wave equation tomography

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GEOPHYSICS, VOL. 57, NO. 1 (JANUARY 1992): P. 15-26, 18 FIGS. Wave-equation tomography Marta Jo Woodward” ABSTRACT The relation between ray-trace and diffraction to- mography is usually obscured by formulation of the two methods in different domains: the former in space, the latter in wavenumber. Here diffraction tomogra- phy is reformulated in the space domain, under the title of wave-equation tomography. With this transfor- mation, wave-equation tomography projects mono- chromatic, scattered wavefields back over source- receiver wavepaths, just as ray-trace tomography projects traveltime delays back over source-receiver raypaths. Derived under the Born approximation, these wavepaths are wave-theoretic backprojection patterns for reflected energy; derived under the Rytov approximation, they are wave-theoretic back-projec- tion patterns for transmitted energy. Differences between ray-trace and wave-equation tomography are examined through comparison of wavepaths and raypaths, followed by their application to a transmission-geometry, synthetic data set. Rytov wave-equation tomography proves superior to ray- trace tomography in dealing with geometrical fre- quency dispersion and finite-aperture data, but inferior in robustness. Where ray-trace tomography assumes linear phase delay and inverts the arrival time of one well-understood event, wave-equation tomography ac- commodatesscattering and inverts all of the signal and noise on an infinite trace simultaneously. Interpreted through the uncertainty relation, these differences lead to a redefinition of Rytov wavepaths as monochro- matic raypaths, and of raypaths as infinite-bandwidth wavepaths (Rytov wavepaths averaged over an infinite bandwidth). The infinite-bandwidth and infinite-time assump- tions of ray-trace and Rytov, wave-equation tomogra- phy are reconciled through the introduction of hand- limited raypaths (Rytov wavepaths averaged over a finite bandwidth). A compromise between rays and waves, bandlimited raypaths are broad backprojection patterns that account for the uncertainty inherent in picking traveltimes from bandlimited data. INTRODUCTION Seismic tomography encompasses a broad group of inver- sion schemes used for imaging seismic velocity fields. De- fined as the reconstruction of a seismic held from integrals over paths through the field, the method is most familiar in its ray-theoretic form as traveltime inversion. In this appli- cation traveltimes are picked for source-geophone experi- ments, then compared to expected traveltimes calculated by tracing rays through an assumed background velocity field. The relation between traveltime and velocity perturbations is linearized with Fermat’s principle, and an updated veloc- ity field is produced by projection of measured traveltime perturbations back through the medium over raypaths. For evenly spacedsource-geophone pairs, the backprojt ( - tion step may be performed analytically: in either the space domain as a generalized inverse Radon transform (Beylklr , 1982; Fawcett and Clayton, 1984) or in the wavenumbrr domain using the projection-slice theorem (Mersereau a 13 Oppenheim, 1974). However, given the irregular spati:tl coverage of the typical seismic experiment, most traveltime inversion is performed numerically in the space domalr , using iterative least squares (Dines and Lytle, 1979; Bishl)o et al.. 1985). Ray-theoretic tomography in this form is kno\si? as ray-trace tomography. Ray-trace tomography works well when two requiremer 1s are met (Wu and Toksoz, 1987). First, because the method relies on the high-frequency assumption of ray theory, t 1’ velocity field being examined must vary slowly on the SC: I? Presented at the 58th Annual International Meeting, Society of Exploration Geophysicists. Manuscript received by the Editor February :‘I, 1990;revised manuscriptreceived April 25, 1991. *Formerly Stanford Exploration Project, Department of Geophysics,Stanford University; presently GECO-PRAKLA NSA, 1325 South Da r7 Ashford, Houston, TX 77077. 0 1992Society of Exploration Geophysicists. All rights reserved. 15

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Page 1: Wave Equation Tomography

GEOPHYSICS, VOL. 57, NO. 1 (JANUARY 1992): P. 15-26, 18 FIGS.

Wave-equation tomography

Marta Jo Woodward”

ABSTRACT

The relation between ray-trace and diffraction to- mography is usually obscured by formulation of the two methods in different domains: the former in space, the latter in wavenumber. Here diffraction tomogra- phy is reformulated in the space domain, under the title of wave-equation tomography. With this transfor- mation, wave-equation tomography projects mono- chromatic, scattered wavefields back over source- receiver wavepaths, just as ray-trace tomography projects traveltime delays back over source-receiver raypaths. Derived under the Born approximation, these wavepaths are wave-theoretic backprojection patterns for reflected energy; derived under the Rytov approximation, they are wave-theoretic back-projec- tion patterns for transmitted energy.

Differences between ray-trace and wave-equation tomography are examined through comparison of wavepaths and raypaths, followed by their application to a transmission-geometry, synthetic data set. Rytov

wave-equation tomography proves superior to ray- trace tomography in dealing with geometrical fre- quency dispersion and finite-aperture data, but inferior in robustness. Where ray-trace tomography assumes linear phase delay and inverts the arrival time of one well-understood event, wave-equation tomography ac- commodates scattering and inverts all of the signal and noise on an infinite trace simultaneously. Interpreted through the uncertainty relation, these differences lead to a redefinition of Rytov wavepaths as monochro- matic raypaths, and of raypaths as infinite-bandwidth wavepaths (Rytov wavepaths averaged over an infinite bandwidth).

