Introduction to Seismic Imagingamalcolm/part2.pdf · 2010-08-24 · Toward a uniﬁed theory of...

Introduction to SeismicImaging

Alison MalcolmDepartment of Earth, Atmospheric and

Planetary SciencesMIT

August 20, 2010

Outline

• IntroductionI Why we image the EarthI How data are collectedI Imaging vs inversionI Underlying physical model

• Data Model• Imaging methods

I KirchhoffI One-way methodsI Reverse-time migrationI Full-waveform inversion

• Comparison of Methods

Approximate Techniques

• KirchhoffI Integral techniqueI Related to X-ray CT imagingI Generalized Radon TransformI Conventionally uses ray theory

• One-wayI Based on a paraxial approximationI Usually computed with finite

differences

One-Way MethodsPhysical Motivation

• downward continuation• imaging condition

Claerbout 71, 85

rs



v/2

s ss s

s ss s s s

r r r r r r r r rr



Claerbout 71, 85

rs


• downward continuation

• imaging condition

r rs

V(x)

s

V(x)

A Data Model

δG(s, r, t) =

∫X

∫T

G0(r, t−t0, x)V(x)∂2t G0(x, t0, s)dxdt0

δG(s, r, ω) = −∫

X

ω2G0(r, ω, x)V(x)G0(x, ω, s)dx

G0(r, t, x)

s

G0(x, t, s)

V(x)

r

A Data Model

δG(s, r, t) =

∫X

∫T


δG(s, r, ω) = −∫

X


⇓∫X

ω2G−(r, ω, x)G−(s, ω, x)V(x)dx

One-Way MethodsApproximating the Wave Equation

Idea (1D, c constant):

(∂2x − ∂2

t )u = (∂x − ∂t)(∂x + ∂t)u

c not constant:

(c(x)2∂2x − ∂2

t )u = (c(x)∂x − ∂t)(c(x)∂x + ∂t)u

− c(x)(∂xc(x))∂xu

c(x) smooth ⇒ better approximation


Taylor (81), Stolk & de Hoop (05)

Multi-dimensional:Lu = f

∂z

(u

∂zu

)=

(0 1

−A 0

)(u

∂zu

)+

(0

−f

)A(z, x, ∂x, ∂t) = ∂2

x − c(z, x)−2∂2t



Diagonalize in smooth background

∂z

(u+

u−

)=

(iB+ 00 iB−

)(u+

u−

)+

(f+f−

)b±(ξ, x, ω) = ±

√ξ2 − c0(x)−2ω2

B± FIOs

Solution:

G0 =

(G+ 00 G−

)

G0 propagator in c0



Diagonalize in smooth background

∂z

(u+

u−

)=

(iB+ 00 iB−

)(u+

u−

)+

(f+f−

)Solution:

G0 =

(G+ 00 G−

)

G0 propagator in c0



Relate u± to u and ∂zu(u+

u−

)= 1

2

((Q+)−1 −HQ+

(Q∗−)−1 HQ−

) (u∂zu

)

q± =

[(ω

c(z, x)

)2

− ‖ξ‖2

]−1/4

Q± differ in sub-principal partsQ± ψDOs

One-Way MethodsModeling Data Stolk & de Hoop (05)

δG(s, r, ω) ≈∫X

ω2Q∗−(r)G−(r, ω, x)V(x)G+(x, ω, s)Q+(s)dxdz

V(x) = 14Q−(z, x)δc(z, x)c0(z, x)

−3Q∗+(z, x)




V(x) = 14Q−(z, x)δc(z, x)c0(z, x)

−3Q∗+(z, x)

Reciprocity:ss r r




V(x) = 14Q−(z, x)δc(z, x)c0(z, x)

−3Q∗+(z, x)

Drop Qs∫X

∫Z

ω2G−(0, r, ω, z, x)G−(0, s, ω, z, x)

(E2E1a)(z, x)dzdx

E2 : b(z, r, s) 7→ δ(t)b(z, r, s)E1 : a(z, x) 7→ δ(r − s)a(z, r+s

2)

One-Way MethodsImaging Claerbout (78) Stolk & de Hoop (06)



rs



d(z, s, r, t) = H(z, 0)∗Q−1∗d(s, r, t)

(H(z, z0))(s, r, t, s0, r0) =

(G−(z, z0))(r, t, r0) ∗ (G−(z, z0))(s, t, s0)

H is an FIO H ∗ H is ΨDO





V(z, x) ≈ d(z, x, x, 0)

V(x) = 14Q−(z)δc(z, x)c0(z, x)

−3Q∗+(z)

