teleseismic frequency band. A source wavefield example is shown in

Figure 3a.

To recover the Green’s function of the receiver wavefield, we first need to generate estimates of

individual event’s source-time function. We do this by applying the processing approach described

in Bostock and Rondenay (1999): i) shift the traces of RP to align the wavefield according to the

calculated direct arrival times, td .a. ; ii) apply a Karhunen-Loeve transform to generate principle com-

ponents; iii) identify the component with the greatest eigenvalue as the source function estimate; iv)

apply an inverse Karhunen-Loeve transform to the source-signature estimate; and v) apply the inverse

of the time shift applied in step i). The estimated source signature is then deconvolved from receiver

wavefields RP , RSV , and RSV leaving an estimate of the receiver wavefield’s Green’s function.

Multi-mode shot-profile teleseismic imaging

Extrapolating the teleseismic source and receiver wavefields developed above in a 2.5D experimen-

tal geometry requires accounting for the kinematics of out-of-plane propagation in the y-direction.

The sole kinematic difference between 2D and 3D acoustic plane-wave propagation through a 2.5D

medium is an extra phase-shift dependent on the magnitude of ray-parameter component, py . A

corollary of this observation is that we can migrate teleseismic wavefields in the x − z plane using the

following in-plane apparent velocity profile,

v2.5D(x , z|py) =v(x , z)

cos(

sin−1 (

v|py|)) . (16)

Imaging different first-order scattering modes requires permuting four parameters: i) source

wavefield extrapolation operator; ii) source velocity function; iii) receiver velocity function; and

13

iv) receiver component. The permutations for each scattering mode are presented in Table 1. The

source wavefield extrapolation operator determines whether we are imaging a forward- or backscat-

tered phase. The rationale for the sign change in the forward-scattering extrapolation operator is that

the scattering generating the receiver wavefield occurred before either the source or receiver wave-

fields were measured. Hence, both S and R wavefields need to be propagated backward in time. This

is done by introducing the forward-scattering extrapolation operator to the source wavefield recursion

relation in equation 6. The remaining three parameters combine to specify which mode is being im-

aged. Scattering modes ending as a P-wave (S-wave) require that the RP (RSV or RS H ) wavefield

component be downward continued at wavespeed α (β). Similarly, scattering modes originating as a

P-wave (S-wave) require that the S wavefield be downward continued at wavespeed α (β).

APPLICATION TO SYNTHETIC DATA

This section presents the results of applying the shot-profile migration algorithm to the synthetic

data set described in Shragge et al. (2001). The structural profile of the synthetic test model is

shown in Figure 4, and consists of a low-velocity crust, a high-velocity mantle, and a remnant un-

derthrust crustal segment. Values for the velocities are found the caption of Figure 4. The data

generated for this model consist of a suite of planes waves with in-plane horizontal slownesses of

px = ± [0.05,0.06,0.07] s/km and zero out-of-plane propagation (i.e., py = 0). Data sections con-

sist of 120 traces at receiver intervals of 3 km. For a more detailed account of the synthetic data

generation and examples, we refer the reader to Shragge et al. (2001).

Shot-profile migration results for this synthetic data set are presented in Figure 5. The migration

velocity profile is exactly the 2D velocity model used to generate the synthetics (c.f., Figure 4). The

P S mode image shown in the upper left panel delineates all parts of the model structure. This im-

14

age also indicates a successfully collapsing of diffraction energy originating from short-wavelength

structure, which illustrates that non-specular reflected/converted energy can be successfully migrated

through this approach. The P P mode image in the top right panel also delineates the crust-mantle

reflector, but poorly recovers the relict crust/mantle interface due to the small reflection coefficients

arising from small material property contrasts. The P S and S Sv mode images in the bottom two pan-

els delineate all parts of the model; however, these images are subject to cross-mode contamination

from modes with the same terminal phase polarization. We do not included images for either scatter-

ing mode P P , which has minimal resolution capabilities, or for mode S P , which has low amplitudes

due to double conversion.

Overall, shot-profile migration of this synthetic data set generates results equal to, or better than,

the results presented in Shragge et al. (2001). However, the imaging improvements are more appro-

priately attributed to the more accurate migration velocity profile (i.e., true 2D vs. an approximate

1D model) than the differing imaging formalisms (i.e., wave-equation migration vs. migration and

material property inversion through a GRT).

