Velocity continuation and the anatomy of residual prestack time migration

Transcription

1 GEOPHYSICS, VOL. 68, NO. 5 SEPTEMBER-OCTOBER 003; P , 11 FIGS / Velocity continuation and the anatomy of residual prestack time migration Sergey Fomel ABSTRACT Velocity continuation is an imaginary continuous process of seismic image transformation in the postmigration domain. It generalizes the concepts of residual and cascaded migrations. Understanding the laws of elocity continuation is crucially important for a successful application of time-migration elocity analysis. These laws predict the changes in the geometry and intensity of reflection eents on migrated images with the change of the migration elocity. In this paper, I derie kinematic and dynamic laws for the case of prestack residual migration from simple geometric principles. The main theoretical result is a decomposition of prestack elocity continuation into three different components corresponding to residual normal moeout, residual dip moeout, and residual zero-offset migration. I analyze the contribution and properties of each of the three components separately. This theory forms the basis for constructing efficient finite-difference and spectral algorithms for timemigration elocity analysis. INTRODUCTION The conentional approach to seismic migration theory Berkhout, 1985; Claerbout, 1985 employs the downward continuation concept. According to this concept, migration extrapolates upgoing reflected waes, recorded on the surface, to the place of their reflection to form an image of subsurface structures. Poststack time migration possesses peculiar properties, which can lead to a different iewpoint on migration. One of the most interesting properties is an ability to decompose the time-migration procedure into a cascade of two or more migrations with smaller migration elocities. This remarkable property is described by Rothman et al as residual migration. Larner and Beasley 1987 generalized the method of residual migration to one of cascaded migration. Cascading finite-difference migrations oercomes the dip limitations of conentional finite-difference algorithms Larner and Beasley, 1987; cascading Stolt-type f -k migrations expands their range of alidity to the case of a ertically arying elocity Beasley et al., Further theoretical generalization sets the number of migrations in a cascade to infinity, making each step in the elocity space infinitesimally small. This leads to a partial differential equation in the time-midpoint-elocity space, discoered by Claerbout Claerbout s equation describes the process of elocity continuation, which fills the elocity space in the same manner as a set of constant-elocity migrations. Slicing in the migration elocity space can sere as a method of elocity analysis for migration with nonconstant elocity Fowler, 1984, 1988; Shurtleff, 1984; Mikulich and Hale, 199. The concept of elocity continuation was introduced in earlier publications Fomel, 1994, Hubral et al and Schleicher et al use the term image waes to describe a similar idea. Adler 00 generalizes it to the case of ariable background elocities under the name Kirchhoff image propagation. The importance of this concept lies in its ability to predict changes in the geometry and intensity of reflection eents on seismic images with the change of migration elocity. Whereas conentional approaches to migration elocity analysis methods take into account only ertical moement of reflectors Deregowski, 1990; Liu and Bleistein, 1995, elocity continuation attempts to describe both ertical and lateral moements, thus proiding for optimal focusing in elocity analysis applications Fomel, 001, 003b. In this paper, I describe the elocity continuation theory for the case of prestack time migration, connecting it with the theory of prestack residual migration Al-Yahya and Fowler, 1986; Etgen, 1990; Stolt, 1996; Saa, 003. By exploiting the mathematical theory of characteristics, a simplified kinematic deriation of the elocity continuation equation leads to a differential equation with correct dynamic properties. In practice, one can accomplish dynamic elocity continuation by integral, finite-difference, or spectral methods. The accompanying paper Fomel, 003b introduces one of the possible numerical implementations and demonstrates its application on a field data example. Manuscript receied by the Editor December 11, 001; reised manuscript receied January 14, 003. Bureau of Economic Geology, Uniersity of Texas at Austin, Uniersity Station, Box X, Austin, Texas sergey.fomel@ beg.utexas.edu. c 003 Society of Exploration Geophysicists. All rights resered. 1650

