Seismic depth migration with the Dirac equation

GEOPHYSICS, VOL. 64, NO. 3 (MAY-JUNE 1999); P. 925–933, 9 FIGS., 1 TABLE.

Seismic depth migration with the Dirac equation

Ketil Hokstad∗ and Rune Mittet‡

ABSTRACT

We demonstrate the applicability of the Dirac equa-tion in seismic wavefield extrapolation by presenting anew explicit one-way prestack depth migration scheme.The method is in principle accurate up to 90◦ from thevertical, and it tolerates lateral velocity variations. Thisis achieved by performing the extrapolation step of mi-gration with the Dirac equation, implemented in thespace-frequency domain. The Dirac equation is an ex-act linearization of the square-root wave equation and isequivalent to keeping infinitely many terms in a Taylorseries or continued-fraction expansion of the square-root operator. An important property of the new methodis that the local velocity and the spatial derivatives decou-ple in separate terms within the extrapolation operator.Therefore, we do not need to precompute and store largetables of convolutional extrapolator coefficients depend-ing on velocity. The main drawback of the explicit scheme

is that evanescent energy must be removed at each depthstep to obtain numerical stability.

We have tested two numerical implementations of themigration scheme. In the first implementation, we per-form depth stepping using the Taylor series approxima-tion and compute spatial derivatives with high-order fi-nite difference operators. In the second implementation,we perform depth stepping with the Rapid expansionmethod and numerical differentiation with the pseudo-spectral method. The imaging condition is a generaliza-tion of Claerbout’s U/D principle.

For both implementations, the impulse response is ac-curate up to 80◦ from the vertical. Using synthetic datafrom a simple fault model, we test the depth migrationscheme in the presence of lateral velocity variations. Theresults show that the proposed migration scheme im-ages dipping reflectors and the fault plane in the correctpositions.

INTRODUCTION

Seismic migration consists of two steps: (1) wavefield extrap-olation and (2) imaging. Migration methods where the wave-field extrapolation step is carried out explicitly (unlike, e.g.,Kirchhoff migration) can be subdivided in two main classes:(1) depth-stepping methods, which extrapolate the wavefieldin the depth direction, and (2) reverse time migration methods,which extrapolate the wavefield backward in time.

Depth-stepping migration schemes are usually implementedby means of one-way first-order wave equations extrapolatingup- and down-going waves separately. Pioneering work on one-way wave propagation was done by Claerbout and Muir, whointroduced the paraxial, (square-root) wave equation and thecontinued-fraction expansion in the seismic industry. The setof one-way wave equations thus obtained are the familiar 5◦,15◦, 45◦, etc. equations (Claerbout, 1985). More recently, one-way schemes based on optimized convolutional extrapolation

Manuscript received by the Editor June 5, 1996; revised manuscript received August 3, 1998.∗IKU Petroleum Research, SINTEF Group, N-7034 Trondheim, Norway. E-mail: [email protected].‡Formerly IKU Petroleum Research, SINTEF Group, N-7034 Trondheim, Norway; currently Sting Research, Rosenborg gt. 17, N-7014 Trondheim,Norway. E-mail: [email protected]© 1999 Society of Exploration Geophysicists. All rights reserved.

operators have been constructed by several authors (Holberg,1988; Blacquiere et al., 1989; Hale, 1991; Sollid and Arntsen,1994; Mittet et al., 1995). Common to most of the methods men-tioned is that they rely on truncated series approximations tothe paraxial wave equation, with low accuracy above a maxi-mum angle of propagation relative to some reference direction(usually the vertical). Presently, one-way migration schemeswith no angular limitations exist only for the special cases ofhomogeneous and horizontally layered media (Kennett, 1983;Claerbout, 1985).

Another interesting application of one-way equations is theimplementation of transparent boundary conditions in finitedifference modeling. One-way methods can be applied bythemselves or in combination with absorbers to reduce artificialreflections from the edges of the computational grid (Lindman,1975; Cerjan et al., 1985).

