Stanford Exploration Project, Report 124, April 4, 2006, pages 31 45

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1 Stanford Exploration Project, Report 14, April 4, 006, pae

2 Stanford Exploration Project, Report 14, April 4, 006, pae Generalized Riemannian wavefield extrapolation Jeff Shrae ABSTRACT Thi paper extend wavefield extrapolation to eneralized Riemannian pace. The key component i the development of a diperion relationhip appropriate for propaatin wavefield on eneralized non-orthoonal mehe. Thi wavenumber contain a number of mixed-domain field in addition to velocity that repreent coordinate ytem eometry. An extended plit-tep Fourier approximation of the extrapolation wavenumber i developed, which provide accurate reult when multiple reference parameter et are ued. Three example are preented that demontrate the validity of the theory. An important conequence i that reater emphai can be placed on eneratin moother computational mehe rather than atifyin retrictive emi-orthoonal criteria. Thi reult hould lead to more accurate and efficient eneralized Riemannian wavefield extrapolation. INTRODUCTION Riemannian wavefield extrapolation (RWE) (Sava and Fomel, 005) eneralize wavefield extrapolation to non-carteian coordinate ytem. The oriinal formulation aumed that coordinate ytem are at leat emi-orthoonal and characterized by an extrapolation direction orthoonal to the other two axe. Thi uppoition reulted in a wave-equation diperion relationhip for the extrapolation wavenumber containin mixed-domain field additional to velocity that encode coordinate ytem eometry. However, emi-orthoonal eometry can be an overly retrictive aertion becaue many computational mehe have reatly varyin mixed-domain coefficient that caue numerical intability durin wavefield extrapolation. Initially, RWE wa deined for dynamic application where wavefield are extrapolated on ray-baed coordinate ytem oriented in the wave propaation direction. Thi approach enerally enerate hih-quality Green function; however, numerical intability (i.e. zero diviion) occur wherever the ray coordinate ytem triplicate. Sava and Fomel (005) addreed thi iue by iteratively moothin the velocity model until coordinate ytem triplication vanih. Thi olution, thouh, i omewhat le than ideal becaue it counter the oriinal purpoe of RWE: coordinate ytem conformal to propaation direction. A econd more eometric RWE application i performin wavefield extrapolation to and from urface of irreular eometry. Shrae and Sava (004) formulate a wave-equation miration from toporaphy tratey that poe wavefield extrapolation directly in locally orthoonal mehe conformal to the acquiition urface. Althouh ucceful in area with loner wavelenth and lower amplitude toporaphy, imain reult deraded in ituation involv- 31

3 3 Shrae SEP 14 in more rued acquiition toporaphy. However, a more eneral obervation i the enetic link between deraded imae quality and the rid compreion/extenion demanded by emiorthoonality. A olution to thee two problem i to extend RWE to include non-orthoonal coordinate ytem. Thi eneralized RWE (GRWE) framework remove the emi-orthoonal contraint to allow propaation in non-orthoonal Riemannian pace. Non-orthoonality introduce two additional term in the GRWE diperion relationhip and make exitin coefficient term lihtly more involved. The non-tationary coefficient in the reultin extrapolation wavenumber can be handled with an extended plit-tep Fourier approach (Stoffa et al., 1990). Importantly, thi olution afford reater flexibility in coordinate ytem dein while facilitatin more rapid meh eneration. Furthermore, reater emphai can be placed on optimizin rid quality by controllin rid cluterin and eneratin moother coefficient field (Shrae, 006b). Thi paper develop the 3D wave-equation diperion relationhip for performin GRWE. I firt dicu eneralized Riemannian eometry and how how the acoutic wave-equation can be formulated in a non-orthoonal Riemannian pace. Subequently, I develop an expreion for a one-way wavefield extrapolation wavenumber and preent the correpondin plit-tep Fourier approximation. I then preent two analytic -D non-orthoonal coordinate ytem to help validate the developed extrapolation wavenumber expreion. The paper conclude with a more realitic example of GRWE enerated Green function etimate throuh a lice of the SEG-EAGE alt model. GENERALIZED RIEMANNIAN GEOMETRY Geometry in a eneralized 3D Riemannian pace i decribed by a ymmetric metric tenor, i j = ji, that relate the eometry in a non-orthoonal coordinate ytem, {x 1, x, x 3 }, to an underlyin Carteian meh, {ξ 1,ξ,ξ 3 } (Guenheimer, 1977). In matrix form, the metric tenor i written, i j = = , (1) where 11, 1,, 13, 3 and 33 are function linkin the two coordinate ytem throuh, 11 = x k ξ 1 x k ξ 1, 13 = x k ξ 1 x k ξ 3, 1 = x k ξ 1 x k ξ, 3 = x k ξ x k ξ 3, = x k ξ x k ξ, 33 = x k ξ 3 x k ξ 3. () (Summation notation - ii = i ued in equation throuhout thi paper.) The aociated (or invere) metric tenor, i j, i defined by i j = i j, where i metric tenor

