Ru-Shan Wu and Ling Chen

Modeling and Imaging Laboratory/IGPPUniversity of California, Santa Cruz

-------------------------------------------------------†Presently at Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China

Prestack depth migration Prestack depth migration in angle-domain in angle-domain

using beamlet decomposition: using beamlet decomposition: Local image matrix and local AVALocal image matrix and local AVA

Beamlet decomposition: Wave field in angle-domain

Local image matrix and local scattering matrix

Effect of acquisition aperture Local AVA: Preliminary tests Conclusion

Outline

True-reflection imaging in angle-domain

Preserving the relative amplitudes of scattered waves w.r.t. incident waves.

Benefits:• Improve image (total strength image) quality,

especially for steep reflectors. • Reduce artifacts (angle-domain filtering).• Provide basis for local AVA and local

inversion

True-amplitude imaging in angle-domain

Amplitude corrections (for ray theory see Hubral et al., Bleistein et al., Xu et al., Audebert et al., ……):

• Transmission loss (boundary reflection and scattering)

• Geometric spreading (for ray method)• Nonuniform information distribution:

Jacobian (Beylkin determinant)• Acquisition aperture effects (hit-count for ray

method)• Intrinsic attenuation (Anelasticity)

True-reflection imaging in angle-domain

for wave-equation based methods

Preserving the relative amplitudes of scattered waves w.r.t. incident waves:

• Nonuniform information distribution: Jacobian

• Acquisition aperture effects (in angle-domain) (including the geometric spreading and hit-count for ray method)

• Transmission and anelastic losses are less important, especially for small-angle reflections

B e am l e t d eco m p o s i ti o n o f th e w a v efi e l d :

n mmnmnz

mnn m

mnz

zxbxu

zxbbuxuzxu

,,

,,,~

w h e r e ),(~

zxb mn - - - - - d e c o m p o s i t i o n v e c t o r s ( a t o m s ) ,

),( zxb mn - - - - - r e c o n s t r u c t i o n v e c t o r s ( a t o m s ) ,

mnz xu , - - - - - c o e f f i c i e n t s o f t h e d e c o m p o s i t i o n a t o m s ,

xn nx - - - - - w i n d o w l o c a t i o n ,

mm - - - - - l o c a l w a v e n u m b e r .

G-D frame atoms

Windowed plane wavesWindowed plane waves

nxi

mn xxgexg m

is a Windowed Plane wave (each beamlet is a windowed plane wave)

Local plane wavesLocal plane waves

Local plane wave: a superposition of windowed plane waves of the same local wavenumber from all neighboring windows:Partial reconstruction of wavefield (mixed domain wavefield: local phase–space):

The corresponding propagating angle:

l

jlzlxi

j xuxxgezxu j ,,,,,

xvjj 1cos

Target area

Source Receiver

in

sc

Local Image Matrix(includes aperture and propagation effects)

High-velocity body

**

),( scinL

Local Scattering Matrix

),( scinS

Point scatterer Planar reflector

shallow

deep

Local image matrix in homogeneous medium(total 201 shots with 176 left-hand receivers )

Local image matrixLocal image matrix: image condition in beamlet domain and mixed domain

Forward-propagated source field:

Backward-propagated scattered field:

j l

jljlSz

S xgxuzxu ,,,,

p q

pqpqRz

R xgxuzxu SS ,,,,

S R

Ra

Sa

a

S

S zxWzxW

kzxL

,,,,,,

coscos,,,,

scin

scin20scin

scin20 coscos k

Local image matrix:

Where Serves as the Jacobian

Ws and WRs are the wave fields in angle-domain by beamlet decomposition

,,,,Re

,,,

scinsinsin

scin

scin, zxLed

zxI

axi zxv

Stacking over frequency to get the final imageIn the local angle-domain:

in sc

,,,, scin zxIzxIThe final image in space domain:

Local Reflection Local Reflection Analysis (AVA):Analysis (AVA):

For planar reflectors: the local image matrix can be represented as:

with

2

2

scinr

scinn

zxI ,,, rn

CDAI (common dip-angle CDAI (common dip-angle image) gathersimage) gathers

Sum up all reflections for a common dip-angle: CDAICDAI gather

r

zxIzxI rnn ,,,,,

Obtain the dip-angle of the local reflectors from CDAI.

