1. Reverse Time Migration of Prism Waves for Salt Flank Delineation
Wei Dai, WesternGeco
Gerard T. Schuster, King Abdullah University of Science and Technology
Sep 25, 2013

2. Outline Introduction and motivation Theory Numerical Results Summary
L model
Salt model
Summary

3. Introduction
Problem: Vertical boundaries (salt flanks) are difficult to image because they are usually not illuminated by primary reflections.
Solution: Prism waves contain valuable information.

4. Conventional Method
When the known boundaries are embedded in the velocity model, conventional RTM can migrate prism waves correctly.

5. Recorded Trace 𝒅 𝒈 𝒔 = 𝒅 𝟏 𝒈 𝒔 + 𝒅 𝟐 (𝒈|𝒔)
Time (s) 2

6. Horizontal Reflector Embedded in the Velocity
Z (km) 3
X (km) 6
Z (km) 3
Conventional RTM Image

7. 𝒎 𝒎𝒊𝒈 (𝒙)= 𝝎 𝝎 𝟐 𝑾 ∗ (𝝎)𝑮 ∗ 𝒙 𝒔 𝑮 ∗ 𝒙 𝒈 𝒅 𝒈 𝒔
Reverse Time Migration Formula
𝒎 𝒎𝒊𝒈 (𝒙)= 𝝎 𝝎 𝟐 𝑾 ∗ (𝝎)𝑮 ∗ 𝒙 𝒔 𝑮 ∗ 𝒙 𝒈 𝒅 𝒈 𝒔
Angular Freq.
Source Spectrum
Green’s functions
Input Data
Z (km) 3 𝒙
𝑮 𝒙 𝒔 = 𝑮 𝒐 𝒙 𝒔 + 𝑮 𝟏 (𝒙|𝒔)
𝑮 𝒙 𝒈 = 𝑮 𝒐 𝒙 𝒈 + 𝑮 𝟏 (𝒙|𝒈)

8. 𝒎 𝒎𝒊𝒈 𝒙 = 𝝎 𝝎 𝟐 𝑾 ∗ 𝝎 𝑮 𝒐 ∗ 𝒙 𝒔 + 𝑮 𝟏 ∗ 𝒙 𝒔 𝑮 𝒐 ∗ 𝒙 𝒈 + 𝑮 𝟏 ∗ 𝒙 𝒈
𝒎 𝒎𝒊𝒈 𝒙 = 𝝎 𝝎 𝟐 𝑾 ∗ 𝝎 𝑮 𝒐 ∗ 𝒙 𝒔 + 𝑮 𝟏 ∗ 𝒙 𝒔 𝑮 𝒐 ∗ 𝒙 𝒈 + 𝑮 𝟏 ∗ 𝒙 𝒈
[ 𝒅 𝟏 𝒈 𝒔 + 𝒅 𝟐 𝒈 𝒔 ]
= 𝝎 𝝎 𝟐 𝑾 ∗ 𝝎 [ 𝑮 𝒐 ∗ 𝒙 𝒔 𝑮 𝒐 ∗ 𝒙 𝒈 𝒅 𝟏 𝒈 𝒔 +
𝑮 𝒐 ∗ 𝒙 𝒔 𝑮 𝒐 ∗ 𝒙 𝒈 𝒅 𝟐 𝒈 𝒔
Ellipses
+ 𝑮 𝟏 ∗ 𝒙 𝒔 𝑮 𝒐 ∗ 𝒙 𝒈 𝒅 𝟏 𝒈 𝒔
+ 𝑮 𝒐 ∗ 𝒙 𝒔 𝑮 𝟏 ∗ 𝒙 𝒈 𝒅 𝟏 𝒈 𝒔
Rabbit Ears
+ 𝑮 𝟏 ∗ 𝒙 𝒔 𝑮 𝒐 ∗ 𝒙 𝒈 𝒅 𝟐 𝒈 𝒔
+ 𝑮 𝒐 ∗ 𝒙 𝒔 𝑮 𝟏 ∗ 𝒙 𝒈 𝒅 𝟐 𝒈 𝒔
Prism Wave Kernels
+ Other terms.]
Z (km) 3
X (km) 6