4 SEP 14 GRWE 33 matrix determinant. The aociated metric tenor i iven by, i j = , (3) and ha the followin metric determinant, = 33 ( 11 1 ) ( 11 1 ). (4) Weihted metric tenor, m i j = i j, i a ueful definition for the followin development. ACOUSTIC WAVE-EQUATION IN 3D GENERALIZED RIEMANNIAN SPACES The acoutic wave-equation for wavefield, U, in a eneralized Riemannian pace i, U = ω (x)u, (5) where the ω i frequency, i the propaation lowne, and i the Laplacian operator, U = 1 ( m i j U ). (6) ξ i ξ j Subtitutin equation 6 into 5 enerate a Helmholtz equation appropriate for propaatin wave throuh a 3D pace, ( 1 m i j U ) = ω U. (7) ξ i ξ j Expandin the derivative term and multiplyin throuh by yield, Definin n j a, m i j ξ i U + m i j U = ω U. (8) ξ j ξ i ξ j n j = mi j ξ i = m1 j ξ 1 lead to a more compact notation of equation 8, + m j ξ + m3 j ξ 3, (9) U n j + m i j U = ω U. (10) ξ j ξ i ξ j Developin a wave-equation diperion relation i achieved by replacin the partial differential operator actin on wavefield U with their Fourier domain dual, ( m i j k ξi in j ) kξj = ω, (11) where k ξi i the Fourier domain dual of differential operator ξ i. Equation 11 repreent the diperion relationhip for wavefield propaation on a eneralized 3-D Riemannian pace.

5 34 Shrae SEP 14 One-way wavefield extrapolation Developin an expreion for the extrapolation wavenumber require iolatin one of the wavenumber in equation 11 (herein aumed to be coordinate ξ 3 ). Rearranin the reult of expandin equation 11 by introducin indicie i, j = 1,,3 yield, k ξ 3 + ( m 13 k ξ1 + m 3 k ξ i ) kξ3 = ω +i ( n 1 k ξ1 + n k ξ ) m 11 k ξ 1 m k ξ m 1 k ξ1 k ξ. (1) An expreion for wavenumber k ξ3 can be obtained by completin the quare, ( ) k ξ3 + (m13 k ξ1 +m 3 k ξ i ) ( ) m = ω k 33 ξ 1 m 11 (m13 ) (13) ( ) ( ) ) ) kξ m (m3 ) k ξ1 k ξ m 1 m13 m 3 +i k ξ1 (n 1 m13 +i k ξ (n m3 n 3. Iolatin wavenumber k ξ3 yield, k ξ3 = a 1 k ξ1 a k ξ +ia 3 ± a 4 ω a 5 k ξ 1 a 6 k ξ a 7 k ξ1 k ξ + a 8 i k ξ1 + a 9 i k ξ a 10 1, (14) where a i are non-tationary coefficient iven by, a = ( ) ( ) ( 33 ) n 1 m13 n3 ( ) n m3 n3 ( ) T. (15) Note that the coefficient contain a mixture of m i j and i j term, and that poitive definite term, a 4,a 5,a 6 and a 10 in equation 14, are quared uch that the familiar Carteian plit-tep Fourier correction i recovered below. Split-Step Fourier Approximation The extrapolation wavenumber defined in equation 14 and 15 cannot be implemented purely in the Fourier domain due to the preence of mixed-domain field (i.e. a function of both ξ 1 and k ξ1 imultaneouly). Thi can be addreed uin an extended verion of the plit-tep Fourier approximation (Stoffa et al., 1990), a popular approach that ue Taylor expanion to eparate k ξ3 into two part: k ξ3 kξ P S 3 + kξ SSF 3. Wavenumber kξ P S 3 and kξ SSF 3 repreent a pure Fourier domain phae-hift and a mixed ω x domain plit-tep correction, repectively. The phae-hift term i iven by, k P S ξ 3 = b 1 k ξ1 b k ξ +ib 3 ± b 4 ω b 5 k ξ 1 b 6 k ξ b 7 k ξ1 k ξ + b 8 i k ξ1 + b 9 i k ξ b 10 1, (16) where b i = b i (ξ 3 ) are reference value of a i = a i (ξ 1,ξ,ξ 3 ). The plit-tep approximation i developed by performin a Taylor expanion about each coefficient a i and evaluatin the

