SEG/New Orleans 2006 Annual Meeting. Non-orthogonal Riemannian wavefield extrapolation Jeff Shragge, Stanford University

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1 Non-orthogonal Riemannian wavefield extrapolation Jeff Shragge, Stanford University SUMMARY Wavefield extrapolation is implemented in non-orthogonal Riemannian spaces. The key component is the development of a dispersion relationship appropriate for propagating wavefields on generalized non-orthogonal meshes. This wavenumer contains a numer of mixed-domain fields in addition to velocity that represent coordinate system geometry. An extended split-step Fourier approximation of the extrapolation wavenumer is developed that provides accurate results when multiple reference parameters sets are used. Three examples are presented that demonstrate the validity of the theory. An important consequence is that greater emphasis can e placed on generating smoother computational meshes, rather than satisfying restrictive semi-orthogonal criteria, leading to more accurate and efficient Riemannian wavefield extrapolation. INTRODUCTION Riemannian wavefield extrapolation (RWE) (Sava and Fomel, 005) generalizes wavefield extrapolation to non-cartesian coordinate systems. The original implimentation assumed that coordinate systems are at least semi-orthogonal and characterized y an extrapolation direction orthogonal to the other two axes. This resulted in a wave-equation dispersion relationship for the extrapolation wavenumer containing mixed-domain fields additional to velocity that encode coordinate system geometry. However, semi-orthogonal geometry can e an overly restrictive assertion ecause many computational meshes have greatly varying mixed-domain coefficients that cause numerical instaility during wavefield extrapolation. Initially, RWE was designed for dynamic applications where wavefields are extrapolated on ray-ased coordinate systems oriented in the wave propagation direction. This approach generally generates high-quality Green s functions; however, numerical instaility (i.e. zero-division) occurs wherever the ray coordinate system triplicates. Sava and Fomel (005) address this issue y iteratively smoothing the velocity model until these triplications vanish. This solution, though, somewhat counters the original purpose of RWE: coordinate systems conformal to the propagation direction. A more geometric RWE application is performing wavefield extrapolation to and from surfaces of irregular geometry. Shragge and Sava (005) formulate a wave-equation migration from topography strategy that poses wavefield extrapolation directly in locally orthogonal meshes conformal to the acquisition surface. Although successful in areas with longer wavelength and lower amplitude topography, imaging results degrade in situations involving more rugged acquisition topography. However, a more general oservation is the genetic link etween degraded image quality and the grid compression/extension demanded y semi-orthogonal geometry. A solution to these prolems is eliminating the semi-orthogonal constraint y allowing wavefield propagation in non-orthogonal Riemannian coordinate spaces. Non-orthogonality introduces two additional terms in the RWE dispersion relationship and makes existing coefficient terms more complex. Non-stationary coefficients in the resulting extrapolation wavenumer can e handled with an extended split-step Fourier approach (Stoffa et al., 990). Importantly, this solution affords greater flexiility in coordinate system design while facilitating more rapid mesh generation. Furthermore, greater emphasis can e placed on optimizing grid quality y controlling grid clustering and generating smoother coefficient fields. This paper develops a D wave-equation dispersion relationship for performing non-orthogonal RWE. I first discuss generalized Riemannian geometry and show how the acoustic wave-equation can e formulated in a non-orthogonal Riemannian space. Susequently, I develop an expression for a one-way wavefield extrapolation wavenumer and present the corresponding split-step Fourier approximation. I then present two analytic -D non-orthogonal coordinate systems to help validate the developed extrapolation wavenumer expressions. The paper