APPLICATION TO FIELD DATA

In this section we show results of applying our migration algorithm to the IRIS-PASSCAL data set

collected by researchers at Oregon State University (Nabelek et al., 1993). A goal of this experiment

was to constrain the geometry of the Juan de Fuca plate as it subducts beneath Oregon coast of North

America (Li, 1996). Forty three-component, broadband instruments were deployed at a total of 69

sites for nine months beginning in May 1993. The sites were situated in central Oregon, USA from

the coast to approximately 300 km inland in a 1D quasilinear array with average station spacing of

5 km. Rondenay et al. (2001) extracted 31 events characterized by high-signal-to-noise ratios. They

15

further extracted source-signature estimates using the procedure described above, and performed

deconvolution to recover estimates of the receiver wavefields Green’s function. We have elected to

migrate the same suite of earthquakes; hence, for a detailed discussion of the choice of events and the

processing applied, we refer the reader to Rondenay et al. (2001).

Wave-equation migration methods generally require regularly sampled wavefields. Teleseismic

acquisition geometry, though, seldom conforms to this requirement. To achieve regular spatial sam-

pling, we sorted the data traces into regularly spaced bins at 2 km intervals. In addition, we applied

a short three-point triangular filter along the data’s offset axis to slightly smooth the high frequency

components of the horizontal wavenumber spectrum. An additional concern is that trace binning can

lead to spatial aliasing phenomena. However, though aliasing problems are usually severe at the sur-

face, the wave-equation approach tends to heal acquisition gaps naturally during extrapolation steps.

Hence, we expect minimal spatial aliasing phenomena to contaminate the images at lithospheric tar-

get depths.

The P-wave and S-wave velocity models used for migration are shown in upper and lower panels

of Figure 6, respectively. Both the P- and S-wave models were adapted from the 1D reference

model used by Rondenay et al. (2001). Based on their imaging results, we incorporated additional

lithospheric structure into the velocity profile, including a slower oceanic mantle, a slower mantle

wedge, and a 10 km thick subducting oceanic crust.

The shot-profile migration results for the CASC-93 data set are presented in Figure 7. The P S

mode image shown in the upper left panel delineates the Juan de Fuca plate subducting initially at a

dip angle of approximately 12◦, and then changing to a steeper dip at 65 km depth. The continental

Moho reflector is imaged at around 40 km depth as a diffuse, but fairly continuous reflector between

100 km to 250 km in horizontal distance. The P S and S S mode images, shown in the upper right and

16

lower left panels, image both the subducting oceanic crust with weaker amplitude and a fairly contin-

uous crustal reflector between both the oceanic crust/mantle wedge and the continental crust/mantle

boundaries. The lower right panel shows a stack of individual mode images. Here, stacking has im-

proved signal-to-noise levels and enhanced the continuity of the crustal reflector nearly to the edge

of the subducting slab.

The shot-profile migration results have imaged a crustal reflector that is fairly continuous from

near the mantle wedge well into the continental interior. Interestingly, the images do not show a

complete absence of reflectivity observed in other Cascadia subduction zone images (Rondenay et

al., 2001; Bostock et al., 2002). One possible cause is the use of different imaging approaches. The

method developed here solely consists of structural/kinematic imaging, and neither includes ampli-

tude weighting factors nor performs inversion for material property perturbations, and any associated

upweighting or downweighting is not incorporated. Applying these additional filters could partially

account for observed differences, and yield more like images.

A second possibility is related to the additional structural complexity included in the migration

velocity profile. In particular, we note that the area where the two images differ the most coincides

with the location where our velocity model is v(x , z). Hence, it is possible that our 2D apparent

migration velocity profiles are more accurate than approximate 1D reference model leading to a

more consistent stacking of individual event images.

DISCUSSION

This paper demonstrates the applicability of teleseismic shot-profile migration through an inter-

pretable migration of the CASC-93 data set. We note that, in general, wave-equation imaging meth-

ods necessarily degrade when wavefields are insufficiently sampled. This will be the case when too

17

few stations record a teleseismic wavefield leading to severe spatial aliasing phenomena. In these

cases we suggest using a Kirchhoff approach naturally amenable to irregular sampling. However,

using a different migration approach does not eliminate aliasing. In cases where there are sufficient

numbers of stations to adequately sample the teleseismic wavefield, but large gaps exist between a

few stations, seismic data regularization approaches can be applied to approximately restore missing

data prior to migration - for example, using Prediction Error Filters (Claerbout, 1999).