6 SEP 14 GRWE 35 reult at tationary reference value b i. The tationary value of k ξ1 and k ξ are aumed to be zero. Thi lead to a plit-tep correction of, kξ SSF 3 = k ξ 3 a 3 (a 3 b 3 ) + k ξ 3 0 a 4 (a 4 b 4 ) + k ξ 3 0 a 10 (a 10 b 10 ), (17) 0 where 0 denote "with repect to a reference medium". The partial differential expreion in equation 17 are, k ξ3 a 3 = b 3, 0 k ξ3 b 4 ω a =, 4 0 b4 ω b10 reultin in the followin plit-tep Fourier correction, k SSF k ξ3 a 10 ξ 3 = i b 3 (a 3 b 3 ) + b 4 ω (a 4 b 4 ) b4 ω b10 = 0 b 10 b 10 ω b 10, (18) b 10 (a 10 b 10 ). (19) b 4 ω b10 Note that we could ue many reference media followed by interpolation imilar to the phaehift plu interpolation (PSPI) technique of Gazda and Suazzero (1984). Importantly, even thouh there are additional a i coefficient in the diperion relationhip, thee can be made mooth throuh meh reularization uch that fewer et of reference parameter are needed to accurately repreent wavenumber k ξ3. In addition, ituation exit where ome coefficient are zero or neliible. For example, the coefficient for a weakly non-orthoonal coordinate ytem (i.e. max { 1, 13, 3 } << max{ 11,, 33 }) within a kinematic approximation reduce to, a / / 33 / / T. (0) Additional pecial cae are preented in Appendix A. EXAMPLE 1 - -D SHEARED CARTESIAN COORDINATES An intructive example i a coordinate ytem formed by a hearin action on a Carteian meh (ee fiure 1). A heared Carteian coordinate ytem i defined by, x 1 x x 3 = 1 0 co θ in θ ξ 1 ξ ξ 3, (1) where θ i the hear anle of the coordinate ytem (θ = 90 i Carteian). Thi ytem reduce to a more workable et of two equation, x1 x 3 = 1 coθ 0 inθ ξ1 ξ 3, ()