2 Velocity Continuation 1651 The paper is organized into two main sections. First, I derie the kinematics of elocity continuation from the first geometric principles. I identify three distinctie terms, corresponding to zero-offset residual migration, residual normal moeout, and residual dip moeout. Each term is analyzed separately to derie an analytical prediction for the changes in the geometry of traeltime cures reflection eents on migrated images with the change of migration elocity. Second, the dynamic behaior of seismic images is described with the help of partial differential equations and their solutions. Reconstruction of the dynamical counterparts for kinematic equations is not unique. Howeer, I show that, with an appropriate selection of additional terms, the image waes corresponding to the elocity continuation process hae the correct dynamic behaior. In particular, a special boundary alue problem with the zero-offset elocity continuation equation produces the solution identical to the conentional Kirchoff time migration. KINEMATICS OF VELOCITY CONTINUATION From the kinematic point of iew, it is conenient to describe a reflector as a locally smooth surface z = zx, where z is the depth, and x is the point on the surface x is a D ector in the 3D problem. The image of the reflector obtained after a common-offset prestack migration with a half-offset h and a constant elocity is the surface z = zx; h;. Appendix A proides the deriations of the partial differential equation describing the image surface in the depth-midpointoffset-elocity space. The purpose of this section is to discuss the laws of kinematic transformations implied by the elocity continuation equation. Later in this paper, I obtain dynamic analogs of the kinematic relationships in order to describe the continuation of migrated sections in the elocity space. The kinematic equation for prestack elocity continuation, deried in Appendix A, takes the following form: = τ + h 3 τ h τ. 1 Here, τ denotes the one-way ertical traeltime τ = z/. The right side of equation 1 consists of three distinctie terms. Each has its own geophysical meaning. The first term is the only one remaining when the half-offset h equals zero. This term corresponds to the procedure of zero-offset residual migration. Setting the traeltime dip to zero eliminates the first and third terms, leaing the second, dip-independent one. One can associate the second term with the process of residual normal moeout. The third term is both dip- and offset-dependent. The process that it describes is residual dip moeout. It is conenient to analyze each of the three processes separately, ealuating their contributions to the cumulatie process of prestack elocity continuation. Kinematics of zero-offset elocity continuation The kinematic equation for zero-offset elocity continuation is = τ. The typical boundary-alue problem associated with it is to find the traeltime surface τ x for a constant elocity, gien the traeltime surface τ 1 x 1 at some other elocity 1. Both surfaces correspond to the reflector images obtained by time migrations with the specified elocities. When the migration elocity approaches zero, poststack time migration approaches the identity operator. Therefore, the case of 1 = 0 corresponds kinematically to the zero-offset poststack migration, and the case of = 0 corresponds to the zero-offset modeling demigration. The ariable x in equation describes both the surface midpoint coordinate and the subsurface image coordinate. One of them is continuously transformed into the other in the elocity continuation process. The appropriate mathematical method of soling the kinematic problem posed aboe is the method of characteristics Courant and Hilbert, The characteristics of equation are the trajectories followed by indiidual points of the reflector image in the elocity continuation process. These trajectories are called elocity rays Fomel, 1994; Liptow and Hubral, 1995; Adler, 00. Velocity rays are defined by the system of ordinary differential equations deried from equation according to the Hamilton-Jacobi theory: dx d = ττ dτ x, d = τ, 3 dτ x d = τ3 x, dτ d = τ + τ τ x, 4 where τ x and τ are the phase-space parameters. An additional constraint for τ x and τ follows from equation, rewritten in the form τ = ττx. 5 The general solution of the system of equations 3 4 takes the parametric form x = A C, τ = B C, 6 τ x = C τ, τ = C τ, 7 where A, B, and C are constant along each indiidual elocity ray. These three constants are determined from the boundary conditions as A = x τ 1 1 = x 0, 8 1 B = τ = τ0, C = τ 1 = τ 0, where τ 0 and x 0 correspond to the zero elocity unmigrated section, while τ 1 and x 1 correspond to the elocity 1. The simple relationship between the midpoint deriatie of the ertical traeltime and the local dip angle α appendix A, = tan α, 11 shows that equations 8 and 9 are precisely equialent to the eident geometric relationships Figure 1 τ 1 x τ 1 tan α = x 0, cos α = τ 0. 1

