Non periodic homogenization for elastic wave …mesoimage.grenoble.cnrs.fr/IMG/pdf/capdeville13.pdfNon periodic homogenization for elastic wave forward and inverse problems in seismology

Non periodic homogenization for elastic waveforward and inverse problems in seismology

Y. Capdeville 1, P. Cupillard 2, J.J. Marigo 4, P. Cance 1, L. Guillot 5,E. Stutzmann 3, J.P. Montagner 3

1Laboratoire de Planetologie et de Geodynamique de Nantes, CNRS, France

2Univ. Nancy; 3IPGP; 4LMS, Ecole Polytechnique; 5CEA

GDR MesoImage 2013, 13/12/2013

INSTITUT DE PHYSIQUE DU GLOBE DE PARIS

IPGP

Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 1

Introduction and motivationsIntroduction and motivations

Introduction : About seismologySeismology mainly gathers

seismic imaging andtomography

Earthquake studies

Seismic risk

...

Most (all ?) of seismology researches need a forward problem to be solvedto link data and models.



The forward problem G (elastic wave equation)

The elastic wave equation is not numerically difficult to solve.Nevertheless, in practice, difficulties come from

medium complexity (anisotropy, anelasticity, fluids, gravity, complexinterfaces, long propagation (> 1000λmin) , small scales, etc)

from the fact that data have a rich content ([1/10s-1000s]) :

Raw data example :07/09/97, Ms=6.8,Venezuela

In general, solving the full wave equation for the whole frequency band isstill difficult.Seismologists use a wide range of tools (Finite Differences, DiscontinuousGalerkin, Spectral elements ...)



Forward modeling : difficult cases

Methods accurately handling discontinuities rely on a mesh ...

figure E. Delavaud

Marmousi

SEAM3D :

All these difficulties are related to small scales / minimum wavelength.Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 4


Introduction and motivations

Seismologists often work with frequency band limited data. → thecorresponding wavefield has a minimum wavelengthObservations indicate that waves of a given wavelength are sensitiveto inhomogeneities of scales much smaller than this wavelength onlyin an effective way. This is one of the reasons why seismic waves canbe used to image the earth.

The earth contains inhomogeneities at all scales.

?



Introduction and motivations

Period range : [150s-1000s]Computed with :

SAW24 and PREMmodels

moho, topography,oceans, rotation,ellipticity



Introduction and motivationsInverse problem, full waveform inversion (FWI)

FWI can only be performed in a limited frequency band.

The best that can be recovered is what is “seen” by the wavefield. Itis an effective version (ρ∗, c∗) of the real earth (ρ, c).

in that case, we know (at best) (ρ∗, c∗), but we look for (ρ, c).

Forward modeling, full waveform modeling

full waveform modeling can only performed in a limited frequencyband.

In many cases the elastic model (ρ, c) we which to propagate incontains details much smaller that the minimum wavelength. Theeffective model (ρ∗, c∗) would be enough to propagate the wavefield.

In that case, we know (ρ, c), but we would like to have (ρ∗, c∗).

Objective

Objective : for a given λ0 = 1/k0, find the upscaling operator

(ρ∗, c∗) = Hk0 (ρ, c)


Homogenization : forward problem

Small scales from the forward modeling point of view

Considering a 3-D medium, for which we can define

a smallest inhomogeneities typical size λh

a minimum wave speed αmin

Solving the wave equation (with FD, SEM ...) for

Ns sources

maximum frequency fmax ⇒ λmin = αmin

fmax

The computing time tc is1 in a smooth 3-D medium (that is λh λmin)

tc ∝ Nsλ−4min

2 in a rough 3-D medium (that is λh λmin)

tc ∝ Nsλ−4h

In a rough medium, the objective of homogenization is tobring back the numerical cost from Nsλ

−4h to Nsλ

−4min



Two scale homogenization : spatial smoothing

We need a smoothing operator :

Fk0 (g)(x) = wk0 ∗ g(x)

such that Fk0 (g) doesn’t contains any variation smaller than λ0 = k−10

Filtering wavelet wk0 example (here in 2-D) :

18 km

20 km

22 km

x1

18 km

20 km

22 km

x2

0.0000

0.0005

0.0010

0.0015

kx2

(1

/m)

0.0000 0.0005 0.0010 0.0015

kx1 (1/m)