9. Migration of Prism Waves
𝒎 𝒎𝒊𝒈 = 𝝎 𝝎 𝟐 𝑾 ∗ 𝝎 𝑮 𝟏 ∗ 𝒙 𝒔 𝑮 𝒐 ∗ 𝒙 𝒈 𝒅 𝟐 𝒈 𝒔
𝑮 𝟏 𝒙 𝒔 = 𝝎 𝟐 𝒎 𝒙 ′ 𝑮 𝒐 𝒙 ′ 𝒔 𝑮 𝒐 𝒙 ′ 𝒙 𝒅𝒙′
Born Modeling
Z (km) 3
X (km) 6

10. Migration of Prism Waves
𝑥 𝑔
𝑥 𝑠
𝑥 2
Z (km)
𝑥 1 3
𝑥 2 ′
𝑥 𝑠 − 𝑥 2 ′ 𝑐 + 𝑥 2 − 𝑥 𝑔 𝑐 = 𝜏 𝑠𝑔
Z (km) 3
X (km) 6

11. Migration of Prism Waves
𝒎 𝒎𝒊𝒈 = 𝝎 𝝎 𝟐 𝑾 ∗ 𝝎 𝑮 𝒐 ∗ 𝒙 𝒔 𝑮 𝟏 ∗ 𝒙 𝒈 𝒅 𝟐 𝒈 𝒔
𝑮 𝟏 𝒙 𝒈 = 𝝎 𝟐 𝒎 𝒙 ′ 𝑮 𝒐 𝒙 ′ 𝒔 𝑮 𝒐 𝒙 ′ 𝒙 𝒅𝒙′
Born Modeling
Z (km) 3
X (km) 6

12. Migration of Prism Waves
Z (km) 3
Z (km) 3
X (km) 6

13. Outline Introduction and motivation Theory Numerical Results Summary
L model
Salt model
Summary

14. The L Model Model size: 301 x 601 Source freq: 20 hz shots: 32
geophones: 601
Z (km) 3
X (km) 6

15. A Shot Gather of the L Model
Time (s)
6.4
X (km) 6

16. Prism Wavepath
Z (km) 3
Z (km) 3
X (km) 6

17. RTM Image /w Smooth Velocity
Z (km) 3
X (km) 6
Z (km) 3
Migration Image of Prism Waves

18. The Salt Model Model size: 601 x 601 Source freq: 20 hz shots: 601
Model size: 601 x 601
Source freq: 20 hz
shots: 601
Z (km)
geophones: 601 6
X (km) 6

19. A Shot Gather of the Salt Model
Time (s) 10
X (km) 6

20. RTM with Smooth Velocity
Migration Velocity
RTM Image
Z (km) 6
X (km) 6
X (km) 6

21. If the Horizontal Reflectors are embedded in the velocity
Migration Velocity
RTM Image
Z (km) 6
X (km) 6
X (km) 6

22. New Method Migration Velocity Conv. RTM Image Prism Wave Image
Z (km) 6
Migration Velocity
Conv. RTM Image
Prism Wave Image
RTM Image
Z (km)
X (km) 6 6
X (km) 6

23. New Method Migration Velocity Prism Wave Image Filtered RTM Image
Z (km) 6
Migration Velocity
X (km) 6
Prism Wave Image
Filtered RTM Image
Z (km) 6
X (km) 6

24. Final Image Horizontal Final Image RTM Image Vertical Z (km) 6 Z (km)
Z (km) 6
Horizontal
Final Image
RTM Image
Vertical
Z (km)
X (km) 6 6
X (km) 6

25. Summary
I propose a new method to migrate prism waves separately.
Avoid the modification of migration velocity.
Reduce cross interference between different waves by migrating different waves in separated steps.
Limitations
Computational cost is doubled.

26. Thanks