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Prestack depth migration in angle-domain using beamlet decomposition: Local image matrix and local AVA. Ru-Shan Wu and Ling Chen Modeling and Imaging Laboratory/IGPP University of California, Santa Cruz ------------------------------------------------------- - PowerPoint PPT Presentation
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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