Recall Q are ΨDOs & H∗H is ΨDO

We have again located the singularities

One-Way MethodsExample

0

5dep

th (

km)

130 135lateral position (km)

Approximate Techniques

• KirchhoffI Integral techniqueI Related to X-ray CT imagingI Generalized Radon TransformI Conventionally uses ray theory

• One-wayI Based on a paraxial approximationI Usually computed with finite

differences

‘Exact’ Techniques

• Reverse-time migration (RTM)I Run wave-equation backwardI Usually computed with finite

differencesI “No” approximations (to the acoustic,

isotropic, linearized wave-equation, for

smooth media assuming single scattering)

• Full-waveform inversion (FWI)I Iterative method to match the entire

waveformI Gives smooth part of velocity model

Reverse-Time MigrationModeling Data

δG(s, r, t) =

∫X

∫T


δG(s, r, ω) = −∫

X


G0(r, t, x)

s

G0(x, t, s)

V(x)

r

Reverse-Time MigrationModeling Data

δG(s, r, t) =

∫X

∫T


δG(s, r, ω) = −∫

X


F : δc 7→ δG is again an FIO

Reverse-Time MigrationForming an image

δG(s, r, ω) = −∫

X


F : δc 7→ δG is again an FIO

F∗ : δG 7→ δc is again an FIO

F∗F is ΨDO for complete coverageStolk et al. (09)

• no direct rays (transversal intersection)• no source-side caustics• c0 ∈ C∞

Reverse-Time MigrationForming an Image

Procedure:Whitmore (83), Loewenthal & Mufti (83), Baysal et al (83)

• back propagate in time


G0(r, t, x)

s

G0(x, t, s)

V(x)

r

Reverse-Time Migrationan Adjoint State Method

Lailly (83,84), Tarantola (84,86,87) Symes (09)

For a fixed source, s,

(c−2∂2t − ∇2)q(x, t; s) =

∫Rs

δG(r, t; s)δ(x − r)dr

q(·, t, ·) = 0 for t > T

receivers act as sources, reversed in time

Im(x) =2

c2(x)

∫ ∫q(x, t; s)∂2

t G0(x, t, s)dtds

Reverse-Time MigrationExample Liu et al (07)

‘Exact’ Techniques

• Reverse-time migration (RTM)I Run wave-equation backwardI Usually computed with finite

differencesI “No” approximations (to the acoustic,

isotropic, linearized wave-equation, for

smooth media assuming single scattering)

• Full-waveform inversion (FWI)I Iterative method to match the entire

waveformI Gives smooth part of velocity model

Full-waveform InversionProblem Setup Tarantola (87), Virieux & Operto (09)

Recall our initial formulation:

Lu := (∇2 −1

c2∂2

t )u = f

u = 0 t < 0

∂zu|z=0 = 0

FWI attempts to solve for c directly given u,f

no splitting of cbut band limited data ⇒ smooth solution

Full-waveform InversionProblem Setup Tarantola (87), Virieux & Operto (09)

Recall our initial formulation:

Lu := (∇2 −1

c2∂2

t )u = f

Compute:

minc‖Lu − d‖L2((S,R)×[0,T])

Full-waveform InversionSome Issues Symes (09)

Compute:

minc‖Lu − d‖L2((S,R)×[0,T])

• Objective function appears non-convexCan we restrict the domain of models sothat it is? e.g. cmin < c < cmax

• What space must c be in for objectivefunction to be differentiable?

• Data are finite bandwidth – we cannotresolve structures on all scales

Full-waveform InversionWork-around 1: Partial Linearization Symes (09)

Idea: Extend δc(x) from X to XExample: δc(x) 7→ δc(x, h) = δ(h)δc(x)

x

s

form image here

r

x + h/2x − h/2 image here

form extended

Claerbout (85) suggested this extension

Full-waveform InversionWork-around 1: Partial Linearization Symes (09)

Idea: Extend δc(x) from X to XExample: δc(x) 7→ δc(x, h) = δ(h)δc(x)

Now extendF[δc] 7→ δG

toF[δc(x, h)] 7→ δG

s.t. F is ‘invertible’ (I − F†F is smoothing)

Find c0 s.t. supp (δc(x, h)) ⊂ {(x, h)|h = 0}Stolk & de Hoop (2001), Stolk et al (2005) show invertibility

Full-waveform InversionWork-around 2: Separation of Scales

Pratt (99), Virieux (09)

minc‖Lu − d‖L2((S,R)×[0,T]) not convex

⇒ ‖cinitial − ctrue‖ must be small

Solve for G(ω) from ωmin to ωmax

No proof optimization converges

Full-waveform InversionWork-around 2: Separation of Scales, example

Virieux (09)

