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Transcript of Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu,...
![Page 1: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/1.jpg)
Decomposition, extrapolation and imaging
of seismic data using beamlets and dreamlets
Ru-Shan Wu, Modeling and Imaging Laboratory, University of California, Santa Cruz
Sanya Symposium, 2011
![Page 2: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/2.jpg)
Outline
• Introduction: Physical wavelet• Time-slicing and depth-slicing of 4-D data• Physical wavelet defined on observation
planes: Dreamlet• Dreamlet and beamlet propagator and
imaging• Applications• Conclusion
![Page 3: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/3.jpg)
Introduction
• Wavefield or seismic data are special data sets. They cannot fill the 4-D space-time in arbitrary ways.
• Wave solutions can only exist on the light cone (hyper-surface) in the 4D Fourier space defined by dispersion relation.
• Physical wavelet is a localized wave solution by extending the light cone into complex causal tube.
• dreamlet can be considered as a type of physical wavelet defined on an observation plane (data plane on the earth surface or extrapolation planes at depth z in the migration/imaging process).
![Page 4: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/4.jpg)
Physical wavelet
• Physical wavelet: localized wave field defined in the 4-dimensional time-space, satisfies the wave equation:– Globally for homogeneous media;– Locally for inhomogeneous media
• Localized by analytic extension to the complex 4-D time-space
• Only exit on the causal tube (nature of wave solution)
![Page 5: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/5.jpg)
Special features of seismic data
• Seismic data are special data sets. They cannot fill the 4-D space-time in arbitrary ways. The time-space distributions must observe causality which is dictated by the wave equation. Wave solutions can only exist on the light cone (hyper-surface) in the 4D Fourier space.
• Often the data are only available on the surface of the earth (the observation plane)
![Page 6: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/6.jpg)
2 2( ) 0t u
4
44
1ˆ( ) ( )
(2 )ip xu x d pe u p
R
0( , ) ( , )x t x x x
4D Fourier domain40( , ) ( , )p p k p p R
4D space-time domain
Wavefield data are solutions from the wave equation:
![Page 7: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/7.jpg)
• where space-time four-vector wavenumber-frequency four-vector by the wave equation absolute value of frequency Lorentz-invariant scalar product
3
3( ) ( )
3( ) [ ( , ) ( , )]
16
= ( )
i t i t
R
ip x
C
du x e u e u
dpe u p
p x p xPp p
0( , )p p p
0p p
p
( , )x t x
0p x p t p x
dp Measure on the light-cone (Minkowski measure)
22| | 0
c
p
![Page 8: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/8.jpg)
Light cone in the Fourier space ( ),
0 (frequency)p
C
C
V
V
planep
![Page 9: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/9.jpg)
24 20 0( , ) : 0, )C p p p p C C p R p
3
316
ddp
p
light cone
Lorentz-invariant measure on C
Light coneWave equation solutions satisfy the dispersion relation (causality)And therefore can only exist on the “light cone”
22 2 2 2
0| | | | 0p p p pc
p p
![Page 10: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/10.jpg)
Space-time light cone(from Wikipedia)
![Page 11: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/11.jpg)
Construction of localized wave solutions
• Kaiser 1994 (Analytic signal transform)• Kiselev and Perel, 2000; Perel and Sidorenko,
2007 (Continuous wavelet transform)
![Page 12: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/12.jpg)
• To construct the wavelets (localized wave solution) (Physical wavelet ), extend from the real space-time to the causal tube in complex space-time,
by applying the analytic-signal transform
where is the unit step function and Is an acoustic wavelet (physical wavelet)
( )
*1
1( ) ( ) 2 ( ) ( )
( ) ( )
ip x iy
C
zC
du x iy u x y dp p y e u p
i idp
p u pk
4R
4 2: 0T x iy y C
( )u x
*( )z p
![Page 13: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/13.jpg)
Analytic Signal Transform and Windowing in the Fourier domain
12( () ) exp ( )z k p y ip xp iy
is an acoustic wavelet of order in the Fourier domain.
