Extended isochron rays in prestack depth (map) migration

Transcription

1 Extended isochron rays in prestack depth (map) migration A.A. Duchkov and M.V. de Hoop Purdue University, 150 N.University st., West Lafayette, IN, (December 15, 2008) Extended isochron rays Running head: Extended isochron rays ABSTRACT Many processes in seismic data analysis and imaging can be identified with solution operators of evolution equations. These include data downward continuation and velocity continuation, where continuation relates to the evolution parameter in such equations. In this paper, we address the question whether isochrons defined by imaging operators can be identified with wavefronts of solutions to an evolution equation. To this end, we consider the double-square-root equation and the space on which its solutions are defined. We refer to this space as the prestack imaging volume, which has coordinates (sub)surface midpoint, offset, (two-way) time and depth. By manipulating the double-square-root operator, we obtain an evolution equation in two-way time describing prestack depth map migration generating extended (pre-stack) images. The propagation of singularities by its solutions is described by extended isochron rays, the corresponding wavefronts of which are extended isochrons. Conventional isochrons are obtained in such a construction by restriction to 1

2 zero subsurface offset. The evolution equation implies a Hamiltonian that generates the extended isochron rays in phase space. 2

3 INTRODUCTION Many processes in seismic data analysis and imaging can be identified with solution operators (propagators) of evolution equations. These include data downward continuation (Clayton (1978); Claerbout (1985); Biondi (2006a)) and velocity continuation (Fomel (1994); Goldin (1994); Hubral et al. (1996); Adler (2002)), where continuation relates to the evolution parameter in such equations. (For a comprehensive theory and framework, see Duchkov et al. (2008); Duchkov and De Hoop (2008).) In this paper, we address the question whether isochrons derived from imaging operators can be identified with wavefronts of solutions to an evolution equation. The role of continuation in imaging, viewed as a mapping of data to an image, is not so obvious. Zero-offset (ZO) depth migration, based on the exploding reflector concept, has been formulated in terms of wave propagation using half-velocities (Lowenthal et al. (1976)) assuming the absence of caustics, while ZO time migration has been formulated in terms of velocity continuation (Fomel, 1994). The geometry of prestack depth migration can be understood in terms of isochrons describing how a surface data sample (viewed as a point source) smears in the subsurface while contributing to an image. Iversen (2004, 2005) studied how these isochrons can be connected via so-called isochron rays. Here, through the introduction of extended isochron rays, we derive a Hamiltonian generating these rays and an associated evolution equation describing prestack depth migration. It is interesting to note that Iversen (2004) proposed different rays connecting isochrons, namely, with what he calls source, receiver and combined parametrizations. We show that it is only the combined parametrization that leads to rays generated by a Hamiltonian. Even when the Hamiltonian is not known one can still determine whether a set of curves 3

4 can be generated by one, namely, by checking an integrability condition. This was done, in the constant background velocity case, for the different parametrizations in Duchkov et al. (2008). Isochron rays were revisited by Silva and Sava (2008) with the purpose to implement a map migration procedure generating common-offset gathers in heterogeneous background velocity models. However, their choice of rays does not admit a Hamiltonian (and associated phase and group velocities) to be constructed. Our Hamiltonian for extended isochron rays implies a method based on a single system of ray tracing equations for prestack depth map migration generating an extended image. Our derivations and constructions make use of a 2n-dimensional prestack imaging volume with coordinates (sub)surface midpoint, offset, (two-way) time and depth (for 3-D seismics, n = 3). This volume appears naturally in the downward-continuation approach to imaging, but is also fundamental in our general geometrical analysis. We consider the double square-root (DSR) equation, an evolution equation in depth. We rewrite the DSR equation in the form of an evolution equation in two-way time. The fronts associated with the solutions to this equation are extended isochrones. Prestack depth (map) migration is then formulated as an initial value problem with the data generating a source in the evolution equation. We can view this fomulation as an extension of the exploding reflector concept from ZO to prestack data using a single evolution equation. An alternative extension, using two wave operators, was proposed by Biondi (2006b). DATA CONTINUATION IN DEPTH: SUMMARY In this section, we summarize data continuation in depth and formulate modelling and imaging, subject to the single scattering approximation, in terms of solving Cauchy initial value problems in depth. The basic operator appearing in these processes is the so-called 4