3 165 Fomel Equation 10 states that the points on a elocity ray correspond to a single reflection point, constrained by the alues of τ 1, 1, and α. As follows from equations 6, the projection of a elocity ray to the time-midpoint plane has the parabolic shape xτ = A + τ B/C, which has been noticed by Chun and Jacewitz On the depth-midpoint plane, the elocity rays hae the circular shape z x = A xb/c A x, described by Liptow and Hubral 1995 as Thales circles. For an example of kinematic continuation by elocity rays, let us consider the case of a point diffractor. If the diffractor location in the subsurface is the point x d, z d, then the reflection traeltime at zero offset is defined from Pythagoras s theorem as the hyperbolic cure τ 0 x 0 = z d + x 0 x d d, 13 where d is half of the actual elocity. Applying equations 6 produces the following mathematical expressions for the elocity rays: x = x d d + x 0 1, 14 τ = τd + x 0 x d 1, 15 where τ d = z d / d. Eliminating x 0 from the system of equations 14 and 15 leads to the expression for the elocity continuation waefront : τx = d d d τd + x x d d. 16 For the case of a point diffractor, the waefront corresponds precisely to the summation path of the residual migration operator Rothman et al., It has a hyperbolic shape when d >undermigration and an elliptic shape when d < oermigration. The waefront collapses to a point when the elocity approaches the actual effectie elocity d. At zero elocity, = 0, the waefront takes the familiar form of the poststack migration hyperbolic summation path. The form of the elocity rays and waefronts is illustrated in Figure a. Another important example is the case of a dipping plane reflector. For simplicity, let us put the origin of the midpoint coordinate x at the point of the plane intersection with the surface of obserations. In this case, the depth of the plane reflector corresponding to the surface point x has the simple expression z p x = x tan α, 17 where α is the dip angle. The zero-offset reflection traeltime τ 0 x 0 is the plane with a changed angle. It can be expressed as τ 0 x 0 = px 0, 18 where p = sin α/ p, and p is half of the actual elocity. Applying formulas 6 leads to the following parametric expression for the elocity rays: x = x 0 1 p, 19 τ = px 0 1 p. 0 FIG. 1. Zero-offset reflection in a constant elocity medium a scheme. Eliminating x 0 from the system of equations 19 and 0 shows that the elocity continuation waefronts are planes with a modified angle: px τx = 1 p. 1 FIG.. Kinematic elocity continuation in the poststack migration domain. Solid lines denote waefronts reflector images for different migration elocities, dashed lines denote elocity rays. a The case of a point diffractor. b The case of a dipping plane reflector.

4 Velocity Continuation 1653 Figure b shows the geometry of the kinematic elocity continuation for the case of a plane reflector. Kinematics of residual NMO The residual normal-moeout NMO differential equation is the second term in equation 1: = h 3 τ. Equation does not depend on the midpoint x. This fact indicates the 1D nature of normal moeout. The general solution of equation is obtained by simple integration. It takes the form τ = C h 1 = τ 1 + h 1 1, 3 where C is an arbitrary elocity-independent constant, and I hae chosen the constants τ 1 and 1 so that τ 1 = τ 1. Equation 3 is applicable only for different from zero. For the case of a point diffractor, equation 3 easily combines with the zero-offset solution 16. The result is a simplified approximate ersion of the prestack residual migration summation path: τx = τd + x x d 1 d + h d 1. 4 Summation paths of the form 4 for a set of diffractors with different depths are plotted in Figures 3 and 4. The parameters chosen in these plots allow a direct comparison with Figures.4 and.5 of Etgen 1990, based on the exact solution and reproduced in Figures 8 and 9. The comparison shows that the approximate solution 4 captures the main features of the prestack residual migration operator, except for the residual dip-moeout DMO cusps appearing in the exact solution when the diffractor depth is smaller than the offset. Neglecting the residual DMO term in residual migration is approximately equialent in accuracy to neglecting the DMO step in conentional processing. Indeed, as follows from the geometric analog of equation 1 deried in Appendix A [equation A-17], dropping the residual DMO term corresponds to the condition tan α tan θ 1, 5 where α is the dip angle, and θ is the reflection angle. As shown by Yilmaz and Claerbout 1980, the conentional processing sequence without the DMO step corresponds to the separable approximation of the double-square-root equation A-4: s r 1 + 1, 6 where t is the reflection traeltime, and s and r are the source and receier coordinates: s = x h, r = x + h. In geometric terms, approximation 6 transforms to cos α cos θ 1 sin α cos θ + 1 sin θ cos α 1. 7 Taking the difference of the two sides of equation 7, one can estimate its accuracy by the first term of the Taylor series for small α and θ. The estimate is 3/4 tan α tan θ Yilmaz and Claerbout, 1980, which agrees qualitatiely with equation 5. Although approximation 4 fails in situations where the DMO correction is necessary, it is significantly more accurate than the 15 approximation of the double-square-root equation, implied in the migration elocity analysis method of Yilmaz and Chambers 1984 and MacKay and Abma 199. The 15 approximation s s + r r 8 FIG. 3. Summation paths of the simplified prestack residual migration for a series of depth diffractors. Residual slowness / d is 1.; half-offset h is 1 km. Compare this figure with Etgen s 1990 Figure.4, reproduced in Figure 8. FIG. 4. Summation paths of the simplified prestack residual migration for a series of depth diffractors. Residual slowness / d is 0.8; offset h is 1 km. Compare this figure with Etgen s 1990 Figure.5, reproduced in Figure 9.

