1. Interferometric Least Squares Migration
Mrinal Sinha* and Gerard T. Schuster
King Abdullah University of Science and Technology
20th October, 2015

2. Outline Motivation Theory of Interferometric LSM
Gradient of Interferometric ILSM
Workflow
Examples
Synthetic Example
Field Data Example
Conclusion

3. Motivation
Problem: Imaging of seismic data with a velocity model which has errors results in poor quality of migration image
Z (km)
1.25
X (km)
2.5
1.5
2.8
(km/s)
Migration Velocity Model
(km/s)
2.8 1
2.5
X (km)
1.25
Z (km)
True Velocity Model
LSM image is defocused because of errors in velocity model
Z(km)
1.25
2.5
X(km)

4. Reference Reflector as a Seismic Guide Star
Target Star
Guide Star
Similarly one can use the knowledge of a reference reflector as a guide star to accurately image other reflectors s g
Z(km)
1.25
2.5
X(km)
Z(km)
1.25
2.5
X(km)

5. Outline Motivation Theory of Interferometric LSM
Gradient of Interferometric ILSM
Workflow
Examples
Synthetic Example
Field Data Example
Conclusion

6. 𝑑 𝑔 𝑠 𝑟𝑒𝑓 = 𝑒 𝑖𝜔(𝜏 𝑟𝑒𝑓 + 𝜏 𝑟𝑒𝑓 𝑙𝑣𝑙 )
Theory
Crosscorrelogram is estimated by cross-correlating the data with reference reflector data
𝑑 𝑔 𝑠 𝑟𝑒𝑓 = 𝑒 𝑖𝜔(𝜏 𝑟𝑒𝑓 + 𝜏 𝑟𝑒𝑓 𝑙𝑣𝑙 )
∅(𝒈|𝒔)= 𝑒 𝑖𝜔( 𝜏 𝑥 0 −𝜏 𝑟𝑒𝑓 )
𝑑 𝑔 𝑠 =𝑒 𝑖𝜔( 𝜏 𝑥 𝜏 𝑥 0 𝑙𝑣𝑙 ) ⊗ =
LVL g s
𝑥 0
LVL g s
ref g s

7. 𝜖=− ∅ 𝒈 𝒔 ∅ 𝒈 𝒔 ∙ ∅ 𝒈 𝒔 𝑜𝑏𝑠 ∅ 𝒈 𝒔 𝑜𝑏𝑠
Theory
Interferometric LSM is used to migrate the crosscorrelograms
A normalized cross-correlation based objective function is minimized
𝜖=− ∅ 𝒈 𝒔 ∅ 𝒈 𝒔 ∙ ∅ 𝒈 𝒔 𝑜𝑏𝑠 ∅ 𝒈 𝒔 𝑜𝑏𝑠

8. Outline Motivation Theory of Interferometric LSM
Gradient of Interferometric ILSM
Workflow
Examples
Synthetic Example
Field Data Example
Conclusion

9. 𝜖=− ∅ 𝒈 𝒔 ∅ 𝒈 𝒔 ∙ ∅ 𝒈 𝒔 𝑜𝑏𝑠 ∅ 𝒈 𝒔 𝑜𝑏𝑠
Gradient Calculation
𝜖=− ∅ 𝒈 𝒔 ∅ 𝒈 𝒔 ∙ ∅ 𝒈 𝒔 𝑜𝑏𝑠 ∅ 𝒈 𝒔 𝑜𝑏𝑠
Objective function
Gradient
where w=< ∅ 𝑔 𝑠 ||∅ 𝑔 𝑠 || , ∅ 𝑜𝑏𝑠 𝑔 𝑠 ||∅ 𝑜𝑏𝑠 𝑔 𝑠 || > and ∅ (𝑔|𝑠)= ∅(𝑔|𝑠) ∅(𝑔|𝑠)
𝜕𝜖 𝜕𝑠(𝑥) =− 𝜕𝑑 𝑔 𝑠 𝜕𝑠 𝑥 [ 𝑑 𝑟𝑒𝑓 𝑔 𝑠 ∗{𝑤 ∅ 𝑔 𝑠 − ∅ 𝒈 𝒔 𝑜𝑏𝑠 }]

10. Interpretation of the Gradient
Weighted Crosscorrelogram Residual
𝜕𝜖 𝜕𝑠(𝑥) =− 𝜕𝑑 𝑔 𝑠 𝜕𝑠 𝑥 [ 𝑑 𝑟𝑒𝑓 𝑔 𝑠 ∗{𝑤 ∅ 𝑔 𝑠 − ∅ 𝒈 𝒔 𝑜𝑏𝑠 }]
Backpropagated residual
Migration kernel s g s g * g s

11. Outline Motivation Theory of Interferometric LSM
Gradient of Interferometric ILSM
Workflow
Examples
Synthetic Example
Field Data Example
Conclusion

12. 𝑔 𝑘+1 = 𝜕𝜖 𝜕𝑠(𝑥) =− 𝜕𝑑 𝑔 𝑠 𝜕𝑠 𝑥 [ 𝑑 𝑟𝑒𝑓 𝑔 𝑠 ∗{𝑤 ∅ 𝑔 𝑠 − ∅ 𝑜𝑏𝑠 𝑔 𝑠 }]
Workflow
1. Identify a reference reflector in the model and data space
X (m)
Z (m)
X (m)
Time (s)
2. Calculate the observed ( ∅ 𝑜𝑏𝑠 𝑔 𝑠 } ) and predicted crosscorrelograms ( ∅ 𝑔 𝑠 )
3. Evaluate the gradient 𝑔 𝑘+1 as given below and use conjugate gradient
𝑔 𝑘+1 = 𝜕𝜖 𝜕𝑠(𝑥) =− 𝜕𝑑 𝑔 𝑠 𝜕𝑠 𝑥 [ 𝑑 𝑟𝑒𝑓 𝑔 𝑠 ∗{𝑤 ∅ 𝑔 𝑠 − ∅ 𝑜𝑏𝑠 𝑔 𝑠 }]