Migrating teleseismic data through a true 2D medium with an approximate 1D migration velocity

profile is a worse approximation for a 2.5D experiment geometry than in the 2D case. This is because

the events with a differing py component have differing apparent velocity profiles. Hence, velocity

model inconsistency could lead to an inappropriate superposition of reflectivity. If the velocity model

error is severe enough, reflectors will be over- or under-migrated more than 1/4 wavelength out-of-

phase and stack destructively. This observation suggests that migration velocity profile accuracy is

important, especially for higher frequency data sets, and highlights the need for developing methods

capable of estimating the lithospheric velocity profiles.

The differences observed between the shot-profile migration results of the CASC 93 data set

and those presented in Rondenay et al.(2001) warrant a discussion of the relative merits of using

an adjoint procedure (i.e., migration) or inverse procedure (i.e., migration plus material property

perturbation inversion) to image the subsurface. However, these two processes are intrinsically related

- the adjoint migration approach is necessarily the first step applied in an inversion. We stress that

the imaging results generated through an adjoint process (i.e., migration) should always be computed

and understood before extending the complexity of the inversion approach to include inversion.

An additional approximation that we employ is the planarity of teleseismic sources. Recall that

this assumption allowed us to reduce the vector teleseismic wavefield to scalar P- and S-wavefield

18

components. This approximation, though, is not a necessary requirement. For example, Zhe and

Greenhalgh (1995) demonstrate that prestack migration is directly applicable to multicomponent data

without a priori knowledge of wavefield polarization. Moreover, they formulate a vector imaging

condition able to generate images for each of the scattering modes described above. Importantly,

incorporating vector wavefield extrapolation and vector imaging condition evaluation into the shot-

profile migration approach could allow earthquake sources nearer than < 30◦ in epicentral distance

to be used for imaging. In turn, this would enhance the imaging frequency bandwidth and lead to a

more complete sampling in epicentral distance and backazimuth.

CONCLUDING REMARKS

This paper demonstrates that the shot-profile migration approach of exploration seismology is directly

applicable to teleseismic wavefields when using planar source wavefields. Our synthetic tests show

that non-specularly scattered energy is still well imaged using this acoustic migration framework.

We have migrated the CASC-93 data set and imaged a crustal reflector that is fairly continuous from

nearly the mantle wedge well into the North American continental interior. Imaging differences

between our and previous studies are conjectured to arise from a combination of the different filters

applied during the imaging process and from the use of more accurate migration velocity profiles.

This latter observation shows the need for establishing techniques capable of performing teleseismic

migration velocity analysis.

Acknowledgments

We would like to thank Biondo Biondi for helpful discussions during the early stages of this work.

We acknowledge the contributions of Michael Bostock and Stephane Rondenay in the preparation of

19

the CASC-93 data set for migration. We also acknowledge the sponsors of the Stanford Exploration

Project for their financial support of this project.

20

REFERENCES

Aprea, C. M., Hildebrand, S., Fehler, M., Steck, L., Baldridge, W. S., Roberts, P., Thurber, C. H., and

Lutter, W. J., 2002, Three-dimensional Kirchhoff migration: Imaging of the Jemez volcanic field

using teleseismic data: Journal of Geophysical Research, 107, no. B10, art. no.–2247.

Ben-Menahem, A., and Singh, S., 1981, Seismic waves and sources: Springer-Velag New York Inc.

Bostock, M. G., and Rondenay, S., 1999, Migration of scattered teleseismic body waves: Geophysical

Journal International, 137, no. 3, 732–746.

Bostock, M. G., Rondenay, S., and Shragge, J., 2001, Multiparameter two-dimensional inversion of

scattered teleseismic body waves 1. Theory for oblique incidence: Journal of Geophysical Re-

search, 106, no. B12, 30771–30782.

Bostock, M., Hyndman, R., Rondenay, S., and Peacock, S., 2002, An inverted continental moho and

the serpentinization of the forearc mantle: Nature, 417, 536–538.

Claerbout, J., 1971, Toward a unified theory of reflector mapping: Geophysics, 36, 467–481.

Claerbout, J., 1985, Imaging the Earth’s Interior: Blackwell Science, Inc., New York.

Claerbout, J., 1999, Geophysical estimation by example: Environmental soundings image enhance-

ment: Stanford Exploration Project, http://sepwww.stanford.edu/sep/prof/.

