1. paul@sep.stanford.edu Amplitude-preserved wave-equation migration Paul Sava & Biondo Biondi SEP108 (pages 1-27)

2. paul@sep.stanford.edu Wave-equation imaging Why? –Complex wavefields –Sharp velocity variation sub-salt What? –Reflectivity function of incidence angle Imaging Migration Velocity Analysis (MVA) Amplitude vs. Angle Analysis (AVA)

3. paul@sep.stanford.edu Angle-Domain Common Image Gathers Applications –imaging –S/G migration (Prucha et at., 1999) –shot-profile migration (Rickett, 2001) –seismic inversion (Prucha et. al., 2001) –MVA –traveltime tomography (Clapp, 2000) –wave-equation MVA (Sava & Biondi, 2000) –C-waves –polarity reversal (Rosales, 2001) –AVA –wave-equation AVA (Gratwick, 2001)

4. paul@sep.stanford.edu Angle-gathers vs. offset-gathers Offset gather Angle gather

5. paul@sep.stanford.edu Agenda ADCIG kinematics image space data space Amplitude-preserved migration general formulation weighting function COMAZ ADCIG amplitudes spatial bandwidth temporal bandwidth RTT Applications true-amplitude migration inversion WEMVA

6. paul@sep.stanford.edu Reflection scheme: global view SourceReceiver V(x,y,z)  

7. paul@sep.stanford.edu Reflection scheme: local view   2h v 

8. paul@sep.stanford.edu ADCIG methods Reflection angleOffset ray-parameter k-domain (RTT) x-domain (slant-stack)

9. paul@sep.stanford.edu ADCIG: example

10. paul@sep.stanford.edu ADCIG methods: comparison Reflection angleOffset ray-parameter indirectly –function of dip directlyReflection angle less sensitivesensitiveInaccurate velocity boundaries data space –mixed with migration image space –separated from migration Computation domain

11. paul@sep.stanford.edu Agenda ADCIG kinematics image space data space Amplitude-preserved migration general formulation weighting function COMAZ ADCIG amplitudes spatial bandwidth temporal bandwidth RTT Applications true-amplitude migration inversion WEMVA

12. paul@sep.stanford.edu Spatial bandwidth khkh kzkz  max  max kzkz  +90-90  max  max

13. paul@sep.stanford.edu Synthetic: ideal gather frequency domainspace domainamplitude

14. paul@sep.stanford.edu Temporal bandwidth imageangle gather dataoffset gather wide frequency band narrow frequency band  kzkz khkh kzkz khkh  khkh kzkz

15. paul@sep.stanford.edu Temporal bandwidth frequency domainspace domainamplitude

16. paul@sep.stanford.edu RTT implementation Two possibilities: –push: loop over input –pull: loop over output khkh kzkz kzkz  angle gather offset gather

17. paul@sep.stanford.edu push RTT offset-gatherangle-gather k-domain x-domain

18. paul@sep.stanford.edu pull RTT offset-gatherangle-gather k-domain x-domain

19. paul@sep.stanford.edu RTT amplitudes

20. paul@sep.stanford.edu Agenda ADCIG kinematics image space data space Amplitude-preserved migration general formulation weighting functions COMAZ ADCIG amplitudes spatial bandwidth temporal bandwidth RTT Applications true-amplitude migration inversion WEMVA

21. paul@sep.stanford.edu Amplitude-preserving migration Definition: the process of recovering the amplitude of the reflectivity vector given –perfect data –infinite bandwidth –infinite aperture

22. paul@sep.stanford.edu Modeling operator L: modeling operator A: Amplitude operator G: Reflection operator i 0 : seismic image r: reflectivity d: seismic data

23. paul@sep.stanford.edu Amplitude operator Clayton & Stolt (1981) L: modeling operator A: amplitude operator G: Reflection operator i 0 : seismic image r: reflectivity d: seismic data

24. paul@sep.stanford.edu Reflection operator L: modeling operator A: amplitude operator G: reflection operator i 0 : seismic image r: reflectivity d: seismic data Clayton & Stolt (1981) Stolt & Benson (1986)

25. paul@sep.stanford.edu Amplitude-preserving operator L: modeling operator A: amplitude operator G: reflection operator i 0 : seismic image r: reflectivity d: seismic data

26. paul@sep.stanford.edu Weighting operator modelingmigration

27. paul@sep.stanford.edu Agenda ADCIG kinematics image space data space Amplitude-preserved migration general formulation weighting functions COMAZ ADCIG amplitudes spatial bandwidth temporal bandwidth RTT Applications true-amplitude migration inversion WEMVA

28. paul@sep.stanford.edu Amplitude correction: the problem frequency domainspace domainamplitude

29. paul@sep.stanford.edu Jacobian: general expression image space data space

30. paul@sep.stanford.edu Jacobian: 2-D, image space  2h v 

31. paul@sep.stanford.edu Jacobian: general expression image space data space

32. paul@sep.stanford.edu Jacobian: 2-D, data space  2h v 

33. paul@sep.stanford.edu Jacobian: 2-D, flat reflectors (Wapenaar et al., 1999)

34. paul@sep.stanford.edu Amplitude correction: the problem frequency domainspace domainamplitude

35. paul@sep.stanford.edu AVA: correct amplitudes frequency domainspace domainamplitude

36. paul@sep.stanford.edu Agenda ADCIG kinematics image space data space Amplitude-preserved migration general formulation weighting function COMAZ ADCIG amplitudes spatial bandwidth temporal bandwidth RTT Applications true-amplitude migration inversion WEMVA

37. paul@sep.stanford.edu COMAZ: stationary-phase view from above 2-DCOMAZ

38. paul@sep.stanford.edu Amplitude component Phase-shift component COMAZ: stationary-phase correction

39. paul@sep.stanford.edu COMAZ: no amplitude corrections

40. paul@sep.stanford.edu COMAZ: all amplitude corrections

41. paul@sep.stanford.edu Agenda ADCIG kinematics image space data space Amplitude-preserved migration general formulation weighting function COMAZ ADCIG amplitudes spatial bandwidth temporal bandwidth RTT Applications true-amplitude migration inversion WEMVA

42. paul@sep.stanford.edu True-amplitude migration L: modeling operator A: amplitude operator G: reflection operator i 0 : seismic image r: reflectivity d: seismic data

43. paul@sep.stanford.edu True-amplitude migration: COMAZ OPERATORS L: modeling W: Jacobian A: amplitude A stat : stationary-phase G: reflection

44. paul@sep.stanford.edu True-amplitude migration: real data

45. paul@sep.stanford.edu Inversion: pseudo-unitary operators InversionMigration

46. paul@sep.stanford.edu Inversion: preconditioned regularization

47. paul@sep.stanford.edu Wave-equation MVA L: Wave-equation MVA m: slowness perturbation d: image perturbation References: SEP100, SEP103, SEP105

48. paul@sep.stanford.edu WEMVA: model

49. paul@sep.stanford.edu WEMVA: correct amplitudes

50. paul@sep.stanford.edu WEMVA: incorrect amplitudes

51. paul@sep.stanford.edu Summary The goal –Reflectivity function of reflection angle The means –correct ADCIG transformations –kinematics –amplitudes –correct migration amplitude

52. paul@sep.stanford.edu Applications true-amplitude migration seismic inversion AVA wave-equation MVA

53. paul@sep.stanford.edu