Fréchet kernels for finite-frequency traveltimes-II. Examples

Fre¨chet kernels for ¢nite-frequency traveltimesöII. Examples

S.-H. Hung, F. A. Dahlen and Guust Nolet

Department of Geosciences, Princeton University, Princeton, NJ 08544, USA. E-mail: [email protected] Accepted 1999 November 9. Received 1999 November 5; in original form 1999 May 14

SUMMARY

3-D Born^Fre¨chet traveltime kernel theory is recast in the context of scalar-wave propagation in a smooth acoustic medium, for simplicity. The predictions of the theory are in excellent agreement with `ground truth' traveltime shifts, measured by cross-correlation of heterogeneous-medium and homogeneous-medium synthetic seismograms, computed using a parallelized pseudospectral code. Scattering, wave-front healing and other ¢nite-frequency di¡raction e¡ects can give rise to cross-correlation traveltime shifts that are in signi¢cant disagreement with geometrical ray theory, whenever the cross-path width of wave-speed heterogeneity is of the same order as the width of the banana^doughnut Fre¨chet kernel surrounding the ray. A concentrated o¡-path slow or fast anomaly can give rise to a larger traveltime shift than one directly on the ray path, by virtue of the hollow-banana character of the kernel. The often intricate 3-D geometry of the sensitivity kernels of P, PP, PcP, PcP2, PcP3, . . . and PzpP waves is

explored, in a series of colourful cross-sections. The geometries of an absolute PP kernel and a di¡erential PP{P kernel are particularly complicated, because of the minimax nature of the surface-re£ected PP wave. The kernel for an overlapping PzpP wave from a shallow-focus source has a banana^doughnut character, like that of an isolated P-wave kernel, even when the teleseismic pulse shape is signi¢cantly distorted by the depth phase interference. A numerically economical representation of the 3-D travel-time sensitivity, based upon the paraxial approximation, is in excellent agreement with the `exact' ray-theoretical Fre¨chet kernel.

Key words: body waves, Fre¨chet derivatives, global seismology, ray theory, tomography, traveltime.

1 INTRODUCTION

This is the second in a series of two back-to-back papers devoted to the analysis of 3-D Fre¨chet kernels for ¢nite-frequency seismic traveltimes, measured by cross-correlation of a broad-band waveform with a spherical-earth synthetic seismogram. In the ¢rst paper (Dahlen et al. 2000, hereafter referred to as Banana^Doughnut I), we used the Born approxi-mation, in conjunction with body wave ray theory, to develop a general procedure for computing such cross-correlation travel-time kernels. In the most general formulation, the Fre¨chet kernel for a body wave phase of interest is expressed as a double sum over all forward-propagating waves from the source and all backward-propagating waves from the receiver, to every single scatterer in the earth. Upon ignoring all but like-type forward scattering, and introducing the Hessians of the forward and backward traveltime ¢elds, we can approximate this numerically intensive double-sum representation by a compact paraxial expansion, which can be computed extremely economically by implementing a single kinematic and dynamic ray trace along each central source-to-receiver ray.

In the present paper (Banana^Doughnut II), we undertake a more quantitative, visual examination of the e¡ects of scattering, front healing and di¡raction upon both seismic wave-forms and cross-correlation traveltimes, restricting attention to scalar-wave propagation in an acoustic medium, for reasons of simplicity. We ¢rst demonstrate the validity of our Born^ Fre¨chet kernel theory by numerical comparison of the predicted traveltime shifts with `ground-truth' shifts measured by cross-correlation of a suite of pseudospectral synthetic seismograms. We then present a pictorial glossary of 3-D, di¡erential and absolute traveltime kernels for a number of commonly observed, direct and re£ected seismic phases in a simple, smooth, acoustic model of the mantle. The validity of the paraxial approximation is assessed by comparison of a number of paraxial kernels with the full, double-sum formulation.

2 ACOUSTIC WAVE PROPAGATION

All of the results obtained in Banana^Doughnut I are appli-cable to an acoustic medium; it is simply necessary to set the rigidity k equal to zero. Many of the complications, which arise

from the proliferation of P, SV, SH?P, SV, SH polarization interactions at a scatterer, disappear; as a result, all of the double sums over body waves that propagate from the source to a scatterer and then on to the receiver are considerably simpli¢ed. To enable the present paper to be easily read without having digested all the grisly details in Banana^Doughnut I, we present an independent derivation of the Fre¨chet kernels of an acoustic wave traveltime here. The ¢nal formulae are identical to those obtained by specialization of the general results to an acoustic medium.

2.1 Equations of motion

We specify the medium in terms of its local density o and acoustic wave speed

c~pi/o, (1)

where i is the incompressibility. The linearized equations of motion governing acoustic wave propagation can be written in terms of these two properties in the form

Ltu~{o{1=p , (2)

Ltp~{oc2(= . u)zoc2m(t)d(r{s) . (3) The unknown quantities u and p are the velocity of the £uid and the associated incremental pressure at point r and time t. The ¢nal term in (3) represents an assumed source of sound at the point s. We presume that this source is some sort of transient pulse that commences at time t~0:

m(t)~0 for t < 0 . (4)

Physically, m(t) can be regarded as the instantaneous rate of change of an in¢nitesimally small volume dV(t) situated at the source point s (Morse & Ingard 1968).

2.2 In¢nite homogeneous medium

In an in¢nite homogeneous medium, with properties

o~constant, c~constant , (5)

eqs (2) and (3) reduce to the classical wave equation +2_p{c{2_L2

tp~{o _m(t)d(r{s) . (6)

Here and elsewhere in this paper, we use a dot to denote di¡erentiation with respect to time. The unique solution to eq. (6) is

p(t)~o _m(t{R/c)_4nR . (7)

The quantity R~kr{sk is the straight-line distance between the source s and receiver r. The acoustic pressure response (7) is a delayed pulse that propagates with speed c and is geometrically attenuated by a factor R{1_{. The shape of the} pressure pulse is the time derivative of m(t), that is, the second derivative of the di¡erential source volume, _m(t)~d V(t).

The analogue of eq. (7) in the frequency domain is

p(u)~o _m(u) exp ({iuR/c)_4nR , (8)

where _m(u) is the Fourier transform of the pressure-response pulse _m(t). Our sign convention is the same as that adopted in Banana^Doughnut I: a factor exp ({iut) appears in the Fourier integral upon transforming from time t to angular frequency u.

2.3 Geometrical ray theory

In a smoothly varying heterogeneous medium, the approximate ray-theoretical or JWKB pressure response is a straightforward generalization of the result (8):

p(u)~ 1 4n X rays (osorcscr)1=2(csR){1_m(u) | exp i({uTzMn/2) . (9)

The subscripts on os, cs and or, cr denote evaluation at the source s and the receiver r, respectively. The summation accounts for the possibility of multipathing, that is, more than one geometrical ray between s and r. The quantity T~Trsis the traveltime of an acoustic wave along each ray, given by the line integral T~ …_r s dl c , (10)

where dl is the di¡erential arclength. Every passage of a wave through a caustic gives rise to a non-geometrical n/2 phase shift; the Maslov index M~Mrsis a monotonically increasing integer that keeps track of the number of caustic passages and attendant n/2 phase shifts along each ray. Finally, R~Rrs is a geometrical attenuation or spreading factor, analogous to the straight-line source^receiver distance R~kr{sk in a homogeneous medium.

The time-domain response corresponding to (9) is p(t)~1 4n X rays(osorcscr) 1=2_(c sR){1_m(M)H (t{T) . (11) Every passage through a caustic acts to Hilbert transform the near-source pressure pulse _m(t); the M-times-transformed pulse is

_m(M)

H (t)~1_nRe …_?

0 _m (u) exp i(utzMn/2) du . (12) The traveltime and number of caustic passages are independent of whether a ray is traced from the source to the receiver or vice versa: Trs~Tsr, Mrs~Msr. The geometrical spreading factor satis¢es a slightly more complicated symmetry relation, namely csRrs~crRsr. These kinematic and dynamical sym-metries together guarantee that the JWKB responses (9) and (11) are invariant under an interchange of the source and receiver, s < r. This is the acoustic version of the principle of source^receiver reciprocity: psr~prs.

2.4 Born approximation

Suppose now that the properties of the medium are subjected to an in¢nitesimal perturbation:

o?ozdo , c?czdc . (13)

A straightforward application of the Born approximation yields the resulting ¢rst-order frequency-domain pressure

perturbation: dp(u)~ _4nu 2 ……… +  X rays0 X rays00 (osorcscr)1=2 |(ccscrR0R00){1  {2  dc c  {(1{kê0. kê00) do_o  

| _m(u) exp i[{u(T0_zT00_)z(M0_zM00_)n/2]_d3_{x . (14)} The integration variable in the representation (14) is the position x of an arbitrary point scatterer in the region of space + where the perturbations (13) are non-zero. The double sum is over all forward rays0 _{from the source s and all backward} rays00 _{from the receiver r to the scatterer x; the unit vectors} kê0and kê00are the wave vectors of the incoming and outgoing waves at x, as illustrated in Fig. 1. The other primed and double-primed variables are the traveltimes T0_~T

xs, T00~Txr, the Maslov indices M0_~M

xs, M00~Mxr, and the geometrical spreading factors R0_~R

xs, R00~Rxr along the forward rays0 and backward rays00_.