Dueker, K. G., and Sheehan, A. F., 1997, Mantle discontinuity structure from midpoint stacks of

converted P to S waves across the Yellowstone hotspot track: Journal of Geophysical Research,

102, no. B4, 8313–8327.

Kennett, B. L. N., 1991, The removal of free surface interactions from three-component seismograms:

Geophysical Journal International, 104, 153–163.

21

Langston, C., 1977, Corvallis, Oregon, crustal and upper mantle structure from teleseismic P and S

waves: Bulletin of the Seismological Society of America, 67, 713–724.

Li, X.-Q., 1996, Deconvolving orbital surface waves for the source duration of large earthquakes and

modeling the recevier functions for the earth structure beneath a broadband seismometer array in

the cascadia subduction zone:.

Nabelek, J., Li, X.-Q., Azevedo, S., Baunmiller, J., Fabritius, A., Leitner, B., Tréhu, A., and Zandt,

G., 1993, A high-resolution image of the Cascadia subduction zone from teleseismic converted

phases recorded by a broadband seismic array: EOS, Transactions of the American Geophysical

Union 74 (43, Fall meeting supplement).

Phinney, R. A., 1964, Structure of the Earth’s Crust from Spectral Behavior of Long-period Body

Waves: Journal of Geophysical Research, 69, no. 14, 2997–3017.

Poppeliers, C., and Pavlis, G. L., 2003, Three-dimensional, prestack, plane wave migration of tele-

seismic P-to-S converted phases: 2. Stacking Multiple Events: Journal of Geophysical Research,

108, no. B5, art. no.–1583.

Rondenay, S., Bostock, M. G., and Shragge, J., 2001, Multiparameter two-dimensional inversion of

scattered teleseismic body waves 3. Application to the Cascadia 1993 data set: Journal of Geo-

physical Research, 106, no. B12, 30795–30807.

Sheehan, A. F., Shearer, P. M., Gilbert, H. J., and Dueker, K. G., 2000, Seismic migration processing

of P-SV converted phases for mantle discontinuity structure beneath the Snake River Plain, western

United States: Journal of Geophysical Research, 105, no. B8, 19055–19065.

Shragge, J., Bostock, M. G., and Rondenay, S., 2001, Multiparameter two-dimensional inversion of

22

scattered teleseismic body waves 2. Numerical examples: Journal of Geophysical Research, 106,

no. B12, 30783–30793.

Stoffa, P. L., Fokkema, J. T., de Luna Freire, R. M., and Kessinger, W. P., 1990, Split-step Fourier

migration: Geophysics, 55, no. 04, 410–421.

VanDecar, J., and Crosson, R., 1990, Determination of teleseismic relative phase arrivals using multi-

channel cross-correlation and least squares: Bulletin of the Seismological Society of America, 80,

150–159.

Wilson, C. K., Jones, C. H., and Gilbert, H. J., 2003, Single-chamber silicic magma system inferred

from shear wave discontinuities of the crust and uppermost mantle, Coso geothermal area, Califor-

nia: Journal of Geophysical Research, 108, no. B5, art. no.–2226.

Zhe, J., and Greenhalgh, S., 1995, Prestack multicomponent migration: Geophysics, 62, 598–613.

23

Scattering Source WF Source Receiver Receiver

Phase Extrapolation Velocity Velocity Component

Operator Function Function

P P e−ikz 1z α α RP

P S e−ikz 1z α β RSV

P P e+ikz 1z α α RP

P S e+ikz 1z α β RSV

S P e+ikz 1z β α RP

S Sv e+ikz 1z β β RSV

S Sh e+ikz 1z β β RS H

Table 1.

24

SS

SS

PS

PS PP

PP

PS

PS Psrc

Psrc

Figure 1.

25

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Tim

eT

ime

Tim

eSurface Position

Tim

e

A

B C

D

Downgoing Wavefield Upgoing Wavefield

z=z1

S(x,z=0,t|s) R(x,z=0,t|s)

R(x,z=z1,t|s)S(x,z=z1,t|s)

Horizontal Distance Horizontal Distance

Horizontal DistanceHorizontal Distance

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Figure 2.

26

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ime

Tim

eSurface Position

Tim

e

A

B C

D

Downgoing Wavefield Upgoing Wavefield

z=z1

p

Horizontal Distance

Horizontal Distance Horizontal Distance

Horizontal Distance

S(x,z=0,t|p) R(x,z=0,t|p)

R(x,z=z1,t|p)S(x,z=z1,t|p)

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Figure 3.

