1. Resolution

2. 0 km 7 km 0 km 3 km m = L d T r = [L L] m 0 km 7 km T

3. Outline Reflection Resolution Reflection Resolution Transmission Resolution Transmission Resolution FWI Resolution FWI Resolution

4. Rayleigh Resolution L D x = min. separation & distinguishable DxDxDxDx z z zL 4 DxDxDxDx

5. Horizontal Resolution Formula Horizontal Resolution Formula For ZO Migration Intersection of Fresnel Zones Horiz. Res. = Far-Offset Traces

6. Horizontal Resolution Formula Horizontal Resolution Formula For ZO Migration This is the one with largest kx This is the one with largest kz

7. Horizontal Resolution Formula Horizontal Resolution Formula For ZO Migration Intersection of Fresnel Zones Horiz. Res. = Far-Offset Traces What is width of minimum Lateral cut at depth z?

8. Fresnel Zone T/2 L z z zL 4

9. Horizontal Resolution Formula Horizontal Resolution Formula For ZO Migration (Farfield) z   L  tan = = L/z ~  x  x  sin = 4 = L/z ~   x 4 L = ~z Goal: Formula for x  Small angle approx. valid when L>>z

10. Vertical Resolution Formula For ZO Migration Intersection of Fresnel Zones roro

11. Outline Reflection Resolution Reflection Resolution Transmission Resolution Transmission Resolution FWI Resolution FWI Resolution

12. Transmission Resolution Formula Range of src-rec pairs with diving waves that visit the Scatterer are limited compared to diffraction resolution where every src-rec pair has a visiting ray.

13. (1) (2) (3) (4) Plug (4) into (3) to get Transmission Resolution Formula

14. (5) (6) (7) (8) Src+rec integration only for src-rec pairs with rays that visit 1 st Fresnel zone

15. Transmission Resolution Formula (8) (9) (10) (11) Fourier transform in y

16. Transmission Resolution Formula Zero- resolution along raypath. Best resolution is perpendicular to ray

17. Transmission Resolution Formula

18. Conclusions LSM:LSM: m = (L L) L d m = (L L) L dTT.. x d(x, ) A(x,x’)  xx’ = m(x’) Smear reflections along fat circles Far-Offset Traces = Better Hor. Res.Far-Offset Traces = Better Hor. Res. Near-Offset Traces = Better Vert. Res.Near-Offset Traces = Better Vert. Res. m = L d m = L dT ZO Migration:ZO Migration: m = (L L) rm = (L L) rT x 4 L = ~ z /4 =  z /4 =  z.

19. Outline Reflection Resolution Reflection Resolution Transmission Resolution Transmission Resolution FWI Resolution FWI Resolution

20. Exercise Create a 6 point scatterer model, with each pair of point scatterers separated by a wavelength. Generate synthetic ZO data and migrate it. Determine if the resolution formula are validated by measuring the apparent x distance between adjacent point scatterers in the image. Use mesh to display image.  x 2 L = ~z 2 km = 100 m

21. Outline Fresnel Zone Stationary Phase and Fourier-Diffraction Rayleigh Resolution

22. Stationary Phase Integration g(x)cos(  (x))dx ~ C’ g(x*)

23. Stationary Phase Integration g(x)e dx ~ C g(x*) i  f (x)

24. Stationary Phase and Fourier-Diffraction g(x)e dx ~ C g(x*) i  f (x) f (x) = t + t gxgxgxgx xsxsxsxs sg x Non-specular g(x) = e itititit specular gsgsgsgs specular g(x)e ~ i  f (x) e i  t gsgsgsgs specular t + t - t + t - gxgxgxgx xsxsxsxs ] data mig. kernel Fermat’s Principle: x* ~

25. Outline Fresnel Zone Stationary Phase and Fourier-Diffraction Rayleigh Resolution

26. DxDxDxDxL/2x0 xg xz D x = min. separation & distinguishable DxDxDxDx Rayleigh Resolution

27. 1-way time (x-x ) + y 22 2c2c2c2c s rsrsrsrs =  rsrsrsrs + T o ZO Migration ZO Migration Smear Reflections along Fat Circles Smear Reflections along Fat Circlesxs r d(x, )  rsrsrsrs s

28. 1-way time ZO Migration ZO Migration Smear Reflections along Fat Circles  rsrsrsrs d(x, ) s  x & Sum

29. 1-way time ZO Migration ZO Migration Smear Reflections along Circles  rsrsrsrs d(x, ) s  r & Sum

30. ZO Migration Resolution ZO Migration Resolution Intersection of Fresnel Zones Vertical Res. = Near-Offset Traces

31. ZO Migration Resolution ZO Migration Resolution Intersection of Fresnel Zones Horiz. Res. = Far-Offset Traces

32. Why is Pt. Scatterer Response of Migration Why is Pt. Scatterer Response of Migration a Blurred Version of Point? Migration: m = L d TMigratedSectionData but d = L r L rL rL rL r Migration Section = Blured Image of r

33. 0 km 7 km 0 km 3 km m = L d T r = [L L] m 0 km 7 km T

34. Outline Fresnel Zone Stationary Phase and Fourier-Diffraction Rayleigh Resolution

35. Aperture Width is also Function of Layer Dip Angle and Depth

36. Vertical Resolution Formula For ZO Migration Intersection of Fresnel Zones Vertical. Res. = Near-Offset Traces What is width of minimum vertical cut at depth z? /4 =  z /4 =  z