The infinite-bandwidth and infinite-time assump- tions of ray-trace and Rytov, wave-equation tomogra- phy are reconciled through the introduction of hand- limited raypaths (Rytov wavepaths averaged over a finite bandwidth). A compromise between rays and waves, bandlimited raypaths are broad backprojection patterns that account for the uncertainty inherent in picking traveltimes from bandlimited data.

INTRODUCTION

Seismic tomography encompasses a broad group of inver- sion schemes used for imaging seismic velocity fields. De- fined as the reconstruction of a seismic held from integrals over paths through the field, the method is most familiar in its ray-theoretic form as traveltime inversion. In this appli- cation traveltimes are picked for source-geophone experi- ments, then compared to expected traveltimes calculated by tracing rays through an assumed background velocity field. The relation between traveltime and velocity perturbations is linearized with Fermat’s principle, and an updated veloc- ity field is produced by projection of measured traveltime perturbations back through the medium over raypaths.

For evenly spaced source-geophone pairs, the backprojt ( - tion step may be performed analytically: in either the space domain as a generalized inverse Radon transform (Beylklr , 1982; Fawcett and Clayton, 1984) or in the wavenumbrr domain using the projection-slice theorem (Mersereau a 13 Oppenheim, 1974). However, given the irregular spati:tl coverage of the typical seismic experiment, most traveltime inversion is performed numerically in the space domalr , using iterative least squares (Dines and Lytle, 1979; Bishl)o et al.. 1985). Ray-theoretic tomography in this form is kno\si? as ray-trace tomography.

Ray-trace tomography works well when two requiremer 1s are met (Wu and Toksoz, 1987). First, because the method relies on the high-frequency assumption of ray theory, t 1’ velocity field being examined must vary slowly on the SC: I?

Presented at the 58th Annual International Meeting, Society of Exploration Geophysicists. Manuscript received by the Editor February :‘I, 1990; revised manuscript received April 25, 1991. *Formerly Stanford Exploration Project, Department of Geophysics, Stanford University; presently GECO-PRAKLA NSA, 1325 South Da r7 Ashford, Houston, TX 77077. 0 1992 Society of Exploration Geophysicists. All rights reserved.

15

Page 2: Wave Equation Tomography

16 Woodward

of the source wavelengths. In this case there is no scattering: phase delay is linear with frequency; the source wavelet is not distorted, and seismic events are completely character- ized by traveltimes. Second, because without scattering rays sample very narrow regions in space, the source-geophone geometry must provide many view angles through the me- dium. When these requirements are not met. wave-theoretic tomography provides a better image.

Wave-theoretic tomography accommodates scattering by replacing traveltime delays with scattered wavefields. Wave- fields are recorded for source-geophone experiments, then compared to expected wavefields calculated by forward modeling through an assumed background velocity field. The wave equation is linearized with either the Born or Rytov approximation, and an updated velocity field is produced by propagation of measured wavefield perturbations back through the medium over wave-propagation paths.

The backpropagation step of wave-theoretic tomography is usually formulated in the frequency-wavenumber domain, under the title of diffraction tomography (Mueller et al., 1979; Mueller, 1980; Devaney, 1982, 1984; Slaney et al., 1984; Wu and Toksoz, 1987). The method solves the problem analytically, for plane-wave scattering and independent monochromatic sources. The problem has also been solved in the time-space domain: as migration by inversion of a generalized Radon transform (Miller et al., 1987). and as nonlinear inversion (Tarantola. 1984, 1987; Mora. 1987). Here wave-theoretic tomography is reformulated for solu- tion by iterative least squares in the frequency-space do- main, under the title of wave-equation tomography (Wood- ward and Rocca, 1988; Woodward, 1989). This reformulation makes wave-equation tomography more flex- ible than diffraction tomography in dealing with irregularly sampled surveys and inhomogeneous background media. More importantly, it encourages physical understanding of the differences between ray and wave inversions through visual comparisons of their space-domain backprojection patterns. Where ray-trace tomography projects traveltime delays back over source-receiver “raypaths,” wave-equa- tion tomography projects monochromatic, scattered wave- fields back over source-receiver “wavepaths.”

The first two sections of this paper develop the equations defining raypaths and wavepaths in a parallel fashion. The next two sections examine the resulting backprojection patterns in the space and wavenumber domains. Wavepaths derived under the Rytov approximation are linked to ray- paths as wave-theoretic trajectories for transmitted energy; those derived under the Born approximation are linked to migration ellipses as wave-theoretic trajectories for reflected energy. The fifth section follows raypaths and Rytov wave- paths through the inversion of a transmission-geometry, synthetic data set. The final section summarizes the differ- ences between ray-trace and Rytov wave-equation tomogra- phy, concluding that raypaths and wavepaths lie at two extremes of the uncertainty relation: the former assuming infinite bandwidth, the latter infinite time Rays and waves are reconciled through the uncertainty relation with the definition of “bandlimited raypaths” as frequency-averaged wavepaths. Bandlimited raypaths are broad backprojection patterns that account for the uncertainty inherent in picking traveltimes from bandlimited data.

RAY-THEORETIC EQUATIONS

Because ray-theoretic tomography relies on the hlk h- frequency approximation of ray theory, its traveltime pi,:ks must represent ray arrivals. Where traveltimes are pic.::d from bandlimited wavelets, this requires either that tie wavelet peaks are undistorted (that phase delay is linear v th frequency) or that Fermat-path first breaks (ray arrivals) .: In be determined. In practical bandlimited applications these requirements are often unmet. Where the velocity f tld varies rapidly on the scale of the source wavelength, wa\ e- lets are distorted by geometrical frequency dispersii n; where events overlap and signal level is low, first breaks r re difficult to pick.