The equivalent time-domain pressure perturbation is dp(t)~{ _4n1  2……… +  X rays0 X rays00 (osorcscr)1=2 |(ccscrR0R00){1  {2  dc c  {(1{kê0. kê00) do o   | m:(M_H0zM00)(t{T0_{T00₎_d3_{x . (15)} All of the primed and double-primed quantities in (14) and (15) are invariant under an interchange of the source and receiver, s < r, except for the wave vectors, which are reversed: kê0?{kê0, kê00?{kê00. The dot product kê0. kê00is una¡ected by this reversal, however, so that the pressure perturbations dp(u) and dp(t) also satisfy the principle of acoustic source^receiver reciprocity: dpsr~dprs.

2.5 Acoustic Fre¨chet kernel

We show in Banana^Doughnut I that a ¢nite-frequency traveltime anomaly measured by cross-correlation of a windowed, observed pressure pulse pobs_{(t)~p(t)zdp(t) with}

a synthetic seismogram p(t) is given, correct to ¢rst order in the perturbations dc and do, by

dT~ …_t2 t1 _p(t)dp(t) dt …_t2 t1 p(t)p(t) dt ~Re …? 0 iu p1(u) dp(u) du …? 0 u 2_{j p(u)j}2_du , (16)

where the asterisk denotes complex conjugation. A negative traveltime shift, dT < 0, corresponds to an advance in the arrival of the observed signal pobs_{(t) with respect to the} syn-thetic signal p(t), whereas a positive traveltime shift, dT > 0, corresponds to a delay. To simplify the integrals in (16), we replace the unperturbed Fourier transform p(u) by the single phase of interest in the sum (9), but retain the summation over all possible singly scattered waves that may arrive during the cross-correlation time window t1¦t¦t2in the representation (14) of the perturbation dp(u). The resulting traveltime shift of an acoustic wave can be written in a form analogous to (77)^(78) in Banana^Doughnut I: dT~ ……… + Kc  dc c  zKo do_o     d3_{x , (17)} where Kc,o~{_2nc1 X rays0 X rays00)c,o R crR0_R00   | …? 0 u

3_{j _m(u)j}2_{sin [u(T}0_zT00_{T){(M0_zM00_{{M)n/2] du} …?

0 u

2_{j _m(u)j}2_du .

(18) The quantities Kcand Koare the 3-D Fre¨chet kernels that relate a measured traveltime shift dT to the fractional perturbations dc/c, do/o in acoustic wave speed and density, respectively. The two factors

)c~1 , )o~1₂(1{kê0. kê00) (19) are normalized acoustic scattering coe¤cients analogous to the 27 elastic scattering coe¤cients )P,SV,SH?P,SV,SH

a,b,o in Banana^

Doughnut I. Scattering o¡ a point heterogeneity in wave speed dc is isotropic, whereas scattering o¡ a density heterogeneity do is identically zero in the forward (kê00~kê0) direction and maximal in the backward (kê00~{kê0) direction, as illustrated in Fig. 2.

s

r

_x

Figure 1. Cartoon cross-section of an acoustic mantle model show-ing the geometrical ray from a buried source s to a surface receiver r. The Born approximation accounts for all singly scattered waves that

propagate along a composite path ray0_{, ray}00_{from the source s to an}

arbitrary point heterogeneity x, and then on to the receiver r. The

incoming and outgoing unit wave vectors at the scatterer x are kê0

and kê00, respectively.

Ωc Ωρ

Figure 2. Perspective plots of the acoustic wave scattering coe¤cients

)c(left) and )o(right). The orthonormal axes are centred upon the

scatterer x; arrows denote the direction of the incoming wave vector kê0.

2.6 Boundaries

The attentive reader will have noted that we have made no mention of the boundary conditions associated with the di¡erential equations (2)^(3). In fact, the above results are strictly valid only for a smooth in¢nite medium with no internal or external boundaries. More generally, the wave-speed and density Fre¨chet kernels (18) must be modi¢ed to account for the partition of incoming wave energy into outgoing transmitted and re£ected energy at each boundary, by introducing the products %, %0 _{and %}00 _{of +(energy)}1=2 acoustic wave re£ection coe¤cients along ray, ray0_{and ray}00_{, as} in Banana^Doughnut I. In all of the spherical earth examples to be presented in this paper, we consider a smooth acoustic `mantle' with no internal discontinuities, and an upper and lower boundary that are both perfectly rigid. The re£ection coe¤cient of an acoustic pressure pulse at a rigid boundary is simply unity,

%~%0_~%00_{~1 , (20)}

so that the kernels (18) are unaltered. If the two bounding surfaces are instead considered to be free, we must account for the reversal in the sign of p(t) and dp(t) upon every re£ection; the kernels are then given by

Kc,o~{_2nc1 X rays0 X rays00 ({1)N{N0_{N00 )c,o_crRR_0R00 | …? 0 u

3_{j _m(u)j}2_{sin [u(T}0_zT00_{T){(M0_zM00_{{M)n/2] du} …?

0 u

2_{j _m(u)j}2_du ,

(21) where N, N0_{and N}00_{are the number of free-surface re£ections} along the three paths ray, ray0_{and ray}00_.

2.7 Paraxial kernel

In the paraxial approximation, we ignore all but forward (kê00~kê0) scattering on nearby, like-type source-to-scatterer-to-receiver ray paths, and we approximate the various quantities in eq. (18) by )c~1 , )o~0 , (22) T0_zT00_{T~1 2qT. (M0zM00) . q , (23) R crR0R00~  jdet (M0_zM00_)j q , (24) M0_zM00_{M~1 2[sig (M0zM00){2] . (25) The matrices M0_~M

msand M00~Mmrare the forward and back-ward 2|2 traveltime Hessians along the central ray s¦î¦r, and q~x{î is the perpendicular distance of a scatterer x away from this ray, as illustrated in Fig. 3(a). The symbols det and sig denote the determinant and the signature, or the number of positive minus the number of negative eigenvalues, of the sum M0_zM00_{, respectively. The paraxial traveltime expansion (23)} is introduced, and the spreading factor and Maslov index relations (24)^(25) are veri¢ed, in Banana^Doughnut I.

A cross-correlation traveltime shift in this approximation is independent of the density, and dependent only upon the 3-D wave-speed perturbation:

dT~ ………

+ K(dc/c) d

3_{x . (26)}

The approximate Fre¨chet kernel K in eq. (26) is given by

K~{_2nc1 q_jdet (M0_zM00_)j …? 0 u 3_{j _m(u)j}2_{sin ' du …?} 0 u 2_{j _m(u)j}2_du , (27)

(a)

(b)

Figure 3. (a) In the paraxial approximation, every scatterer x is perpendicularly projected onto the nearest point m on the central geometrical ray from the source s to the receiver r. The di¡erence vector is q~x{î. (b) In many circumstances, including the two illustrated here, a scatterer can be projected onto more than one paraxial point î. (Left) A scatterer in the vicinity of the surface re£ection point of a PP wave. (Right) A scatterer in the vicinity of the core^mantle boundary (CMB) re£ection point of a PcP wave.

where

'~1₂uqT_{. (M}0_zM00_{) . q{[sig (M}0_zM00_{){2]n/4 .}

(28) Since ({1)N0_zN00_{N

in the free-surface representation (21) is approximated by 1, the result (26)^(28) is applicable, regard-less of the character of the upper and lower boundaries. We shall, in what follows, refer to Kcand Koin (18) as the `exact' ray-theoretical kernels, and we shall refer to K in (27) as the paraxial kernel. In the case of a compound ray such as PP or PcP, there may be more than one perpendicular projection point î for some scatterers x, as illustrated in Fig. 3(b); the paraxial kernel K in such instances is a sum over all such central ray points î.

2.8 Di¡erential kernel

The ¢rst-order perturbation d(*T) in a di¡erential traveltime,

*T~TB{TA, (29)

measured by cross-correlation of two observed body wave arrivals pobs

A (t)~pA(t)zdpA(t) and pobsB (t)~pB(t)zdpB(t) is likewise a linear functional of the 3-D wave-speed and density heterogeneity: d(*T)~ ……… + Kc  dc c  zKo do_o     d3_{x . (30)}

As in Banana^Doughnut I, we restrict attention to the only case of practical interest, in which the Maslov indices and, therefore, the pulse shapes of the two phases are identical: MA~MB[ _m(MHA)(t)~ _m(MHB)(t) . (31) Each of the di¡erential Fre¨chet kernels Kc,o in (23) is then simply the di¡erence of the kernels for the individual phases, i.e. KB{A

c,o ~Kc,oB{Kc,oA if MA~MB. (32)

A similar remark obviously applies to the paraxial kernels: d(*T)~ ……… +K(dc/c) d 3_{x , (33)} where KB{A_~KB_{KA _{if M} A~MB. (34)

The physically plausible results (32) and (34) reduce the problem of computing di¡erential Fre¨chet kernels to pointwise subtraction.