5 double-square-root (DSR) operator, P DSR = P DSR (z, x s, x r, D s, D r, D t ) (Belonosova and Alekseev (1967); Clayton (1978); Claerbout (1985)). Here, D t corresponds in the Fourier domain with multiplication by frequency, ω, D r corresponds in the Fourier domain with multiplication by wavevector, k r, and D s corresponds in the Fourier domain with multiplication by wavevector, k s. The DSR operator can be viewed as a pseudodifferential operator. Its principal symbol, denoted here by a subscript 1, determines the associated propagation of singularities and ray geometry, that is, the kinematics. To guarantee dynamically correct propagation, one also needs the so-called subprincipal part of the symbol (De Hoop (1996);?);?); Stolk and De Hoop (2005)). We suppress this contribution in the presentation here, the focus being on the geometry. The principal symbol of the DSR operator is given by the standard expression, P DSR 1 (z, x s, x r, k s, k r, ω) = ω 1 c(z, x s ) 2 k s 2 ω 2 + ω 1 c(z, x r ) 2 k r 2 ω 2, (1) while considering ω > 0; z [0, Z] denotes depth, where Z denotes the maximum depth considered. Modelling: Upward continuation Seismic reflection data can be modelled in the Born approximation with an inhomogeneous DSR equation (Stolk and De Hoop (2005), suppressing the (de)composition operators in the flux normalization): ( )[ z ip DSR (z, x s, x r, D s, D r, D t )]u = δ(t)e(z, x s, x r ), u z=z = 0, (2) where E(z, x s, x r ) = δ(x r x s ) 1 2 (c 3 δc) ( z, 1 2 (x r + x s ) ) ; (3) 5

6 δc(z, x) is the function containing the rapid velocity variations representative of the scatterers, superimposed on a smooth background model described by the function c(z, x). Equation (2) is solved in the direction of decreasing z (upward, and forward in time). The data are then modelled by the solution of (2) restricted to z = 0: u(z = 0, x s, x r, t). The corresponding solution operator is schematically illustrated in Fig. 1 (b). In general, we denote the propagator associated with the left-hand side of (2), from depth z to depth z, by H ( z,z). Throughout, we assume that the DSR condition holds: For all source-receiver combinations present in the acquisition geometry the source rays (each connecting a scattering or image point to a source) and receiver rays (each connecting a scattering or image point to a receiver) become nowhere horizontal, see Fig. 1 (a). We note that the DSR condition can be generalized to hold with respect to a curvilinear coordinate system defining a pseudodepth Stolk et al. (2008). DSR rays The propagation of singularities by solutions of (2) is governed by the Hamiltonian, H DSR (z, x s, x r, k z, k s, k r, ω) = (k z P DSR 1 (z, x s, x r, k s, k r, ω)), (4) cf. (1). We denote the DSR rays by X = (x s (z,.), x r (z,.), t(z,.)), K = (k s (z,.), k r (z,.), ω(z,.)), (5) which solve the Hamilton system d(x s, x r, t) dz = HDSR (k s, k r, ω), d(k s, k r, ω) = HDSR dz (x s, x r, t). Here, depth is the evolution parameter. Slowness vectors are obtained from the wavevectors through k/ω. In Fig. 2 (a) we show the typical representation of the geometry as a couple 6

7 of rays in two copies of physical space, viz. one with coordinates (z, x s ) and one with coordinates (z, x r ). In Fig. 2 (b) we show a DSR ray as a single curve defined by (5) in the extended volume with coordinates (x s, x r, z, t), which we will refer to as the prestack imaging volume. A DSR ray considered as a curve in the prestack imaging volume (see Fig. 2 (b)) satisfies the DSR condition if it does not turn in the z or in the t direction. Thus t(z,.) in (5) is a monotonous function in z. This in turn implies that each DSR ray intersects planes z = const and t = const at most once. Thus any of these planes can be chosen to pose an initial value problem for data continuation, and either z or t can be used as an evolution parameter. Imaging: Downward continuation Imaging reflection data, d = d(x s, x r, t), can be formulated as solving the initial value problem, [ z ip DSR (z, x s, x r, D s, D r, D t )]u = 0, u(0, x s, x r, t) = d(x s, x r, t), (6) now to be solved in the direction of increasing z (downward, and backward in time). The image at (z, x) is obtained upon subjecting the solution of (6) to the imaging conditions: u(z, x, x, t = 0). We have that u(z,.,.,.) = H (0,z) d, where denotes taking the adjoint. Through the above formulation, imaging is described to act in the prestack imaging volume as a composition of a continuation operator (in depth) and restriction operators (imaging conditions). 7