7 36 Shrae SEP 14 Fiure 1: -D heared Carteian coordinate. Left panel: Phyical domain repreented by heared Carteian coordinate defined by {x 1, x 3 }; Riht panel: GRWE domain choen to be the unit quare {ξ 1,ξ }. jeff1-dexamp NR X 3 θ Phyical Space X 1 ξ 3 Canonical Space ξ 1 that ha a metric tenor i j iven by, i j = xk ξ 1 x k ξ 1 x k x k ξ 1 ξ 3 x k x k ξ 1 ξ 3 x k x k ξ 3 ξ 3 11 = = 1 coθ coθ 1, (3) with a determinant = in θ and an aociated metric tenor i j iven by, i j = 1 in θ 1 co θ co θ 1. (4) Note that becaue the tenor in equation 4 i coordinate invariant, equation 10 implifie to, U = i j and enerate the followin diperion relation, U ξ i ξ j = ω U, (5) i j k ξi k ξj = ω. (6) Expandin out thee term lead to an expreion for wavenumber k ξ3, ( k ξ3 = k ξ 1 ± ω ( ) ) kξ 1. (7) Subtitutin the value of the aociated metric tenor in equation 4 into equation 7 yield, k ξ3 = coθ k ξ1 ± inθ ω k ξ 1. (8) A numerical tet uin a Carteian coordinate ytem heared at 5 from vertical i hown in fiure. The backround velocity model i 1500 m 1 and the zero-offet data conit of 4 flat plane-wave t = 0., 0.4, 0.6 and 0.8. A expected, the zero-offet miration reult imae reflector at depth z=300, 600, 900, and 100 m. Note that the propaation ha created boundary artifact: thoe on the left are reflection due to a truncated coordinate ytem while thoe on the riht are hyperbolic diffraction caued by truncated plane-wave.

8 SEP 14 GRWE 37 Fiure : Sheared Carteian coordinate ytem tet. Coordinate ytem hear anle and velocity are θ = 5 and 1500 m 1, repectively. Zerooffet data conit of 4 flat planewave impule at t = 0.,0.4,0.6 and 0.8 that are correctly imaed at depth z = 300,600,900, and 100 m. jeff1-ray0 ER EXAMPLE - POLAR ELLIPSOIDAL A econd intructive example i a tretched polar coordinate ytem (ee fiure 3). A polar ellipoidal coordinate ytem i pecified by, x1 x 3 a(ξ3 )ξ = 1 coξ 3 a(ξ 3 )ξ 1 inξ 3. (9) Parameter a = a(ξ 3 ) i a mooth function controllin coordinate ytem ellipticity and ha curvature parameter b = a ξ 3 and c = a. The metric tenor i j i, ξ 3 i j = a ξ 1 a b ξ 1 a b ξ 1 (b + a ), (30) with determinant = a 4 ξ1. The aociated metric and weihted aociated metric tenor are iven by, i j = b +a b a 4 a 3 ξ 1 b 1 a 3 ξ 1 a ξ1 and m i j = ξ1 (b +a ) a b a b a 1 ξ 1. (31) Tenor i j and m i j pecify a wavenumber appropriate for extrapolatin wavefield on a -D non-orthoonal meh (ee equation A-7). However, becaue the coordinate ytem i patially variant, we mut alo compute the n i field: n 1 = a +b ac and n a 3 = 0. Inertin thee value yield the followin extrapolation wavenumber k ξ3, k ξ3 = ξ ( 1b a a k ξ 1 ± a ξ1 ω ξ1 k ξ 1 ik ξ1 ξ + b ) ac 1 a. (3)

9 38 Shrae SEP Fiure 3: Polar ellipoidal coordinate ytem example. jeff1-dex NR Depth (m) Ditance (m) The kinematic verion of equatio i, b k ξ3 = ξ 1 a k ξ 1 ± a ω k ξ1, (33) while the orthoonal polar cae (i.e. a = 1) recover the followin, k ξ3 = ±ξ 1 ω k ξ 1. (34) Fiure 4 how an ellipoidal polar coordinate ytem defined by a = ξ ξ 3. The upper left panel how a v(z) = z velocity function overlain by a coordinate ytem meh. The upper riht panel preent velocity model a mapped into the GRWE domain. The data ued in thi tet conited of 4 flat plane-wave. Given thi experimental etup, propaatin flat plane-wave hould not bend in the Carteian domain becaue of the v = v(z) velocity model, even thouh there i velocity variation acro each extrapolation tep in the GRWE domain. Hence, the impule have curvature in the GRWE domain (lower riht panel). The lower left panel how the GRWE domain imain reult mapped back to a Carteian domain. Conitent with theory, the flat plane-wave are imaed a flat reflector. Note that the ede effect are aain caued by coordinate ytem and plane-wave truncation. EXAMPLE 3 - GRWE GREEN S FUNCTION GENERATION The final tet ue GRWE to enerate etimate of Green function. The tet velocity model i a lice of the SEG-EAGE alt model (ee fiure 5). The contrat between the alt body and ediment velocitie caue the wavefield to exhibit complicated propaation includin inificant wavefield triplication and multipathin. The upper left panel how the velocity model with an overlain coordinate meh enerated by the differential method dicued in Shrae (006a). The meh i a ray-baed coordinate ytem becaue the firt and lat extrapolation tep are formed by the 0.04 and.5 travel-time iochron from a firt-arrival Eikonal equation olution. The velocity model in the GRWE domain i illutrated in the upper riht panel. The lower riht panel how the impule repone tet in the GRWE domain. The 7 impulive wave conform fairly well to the travel-time tep, except where they enter the alt body to