Reverse time migration methods rely on the invariance ofthe seismic wave equations with respect to time reversal, and

925

926 Hokstad and Mittet

are implemented by running wave equations backwards intime (Chen and McMechan, 1991; Mufti et al., 1996). At eachdiscrete time step, seismic measurements are introduced asboundary conditions. Reverse time migration schemes are usu-ally based on two-way second-order wave equations. Conse-quently, there is no restriction with respect to maximum prop-agation angle. The reverse time migration method can easilybe extended to the elastic, viscoelastic, and anisotropic case.If the wavepropagation is based on finite difference modeling(Hokstad et al., 1998), there are no restrictions with respect tocomplexity of the velocity model.

The use of two-way second-order wave equations is also themain drawback of reverse time migration. It may be difficultto properly control the propagation of waves in the velocitymodel. Spurious artifacts with high dips often appear in theimages, in particular for full elastic implementations. Also, 3-Dfinite difference–based prestack elastic reverse-time migration(Mittet et al., 1997) is very expensive in terms of memory useand CPU time.

The major purpose of this paper is to introduce a new (to theseismic industry) wave equation which offers the most desiredproperties of both traditional depth-stepping and reverse timemigration: (1) one-way wavefield extrapolation with a first-order equation, and (2) accurate wave propagation up to 90◦

from the vertical in velocity models with arbitrary lateral veloc-ity variations. We achieve this by doing wave propagation withthe Dirac equation, which is one of the most important equa-tions in relativistic (high-energy) quantum mechanics (Bjorkenand Drell, 1964; Itzykson and Zuber, 1980). Generally, theDirac equation may be used for one-way wave propagationpurposes in both prestack and poststack migration schemes.Recent theoretical investigations (unpublished) indicate thatit may be extended to the anisotropic and elastic cases, as well.Numerical implementations of the wavepropagation may beexplicit or implicit finite differencing, pseudospectral methods,or ray-based techniques.

In this paper, however, we limit ourselves to the construc-tion of an explicit prestack depth migration scheme, operatingin source/receiver coordinates. The wave field extrapolationis performed in the frequency-space ( f -x) domain. The equa-tions are given in three spatial dimensions, although they havebeen implemented and tested in two dimensions only.

In the derivation of the wavefield extrapolating equation,we start from a linearization of the paraxial wave equationin terms of unknown coefficients. This is equivalent to Diracsoriginal problem (see the Appendix). The initially unknowncoefficients may be identified as a particular set of 2× 2 ma-trices, and the linearized wave equation can be recognized asa variety of the general Dirac equation. The Dirac equationprovides an exact linearization of the paraxial wave equation,and in the language of seismic migration, it may be viewed asthe “90◦ equation.”

We demonstrate applicability of the Dirac equation in seis-mic migration by means of two alternative numerical imple-mentations. In the first implementation, we compute spatialderivatives with local finite-difference operators and performdepth stepping with the Taylor series approximation. In the sec-ond implementation, we perform spatial differentiation withthe pseudospectral method and depth stepping with the rapidexpansion method (Tal-Ezer, 1986; Tal-Ezer et al., 1987), whichgives infinite accuracy in both the lateral and vertical directions.

For both implementations, we check the migration impulse re-sponse. Using synthetic data from a simple fault model, we testthe performance of the depth migration scheme in the presenceof lateral velocity variations and compare the results with theimage obtained by reverse time migration.

THE DIRAC EQUATION

Linearization of the paraxial wave equation

In the f -x domain, the acoustic wave equation in a source-free medium with constant density is(∇2 + k2

0

)P(x; k0) = 0, (1)

where k0=ω/c(x), ω is the angular frequency, c(x) is the lo-cal propagation velocity, and P is the pressure at position x.For simplicity, the dependency on k0 and x= (x, y, z) will besuppressed when not explicitly required.

In one spatial dimension, the second-order differential op-erator of the acoustic wave equation can be factorized into aproduct of first-order operators:(

∂2z + k2

0

)P = (∂z + ik0)(∂z − ik0)P = 0, (2)

which leads to first-order differential equations,

∂z P = ±ik0 P, (3)

satisfied by up- and down-going waves. Such a simple relationdoes not exist in higher spatial dimensions. When construct-ing one-way wavefield extrapolators in three (and two) dimen-sions, it is common to approximate the paraxial wave equation,which formally can be written

∂z P = ±i√

k20 + ∂2

x + ∂2y P, (4)

by some truncated series expansion, such as a Taylor series orcontinued fraction.