8 DATA CONTINUATION IN TWO-WAY TRAVELTIME Here, we discuss an alternative form of data continuation, namely in two-way traveltime rather than depth. To begin with, let, for given (z, x s, x r, k s, k r ), Θ denote the mapping ω k z = P DSR 1 (z, x s, x r, k s, k r, ω) (cf. (4)). Under the DSR condition there exists an inverse mapping, Θ 1 : k z ω = Θ 1 (z, x s, x r, k z, k s, k r ), that solves the equation (Stolk and De Hoop, 2005, Lemma 4.1): k z = P DSR 1 (z, x s, x r, k s, k r, Θ 1 (z, x s, x r, k z, k s, k r )). (7) The inverse mapping also defines the principal symbol, P T W T 1, of a pseudodifferential operator: P T W T 1 (x s, x r, z, k s, k r, k z ) = Θ 1 (z, x s, x r, k z, k s, k r ) = c rc s c 2 k z c k 2 z ( k r 2 k s 2 ) c 2 2 c 2 sc 2 r + kz 2 ( k r 2 c 2 r k s 2 c 2 s) c 2, (8) where c r = c(z, x r ), c s = c(z, x s ), c 2 = c(z, x s ) 2 c(z, x r ) 2 and c 2 + = c(z, x s ) 2 + c(z, x r ) 2. We note that formula (8) admits the limit c 2 0 to be taken. Remark. It appears often useful to transform coordinates from subsurface lateral source and receiver coordinates to subsurface midpoint and offset coordinates, x r = x + h, x s = x h, k s = 1 2 (k x k h ), k r = 1 2 (k x + k h ). (9) This coordinate transform can be applied to the symbol of the DSR operator (Shubin, 2001, Thm. 4.2); we get P DSR 1 (z, x, h, k x, k h, ω) = P DSR 1 (z, x s (x, h), x r (x, h), k s (k x, k h ), k r (k x, k h ), ω). (10) 8

9 Then P T W T 1 (z, x, h, k z, k x, k h ) = c rc s c 2 k z c k x, k h kz 2 c 2 4c 2 sc 2 r ( k x 2 + k h 2 ) k 2 z where, now, c s = c(z, x h) and c r = c(z, x + h). (c 2 )2 + 2 k x, k h kz 2 c 2 + c2, (11) Modelling: Forward continuation We now consider the initial value problem, [ t ip T W T 1 (x s, x r, z, D s, D r, D z )]ũ = 0, ũ(z, x s, x r, 0) = E(z, x s, x r ) (12) (cf. (3)) to be solved in the direction of increasing two-way time t (thus decreasing z). We note that the solution at z = 0 (to leading asymptotic order) models the reflection data, that is, ũ(0, x s, x r, t) = u(0, x s, x r, t). Equation (12) generalizes the notion of exploding reflector modelling zero-offset reflection data to exploding extended reflector modelling pre-stack reflection data. DSR rays revisited The propagation of singularities by solutions of (12) is governed by the Hamiltonian, H T W T (z, x s, x r, ω, k z, k s, k r ) = ω P T W T 1 (z, x s, x r, k z, k s, k r )). (13) Now, we denote the DSR rays by X = (z(t,.), x s (t,.), x r (t,.)), K = (kz (t,.), k s (t,.), k r (t,.)), (14) which solve the Hamilton system d(z, x s, x r ) dt = HT W T (k z, k s, k r ), d(k z, k s, k r ) = HT W T dt (z, x s, x r ). 9