•the AST can be looked as a windowing in the Fourier domain (windowed Fourier transform)
![Page 14: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/14.jpg)
Space localization at t=0( ,0) : 3,10,15,100r
r
![Page 15: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/15.jpg)
Time localization at r=0(real part- solid; imag- dashed)
(0, ) : 3,10,15,100t
t
![Page 16: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/16.jpg)
Wavefield data on planes:
Data acquisition plane on surface Extrapolation planes during
migration/imaging
![Page 17: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/17.jpg)
z
Surface
Extrapolated planes
•Data acquisition on the surface•Wave field downward continuation •Depth migration by downward-continuation or
Survey sinking + Imaging condition
![Page 18: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/18.jpg)
Two different decomposition schemes
• For Time-slices: All the space-axis are symmetric
• Depth-slices: Time-axis and space axes are different and need to be treated differently
![Page 19: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/19.jpg)
Time-slicing in 4-D
A time-slice
![Page 20: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/20.jpg)
Depth-slicing in 4D
Depth
(x)
A depth slice
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Two different decomposition schemes
• For time-slices: All the space-axis are symmetric: e.g. Curvelet
• Depth-slices: Time-axis and space axes are different and need to be treated differently: e.g. Pulsed-beam; wavepacket; Dreamlet (Drumbeat-beamlet)
![Page 22: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/22.jpg)
Dreamlet: A type of physical wavelet defined on observation planes
(data planes)
Wu et al., 2008; 2009; 2011 (SEG abstracts)
![Page 23: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/23.jpg)
Dreamlet (localized time-space solution of wave equation)
• Dreamlet: Physical wavelet on a plane x=(x,y)
• Time-space wavelet (directional wavepacket, “pulsed beam”)
( , ) ( , ) ( ) ( )tt x xd x t d x t g t b x
( , )( , )( , ) ( , , )t x t x zd x t d x z t 2
2
c
through dispersion relation:
( )tg t :Drumbeat; ( )xb x :Beamlet
![Page 24: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/24.jpg)
Construction of dreamlet atoms:Drumbeat (t-f atom) beamlet (x-k atom)
( ))
( )
( ) (
i t i tt
i xx
g eW W
B
e
b e
Windowing in frequency and horizontal wavenumber domains
Windowing on the light-cone (through the dispersion relation)
( , )( , )
( ) ( )
( , ) ( , , )
, ,
t x t x z
i t i x z
d x t d x z t
D e
![Page 25: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/25.jpg)
Dreamlet = Wavepacket Windowing on the light cone
0 (= : frequency)p
CCVV
planep
xk
zk
2 2 2( / )c
![Page 26: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/26.jpg)
Integration on the light-cone
• On the light cone we have ( and k as variables)2 2 2 2 2z x yK k K K p C
2 2 ( ) ( )( )
2z z
z
K KK
New measure on the light-cone2
3 2 216 | |d
d dkdp
k
ξ
ξ
The integration on the light cone for wave solution:
( )= ( )ip xdC
u x dp e u p
ξ
![Page 27: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/27.jpg)
Discrete wavelet atoms obtained by windowing on the light-cone
Dreamlets2 2( ) ( ) ( , )t xp k d ξ
Discrete wavelet transform (Orthogonal or sparse frame)
vs. Continuous wavelet transform
![Page 28: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/28.jpg)
The window defined on the observation plane (red segment) and window for the whole space (green disk).
![Page 29: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/29.jpg)
Examples of dreamlet decomposition on seismic data
![Page 30: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/30.jpg)
The poststack data of SEG 2D salt model
![Page 31: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/31.jpg)
Dreamlet decomposition of the SEG salt data by local exponential frames
x
t-f
![Page 32: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/32.jpg)
Dreamlet decomposition of the SEG salt data using different thresholds: 1%
x
f - t
![Page 33: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/33.jpg)
Dreamlet decomposition of the SEG salt data using different thresholds: 2%
![Page 34: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/34.jpg)
Dreamlet decomposition of the SEG salt data using different thresholds: 3%
![Page 35: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/35.jpg)
Dreamlet decomposition of the SEG salt data using different thresholds: 4%
![Page 36: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/36.jpg)
Compression Ratio (CR) for Dreamlet decomposition of seismic data
Figure 1: Comparison Ratios of different decomposition methods (SEG/EAGE salt model poststack data).
Dreamlets
Curvelets
Beamlets(Local-cosine basis)
![Page 37: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/37.jpg)
Features of Dreamlet and Beamlet
• Different levels of localization• Wave data decomposition and compression• Wave propagation, scattering and imaging • Imaging in compressed domain• Other applications: Illumination, resolution,
velocity analysis and tomography, demultiples
![Page 38: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/38.jpg)
Beamlet Localization (space-direction)
Figure 4: Spreading of beamlet ( )propagation. Top is the beamlet of , and bottom . 8 0
39Hz
![Page 39: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/39.jpg)
Dreamlet localization (t-f-x-k)
39 , 8Hz 39 , 0Hz Figure 3: Snapshots of a single dreamlet propagation. On the left is the dreamlet of ,and on the right, .