concludes with the example of RWE-generated Green s function estimates through a SEG- EAGE model section. GENERALIZED RIEMANNIAN GEOMETRY Geometry in a generalized D Riemannian space is descried y a symmetric metric tensor, g ij g ji, that relates the geometry in a non-orthogonal coordinate system, {x, x, x }, to an underlying Cartesian mesh, {ξ,ξ,ξ }. The matrix-form metric tensor is, gij g g g g g g g g g g g g, () g g g g where g, g, g, g, g and are functions linking the two coordinate systems through, ( e.g. () g ij x k x k g x ) k x k ξ ξ (Summation notation - e.g. g ii g + g + - is used throughout this paper.) The associated (or inverse) metric tensor, g ij, is defined y g ij g g ij, where g is metric tensor matrix determinant. The associated metric tensor is given y, g ij g g g g g g g g g g g g g g g g g g g, g g g g g g g g g g g () and has the following metric determinant, g (g g g ) g g + g g g g g (g g g ). (4) Weighted metric tensor, m ij g g ij, is a useful definition for the following development. GENERALIZED -D ACOUSTIC WAVE-EQUATION The acoustic wave-equation for wavefield, U, in a generalized Riemannian space is, U ω s (x)u, (5) where the ω is frequency, s is the propagation slowness, and is the Laplacian operator, U ( ) ij U m. (6) g Sustituting equation 6 into 5 generates a Helmholtz equation appropriate for propagating waves through a D space, ( ) ij U m ω s U. (7) g Expanding derivative terms and multiplying y g yields, m ij ξ i U ξ j + m ij U g ω s U. (8) 6

2 Defining n j as, n j mij ξ i m j ξ + m j ξ leads to a more compact notation of equation 8, + m j ξ, (9) U n j + m ij U g ω s U. (0) ξ j The -D wave-equation dispersion relation is developed y replacing the partial differential operators acting on wavefield U with their Fourier-domain duals, (m ij k ξi in j ) k ξj g ω s, () where k ξi is the dual of differential operator ξ. Equation represents the dispersion relationship for wavefield propagation on a i generalized -D Riemannian space. One-way wavefield extrapolation Developing an expression for the extrapolation wavenumer requires isolating one of the wavenumers in equation (herein assumed to e coordinate ξ ). Introducing indicies i, j,,, into equation and rearranging terms yields, m kξ + (m k ξ + m ) k ξ i n k ξ ) g ω s + i (n k ξ + n k ξ m kξ m k ξ m k ξ k ξ. () An expression for wavenumer k ξ can e otained y completing the square and isolating the k ξ contriutions, k ξ a k ξ a k ξ + ia () ± a4 ω a5 k ξ a 6 k ξ a 7 k ξ k ξ + a 8 i k ξ + a 9 i k ξ a0, where a i are non-stationary coefficients given y, a g g n m s g ( g ) g ( g g g n ( ) m m n n ( m ) m m n ( m ) g )... T n m. (4) where i i (ξ ) are reference values of a i a i (ξ,ξ,ξ ). The split-step approximation is developed y performing a Taylor expansion aout each coefficient a i and evaluating the results at stationary reference values i. The stationary values of k ξ and k ξ are assumed to e zero. This leads to a split-step correction of, kξ SSF k ξ ( ) ( ) k a a + ξ ( ) 0 a a a a0 0, (6) 0 0 where 0 denotes "with respect to a reference medium". The partial differential expressions in equation 6 are, a, 0 a 4 ω, ω 0 a 0, ω 0 (7) resulting in the following split-step Fourier correction, kξ SSF i (a ) + 4 ω (a 4 4 ) 4 ω 0 0 (a 0 0 ). (8) 4 ω 0 Note that we could use many reference media followed y interpolation similar to the phase-shift plus interpolation (PSPI) technique of Gazdag and Sguazzero (984). Importantly, even though there are additional a i coefficients in the dispersion relationship, these can e made smooth through mesh regularization such that fewer sets of reference parameters are needed to accurately represent wavenumer k ξ. In addition, situations exist where some coefficients are zero or negligile. For example, four coefficients adequately descrie a weakly non-orthogonal coordinate system (i.e. max {g, g, g } << max{g, g, }) within a kinematic (i.e. no imaginary terms) approximation. EXAMPLE - -D SHEARED CARTESIAN COORDINATES An instructive example is a coordinate system is a -D sheared Cartesian mesh defined y, x cosθ x 0 sinθ ξ ξ, (9) where θ is the shear angle of the coordinate system (θ 90 is Cartesian). This system has a metric tensor g ij given y, xk x k gij ξ ξ x k ξ x k ξ x k x k ξ ξ x k x k ξ ξ cosθ cosθ, (0) Note that the coefficients contain a mixture of m ij and g ij terms, and that positive definite terms, a 4,a 5,a 6 and a 0 in equation, are squared such that the familiar Cartesian split-step Fourier correction is recovered elow. Split-Step Fourier Approximation The extrapolation wavenumer defined in equations and 4 cannot e implemented purely in the Fourier domain due to the presence of mixed-domain fields (i.e. a function of oth ξ and k ξ simultaneously). This can e addressed using an extended version of the split-step Fourier approximation (Stoffa et al., 990) that uses Taylor expansions to separate k ξ into two parts: k ξ kξ P S +k SSF ξ. Wavenumers k P S ξ and k SSF ξ represent a pure Fourier domain phase-shift and a mixed ω x domain split-step correction, respectively. The phase-shift term is given y, k P S ξ k ξ k ξ + i (5) ± 4 ω 5 k ξ 6 k ξ 7 k ξ k ξ + 8 i k ξ + 9 i k ξ 0, with determinant g sin θ and associated metric tensor g ij, g ij sin θ cosθ cosθ. () Note that ecause the tensor in equation is spatially invariant, equation 0 simplifies to, U g ij and generates the following dispersion relation, U ω s U, () g ij k ξi k ξj ω s. () Expanding terms leads to an expression for wavenumer k ξ, ( k ξ g k ξ ± s ω g ( ) g ) kξ. (4) 7

3 Sustituting the values of the associated metric tensor in equation into equation 4 yields, k ξ cosθ k ξ ± sinθ s ω k ξ. (5) A numerical test using a Cartesian coordinate system sheared at 5 from vertical is shown in figure. The ackground velocity model is 500 ms and the zero-offset data consist of 4 flat plane-waves t 0., 0.4, 0.6 and 0.8 s. As expected, the zero-offset migration results image reflectors at depths z00, 600, 900, and 00 m. Note that the propagation has created oundary artifacts: those on the left are reflections due to a truncated coordinate system while those on the right are hyperolic diffractions caused y truncated plane-waves. Depth (m) Distance (m) Figure : Polar ellipsoidal coordinate system example. Figure : Sheared Cartesian coordinate system test. Coordinate system shear angle and velocity are θ 5 and 500 ms, respectively. Zero-offset data consist of 4 flat plane-wave impulses at t 0.,0.4,0.6 and 0.8 s that are correctly imaged at depths z 00, 600, 900, and 00 m. EXAMPLE - POLAR ELLIPSOIDAL A second instructive example is a stretched polar coordinate system (see figure ). A polar ellipsoidal coordinate system is specified y, x a(ξ )ξ cosξ. (6) x a(ξ )ξ sinξ Parameter a a(ξ ) is a smooth function controlling coordinate system ellipticity and has curvature parameters ξ a and c a ξ. The metric tensor g ij is, gij a ξ a ξ a ξ ( + a ), (7) with determinant g a 4 ξ. The associated metric and weighted associated metric tensors are given y, g ij +a a 4 a ξ a ξ a ξ and m ij ξ ( +a ) a a a ξ (8) Tensors g ij and m ij specify a wavenumer appropriate for extrapolating wavefields on a -D non-orthogonal mesh. However, ecause the coordinate system is spatially variant, we must also. Figure : Ellipsoidal polar coordinate system test example. Upper left: v(z) z velocity function overlain y a polar ellipsoidal coordinate system defined y parameter a + 0.ξ 0.05ξ. Upper right: velocity model in the RWE domain. Bottom right: Imaged reflectors in RWE domain. Bottom left: the RWE domain image mapped to a Cartesian mesh. compute the n i fields: n a + ac a and n 0. Inserting these values yields the following extrapolation wavenumer k ξ, k ξ ξ ( a a k ξ ± a ξ s ω ξ k ξ ik ξ ξ + ) ac a. (9) Note that the kinematic version of equation 9 is, k ξ ξ a k ξ ± a s ω k ξ, (0) Figure shows an ellipsoidal polar coordinate system defined y a + 0.ξ 0.05ξ. The upper left panel shows a v(z) z velocity function overlain y the coordinate system mesh. The upper right panel presents velocity model as mapped into the RWE domain. The data used in this test consisted of 4 flat plane-waves. Given this experimental setup, propagated flat plane-waves should not end in the Cartesian domain ecause of the v v(z) velocity model, even though there is velocity variation across each extrapolation step in the RWE domain. Hence, the impulses have curvature in the RWE domain (lower right panel). The lower left panel shows the RWE domain imaging results mapped ack to a Cartesian domain. Consistent with theory, the flat planewaves are imaged as flat reflectors. Note that the edge effects are again caused y coordinate system and plane-wave truncation. 8