In this paper, we take another approach where we avoidseries expansions of the square root operator. We construct aone-way wavefield extrapolator by linearizing the square rootoperator in terms of unknown coefficients. We write√

k0 + ∂2x + ∂2

y = α0k0 − α1∂x − α2∂y, (5)

and hence,

∂z9 = ±i(α0k0 − α1∂x − α2∂y)9, (6)

where the wavefield is denoted by 9 =9(x; k0). Conditions todetermine the unknown coefficients are obtained by squaringequation (5) or, equivalently, by requiring that the wavefield9satisfies the acoustic wave equation (1). Squaring and collectingterms, we find that the coefficients must obey

αµαν + αναµ = 2δµν, (7)

where µ, ν= 0, 1, or 2, and δµν is the Kronecker delta func-tion. These conditions can not be fulfilled by ordinary num-bers which obey the commutative law. Instead we must allowthe coefficients to be matrices. Equation (7) can be recognizedas the algebra of the non-Abelian group SU(2). The simplestchoice of a set of matrices which satisfy equation (7) are thegenerators of SU(2), which are usually called the Pauli matri-ces. The Hermitian Pauli matrices, conventionally denoted by

Depth Migration with the Dirac Equation 927

σi , are (Itzykson and Zuber, 1980)

σ1 =(0 1

1 0

), σ2 =

( 0 i

−i 0

),

(8)

σ3 =(1 0

0 −1

).

A good choice for the purpose of this paper is to take

α0 = σ3, α1 = σ2 = iσ3σ1,

α2 = σ1 = iσ3σ2.

(9)

Having identified the coefficients αµ, we may now write downthe differential equation for Ψ. It can be shown that the choiceof sign in equation (6) is just a matter of taste because bothoptions give the same set of solutions. In the following, weonly use the positive sign. Collecting terms, we get

∂zΨ = iGΨ, (10)

where the linear first-order matrix differential operator iG isgiven by

iG =(

ik0 ∂x − i∂y

−∂x − i∂y −ik0

), (11)

and Ψ= (9 A9B)T is a two-component column vector in someabstract vector space. When ∂xΨ= 0 (i.e., for plane wavespropagating in the z-direction), equation (10) reduces to equa-tion (2).

In the derivation of equation (10), we have not made anyapproximations. Hence, equation (10) is exact for waves prop-agating at angles up to 90◦ from the vertical.

Equations (6) and (10) can be recognized as the Dirac equa-tion, which in relativistic quantum mechanics may representthe quantum state of a high-energy uncharged zero-mass ferm-ion (e.g., a neutrino; see the Appendix).

Eigenvalues and eigenvectors

In a homogeneous medium, equation (10) may be trans-formed to the frequency-wavenumber ( f -k) domain. Applyinga 3-D spatial Fourier transform, we get

GΨ = kzΨ, (12)

where

G = G(kx , ky, k0) =(

k0 kx − iky

−kx − iky −k0

). (13)

Equation (13) has the form of an eigenvalue problem for kz .Computing the eigenvalues, we obtain

kz = ±√

k20 − k2

x − k2y, (14)

as desired. The matrix G, in contrast to the Pauli matricesσi , isnot Hermitian. By a theorem in linear algebra, this correspondsto the fact that kz is imaginary in the evanescent region. The

eigenvectors are

Ψ+ = N

1

−(kx + iky)k0 + |kz|

, Ψ− = N

−(kx − iky)k0 + |kz|

1

,(15)

for the positive and negative eigenvalues, respectively. N is anormalization factor:

N = N(kx , ky, k0) = 1

/√1+ k2

x + k2y

(k0 + |kz|)2. (16)

The eigenvectors has the following properties:ψ A+ andψ B

− aresymmetrical with respect to horizontal wavenumber, whereasψ B+ and ψ A

− are antisymmetrical. For plane waves propagatingat small angles from the vertical, |ψ A

+|� |ψ B+ | and |ψ B

− |� |ψ A−|.

Therefore,ψ A+ andψ B

− are called the large or symmetrical com-ponents, and ψ B

+ and ψ A− are called the small or antisymmetri-

cal components. In the limit of plane waves propagating at 90◦

from the vertical, the “large” and “small” components are ofequal magnitude.