CRAI (common reflection-angle CRAI (common reflection-angle image) gathersimage) gathers

Sum up reflected energy for a common reflection-angle for all possible dip-angles: CRAI.CRAI.

Performing local AVA from CRAI gathers.The calculation of local reflection coefficients:

n

zxIN

zxR rnr ,,,1

,,

Local AVA for an oblique interface in homogeneous background

Local image matrices at a point on the middle of dipping interface 14° obtained from 80 shots with a two-side receiver array (513 receivers).

The dotted line corresponds to the theoretical prediction without aperture effect.

Obtain the dip-angle of local reflectors from the CDAI gathers

CDAI gathers for a local reflector at its central location

. Angle-dependent reflection coefficients at the interface using 256 shots with 513 two-side detectors for the horizontal layered model

with different velocity contrasts: (a) 10%; (b) 25%; (c) 50%; (d)150%

Calculated reflection coefficients from CRAI gathers

Dotted: synthetic; Red: 513 points two-sidesBlue: 257 points one-side; Green: 129 points two-sides

Angle-dependent reflection coefficients at the interface obtained from LIM

in case of 10% velocity contrast for the horizontal layered model

Local image matrix and the local scattering matrix

The local image matrix has the acquisition-aperture and propagation effects included. The purpose of the imaging/inversion is to recover the real local scattering matrix and obtain the local reflection coefficients. To achieve the true-reflection imaging, we need to estimate the acquisition-aperture effect and apply the correction.

Acquisition-Aperture Efficacy(Effect of the source-receiver configuration)

• Acquisition-Aperture Efficacy (AAE) Matrix

• Acquisition-Aperture Dip-response function

• Aperture corrections

Target area

Source Receiver

in

sc

Acquisition-Aperture Efficacy:

(includes propagation effects)

Overburdenstructures

**

),( scinE

Assume scatteringCoefficients as 1

With unit impulses at both source and receivers, the local acquisition-aperture efficacy matrix is obtained as:

Acquisition-aperture efficacy matrixAcquisition-aperture efficacy matrix

Where G’s are the Green’s functions in beamlet domain

2

12

*

2

,,,ˆ

,,,ˆ,,,

S

SR q

qRpqz

S llSjlzpjz

xxgxxG

xxgxxGxE

Acquisition-apertureAcquisition-aperture dip-response functiondip-response function

Acquisition-aperture dip-response as a function of dip-angle of local interface (reflector), which reduce the AAE matrix into a vector:

r

xExE rnznz

,,,,,

with

2

2

scinr

scinn

Acquisition-Aperture Dip-Response(Acquisition Configuration Response)

**

*S2

S3

S1

Acquisition-Dip-Response (horizontal reflector) from all the 325 shots

Acquisition-Dip-Response (45 down from horizontal) from all the 325 shots

image by common-shot prestack G-D migration

G-D beamlet prestackmigration image

Acquisition-DipResponse for 45o

reflectors

Improved image afterDirectional illumination

correction

Conclusion• Local image matrix can be obtained from

the local incident and scattered plane waves based on beamlet decomposition

• The goal of true-reflection imaging in angle-domain is to remove the acquisition aperture effect and propagation effect through directional illumination analysis and the corresponding corrections

Conclusion (continued)

• CDAI and CRAI gathers can be deduced from local image matrices (after corrections)

• CDAI gathers can be used to determine the dip-angles of local reflectors

• CRAI gathers can be used for local AVA analysis (and further for local inversion)

Acknowlegement

We thank the support from WTOPI Research Consortium at UCSC

We thank the support from DOE Project at UCSC

___________________________________________Welcome to visit our Consortium booth #2745