0 1

wk0(x) |wk0|(k)



Small scale inhomogeneities for layered media

We know since the work of Backus (1962) that the effective medium isnot just a low pass filtered version of the original medium.

nV ∗SH

2 = V 2S,1 + V 2

S,2 + ...+ V 2S,n

n

V ∗SV

2 =1

V 2S,1

+1

V 2S,2

+ ...+1

V 2S,n

λ

L

λ >> L

(the density is constant)



Small scale inhomogeneities for layered media

Backus (1962) shown that, from a finely layered media (1-D media),described by the A,C ,F , L,N elastic parameters for TI media, forλmin >> 1/k0, the effective parameters A∗,C∗,F ∗, L∗,N∗ can becomputed as :

1

C∗ = Fk0 (1

C)

1

L∗= Fk0 (

1

L)

A∗ − F ∗2

C∗ = Fk0 (A− F 2

C)

F ∗

C∗ = Fk0 (F

C)

N∗ = Fk0 (N)

Note that, in the isotropic case : A = C = λ+ 2µ, L = N = µ and F = λ.

We therefore already know that

Hk0 (ρ, c) 6= Fk0 (ρ, c)



For higher dimension (2-D, 3-D), the effect of small scales has beenstudied for long

in mechanics with the two scale homogenization of periodic media(e.g. Auriault & Sanchez-Palencia (1977), Bensoussan & al (1977),Sanchez-Palencia (1980) ...).

with numerical upscaling (HMM (B. Engquist (2003)) and others ...)

But our problem is neither random or with scale separation ...



Two scale homogenization : the non-periodic case

Two scale homogenization is a classical asymptotic solution to takeinto account small inhomogeneities ;

it is mostly developed for periodic or stochastic media ;

we propose an extension to the non-periodic deterministic case

The solution to the wave equation u is sought as an asymptotic series :

u(x, t) =∑i≥0

ε0iuε0,i (x,

x

ε0, t)

where

ε0 =λ0

λmin

λ0 is user defined. For scales λ

λ > λ0 are the macroscopic scales

λ < λ0 are the microscopic scales



Two scale homogenization : the non-periodic case

Compared to the periodic case where the “cell” property c are easy tobuild from the original periodic property cε(x) :

c(y) = cε(εy)

where

y =x

εand ε =

`

λmin,

the difficulty of the non-periodic case is to build, for each x , the “cell”property

cε0 (x, y)

from the original c(x) such that the results make sens.



Two non periodic scale homogenization : workflow

Step 1. For a given complex elastic model defined by its elastic tensorc(x), solve the following set of problems (6 in 3-D) (the cell problem) tofind the initial guess correctors χkl

s :

∇ · c :(∇χkl

s

)= Fkl

Fkl = −∇ · (c : (ek ⊗ el))

+ periodic boundary conditions

Ω6

Ω6

Ω

Ω Ω

ΩΩΩ

Ω Ω

Ω1 2 3

4 5

7 8 9

This is a simple static elastic equation with a set of loading. We havecurrently two solvers to solve the above equations in 2D and 3D :

one based on a finite element method. Optimal but needs a finiteelement mesh (based on tetrahedron).

one based on FFT. Easy to implement. Does not need a mesh. Canbe a problem for strongly discontinuous media.




Step 2. Once the initial corrector guess χkls is obtained :

Compute quantities :

Gs(x) = I +1

2

(∇χs + T∇χs

)Hs(x) = c : Gs

The ε0 effective tensor can be directly be obtained as

cε0,∗(x) = Fk0 (Hs) : Fk0 (Gs)−1(x)

( ε0 = λ0/λmin = (k0λmin)−1)

the effective density is simply :

ρε0,∗ = Fk0 (ρ)

final two scale quantities cε0 (x, y) and then Gε0 (x, x/ε0),Hε0 (x, x/ε0), χε0 (x, x/ε0) can also be built.