5 1654 Fomel corresponds geometrically to the equation cos α cos θ 3 + cos α cos θ. 9 Its estimated accuracy from the first term of the Taylor series is 1/8 tan α + 1/8 tan θ. Unlike the separable approximation, which is accurate separately for zero offset and zero dip, the 15 approximation fails at zero offset in the case of a steep dip and at zero dip in the case of a large offset. Kinematics of residual DMO The partial differential equation for kinematic residual DMO is the third term in equation 1: = h. 30 τ It is more conenient to consider the residual DMO process coupled with residual NMO. Etgen 1990 describes this procedure as the cascade of inerse DMO with the initial elocity 0, residual NMO, and DMO with the updated elocity 1.The kinematic equation for residual NMO+DMO is the sum of the two terms in equation 1: = h 3 τ The deriation of the residual DMO + NMO kinematics is detailed in Appendix B. Figure 5 illustrates it with the theoretical impulse response cures. Figure 6 compares the theoretical cures with the result of an actual cascade of the inerse DMO, residual NMO, and DMO operators. Figure 7 illustrates the residual NMO+DMO elocity continuation for two particularly interesting cases. Figure 7a shows the continuation for a point diffractor. One can see that when the elocity error is large, focusing of the elocity rays forms a distinctie loop on the zero-offset hyperbola. Figure 7b illustrates the case of a plane dipping reflector. The image of the reflector shifts both ertically and laterally with the change in NMO elocity. FIG. 5. Theoretical kinematics of the residual NMO+DMO impulse responses for three impulses. Left The elocity ratio 1 / 0 is Right The elocity ratio 1 / 0 is In both cases, the half-offset h is 1 km. FIG. 6. The result of residual NMO+DMO cascading inerse DMO, residual NMO, and DMO for three impulses. Left The elocity ratio 1 / 0 is Right The elocity ratio 1 / 0 is In both cases, the half-offset h is 1 km.

6 Velocity Continuation 1655 The full residual migration operator is the chain of residual zero-offset migration and residual NMO+DMO. I illustrate the kinematics of this operator in Figures 8 and 9, which are designed to match Etgen s 1990 Figures.4 and.5. A comparison with Figures 3 and 4 shows that including the residual DMO term affects the images of objects with the depth smaller than the half-offset h. This term complicates the residual migration operator with cusps. FROM KINEMATICS TO DYNAMICS The theory of characteristics Courant and Hilbert, 1989 states that if a partial differential equation has the form n ij ξ 1,...,ξ n P ξ i, j = 1 i ξ j + F ξ 1,...,ξ n,p, P,..., P = 0, 3 ξ 1 ξ n where F is some arbitrary function, and if the eigenalues of the matrix Λ are nonzero, and one of them is different in sign from the others, then equation 3 describes a wae-type process, and its kinematic counterpart is the characteristic equation n i, j = 1 ij ξ 1,...,ξ n ψ ξ i ψ ξ j = 0 33 Dynamics of zero-offset elocity continuation In the case of zero-offset elocity continuation, the characteristic equation is reconstructed from equation to hae the form ψ ψ + t ψ = 0, 36 where τ is replaced by t according to equation 35. According to equation 3, the corresponding dynamic equation is P +t P + F x,t,,p, P, P, P =0, 37 where the function F remains to be defined. The simplest case of F equal to zero corresponds to Claerbout s elocity continuation equation Claerbout, 1986, deried in a different way. Lein 1986a proides the dispersion-relation deriation, conceptually analogous to applying the method of characteristics. In high-frequency asymptotics, the waefield P can be represented by the ray-theoretical WKBJ approximation, Pt, x, Ax,ft τx,, 38 where A is the amplitude, f is the short high-frequency waelet, and the function τ satisfies the kinematic equation. with the characteristic surface ψξ 1,...,ψ n =0 34 corresponding to the waefront. In elocity continuation problems, it is appropriate to choose the ariable ξ 1 to denote the time t,ξ to denote the elocity, and the rest of the ξ-ariables to denote one or two lateral coordinates x. Without loss of generality, let us set the characteristic surface to be ψ = t τx; = 0, 35 and use the theory of characteristics to reconstruct the main second-order part of the dynamic differential equation from the corresponding kinematic equations. As in the preceding section, it is conenient to consider separately the three different components of the prestack elocity continuation process. FIG. 8. Summation paths of prestack residual migration for a series of depth diffractors. Residual slowness / d is 1.; half-offset h is 1 km. This figure reproduces Etgen s 1990 Figure.4. FIG. 7. Kinematic elocity continuation for residual NMO+DMO. Solid lines denote waefronts zero-offset traeltime cures dashed lines denote elocity rays. a The case of a point diffractor; the elocity ratio 1 / 0 changes from 0.9 to 1.1. b The case of a dipping plane reflector; the elocity ratio 1 / 0 changes from 0.8 to 1.. In both cases, the half-offset h is km.