Outline

• IntroductionI Why we image the EarthI How data are collectedI Imaging vs inversionI Underlying physical model

• Data Model• Imaging methods

I KirchhoffI One-way methodsI Reverse-time migrationI Full-waveform inversion

• Comparison of Methods

Comparison of MethodsKirchhoff vs One-way Fehler & Huang (02)

Kirchhoff


Phase-shift


Split-step

Comparison of MethodsKirchhoff vs RTM Zhu & Lines (98)


Kirchhoff


Reverse-time

Comparison of MethodsOne-Way vs RTM Farmer (06)

Model


One-way


Reverse-time


One-way


Reverse-time

Comparison of MethodsReal Data Farmer 2006

Kirchhoff


One-way


Reverse-time

Summary of Methods

• Kirchhoff requires a lot of rays(or a simple model)

• One-way doesn’t handle turning waves

• RTM separation of up/down-going wavesis challenging

c0 is key• FWI optimization doesn’t convergence

from an arbitrary starting model

. . . and we never spoke about the source . . .

. . . or multiple-scattering . . .

Edip Baysal, Dan D. Kosloff, and John W. C. Sherwood.

Reverse time migration.Geophysics, 48(11):1514–1524, 1983.

J. F. Claerbout.

Imaging the Earth’s Interior.Blackwell Sci. Publ., 1985.

Jon F. Claerbout.

Toward a unified theory of reflector mapping.Geophysics, 36(3):467–481, 1971.

P. A. Farmer.

Reverse time migration: Pushing beyond wave equation.EAGE Annual Meeting, 2006.

Michael C. Fehler and Lianjie Huang.

Modern imaging using seismic reflection data.Annual Review of Earth and Planetary Sciences, 30(1):259–284, 2002.

P. Lailly.

The Seismic Inverse Problem as a Sequence of Before-Stack Migrations, pages 206–220.Conference on Inverse Scattering: Theory and Applications. SIAM, Philadelphia, 1983.

P. Lailly.

Migration Methods: Partial but Efficient Solutions to the Seismic Inverse Problem.Inverse Problems on Acoustic and Elastic Waves. SIAM, Philadelphia, 1984.

Faqi Liu, Guanquan Zhang, Scott A. Morton, and Jacques P. Leveille.

Reverse-time migration using one-way wavefield imaging condition.SEG Technical Program Expanded Abstracts, 26(1):2170–2174, 2007.

Dan Loewenthal and Irshad R. Mufti.

Reversed time migration in spatial frequency domain.Geophysics, 48(5):627–635, 1983.

C. C. Stolk and M. V. de Hoop.

Modeling of seismic data in the downward continuation approach.SIAM J. Appl. Math., 65(4):1388 – 1406, 2005.

C. C. Stolk and M. V. de Hoop.

Seismic inverse scattering in the downward continuation approach.Wave Motion, 43(7):579–598, 2006.

Christiaan C. Stolk, Maarten V. de Hoop, and William W. Symes.

Kinematics of shot-geophone migration.Geophysics, 74(6):WCA19–WCA34, 2009.

W. W. Symes.

The seismic reflection inverse problem.Inverse Problems, 25:123008, 2009.

A. Tarantola.

Inverse Problem Theory.Elsevier, 3 edition, 1987.

Albert Tarantola.

Inversion of seismic reflection data in the acoustic approximation.Geophysics, 49(8):1259–1266, 1984.

Albert Tarantola.

A strategy for nonlinear elastic inversion of seismic reflection data.Geophysics, 51(10):1893–1903, 1986.

M. E. Taylor.

Pseudodifferential Operators.Princton University Press, Princton, NJ, 1981.

J. Virieux and S. Operto.

An overview of full-waveform inversion in exploration geophysics.Geophysics, 74(6):WCC1–WCC26, 2009.

N. D. Whitmore.

Iterative depth migration by backward time propagation.In Expanded Abstracts, page S10.1. Society of Exploration Geophysicists, 1983.

Jinming Zhu and Larry R. Lines.

Comparison of kirchhoff and reverse-time migration methods with applications to prestack depthimaging of complex structures.Geophysics, 63(4):1166–1176, 1998.

[10, 11, 2] [17, 12, 8] [3, 19, 9] [1, 6, 7] [15, 16, 14] [13, 5, 20] [4, 18]

Introduction to Seismic Imagingamalcolm/part2.pdf · 2010-08-24 · Toward a uniﬁed theory of...

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