27

Horizontal Distance (km)

Tim

e (s

)Synthetic Structural Model

0 50 100 150 200 250 300

0

20

40

60

80

100

Figure 4.

28

Horizontal Distance (km)

Dep

th (

km)

Forward−scattered P−S

0 100 200

0

50

100

Horizontal Distance (km)

Dep

th (

km)

Backscattered P−P

0 100 200

0

50

100

Horizontal Distance (km)

Dep

th (

km)

Backscattered P−S

0 100 200

0

50

100

Horizontal Distance (km)

Dep

th (

km)

Backscattered S−S

0 100 200

0

50

100

Figure 5.

29

Horizontal Distance (km)

Dep

th (

km)

P−wave velocity model

−100 0 100 200 300 400

0

50

100

6500

7000

7500

8000

Horizontal Distance (km)

Dep

th (

km)

S−wave velocity model

−100 0 100 200 300 400

0

50

100

3400

3600

3800

4000

4200

4400

4600

4800

Figure 6.

30

Horizontal Distance (km)

Dep

th (

km)

Forward−scattered P−S

0 50 100 150 200

0

50

100

Horizontal Distance (km)

Dep

th (

km)

Backscattered P−S

0 50 100 150 200

0

50

100

Horizontal Distance (km)

Dep

th (

km)

Backscattered S−S

0 50 100 150 200

0

50

100

Horizontal Distance (km)

Dep

th (

km)

All Mode Stack

0 50 100 150 200

0

50

100

Figure 7.

31

LIST OF TABLES

1 Migration parameters required to migrate each first-order scattering mode.

32

LIST OF FIGURES

1 Ray paths of forward and backscattered phases generated by an impinging teleseismic P

wave front. Solid lines mark P ray paths with S rays marked by dashed lines. Uppercase letters de-

note down-going phases with lowercase letters symbolizing up-going phases. The idealized vertical

and radial seismograms indicate the approximate arrival and amplitude relationships for some of the

arrivals in the ray diagram above. The diagram demonstrates the natural separation of P and S phases

from near vertically incident waves onto the vertical and horizontal components.

2 Idealized experiment of a point scatterer in a half-space: a) Downgoing source wavefield

S(x , z = 0, t |s) for a shot point located at s = s(x0, z = 0); b) Source wavefield S (x , z = z1, t |s) at

depth z = z1; c) Receiver wavefield R (x , z = 1, t |s) generated through interaction of wavefield in

panel b) and earth structure; and d) the resulting receiver wavefield measured at the free-surface,

R (x , z = 0, t |s). The adjoint of this process, shot-profile migration requires propagating the wave-

fields in panels a) and d) into the subsurface to form the wavefields in panels b) and c), and creating

an image by correlating the wavefields in b) and c).

3 Teleseismic wavefield backscattering imaging example for a point-scatterer in a half-space

of constant velocity: a) Downgoing plane-wave source S(x , z = 0, t |p) with ray-parameter p; b)

Source wavefield S (x , z = z1, t |p) at depth z = z1; c) Receiver wavefield R (x , z = 1, t |p) generated

through interaction of wavefield in panel b) and earth structure; and d) the resulting receiver wavefield

as measured at the free-surface, R (x , z = 0, t |p). The adjoint process - i.e., shot-profile migration -

requires propagating the wavefields in panels a) and d) into the subsurface to form the wavefields in

panels b) and c), and creating an image by correlating the wavefields in b) and c).

4 Shear velocity profile of the lithospheric model used for synthetic test of shot-profile mi-

gration algorithm. The model consists of three components: a low-velocity crust (α=6.2 km/s, β=3.6

33

km/s), a high-velocity mantle (α=8.0 km/s, β=4.5 km/s) and a remnant underthrust crustal segment

(α=8.0 km/s, β=4.5 km/s)

5 Shot-profile migration results for the 4 effective scattering modes. Reflections from slow-

to-fast interfaces (dark-to-white in Fig. 4) show up as white reflectors, while the fast-to-slow (white-

to-dark) are imaged as black. Top left: P S; Top right: P P; Bottom left: P S; and Bottom right: S S.

6 Cascadia velocity models used for shot-profile migration. Top: P-wave velocity model; and

Bottom: S-wave velocity model.

7 Shot-profile migration results for two scattering modes, and the resulting stack. Top Left:

P S; Top Right: P S; Lower Left: S S and Bottom Right: stack of the other three images.

34