Step 3.

the order 0 uε0,0 solution can be obtained solving the effective waveequation :

ρ∗,ε0∂ttuε0,0 −∇ · 〈σε0,0〉 = f

〈σε0,0〉 = c∗,ε0 : ε(uε0,0

)Once the order solution uε0,0, we can access to the order 0 stressand deformation with :

σε0,0(x) = Hε0 (x, x/ε0) : ε(uε0,0

)(x)

εε0,0(x) = Gε0 (x, x/ε0) : ε(uε0,0

)(x)

the partial order 1 displacement (cite effect) is then obtained as :

u(x) = uε0,0 + ε0χε0 (x, x/ε0) : ε(uε0,0) + O(ε0)

(in practice, often in O(ε20) because 〈u1,ε0〉 is small)



A 2-D random example

The inner square is a300×300 homogeneoussquares cells (100×100m2). In each cell, thedensity, λ and µ aredetermine randomly withcontrast up to 50%.




The source is an explosion (1.5Hz of central frequency, 3.6Hz.λmin ' 0.8km.)



A 2-D random example : 3 intuitive solutions

-1

0

1

x component

-0.2

0

0.2

y component

-1

0

1

-0.2

0

0.2

5 10 15time (s)

-1

0

1

5 10 15time (s)

-0.2

0

0.2

(i)

(ii)

(iii)

(i) velocity average(ii) elastic parame-ter average(iii) non-matchingmesh




Order 0 homogenized model




Order 0 homogenized model




Black line : refer-ence solution

Red line :

Left panels :uε0,0 for c∗,ε0

Right panels :u obtained ina velocitylow-passfilteredmediumV ∗

p = Fk0 (Vp)V ∗

s = Fk0 (Vs)



A 2-D random example : source at the center of the square




The homogeneization theory at the order 0 allows to correct the momenttensor for interactions with inhomogeneities within the nearfield of thesource :

M∗,ε0 = tGε0 (x0, x0/ε0) : M

For our example :




-1

-0.5

0

0.5referencehomogenizedelastic tensor average

x1 xomponent

5 10 15time (s)

-0.4

-0.2

0

0.2

0.4

0.6

x2 component



Marmousi2 example

Homogenization allows for simple meshes, even for complex media.

Mesh for the original model Mesh for the homogenized model


Homogenization : 3D

3D homogenization

bc

ld

32km

λmin :

λdom :

A 3D homogenization tool is ready andunder testing (P. Cupillard, post-docfunded by the ANR meme)

Dis

plac

emen

t

10 20 30

Time (s)

Dis

plac

emen

t

10 20 30

Time (s)

Dis

plac

emen

t

10 20 30

Time (s)10 20 30

Time (s)10 20 30

Time (s)10 20 30

Time (s)10 20 30

Time (s)10 20 30

Time (s)10 20 30

Time (s)

x component y component z component

average

order 0

reference


Homogenization : 3D

Some remarks compared to other homogenizationtechniques

Compared to more classical techniques, such asperiodic two scale homogenizationHMM methods (Heterogeneous Multi-Scale Methods of B. Engquist)others ...,

the proposed technique fails to achieve one of the main objective : havingnumerical cost for the homogenization problem depending on themacro-scale and not on the micro-scale problem.Nevertheless, our objective is to have a wave propagation cost (thousandsof time steps and sources) linked to the macro-scale size problem, andthis objective is fulfill.For λh smallest inhomogeneities typical size,

Initial cost (in 3D) :

Nsλ−4h

New cost :

λ−3h + Nsλ

−4min

with λh λmin, the new cost is interesting.


Homogenization : inverse problem

Small scales from the inverse problem point of view

Inverse problem, full waveform inversion (FWI)

FWI can only be performed in a limited frequency band.

The best that can be recovered is what is “seen” by the wavefield. Itis an effective version (ρ∗, c∗) of the real earth (ρ, c).

We expect a FWI result to be at best an effective medium.As a first step, we test this intuitive idea in the layered media simplecase.



Residual homogenization of layered media

Non-periodic homogenization of layered media is simple because ananalytically solution exists to the cell equation. In such case, theBackus parameters are involved (still with ε0 = (k0λmin)−1) :

ρε0∗ = Fk0 (ρ)

1

C ε0∗ = Fk0

(1

C

)1

Lε0∗ = Fk0

(1

L

)Aε0∗ − (F ε0∗)2

C ε0∗ = Fk0

(A− F 2

C

)F ε0∗

C ε0∗ = Fk0

(F

C

)Nε0∗ = Fk0 (N)

if we define the Backus parameters p(r) = (p1, ..., p6)(r) with

p1(r) = ρ(r)

p2(r) =1

C(r)

p3(r) =1

L(r)

p4(r) =

(A− F 2

C

)(r)

p5(r) =F

C(r)

p6(r) = N(r)

thenpε0∗ = Fk0 (p)



Residual homogenization of layered media

We introduce the notion of residual homogenization, that allows tohomogenize only the “difference” between a reference model andgiven model.

pε0∗(r) = pref(r) + Fk0 (p− pref) (r)

where

pref is the Backus parameter vector of the “reference” model. The“reference” model can be anything, related or not to the model pand with or without small scales and discontinuities.