Given these limitations, the source-geophone travelt rle integrals of ray-theoretic tomography are:

r(gls1 = I n,(r)l[rls. g, w(r)1 dr. (1)

Here the source-geophone pair is indicated by s, g; r is t?e space-coordinate vector; u’ is inverse velocity or slownc~~s; and f. is the raypath from s to g through ~$3 (1 along tie raypath. 0 elsewhere). The expected source-geophone tr iv- eltime integrals, calculated by ray-tracing though the first- guess background velocity field, are:

Here MJ~ is the background slowness field and Lo the rayf: th through that field. Because L is a function of ~3, the acou,; ic tomography problem is nonlinear. To create a linear relat i n-t between Ar and AM,. ray theory invokes Fermat’s print [lle and approximates L by L,, producing

At(&) = Aw(r)Lo(rls, g) dr. (3)

The backprojection step of ray-theoretic tomography n- volves the solution of the system of linear equations reslllt- ing from consideration of a number of source-geoph5)le pairs: I,“Aw = At. The nonlinear part of the problenl is attacked iteratively, for successively updated backgrol I Id velocity fields.

WAVE-THEORETIC EQUATIONS

Because wave-theoretic tomography is full waveform n- version, it makes no assumptions about the characterizat i >n of an event by a single traveltime pick. However, it di:es assume that the source wavelet is sufficiently well known tor calculation of a background wavefield, and that the recorr I :d seismic traces are complete and free of noise. Given these assumptions, the wave-theoretic equivalent of equation (3) may be generated by linearizing the scalar wave-equal I )n with either the first-order Born or the first-order Ry. IV approximation.

Born

Where ray-trace tomography creates a linear relatur between velocity and traveltime perturbations, the B )rn

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Wave-equation Tomography 17

approximation creates a linear relation between velocity and wavefield-amplitude perturbations: AT(w) = q’(w) - q,,(w). These are complex amplitudes (V = Ae’*), measured in the temporal-frequency domain for source-geophone pairs. The Born approximation begins with the wave equation written as:

A’Ws) = O(rFo[glr, vo(r)l

x {To[r/s, vo(r)l + AWrls, O(r)l} dr (4)

(Slaney et al., 1984). 0 is the object function, or the perturbed velocity field expressed as:

O(r) = ki(r)[ 1 - vjj(r)/v’(r)]

= 2ki(r)Av(r)/u(r). (5)

Go is the Green’s function or impulse response for the background medium: for constant velocity and two or three dimensions,

e’w Go(r) =q or $ Hi,“(k0lrl), (6)

where k. is the background wavenumber W/ZI~, and Hh” is a zero-order Hankel function of the first kind.

Equation (4) is usually interpreted by Huygens’s construc- tion. The anomalous wavefield at a specific geophone is generated by superposition: each point in the medium acts as a scatterer, emitting an impulse response scaled by the product of the wavefield and the object function at that point. By using reciprocity to replace G,(rlg) with G,(gjr), then regrouping terms and introducing the concept of a “wavepath” Y,

A*(&) = Au(r) 7’(r) %r/s, g, u(r)1 dr

Lf(r/s, g) = 2k,:(r)Go(rlg){~lo(rls) + AWrls, O(r)]}, (7)

this interpretation can be altered to resemble that of equation (2). Now the scattered complex amplitudes of wave-theo- retic tomography are revealed as integrals through a per- turbed velocity field over monochromatic wavepaths %-just as the traveltime delays of ray-theoretic tomography are integrals through the perturbed velocity field over ray- paths L.

As in the ray-trace application, Y (specifically A\u) is a function of AZ), and the problem is nonlinear. Under the Born approximation the equation is linearized by assuming the wavepath L!Z to be independent of the velocity perturba- tion (Aul +Z Ur,), yielding the monochromatic analog of equation (3):

A’Ws) = Au(r) - Y0(r/s, g) dr z!(r)

For a point source at s, TCj is the Green’s function for the background medium, and the “Born wavepath” is

%)(rls, g) = 2koZ(r)Go(rJg)Go(rls). (9)

Rytov

The first-order Rytov approximation creates a linear rel;l- tion between velocity and wavefield-phase perturbation ;: A@.(o) = In [q(w)] - In [Ur,(w)]. As with the Born appro,;i- mation, these complex phases are measured in the tempor tI- frequency domain for source-geophone pairs. The Ryt IV approximation begins with the wave equation written as:

x {[V(A@(rls, O(r))]’ + O(r)} dr (1:)

(Slaney et al., 1984). This equation is more difficult to interpret physically than the Born equivalent. Howevc,r, under the Rytov approximation [V(A@)I’ =C 0, and t 1: equation becomes

Awls) = s O(r) Go(glr)‘J’o(r/s) dr,

To (ids) (1 I)

This is just the Born equation with Aq replaced by ‘PoAl C . [The Rytov formula reduces to the Born formula in t I? weak-scattering limit, where *,A@ = A\V (Devaney, 1981 I Following the Born development, equation (I 1) can ): rewritten as:

@(g/s) = /

AT!(r) - 30(r/s, g) dr 4r)

(I !)

