Multisource Least-Squares Extended Reverse-time Migration ... · Multisource Least-Squares Extended...

Background Theory and Implementation Numerical tests Conclusion and Discussion

Multisource Least-Squares

Extended Reverse-time Migration with

Preconditioning Guided Gradient Method

Yujin Liu[1][2] and William W. Symes[1]

1TRIP, Rice University

2LEON, China University of Petroleum (Huadong)

TRIP Annual Meeting 2013

Outline

1 BackgroundERTM vs LSERTMPhase EncodingPreconditioning

2 Theory and ImplementationERTM with Phase EncodingPreconditioner: approximated diagonal of HessianInversion Scheme: PGG method

3 Numerical testsSalt Model

4 Conclusion and Discussion

Outline

ERTM vs LSERTM

ERTM (Extended Reverse-Time Migraion):

is only adjoint of linearized extended Born Modeling(LEBM) operatorprovides extended image with limited resolution andimbalanced amplitudes

LSERTM (Least-Squares ERTM):

approximates inverse of LEBM operator usingiteration methodsprovides extended image with high resolution andbalanced amplitudeis more reliable in velocity analysis and AVO/AVAis more expensive!

ERTM vs LSERTM

is only adjoint of linearized extended Born Modeling(LEBM) operator

provides extended image with limited resolution andimbalanced amplitudes

ERTM vs LSERTM

approximates inverse of LEBM operator usingiteration methods

provides extended image with high resolution andbalanced amplitudeis more reliable in velocity analysis and AVO/AVAis more expensive!

ERTM vs LSERTM

approximates inverse of LEBM operator usingiteration methodsprovides extended image with high resolution andbalanced amplitude

is more reliable in velocity analysis and AVO/AVAis more expensive!

ERTM vs LSERTM

approximates inverse of LEBM operator usingiteration methodsprovides extended image with high resolution andbalanced amplitudeis more reliable in velocity analysis and AVO/AVA

is more expensive!

ERTM vs LSERTM

Review of Phase Encoding

Main idea: Encode all of shot gathers into one or several super-shotgathers with designed encoding functions so as to solve a smallernumber of wave equations.

Random phase encoding (Morton,1998; Romero,2000; Kreb et.al 2009;Tang, 2009; ...)

Plane-wave phase encoding (Zhang, 2005; Liu, 2006; Tang, 2009; ...)

Amplitude encoding (Godwin el al. 2010)

Deterministic source encoding (Symes, 2010; Gao, 2010)

Review of Phase Encoding

Main idea: Encode all of shot gathers into one or several super-shotgathers with designed encoding functions so as to solve a smallernumber of wave equations.

Random phase encoding (Morton,1998; Romero,2000; Kreb et.al 2009;Tang, 2009; ...)

Plane-wave phase encoding (Zhang, 2005; Liu, 2006; Tang, 2009; ...)

Amplitude encoding (Godwin el al. 2010)

Deterministic source encoding (Symes, 2010; Gao, 2010)

Review of Preconditioning

Main idea: Using preconditioner to accelerate the convergent rate ofiteration

Approximated diagonal of Hessian (Pratt, 1999; Shin, 2001; Tang, 2010; ...)

Structure-oriented filter (Prucha, 2002; Clapp, 2005)

Deblurring filter (Aoki et al., 2009)

Sparsity promotion (Herrmann et al. 2009)

Image-guided filter (Ma et al., 2010)

Review of Preconditioning

Main idea: Using preconditioner to accelerate the convergent rate ofiteration

Approximated diagonal of Hessian (Pratt, 1999; Shin, 2001; Tang, 2010; ...)

Structure-oriented filter (Prucha, 2002; Clapp, 2005)

Deblurring filter (Aoki et al., 2009)

Sparsity promotion (Herrmann et al. 2009)

Image-guided filter (Ma et al., 2010)

Outline

Linearized Born approximation

Acoustic constant density(ACD) wave equation:

(∇2 + ω2m(x, z))u(x, z, w) = −f(w)δ(x− xs) (1)

Extended ACD wave equation:

∇2u(x, z, w) + ω2

∫dym(x,y, z)u(y, z, w) = −f(w)δ(x− xs) (2)

Split the extended model into two parts:

m(x,y, z) = b(x, z)δ(x− y) + r(x,y, z) (3)

Linearized approximation:

(∇2 + ω2b(x, z))u0(x, z, w) = f(w)δ(x− xs) (4)

(∇2 + ω2b(x, z))δu(x, z, w) = −ω2

∫dyr(x,y, z)u0(y, z, w) (5)

LEBM operator and its adjoint

Linearized extended Born modeling (LEBM):

d(xr,xs, ω) = −ω2f(ω)

∫dxdhG(xr,x + h, ω)r(x,h)G(x− h,xs, ω)

Extended reverse time migration (ERTM):

r(x,h) = −∫

dxsdxrdω ω2f∗(ω)G∗(xs,x−h, ω)G∗(x+h,xr, ω)d(xr,xs, ω)

Encoded LEBM and ERTM

Encoding function:

α(xs, ps) =1√Nγ(xs, ps)

(ps: realization index; N : realization number; γ random sequence of signs.)