10 Here, two-way time is the evolution parameter. Remark (c(z) models). In the case of a vertically inhomogeneous velocity model, we have c(z, x r ) = c(z, x s ) = c(z). We could use the expressions above, but it is more straightforward to rederive the Hamiltonian directly from (4) and (1), in x s, x r coordinates, that is, H T W T (z, x s, x r, ω, k z, k s, k r ) ( c(z) = ± ω k z 2 k 4 z ( k s 2 k r 2 ) 2 + 2k 2 ) z ( k r 2 + k s 2 ) + 1. (15) Transforming the Hamiltonian from subsurface lateral source and receiver coordinates to subsurface midpoint and offset coordinates, yields H T W T c(z) (z, x, h, ω, k z, k x, k h ) = ω k z 2 k 4 z k x, k h 2 + k 2 z ( k x 2 + k h 2 ) + 1. (16) We note that for constant background velocity, c(z) = v say, equation (15) reduces to (Sava, 2003, (3)); equation (16) reduces to the Fourier domain counterpart of (Fomel, 2003, (A-10)). Imaging: Backward continuation Imaging reflection data, d = d(x s, x r, t), can be formulated as solving the initial value problem, ( )[ t ip T W T 1 (x s, x r, z, D s, D r, D z )]ũ = δ(z)d(x s, x r, t), ũ t=t = 0, (17) to be solved in the direction of decreasing two-way time t (thus increasing z). The extended image at (z, x s, x r ) is obtained upon subjecting the solution to the imaging condition: ũ(z, x s, x r, t = 0). We have that ũ(.,.,., 0) = T 0 H (t,0) d(.,., t ) dt if H ( t,t) is the propagator, 10

11 from time t to time t, associated with the left-hand side of (12). Continuation in two-way time is schematically illustrated in Fig. 3. Remark (exploding reflectors). We use the Hamiltonians for data continuation in two-way traveltime and depth to revisit the exploding reflector model used in the early development of seismic imaging. For a vertically inhomogeneous velocity model, c(z), the Hamiltonian H T W T in (16) does not depend on h, and thus the phase variable k h remains constant in the course of downward continuation ( dk h dt = HT W T h = 0). For zero sourcereceiver offset (ZO) surface data, k h = 0 (this follows immediately from the symmetry of common midpoint gathers) and it remains zero for all t. Then equation (16) reduces to the Hamiltonian for zero-offset data modelling in the exploding reflector approach by Lowenthal et al. (1976); Claerbout (1985); Cheng and Coen (1984): c(z) ω k z 1 + k x 2 2 kz 2 = H ZO (x, z, ω, k x, k z ), (18) where we recognize the half-velocity 1 2c(z) typical for the exploding reflector model. We note that we describe, here, the exploding reflector model by a first-order evolution equation instead of a second-order wave equation. EXTENDED ISOCHRON RAYS Isochrons form the impulse response of an imaging operator, that is, they are the fronts in extended image space with coordinates, here, (z, x, h), that originate from a particular data sample. We first consider the ray-geometrical impulse response associated with equation (17), which may be constructed by shooting a fan of rays originating at a point source. Using the Hamiltonian in (16), we generate the backward flow (equivalent to (14) but time 11

12 reversed) X(t) = (z(t,.), x(t,.), h(t,.)), K(t) = (kz (t,.), k x (t,.), k h (t,.)), (19) supplemented with the initial conditions: (z(t 0,.), x(t 0,.), h(t 0,.)) = X 0 = (0, x 0, h 0 ), (k z (t 0,.), k x (t 0,.), k h (t 0,.)) = K 0 = (k z0, k x0, k h0 ). While letting K 0 vary in all possible directions, we connect the endpoints, X(t = 0), to form an extended isochron corresponding with two-way time t 0, see Fig. 4 (a) where (z 0, x 0, h 0 ) = (0, 0,.5) (two-dimensional seismics) and t 0 = 1, 3, 5. By increasing t 0 we trace extended isochron rays, see the thin black lines in Fig. 4 (b). Essentially, we have obtained a Hamilton flow description of map migration. We note that H T W T is anisotropic even though the underlying wave equation has been taken in an isotropic medium. The isochron ray tangent vectors define the group velocity being normal to the slowness surface defined by H T W T = 0. Restriction: Conventional isochron rays Conventional isochrons are defined in image space with coordinates (z, x). Functions defined on the image space are obtained from functions defined on the prestack imaging volume by restricting h to zero. This is the way how isochrons are obtained from extended isochrons. The restriction is illustrated in Fig. 4 (b): The fat grey curves are conventional isochrons (t 0 = 1, 3, 5) while the fat black line is a conventional isochron ray. We note that extended isochrons exist for every (positive) t 0, while they will not intersect the h = 0 plane for t 0 less then the direct-wave traveltime from source to receiver. Indeed, conventional isochrons 12