%(rls, g) = 2k,?(r) Go(r~g)*o(rls)

u'o(sls) ’ and the scattered complex phases again viewed as integn Ii through the perturbed velocity field over wavepaths Zo. F )r

a point source at s, To is the Green’s function for tl,‘: background medium, and the “Rytov wavepath” is

2o(r/s, g) = Xi(r) Go (rIdGo (rls)

Go(&) ’ (I ‘1

Where ray-theoretic tomography forms a system of equ I- tions through consideration of a number of sources at (I geophones, wave-theoretic tomography forms a syste II of equations-LJ,,A,,, = AVr or LfoA,,iV = AQ, throul11 consideration of a number of sources, geophones, and fr :. quencies. The extra dimension of information in wav :. theoretic as compared to ray-theoretic tomography w II become apparent later. As with ray-theoretic tomograph .I the nonlinear part of the problem is attacked iteratively.

WAVEPATHS VERSUS RAYPATHS: SPACE DOMAIN

Figures la and lb show Rytov and Born examples I 1’ two-dimensional wavepaths for a homogeneous backgrour t velocity field. Their familiar raypath analog is shown r Figure 2. Unlike the raypath, the wavepaths are monochrc 1 matic, complex, and infinite in spatial extent. While tt t complex absolute values of the wavepaths decay with di tance from the source and receiver, the phase of the patterr ! oscillates from 7r to -7~. Because the Born and Rytcr approximations are based on scattering theory, contours c 1’

Page 4: Wave Equation Tomography

18 Woodward

constant phase on the wavepaths yield confocal ellipses: curves of constant source-scatterer-receiver traveltimes, with the source and receiver located at the foci. The equa- tion describing these ellipses is:

+-=I; a2-c2 a2

a > c, (14)

where a is the semimajor axis (half the scattering path from source to receiver) and c is half the source-receiver offset. Beyond these observations, the fine structure of the wave- paths depends on whether they are Born- or Rytov-gener- ated patterns.

Rytov wavepaths

Phase and amplitude separate naturally in the Rytov wavepath. When multiplied by AVIV, the imaginary part of the wavepath yields the time-delay-like phase delay A+(w), the real part the log of the amplitude ratio In [A(o)lA This natural separation underscores the parallelism between Rytov and transmission ray-trace tomography. Reinter- preted as IV In (AlA,)lk,, =C I and IV(A$)lk,I 5 I, the Rytov approximation is compatible with transmission ray theory in requiring both the amount of scattering per wave- length and the scattering angle to be small (Chemov, 1960).

In full waveform inversion, these restrictions are best met by forward-scattered energy in transmission-geometry imple- mentations: applications where traveltime delays accumu- late through a velocity perturbation. In fact, tomography under the Rytov and eikonal approximations are equivalent in the very-short-wavelength limit (Devaney, 1981).

For the two-dimensional Rytov wavepaths, the imaginary parts of the patterns pass through zeros at the boundaries between the first, second, third, etc., Fresnel zones. Since a scatterer within the first Fresnel zone generates a wavefield reaching the geophone within a half wavelength of the source wavefield, a low-velocity scatterer in this zone produces a phase delay, a high-velocity scatterer a phase advance. A scatterer in the second Fresnel zone generates a wavefield reaching the geophone between a half and a full wavelength behind the source wavefield: a low velocity scatterer pro- duces a phase advance, a high velocity scatterer a phase delay. Similar arguments about amplitudes explain the oscil- lations of the real parts of the Rytov wavepaths. For the imaginary, three-dimensional wavepath slice in Figure 3, there is an additional, paradoxical zero on the source- receiver Fermat path. Since a point on this curve scatters energy in-phase with the background wavefield, it yields an amplitude perturbation but no phase perturbation at the receiver.

a) Rytov 2-D 5 & 10 Hz navepaths

b) Born 2-D 5 & 10 Hz wavepaths

FIG. I. Frequency-space domain wavepaths for a constant-velocity background field. (a) Rytov 5 and IO Hz wavepaths. (b) Born 5 and IO Hz wavepaths. The velocity field is 2000 m/s; the source and receiver are separated by 2000 m. White is positive; black is negative; grey is zero.

Page 5: Wave Equation Tomography

Wave-equation Tomography 19

Born wavepaths

Phase and amplitude fail to separate in the Born wavepath: when multiplied by the object function, the real part yields a[V(o) - qO(o)], the imaginary part 9[q(w) - qO(o)]. Without the Rytov wavepath’s normalization factor of Go(gls), Born-wavepath zero crossings shift in a complex way with source-receiver separation and frequency. Born wavepaths are more naturally examined in the time-space domain of prestack migration. Figure 4 shows the constant- velocity wavepaths of Figure lb, Fourier transformed into this domain and sliced at two times. The patterns are recognizable as the ellipses over which time arrivals are smeared in prestack migration (Schneider, 1971). While Born and Rytov wavepaths look alike in this domain (except for a time shift), the Born data parameters are much more suited to time than the Rytov parameters. Since the Fourier transform of A*(o) is the difference between the measured and expected seismic traces, Born tomography implemented in time-space corresponds to the projection of unexpected time arrivals back over isochronal scattering ellipses (Miller et al., 1987). Rytov tomography implemented in time-space corresponds to the projection of the Fourier transform of complex-phase perturbations back over these same patterns, a much less intuitively meaningful operation. The suitability of the Born approximation for backscattered energy and reflection geometries arises from its weak-scattering as-

Ltl

FIG. 2. Raypath analog of the wavepaths in Figure 1.

sumption: A* < Vr,, (or equivalently, IAAIAo[ * 1 and iA+\ < 1). By requiring both the total amount of scattering and the total change in phase to be small, the method trades the ability to handle large cumulative phase delays for the ability to handle large scattering angles (Chernov, 1960).