10 SEP 14 GRWE 39 Fiure 4: Ellipoidal polar coordinate ytem tet example. Upper left: v(z) = z velocity function overlain by a polar ellipoidal coordinate ytem defined by parameter a = 1+0.ξ ξ3. Upper riht: velocity model in the GRWE domain. Bottom riht: Imaed reflector in GRWE domain. Bottom left: the GRWE domain imae mapped to a Carteian meh. jeff1-polar ER

11 40 Shrae SEP 14 the lower left of the imae. The miration reult mapped back to Carteian pace are hown in the lower left panel. The wavefield to the left of the hot point i fairly complicated and the enery in the alt body and the correpondin upward refracted (and perhap reflected?) wavefield are tronly preent. Fiure 6 preent a comparion tet between GRWE and Carteian extrapolation. Beneath and to the riht of the hot point the wavefield are fairly imilar except for a phae-chane. However, they are inificantly different to the left becaue Carteian-baed extrapolation cannot propaate enery laterally with the ame accuracy and upward at all. Hence, thi enery i abent from the wavefield in the lower panel. CONCLUDING REMARKS Thi paper extend the theory of Riemannian wavefield extrapolation to eneralized 3D nonorthoonal coordinate ytem. The extrapolation wavenumber decouple from the other wavenumber allowin u to ue an extended plit-tep Fourier approximate olution. The example indicate that wavefield can be extrapolated on non-orthoonal coordinate mehe. Thi eneralization allow uer to dein mehe that have moother mixed-domain field that hould lead to more accurate and efficient GRWE. ACKNOWLEDGEMENTS I acknowlede the contribution of Paul Sava and Serey Fomel in layin the roundwork for the current theory and thank to Brad Artman for enlihtenin dicuion. REFERENCES Gazda, J. and P. Suazzero, 1984, Miration of eimic data by phae-hift plu interpolation: Geophyic, 69, Guenheimer, H., 1977, Differential Geometry: Dover Publication, Inc., New York. Sava, P. and S. Fomel, 005, Riemannian wavefield extrapolation: Geophyic, 70, T45 T56. Shrae, J. and P. Sava, 004, Incorporatin toporaphy into wave-equation imain throuh conformal mappin: SEP 117, 7 4. Shrae, J., 006a, Generalized riemannian wavefield extrapolation: SEP 14. Shrae, J., 006b, Structured meh eneration uin differential method: SEP 14. Stoffa, P. L., J. T. Fokkema, R. M. de Luna Freire, and W. P. Keiner, 1990, Split-tep Fourier miration: Geophyic, 55, no. 04,

12 SEP 14 GRWE 41 Fiure 5: Example of GRWE enerated Green Function on tructured non-orthoonal meh for a lice throuh the SEG-EAGE Salt data et velocity model. Top left: Salt model in phyical pace with an overlain ray-baed meh. Top riht: Velocity model in the GRWE domain. Bottom riht: Wavefield propaated in ray-coordinate throuh velocity model hown in the upper riht. Bottom left: Wavefield in bottom riht interpolated back to Carteian pace. jeff1-spexamp ER