![Page 40: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/40.jpg)
Beamlet localization (Space-direction localization)
• Space localization Local perturbation theory: Beamlet propagator – Efficient migration algorithm in strongly heterogeneous
media• Direction localization Local angle domain analysis: – Local imaging matrix and angle gathers– Energy-flux Green’s function– Directional illumination analysis (DIA)– Local resolution analysis – Local inversion
![Page 41: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/41.jpg)
SEG 2D Salt model
![Page 42: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/42.jpg)
Local perturbation vs. global perturbation
Global references and global perturbations Local references and local perturbations
![Page 43: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/43.jpg)
Illumination analysis andTrue-reflection imaging
• Directional illumination analysis• Acquisition-aperture correction in the local
dip-angle domain with beamlet migration
![Page 44: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/44.jpg)
image by common-shot prestack G-D migrationimage by common-shot prestack G-D migration
Total Acquisition-Dip-Response intensity from all the 325 shotsTotal Acquisition-Dip-Response intensity from all the 325 shots
Total illumination intensity from all the 325 shotsTotal illumination intensity from all the 325 shots
![Page 45: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/45.jpg)
Acquisition-Dip-Response (horizontal) from all the 325 shotsAcquisition-Dip-Response (horizontal) from all the 325 shots
Acquisition-Dip-Response (45 down from horizontal) from all the 325 shots
Acquisition-Dip-Response (45 down from horizontal) from all the 325 shots
image by common-shot prestack G-D migrationimage by common-shot prestack G-D migration
![Page 46: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/46.jpg)
速度模型 (Velocity model on slice C of the SEG 3D salt model)
Example of 3D true-reflection beamlet migration(see Mao and Wu)
![Page 47: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/47.jpg)
True-reflection image (right)
vs. standard migration (left)
普通成像 ( 左 ) 和真反射成像 ( 右 ) 的对比
![Page 48: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/48.jpg)
Dreamlet localization (Full phase-space localization)
• Efficient seismic data decomposition (Ideal decomposition)
• Dreamlet propagator and migration – Link to fast asymptotic wave-packet propagation – Imaging in the compressed domain
![Page 49: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/49.jpg)
Changes of dreamlet coefficients with depth during Shot-domain prestack migration
Scattered field (data)
Source field
Scattered field (high-compression)
CR=5.6
CR=15.2
![Page 50: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/50.jpg)
Coefficient changes during dreamlet survey-sinking prestack depth migration
Variation of dreamlet coefficient amount during migration. The black line isfor the survey sinking dreamlet coefficients using sunk data.
Full data
Sunk data
![Page 51: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/51.jpg)
Conclusion
• Wave solutions can only exist on the light cone in the 4D Fourier space defined by the dispersion relation
• Physical wavelet defined by Kaiser is a localized wave solution by extending the light cone into complex causal tube. The effect is windowing on the light-cone.
• Dreamlet can be considered as a type of discrete physical wavelet defined on an observation plane
![Page 52: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/52.jpg)
Conclusion-continued
• Curvelet is good for decomposition of time-slice 4-D data; while dreamlet is good for depth-slice 4-D data.
• Causality (or dispersion relation) built into the wavelet (dreamlet) and propagator is a distinctive feature of physical wavelet which is advantageous for applications in wave data decomposition, propagation and imaging.
![Page 53: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/53.jpg)
Conclusion-continued
• The applications in illumination, true-reflection imaging, local angle domain analysis, imaging in compressed domain are only in the beginning.
![Page 54: Decomposition, extrapolation and imaging of seismic data using beamlets and dreamlets Ru-Shan Wu, Modeling and Imaging Laboratory, University of California,](https://reader030.fdocuments.in/reader030/viewer/2022032517/56649cbb5503460f94983a44/html5/thumbnails/54.jpg)
Acknowledgments
• This is a Group effort mainly conducted in the Modeling and Imaging Lab at UCSC. I thank all my colleagues and students. Bangyu Wu, Yu Geng and Jian Mao directly involved in the work of this talk.
• I am grateful to Chuck Mosher for initiating the study of wavelet transform on wave propagation and the continuous interaction with our group. I thank Jinghuai Gao for the collaboration, Dr. Howard Haber and Dr. Gerald Kaiser for their discussions and comments.
• This work is supported by WTOPI (Wavelet Transform On Propagation and Imaging for seismic exploration) Project at University of California, Santa Cruz.