Kinematics of 2D specularly-reflected

and diffracted multiples in data

space and image spaceGabriel Alvarez

Stanford University

2

Goal

Understand how specularly-reflected and diffracted2D multiples map to subsurface image gatherswhen migrated with wave equation migration.

3

The Problem

Moveout-based multiple attenuation algorithms inimage space benefit from the power of migration to handle the complex wave propagation of the primaries.

The question remains: What is the moveout of the multiples in image space?

4

Outline• Moveout of 2D diffracted and specularly-reflected multiples in data space

• Mapping of multiples from CMPs to Subsurface Offset Domain Common Image Gathers (SODCIGs)

• Mapping of multiples from SODCIGs to Angle Domain Common Image Gathers (ADCIGs)

• Discussion and conclusions

5

Surface vs. Subsurface OffsetmD hDhD

2

6

Data Space vs. Image SpaceTi

me

m D

hD

Data Space

CMPsD

epth

m ξhξ

Image Space

SODCIGs

Dep

th

m ξ

Image Space

ADCIGs

7

Moveout of speculary-reflected multiplesFlat water-bottom:

ts1 tr1V

mD hDhD

ZDts2 tr2

ZDts2 tr2

8

Moveout of speculary-reflected multiplesDipping water-bottom:

V

mD hD

φ

hD

2φ

ZD

9

Moveout of Diffracted MultiplesFlat water-bottom:

ts1 tr1V

mD hDhD

ZDts2 tr2

ZDts2 tr2

Xd

10

Moveout of Diffracted MultipleDipping water-bottom:

V

mD hD

φ

2φ

hD

Xd

ZD

αs: takeoff angle of the source ray

11

Moveout Comparison

Dipping water-bottom

12

Moveout of Multiples inSubsurface Offset DomainCommon Image Gathers

(SODCIGs)

13

Image Coordinates of Non-diffracted MultipleFlat water-bottom:

V1

mD hD

αs αr

hD

V2=ρV1 βrβs

hξmξ hξ

212

ρξ Dhh

222 412 DDD ZhZz ρρ

ξ

Dmm ξ

14

SODCIG

Specularly-reflected multiple. Flat water-bottom

Half-subsurface offset (m)0-200-400 400200

Depth (m

)

1000

1200

1400

15

SODCIG

Specularly-reflected multiple. Flat water-bottom

Half-subsurface offset (m)0-200-400 400200

Depth (m

)

1000

1200

1400

16

Image Coordinates of Non-diffracted Multiple

ts1 tr1

V1

mD

αr-φ

αs+φ

αr

φ

V2

βr-φ

βs+φts2~

~tr2

hD hD

mξhξ hξ

rsss sinsin4sinsin2 2211

ββρφααξ rsrsD ttttVhh ~ ~

ss coscos21

βραξ sst ttVz ~

rsss sinsin4sinsin2 2211

ββρφααξ rsrsD ttttVmm ~ ~

17

Constant subsurface-offset section

Specularly-reflected multiple from a dipping water-bottom

Horizontal position (m)160014001200 2000

Depth (m

)

800

1200

1600

1800

18

SODCIG

Specularly-reflected multiple from a dipping water-bottom

Half-subsurface offset (m)0-400-800 800400

Depth (m

)

600

1000

1400

19

Image Coordinates of Diffracted Multiple

rsrs sinsinsinsin2 2211

ααρααξ rsrsD ttttVhh ~ ~

rcos2

βρξ rD VtZz ~

rs2

rs sinsinsinsin2 2211

ααρααξ rsrsD ttttVmm ~ ~

ts1

ts2

tr1

tr2

V1

V2

mD hD

hξ

βr

αs

βs

αr

mξ

Zdiff

Xdiff

hD

~~

hξ

20

Constant subsurface-offset sections

Diffracted multiple from a flat water-bottom

Horizontal position (m)260024002000 3000

Depth (m

)

800

1200

1600

2800Horizontal position (m)

260024002000 3000

Depth (m

)

800

1200

1600

2800

Half-subsurface offset 0 m Half-subsurface offset -200 m

21

SODCIGs

Diffracted multiple from a flat water-bottom

Half-subsurface offset (m)0-400 400

Depth (m

)

1000

1600

1400

1200

Half-subsurface offset (m)0-400 400

Depth (m

)

