Multi-source Least-squares Migration with Topography
Dongliang Zhang and Gerard SchusterKing Abdullah University of Science and Technology
Outline
Summary
TheoryUse ghost extrapolation to reduce stair-step diffractions from irregular surfaces
Numerical ExampleTests on Marmousi model and Foothills model
MotivationIrregular surface problems
Outline
Summary
TheoryUse ghost extrapolation to reduce stair-step diffractions from irregular surfaces
Numerical ExampleTests on Marmousi model and Foothills model
MotivationIrregular surface problems
Irregular Surface Problems
Datuming the data from irregular surface to flat surface
Motivation
Problem: Irregular Surface
Using Ghost extrapolation
Motivation
RTM migrates directly from the irregular surface
Air
Surface
Stair step
Subsurface
Solution: Ghost RTM
Outline
Summary
TheoryUse ghost extrapolation to reduce stair-step diffractions from irregular surfaces
Numerical ExampleTests on Marmousi model and Foothills model
MotivationIrregular surface problems
Least-squares Migration
𝐝=L𝐦f(m)+regularization term
g)
m𝒌+𝟏= m𝒌−𝜶 g𝒌
𝜶=(g𝒌)𝐓 g𝒌
(Lg𝒌)𝐓 Lg𝒌
Workflow of Multisource LSM with Topography
1. Forward modeling with topography to calculate the data residual
3. Update the reflectivity using the conjugate gradient method
2. Calculate gradient (RTM image) of data residual with topography
• Blended encoded shot gathers
Forward Modeling with Topography
Difficulty :Implement free surface boundary condition
Calculate the pressure on the points near by the free surface
Acoustic equation:
2 2 2
2 2 2 2
1
0
P P Px z v tP on the surface
Ghost point
Ghost Extrapolation
Zi,j
Zi-1,j
Zi-2,j
Zi+1,j
Zi+2,j
Surface
Zb
3 2P(z)=az +bz +cz+d
b
i-2,j i-2,j i-1,j i-1,j
i,j i,j
P(z )=P P(z )=PP(z )=P P(z )=0
G Gi+1,j i+2,jP =P(Δz) P =P(2Δz)
1 )i-2,j i-1,j i,jG Gi+1,j i+2,j
2
2 21 4 5- P + P - P1
4 1+ P - P3 1∂ P≈ (∂z 2 3 2Δz 2
Taylor Series
Extrapolation in z direction Extrapolation in x direction
Ghost Extrapolation
2 2 2
2 2 2 2
1P P Px z v t
Example of Dipping Surface
Surface Air
Surface
Stair step
Subsurface
0 X (km) 2
ZoomModel0
1.5
Z (k
m)
Mirror imageCommon Shot Gather
Pi-1,j
Pi-2,j
Pi+2,j=-Pi-2,j
Pi+1,j=-Pi-1,j
Air
Zero velocity layer
V=0
Subsurface
Air
Ghost extrapolation
0 X (km) 2
0
1.5
Z (k
m)
Zoom ViewsConventional method New method
Outline
Summary
TheoryUse ghost extrapolation to reduce stair-step diffractions from irregular surfaces
Numerical ExampleTests on Marmousi model and Foothills model
MotivationIrregular surface problems
0 X (km) 2
Grids size: 201 x 400 dx=dz=5 m Peak Freq.: 25 Hz Shots: 200 Receiver: 400 Max difference of elevation: 180 m
Marmousi Model
0
1
Z (k
m)
0
1
V
(km
/s)
Migration Velocity
Reflectivity Model
Marmousi Model
0 X (km) 2
0
1
Z (k
m)
0
1
Z (k
m)
Ghost FD
0 X (km) 2
Common Shot Gather0
2
T (s
)
0 X (km) 2
Ghost LSRTM Image
Ghost FD
Marmousi Model
Ghost FD Conventional FD
Conventional FD
LSRTM Image
RTM ImageGhost RTM Image
0 X (km) 2
0
1
Z (k
m)
0
1
Z (k
m)
Zoom Views
Ghost FD
Ghost LSRTM Image
Ghost FD Conventional FD
LSRTM Image
RTM ImageGhost RTM Image
Conventional FD
0 X (km) 8
Grids size: 333 x 833 dx=dz=10 m Peak Freq.: 15 Hz Shots: 208 Receiver: 833 Max difference of elevation: 500 m
Foothills Model
0
3
Z (k
m)
0
6
V
(km
/s)
Migration Velocity
Reflectivity Model
0 X (km) 8 0 X (km) 2
Common Shot Gather
Ghost FD
Foothills Model0
3
Z (k
m)
0
3
Z (k
m)
0
2T
(s)
0 X (km) 8
Ghost LSRTM Image
0 X (km) 8
LSRTM Image
Ghost FD
Ghost FD
Ghost RTM Image
Conventional FD
RTM Image
Conventional FD
Foothills Model0
3
Z (k
m)
0
3
Z (k
m)
Ghost LSRTM Image LSRTM Image
Ghost FD
Ghost FD
Ghost RTM Image
Conventional FD
RTM Image
Conventional FD
Zoom Views
Summary• MLSM can produce high quality images efficiently:
MLSM with topography produces high quality image,
multi-source saves the computational time
Ghost extrapolation can reduce stair-step diffraction artifacts
• Future work:Using 2D ghost extrapolation
Test on field data
High accuracy for the free surface boundary condition
Elastic
Thank you!