2.9 Reduction to ray theory

In the limit of high frequency, u??, it is possible to evaluate the 2-D integral over the transverse coordinates q~(q1, q2) in the paraxial representation (26) by the method of stationary phase. The resulting traveltime shift is exactly that predicted by geometrical ray theory:

dTray~{ …_r

s c

{2_{dc dl . (35)}

As is well known, Fermat's principle enables the integration in (35) to be carried out along the unperturbed ray.

3 NUMERICAL VALIDATION

We begin by considering an extremely simple exampleöa single, smooth, spherical, cosine-bell `inclusion' in an other-wise in¢nite homogeneous medium. The anomaly is situated at the centre of a 4000|4000|4000 km3 _{cube, as depicted in} Fig. 4; an explosive point source is detonated at a point s near one of the corners of the cube, and the wave¢eld and traveltimes are sampled at a fan-shaped array of equidistant receivers r in the vicinity of the diagonally opposite corner. We denote the position of an individual receiver r by its azimuth 00, 50, . . . , 850, 900; the middle or 450 receiver lies along the straight-ray path through the spherically symmetric anomaly. The linear distance from the source to each of the 19 receivers is kr{sk~4000 km; the centre of the anomaly is halfway between the source and the 450 receiver.

We investigate the e¡ect of both `slow' and `fast' anomalies, having do~0 and a wave-speed perturbation of the form

dc~ dc0[1zcos(2nr/a)] if r¦a/2

0 if r§a/2

(

, (36)

where r is the radial distance from the centre. We also consider a pure density anomaly, having dc~0 and the same cosine-bell shape:

do~ do0[1zcos(2nr/a)] if r¦a/2

0 if r§a/2

(

. (37)

The background wave speed and density are c~8 km and o~3300 kg m{3_{, respectively.} 0 1000 2000 3000 4000 0 1000 2000 3000 4000 0 1000 2000 3000 4000 x (km) y (km) z (km) s r 45° 90° 0°

Figure 4. Model con¢guration and source^receiver geometry used in numerical simulation of 3-D scalar-wave propagation. The shaded isosurface represents a smooth, spherically symmetric wave-speed or density anomaly embedded in an otherwise homogeneous background medium. Solid lines are the unperturbed straight rays between the source s, denoted by an asterisk, and an array of equidistant receivers r, denoted by open circles. The azimuth of the receivers varies from 00 to 900, so that some of the rays sample the anomaly, whereas others do not.

3.1 Source-time function

The time variation of the di¡erential source volume in eq. (3) is assumed to be a Gaussian, of the form

m(t)~ exp {4n2 t q{ 1 2  2 " # . (38)

The resulting pressure response in the background homogeneous medium is an acausal full-cycle pulse,

_m(t)~{8n2_q{2 _t{1 2q   exp {4n2 t q{ 1 2  2 " # , (39)

that is ¢rst manifestly di¡erent from zero at t~0. The associated power spectrum

j _m(u)j2_~(u2_q2_{/4n) exp ({u}2_q2_/8n2_{) (40)} has its centroid and its maximum at

ucen~  2 p (2n/q) , umax~2  2/n p (2n/q) , (41)

respectively. We shall, in the discussion that follows, refer to q as the characteristic period of the pulse (39), and to j~cq as the associated characteristic wavelength. It is, however, clear from (41) that the dominant or `visual' period and wavelength are about 1:5 times shorter than this:

qvis/q&jvis/j&1/  2 p &1 2  n/2 p . (42)

Since _m(t) is a full-cycle (up, then twice as far down, then back up) rather than a half-cycle (up, then back down) pulse, the pressure response p(t) simulates a ground-velocity rather than a ground-displacement body wave seismogram.

3.2 Pseudospectral synthetic seismograms

A parallelized pseudospectral method was used to solve eqs (2)^(3) numerically. The anisotropic, elastic code of Hung & Forsyth (1998) was adapted for this purpose. In this method, the wave¢eld variables p and u are represented as discrete 3-D Fourier expansions, enabling the spatial derivatives +p and = . u to be computed by multiplication in the wavenumber domain. Grid dispersion is minimal in comparison to ¢nite di¡erence methods, which use only a few neighbouring nodes to approximate the spatial derivatives. A conventional fourth-order Runge^Kutta scheme was used to advance p and u in time. An absorbing boundary condition described by Cerjan et al. (1985) was utilized on each of the eight faces of the

cube, to suppress arti¢cial re£ections. The pressure response at receivers r between gridpoints was computed by means of a 3-D bilinear interpolation from the eight adjacent nodal values. Computer memory and time considerations dictated the use of a relatively long-period source pulse, q~50 s (qvis~30^35 s). The numerical implementation was fully tested by comparison with the exact analytical solution (7) in the background homogeneous medium.

Table 1 lists the parameters utilized for a suite of `single-spherical-scatterer' numerical validation experiments. The width a and average fractional amplitude dc0/c of each of the wave-speed anomalies (36) have been chosen to ensure that the maximum geometrical advance or delay (35) is always the same, dTray~+3 s. The corresponding ray-theoretical traveltime shift for a density anomaly (37) is, of course, zero. The most critical parameter governing the validity of ray theory is the ratio a/j of the characteristic scale length of the 3-D hetero-geneity to the characteristic wavelength of the wave. Roughly speaking, we expect ray theory to pertain whenever this ratio signi¢cantly exceeds unity, a/j&1.

Fig. 5 shows a suite of synthetic waveforms for a relatively concentrated, relatively strong wave-speed anomaly (Case 1: a/j~1:5, dc0/c~+4 per cent) and a broader, more subdued one (Case 3: a/j~10, dc0/c~+1.5 per cent); the correspond-ing seismograms in the background homogeneous medium are superimposed for comparison. The waveform perturbations produced by a strong, concentrated anomaly are extremely subtle. Clearly, no matter how the traveltime shifts at the receivers in the vicinity of 450 are measured, they will be much less than those predicted by geometrical ray theory, dTray~+3 s. On the other hand, the arrivals near 450 in the case of a broad, subdued anomaly are either advanced or delayed by about this expected amount. Other ray-theoretical e¡ects are also visible in this latter case; in particular, the maximum amplitude of the 450 pulse is increased in the case of a slow anomaly (dc < 0) and decreased in the case of a fast one (dc > 0), as a result of geometrical focusing and defocusing, respectively. Synthetic seismograms were also computed for a suite of models having a density heterogeneity do rather than a wave-speed hetero-geneity dc (Cases 4^6). The resulting waveform perturbations were all indiscernible to the naked eye at the level of resolution shown here.

3.3 Fre¨chet kernel geometry

In this simple case of a homogeneous background medium, there is a single straight ray0_{from the source s and a single}

Table 1. Summary of parameters used in the numerical validation experiments: q is the characteristic period of the unperturbed pressure pulse _m(t) and j~cq is the associated characteristic wavelength; a is the characteristic width of the bell-shaped anomaly and dc0/c and do0/o are the average fractional perturbations; L~4000 km is the invariant distance between the source s and the receiver r.

Case q (s) j (km) a (km) L (km) dc0/c (%) do0/o (%) a/j L/j L/a dTray(s)

1 50 400 600 4000 +4 = 1:5 10 6:67 +3 2 50 400 1000 4000 +2:4 = 2:5 10 4 +3 3 20 160 1600 4000 +1:5 = 10 25 2:5 +3 4 50 400 600 4000 = +4 1:5 10 6:67 0 5 50 400 1000 4000 = +2:4 2:5 10 4 0 6 20 160 1600 4000 = +1:5 10 25 2:5 0

straight ray00_{from the receiver r to the scatterer x. The} travel-times, Maslov indices and geometrical spreading factors in eq. (18) are given by

T~kr{sk_c , T0_~kx{sk c , T00~ kx{rk c , (43) M~M0_~M00_{~0 , (44)} R~kr{sk , R0_{~kx{sk , R}00_{~kx{rk . (45)}

The resulting `exact' Fre¨chet traveltime kernels Kcand Kofor a q~50 s acoustic wave are depicted in Fig. 6. The receiver r is at azimuth 450; the superimposed black circles show the dc/c~0, +2, +4, +6 per cent contours of an a~1:5j, dc0/c~+4 per cent (i.e. jdcmaxj/c~8 per cent) anomaly for comparison. The banana^doughnut character of both kernels is evidentöalthough perhaps `cigar^doughnut' would be a more apt metaphor in this straight-ray instance! In a ray-plane cross-section, both Kcand Koexhibit an elliptical shape,