The eigenvectors are not orthogonal, that is

ψ†−ψ+ 6= 0, (17)

where the dagger denotes Hermitian conjugation. However,the eigenvectors are orthogonal with respect to the redefinedinnerproduct:

ψ−ψ+ = ψ†−σ3ψ+ = 0. (18)

Up-going and down-going waves

Equation (10) is a first order differential equation which canbe applied for depth extrapolation of both up- and down-goingwaves. The properties of being up-going or down-going is at-tached to the combined properties of the Dirac equation (10)and the eigenvectors ψ± given in equation (15). It is easy toshow that when combining equations (12) and (15) to elimi-nate the small components, we recover the familiar 15◦, 45◦,etc. equations. If equation (10) is to be useful in seismic depthmigration, we must relate the seismic wave field

P = U + D, (19)

where P is the total pressure field, and U and D are the up-and down-going parts of P , to the eigenvectorsψ±. To achievethis, we expand U and D with the eigenvectors as basis. In thef -x domain, we write,

ΨD(x, y, zs; k0) = 1(2π)2

∫dkx dky ei(kx x+ky y)

×ψ+(kx , ky, k0)D(zs; kx , ky, k0),

(20)

ΨU (x, y, zr ; k0) = 1(2π)2

∫dkx dky ei(kx x+ky y)

×ψ−(kx , ky, k0)U(zr ; kx , ky, k0),

(21)

where zs and zr are the source and receiver depths, respectively.Equations (20) and (21) are used to construct initial conditionsfor the depth stepping from the seismic data U and D.


Numerical stability

Formally, the solution to equation (10) can be written

Ψ(x, y, z +1z; k0) = eiG1zΨ(x, y, z; k0). (22)

In a numerical implementation, the operator exponential mustbe replaced by some series approximation. This will be consid-ered in the next section.

The stability condition for recursive application of equa-tion (22) is

|eiG1z| ≤ 1. (23)

Hence, the solution is stable if the eigenvalues of G (i.e., kz)are real. A marching solution in the depth direction becomesunstable in the evanescent region where kz is imaginary. Equa-tion (22) is numerically stable if the evanescent energy is sup-pressed at each depth step.

PRESTACK DEPTH MIGRATION

The process of migration consists of two parts: wavefield ex-trapolation and imaging. Generally, the extrapolation step canbe carried out by almost any numerical approximation to equa-tion (22). We have implemented and tested two possible solu-tions: the familiar Taylor series method and the rapid expansionmethod. The imaging step is performed by a generalization ofClaerbout’s (1971) U/D principle.

Taylor series method

In the Taylor series approximation, the depth stepping equa-tion is

Ψ(z +1z) =N∑

n=0

∂nz Ψ(z)

n!1zn, (24)

where Ψ equals ΨU or ΨD . The z-derivatives of Ψ can becomputed recursively by repeated applications of the matrixoperator

∂zΨ = iGΨ, (25)

∂nz Ψ = iG∂n−1

z Ψ, (26)

where ∂nz denotes the nth derivative with respect to z and the

matrix differential operator iG is given by equation (11). In thelimit N→∞, we recover the formal solution (22). An impor-tant property of the operator iG is that the local velocity and thespatial derivatives are decoupled in separate terms within theoperator. This makes the migration scheme easy to implementbecause we can use the same operator for spatial differenti-ation for all values of k0. We avoid the use of large tables ofprecomputed operator coefficients depending on velocity (ork0). With the Taylor series method, we compute the horizontalderivatives in iG using high-order coarse-grid finite-differenceoperators, as proposed by Holberg (1987). The finite differencemethod was implemented on a regular grid staggered in the x-direction, with operator half-length Lx ' N . Using short localoperators, we get an implementation with finite accuracy, butwhich can be optimized with respect to computational perfor-mance.