Inhomogeneous background velocity field

Figure 5 shows examples of the imaginary parts of Rytov wavepaths in inhomogeneous media. The two-dimensional, 15 Hz wavepaths encompass their expected semicircular, refracted, and direct-plus-reflected-plus-refracted raypath equivalents. Because the direct-plus-reflected-plus-refracted wavepath contains three overlapping events, its zeros no longer fall on Fresnel-zone boundaries. The interior regions of the overlapping ellipses change sign from frequency to frequency, as determined by interference between the three events. This geometry does not fit the transmission geometry applications for which Rytov tomography is best suited. Where multiple waves are considered simultaneously, the linear relation between cumulative phase delay and velocity perturbation breaks down (Keller, 1969). For this applica- tion, the different wavefields would have to be separated before Rytov inversion.

WAVEPATHS VERSUS RAYPATHS: WAVENUMBER DOMAIN

Propagating energy

Figure 6 shows the two-dimensional wavenumber-ampli- tude spectra of the 10 Hz wavepath and raypath of Figures 1 and 2. The gross structure of the spectra is easily related to the space-domain patterns: both ray-theoretic and wave- theoretic tomography are most sensitive to spatial frequen- cies representing velocity variations paralleling the source- geophone axis and least sensitive to those perpendicular to the axis. The fine structure of the wavepath spectra requires more scrutiny.

Devaney (1984) and Wu and Toksijz (1987) show how to construct spectra similar to those in Figure 6 from plane- wave scattering arguments. Adapted from their work, Figure 7 illustrates two plane waves: emanating from source s, scattering at point a or b, and arriving at geophone g. For source and geophone plane waves described by k, and k,,

FIG. 3. Slice through a 3-D, 5 Hz Rytov wavepath in the constant-velocity background field of Figure I.

FIG. 4. Time-space domain wavepaths (S{L!$}) for the con- stant-velocity background field of Figure 1.

Page 6: Wave Equation Tomography

20 Woodward

the scatterer’s wavenumber is k, - k, . In plane rectangular where oS and og describe the angles k, and k, make with the coordinates the wavenumber becomes: x-axis, respectively. For a single source-geophone experi-

k,=2zsin (y) sin(T) (15)

ment, the possible combinations of plane-wave angles are constrained by the appropriate scattering ellipses. Parame- terizing CY~ in equation (15) in terms of equation (14)‘s u and c produces Figure 8. The left and right panels relate the scattering ellipses of Figure 1 to the wavepath spectrum of Figure 6. The wavepath region on the source-receiver axis corresponds to a flat ellipse of eccentricity l(a = 1) and is sensitive to scatterers with wavenumbers outlining the cir- cular holes. The wavepath region surrounding the source- receiver axis corresponds to ellipses of decreasing eccentric- ity (increasing a) and is sensitive to higher wavenumbers as labeled. The wavepath has a maximum spatial frequency twice that of the source wavefield (2ko), approached as a limit by the spacing of the most distant confocal ellipses.

Evanescent energy

The characteristic holes in the spatial-amplitude spectra of the monochromatic wavepaths arise from a causality (source/sink) condition placed on the source and geophone Green’s functions in equation (6). Replacing the source Green’s function with its complex conjugate yields the wavepaths and spatial-amplitude spectra shown in Figure 9. The familiar ellipses (the loci of points the sum of whose distances from source and geophone is constant) have been

FIG. 5. Imaginary parts of Rytov, frequency-space domain wavepaths for three inhomogeneous background-velocity fields, along with their raypath analogs. Top panels: refrac- tion through a vertical gradient. Middle panels: transmission through three layers. Bottom panels: reflection above an interface (with direct, reflected, and refracted arrivals).

FIG. 7. Plane-wave scattering: propagating energy.

FIG. 6. Two-dimensional wavenumber-amplitude spectra, origins in the centers. Left panel: 10 Hz wavepath. Right

FIG. 8. Diagram relating wavepath scattering ellipses to

panel: raypath. (k,, = 2ke, k, = 2n * IO/Us.) wavepath spatial-amplitude spectra through the semimajor axis parameter a; a > 1. (The foci half-offset c equals 1.)

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Wave-equation Tomography 21

replaced by hyperbolas (the loci of points the difference of whose distances from source and geophone is constant):

x2 z2 -++-_= 1; a2-c2 a2

a < c. (16)

This transformation physically results from the replacement of an exploding, causal source by an imploding, anticausal source. Just as energy from an exploding source that is scattered by points along a single ellipse reaches a geophone in-phase, energy from an imploding source scattered by points along a single hyperbola reaches a geophone in-phase. Alternatively, Figure 6 represents propagating energy and Figure 9 evanescent energy. Anticausal scattering and the relation between space-domain hyperbolas and wavenum- ber-domain holes are diagramed in Figures 10 and 11.

