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

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

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)

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)

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

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 0 + 𝜏 𝑥 0 𝑙𝑣𝑙 ) ⊗ = LVL g s 𝑥 0 LVL g s ref g s

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

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

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 }]

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

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

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 }]

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

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) 0 X (km) 2.5 0 X (km) 2.5

Reflectivity Model 1.25 Reference layer Z (km) 0 2.5 X (km)

LSM image Z (km) 1.25 0 2.5 X (km)

Interferometric LSM image 1.25 Z (km) 0 2.5 X (km)

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

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)

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Thank You