7 1656 Fomel Substituting approximation 38 into the dynamic elocity continuation equation 37, collecting the leading-order terms, and neglecting the F function leads to the partial differential equation for amplitude transport: A = τ A + τ A. 39 The general solution of equation 39 follows from the theory of characteristics. It takes the form Ax,= Ax 0,0 exp uτ τ 0 du, 40 where the integral corresponds to the curilinear integration along the corresponding elocity ray, and x 0 corresponds to the starting point of the ray. In the case of a plane dipping reflector, the image of the reflector remains plane in the elocity continuation process. Therefore, the second traeltime deriatie τ/ in equation 40 equals zero, and the exponential is equal to one. This means that the amplitude of the image does not change with the elocity along the elocity rays. This fact does not agree with the theory of conentional post-stack migration, which suggests downscaling the image by the cosine factor τ 0 /τ Chun and Jacewitz, 1981; Lein, 1986b. The simplest way to include the cosine factor in the elocity continuation equation is to set the function F to be 1/t P/. The resulting differential equation P + t P + 1 P t = 0 41 has the amplitude transport Ax,= τ 0 τ Ax 0,0 exp uτ τ 0 du, 4 corresponding to the differential equation A = τ A + τ A A 1 τ. 43 FIG. 9. Summation paths of prestack residual migration for a series of depth diffractors. Residual slowness / d is 0.8; half-offset h is 1 km. This figure reproduces Etgen s 1990 Figure.5. Appendix C proes that the time-and-space solution of the dynamic elocity continuation equation 41 coincides with the conentional Kirchhoff migration operator. Dynamics of residual NMO According to the theory of characteristics, described in the beginning of this section, the kinematic residual NMO equation corresponds to the dynamic equation of the form P + h P + Fh,t,,P= t with the undetermined function F. In the case of F = 0, the general solution is easily found to be Pt, h,=φ t + h, 45 where φ is an arbitrary smooth function. The combination of dynamic equations 44 and 41 leads to an approximate prestack elocity continuation with the residual DMO effect neglected. To accomplish the combination, one can simply add the term h / 3 t P/ from equation 44 to the left side of equation 41. This addition changes the kinematics of elocity continuation, but does not change the amplitude properties embedded in the transport equation 4. Dunkin and Lein 1973 and Hale 1983 adocate using an amplitude correction term in the NMO step. This term can be easily added by selecting an appropriate function F in equation 44. The choice F = h / 3 t P results in the equation P + h 3 t t P + P = 0 46 with the general solution Pt, h,= 1 t φ t + h, 47 which has the Dunkin-Lein amplitude correction term. Dynamics of residual DMO The case of residual DMO complicates the building of a dynamic equation because of the essential nonlinearity of the kinematic equation 30. One possible way to linearize the problem is to increase the order of the equation. In this case, the resultant dynamic equation would include a term that has the second-order deriatie with respect to elocity. Such an equation describes two different modes of wae propagation and requires additional initial conditions to separate them. Another possible way to linearize equation 30 is to approximate it at small dip angles. In this case, the dynamic equation would contain only the first-order deriatie with respect to the elocity and high-order deriaties with respect to the other parameters. The third, and probably the most attractie, method is to change the domain of consideration. For example, one could switch from the common-offset domain to the domain of offset dip. This method implies a transformation similar to slant stacking of common-midpoint gathers in the postmigration domain in order to obtain the local offset dip information. Equation 30 transforms, with the help of the results from