480 500 520 540 560 time (s) 0º 5º 10º 15º 20º 25º 30º 35º 40º 45º 50º 55º 60º 65º 70º 75º 80º 85º 90º 495 500 505 510 515 520 525 time (s) 480 500 520 540 560 time (s) 0º 5º 10º 15º 20º 25º 30º 35º 40º 45º 50º 55º 60º 65º 70º 75º 80º 85º 90º 495 500 505 510 515 520 525 time (s) (a) Case 1, δc/c < 0 (b) Case 1, δc/c > 0 (c) Case 3, δc/c < 0 (d) Case 3, δc/c > 0

Figure 5. Synthetic pressure-response seismograms p(t), computed using the pseudospectral method. All of the receivers are situated at the same distance from the source, kr{sk~4000 km; the azimuths 00, 50, . . . , 850, 900 are indicated on the left. Identical solid lines show the unperturbed seismograms in the homogeneous background medium; dashed lines show the e¡ect of a spherically symmetric, slow (top) or fast (bottom) perturbation in the acoustic wave speed. (a) Case 1: a/j~1:5, dc0/c~{4 per cent. (b) Case 1: a/j~1:5, dc0/c~z4 per cent. (c) Case 3: a/j~10, dc0/c~{1:5 per cent. (d) Case 3: a/j~10, dc0/c~z1:5 per cent. It is evident that cross-correlation is the only viable means of measuring the traveltime shift dT of such a long-period waveform. The e¡ect of wave-speed heterogeneity dc di¡ers for di¡erent segments of the pulse; traditional seismological methods that focus upon a single characteristic such as an analyst-picked onset or the maximum would be quite arbitrary, as well as prone to errors due to noise.

whereas in a cross-section perpendicular to the ray path, they are circularly symmetric, with a yellow hollow interior £anked by red (Kc,o< 0) side lobes. This yellow-to-red region can be regarded as the (broad-band) ¢rst Fresnel zone, within which

0¦u(T0_zT00_{{T)¦n , (46)}

where u&ucen&umax is the dominant frequency of the pulse (39). The fringing green (Kc,o> 0) sidelobes lie within

the second Fresnel zone, where n¦u(T0_zT00_{{T)¦2n. The} absolute magnitude of the kernels is signi¢cantly reduced in this as well as the surrounding higher-order Fresnel zones as a result of destructive interference among adjacent frequencies u and uzdu in the integral (18).

Finally, we note that the overall magnitude of the density kernel is everywhere much less than that of the wave-speed kernel: jKoj%jKcj. The reason for this is evident: destructive 0 1000 2000 3000 4000 5000

vertical distance (km)

horizontal distance (km)

traveltime kernel K

(10

_s/km

₎

distance (km)

distance (km)

vertical distance (km)

horizontal distance (km)

traveltime kernel K

(10

_s/km

₎

distance (km)

distance (km)

ray plane cross sections

perpendicular cross sections

(a) wave-speed anomaly (a/

λ

= 1.5)

(b) density anomaly (a/

λ

= 1.5)

Figure 6. Cross-sections of the 3-D traveltime Fre¨chet kernels in a homogeneous background medium for a receiver at a distance kr{sk~4000 km

and an azimuth of 450. The characteristic period of the Gaussian-derivative wavelet _m(t) is q~50 s. (a) Sensitivity Kcto a fractional change dc/c in

wave speed. (b) Sensitivity Koto a fractional change do/o in density. Left panels show a ray-plane cross-section passing through the source s and

receiver r; right panels show a path-perpendicular cross-section, through the source^receiver midpoint. Circular contours show the location and size of a Case 1 or Case 4 anomaly. The outer circle, of radius a~1:5 j, is where the smooth, cosine-bell perturbation (36) or (37) vanishes.

interference limits the contribution of all scattering interactions that are not nearly forward (kê00&kê0), and forward scattering o¡ a density heterogeneity is extremely ine¤cient:

)o~1₂(1{kê0. kê00)&0 . (47) The upshot of this is hardly a surpriseöa traveltime shift dT is much less sensitive to density heterogeneity do than to wave-speed heterogeneity dc. Homogeneous-medium Fre¨chet kernels with properties similar to those presented here have been discussed previously by Woodward (1992), Yomogida (1992) and Vasco & Majer (1993).

3.4 Traveltime comparison

Synthetic traveltime shifts dT were measured for all of the cases considered by cross-correlation of the perturbed and unperturbed pseudospectral seismograms. The time window t1¦t¦t2was chosen conservatively, to ensure that we always included essentially the entire two-sided pulse _m(t). The values of dT were determined by least-squares ¢tting of a parabola to the digital cross-correlagram in the vicinity of its maximum. Fig. 7 compares the measured cross-correlation traveltime shifts with the theoretical predictions of both Born^Fre¨chet kernel theory (17)^(18) and geometrical ray theory (35). As expected on the basis of the above discussion, a slow

0 1 2 3 0º 30º 60º 90º traveltime delay δ T (seconds) receiver azimuth 0 1 2 3 0º 30º 60º 90º receiver azimuth 0 1 2 3 0º 30º 60º 90º receiver azimuth -3 -2 -1 0 0º 30º 60º 90º traveltime delay δ T (seconds) receiver azimuth -3 -2 -1 0 0º 30º 60º 90º receiver azimuth -3 -2 -1 0 0º 30º 60º 90º receiver azimuth 0 1 2 3 0º 30º 60º 90º

traveltime delay δT (seconds)

receiver azimuth 0 1 2 3 0º 30º 60º 90º receiver azimuth 0 1 2 3 0º 30º 60º 90º receiver azimuth (a) δc/c < 0 (b) δc/c > 0 (c) δρ/ρ < 0

Figure 7. Traveltime anomaly dT versus receiver azimuth. Solid circles are the traveltimes measured by cross-correlation of the pseudospectral

synthetic seismograms; un¢lled inverted triangles are the theoretical traveltimes computed using the Born Fre¨chet kernels Kcand Ko; smooth curve

shows the Fermat straight-ray approximation. (a) Slow wave-speed anomalies: Cases 1 to 3 (left to right). (b) Fast wave-speed anomalies: Cases 1 to 3 (left to right). (c) Density anomalies: Cases 4 to 6 (left to right). Wave-front healing and di¡raction reduce the magnitudes of the traveltime shifts produced by both a concentrated (Case 1) and a quasi-concentrated (Case 2) wave-speed anomaly. The traveltime shifts produced by a smoother (Case 3) anomaly are, in contrast, in good agreement with geometrical ray theory. A modest density anomaly produces an imperceptible traveltime shift, no matter how it is measured.

or fast concentrated wave-speed anomaly (Case 1: a/j~1:5, dc0/c~+4 per cent) gives rise to a relatively small travel-time shift, whereas a broader anomaly (Case 3: a/j~10, dc0/c~+1:5 per cent) gives rise to a more substantial shift that is in good agreement with ray theory (dT&dTray). An `intermediate' wave-speed anomaly (Case 2: a/j~2:5, dc0/c~+2:4 per cent) gives rise to an `intermediate' traveltime shift, whose maximum at the 450 receiver is slightly reduced from the ray-theoretical maximum, dTray~+3 s, as a result of di¡raction and wave-front healing. All three density anomalies (Cases 4^6) give rise to a negligible traveltime shift, dT&0, as expected.

In every case, the measured traveltime shift is in excellent agreement with the shift computed by integration with the Fre¨chet kernel Kc or Ko. This is the most important ¢nding of the present numerical study, since it validates our Born^ Fre¨chet kernel theory, both in situations where ray theory is applicable, as well as in situations where it is not. In the example considered here, the lateral refraction of geometrical rays is relatively slight, so there are no caustics or triplications in the vicinity of any of the receivers r. In addition, the maximum traveltime anomalies are substantially less than the characteristic period of the waves (jdTrayj&qvis/10), so there is no possibility of confusion due to a cycle skip. This is the proper province of the Born approximation.

It is noteworthy that the maximum absolute traveltime shift produced by a concentrated wave-speed anomaly is not observed at the 450 receiver; rather, there is a local minimum in jdTj at 450, with maxima at azimuths 350 and 550. This is an extremely unintuitive result on the basis of naive ray-theoretical considerations; however, it is a natural consequence of the banana^doughnut character of the wave-speed kernel Kc. A su¤ciently small heterogeneity dc0/c can ¢t inside the yellow doughnut hole of the 450 kernel, and thus give rise to a negligible traveltime shift dT. At 350 and 550 the anomaly dc0/c is situated within the red doughnut itself, where jKcj is maximal. In this way, a concentrated o¡-path anomaly can give rise to a larger traveltime shift than one directly on the ray path. Inspection of the synthetic seismograms in Fig. 5 suggests that the minimum at 450 would not be present if dT were measured by hand-picking the `onset' or ¢rst `break' of the arrivals. In a cross-correlation measurement, however, the entire pulse contributes to the traveltime shift. The principal di¡erence between the perturbed and unperturbed seismo-grams at 450 is the steeper downswing following the initial upswing in the case of a slow anomaly, dc < 0, and the more gradual downswing in the case of a fast anomaly, dc > 0. In essence, di¡raction by a slow or fast anomaly shifts a fraction of the total energy to earlier or later times in the downswing, leading to a reducedöin fact locally minimalöabsolute traveltime shift jdTj.