13 do not exist for t 0 < 1 in the model example shown in Fig. 4 (b). We note that this fact causes some complications with initializing conventional isochron rays, see Iversen (2004). Extended isochron rays are solutions to Hamilton equations as described in (19). We now establish the connection between extended isochron rays and conventional isochron rays (in image space) for common (source-receiver) offset migration as defined in Iversen (2004). For simplicity, we will discuss the case of two-dimenional seismics when x and h are one-dimensional. As was mentioned earlier, the direction of an extended isochron ray is determined by the wavevector K 0 = (k z0, k x0, k h0 ). The components of this vector are subject to the constraint that H T W T = 0. We then fix k x0, which constraint can be imposed under the common offset restriction. With these two constraints, we have only one degree of freedom left for the three components of K 0. The resulting one-parameter family of extended isochron rays, for a constant velocity case, is shown in Fig. 4 (b) by thin lines. This family of rays intersects the image space at h = 0 along a smooth curve, illustrated in Fig. 4 (b) by a fat line. This smooth curve is a conventional isochron ray. It is not at all immediate, however, that the curve obtained by connecting the intersection points of extended isochron rays form a ray again. First, we need to to check whether this curve can be parameterized by two-way time, t 0 (the natural evolution parameter for isochron rays). Second, we need to check that varying k x0 provides us with an integrable family of curves. We have carried out these checks for the constant velocity case (Duchkov et al. (2008)) shown in Fig. 4 (b). Indeed, in this case the fat line constructed as described above coincides with a conventional isochron ray in the combined parametrization as introduced by Iversen (2004). On the other hand, we argue that the rays as proposed in Silva and Sava (2008) are not related to solutions to any Hamilton system and hence should not be called rays. 13

14 Extended isochron rays in the presence of caustics We consider a velocity model with a low-velocity lens imbedded in homogeneous medium: c(x, z) = 1.4 e 9(x2 +(1 z) 2). (20) This model generates caustics in the propagating wavefield but still satisfies the DSR condition: The low velocity lens is not strong enough to produce turning rays for comparatively short offsets. Conventional (common-offset) isochrons were constructed for this medium in (Stolk, 2002, Fig. 5, bottom). Isochron shown in that figure corresponds to initial parameters (t 0, x 0, h 0, z 0 ) = (4.73, 0,.2, 0) and has a rather complicated form. The extended isochron surface for these initial data is shown in Fig. 5 (a) and a few extended isochron rays are shown by thin lines in Fig. 5 (b). Besides the fact that extended isochrons have now complicated forms, the way we construct conventional isochrons and isochron rays remains the same. A conventional isochron is an intersection of the extended isochron surface in Fig. 5 (a) with an image plane corresponding to h = 0. This intersection is shown as grey curve in Fig. 5 (b). A better view of this curve is shown in Fig. 6 labelled as t 0 = Another two conventional isochrons are shown with grey curves as well for t 0 = 2 and t 0 = 3.8. One can see that for small two-way times conventional isochron is a connected curve (t 0 = 2). Then a second piece appear around t 0 = 3.8 and for two-way time t 0 = 4.72 one can see 9 smooth branches making up two piece-wise disjoint figures. For obtaining a conventional isochron ray (Iversen, 2004, combined parametrization) we shoot a one-parameter family of extended isochron rays corresponding to a fixed k x0 as shown in Fig. 5 (b) by thin black lines, for k x0 =.2. The thick black line corresponds to a 14

