an introduction to modeling, imaging and full waveform ... · pdf filean introduction to...

An Introduction to Modeling, Imaging and Full Waveform Inversion

Pengliang Yang

ISTerre, Univ. Grenoble Alpes, [email protected]

http://seiscope2.osug.fr

2016 Madagascar school, Zurich

SEISCOPE

Yang et al. (ISTerre) Introduction to modeling, imaging and FWI Madagascar 1 / 28

http://seiscope2.osug.fr

Outline SEISCOPE

1 The wave equation

2 Forward modeling

3 Absorbing boundary condition

4 Seismic imaging/migration

5 Full waveform inversion (FWI)


Outline SEISCOPE

1 The wave equation

2 Forward modeling





The wave equation SEISCOPE

Particle velocity vector v = [vx , vy , vz ]T ; stress vector σ = [σxx , σyy , σzz , σyz , σxz , σxy ]T

Newton’s law: ρ∂tvi = ∂jσij , i , j ∈ x , y , z = 1, 2, 3Generalized Hooke’s law: σij = cijklεkl

Voigt indexing: (11)→ 1, (22)→ 2, (33)→ 3, (23) = (32)→ 4, (13) = (31)→ 5,(12) = (21)→ 6

ρ∂tv = Dσ + fv

∂tσ = CDTv + fσ,C =

c11 c12 c13 c14 c15 c16

· c22 c23 c24 c25 c26

· · c33 c34 c35 c36

· · · c44 c45 c46

· SYM · · c55 c56

· · · · · c66

,DT =

∂x 0 00 ∂y 00 0 ∂z0 ∂z ∂y∂z 0 ∂x∂y ∂x 0

(1)

Triclinic: 21 independent coefficients in C

Monoclinic: 13 independent coefficients in C

Orthorombic: 9 independent coefficients in C

Transverse isotropic: 5 independent coefficients in C


Isotropic elastic medium:

C =

λ+ 2µ µ µ 0 0 0µ λ+ 2µ µ 0 0 0µ µ λ+ 2µ 0 0 00 0 0 µ 0 00 0 0 0 µ 00 0 0 0 0 µ

(2)

Acoustic medium (µ = 0): pressure p = (σxx + σyy + σzz)/3ρ∂tv = ∇p∂tp = κ∇ · v, κ = ρc2

(3)

1st order equation system → 2nd order equation

1

c2∂ttp − ρ∇ ·

(1

ρ∇p)

= f (4)

Constant density:

(1

c2∂tt −∇2)p = f ,∇2 = ∂xx + ∂yy + ∂zz (5)


Outline SEISCOPE

1 The wave equation

2 Forward modeling





Forward modeling SEISCOPE

Modeling=Simulate wave propagation by solving wave equation numerically

Which numerical method?

1 Finite difference method (FDM)

2 Finite element method (FEM) → Spectral element method (SEM)

3 Fourier/spectral method: pseudo spectral method (PSM), lowrank finite difference

Tradeoff between computation and accuracy?

FDM outperforms FEM and PSM due to excellent efficiency;

SEM, PSM, lowrank FD arrive at spectral accuracy in space;

FEM, SEM are more flexible to handle complex topography, generally moreexpensive than FDM;

In which domain: Time domain or Frequency domain?


Numerical methods: Recap SEISCOPE

1. Finite difference method (FDM) in time domain

direct discretization of the original differential equation, both in time and in space

explicit time integration: n→ n + 1, ( 1c2 ∂tt − ∂xx)p = f

pn+1j − 2pn

j + pn−1j

∆t2c2− ∂xxpn = f n ⇒ pn+1

j = 2pnj − pn−1

j +∆t2c2

∆x2(pn

j+1 − 2pnj + pn

j−1)

(6)

implicit time integration: n→ n + 1, matrix inversion

pn+1 − 2pn + pn−1

∆t2c2− ∂xxpn+1 = f n (7)

⇒ Apn+1 = b,A =

2 + α2 −1

−1. . .

. . .

. . .. . . −1−1 2 + αM

, αj =∆x2

c2j ∆t2

(8)


1. Finite difference method (FDM) in frequency domain: solve Helmholz equation

direct discretization of the original differential equation, both in frequency and space

implicit scheme with matrix inversion p = A−1 f

(1

c2∂tt − ∂xx)p = f ⇒ −ω

2

c2p − ∂xx p = f ⇒ −ω

2

c2j

pj −pj+1 − 2pj + pj−1

∆x2= f (9)

⇒

. . .

. . .. . .

· · · − 1∆x2

2∆x2 − ω2

cj2 − 1

∆x2 · · ·. . .

. . .. . .

︸︷︷︸

A

...pn−1

pn

pn+1

...

︸︷︷︸

p

=

...

f...