We note ¢nally that at the receivers in the vicinity of 200 and 700, the traveltime shifts produced by a concentrated anomaly are actually of the `opposite' signöfast, dT < 0, for a slow anomaly, dc < 0, and slow, dT > 0, for a fast anomaly, dc > 0. This result is particularly perplexing on the basis of naive considerations, since in the latter case it appears to violate the principle of causality. The Fre¨chet kernel provides a ready explanation: the centre of the anomaly is in this case situated in the second (green) Fresnel zone, where Kc is positive. Blow-ups of the synthetic seismograms in Fig. 5 reveal that the ¢rst `breaks' do respect causality, as

of course they must. The principal discrepancy between the 200 and 700 perturbed and unperturbed seismograms is in the tails.

3.5 Wielandt e¡ect

A previous numerical investigation of scattering and di¡raction by a single spherical inclusion has been conducted by Wielandt (1987). He considered a sphere with a constant wave-speed perturbation dc; this problem of a hard acoustic scatterer has a well-known analytical solution (Sommerfeld 1949). An auto-mated picking method was used to measure synthetic travel-times rather than cross-correlation; the `onset' of every pulse was de¢ned to be the point at which its amplitude ¢rst exceeded 15 per cent of its peak value. Using this technique, Wielandt observed a large and consistent di¡erence in the character of the measured traveltimes, depending upon the sign of the anomaly. Positive traveltime shifts produced by a slow (dc < 0) sphere were signi¢cantly reduced by wave-front healing, whereas negative shifts produced by a fast (dc > 0) sphere were in much better agreement with ray theory. The physical explanation for this slow^fast asymmetry is straightforward: waves di¡racted around the boundary of a slow sphere are able to arrive signi-¢cantly before the straight-ray geometrical arrival, whereas no such faster di¡racted path exists in the case of a fast anomaly. The upshot of such a di¡raction-induced asymmetry in tele-seismic traveltimes would be a bias towards faster wave speeds in a 3-D inversion. This Wielandt e¡ect has subsequently been studied by a number of investigators, in both 2-D and 3-D pseudo-random models, using a variety of ¢nite di¡erence, ray-theoretical, eikonal and Fourier wave-front-migration methods (e.g. MÏller et al. 1992; Roth et al. 1993; Nolet & Moser 1993; Witte et al. 1996; Gudmundsson 1996). There is a very slight slow^fast asymmetry in the cross-correlation traveltimes (solid circles) in Fig. 7, but the e¡ect is of variable sign, depending upon the circumstances, and much smaller than that observed for picked times by Wielandt. This is in accordance with Born^ Fre¨chet kernel theory, which predicts that there should be no slow^fast asymmetry whatsoever for slight traveltime shifts measured by cross-correlation. A change in the sign of the wave-speed anomaly, dc?{dc, in eq. (17) or (26) simply changes the sign of the traveltime anomaly, dT?{dT. Nolet & Dahlen (1999) ¢nd an analogous slow^fast asymmetry, which is more signi¢cant for larger values of the fractional traveltime anomaly jdTj/qvis, in their analysis of the healing of a wave front with an initial delay or advance in a homogeneous medium. The slow^fast asymmetry and the associated error in the Born^Fre¨chet linearization (17)^(18) are also likely to be larger in the present case for larger-amplitude anomalies +dc/c.

4 RAY-THEORETICAL KERNEL

In this section, we present a number of examples of 3-D Fre¨chet kernels Kc for a slightly more realistic seismological situationöa smooth, spherically symmetric mantle model with a radial wave-speed pro¢le given by

c(r)~(r/a)[c0zc0a ln (a/r)] in b¦r¦a , (48) where

a~6371 km , b~3471 km (49)

and

c0~4:5 km s{1, c0~0:0025 s{1. (50) The parameters (49)^(50) have been chosen to simulate the propagation of shear rather than compressional waves; the wave speed increases monotonically from its surface value c0~4:5 km s{1at a~6371 km to a maximum cmax~7:7 km s{1 at the core^mantle boundary b~3471 km, as illustrated in Fig. 8. Because of the rigid upper boundary conditions (20), the results presented here are best regarded as kernels for horizontally polarized SH, SSSH, ScSSH, . . . waves; in the spirit of acoustic correctness, however, we shall refer to the various waves and associated kernels as P, PP, PcP, . . . . The particular analytical form of the wave-speed pro¢le (48) was chosen to expedite the two-point ray tracing from the source s and receiver r to an arbitrary scatterer x. The details of the quasi-analytical procedures used to trace rays are described in Appendix A.

We restrict attention for the time being to a source s situated upon the earth's surface. Fig. 9 summarizes the various scattering paths that must then be taken into account in computing a ray-theoretical Fre¨chet kernel Kc. Each ray0or ray00_{in the double sum (18) is characterized by the number of} re£ections it experiences o¡ either the upper or lower surface between the source s or the receiver r and the scatterer x. A re£ection o¡ the core^mantle boundary may either arrive directly at the scatterer x from below, or proceed on to re£ect again o¡ the top surface to arrive at x from above. A re£ection o¡ the upper surface that does not encounter the core^mantle boundary is a special case, since depending upon its location, a scatterer x may have either no or two such incoming rays0_or rays00_{. A scatterer situated above the source-to-receiver caustic} has two incoming surface-re£ected rays0_{, one that has passed} through the caustic and arrives from below, and another that

has not yet passed through the caustic and arrives from above. Similar remarks apply to the backward rays00_{and the} receiver-to-source caustic, so that these two caustics partition `scatterer space' into four distinct regions, as illustrated in Fig. 10. In all the examples considered here, we have summed all rays0<sub>and</sub> rays00 <sub>having two or fewer re£ections o¡ the upper surface</sub> and the core^mantle boundary. This is purposeful overkill, since our intent is to compute bona ¢de `exact' (ray0_zray00₎ Fre¨chet kernels Kc. In practice, the only signi¢cant contri-butions to Kc come from like-type scattering paths with traveltimes T0_zT00_&T.

To display the kernels, we use the spherical polar coordinate system depicted in Fig. 11. The source s and receiver r are both situated on the equator, at colatitude h~900, and longitudes ~0 and ~*, respectively. The quantity * is the angular epicentral distance; geometrical rays between s and r are con-¢ned to the equatorial source^receiver plane. To visualize the inherently 3-D kernels on the printed page, we resort to 2-D cross-sections, at ¢xed colatitude h, longitude  or depth h~a{r.

4.1 P wave

In Fig. 12 we display a number of cross-sectional views of the wave-speed kernel Kcfor a direct P wave at an epicentral distance *~600. Two di¡erent characteristic periods are considered, q~10 s and q~20 s. Fig. 12(a) depicts the two ray-plane cross-sections at constant colatitude h~900; Fig. 12(b) depicts several sections that cut quasi-perpendicularly across the ray path, at constant longitude, ~300 and ~450; Fig. 12(c) displays the kernel for a q~20 s wave on three vertical slices parallel to the ray plane, at colatitudes 900+10, 900+20, 900+30. The now familiar banana^doughnut character of the kernel is again apparent. The sensitivity is yellow, or identically zero, everywhere along the geometrical ray path; the maxi-mum sensitivity is in the red outer part of the ¢rst Fresnel zone, where Kc< 0. This in turn is surrounded by a faint green second Fresnel zone, where Kc> 0. The black curves in Fig. 12(a) show the radial dependence of the sensitivity along a line passing through the turning point at ~300. These 1-D representations illustrate the zero sensitivity along the ray, and the surrounding zones where Kc< 0 and Kc> 0, particularly clearly. For a ¢xed source^receiver geometry, the cross-path extent or `fatness' of the kernel Kcvaries as the square rootpq of the characteristic period of the wave; the q~20 s kernel is for this reasonp2times as `fat' as the q~10 s kernel. The kernel for an even shorter-period wave, q%20 s, would be an extremely slender hollow banana. Finally, we note that the acoustic wave kernels presented here agree extremely well with the SH-wave kernels computed by Marquering et al. (1999) using surface wave rather than body wave summation. We have not attempted a more quantitative comparison, inasmuch as the wave-speed pro¢les and the source time functions are slightly di¡erent in the two studies.

4.2 PP wave

The traveltimes of PP and SS surface re£ections are widely used in seismic tomography because of their unique ability to probe the structure of the upper mantle beneath the bounce point, in regions that may not be well sampled by subsource and subreceiver rays (e.g. Kuo et al. 1987; Sheehan & Solomon 4000 4500 5000 5500 6000 radius (km) 0 2 4 6 8 10 wave speed (km/s) 3471 6371 0 500 1000 1500 2000 2500 2900 depth (km) CMB

Figure 8. Variation of acoustic wave speed c(r) versus depth h~a{r in the model (48)^(50). The core^mantle boundary (CMB) is situated at a depth h~2900 km.