Rapid expansion method

The rapid expansion method was presented by Tal-Ezer(1986). It was used by Tal-Ezer et al. (1987) to obtain a time-

stepping solution to the seismic forward-modeling problem.Here, we use the method to obtain a depth-stepping solu-tion. In the rapid expansion method, the exponential func-tion is approximated by a finite series in modified Chebyshevpolynomials:

e1z0 ' HM (1z0) =M∑

k=0

Ck Jk(1z R)Qk

(0

R

), (27)

where C0= 1 and Ck = 2 for k 6= 0, Jk is the kth-order Besselfunction, and Qk(ξ) are modified Chebyshev polynomials thatsatisfy the recursion relation

Qk+1 = 2ξQk + Qk−1, (28)

with Q0= 1 and Q1= ξ . Substituting the series expansion (27)for the operator exponential in equation (22), we get

Ψ(z +1z) =M∑

k=0

Ck Jk(1z R)Φk(z), (29)

where

Φ0(z) = Ψ(z),

Φ1(z) = iGR

Ψ(z),

Φk+1(z) = 2iGR

Φk(z)+Φk−1(z),

(30)

and iG is given by equation (11). To insure convergency, 1z Rmust be larger than the range of eigenvalues of G, which means1z R ≥max{kz} and M >1z R (Tal-Ezer, 1986). With the rapidexpansion method, we compute spatial derivatives in the hor-izontal direction by the pseudospectral method,

∂xΨ = F−1{ikxF{Ψ}}, (31)

where F and F−1 denotes the forward and inverse spatialFourier transform. Combining the pseudospectral method andrapid expansion method, we obtain a well-balanced implemen-tation with infinite accuracy in both the lateral and the verticaldirections (Tal-Ezer, 1986).

Imaging

We use an imaging condition which generalizes Claerbout’s(1971) U/D principle,

IC (x, y, z) =∫

dωU D∗

DD∗, (32)

in a way consistent with the definition of ΨU and ΨD . We definethe image at depth z= (k − 1)1z by

I (x, y, z) =∫

dωΨ†Dσ1ΨU

Ψ†DΨD

. (33)

If ΨD is a plane wave in the vertical direction (i.e., kx = ky = 0),definition (33) reduces to equation (32).

The spectral division in equation (33) can be numericallyunstable at high and low frequencies where, usually, the nu-merator and denominator are both close to zero. To insurenumerical stability, we use the modified imaging equation

I (x, y, z) =∫

dωΨ†Dσ1ΨU∫dωΨ†DΨD

, (34)

where the denominator is a geometrical spreading correction.


Impulse response

We computed the migration impulse response in a dis-cretized homogeneous macro model with spatial sampling in-terval 1x =1z= 10.0 m and velocity equal to 1500 m/s. Themaximum temporal frequency was 35 Hz. Figure 1 shows themigration impulse response computed with the Taylor seriesmethod and high-order finite differences. The number of termsN in the Taylor series and the halflength Lx of the finite dif-ference operator was N = Lx = 8. Figure 2 shows the migrationimpulse response computed with the rapid expansion method,keeping the first five terms in the Chebyshev series [equa-tion (29)], and pseudospectral differentiation. For both meth-ods, the impulse response is good up to approximately 80◦ fromthe vertical, where the image gradually vanishes.

FAULT MODEL EXAMPLE

To compute the impulse response is an important exercisewhich shows the accuracy of the depth migration scheme atlarge angles from the vertical. More important, however, is totest the behavior of the proposed scheme in the presence oflateral velocity variations. To accomplish this, we generateda simple acoustic fault model with a simulated unconformity.The model was discretized on a regular grid with 401× 281nodes, and node spacing 1x =1z= 12.5 m. A buffer zone 40grid points wide was included along the edges of the model toprevent edge reflections. The geometry of the model is shownin Figure 3, and the physical parameters of the model aregiven in Table 1. We modeled 10 synthetic shot records with ahigh-order finite-difference solution to the two-way acousticwave equation (Holberg, 1987). The shot interval was 200 m,

FIG. 1. Migration impulse response computed with the Taylorseries method and high-order finite differences.

FIG. 2. Migration impulse response computed with the Rapidexpansion method and pseudospectral differentiation.

and the source was spherically symmetrical with maximumfrequency 40 Hz. Data were recorded on a simulated ocean-bottom hydrophone cable 125 m below the source, with 3.0 kmaperture and receiver spacing equal to 12.5 m. The source andreceiver configuration is indicated in Figure 3. Attenuation offree surface effects is an important field of research but, be-cause it is not a topic of this paper, the data were modeledwithout free surface multiples. Figure 4 shows the shot recordfor shot position 7 after removal of the direct wave.