1000

1600

1400

1200

Half-subsurface offset (m)0-400 400

Depth (m

)1000

1600

1400

1200

22

Image Coordinates of Diffracted Multiple

rsrs sinsinsinsin2 2211

ββρααξ rsrsD ttttVhh ~ ~

ss coscos21

βραξ st ttVz ~

rsrs sinsinsinsin2 2211

ββρααξ rsrsD ttttVmm ~ ~

ts1

tr1

V1

V2

mD

hξ

αr-φ

βr-φ

αs+φ

βs+φ

αr

φ

ts2~

~tr2

mξ

hD

Zdiff

hξ

hD

23

Constant subsurface offset sections

Diffracted multiple from a flat water-bottom

Half-subsurface offset 0 m Half-subsurface offset -200 m

Horizontal position (m)200018001600 2400

Depth (m

)

1200

1400

1800

2200

1600

Horizontal position (m)200018001600 2400

Depth (m

)

1200

1400

1800

2200

1600

24

SODCIGs

Diffracted multiple from a dipping water-bottom

Half-subsurface offset (m)0-800 800

Depth (m

)1000

1800

1400

1200

1600

Half-subsurface offset (m)0-800 800

Depth (m

)

1000

1600

1400

1200

1800

Half-subsurface offset (m)0-800 800

Depth (m

)

1000

1600

1400

1200

1800

25

Moveout of Multiples inAngle-Domain

Common-Image-Gathers(ADCIGs)

26

ADCIG for specularly-reflected multiple

γρ

ργργρ

γ

γ

ξξ 22

222

sin

1tancos11

0zz

ts2tr2

(xrξ,zrξ) (xsξ,zsξ)hξ

βrβs

(xγξ,zγξ)

2γ

mξ

~ ~

hξ

2sr ββ

γ

γξξξ γtanhzz

ξξ γmm

27

ADCIG

Specularly-reflected multiple from a flat water-bottom

Half-aperture angle (degrees)20100 4030

Depth (m

)

1200

1400

1600

28

ADDCIG

Specularly-reflected multiple from a dipping water-bottom

Half-aperture angle (degrees)0-20-40 4020

Depth (m

)

1000

1400

1800

29

ADCIGs

Diffracted multiple from a flat water-bottom

Half-aperture angle (degrees)0-40 40

Depth (m

)

1200

1600

1400

Half-aperture angle (degrees)0-40 40

Depth (m

)

1200

1600

1400

0-40 40D

epth (m)

1200

1600

1400

Half-aperture angle (degrees)

30

ADCIGs

Diffracted multiple from a dipping water-bottom

0-40 40D

epth (m)

1400

2000

1600

Half-aperture angle (degrees)

Half-aperture angle (degrees)0-40 40

Depth (m

)

1400

2000

1600

Half-aperture angle (degrees)0-40 40

Depth (m

)

1400

2000

1600

31

Discussion and Conclusions

32

Discussion

Water-bottom, specularly-reflected multiples, migrated with sediment velocity migrate to negative subsurface offsets.

V

mD hDhD

ZD

ZD

2hξ<0

33

Discussion

On the other hand, primaries, migrated with slower velocitiesmap to positive subsurface offsets.

V

mD hDhD

ZD

ZD2hξ>0

34

Discussion

Diffracted multiples may map to positive subsurfaceoffsets in SODCIGs even if migrated with faster velocities.

V

mD hDhD

ZD

2hξ<0

V

mD hDhD

ZD

ZD

2hξ>0

35

ConclusionsSpecularly-reflected water-bottom multiples migrate as primaries at twice the water depthand with twice the dip.

Diffracted multiples do not migrate like primariesbut their moveout in both SODCIGs and ADCIGs can be computed if the location of thediffractor is known.

36

ConclusionsBetter understanding of the moveout of the multiples in SODCIGs and ADCIGs will help in designing more accurate Radon transforms to attenuate the multiples in image space.

37

Thank you for your attention.

I will be happy to entertain your questions.

38

From SODCIGs to ADCIGs

(xrξ,zrξ) (xsξ,zsξ)

βr

βs

mξhξ

2γ

A C

(mξγ,zξγ)D

BF

Ehξ

2sr ββ

γ

γξξξ γtanhzz

ξξ γmm