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 P

x 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 )=P

P(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,j

G Gi+1,j i+2,j

2

2 2

1 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 P

x z v t

Example of Dipping Surface

Surface Air

Surface

Stair step

Subsurface

0 X (km) 2

ZoomModel0

1.5

Z (

km

)

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 (

km

)

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 (

km

)

0

1

V

(km

/s)

Migration Velocity

Reflectivity Model

Marmousi Model

0 X (km) 2

0

1

Z (

km

)

0

1

Z (

km

)

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 (

km

)

0

1

Z (

km

)

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 (

km

)

0

6

V

(km

/s)

Migration Velocity

Reflectivity Model

0 X (km) 8 0 X (km) 2

Common Shot Gather

Ghost FD

Foothills Model

0

3

Z (

km

)

0

3

Z (

km

)

0

2

T (

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 Model

0

3

Z (

km

)

0

3

Z (

km

)

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!