4 EXAMPLE - RWE GREEN S FUNCTION GENERATION The final test uses RWE to generate Green s function estimates. The test velocity model is a slice of the SEG-EAGE salt model (see figure 4). Velocity contrasts etween the salt ody and sediment cause complex wavefield propagation including significant wavefield triplication and multipathing. The upper left panel shows the velocity model with an overlain coordinate mesh generated y differential meshing (Liseikin, 004). The mesh is a traveltime-ased coordinate system where the first and last extrapolation steps are formed y the 0.04 s and.5 s Eikonal equation isochrons. The velocity model in the RWE domain is illustrated in the upper right. The lower right panel shows the impulse response test in the RWE domain. The 7 impulsive waves conform fairly well to the traveltime steps, except where they enter the salt ody to the lower left of the image. The migration results mapped ack to Cartesian space are shown in the lower left panel. The wavefield to the left of the shot point is fairly complicated and the energy in the salt ody and the corresponding upward refracted (and perhaps reflected?) wavefields are strongly present. Figure 4: Example of RWE generated Green s Functions on structured non-orthogonal mesh for a slice through the SEG-EAGE Salt data set velocity model. Top left: Salt model in physical space with an overlain ray-ased mesh. Top right: Velocity model in the RWE domain. Bottom right: Wavefield propagated in ray-coordinates through velocity model shown in the upper right. Bottom left: Wavefield in ottom right interpolated ack to Cartesian space. Figure 5 presents a comparison test etween RWE and Cartesian extrapolation. Beneath and right of the shot point the wavefields are fairly similar except for a phase-change. However, they are significantly different to the left ecause Cartesian-ased extrapolation neither propagates energy laterally with the same accuracy nor upward at all. Hence, this energy is asent from the wavefield in the lower panel. CONCLUDING REMARKS This paper discusses Riemannian wavefield extrapolation in D non-orthogonal coordinate systems. The extrapolation wavenumer is decoupled from the other wavenumers allowing for an extended split-step Fourier approximation. Examples indicate that wavefields can e extrapolated on non-orthogonal coordinate meshes. Hence, users can concentrate more on mesh design to optimizing mixed-domain field smoothness leading to more accurate RWE. ACKNOWLEDGEMENTS Thanks to Paul Sava and Sergey Fomel for helpful discussions. REFERENCES Gazdag, J. and P. Sguazzero, 984, Migration of seismic data y phase-shift plus interpolation: Geophysics, 49, 4. Liseikin, V., 004, A Computational Differential Geometry Approach to Grid Generation: Springer-Verlag, Berlin. Sava, P. and S. Fomel, 005, Riemannian wavefield extrapolation: Geophysics, 70, T45 T56. Shragge, J. and P. Sava, 005, Wave-equation migration from topography: 75th Ann. Internat. Mtg., Soc. of Expl. Geophys., Expanded Astracts, Stoffa, P. L., J. T. Fokkema, R. M. de Luna Freire, and W. P. Kessinger, 990, Split-step Fourier migration: Geophysics, 55, no. 04, Figure 5: Results of a comparison test etween generalized Riemannian wavefield extrapolation (top panel) and Cartesian-ased extrapolation (ottom panel). 9

5 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list sumitted y the author. Reference lists for the 006 SEG Technical Program Expanded Astracts have een copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the We. REFERENCES Gazdag, J., and P. Sguazzero, 984, Migration of seismic data y phase-shift plus interpolation: Geophysics, 49, 4. Liseikin, V., 004, A Computational Differential Geometry Approach to Grid Generation: Springer-Verlag. Sava, P., and S. Fomel, 005, Riemannian wavefield extrapolation: Geophysics, 70, T45 T56. Shragge, J., and P. Sava, 005, Wave-equation migration from topography: 75th Annual International Meeting, SEG, Expanded Astracts, Stoffa, P. L., J. T. Fokkema, R. M. de Luna Freire, and W. P. Kessinger, 990, Split-step Fourier migration: Geophysics, 55,