13. Outline Motivation Theory of Interferometric LSM
Gradient of Interferometric ILSM
Workflow
Examples
Synthetic Example
Field Data Example
Conclusion

14. Migration Velocity Model
Velocity Models
True Velocity Model
Migration Velocity Model 1
2.8
(km/s)
1.5
2.8
(km/s)
1.25
1.25
Z (km)
Z (km)
X (km)
X (km)

15. Reflectivity Model
1.25
Reference layer
Z (km)
X (km)

16. LSM image
Z (km)
1.25
X (km)

17. Interferometric LSM image
1.25
Z (km)
X (km)

18. Outline Motivation Theory of Interferometric LSM
Gradient of Interferometric ILSM
Workflow
Examples
Synthetic Example
Field Data Example
Conclusion

19. Field Data Example CSG Acquisition 496 shots at an interval of 37.5 m
Cable length = 6 km
480 receivers at an interval of 12.5 m
Recording time = 10 s
CSG
Time (s) 10 6
Offset (km)

20. Field Data Example
To test ILSM low-velocity anomalies are added to the water layer
1.3
2.8
(km/s) 1 2 4 8 12 16
X (km)
Z (km)

21. Field Data Example KM image (True Velocity) Reference Reflector 4
X (km) 2 1 8 12 16
Reference Reflector
Z (km)

22. Field Data Example LSM image (True Velocity) 1 Z (km) 2 4 8 12 16
X (km)

23. Field Data Example LSM image (Wrong Velocity) 4 X (km) 2 1 8 12 16
Z (km)

24. Interferometric LSM image
Field Data Example
Interferometric LSM image
(Wrong Velocity) 1
Z (km) 2 4 8 12 16
X (km)

25. Field Data Example LSM image (True Velocity) 1 Z (km) 2 4 8 12 16
X (km)

26. Field Data Example
LSM image
(True Velocity)
0.4
Z (km) 1 7 11
X (km)

27. Field Data Example
LSM image
(Wrong Velocity) 7 11
X (km) 1
0.4
Z (km)

28. Interferometric LSM image
Field Data Example
Interferometric LSM image
(Wrong Velocity) 7 11
X (km) 1
0.4
Z (km)

29. Field Data Example
LSM image
(True Velocity)
0.4
Z (km) 1 7 11
X (km)

30. Field Data Example LSM image (True Velocity) 1 Z (km) 2 4 8 12 16
X (km)

31. Field Data Example
LSM image
(True Velocity)
0.4
Z (km) 1 13 17
X (km)

32. Field Data Example LSM image (Wrong Velocity) 0.4 Z (km) 1 13 17
X (km) 1
0.4
Z (km)

33. Interferometric LSM image
Field Data Example
Interferometric LSM image
(Wrong Velocity) 13 17
X (km) 1
0.4
Z (km)

34. Field Data Example
LSM image
(True Velocity)
0.4
Z (km) 1 13 17
X (km)

35. Field Data Example LSM image (True Velocity) 1 Z (km) 2 4 8 12 16
X (km)

36. LSM image
(True Velocity) 1
Z (km)
1.5 6
X (km)

37. Interferometric LSM image (Wrong Velocity)1
Z (km)
1.5 6
X (km)

38. Why it fails ?? 𝜏 𝑟𝑒𝑓 𝑙𝑣𝑙 > 𝜏 𝑥0 𝑙𝑣𝑙 𝜏 𝑟𝑒𝑓 𝑙𝑣𝑙 ≈ 𝜏 𝑥0 𝑙𝑣𝑙
𝜏 𝑟𝑒𝑓 𝑙𝑣𝑙 > 𝜏 𝑥0 𝑙𝑣𝑙
𝜏 𝑟𝑒𝑓 𝑙𝑣𝑙 ≈ 𝜏 𝑥0 𝑙𝑣𝑙
LVL g s
ref
Phase associated with the raypath in the LVL does not cancel out
𝑥 0

39. Field Data Example LSM image (True Velocity) 1 Z (km) 2 4 8 12 16
X (km)

40. LSM image
(True Velocity) 1
Z (km)
1.5 6
X (km)

41. Interferometric LSM image (Wrong Velocity)1
Z (km)
1.5 6
X (km)

42. Outline Motivation Theory of Interferometric LSM
Gradient of Interferometric ILSM
Workflow
Examples
Synthetic Example
Field Data Example
Conclusion

43. Interferometric LSM image
Conclusion
ILSM is used to mitigate the defocusing caused in the migration image due to statics
Defocusing caused by errors in velocity model above the reference layer is mitigated
Reflectors close to the reference reflector are better imaged
Z (km)
1.25
2.5
X (km)
LSM image
Z (km)
1.25
2.5
X (km)
Interferometric LSM image

44. Conclusions Limitations
Identification of the reference reflector and its corresponding reference reflection is crucial
Reflectors located far away from the reference reflector do not get focused accurately

45. Acknowledgement SEG for providing the opportunity
Sponsors of the CSIM consortium
KAUST Supercomputing Laboratory for the HPC support
The audience for their kind attention

46. Thank You