We imaged the ten shot records with the proposed depth mi-gration scheme, performing the depth stepping by the rapid

Table 1. Physical parameters of the fault model.

Layer Density (kg/m3) Velocity (m/s)

1 1000.0 1500.02 1700.0 1600.03 1800.0 1700.04 1900.0 1800.05 2000.0 1900.06 2050.0 2000.07 2100.0 2200.08 2200.0 2400.09 2400.0 2700.0

FIG. 3. Geometry of the fault model. Source positions are in-dicated with stars and receivers with triangles. The physicalparameters of the model are given in Table 1.

FIG. 4. Shot record for source location 7 after removal of thedirect wave.


expansion method and pseudospectral differentiation. Forcomparison, the data was also imaged by two-way reverse-time migration, where the wave propagation is performed byfinite differences. In both cases, the macro velocity model wascreated by smoothing the true velocity model with a low-pass

FIG. 5. Time sections from depth stepping of the down-going waves 9D at depths 0.25, 0.5, 0.75, and 1.0 km.

filter rejecting wavenumbers larger than 10% of the Nyquistwavenumber. The density was taken to be constant.

In Figures 5 and 6, we show extrapolated sections from thedepth stepping for source location 7. The time sections arerecorded at depths 0.25, 0.50, 0.75, and 1.0 km relative to the


source depth. Figure 5 shows the two components of the vec-tor ΨD , and Figure 6 shows the components of ΨU . The non-causal events visible in the sections are due to removal of theevanescent energy from the data. For ΨD , the upper compo-

FIG. 6. Time sections from depth stepping of the up-going waves 9U at depths 0.25, 0.5, 0.75, and 1.0 km.

nent (ΨAD) is symmetrical with respect to zero offset, whereas

the lower component (ΨBD) is antisymmetrical. For ΨU we have

the oposite situation. These symmetry properties follow imme-diately from the properties of the eigenvectors ψ+ and ψ− in


equation (15). The large (symmetric) and small (antisymmet-ric) components are plotted with different scaling because theydiffer by one order of magnitude.

Figure 7 shows the raw image obtained by depth stepping ofa single shot record (shot 7). Dipping reflectors are correctlypositioned, and parts of the fault plane can be identified as well.The migration smiles and the smearing at both edges of theshallow plane reflectors are caused by limited aperture. Aftermigration of all shot records, we muted and stacked the partialimages. Figure 8 shows the result from stacking the ten partialimages obtained by depth stepping. Figure 9 is the correspond-ing result from reverse time migration. The fault plane partlyvanishes in the stacking of partial images since it is only illumi-nated from the rightmost shot positions. The proposed depthmigration scheme and reverse time migration gives similarresults. The fault plane is, however, better imaged by reversetime migration. There are two reasons for this. First, the reversetime migration scheme which is based on a high-order finitedifference solution to the two-way wave equation, is optimalfor wave propagation at large angles from the vertical. Second,for stable depth extrapolation, we must filter out evanescentenergy at each depth step. The filter was applied from theevanescent limit as defined by the maximum velocity at eachdepth level, at the cost of accuracy in low-velocity regions.

FIG. 7. Image of shot 7 obtained by depth stepping.

FIG. 8. Stack of ten partial images obtained by depth stepping.

CONCLUSIONS

We have derived and tested a new explicit prestack seis-mic depth migration scheme, operating in source/receiver co-ordinates. One-way wavefield extrapolation is performed withthe Dirac equation implemented in the space-frequency do-main. Theoretically, the wavefield extrapolation is accurate upto 90◦ from the vertical. An important property of the migra-tion scheme is that spatial derivatives and the local velocityare decoupled in separate terms within the extrapolation oper-ator. Tables of precomputed operator coefficients dependingon velocity are not needed.

We have tested two numerical implementations of the sch-eme, performing the depth stepping with the Taylor series ap-proximation and the rapid expansion method, respectively. Theimaging condition is a generalization of Claerbout’s (1971)U/D principle. Numerical experiments show that for both im-plementations, the migration impulse response is accurate upto 80◦ from the vertical.

