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

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

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).

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)

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)

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:

• 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

Light cone in the Fourier space ( ),

0 (frequency)p

C

C

V

V

planep

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

Space-time light cone(from Wikipedia)

Construction of localized wave solutions

• Kaiser 1994 (Analytic signal transform)• Kiselev and Perel, 2000; Perel and Sidorenko,

2007 (Continuous wavelet transform)

• 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

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)

Space localization at t=0( ,0) : 3,10,15,100r

r

Time localization at r=0(real part- solid; imag- dashed)

(0, ) : 3,10,15,100t

t

Wavefield data on planes:

Data acquisition plane on surface Extrapolation planes during

migration/imaging

z

Surface

Extrapolated planes

•Data acquisition on the surface•Wave field downward continuation •Depth migration by downward-continuation or

Survey sinking + Imaging condition

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

Time-slicing in 4-D

A time-slice

Depth-slicing in 4D

Depth

(x)

A depth slice

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)

Dreamlet: A type of physical wavelet defined on observation planes

(data planes)

Wu et al., 2008; 2009; 2011 (SEG abstracts)

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

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

Dreamlet = Wavepacket Windowing on the light cone

0 (= : frequency)p

CCVV

planep

xk

zk

2 2 2( / )c

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

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

The window defined on the observation plane (red segment) and window for the whole space (green disk).

Examples of dreamlet decomposition on seismic data

The poststack data of SEG 2D salt model

Dreamlet decomposition of the SEG salt data by local exponential frames

x

t-f

Dreamlet decomposition of the SEG salt data using different thresholds: 1%

x

f - t

Dreamlet decomposition of the SEG salt data using different thresholds: 2%

Dreamlet decomposition of the SEG salt data using different thresholds: 3%

Dreamlet decomposition of the SEG salt data using different thresholds: 4%

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)

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

Beamlet Localization (space-direction)

Figure 4: Spreading of beamlet ( )propagation. Top is the beamlet of , and bottom . 8 0

39Hz

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, .

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

SEG 2D Salt model

Local perturbation vs. global perturbation

Global references and global perturbations Local references and local perturbations

Illumination analysis andTrue-reflection imaging

• Directional illumination analysis• Acquisition-aperture correction in the local

dip-angle domain with beamlet migration

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

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

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)

True-reflection image (right)

vs. standard migration (left)

普通成像 ( 左 ) 和真反射成像 ( 右 ) 的对比

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

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

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

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

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.

Conclusion-continued

• The applications in illumination, true-reflection imaging, local angle domain analysis, imaging in compressed domain are only in the beginning.

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.