︸︷︷︸

f

(10)

very efficient for multiple sources due to only 1 time of matrix inversion:[p1, p2, . . .] = A−1 [f1, f2, . . .]

easy to incorporate attenuation c(ω) = c0(1− isign(ω)2Q

) → more diagonal dominantfor stable matrix inversion

filling the matrix in multifrontal parallel solver: MUMPS, umfpack, ...



2. Finite element method (FEM): matrix inversion at each time step⇒ solving the weak form of the differential equation, test/basis function φj ∈ [0, 1]Second order wave equation with displacement∫ 1

0

(∂ttu(x , t)− c2(x)∂xxu(x , t)− f (x , t)

)φj(x)dx = 0 (11)

⇒ state variable u(x , t) =∑N

i=1 ui (t)φi (x): polynomial expansion separable over spaceand time∑

i

∂ttui

∫ 1

0

φiφjdx︸︷︷︸Mji

+c2∑i

ui

∫ 1

0

∂xφi∂xφjdx︸︷︷︸Kji

=

∫ 1

0

f φjdx ⇒ M∂ttu + Ku = f ′ (12)

Mun+1 − 2un + un−1

∆t2+ Kun = f ′ ⇒ un+1 = 2un − un−1 + M−1(f ′ − Kun) (13)

SEM (Komatisch et al., 1998): specific mesh, diagonal mass matrix M−1 forefficient wave extrapolation by choosing GLL collocation points



3. Fourier/spectral methods: computing spatial derivative using Fourier transform

Pseudo spectral method (PSM):

f (x) =

∫f (k)e ikxdk,⇒ ∂x f =

∫ikf (k)e ikxdk, ∂xx f =

∫−k2f (k)e ikxdk (14)

1

c2∂ttp = (∂xx + ∂zz)p ⇒ pn+1 = 2pn − pn−1 + ∆t2c2F−1[−(k2

x + k2z )Fpn] (15)

Lowrank finite difference method (Fomel et al., 2013):

p(x, t + ∆t) =

∫p(k, t)e iφ(x,k,∆t)dk,=

∫e ik·xW (x, k)p(k, t)dk, (16)

W (x, k) = e i [φ(x,k,∆t)−k·x],W (x, k) ≈M∑

m=1

N∑n=1

W (x, km)amnW (xn, k) (17)

Approximate lowrank decomposition leads to

p(x, t + ∆t) ≈M∑

m=1

W (x, km)N∑

n=1

amn

∫e ik·xW (xn, k)p(k, t)dk, (18)


Outline SEISCOPE

1 The wave equation

2 Forward modeling





Absorbing boundary condition (ABC) SEISCOPE

PA (Paraxial approximation, radiation) boundary condition (Clayton and Engquist, 1977):→ cancel the refection on the boundary using waves propagating in opposite directionaccording to pseudo differential operator decomposition

(1

c2∂tt − ∂xx − ∂zz)p = −(∂x +

1

c∂t√

1− S2)(∂x −1

c∂t√

1− S2)p = 0, S =∂z∂t/c

(19)

1st order approximation:√

1− S2 ≈ 1, perfect in 1D S = 0, accurate within aspecific range of angles in multi-dimensions!

(1

c2∂tt−∂xx)p = (

1

c∂t−∂x)(

1

c∂t+∂x)p = 0⇒ x ∈ [0, L]

1c∂tp(0, t)− ∂xp(0, t) = 0

1c∂tp(L, t) + ∂xp(L, t) = 0

(20)

2nd order approximation:√

1− S2 ≈ 1− 12S2(

∂x −∂tc

+c∂zz2∂t

)p = 0⇒ ∂xtp −

1

c∂ttp +

c

2∂zzp = 0(x = 0) (21)


Sponge layers/Gaussian taper boundary condition (Cerjan et al., 1985):→ multiply an exponential decaying factor e−dx∆t in boundary layers at each time step

easy to implement, stable in any anisotropic medium!

a good choice for the damping profile to achieve PML-like effect:

dx = −3vmax

2Llog(Rc)

(xL

)2

,Rc = 10−3 ∼ 10−6 (22)

PML(Perfectly matched layer)(Berenger, 1994): 1st order wave equation system→ coordinate transformation in complex-valued anisotropic medium

x = x +1

iω

∫ x

0

dx(s)ds ⇒ ∂x =1

1 + dxiω

∂x , (23)

intrinsically unstable in anisotropic medium for long duration simulation!

Frequency domain multiplication → Time domain convolution, implemented byrecursion with memory variables

∂tv = 1ρ∇p

∂tp = κ∇ · v⇒

∂tvi = 1

ρ∂ip − divi , i ∈ x , y , z

∂tp = κ(∂xvx + ∂yvy + ∂zvz)− (dx + dy + dz)p(24)


Outline SEISCOPE

1 The wave equation

2 Forward modeling





Imaging condition (IC) SEISCOPE

Deconvolution (U/D) IC (Claerbout, 1971): Recorded signal (upcoming wavefield atreceivers) = Earth’s reflectivity * downgoing source wavelet (downgoing wavefield).