8 Velocity Continuation 1657 Appendix A, to the form with and 3 = τ sin θ cos α sin θ, 48 1 cos α = 1 +, 49 1 sin α = For a constant offset dip tan θ = /, the dynamic analog of equation 48 is the third-order partial differential equation cot θ 3 P 3 P 3 +t 3 P + t 3 P = Equation 51 does not strictly comply with the theory of second-order linear differential equations. Its properties and practical applicability require further research. CONCLUSIONS I hae deried kinematic and dynamic equations for residual time migration in the form of a continuous elocity continuation process. This deriation explicitly decomposes prestack elocity continuation into three parts corresponding to zerooffset continuation, residual NMO, and residual DMO. These three parts can be treated separately both for simplicity of theoretical analysis and for practical purposes. It is important to note that in the case of a 3D migration, all three components of elocity continuation hae different dimensionality. Zerooffset continuation is fully 3D. It can be split into two D continuations inline the inline and crossline directions. Residual DMO is a D common-azimuth process. Residual NMO is a 1D single-trace procedure. The dynamic properties of zero-offset elocity continuation are precisely equialent to those of conentional poststack migration methods such as Kirchhoff migration. Moreoer, the Kirchhoff migration operator coincides with the integral solution of the elocity continuation differential equation for continuation from the zero elocity plane. This rigorous theory of elocity continuation gies us new insights into the methods of prestack migration elocity analysis. Extensions to the case of depth migration in a ariable elocity background are deeloped by Liu and McMechan 1996 and Adler 00. A practical application of elocity continuation to migration elocity analysis is demonstrated in the companion paper Fomel, 003b, where the general theory is used to design efficient and practical algorithms. ACKNOWLEDGMENTS This work was completed when the author was a member of the Stanford Exploration Project SEP at Stanford Uniersity. The financial support was proided by the SEP sponsors. I thank Bee Bednar, Biondo Biondi, Jon Claerbout, Sergey Goldin, Bill Harlan, Daid Lumley, and Bill Symes for useful and stimulating discussions. Paul Fowler, Hugh Geiger, Samuel Gray, and one anonymous reiewer proided aluable suggestions that improed the quality of the paper. 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9 1658 Fomel Rothman, D. H., Lein, S. A., and Rocca, F., 1985, Residual migration Applications and limitations: Geophysics, 50, Saa, P., 003, Prestack residual migration in the frequency domain: Geophysics, 68, Saa, P., and Fomel, S., 003, Angle-domain common-image gathers by waefield continuation methods: Geophysics, 68, Schleicher, J., Hubral, P., Hocht, G., and Liptow, F., 1997, Seismic constant-elocity remigration: Geophysics, 6, Schneider, W. A., 1978, Integral formulation for migration in twodimensions and three-dimensions: Geophysics, 43, Shurtleff, R. N., 1984, An f-k procedure for prestack migration and migration elocity analysis: Presented at the 46th Ann. Mtg., Eur. Assn. Geosci. Eng. Stolt, R. H., 1978, Migration by Fourier transform: Geophysics, 43, , A prestack residual time migration operator: Geophysics, 61, Tygel, M., Schleicher, J., and Hubral, P., 1994, Pulse distortion in depth migration: Geophysics, 59, Yilmaz, O., and Chambers, R. E., 1984, Migration elocity analysis by wae-field extrapolation: Geophysics, 49, Yilmaz, O., and Claerbout, J. F., 1980, Prestack partial migration: Geophysics, 45, Yilmaz, O., 1979, Prestack partial migration: Ph.D. diss., Stanford Uniersity. APPENDIX A DERIVING THE KINEMATIC EQUATIONS The main goal of this appendix is to derie the partial differential equation describing the image surface in a depthmidpoint-offset-elocity space. z = The deriation starts with obsering a simple geometry of reflection in a constant-elocity medium, shown in Figure A-1. The well-known equations for the apparent slowness, s = sin α 1, A-1 r = sin α, A- relate the first-order traeltime deriaties for the reflected waes to the emergence angles of the incident and reflected rays. Here, s stands for the source location at the surface, r is the receier location, t is the reflection traeltime, is the constant elocity, and α 1 and α are the angles shown in Figure A-1. Considering the traeltime deriatie with respect to the depth of the obseration surface z shows that the contributions of the two branches of the reflected ray, added together, form the equation z = cos α 1 + cos α. A-3 It is worth mentioning that the elimination of angles from equations A-1, A-, and A-3 leads to the famous doublesquare-root equation, 1 s + 1 r, A-4 published in the Russian literature by Belonosoa and Aleksee 1967 and commonly used in the form of a pseudodifferential dispersion relation Clayton, 1978; Claerbout, 1985 for prestack migration Yilmaz, 1979; Popoici, Considered locally, equation A-4 is independent of the constant elocity assumption and enables recursie prestack downward continuation of reflected