For the C parameter, it gives :

1

C ε0∗ =1

Cref+ Fk0

(1

C− 1

Cref

)



Residual homogenization in layered media

6000 6200 6371

4

5

km

/s

prem

test1homo (Vsh)

homo (Vsv)

Vs

6000 6200 6371

6

7

8

9

10

homo (Vph)

homo (Vpv)

Vp

6000 6200 6371radius (km)

3

4

10

00

kg

/m3

6000 6200 6371radius (km)

0.7

0.8

0.9

1

1.1

ηρ



Residual homogenization in layered media

3000 3500 4000 4500 5000time (s)

0

premtest1homogenized

3000 3500 4000 4500 5000time (s)

0



Full waveform inversion in layered media

To test the idea that a FWI can recover at best an homogenized model

we choose a reference model (e.g. PREM) : pref

layered earth model parameters : Backus residuals with respect topref,

δp = p− pref

vertical (depth) discretization : polynomial of degree N. Defined byvalues at discreet radius r0, ..., rN (rN=free surface).

Finally, the model vector m is

m = (δp1(r0), δp2(r0), ..., δp6(r0), δp1(r1), ..., δpj(rk), ..., δp6(rN)) ,

5000 5200 5400 5600 5800 6000 6200 6371radius (km)

-0.2

0

0.2

0.4

0.6

0.8

1




Effective Backus residuals arecontinuous

5800 6000 6200 6371radius (km)

0

P3

Effective velocity residuals arenot ...

6000 6050 6100 6150 6200radius (km)

-300

-200

-100

0

100

Vs

(m/s

)

Vsh Vsvδ

δ

δ




To test the idea that a FWI can recover at best an homogenized model

cost function to be minimized :

Φ(m) = T [g(m)− d]C−1d [g(m)− d]

with

m model parameters ;

g(m) : forward problem in model m. Computed with normal modesummation ;

d data set. Computed with normal mode summation in the targetmodel + 3% random noise ;



Gauss-Newton iterative least square minimization

mi+1 = mi +(tGiCi

d

−1Gi + λi

)−1 [tGiCi

d

−1(di − gi (mi ))

],

where

mi model at iteration i

Gi partial derivative matrix. Computed with normal mode (of modelmi ) full coupling. No approximation.

λi damping at iteration i




m parameter set :

we use the Backus parameters set pi (r) (p1 = ρ ; p2 = 1C ; p3 = 1

L ;

p4 = A− F 2

C ; p5 = FC ; p6 = N)

we invert relative we respect to a reference model (here PREM), pref

δp = p− pref

we use a radial Lagrange polynomials for the vertical parameters,which can be defined by values at given set of radius r1...rN .

m = (δp1(r1), ..., δp6(r1), δp1(r2), ..., δp6(rN))

d data to be inverted

Synthetic data d data to be inverted are computed with normalmodes in TEST1 model ;

a random noise of 3% amplitude is added to the data ;

we use a single 3 components receiver in the period band[33s-1000s], two sources depth (10 km and 700 km) and anepicentral distance of 130.



Full waveform inversion in layered media : raw result

grey line :referencemodel

red line :target model

black lines :inversionresults for 8differentrealizations ofthe 3%random noise

6

7

8

9

10

km

/s

5800 6000 6200 6400

Vpv

3

4

5

6km

/s

5800 6000 6200 6400

Vsv

3

4

kg

/m3

5800 6000 6200 6400

km

ρ

6

7

8

9

10

km

/s

5800 6000 6200 6400

Vph

3

4

5

6

km

/s

5800 6000 6200 6400

Vsh

0.5

1.0

1.5

5800 6000 6200 6400

km

η



Full waveform inversion : raw result in the spectral domain

red line :target model

black lines :inversionresults for 8differentrealizations ofthe 3%random noise

p1 = ρp2 = 1

Cp3 = 1

L

p4 = A− F 2

C

p5(r) = FC

p6 = N

50 100 150 200

0

δp1

50 100 150 200

0

δp2

50 100 150 200

Wave number index

0

δp3

50 100 150 200

0

δp4

50 100 150 200

0

δp5

50 100 150 200

Wave number index

0

δp6



Full waveform inversion : homogenized result

grey line :referencemodel

red line :homogenizedtarget model

black lines :homogenizedinversionresults for 8differentrealizations ofthe 3%random noise