INVERSION

This section follows the differences between raypaths and Rytov wavepaths through inversion of a synthetic data set. The example is limited to wave-equation tomography under the Rytov approximation, as Rytov’s phase delays are both more similar than Born’s scattered amplitudes to the travel- time delays of ray theory and more appropriate for the transmission geometry of the test experiment. (See Orista- glio, 1985; Beydoun and Tarantola, 1988; and Lo et al., 1988, for Born-Rytov comparisons.) The upper panel in Figure 12

FIG. 9. Evanescent, two-dimensional, frequency-space do- main wavepath for a constant-velocity background field, along with its spatial-amplitude spectrum. The wavepath is formed by replacement of the source Green’s function in equation (13) by iH,$2)(kolr1)/4. (Hh2’ is a zero-order Hankel function of the second kind.) The velocity and axes are the same as those of Figures 1 and 6.

shows the velocity field used for the two-dimensional exper- iment: an anomalous circular region 500 m in diameter and 5-percent slower than a 2000 m/s background field. A single shot was positioned on the surface directly above the anom- aly; multiple geophones were positioned every 40 m at a depth of 2000 m, up to a maximum offset of 1480 m.

Rytov data

As shown elsewhere (Woodward, 1989), the Rytov ap- proximation is accurate for this example, thereby separating linearization problems from experimental-geometry effects. The data for the Rytov inversion was calculated from the shot profiles shown in the lower panels of Figure 12; the resulting phase delays and log amplitude ratios for offsets of 0,360, and 1060 m appear in Figure 13. The phase delays are plotted as time delays, having been normalized by fre- quency. Because the phase delays are small, phase unwrap- ping was not the problem that it sometimes becomes in more complicated applications (Tribolet, 1977; Kaveh et al., 1984; Soumekh, 1988).

Figure 14 illustrates the physical meaning of imaginary Rytov wavepaths as forward-modeling tools. The leftmost panels in Figure 14 superimpose 5, 10, 15,20, 25, and 30 Hz first-Fresnel zones over the circular anomaly, for the three different offsets of Figure 13. Given that Fresnel-zone boundaries are equivalent to imaginary Rytov-wavepath zero crossings, the absolute maxima in Figure 13 can be

FIG. 10. Plane-wave scattering: evanescent energy.

FIG. 11. Diagram relating evanescent wavepath scattering ellipses to evanescent wavepath spatial-amplitude spectra through the semimajor axis parameter a; a < 1. (The foci half-offset c equals 1.) /-

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22 Woodward

predicted by inspection of these diagrams: for each offset, they occur at that frequency for which the first Fresnel zone just encompasses the anomaly. When the anomaly protrudes into the second Fresnel zone, it underlies a negative portion of the wavepath and contributes to a phase perturbation of opposite sign. In a similar manner, several relative maxima and minima in Figure 13 can be predicted by inspection of the rightmost panels in Figure 14. Here the boundaries of the first five Fresnel zones for several frequencies are shown superimposed on the anomaly, the sign of each wavepath alternating from positive to negative from the inside out. When the edge of the anomaly just grazes the inside bound- ary of a negative oscillation, it produces a relative minimum; when the edge grazes the inside boundary of a positive oscillation, it produces a relative maximum. The plots illus-

-1000 = b-4 1 Ix-m

-0 El0

-0 N-

Velocity field

-1000 1000

h

;y .

b

N-

Background wavefield

-1000 1000

rl,

cu-

Full wavefield

FIG. 12. Two-dimensional transmission seismic experiment: velocity model with 5-percent slow circular velocity anom- aly; shot profile through the background-velocity model; shot profile through the full model.

trate relative minima and maxima at 1 I and 17 Hz for the 360 m offset trace, and at 8 Hz for the 1060 m offset trace.

Ray data

Since weak scattering is apparent in the finite-difference data set of Figure 12, the linear phase-delay assumption of ray theory is slightly violated by this experiment. To prevent this linearization problem from interfering with the experi- mental geometry effects, ray traveltimes were not picked from the data set. Instead, traveltimes were calculated by tracing straight rays through the model. (The issue of event picking in the presence of geometrical frequency disper- sion-nonlinear phase delay-is discussed below, and in Wielandt, 1987.) Figure 15 shows these traveltime delays, along with a raypath diagram for a geophone at 1000 m offset. The time delays agree with the high-frequency delays in the Rytov data.

Results

Figure 16 shows the results for: ray; monochromatic 5 Hz; monochromatic I5 Hz; monochromatic 25 Hz, and multifre- quency 5, 10, 15,20, and 25 Hz inversions. All the examples were computed using LSQR, the conjugate gradient, linear system solver of Paige and Saunders (1982). The elongation of the images in the z-direction arises from the familiar finite-aperture problem of tomography (Menke, 1984): with- out horizontally placed shots and geophones, the spectral patterns of Figure 6 never sweep across the k, axis. The deterioration in the monochromatic images from the low- frequency 5 Hz to the high-frequency 25 Hz-and on through the asymptotically high-frequency ray image- arises from the decreasing match between the central reso- lution vectors of the respective inversions and the anomaly’s size and shape. (See Woodward, 1989, for a singular-value decomposition analysis of the example.) More simply ex-

h

Z-4 O-

36 \

\

\ 2 . .

-A---_--___

‘zio- :. ..-. _.--- ______________.._

, I I I c

5 10 15 20 25 30 35 Frequency (Hz)

FIG. 13. Rytov data: complex-phase delay versus frequency, imaginary part normalized by frequency. The solid, dashed and dotted lines indicate offsets of 0, 360, and 1060 m, respectively.

