Weighted Stacking of 3-D Converted-wave Data for

Birefringent Media

Richard Bale

Introduction

• Shear-wave splitting (birefringence) is a “litmus test” for azimuthal anisotropy

• Used to characterize fractured media • Can degrade image if uncorrected • Causes acquisition footprint for 3-D

Radial and Transverse DMO Stacks

Transverse @ 2200 msRadial @ 2172 ms

1040 1060 1080 11101020 1040 1060 1080 11101020960

940

920

900

880

860

960

940

920

900

880

860

In-lines In-lines

CrossLines

CrossLines

Background

• Estimation of principal axes – e.g. using transverse polarity from 3-D data

• Layer stripping to remove anisotropy – needs estimates of S1-S2 transmission

• 2-D: rotation after stack

• 3-D: must rotate before stack– …but is that all?

Converted Wave Splitting

Legend:PSVS1S2

AzimuthallyAnisotropic

Layer

FractureDirection

RadialDirection

X

Y

Shot Receiver

ConversionPoint

Surface Geometry

X

Y

S1

S2Radial

UPS

Receiver

Shot

The 3-D Splitting Equation

0

)()(

)(0

0)()(

)(

)(2

1

2

1

PS

i

i U

ef

ef

Y

XRR

cossin

sincosRwhere: is a rotation matrix,

Radialconverted wave

Projection onto S1 & S2

S1 & S2propagationRecording on

X and Y

)()(

)(

)(

)(2

1

2

1

2

1

PSi

i

PS

PS Uef

ef

U

U

and:

)()sin(

)()cos()(

2

1

PS

PS

U

UR

Multi-azimuth CCP Binning

X

CCP Bin

Receivers

Shots

1

1,2

1,1

S

S

A

A

2

2,2

2,1

S

S

A

A

3 N….

NS

NS

A

A

,2

,1 ….

)()sin(

)()cos(

)(

)()(

)(

)(

2

1

,2

,1

PSi

PSi

i

iT

iS

iS

U

U

Y

X

A

AR

Acquisition- dependent amplitudes

Least Squares Stacking for S1 and S2

• We have two (decoupled) least squares problems, for UPS1 and UPS2

• Weighted stacking equations:

N

ii

N

iiSi

PS

AU

1

2

1,1

1

)(cos

)()cos()(ˆ

N

ii

N

iiSi

PS

AU

1

2

1,2

2

)(sin

)()sin()(ˆ

S1 Effective Fold S2 Effective Fold

Orthogonal Acquisition1200

1000

800

Y (meters)

600

400

200

00 200 400 600 800 1000 1200

X (meters)

ShotLines

ReceiverLines

Isotropic ACCP Fold

Cross-line Number

In-l

ine

Num

ber

S1 Effective ACCP Fold: =0°

Cross-line Number

In-l

ine

Num

ber

S1 Effective ACCP Fold: =45°

Cross-line Number

In-l

ine

Num

ber

Gryphon: Acquisition Geometry

1013 1050 1100

In-lines

950

Crosslines900

860

30

20

10

0

CCP Stacking Fold

Cables

Shot Lines

400m

400m

Effective Fold Maps

1013 1050 1100

In-lines

950

Crosslines900

860

S1 Norm:cos2(i-) S2 Norm:sin2(i-)

1013 1050 1100

In-lines

950

Crosslines900

860

TransverseRadial

Radial and Transverse DMO Stacks

1040 1060 1080 11101020

2.0

2.1

2.2

2.3

2.4

2.0

2.1

2.2

2.3

2.4

Crossline895

Crossline922

1040 1060 1080 11101020

2.0

2.1

2.2

2.3

2.4

2.0

2.1

2.2

2.3

2.4

S2S1

Least-squares S1 and S2 DMO Stacks

1040 1060 1080 11101020

2.0

2.1

2.2

2.3

2.4

2.0

2.1

2.2

2.3

2.4

Crossline895

Crossline922

1040 1060 1080 11101020

2.0

2.1

2.2

2.3

2.4

2.0

2.1

2.2

2.3

2.4

Least-squares S1 and S2 DMO Stacks

S2 @ 2200 msS1 @ 2172 ms

1040 1060 1080 11101020 1040 1060 1080 11101020960

940

920

900

880

860

960

940

920

900

880

860

In-lines In-lines

CrossLines

CrossLines

Radial and Transverse DMO Stacks

Transverse @ 2200 msRadial @ 2172 ms

1040 1060 1080 11101020 1040 1060 1080 11101020960

940

920

900

880

860

960

940

920

900

880

860

In-lines In-lines

CrossLines

CrossLines

Offset Dependence• Converted wave amplitudes strongly depend

on angle of incidence• For small angles (<20°):

for ray-parameter p, local compressional velocity VP, angle of incidence , and local shear velocity VS

(Stewart, Zhang, and Guthoff; 1995)

P

SSS

PS

Vp

RpVR

sin

4

Angle Gathers: Alba Radial Component

-60 -40 -20 0 20 40 60

MaximumAmplitude in Window

Linear Sampling Sin() Sampling

Tim

e (s

ec.)

Tim

e (sec.)

Offset Dependent Least Squares Stacking for S1 and S2

Updated weighted stacking equations:

N

ii

i

i

N

iiSi

i

i

SS

rtr

tArtr

ConsttU

1

2

2

10,1

01

)(cos)(

)()cos()(

.)(ˆ

N

ii

i

i

N

iiSi

i

i

SS

rtr

tArtr

ConsttU

1

2

2

10,2

02

)(sin)(

)()sin()(

.)(ˆ

S2 Effective Fold

S1 Effective Fold

t0 = 2-way zero-offset timeri = offsett(ri) = 2-way time

S1 Effective ACCP Fold: =45°

Cross-line Number

In-l

ine

Num

ber

Offset weighted S1 Eff. ACCP Fold: =45°

Cross-line Number

In-l

ine

Num

ber

Conclusions

• Preliminary work done on a method for 3-D S1 and S2 imaging

• Offset dependence can be included

• Effective fold maps include azimuth and offset effects for S1 and S2

• Initial results show increased resolution, and less acquisition footprint

Future Work

• Test method on synthetic and field data• Fractured reservoir imaging: consider

conversions within birefringent medium• Long offset issues:

– Orthogonality

– 2-term AVO fit: IS, (IS1, IS2?), parameterization

• Multiple birefringent layers• PS1-PS2 Migration?

Acknowledgements

• Kerr-McGee North Sea (UK) Ltd., and the Gryphon partners

• Chevron and the Alba partners

• WesternGeco– In particular Tony Probert and Gabriela Dumitru

• The CREWES sponsors