1991; Woodward & Masters 1991; Neele & Snieder 1992; Paulssen & Stutzman 1996; Neele et al. 1997; Neele & de Regt 1999; Shearer et al. 1999). In Fig. 13, we display a number of cross-sections through the 3-D Fre¨chet kernel Kcfor a q~20 s PP wave at an epicentral distance *~600. Visual comparison with the surface wave-sum SS kernel of Marquering et al. (1999) shows that its previously rather poorly understood geometrical complexity is faithfully reproduced. The present formulation provides a simple physical interpretation of the principal features of this complicated 3-D geometry, in terms of the traveltimes T, T0_{, T}00_{and Maslov indices M, M}0_{, M}00_of the contributing body waves.

The PP wave from s to r passes through the source-to-receiver caustic, where it experiences a non-geometrical n/2 phase shift, at ~2*/3~400. The backward wave from r to s likewise passes through the receiver-to-source caustic at ~*/3~200. The presence of these two caustics is responsible for the fundamental change in the character of the kernel along the geometrical PP ray. This change is most obvious in the ray-plane cross-section exhibited in Fig. 13(a): the traveltime sensitivity is identically zero (yellow) between the source and the receiver-to-source caustic, 00 <  < 200, and between

the receiver and the source-to-receiver caustic, 400 <  < 600, whereas it is minimal (red) between the two caustics, 200 <  < 400. These di¡erences are produced by the jumps in the source-to-scatterer and receiver-to-scatterer Maslov indices: M0_zM00_{M~ 0 if 00 <  < 200 {1 if 200 <  < 400 0 if 400 <  < 600 8 > < > : . (51)

The oscillatory term in the numerator of (21) changes character as a result of these jumps,

sin u(T0_zT00_{{T)? cos u(T}0_zT00_{T)

? sin u(T0_zT00_{{T) , (52)}

as we move along, or very nearly along, the ray from the source s to the receiver r.

In Fig. 13(b) we display a series of quasi-perpendicular cross-sections of the kernel Kc, at equally spaced longitudes x x x x x x x x x x x x x no reflections

+

one surface reflection one CMB reflection

+

two surface reflections two CMB reflections

+

..

.

Figure 9. Pictorial glossary of the possible composite ray paths from a surface source s to a surface receiver r. Each path has a single non-Snell interaction at a (usually buried but possibly sur¢cial) scatterer x. Solid and dotted lines distinguish cases in which there are multiple single-scattering

paths from s to r, with a given number of re£ections o¡ the free surface or the core^mantle boundary (CMB). In principle, all of these ray0_{, ray}00

combinations must be accounted for in evaluating the double sum (18).

~300, 350, 400, 450, 500, 550; in Fig. 13(c) we show several constant-depth slices at h~20, 100, 200, 400 km; and in Fig. 13(d) we show a number of vertical slices parallel to the ray plane, at colatitudes 900+10, 900+20, 900+30. The traveltime sensitivity is essentially zero (yellow) at all points x situated below the source-to-receiver and the receiver-to-source caustics, because no like-type (single surface re£ection) rays0 _{or rays}00 _{can be traced from s or r to this region of} `scatterer space', as discussed above. The ~400 section passes directly through the caustic; the saturated `red-brick' character of Kc expresses the strong sensitivity of a PP traveltime shift dT to wave-speed heterogeneity dc/c at this point. The saddle-shaped character of the near-surface depth slices, h~20, 100 km, is a consequence of the minimax nature of the PP wave. Waves that scatter o¡ in-plane heterogeneity dc/c to the `east' or `west' of the surface bounce point ~300 travel deeper and therefore arrive earlier than the geometrical PP

wave, whereas those that scatter o¡ heterogeneity to the `north' or `south' take an out-of-plane detour and therefore arrive later than the PP wave. At deeper depths, h~200 km, it becomes evident that there are in reality two saddles, one with its `stirrups' dipping towards the ray plane and the other with them dipping away.

In summary, the traveltime Fre¨chet kernel of a PP or SS wave is a wondrously shaggy 3-D beast, characterized by strong positive (Kc> 0) as well as negative (Kc< 0) sensitivity to o¡-path wave-speed heterogeneity far away from the geo-metrical ray. In most contemporary tomographic studies, this complicated 3-D dependence upon dc/c is replaced by a 1-D line integral (35) along the ray. Even more crudely, a measured traveltime shift dTPP or dTSS is often considered to be a vertically averaged, near-surface advance or delay accumulated just beneath the surface re£ection point. It is evident from Fig. 13 that any such pointwise interpretation is a considerable approximation!

4.3 PcP wave

The traveltimes of single PcP and ScS core re£ections and multiple PcP2, PcP3, . . . and ScS2, ScS3, . . . reverberations provide another rich source of tomographic data, which can be used to constrain mantle heterogeneity in the corridor between a source s and a receiver r (e.g. Sipkin & Jordan 1976, 1980; Katzman et al. 1998). Fig. 14 shows the 3-D Fre¨chet kernel Kc of a q~20 s PcP wave, recorded at an epicentral distance *~440. In essence, Kcis a hollow banana that is folded over itself at the core re£ection point, ~220. The ray-plane (h~900) cross-section is depicted in Fig. 14(a); the black curve shows the depth variation of the sensitivity along a line through the re£ection point. Fig. 14(b) shows two longitudinal slices, at ~220 and ~330, and Fig. 14(c) shows four depth slices, at h~2850, h~2800, h~2700 and h~2500 km. The maximum sensitivity is not located right on the core^mantle boundary, at a depth h~2900 km, but rather at the `cross-over point' of the folded banana skins, approximately 200^250 km above. The interior `¢ssioning doughnut' is due to the constructive interference of like-type scattered waves within the ¢rst Fresnel zone; however, the fringing green and red ellipses are the result of unlike-type P-to-P scattering paths, which happen to have a traveltime T0_zT00 _{similar to the time T of the PcP wave.} The bowl-shaped locus of these P-to-P scatterers x is barely visibleötry squintingöin Fig. 14(a). The ¢nal view, Fig. 14(d), shows the o¡-path structure on a series of colatitudinal slices, at 900+10, 900+20 and 900+30.

5 PARAXIAL APPROXIMATION

In the above three examples, as well as all other cases we have investigated, the dominant contribution to the `exact' traveltime kernels Kccomes from like-type forward-scattering (kê00&kê0) paths in the vicinity of the unperturbed ray path. An unlike-type composite path is a forward-and-backward pair ray0_{, ray}00_{that does not have the same total number of surface} and core^mantle boundary re£ections as the central ray. These make relatively minor contributions to the double sum (18) in only two situations.

s r

s r

no one-bounce rays′ and rays″ two one-bounce rays′ and two one-bounce rays″

no one-bounce rays ′

and two one-bounce rays ″

caustic caustic

two one-bounce rays

′

and no one-bounce rays

″

Figure 10. (Top) Family of PP rays shot from a surface source s and receiver r. The source-to-receiver and receiver-to-source caustics are the loci where adjacent rays cross. (Bottom) Dashed lines tangent to the surface of the earth at s and r denote these two bowl-shaped caustic surfaces. The caustics subdivide the space of single scatterers x into four regions, as shown.

φ = 0 N r φ = ∆ θ = π/2 s ∆

Figure 11. Schematic depiction of the equatorial coordinate system

used to depict the `exact' and paraxial Fre¨chet kernels Kcand K. The

North Pole is denoted by N.

(1) At points x in the vicinity of a surface re£ection point, the direct source-to-scatterer-to-receiver traveltime T0_zT00 _{can be} very nearly equal to the re£ection time T. This gives rise to small modi¢cations in the kernels Kc of re£ected phases such as PP and PcP in thin `boundary layers' near the re£ecting boundaries. (2) In addition, there may be occasional `accidental' travel-time coincidences T0_zT00_{&T such as the one that gives rise to}

the nearly invisible bowl-shaped £ange on the PcP-wave kernel in Figs 14(a) and (c).

Both of these e¡ects are ignored in the paraxial approximation (26)^(28). We investigate the veracity of this approximation by comparing the paraxial kernels K for a P, PP and PcP wave with the corresponding `exact' kernels Kcin this section.

60˚ 50˚ 40˚ 30˚ 20˚ 10˚ 0˚ 4000 5000 6000 85˚ 90˚ 95˚ 4000 5000 6000 85˚ 90˚ 95˚ 4000 5000 6000 60˚ 50˚ 40˚ 30˚ 20˚ 10˚ 0˚ 4000 5000 6000 85˚ 90˚ 95˚ 4000 5000 6000 85˚ 90˚ 95˚ 4000 5000 6000 –3 +3 traveltime kernel K (10-6 s/km3) 60˚ 40˚ 20˚ 0˚ 4000 5000 6000 60˚ 40˚ 20˚ 0˚ 60˚ 40˚ 20˚ 0˚ 400 0 5000 6000

(a) ray-plane cross sections (

θ

=90º)

∆=60º τ=10 s

(b) perpendicular cross sections

at φ=30º at φ=45º

∆=60º τ=20 s at φ=30º at φ=45º

(c) vertical cross sections off ray-plane

at θ=90º+_1º at θ=90º+_2º at θ=90º+_3º

Figure 12. 2-D cross-sections through the 3-D `exact' Fre¨chet kernel Kcfor a teleseismic P wave at an epicentral distance *~600. (a) Ray-plane

cross-sections; solid lines show the variation of Kcwith depth on a line through the turning point. (b) Longitudinal cross-sections at ~300 and

~450. Top panel shows the kernel for a q~10 s wave and bottom panel shows the kernel for a q~20 s wave in both cases. (c) Vertical cross-sections through the q~20 s kernel at distances +10, +20 and +30 o¡ the ray plane.