U(x, t) = R(x) ∗ D(x, t)⇒ R(x) =U(x, t)

D(x, t), unstable (25)

Zero-lag crosscorrelation IC → illumination compensation

I (x) =nt∑i=1

U(x, ti )D(x, ti )⇒

I ′(x) =

∑nti=1 U(x, ti )D(x, ti )∑nt

i=1 D2(x, ti )

=U1D1 + · · ·+ UiDimax |·| + · · ·+ UntDnt

D1D1 + · · ·+ DiDimax |·| + · · ·+ DntDnt

(26)

requires simultaneous accesing to the source and receiver wavefields

Excitation amplitude IC (Chang and McMechan, 1986): taking maximum absoluteof downgoing wavefield at excitation time ti

I (x) =U(x, ti )

max |D(x, ti )|(27)

knowing ti is enough, no need for storing or reconstructing source wavefield


Imaging condition (IC)→ Angle gathers SEISCOPE

Extended IC (Sava and Fomel, 2006): redundant dimensions to evaluate energy focusing

Conventional IC:I (x) = R(x, t = 0),R(x, t) = U(x, t) ∗ D(x, t)

Space-shift IC: h = 0→ conventional IC

I (x, h) = R(x, h, t = 0),R(x, h, t) = U(x + h, t) ∗ D(x− h, t) (28)

Time-shift IC: τ = 0→ conventional IC

I (x, τ) = R(x, τ, t = 0),R(x, τ, t) = U(x, t + τ) ∗ D(x, t − τ) (29)

Practical usage in RTM and FWI: horizontal subsurface offset shift, (x ± h, y , z)

Implementated in frequency domain via Fourier transform

R(m, h) =∑ω

U(x + h, ω)D∗(x− h, ω) R(m, τ) =∑ω

U(x, ω)D∗(x, ω)e2iωτ (30)

Excitation amplitude IC+ extended time/space shift ⇒ an easier way to obtainangle gathers with low computation and memory cost? R(x, τ)⇒ R(x, θ)


Outline SEISCOPE

1 The wave equation

2 Forward modeling





FWI=Modeling engine+Optimization SEISCOPE

Objective=Waveform misfit

minm

C(m) =1

2‖dcal(m)− dobs‖ (31)

Given an initial model m0, iteratively update the model

mk+1 = mk + γk∆mk (32)

using different optimization methods: ∆mk = −H−1∇mC

Steepest descent method (SDM): ∆mk = −∇mC

Nonlinear conjugate gradient (NLCG) (Pica et al., 1990):∆mk = dk = −∇mC + βdk−1

Quasi-Newton method, L-BFGS (Brossier, 2011): H ≈ Ha approximate Hessianusing previously memorized gradients ∇C |k ,∇C |k−1,∇C |k−2, · · ·Newton method, Truncated Newton (Santosa and Symes, 1988; Metivier et al.,2014): compute Hessian vector product using CG method within few iterations

Gauss-Newton (Shin et al., 2001): Ha = J†J, approximate Hessian using first orderscattering information


Frequency domain FWI SEISCOPE

Frequency/Fourier domain FWI (Pratt et al., 1998): −( 1v2ω

2 +∇2)p = f

C(m) =1

2

∑s

∑r

∑ω

|d s,rcal (m, ω)− d s,r

obs(ω)|2, d s,rcal (m, ω) = Rs,r p (33)

Multiscale approach: from low frequency to high frequency

Extremely efficient for multisource problem: same A−1 for any source fi ,[p1, p2, . . .] = A−1 [f1, f2, . . .]

Easy for incorporating seismic attenuation: complex-valued velocity

Super-big matrix inversion requiring large storage in 3-D

No way for data windowing and selective inversion

→ Laplacian-Fourier domain (Shin and Cha, 2009): Fourier domain with dampedtraces


Time domain FWI SEISCOPE

Time domain: Convenient for data pre-processing: windowing, filtering, denoising

C(m) =1

2

∑s

∑r

∫ T

0

dt|d s,rcal (m, t)− d s,r

obs(t)|2, d s,rcal (m, t) = Rs,rp (34)

Lagrange functional with model parameter m ≡ c

L(m, p, p) = C(m) + 〈p, ( 1

c2∂tt −∇2)p − f 〉Ω×[0,T ] (35)

1 forward wave equation:

∂L

∂p= 0⇒ (

1

c2∂tt −∇2)p = f (36)

2 adjoint state equation: different misfit C only change adjoint source

∂L

∂p= 0⇒ (

1

c2∂tt −∇2)p = −∂C

∂p= −R†(Rp − dobs) (37)