Page 9: Wave Equation Tomography

Wave-equation Tomography 2. ‘B

plained in terms of wavepath and raypath pictures, it is easier to construct the localized anomaly from a linear combination of broad ellipsoids than from a linear combina- tion of narrow lines-given the narrow range of viewing angles available. All the monochromatic and ray inversions suffer from strong amplitudes at the source, the most highly illuminated area in the model. By using scattering informa- tion (the nonlinearity in the phase delays), the multifre- quency inversion offers the sharpest image and finally moves energy away from the source, truly localizing the anomaly in the center of the model.

BANDLIMITED RAYPATHS

The preceding sections have shown that multifrequency Rytov wave-equation tomography is superior to ray-trace tomography in dealing with geometrical frequency disper- sion and inverting finite-aperture data. The method achieves this superiority because it is monochromatic, representing one extreme of the uncertainty relation: Athw 2 112. Since Ao is infinitely small, A.t is infinitely long and the entire seismic coda is utilized along with the first arrivals. The method accounts for scattered energy arriving at any timefrom any distance and its wavepath backprojection patterns cover the entire x, z plane. Unfortunately, wave-equation tomography’s superiority is gained at the expense of robust- ness. Because it is monochromatic, modeled events cannot simply be windowed from unmodeled events in time all of the signal and noise on a trace must be dealt with simulta- neously. For any seismic experiment, the validity of the

-0 fl0

-0 l-4”

-0 00 -0 Nd

-0 a0 -0 Nd

-1000 Offset (m)

1000 -1000 Offeet (m)

1000

FIG. 14. Imaginary Rytov wavepaths superimposed on a contoured outline of the circular velocity anomaly. Left panels: first-Fresnel zones for 5, 10, 15, 20, and 25 Hz at offsets 0,360, and 1060 m. The ~-HZ first-Fresnel zone is the widest; the 25Hz zone the narrowest. Right panels: the boundaries of the first five Fresnel zones for several frequen- cies and offsets.

monochromatic assumption depends on the knowledge 11‘ the source, the quality of the data, and whether modeht events can be separated from unmodeled events befolf: inversion.

Ray-trace tomography is more robust than wave-equatic IL tomography because it selects as signal the time delay of om: well-understood event. While in theory the fundament 11 assumption of ray-trace tomography is that of high fr.:. quency, in bandlimited practice it is more often that of ro geometrical frequency dispersion: that phase delay is a I’ proximately linear with frequency and described by : II average time delay picked from a wavelet peak. Instead 4 I’ inverting amplitude changes and phase delays for a singIt: high frequency, ray-trace tomography actually inverts dieslope of a phase-delay versus frequency curve that is a.;. sumed linear over an infinite bandwidth. Behavior over it broad range of frequencies is characterized by a singIf: measurement; scattering is forbidden and energy travel,, along the fastest source-receiver path. Ray-trace tomogr I. phy represents the other extreme of the uncertainty relatio I. where At is infinitely short and Aw infinitely large. Raypatl, are compact because ray-theoretic tomography is essential !’ infinite bandwidth. [The broadband nature of ray theory hi:,, also been pointed out by Foreman (1989).]

In real seismic applications phase delay is rarely linear ar (I bandwidth is never infinite. The uncertainty inherent II picking a peak from a distorted, bandlimited wavelet I; usually incorporated in ray-trace tomography only inc i. rectly. Algorithms acknowledge that raypaths are far highs ‘. in wavenumber than the model being inverted by smoothirlj: the inversion in several ways. The final cell representaticsih can be bandpassed; the model can be parameterized in tern,‘; of smooth basis functions instead of cells, reducing ttic, high-wavenumber indeterminacy and expense of the pro ). lem at the outset (Dziewonski et al., 1977; Van Trier, 1988 ); the indeterminacy can be removed by imposing smoothne ‘I; constraints on the model during the inversion with damps (1 least squares (Menke, 1984a; Sword, 1988); the result can lil, smoothed by broadening the backprojection raypaths ther I.

A 1000 m

-0

a0 -0 *- r.zl 0

-1000 1000 Offset (m)

FIG. 15. Ray-data traveltime delays for all offsets, and dieraypath for a geophone at 1000 m offset.

Page 10: Wave Equation Tomography

24 Woodward

selves with convolutional quelling (a specialized form of

weighted, damped least squares; Meyerholtz et al., 1989).

Transmission-geometry Rytov wavepaths provide a way

of incorporating bandwidth and uncertainty information into

ray-trace tomography in a more direct and physically rea-

sonable fashion. Ray tomography can be redefined as Rytov

tomography under a nondispersive constraint: with g(A@)

and %(A@) assumed to be linear in frequency and zero,

respectively, over a specified bandwidth. Under this defini-

tion the traveltime delay picked for an event becomes a

normalized average of imaginary phase perturbations over

frequency:

At =& I

~mrr .a[A@(w)] d w

. (17)

0 Wmln

The appropriate backprojection pattern becomes a similar

normalized average of imaginary wavepaths over frequency:

I

I

wrndr LA,(r) = -

J+[Y”(w, r)] dw.

A0 (18)

” Wrnl”

Figure I7 shows examples of the “bandlimited raypaths”

defined by equation (l8), formed by numerically integrating

the imaginary, monochromatic Rytov wavepaths of Figures

1 and 5 from 5 to 25 Hz, 15 to 2.5 Hz, and 20 to 25 Hz. The

integrations cancel the rapidly oscillating outer regions of the

wavepaths while adding the smooth, first-Fresnel zones.

