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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI6-6 1 Illumination, resolution, and incidence-angle in PSDM: A tutorial Isabelle Lecomte NORSAR, R&D Seismic Modelling, P.O.Box 53, 2027 Kjeller, Norway ?

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI6-6 2 Hubble telescope: space-variant PSF* *Point-Spread Functions Space-variant PSF!

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI6-6 3 Point-Spread Functions in Marmousi* *Marmousi model courtesy of IFP Seismics: PSF may be very space-variant!

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI6-6 4 Resolution, illumination, …etc! *http://www.lenna.org, **Liner (2000), and Monk (2002) * Acoustic/elastic impedance Reflection ~ contrasts! ** PSDM … at best! ! Not 1D convolution!

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI6-6 5 Content Introduction Image formation in PSDM Scattering wavenumber: the key! Resolution Illumination Examples Controlling imaging Conclusions

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI6-6 6 Waves! Scattering Imaging in PSDM: K is the key! Getting data Incident wave Waves! Imaging Key information: Scattering Wavenumber! Imaging 2 Focusing ? Back propagation Waves! 1 Wave propagation corrections G,G: GF (*) ● : GF-node (*) GF: Green’s Function Migration

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI6-6 7 Scattering isochrones Common shot (x = 0) ■ Common offset (0 m) ■ Model: constant velocity Data: point scatterer data ellipse circle ■ ● point scatterer PSDM

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI6-6 8 PSDM and point scatterer ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ■■■■■■■■■■■■■■■ ● Common offset (0 m) Common shot (x = 0) 1 trace ∑ traces Same point scatterer… …different PSDM images!

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI6-6 9 PSF and PSDM: why? scattering structures = set of point scatterers (e.g., exploding reflector concept, etc) PSDM(point scatterer) = Point-Spread Function If PSF known: PSDM image = Reflectivity * PSF Question 1: how to get PSF without generating synthetic point scatterers at each image point? Question 2: how to use PSF to understand and improve PSDM?

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI Content Introduction Image formation in PSDM Scattering wavenumber: the key! Resolution Illumination Examples Controlling imaging Conclusions

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI Methods: ”ray-tracing” based Green’s functions –Paraxial ray tracing –Wavefront Construction –Eikonal solver PSDM (~Kirchhoff) –Diffraction Stack (DS) –Local Imaging (LI) 1 GF-node only! ”SimPLI” (*) –Simulated Prestack Local Imaging No seismic records needed! (*) patent pending ●

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI Scattering Wavenumber K Incident wavenumber Definition at a local “Scattering Object” (diffraction, reflection,..) scattered wavenumber Easy to calculate with ray tracing and similar Calculation performed in the PSDM velocity model

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI K: which parameters? source sgeophone g - V s : incident wave velocity - V g : scattered wave velocity - ŝ and ĝ: unit vectors - frequency - V P : P-velocity - V S : S-velocity If V s = V g (no wave conversion) ”incidence” angle = 0 ║ĝ – ŝ║ = 2 ●● ŝĝ K ● ŝ ĝ K ”incidence” angle ≠ 0 ║ĝ – ŝ║ < 2

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI From K to PSF using FFT 2D FFT -1 PSF Z X Green’s Functions at one GF-node ● Marmousi ● -Kx max.+Kx max. -Kz max. 0. max./2 0 module no data! ● 2D FFT ●

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI K and scattering isochrones K corresponds to a local plane wavefront approximation of the scattering isochrone K is perpendicular to the scattering isochrone ║K ║ = f( ) : pulse effect [K][K] PSF

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI Content Introduction Image formation in PSDM Scattering wavenumber: the key! Resolution Illumination Examples Controlling imaging Conclusions

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI Inverse problem 2 Direct problem 1 Resolution of an inverse problem! Your model! Generalized Inverse d: data m: parameters obs.: observed est.: estimated Data independent! Resolution!