3 update the model based on the gradient

∇mC =∂L

∂m= − 2

c3

∫ T

0

dtp∂ttp (38)


3D viscoelastic FWI in time domain (Yang et al., 2016a) SEISCOPE

1. Forward equation: ρ∂tv = Dσ + fv

∂tσ = CDTv − C :: Γ∑L`=1 y`ξ` + fσ

∂tξ` + ω`ξ` = ω`DTv, ` = 1, · · · , L,

(39)

2. Adjoint equation:ρ∂t v = Dσ +

∑L`=1 Dξ` + ∆dv

∂tσ = CDT v + C∆dσ

∂t ξ` − ω`ξ` = ω`y`(C :: Γ)C−1σ, ` = 1, · · · , L.(40)

where

Γ =

Q−111 Q−1

12 Q−113 Q−1

14 Q−115 Q−1

16

Q−121 Q−1

22 Q−123 Q−1

24 Q−125 Q−1

26

Q−131 Q−1

32 Q−133 Q−1

34 Q−135 Q−1

36

Q−141 Q−1

42 Q−143 Q−1

44 Q−145 Q−1

46

Q−151 Q−1

52 Q−153 Q−1

54 Q−155 Q−1

56

Q−161 Q−1

62 Q−163 Q−1

64 Q−165 Q−1

66

(41)


3. Gradient expression (Yang et al., 2016a):

∂χ

∂ρ=

∫ T

0

dtv†∂tv,

∂χ

∂CIJ=−

∫ T

0

dtσ†C−1 ∂C

∂CIJC−1(∂tσ − fσ),

∂χ

∂Q−1IJ

=

∫ T

0

dtσ†C−1(C ::∂Γ

∂Q−1IJ

)(L∑`=1

y`ξ`),∂χ

∂QIJ= −Q−2

IJ

∂χ

∂Q−1IJ

. (42)

4. Constant-Q approximation by Least-squares method (Blanch et al., 1995; Yang et al.,2016a)

minγ`

χQ1 , χQ

1 =

∫ ωmax

ωmin

(L∑`=1

γ`ω`ω

ω2 + ω2`

− γ−1)2dω. (43)

in whichY IJ` (x) = y`Q

−1IJ (x) with y` = γ γ`. (44)


Accessing incident wavefield backwards SEISCOPE

1n

n N

2n

1n N 1/2 1( , )n np v

1/2 1( , )n np v

1/2( , )N Np v

3/2 1( , )N Np v

forward simulation

backward reconstruction

adjoint wavefieldforward wavefield

crosscorrelation

n N

1n N

1n

2n 3n

3n

adjoint simulation


Accessing incident wavefield backwards SEISCOPE

It is mandatory to access incident wavefield backwards to build FWI gradient (or applyimaging condition in RTM)

1 reading the stored forward wavefield from the disk, even with data compression (Sunand Fu, 2013; Boehm et al., 2016)Reading disk memory is extremely slow for I/O intensive problems!

2 inferring from the final snapshots and the saved boundaries via reverse propagation(RP) (Tromp et al., 2005; Dussaud et al., 2008; Clapp, 2008; Yang et al., 2014):Efficient: Computation is much faster compared with disk accessing; may involvelarge 3D boundary storage, which can be significantly reduced using interpolation(Yang et al., 2016c);

3 remodeling using checkpointing (Griewank, 1992; Griewank and Walther, 2000;Symes, 2007; Anderson et al., 2012) from stored state to another state: Muchbetter than disking reading, more expensive compared with reverse propagation dueto repeated modeling!

4 checkpointing-assisted reverse-forward simulation (CARFS) =RP+checkpointing(Yang et al., 2016b) works in both attenuating and non-attenuating medium!


Reverse propagation SEISCOPE

Reverse propagation based on the final snapshot and stored boundaries:

forward modeling:pn+1 = 2pn − pn−1 + ∆t2v 2∇2pn (45)

store boundary at each forward timestepping

reverse propagation:pn−1 = 2pn − pn+1 + ∆t2v 2∇2pn (46)

inject boundary at each backward timestepping


Exercises SEISCOPE

Goal:1 understand the fundamental gadgets in Madagascar2 write codes within the framework of Madagascar3 reproduce nummerical experiments using SConstruct

Method: FDM

Steps:

1 forward modeling with Clayton-Enquist ABC2 wavefield reconstruction by RP+stored boundaries3 full waveform inversion

Where is the exercise?1 RSFSRC/book/xjtu/modeling2fwi2 https://yangpl.wordpress.com/activities/


Acknowledgment SEISCOPE

Seiscope consortium for financial support on the travel

Community effort for developing Madagascar: a great tool

Sergy Fomel for the invitation

ETH Zurich for providing the sharing opportunity

Thanks for your attention!


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