5.1 P wave

Fig. 15 shows the paraxial Fre¨chet kernel K for a q~10 s and a q~20 s P wave at an angular epicentral distance *~600. The

source^receiver geometry, all of the cross-sectional views, the plotting format and the colour scale are all identical to those in Fig. 12. It is evident that the agreement between the `exact' and approximate kernels is excellent. The construction of the

Figure 13. 2-D cross-sections through the 3-D `exact' Fre¨chet kernel Kcfor a minimax PP wave, with a characteristic period q~20 s at an

epi-central distance *~600. (a) Ray-plane slice at h~900. (b) Longitudinal slices at ~300, 350, 400, 450, 500, 550. (c) Constant-depth slices at h~20, 100, 200, 400 km; the corresponding radii are r~6351, 6271, 6171, 5971 km. (d) Vertical slices at distances +10, +20, +30 o¡ the ray plane.

paraxial kernel K of a P wave is extremely straightforwardö every scatterer x is projected onto a single point î on the central source-to-receiver ray.

It is immaterial in the context of the paraxial approximation whether the leading factor of c in eq. (27) is evaluated at the position of the scatterer x or at the projection point î. The former alternative is preferable, because it leads to a better agreement between the paraxial and `exact' kernels K and Kc. In particular, the slightly larger sensitivity in the uppermost banana `skin' is more accurately reproduced. This subtle di¡erence between the upper and lower sensitivities may be most easily seen by comparing the black curves, showing the

radial dependence of K and Kcalong a line through the turning point. The slightly darker red and fringing green at the top of the doughnut cross-sections in Figs 15(b) and 12(b) is also evident. If c(x) were replaced by c(î) in eq. (27), the paraxial doughnut would be circularly symmetric.

5.2 PP wave

In Fig. 16, we illustrate the paraxial traveltime kernel K for a q~20 s PP wave at an epicentral distance *~600 for comparison with Fig. 13. In this case, every scatterer x must be projected onto two points î, one on each leg of the

Figure 14. 2-D cross-sections through the 3-D `exact' Fre¨chet kernel Kcfor a q~20 s PcP wave at an epicentral distance *~440. (a) Ray-plane

cross-section at h~900; black curve shows the depth variation of Kcon a plumb line through the CMB bounce point. (b) Longitudinal cross-sections

at ~220, 330. (c) Constant-depth cross-sections at h~2850, 2800, 2700, 2500 km (i.e. 50, 100, 200, 400 km above the core^mantle boundary). (d) Vertical cross-sections at distances +10, +20, +30 o¡ the ray plane.

central geometrical ray. The sum of forward and backward traveltime Hessians M0_zM00 _{is divergent at both the} source-to-receiver and receiver-to-source caustics, ~2*/3~400 and ~*/3~200, as we discuss in Appendix A5. The associated jump discontinuities in the signature,

sig (M0_zM00_)~ 2 if 00 <  < 200_{0 if 200 <  < 400} 2 if 400 <  < 600 8

<

: , (53)

are responsible for the characteristic zero-to-maximal-to-zero sensitivity variations (51) along the ray:

sin1₂uqT_{. (M}0_zM00_{) . q? cos}1 2uqT. (M0zM00) . q ? sin1₂uqT_{. (M}0_zM00_{) . q . (54)} 60˚ 50˚ 40˚ 30˚ 20˚ 10˚ 0˚ 4000 5000 6000 85˚ 90˚ 95˚ 4000 5000 6000 85˚ 90˚ 95˚ 4000 5000 6000 60˚ 50˚ 40˚ 30˚ 20˚ 10˚ 0˚ 4000 5000 6000 85˚ 90˚ 95˚ 4000 5000 6000 85˚ 90˚ 95˚ 4000 5000 6000 –3 +3 traveltime kernel K (10-6 s/km3) 60˚ 40˚ 20˚ 0˚ 4000 5000 6000 60˚ 40˚ 20˚ 0˚ 60˚ 40˚ 20˚ 0˚ 4000 5000 6000

(a) ray-plane cross sections (

θ

=90º)

∆=60º τ=10 s

(b) perpendicular cross sections

at φ=30º at φ=45º

∆=60º τ=20 s at φ=30º at φ=45º

(c) vertical cross sections off ray-plane

at θ=90º+_1º at θ=90º+_2º at θ=90º+_3º

Figure 15. 2-D cross-sections through the 3-D paraxial Fre¨chet kernel K for a teleseismic P wave at an epicentral distance *~600. Compare with the

corresponding views of the `exact' kernel Kcin Fig. 12.

60˚ 50˚ 40˚ 30˚ 20˚ 10˚ 0˚ 4000 5000 6000 0˚ 10˚ 20˚ 30˚ 40˚ 50˚ 60˚ 100˚ 90˚ 80˚ 100˚ 90˚ 80˚ 100˚ 90˚ 80˚ 0˚ 10˚ 20˚ 30˚ 40˚ 50˚ 60˚ 100˚ 90˚ 80˚ –4 +4 traveltime kernel K (10-6 s/km3) 85˚ 90˚ 95˚ 5000 6000 85˚ 90˚ 95˚ 5000 6000 85˚ 90˚ 95˚ 5000 6000 85˚ 90˚ 95˚ 5000 6000 85˚ 90˚ 95˚ 5000 6000 85˚ 90˚ 95˚ 5000 6000 60˚ 40˚ 20˚ 0˚ 4000 5000 6000 60˚ 40˚ 20˚ 0˚ 60˚ 40˚ 20˚ 0˚ 4000 5000 6000

(a) ray-plane cross sections (

(b) perpendicular cross sections

at φ=30º at φ=35º

at φ=40º at φ=45º

at φ=50º at φ=55º

(c) constant-depth cross sections

at depth=20 km

at depth=100 km

at depth=200 km

at depth=400 km

(d) vertical cross sections off ray-plane

at θ=90º+_1º at θ=90º+_2º at θ=90º+_3º

Figure 16. 2-D cross-sections through the 3-D paraxial Fre¨chet kernel K for a PP wave at an epicentral distance *~600. Compare with the

corresponding views of the `exact' kernel Kcin Fig. 13.

The in-plane variation of the paraxial kernel (27)^(28) in the vicinity of the two caustics is extremely singular:

K* {  ? p 2nc …_? 0 u 3_{j _m(u)j}2_sin1 2?uq21du …_? 0 u 2_{j _m(u)j}2_du :200, ;400 {  ? p 2nc …? 0 u 3_{j _m(u)j}2_cos1 2?uq21du …? 0 u 2_{j _m(u)j}2_du ;200, :400 8 > > > > > > > > > > > > < > > > > > > > > > > > > : , (55) where : and ; specify whether the caustics are approached from the source or receiver side, respectively. The leading factor of p? in (55) renders the traveltime shift dTPP very sensitive to heterogeneity dc in the vicinity of the two caustics; however, the in¢nitely oscillatory terms shrink the width of the paraxial kernel to zero at these points, as illustrated in Fig. 16(a).

In general, the paraxial PP kernel K is in very good agree-ment with the `exact' kernel Kc. The principal discrepancies occur in the vicinity of the source-to-receiver and receiver-to-source caustics, and in a thin boundary layer near the upper surface of the earth, particularly in the tails of the saddle. Most of these discrepancies are due to the neglect of unlike-type scattering paths; for example, at scatterers x near the surface re£ection point, both no-bounce P-to-P and two-bounce Pp-to-pP rays contribute to Kcbut not to K. The `swallowtails' extending down into the earth, beneath r~6000 km, in the perpendicular cross-sections at ~300 and ~350 in Fig. 16(b) are another clear di¡erence. The `exact' kernel Kcin Fig. 13(b) is essentially zero at all scatterers x beneath the two caustic surfaces, as we have seen.

5.3 PcP wave

Fig. 17 shows the paraxial kernel K for a q~20 s PcP wave at an epicentral distance *~440; every scatterer is again pro-jected onto two points î, one on each leg of the compound ray. The agreement with the `exact' wave-speed kernel Kcdepicted in Fig. 14 is seen to be excellent. Close inspection reveals a slight discrepancy in the vicinity of the core^mantle boundary re£ection point, where no-bounce P-to-P and two-bounce PcP-to-PcP scattering paths contribute to Kc, but not to K. The bowl-shaped £ange in Figs 14(a) and (c) is also absent in Figs 17(a) and (c), as noted previously.