waes in heterogeneous isotropic media. Introducing the midpoint coordinate x = s + r/ and halfoffset h = r s/, one can apply the chain rule and elementary trigonometric equalities to formulas A-1 and A- and transform these formulas to = s + r = r s sin α cos θ =, A-5 cos α sin θ =, A-6 where α = α 1 + α / is the dip angle, and θ = α α 1 / is the reflection angle Clayton, 1978; Claerbout, Equation A-3 transforms analogously to z cos α cos θ =. A-7 This form of equation A-3 is used to describe the stretching factor of the waeform distortion in depth migration Tygel et al., Diiding equations A-5 and A-6 by equation A-7 leads to z = tan α, A-8 z = tan θ. A-9 FIG. A-1. Reflection rays in a constant elocity medium a scheme. Equation A-9 is the basis of the angle-gather construction of Saa and Fomel 003. Substituting formulas A-8 and A-9 into equation A-7 yields yet another form of the doublesquare-root equation: z = 1 + z 1 + z 1, A-10

10 Velocity Continuation 1659 which is analogous to the dispersion relationship of Stolt prestack migration Stolt, The law of sines in the triangle formed by the incident and reflected ray leads to the explicit relationship between the traeltime and the offset: t = h cos α 1 + cos α sinα α 1 = h cos α sin θ. A-11 An algebraic combination of formulas A-11, A-5, and A-6 forms the basic kinematic equation of the offset continuation theory Fomel, 003a: t + 4h = ht 4 +. A-1 Differentiating A-11 with respect to the elocity yields = hcos α sin θ. A-13 Finally, diiding equation A-13 by equation A-7 produces z = h cos θ sin θ. A-14 Equation A-14 can be written in a ariety of ways with the help of an explicit geometric relationship between the halfoffset h and the depth z, sin θ cos θ h = z cos α sin θ, A-15 which follows directly from the trigonometry of the triangle in Figure A-1 Fomel, 003a. For example, equation A-14 can be transformed to the form obtained by Liu and Bleistein 1995: z = z cos α sin θ = z cos α 1 cos α. A-16 In order to separate different factors contributing to the elocity continuation process, one can transform this equation to the form z = z cos α + h z 1 tan α tan θ + h z z z. = z z A-17 Rewritten in terms of the ertical traeltime τ = z/, it further transforms to equation = τ + h 1 4, A-18 3 τ equialent to equation 1 in the main text. Yet another form of the kinematic elocity continuation equation follows from eliminating the reflection angle θ from equations A-14 and A-15. The resultant expression takes the following form: z = z + h z + h sin α + z cos α = z cos α + h. A-19 z + h sin α + z APPENDIX B DERIVATION OF THE RESIDUAL DMO KINEMATICS This appendix deries the kinematical laws for the residual NMO+DMO transformation in the prestack offset continuation process. The direct solution of equation 31 is nontriial. A simpler way to obtain this solution is to decompose residual NMO+DMO into three steps and to ealuate their contributions separately. Let the initial data be the zero-offset reflection eent τ 0 x 0. The first step of the residual NMO+DMO is the inerse DMO operator. One can ealuate the effect of this operator by means of the offset continuation concept Fomel, 003a. According to this concept, each point of the input traeltime cure τ 0 x 0 traels with the change of the offset from zero to h along a special trajectory, which I call a time ray. Time rays are parabolic cures of the form xτ = x 0 + τ τ0 x 0 τ 0 x 0 τ 0 x 0, B-1 with the final points constrained by the equation h = τ τ τ 0 x 0 τ 0 x 0 τ 0 x 0, B- where τ 0 x 0 is the deriatie of τ 0 x 0. The second step of the cumulatie residual NMO+DMO process is the residual normal moeout. According to equation 3, residual NMO is a one-trace operation transforming the traeltime τ to τ 1 as follows: where τ 1 = τ + h d, 1 d = B-3. B-4 The third step is DMO corresponding to the new elocity 1. DMO is the offset continuation from h to zero offset along the redefined time rays Fomel, 003a x τ = x + hx τ τ1 H 1 τ, B-5 where H = 1 /, and X = 1 /. The end points of the time rays B-5 are defined by the equation τ τ 1 H = τ 1 hx. B-6 The partial deriaties of the common-offset traeltimes are constrained by the offset continuation kinematic equation hh X = τ 1 H, B-7

