1. © 2013, PARADIGM. ALL RIGHTS RESERVED. Long Offset Moveout Approximation in Layered Elastic Orthorhombic Media Zvi Koren and Igor Ravve

2. © 2013, PARADIGM. ALL RIGHTS RESERVED. Locally 1D Orthorhombic Layered Model  Multi-layer orthorhombic structure  Different azimuthal orientation at each layer  Common vertical axis  Wave type: Compressional  In 1D model, the magnitude and the azimuth of horizontal slowness are the same for all layers 2

3. © 2013, PARADIGM. ALL RIGHTS RESERVED. Layer Parameters  Thickness and Vertical compressional velocity:  Vertical shear-to-compression velocity ratio:  Thomsen-like parameters:  Azimuthal orientation: 3

4. © 2013, PARADIGM. ALL RIGHTS RESERVED. Effective Model  Effective model is presented by a single layer with azimuthal anisotropy that yields the same moveout as the original layered medium, for any magnitude and azimuth of the horizontal slowness 4

5. © 2013, PARADIGM. ALL RIGHTS RESERVED. Direct & Parametric Presentation of Azimuthally Dependent Normal Moveout  Direct NMO vs. surface azimuth/offset:  Direct NMO vs. phase velocity azimuth/offset:  NMO vs. phase velocity azimuth/horizontal slowness: 5 Lengthwise – offset along phase azimuth Transverse – offset normal to Lengthwise

6. © 2013, PARADIGM. ALL RIGHTS RESERVED. Classical Moveout Approximation for ORT  Approximation for VTI, Tsvankin & Thomsen (1994)  Choices for effective horizontal velocity:  Alkhalifah and Tsvankin, 1995: 6 works for azimuthal anisotropy as well

7. © 2013, PARADIGM. ALL RIGHTS RESERVED. Why Parametric Approximation?  Fractional moveout approximation has the asymptotic correction factor in the denominator of the nonhyperbolic term, A  While it works perfectly for compressional waves in VTI layered medium, it may lead to negative correction factor A for ORT medium  It never happens in ORT planes of symmetry, but may occur for some azimuths in between 7

8. © 2013, PARADIGM. ALL RIGHTS RESERVED. Effective Model for Short Offsets  Short-offset effective model has 8 coefficients: 3low-order coefficients: fast and slow NMO velocities and slow azimuth 5high order coefficients: three effective anellipticities and two additional azimuths 8

9. © 2013, PARADIGM. ALL RIGHTS RESERVED. Short-Offset Model for Fixed Phase Azimuth  Generally (for any azimuth), there are eight short- offset moveout coefficients  For a fixed phase velocity azimuth, there are only one low-order and one high-order short-offset coefficients 9

10. © 2013, PARADIGM. ALL RIGHTS RESERVED. Effective Model for Fixed Azimuth  The short-offset coefficients are related to power series expansion for infinitesimal horizontal slowness  To accurately describe the moveout, expansion coefficients are needed also in the proximity of the critical slowness (long-offset parameters) 10

11. © 2013, PARADIGM. ALL RIGHTS RESERVED. Long-Offset Parameters: Per Azimuth  Short-offset parameters are computed in the proximity of the vertical direction, it has no azimuth  Long-offset parameters are computed for a proximity of a horizontal direction, characterized by a fixed azimuth. We compute them per azimuth 11

12. © 2013, PARADIGM. ALL RIGHTS RESERVED. Separation of Long-Offset Parameters  Two long-offset coefficients enforce convergence of moveout to correct asymptote for infinite offsets  Unlike short-offset parameters related to all layers, the two long-offset parameters are separated  The first is related to the “fast” layer (with fastest horizontal velocity for given phase azimuth) and controls the tilt of the asymptote  The second is related to all other (“slow”) layers and controls the elevation of the asymptote 12

13. © 2013, PARADIGM. ALL RIGHTS RESERVED. Gluing (Combining) the Coefficients  With the given short-offset and long-offset coefficients, we can “glue” them into a unique continuous function for the whole feasible range of the horizontal slowness  Expansions of the synthetized function into a power series for the infinitesimal horizontal slowness and in the proximity of the critical slowness yield the required computed coefficients 13

14. © 2013, PARADIGM. ALL RIGHTS RESERVED. Short-Offset Expansions  Expansions of three moveout components for infinitesimal horizontal slowness 14

15. © 2013, PARADIGM. ALL RIGHTS RESERVED. Short-Offset Coefficients 15

16. © 2013, PARADIGM. ALL RIGHTS RESERVED. Parameters of Global Effective Model 8 global effective parameters, needed to compute short-offset coefficients, depend on elastic properties, azimuthal orientations and thickness of the layers in a multi-layer structure 3 low-order parameters 5 high-order parameters 16

17. © 2013, PARADIGM. ALL RIGHTS RESERVED. Fourth Order NMO Velocity vs. Phase Azimuth 17 Using the parametric forms of moveout components we derived the fourth order NMO velocity

18. © 2013, PARADIGM. ALL RIGHTS RESERVED. Fourth Order Velocity Kernel  Kernel has a bilinear form:  L is a row vector of length 5 that includes the 4 th order effective parameters,  P is a column vector of length 7 that depends on the phase velocity azimuth alone,  L and M are independent of  M is a matrix of dimension 5x7, whose components depend on the three low order parameters only, in a nonlinear way 18

19. © 2013, PARADIGM. ALL RIGHTS RESERVED. 5x7 Matrix Needed for Fourth-Order Velocity (Columns 1 to 3) 19

20. © 2013, PARADIGM. ALL RIGHTS RESERVED. 5x7 Matrix (Columns 4 to 7) 20

21. © 2013, PARADIGM. ALL RIGHTS RESERVED. Long-Offset Expansions  Expansions of three moveout components in the proximity of critical slowness  Unbounded term includes the small value in the denominator 21

22. © 2013, PARADIGM. ALL RIGHTS RESERVED. How We Compute Contribution of Slow Layers  Contribution of “slow” layers in the lengthwise and transverse offset components and traveltime is computed per slowness azimuth: Assume in the “fast” layer propagation occurs in the horizontal plane, zenith angle 90 deg. Applying Snell’s law, compute zenith angle of the phase velocity for each “slow” layers Given phase velocity direction, compute for each “slow” layer 22

23. © 2013, PARADIGM. ALL RIGHTS RESERVED. Contribution of Fast Layer into Moveout  We assume that the phase velocity direction in the local orthorhombic frame of the “fast” layer is, where the vertical component is infinitesimal, and the horizontal components are  is the local ORT axis of the “fast” layer 23

24. © 2013, PARADIGM. ALL RIGHTS RESERVED. Contribution of Fast Layer (Continued)  The horizontal slowness in the “fast” layer is  Performing the infinitesimal analysis, we obtain the phase velocity, the polarization vector, the ray velocity components, the components of the lateral propagation and the traveltime vs. the infinitesimal parameter 24

25. © 2013, PARADIGM. ALL RIGHTS RESERVED. From Coefficients at Two Ends of Slowness Interval to Combined Continuous Moveout  The offset components and the traveltime are approximated for the whole feasible range of horizontal slowness with continuous functions  Coefficients of continuous functions are obtained by combining short- and long-offset coefficients 25

26. © 2013, PARADIGM. ALL RIGHTS RESERVED. Combined Moveout Functions  The moveout approximation functions are 26

27. © 2013, PARADIGM. ALL RIGHTS RESERVED. Relationship between Short & Long Offset Coeff. and Parameters of Combined Functions 27 lengthwise offset component transverse offset component traveltime

28. © 2013, PARADIGM. ALL RIGHTS RESERVED. Intercept Time  With the parametric functions, we obtain the moveout approximation in domain in a straightforward way  The horizontal slowness has no transverse component while the offset has both components  The intercept time simplifies to 28

29. © 2013, PARADIGM. ALL RIGHTS RESERVED. Test for Multi-Layer Structure 29 #δ1δ1 δ2δ2 δ3δ3 ε1ε1 ε2ε2 γ1γ1 γ2γ2 V, km/sfΔz, kmφ ax o 10.15 00.20 0.06 2.00.720.5VTI 20.12-0.080.070.18-0.150.03-0.033.00.740.520 30.09-0.10-0.060.16-0.120.10-0.084.80.750.5110 40.13-0.08-0.070.15-0.140.08-0.093.00.760.560 50.16-0.170.050.12-0.170.05-0.063.50.780.5140