15 conventional isochron ray: it is an intersection of a fan of extended isochron rays with h = 0 plane. A more complicated case of a conventional isochron ray, corresponding to k x0 =.02, is shown in Fig. 6. We observe that it consists of two disjoint branches (thick black lines) that can appear when extended isochron rays intersect the plane h = 0 more than once. Second branch of the conventional isochron ray appears around t 0 = 3.8 when conventional isocnron splits onto two pieces. CONCLUSION We have introduced the notion of extended isochron rays. These rays are defined in a prestack imaging volume with coordinates (sub)surface midpoint, offset, (two-way) time and depth, and are generated by a Hamiltonian. Our construction of this Hamiltonian depends on the DSR condition, which can be generally formulated in terms of curvilinear (sub)surface coordinates (Stolk et al. (2008)). Extended isochrons appear as wavefronts associated with solutions to an evolution equation. We establish how conventional isochrons are obtained from extended isochrons through a restriction to zero subsurface offset. The initialization of extended isochron rays appears as natural as the initialization of rays in geometrical acoustics, unlike the initialization of conventional isochron rays as described in (Iversen, 2004, 2005). In these papers, also, no Hamiltonian was obtained. Indeed, we argue that, through data continuation in (pseudo)depth or two-way time, the prestack imaging volume appears natural in the (geometrical) analysis of imaging operators. The results presented in this paper imply a formulation of prestack map depth migration via the extended image in terms of a Hamiltonian flow. Such a formulation is amenable to a curvelet decomposition of the corresponding imaging operator (Douma and De Hoop 15

16 (2007); de Hoop et al. (2008)). ACKNOWLEDGMENTS The authors would like to thank the members, BP, ConocoPhillips, ExxonMobil, Statoil- Hydro, and Total, of the Geo-Mathematical Imaging Group for financial support. 16

17 REFERENCES Adler, F., 2002, Kirchhoff image propagation: Geophysics, 67, Belonosova, A. and A. Alekseev, 1967, About one formulation of the inverse kinematic problem of seismics for a two-dimensional continuously heterogeneous medium: Some methods and algorithms for interpretation of geophysical data (in Russian), , Nauka. Biondi, B., 2006a, 3D seismic imaging, volume 14: Investigations in Geophysics Series, SEG., 2006b, Prestack exploding-reflectors modeling for migration velocity analysis: Expanded Abstracts, SEG, 25 (1), Cheng, G. and S. Coen, 1984, The relationship between Born inversion and migration of common-midpoint stacked data: Geophysics, 49, Claerbout, J., 1985, Imaging the earth s interior: Blackwell Scientific Publishing. Clayton, R., 1978, Common midpoint migration: Technical Report, Stanford University, SEP-14. De Hoop, M., 1996, Generalization of the Bremmer coupling series: J. Math. Phys., 37, de Hoop, M., H. Smith, G. Uhlmann, and R. van der Hilst, 2008, Seismic imaging with the generalized radon transform: A curvelet transform perspective: Inverse Problems, in print. Douma, H. and M. De Hoop, 2007, Leading-order seismic imaging using curvelets: Geophysics, 72, S231 S248. Duchkov, A. and M. De Hoop, 2008, Velocity continuation in the downward continuation approach to seismic imaging: J. Geoph. Int., in print. 17

18 Duchkov, A., M. De Hoop, and A. Sá Barreto, 2008, Evolution-equation approach to seismic image, and data, continuation: Wave Motion, 45, Fomel, S., 1994, Method of velocity continuation in the problem of seismic time migration: Russian Geology and Geophysics, 35, No. 5, , 2003, Velocity continuation and the anatomy of residual prestack time migration: Geophysics, 68, Goldin, S., 1994, Superposition and continuation of operators used in seismic imaging: Russian Geology and Geophysics, No. 9, Hubral, P., M. Tygel, and J. Schleicher, 1996, Seismic image waves: Geophys. J. Int., 125, Iversen, E., 2004, The isochron ray in seismic modeling and imaging: Geophysics, 69, , 2005, Tangent vectors of isochron rays and velocity rays expressed in global cartesian coordinates: Stud. Geophys. Geod., 49, Lowenthal, D., L. Lu, R. Roberson, and J. Sherwood, 1976, The wave equation applied to migration: Geophysical Prospecting, 24, Sava, P., 2003, Prestack residual migration in the frequency domain: Geophysics, 68, Shubin, M. A., 2001, Pseudodifferential operators and spectral theory: Springer. Silva, E. and P. Sava, 2008, Modeling and imaging with isochron rays: Technical Report, Colorado School of Mines, CWP-602, Stolk, C., 2002, Microlocal analysis of the scattering angle transform: Communications in Partial Differential Equations, 27 (9-10), Stolk, C. and M. De Hoop, 2005, Modeling of seismic data in the downward continuation 18