11 1660 Fomel which is equialent to equation A-1 in Appendix A. Additionally, as follows from equations B-3 and the ray inariant equations from Fomel 003a, τ 1 X = τ = τ τ 0 x 0 τ 0 x 0. B-8 Substituting equations B-1 B-4 and B-7 B-8 into equations B-5 and B-6 and performing the algebraic simplifications yields the parametric expressions for elocity rays of the residual NMO+DMO process: x d = x 0 + h τ 0 x 0 1 T T τd = τ 1 d T d, T d, where the function T h,τ 0 x 0,τ 0 x 0 is defined by B-9 T h,τ,τ x = τ + τ +4h τx, B-10 T d = T h,τ1 d,τ 0 x 0Th,τ 0 x 0,τ 0 x 0, B-11 and τ 1 d = τ 0T + dh. B-1 The last step of the cascade of inerse DMO, residual NMO, and DMO is illustrated in Figure B-1. The three plots in the figure show the offset continuation to zero offset of the inerse DMO impulse response shifted by the residual NMO operator. The middle plot corresponds to zero NMO shift, for which the DMO step collapses the waefront back to a point. Both positie top plot and negatie bottom plot NMO shifts result in the formation of the specific triangular impulse response of the residual NMO+DMO operator. As noticed by Etgen 1990, the size of the triangular operators dramatically decreases with the time increase. For large times pseudodepths of the initial impulses, the operator collapses to a point corresponding to the pure NMO shift. FIG. B-1. Kinematic residual NMO+DMO operators constructed by the cascade of inerse DMO, residual NMO, and DMO. The impulse response of inerse DMO is shifted by the residual NMO procedure. Offset continuation back to zero offset forms the impulse response of the residual NMO+DMO operator. Solid lines denote traeltime cures; dashed lines denote the offset continuation trajectories time rays. Top 1 / 0 = 1.. Middle 1 / 0 = 1; the inerse DMO impulse response collapses back to the initial impulse. Bottom 1 / 0 = 0.8. The half-offset h in all three plots is 1 km. APPENDIX C INTEGRAL VELOCITY CONTINUATION AND KIRCHHOFF MIGRATION The main goal of this appendix is to proe the equialence between the result of zero-offset elocity continuation from zero elocity and conentional poststack migration. After soling the elocity continuation problem in the frequency domain, I transform the solution back to the time- and-space domain and compare it with the conentional Kirchhoff migration operator Schneider, The frequency-domain solution has its own alue, because it forms the basis for an efficient spectral algorithm for elocity continuation Fomel, 003b. Zero-offset migration based on elocity continuation is the solution of the boundary problem for equation 41 with the boundary condition P =0 = P 0, C-1 where P 0 t 0, x 0 is the zero-offset seismic section, and Pt, x, is the continued waefield. In order to find the solution of the boundary problem composed of equations 41 and C-1, it is conenient to apply the function transformation Rt, x,=tpt, x,, the time coordinate transformation σ = t / and, finally, the double Fourier transform oer

12 Velocity Continuation 1661 the squared time coordinate σ and the spatial coordinate x: ˆR = Pt,x, expi σ ikxt dtdx. C- With the change of domain, equation 41 transforms to the ordinary differential equation d ˆR d = i k ˆR, C-3 and the boundary condition C-1 transforms to the initial alue condition where ˆR 0 = ˆR0 = ˆR 0, P 0 t 0,x 0 expi σ 0 ikx 0 t 0 dt 0dx 0, C-4 C-5 and σ 0 = t0 /. The unique solution of the initial alue Cauchy problem C-3 C-4 is easily found to be ˆR = ˆR 0 exp i k. C-6 We can see that, in the transformed domain, elocity continuation is a unitary phase-shift operator. An immediate consequence of this remarkable fact is the cascaded migration decomposition of poststack migration Larner and Beasley, 1987: exp i k 1 + +n =exp i k 1 exp i k n. C-7 Analogously, 3D poststack migration is decomposed into the two-pass procedure Jakubowicz and Lein, 1983: exp i k 1 + k = exp i k 1 exp i k. C-8 The inerse double Fourier transform of both sides of equality C-6 yields the integral conolution operator Pt, x,= P 0 t 0,x 0 Kt 0,x 0 ;t,x,dt 0 dx 0, C-9 with the kernel K defined by K = t / 0 t exp i k π m+1 + ikx x 0 i t t 0 dkd, C-10 where m is the number of dimensions in x and k m equals 1 or. The inner integral on the waenumber axis k in formula C- 10 is a known table integral Gradshtein and Ryzhik, Ealuating this integral simplifies equation C-10 to the form / t 0 t K = i m/ m/+1 [ m i exp t 0 t x x 0 ] d. C-11 The term i m/ is the spectrum of the anti-causal deriatie operator d/dσ of the order m/. Noting the equialence m/ 1 m/ 1 m/ m/ = =, C-1 σ t t which is exact in the 3D case m = and asymptotically correct in the D case m = 1, and applying the conolution theorem, we can transform operator C-9 to the form 1 cos α Pt, x, = π m/ 0 ρ m/ m/ ρ P 0, x 0 dx 0, C-13 where ρ = t + x x 0, and cos α = t 0 /t. Operator C-13 coincides with the Kirchhoff operator of conentional poststack time migration Schneider, 1978.