Encoded seismic data:

dobs(xr, ps, ω) =

∫dxs α(xs, ps)dobs(xr,xs, ω)

Encoded source wavefield:

S(x, ps, ω) =

∫dxsα(xs, ps)f(ω)G(x,xs, ω)

Encoded LEBM and ERTM

Encoded LEBM:

d(xr, ps, ω) = −ω2

∫dxdhG(xr,x + h, ω)r(x,h)S(x− h, ps, ω)

Encoded ERTM:

r(x,h) = −∫

dpsdxrdω ω2S∗(ps,x− h, ω)G∗(x + h,xr, ω)d(xr, ps, ω)

Preconditioner: approximated diagonal of Hessian

Diagonal of Hessian in the subsurface offset domain (Valenciano, 2006):

D(x,h) =

∫dxsdxrdω ω4|f(ω)|2 |G(x− h,xs, ω)|2|G(x + h,xr, ω)|2.

Encode receiver wavefield with β(xr, pr) = 1Nγ(xr, pr),

R(x, pr, ω) =

∫dxr β(xr, pr)G(x,xr, ω).

Encoded Diagonal of Hessian:

D(x,h, ps, pr) =

∫dω ω4|S(x− h, ps, ω)R(x + h, pr, ω)|2.

More approximation:

DSS(x,h, ps) =

∫ω ω4|S(x− h, ps, ω)|2|S(x + h, ps, ω)|2.

Observation:

It has shown that this approximation seems to be accurate enough as apreconditioner in least-squares inversion (Tang, 2010). Here we extend this ideainto subsurface offset domain.

Inversion Scheme: PGG Method

Encoded LEBM can be written in a compact form:

d = Lm

Encoded Least-squares extended reverse-time migration:

minmJLSM [m] =1

2‖ Lm− dobs ‖p +

2‖m ‖p

where ‖ · ‖p denotes `p norm with 1 ≤ p ≤ 2.

Equivalent form:

minmJLSM [m] =1

2‖Wr(Lm− dobs) ‖2 +

2‖Wmm ‖2

Methods:

Iteratively Reweighted Least-Squares (IRLS) method (Claerbout, 1992)

Non-linear inverse problemNeed to calculate weighting matrix at the outer loop of CG

Conjugate Guided Gradient (CGG) method (Ji, 2006)

A variant of IRLSLinear inverse problemOnly one calculation of L and LT is needed at each iteration

Preconditioning Guided Gradient (PGG) method

Updated version of CGGIncorporates preconditioner into CGGAdapts to phase encoding scheme

Algorithm 1 MLSERTM with PGG

1: for k = 0 · · ·niter do2: generate random sequence of signs γ3: encode sources and data to get L and dobs

4: rk = Lmk − dobs

5: compute Wkr

6: compute Wkm

7: dmk = WT,km D−1

SSLTWT,k

8: drk = Ldmk

9: αk =〈drk,rk〉〈drk,drk〉

10: mk+1 = mk+1 − αkdmk

11: end for

Outline

Reflectivity imaging test

Figure: Velocity model

Figure: Reflectivity

Figure: RTM

Figure: RTM after Laplacian filter

Figure: Approximated diagonal of Hessian

Figure: LSRTM without preconditioning

Figure: LSRTM with preconditioning

Figure: LSRTM with preconditioning and `1.5 norm on model

Figure: LSRTM with preconditioning and `1 norm on model

Extended reflectivity imaging test

Figure: Approximated diagonal Hessian in subsurface offset domain

Figure: LSERTM with preconditioning

Figure: LSERTM with preconditioning and `1.5 norm on model

Outline

Conclusion and Discussion

LSERTM provides reflectivity image with higher resolution andmore balanced amplitude, which is useful in the followingmigration velocity analysis and AVO/AVA analysis;

Seismic data is compressed greatly with the help of phaseencoding so that the efficiency of seismic imaging is improveddramatically;

Using approximated diagonal of Hessian as preconditioner canimprove the convergent rate of LSM;

A modified CGG inversion scheme namely PGG is proposed tosolve `p norm problem flexibly and efficiently;

Sparsity seems to be a good prior information in suppressingcrosstalk introduced by phase encoding, especially in imaging ofextended reflectivity.

Acknowledgments

Thank CSC for supporting my visit to TRIP!

Thank TRIP for hosting me!

Thank the sponsors of TRIP for their support!

Thank you!

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