6 DIFFERENTIAL KERNEL

Di¡erential traveltime shifts d(*T) are widely used in both global and regional tomographic studies because they are thought to be particularly sensitive to wave-speed variations dc in a particular region within the earth. For example, the relative traveltime variations of teleseismic P waves at a number of closely spaced stations are frequently used to invert for the wave-speed heterogeneity dc beneath a seismic network or array; the e¡ect of near-source and deep-mantle anomalies dc is considered to be negligible, because those portions of

the associated ray paths are virtually identical (e.g. Aki et al. 1977; Humphreys et al. 1984; VanDecar & Crosson 1990). Di¡erential PP^P and SS^S traveltime shifts at a single station are likewise often ascribed to heterogeneity dc in the vicinity of the surface re£ection, on the grounds that the near-source and near-receiver ray paths are similar (e.g. Kuo et al. 1987; Sheehan & Solomon 1991; Woodward & Masters 1991). Another popular single-station technique makes use of di¡er-ential traveltime shifts between successive multiple PcP or ScS reverberations; a sequence PcP2^PcP1, PcP3^PcP2, . . . or ScS2^ScS1, ScS3^ScS2, . . . of such measurements can be used to constrain the heterogeneity dc in a 2-D corridor between a source s and receiver r (e.g. Sipkin & Jordan 1980; Katzman et al. 1998). We present a number of 3-D Fre¨chet kernels KB{A_~KB_{KA_{for such di¡erential traveltime measurements} d(TB{TA) in Fig. 18; all of the illustrated di¡erential kernels have been computed using the identical-pulse-shape paraxial approximation (33)^(34).

The left side of Fig. 18(a) shows a ray-plane cross-section of the paraxial sensitivity kernel for a di¡erential traveltime measurement made by cross-correlating two q~20 s P waves, recorded at a pair of closely spaced stations situated on the same azimuth, at epicentral distances *~610 and *~600. The sensitivity is very slight over the ¢rst two-thirds of the ray path, 00 <  < 400, as expected. The Fresnel zones of the individual waves overlap beneath the array; the red and blue banana `skins' delineate the maximum sensitivity of the 610 and 600 arrivals, respectively. These two regions of maximal sensitivity are of opposite sign, since the measured quantity is the shift in the traveltime di¡erence d(T610{T600). In the ray-theoretical limit, the kernel would consist of a single red and a single blue `spaghetti stalk', one along each of the constituent rays.

The right side of Fig. 18(a) shows the ray-plane sensitivity of a q~20 s PP^P di¡erential traveltime measurement, at an epicentral distance *~600. The 3-D Fre¨chet kernel is simply the di¡erence of the PP and P kernels: KPP{P_~KPP_{KP_. Near-source and near-receiver heterogeneity dc whose lateral scale is wider than about 50 will tend to be cancelled by the rapidly alternating zones of red and blue sensitivity; however, heterogeneity that is signi¢cantly narrower than this may not be subject to this cancellation. Kernels analogous to those in Fig. 18(a) were presented by Marquering et al. (1999). Their di¡erential kernels, which are computed by means of surface wave summation, are very similar to those shown here.

Figs 18(b) and (c) depict a sequence of kernels for the di¡erential traveltimes of q~20 s PcP2^PcP1, PcP3^PcP2and PcP4^PcP3waves, at an epicentral distance *~480. The ¢nite-frequency sensitivity of these low-ray-parameter waves cannot be computed by surface wave summation. The maximum ray-plane sensitivity in each panel of Fig. 18(b) is in the vicinity of the source s and the receiver r. As in the case of PP^P, the alternating red and blue zones of sensitivity will tend to cancel the e¡ect of su¤ciently smooth subsource and subreceiver heterogeneity dc. The intensity of the alternating zones decreases from left (PcP2^PcP1) to right (PcP4^PcP3), showing that the cancellation e¡ect is more pronounced for the more steeply incident waves, whose near-source and near-receiver geometrical ray paths are closer together. The alternating zones of negative (orange) and positive (green) sensitivity in the upper mantle are indicative of the degree of lateral coverage and resolution in the source-to-receiver corridor. It is note-worthy that these shallow minima and maxima are not centred

upon the surface re£ection pointsöfor the same reason that a single-phase kernel is a hollow banana. The o¡-path sensitivity at a depth h~400 km extends to +50, as illustrated in Fig. 18(c). As we have noted earlier, this cross-path width scales as the square rootpqof the characteristic period of the cross-correlated waves.

In a recent tomographic study, Katzman et al. (1998) used 2-D multiple-ScSSH Fre¨chet kernels K2D, computed by Zhao & Jordan (1998) by means of whole-earth normal-mode sum-mation, to image the heterogeneity in upper mantle shear wave speed along a corridor between Tonga and Hawaii. Such 2-D traveltime kernels are appropriate only if the heterogeneity

is cylindrically symmetric or quasi-symmetric in the cross-path direction (Marquering et al. 1999). A 3-D rather than 2-D analysis of the Tonga^Hawaii corridor might lead to di¡erent conclusions, inasmuch as the kernels K and K2D are fundamentally di¡erentömost notably, the latter do not exhibit zero sensitivity along the geometrical ray.

7 DEPTH PHASE INTERFERENCE

In all of the examples presented so far in this paper, the various phases examined, P, PP, PcP, PcP2, . . . , have been

44˚ 33˚ 22˚ 11˚ 0˚ 4000 5000 6000 85˚ 90˚ 95˚ 4000 5000 6000 85˚ 90˚ 95˚ 4000 5000 6000 0˚ 11˚ 22˚ 33˚ 44˚ 100˚ 90˚ 80˚ 0˚ 11˚ 22˚ 33˚ 44˚ 0˚ 11˚ 22˚ 33˚ 44˚ 0˚ 11˚ 22˚ 33˚ 44˚ 100˚ 90˚ 80˚ –1.5 +1.5 traveltime kernel K (10-6 s/km3) 44˚ 33˚ 22˚ 11˚ 0˚ 4000 5000 6000 44˚ 33˚ 22˚ 11˚ 0˚ 44˚ 33˚ 22˚ 11˚ 0˚ 4000 5000 6000

(a) ray-plane cross sections (

θ

=90º)

∆=44º τ=20 s

(b) perpendicular cross sections

at φ=22º at φ=33º

(c) constant-depth cross sections

at depth=2850 km at depth=2800 km at depth=2700 km at depth=2500 km

(d) vertical cross sections off ray-plane

at θ=90º+_1º at θ=90º+_2º at θ=90º+_3º

Figure 17. 2-D cross-sections through the 3-D paraxial Fre¨chet kernel K for a PcP wave at an epicentral distance *~440. Compare with the

corresponding views of the `exact' kernel Kcin Fig. 14.

well separated in the time domain. In a more complicated earth model, with a Mohorovic­ic¨ or other internal discontinuities, there may be more than one signi¢cant body wave phase that arrives within the cross-correlation time interval t1¦t¦t2. We

develop a general procedure that can be used to construct 3-D Fre¨chet kernels for such interfering or overlapping phases in Banana^Doughnut I. To illustrate this extended theory, we consider an especially simple case here.

60˚ 50˚ 40˚ 30˚ 20˚ 10˚ 0˚ 4000 5000 6000 60˚ 50˚ 40˚ 30˚ 20˚ 10˚ 4000 5000 6000 6000 –4 +4 differential kernel K (x10-6 _s/km3₎ 48˚ 36˚ 24˚ 12˚ 0˚ 4000 5000 6000 48˚ 36˚ 30˚ 12˚ 0˚ 48˚ 36˚ 24˚ 12˚ 0˚ 4000 5000 6000 0˚ 12˚ 24˚ 36˚ 48˚ 100˚ 90˚ 80˚ 0˚ 12˚ 24˚ 36˚ 48˚ 0˚ 12˚ 24˚ 36˚ 48˚ 100˚ 90˚ 80˚ –1.5 +1.5 differential kernel K (10-6 _s/km3₎

(a) ray-plane cross sections (

θ

=90º)

P(∆=61º)-P(∆=60º) τ=20 s PP-P(∆=60º) τ=20 s

(b) ray-plane cross sections (

θ

=90º)

PcP₂-PcP₁(∆=48º) τ=20 s PcP₃-PcP₂(∆=48º) τ=20 s PcP₄-PcP₃(∆=48º) τ=20 s

(c) constant-depth cross sections

at depth=400 km at depth=400 km at depth=400 km

Figure 18. Di¡erential traveltime kernels for q~20 s waves computed using the paraxial approximation. (a) (Left) Ray-plane cross-section showing the sensitivity of the di¡erential P-wave traveltime between two stations, located on the same azimuth, at epicentral distances *~600 and *~610. (Right) Sensitivity of a PP^P di¡erential traveltime measurement at an epicentral distance *~600. (b) Ray-plane cross-sections showing the

sensitivity of PcP2^PcP1, PcP3^PcP2 and PcP4^PcP3 di¡erential traveltime measurements at an epicentral distance *~480. (c) Upper-mantle

(h~400 km, r~5971 km) constant-depth slices for the same PcP2^PcP1, PcP3^PcP2and PcP4^PcP3di¡erential traveltimes.