7

8

9

10

km

/s

5800 6000 6200 6400

Vpv

4

5

km

/s

5800 6000 6200 6400

Vsv

3.0

3.5

4.0

kg

/m3

5800 6000 6200 6400

km

ρ

7

8

9

10

km

/s

5800 6000 6200 6400

Vph

4

5

km

/s

5800 6000 6200 6400

Vsh

0.8

0.9

1.0

1.1

1.2

5800 6000 6200 6400

km

η



Full waveform inversion : the inversion is multi-scale

The inversion is designed such that the frequency band increases alongwith the iteration number.Results for the δp3 residual (p3 = 1

L ) in the spectral domain for threeiterations and therefore for 3 different frequency bands.

0 50 100 150

0

0 50 100 150

0

0 50 100 150

0

wave number index

red line : target model ; black lines : inversion results for 8 differentrealizations of the 3% random noise



Full waveform inversion : effect of a wrong a priori crust

3.03.23.43.63.84.04.24.44.64.8

Vs (

km

/s)

6150 6200 6250 6300 6350

a

3.03.23.43.63.84.04.24.44.64.8

Vs (

km

/s)

6150 6200 6250 6300 6350

b

3.03.23.43.63.84.04.24.44.64.8

Vs (

km

/s)

6150 6200 6250 6300 6350

Radius (km)

c

3.03.23.43.63.84.04.24.44.64.85.05.2

6150 6200 6250 6300 6350

d

3.03.23.43.63.84.04.24.44.64.85.05.2

6150 6200 6250 6300 6350

e

3.03.23.43.63.84.04.24.44.64.85.05.2

6150 6200 6250 6300 6350

Radius (km)

f



Conclusions and perspectives

Conclusions

So far, homogenization technique, by removing small scales, is usefulfor the forward problem (by easing the meshing problem forexample).

for the inverse problem, homogenization can allow to build amulti-scale inversion and a consistant parametrisation.

an elastic model obtained from a FWI is indeed an homogenizedversion of the traget model (at least in the layered case)

perspectives

develop the down-scaling inverse problem (ongoing project)

develop the higher dimensions (than layered) inverse problem (soon)

up to what extent homogenization can lead to unicity of the inverseproblem solution ?


Homogenization : Topography

Two scale homogenization : high frequency topography

Γ

Γs

Matching zone

vi, τ i

ui,σi

`

`

Γ : free surface ;

Γs : effective freesurface ;

vi , τ i asymptoticexpansion validclose to the freesurface ;

ui ,σi asymptoticexpansion validaway from the freesurface ;

ε =`

λmin 1



Matching asymptotic

Matching the two asymptotic expansions gives :

order 0 : any smooth free Γs surface can replace the original freesurface Γ ;

order 1 : an optimum smooth free surface exists and the free surfacehas to be replace by a DtN condition.

DtN condition :

σ · n = ε−hu + ∂xp ((b2 + h)σppp)

+ O(ε2)

where p the tangent vector to the effective free surface Γs ; b2 and h aresmooth surface coefficient. A finite element problem needs to be solvedto compute them. For the non periodic case, for homogeneous elasticproperties we find b2 + h = 0. In that case :

σ · n = −εhu + O(ε2)



Effective topography example

19 km

20 km

19 km

20 km

19 km 20 km 21 km

0 10 20 30 40(km)

-100

0

100

Am

pli

tude

(m)

fast topography

effective topography (1km)

average topography (1km)

x2



Traces for two topography and two ε0

top

og

rap

hy

0referenceorder 1average

=1 =0.5

10 15

top

og

rap

hy

1

10 15

ε ε0 0

time (s)

λmin ' 1km, epicentral distance 35 kmY. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 48

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Non periodic homogenization for elastic wave …mesoimage.grenoble.cnrs.fr/IMG/pdf/capdeville13.pdfNon periodic homogenization for elastic wave forward and inverse problems in seismology