(The bandlimited equivalent of the reflection-geometry

wavepath in Figure 5 is omitted: since a linear relation

between phase delay and frequency does not hold for this

case, a bandlimited transmission raypath cannot be defined.)

FIG. 16. Inversion results.

Three observations can be made about these bandlimited

raypaths.

First, they graphically illustrate how the extra information

in full waveform inversion is lost when an event is specified

by a single time pick. Windowing an event in the time

domain smooths the event in the frequency domain: only

when the medium is nondispersive is the discarded high-

wavenumber information redundant.

Second, because they take geometrical spreading into

account, the bandlimited raypaths are high in amplitude at

the source and receiver and low in amplitude elsewhere.

This weighting is an expression of wavefront healing (Claer-

bout, 1985). In contrast to traditional ray-trace tomography,

a velocity perturbation close to the source or receiver will

have a much larger impact on the recorded signal than a

more distant perturbation of similar magnitude.

Third, and most importantly, the breadth of the bandlim-

ited raypaths depends inversely on the width of the fre-

quency band summed over and not on the central frequency

of the band. Imaginary Rytov wavepaths can be defined as

“monochromatic raypaths”; bandlimited raypaths can be

imagined as collapsing to traditionally narrow raypath pat-

terns for infinite bandwidth. The inverse relation between

bandlimited-raypath width and temporal-frequency band-

FIG. 17. Two-dimensional bandlimited raypaths for the con- stant, vertical-gradient and layered background-velocity fields of Figures I and 5.

Page 11: Wave Equation Tomography

Wave-equation Tomography 2!$

width is shown in more detail in Figure 18, with plots of half cross-sections through 3-D bandlimited raypaths for dif- ferent combinations of o,,,,” and wmax. While the relation is demonstrated more mathematically in Appendix A, it can be derived most intuitively from the uncertainty relation. In the uncertainty relation Ar corresponds to the time window of the trace examined (i.e., for traveltime picks. the sample rate). Since it dictates the averaging of frequency informa- tion in the Fourier domain, it is inversely proportional to the bandwidth of the applicable bandlimited raypath. Because it also limits the distance detectable scatterers can stray from the Fermat path, it is directly proportional to the bandlimited raypath’s width.

CONCLUSIONS

This paper clarifies the differences between ray and wave tomography through pictorial comparisons of their respec- tive backprojection patterns. These comparisons underscore the parallelism between ray-trace and Rytov, wave-equation tomography, and suggest ways of modifying rays to reflect the bandlimited nature of most seismic data. In contemplat- ing any ray inversion, thought must be given to what picked traveltimes represent: asymptotically high-frequency ray arrivals, or group velocities characteristic of the source bandwidth. Very preliminary results suggest traveltimes picked from wavelet peaks may be best modeled with bandlimited raypaths, formed by narrowly averaging mono- chromatic wavepaths around the dominant source frequency

I 0 6i)O 1600 Half-width x (m)

L h: 55-75 Hz -3

.-l I

i, 600 1600 Half-width x (m)

b GO id00 0 500 lciO0 Half-width x (m) Half-width x (m)

FIG. 18. Bandlimited-raypath width. Half cross-sections through 3-D constant-velocity bandlimited raypaths (dotted lines) and their envelopes (solid lines) for four different combinations of mm,” and mrnax. The cross-section is taken midway between the source and receiver: the source-re- ceiver separation is 2000 m; and the background velocity is 2000 m/s.

(Woodward, 1989 p. 55). This frequency dependence sup- gests the simultaneous inversion of separate bandregions b,’ a modified ray-trace technique.

ACKNOWLEDGMENTS

I thank the sponsors of the Stanford Exploration Projec 1 and Jon Claerbout for their generous support. I also than-: Fabio Rocca, who first intuited the existence of some lini; between my early fat-ray ideas and Rytov diffraction tomo& - raphy. His guidance and encouragement were indispensabli: in the formalization and completion of this work. Finally, I must acknowledge the inspiration of this research by J. C Hagedoorn’s seismic-beam discussion, in his classic papt I on seismic reflection interpretation (1954).

REFERENCES

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APPENDIX A

BANDLIMITED-RAY WIDTH

The equation describing the envelope (A) of the bandlim- ited raypaths show in Figure I8 is:

2 [I - cos (AtA@)] - 2 ho sin (AhrAw)

cos (AtAm) + “1,x + Wiin] (A-1)

Here b is the product of the Green’s function geometrical- spreading terms in equations (6) and (13):

4rrd b=

x2 f d214’ (A-2)

where d is the source-receiver separation and .Y is the offset from the source-receiver axis. At is the time delay between the direct and x-scattered arrivals:

At=; (2+2+;-d). (A-3)

The width of the bandlimited raypath is determined by the term dominating A’s behavior for small x. In equation (1’~.I) this is the factor multiplying 2/At4: 1 - cos (Athw). I he first zero of this term marks the boundary of the first n,;iin lobe in A, echoing the uncertainty relation discussion H i.h: At& = 27~. The simple inverse relation between At and \w yields a more complicated inverse relation when solved ‘or the half-width X:

x = v’: (E+ d). (1lL.4)

For large Aw (or d >> ~zJ/Aw) this simplifies to:

J ml6 x= -.

A0 (. i-5)