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI [K] for [5-60] Hz s = [0-10] º K and resolution: wavenumber coverage KxKx KZKZ Lateral resolution ~ 2 / K X Vertical resolution ~ 2 / K Z 1 Marmousi model Courtesy of IFP

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI K and PSF: no data! PSF K K PSDM of point scatterer and PSF PSDM – data from point scatterer Common offset (0 m) Common shot (x = 0) high R low R K mean

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI Content Introduction Image formation in PSDM Scattering wavenumber: the key! Resolution Illumination Examples Controlling imaging Conclusions

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI K and reflection ”P-to-S” reflection ”P-to-P” reflection Reflector From source incident ray In the PSDM velocity model: - A given couple (k s,k g ) may correspond to an actual reflection. - it is the case IF there is a reflector perpendicular to K at the GF-node. s incidence angle ss ss To geophone reflected ray g scattering angle gg gg

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI [K] for [5-60] Hz s = [0-10] º Illuminated dips 2 ~ 45 º ~ 25 º K and illumination: dip Marmousi Model Courtesy of IFP Marmousi model Courtesy of IFP

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI Content Introduction Image formation in PSDM Scattering wavenumber: the key! Resolution Illumination Examples Controlling imaging Conclusions

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI Hz 120 Hz [K] Playing with the pulse Spectrum Target model (Vp)Reflectivity10 Hz SimPLI 20 Hz SimPLI 30 Hz SimPLI 40 Hz SimPLI

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI Fault 4 km offset Reflectivity = 1 FFT +1 “Green’s Functions” Illumination and resolution: illustration Fault [K] incl. 20 Hz pulse 0 km offset Fault PSF FFT -1 PSF FFT -1 SimPLI – 0 km offset FFT -1 SimPLI – 4 km offset Fault

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI Incidence-angle in PSDM SimPLI Image: 00°-05°SimPLI Image: 05°-15°SimPLI Image: 15°-25°SimPLI Image: 25°-35°SimPLI Image: 35°-45° [K] Filter : 00°-05°[K] Filter : 05°-15°[K] Filter : 15°-25°[K] Filter : 25°-35°[K] Filter: 35°-45°Reflectivity : 00°-05°Reflectivity : 05°-15°Reflectivity : 15°-25°Reflectivity: 25°-35°Reflectivity: 35°-45° Final SimPLI Image – 20 Hz Σ

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI Overburden effects ● A KPSF Good resolution Good illumination ● B KPSF Poor resolution Bad illumination Not illuminated!

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI PSDM images: not a simple 1D convolution! Elastic impedance (x,z) KXKX KZKZ “1D” PSDM ! No illumination effects! KXKX KZKZ 2D Filter: 0 km offset KXKX KZKZ 2D Filter: 4 km offset This is PSDM effects! Function of survey, overburden, pulse, wave- phases, local velocity.

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI Content Introduction Image formation in PSDM Scattering wavenumber: the key! Resolution Illumination Examples Controlling imaging Conclusions

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI Image and survey sampling K PSF SimPLI shot: 12.5 m K PSF SimPLI shot: 125 m K PSF SimPLI shot: 625 m

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI ”blind!” automatic corrections Controlling imaging: check local K! Irregular Sampling! Blind!Controlled!

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI Conclusions Define your PSDM velocity model… –Should be smooth in the imaging zone… –… but can have layers with contrast outside! …then use the scattering wavenumbers! –Prior or after imaging –Survey planning mode –Resolution/illumination analyses –Controlling and improving imaging –Understanding image formation –Testing the validity of interpretation results Flexible and fast! –Ray tracing based –FFT

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2006 SEG 76 th Annual Meeting – 10/5/2006 – SPMI Acknowledgements Research Council of Norway (projects /420, /43, and /420) Statoil (Gullfaks), IFP (Marmousi), Seismic Unix, and the “Svalex” project (www.svalex.net, Storvola) Håvar Gjøystdal, Åsmund Drottning and Ludovic Pochon-Guerin. Thanks

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