30. © 2013, PARADIGM. ALL RIGHTS RESERVED. Lengthwise and Transverse Offset Components vs. Slowness for Constant Phase Azimuth 30

31. © 2013, PARADIGM. ALL RIGHTS RESERVED. Lag between Acquisition Azimuth and Phase Velocity Azimuth 31 Const. reflection angle, varying azimuthConst. phase azm, varying reflection angle

32. © 2013, PARADIGM. ALL RIGHTS RESERVED. Traveltime vs. Slowness & its Error for Constant Phase Velocity Azimuth 32 traveltime vs. slownesserror of traveltime vs. slowness

33. © 2013, PARADIGM. ALL RIGHTS RESERVED. Traveltime vs. Offset for Constant Phase Azimuth: Parametric Model and Alkhalifah Strong & Weak 33 traveltime vs. offseterror of traveltime vs. offset

34. © 2013, PARADIGM. ALL RIGHTS RESERVED. Lengthwise and Transverse Offset Components vs. Phase Azimuth for Constant Reflection Angle 34 lengthwise & transverse offset componentserror of offset components

35. © 2013, PARADIGM. ALL RIGHTS RESERVED. Traveltime vs. Phase Azimuth for Constant Refl. Angle: Parametric and Alkhalifah Strong & Weak 35 traveltime vs. phase velocity azimutherror of traveltime vs. phase azimuth

36. © 2013, PARADIGM. ALL RIGHTS RESERVED. Effective Velocities and Effective Anellipticity Diagrams vs. Phase Velocity Azimuth 36 effective NMO velocities V 2 & V 4 and fast layer’s horizontal ray velocity exact effective anellipticity (S) and its weak anisotropy approximation (W)

37. © 2013, PARADIGM. ALL RIGHTS RESERVED. Effective Anellipticity, Critical Reflection Angle and Horizontal Slowness vs. Phase Velocity Azimuth 37 effective anellipticity and its weak anisotropy approximation vs. azimuth slowness for constant reflection angle and critical reflection angle vs. azimuth

38. © 2013, PARADIGM. ALL RIGHTS RESERVED. Updating 3D Background Azimuthally Anisotropic Layered Model  Input: Background 3D layered orthorhombic model Migrated image and 3D Common Image Gathers o Surface azimuth and offset domain o Subsurface phase azimuth and opening angle or horizontal slowness  Residual moveout analysis: From azimuthally dependent residual moveout, defined for each horizon, we obtain eight residual global effective parameters per horizon 38

39. © 2013, PARADIGM. ALL RIGHTS RESERVED. 39 Generalized Dix Inversion  Compute the global effective parameters of the background orthorhombic layered model  Add the global residual effective parameters to the global background effective parameters, and obtain the updated global effective parameters, attached to horizons  Dix inversion: For each layer, use the top and bottom global effective parameters, related to horizons, to obtain the local effective parameters related to individual layers  For each layer, compute local interval velocity parameters (non-unique solution, may require additional assumptions)

40. © 2013, PARADIGM. ALL RIGHTS RESERVED. Conclusions  We derived new asymptotic correction of the moveout approximation for ORT layered media  The approximation has the same power series expansion of the moveout components as the moveout of the original multi-layer package for infinitesimal slowness and nearly critical slowness 40

41. © 2013, PARADIGM. ALL RIGHTS RESERVED. Conclusions (continued)  For infinitesimal slowness, we keep two terms of the moveout series per azimuth (and vertical time)  For nearly critical slowness, we keep two terms of the moveout series per azimuth as well  One long-offset term characterizes the propagation through the layer with the fastest horizontal velocity, while the other term describes the propagation through the “slower” layers 41

42. © 2013, PARADIGM. ALL RIGHTS RESERVED. Conclusions (continued)  The approximation is parametric: lengthwise and transverse offset components and traveltime are functions of horizontal slowness and its azimuth  Parametric functions allow approximating the moveout in both t-x and tau-p domains  For wide opening angles the asymptotic correction terms are essential to match the exact ray tracing 42