19 approach: SIAM J. Appl. Math., 65, Stolk, C., M. De Hoop, and W. Symes, 2008, Kinematics of shot-geophone migration: Geophysics, submitted. 19

20 LIST OF FIGURES 1 Upward/downward data continuation. (a) Upward/downward data continuation as solutions to initial value problems (2), (6) with the homogeneous DSR equation; (b) DSR modelling operator that solves initial problem (2) with the inhomogeneous DSR equation. 2 Geometry of the DSR operator for up/downward data continuation. (a) DSR ray viewed as a pair of conventional rays; (b) DSR ray viewed as solution to the DSR Hamilton equations (cf. (5)) with X defining position on a ray and K defining its orientation. 3 Forward/backward two-way time continuation for modelling and imaging. (a) Modelling as a solution of initial problem (12); (b) imaging as a solution of initial problem (17). 4 Notion of (extended) isochrons. (a) Extended isochrons for two-way traveltimes t 0 = 1, 3, 5; (b) Conventional isochrons (thick grey lines) and isochron rays (thick black lines) in the image space obtained from extended isochrons and rays by restricting h to zero. 5 Extended isochrons and rays in case of caustics. (a) Extended isochron for two-way traveltime t 0 = 4.72; (b) Conventional isochron (thick grey line) and isochron ray (thick black line) in the image space corresponding to slice h = 0; corresponding extended isochron rays are shown by thin black lines. 6 Conventional (common-offset) isochrons and an isochron ray in the presence of caustics. Grey lines - conventional isocrons as intersection of extended isochron surfaces with h = 0 plane. They correspond to two-way time t 0 = 2, 3.8 and 4.7 seconds (from top to bottom); note that the lowest one is identical to the one in (Stolk (2002),Fig. 5, bottom). Thick black lines - conventional isochron ray as intersection of a fan of extended isochron rays (for fixed k x0 =.02) with h = 0 plane. 20

21 Figure 1: Upward/downward data continuation. (a) Upward/downward data continuation as solutions to initial value problems (2), (6) with the homogeneous DSR equation; (b) DSR modelling operator that solves initial problem (2) with the inhomogeneous DSR equation. Duchkov & De Hoop Extended isochron rays 21

22 Figure 2: Geometry of the DSR operator for up/downward data continuation. (a) DSR ray viewed as a pair of conventional rays; (b) DSR ray viewed as solution to the DSR Hamilton equations (cf. (5)) with X defining position on a ray and K defining its orientation. Duchkov & De Hoop Extended isochron rays 22

23 Figure 3: Forward/backward two-way time continuation for modelling and imaging. (a) Modelling as a solution of initial problem (12); (b) imaging as a solution of initial problem (17). Duchkov & De Hoop Extended isochron rays 23

24 Figure 4: Notion of (extended) isochrons. (a) Extended isochrons for two-way traveltimes t 0 = 1, 3, 5; (b) Conventional isochrons (thick grey lines) and isochron rays (thick black lines) in the image space obtained from extended isochrons and rays by restricting h to zero. Duchkov & De Hoop Extended isochron rays 24

25 Figure 5: Extended isochrons and rays in case of caustics. (a) Extended isochron for twoway traveltime t 0 = 4.72; (b) Conventional isochron (thick grey line) and isochron ray (thick black line) in the image space corresponding to slice h = 0; corresponding extended isochron rays are shown by thin black lines. Duchkov & De Hoop Extended isochron rays 25

26 Figure 6: Conventional (common-offset) isochrons and an isochron ray in the presence of caustics. Grey lines - conventional isocrons as intersection of extended isochron surfaces with h = 0 plane. They correspond to two-way time t 0 = 2, 3.8 and 4.7 seconds (from top to bottom); note that the lowest one is identical to the one in (Stolk (2002),Fig. 5, bottom). Thick black lines - conventional isochron ray as intersection of a fan of extended isochron rays (for fixed k x0 =.02) with h = 0 plane. Duchkov & De Hoop Extended isochron rays 26