13 4 Shrae SEP 14 Fiure 6: Reult of a comparion tet between eneralized Riemannian wavefield extrapolation (top panel) and Carteian-baed extrapolation (bottom panel). jeff1-spcompare ER

14 SEP 14 GRWE 43 APPENDIX A The extrapolation wavenumber developed in equation 14 i appropriate for any non-orthoonal Riemannian eometry. However, there are a number of ituation where ymmetry or partial orthoonality are preent. Moreover, one may wih to make a kinematic approximation where all of the imainary component of the wavenumber are inored. Thee ituation are dicued in thi Appendix. 3D Semi-orthoonal Coordinate Sytem - Semi-orthoonal coordinate ytem occur where one coordinate i orthoonal to the other two coordinate (Sava and Fomel, 005). In thee cae the m 13 and m 3 component of the weihted metric tenor are identically zero, which lead to the followin extrapolation wavenumber, k ξ3 = ia 3 ± a 4 ω a 5 k ξ 1 a 6 k ξ a 7 k ξ1 k ξ +ia 8 k ξ1 +ia 9 k ξ a 10 1, (A-1) where, a = n 1 n T. (A-) which are the coefficient recovered by Sava and Fomel (005). 3-D Kinematic Coordinate Sytem - Wave-equation miration amplitude are enerally inexact in laterally variant media - even in a Carteian baed ytem. Hence, one beneficial approximation that reduce computational cot i to conider only the kinematic term in equation 14. Thi approximate enerate the followin extrapolation wavenumber, k ξ3 = a 1 k ξ1 + a k ξ ± a 4 ω a 5 k ξ 1 a 6 k ξ a 7 k ξ1 k ξ a 10 1, (A-3) where, a = ( ) 33 ( 3 33 ) ( ) T. (A-4) 3-D Kinematic Semi-orthoonal coordinate ytem - Combinin the two above retriction yield the followin extrapolation wavenumber, k ξ3 = ± a 4 ω a 5 k ξ 1 a 6 k ξ a 7 k ξ1 k ξ a 10 1, (A-5) where, a = ( ) 33 ( 3 33 ) ( 33 ) 0 0 T. (A-6)

15 44 Shrae SEP 14 Note that the expreion = 1, and that component of the metric tenor are inificantly implified. -D Non-orthoonal coordinate ytem - Two-dimenional ituation are handled by identifyin ξ = 0. Hence, all derivative in the aociated metric tenor i j with repect coordinate ξ are identically zero. Hence, a -D non-orthoonal coordinate ytem can be repreented by where, a = k ξ3 = a 1 k ξ1 +ia 3 ± a 4 ω a 5 k ξ 1 + a 8 ik ξ1 a 10 1, (A-7) ( ) n 1 0 T. (A-8) -D Non-orthoonal Kinematic Coordinate Sytem - Two-dimenional kinematic ituation are handled throuh identity ξ = 0. Aain, all derivative in the aociated metric tenor i j with repect coordinate ξ are identically zero, and the -D non-orthoonal kinematic extrapolation wavenumber i where, a = k ξ3 = a 1 k ξ1 ± a 4 ω a 5 k ξ 1 a 10 1, (A-9) ( ) T. (A-10) -D Orthoonal Coordinate Sytem - Two-dimenional ituation are handled with ξ = 13 = 0. Accordinly, all derivative in the aociated metric tenor i j with repect coordinate ξ are identically zero, and the -D non-orthoonal coordinate ytem i repreented by where, a = 0 0 k ξ3 = ia 3 ± a 4 ω a 5 k ξ 1 +ia 8 k ξ1 a 10 1, (A-11) ( ) n 1 0 T. (A-1) -D Orthoonal Kinematic Coordinate Sytem - The above two approximation can be combined to yield the followin extrapolation wavenumber for -D orthoonal kinematic coordinate ytem, where, a = k ξ3 = ± a 4 ω a 5 k ξ 1 a 10 1, (A-13) ( ) T. (A-14)