Using synthetic data from a fault model with a simulatedunconformity, we have tested the performance of the depthmigration scheme in the presence of lateral velocity variations.The results shows that the new migration scheme images dip-ping reflectors in the right position. In some partial imagesobtained by migration of single shots, parts of the fault planecan be identified as well. The result was compared with the im-age obtained by reverse time migration. The comparison showsthat the fault plane is better imaged by reverse time migration.

The main purpose of the paper is to introduce the Diracequation in the seismic industry and to demonstrate that thisequation is applicable to prestack depth migration. We have notyet emphasized optimization with respect to numerical speedand memory use. Efficiency considerations as well as alterna-tive implementations and applications of the Dirac equation,such as implicit schemes and poststack migration, will be leftto further work.

ACKNOWLEDGMENTS

This work was performed under a scholarship sponsoredby The Norwegian Research Council and Petroleum Geo-Services. We are grateful for their financial support.

FIG. 9. Stack of ten partial images obtained by reverse timemigration.


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APPENDIX

SEISMIC MIGRATION AND QUANTUM MECHANICS

There are numerous examples of theoretical problems fromdiverse fields of science which are expressed in terms of thesame partial differential equations and which may be analyzedusing the same mathematical methods. Well-known examplesare the scalar wave equation and the Born approximation.Therefore, it is not surprising that equations originating inquantum mechanics can be recognized in other branches ofphysics as well.

We give two qualitative arguments why the Dirac equation,which is one of the most important equations in relativisticquantum mechanics, should be applicable to seismic problems.First, it is well known that the 15◦ equation is mathematicallyequivalent to the Schrodinger equation where depth z in the 15◦

equation has the same role as time t in the Schrodinger equa-tion (Claerbout, 1985). Moreover, it was noticed by Claerbout(1971) that the 45◦ equation is mathematically equivalent to theSchrodinger equation plus the lowest order relativistic correc-tion. On the other hand, from quantum mechanics, we knowthat the Schrodinger equation is the low-energy limit of theDirac equation. Hence, making the correspondence t↔ z andlow/high energy↔ small/large angle, it is clear that the Diracequation could be a 90◦ equation for seismic one-way wavepropagation problems.

The second argument is based on Dirac’s first interpretationof his equation (Itzykson and Zuber, 1980) which was derivedin the 1920s. He found that the Dirac equation has two typesof solutions with positive and negative energy, respectively.Dirac interpreted the two types of solutions as positive en-ergy electrons propagating forward in time and negative energyelectrons propagating backward in time. With the correspon-dence t↔ z and positive/negative energy↔ positive/negativevertical wavenumber (kz), we get a second indication that theDirac equation can be applicable to seismic problems. In mod-ern quantum field theory, the negative energy solutions of theDirac equation are interpreted as positrons, the antiparticle ofthe electron.

In this paper, we linearize the paraxial wave equation (4) bywriting

i√

k20 + ∂2

x + ∂2yΨ = i (α0k0 −α1∂x −α2∂y)Ψ, (A-1)

which leads to the Dirac equation in the form of equation (10).Dirac’s original problem was to linearize the square root equa-tion

ih∂tΨ =√−c2h2∇2 +m2c4Ψ, (A-2)

which is the quantum mechanical equation for the energy of anoninteracting relativistic particle,

E =√

p2c2 +m2c4, (A-3)

where E is the total energy, p is the length of the mo-mentum vector, m is the mass, and c is the velocity oflight. Equation (A-2) is mathematically equivalent to equa-tion (A-1) if we take m= 0 and interchange the meaning of tand z.

The Dirac equation was originally derived for an electronwith nonzero mass m. Thus, Ψ must be a four-component vec-tor, and the factorization in equation (A-2) must be carriedout with a set of four 4× 4 matrices. For a zero-mass fermion,2× 2 matrices are sufficient. The Dirac equation for a zero-mass fermion is sometimes called the Weyl equation.

In quantum mechanics, the multicomponent structure of Ψintroduces a new physical degree of freedom called spin orintrinsic angular momentum. For our purpose, the Dirac equa-tion is merely a convenient way to linearize the paraxial waveequation. We do not attach any particular physical meaningto the column vector nature of the wavefield Ψ, nor do wepay any attention to the relativistic covariance of the Diracequation.

Seismic depth migration with the Dirac equation

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