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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.
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 2
Introduction and motivationsIntroduction and motivations
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 ...)
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 3
Introduction and motivationsIntroduction and motivations
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 motivationsIntroduction and motivations
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.
?
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 5
Introduction and motivationsIntroduction and motivations
Introduction and motivations
Period range : [150s-1000s]Computed with :
SAW24 and PREMmodels
moho, topography,oceans, rotation,ellipticity
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 6
Introduction and motivationsIntroduction and motivations
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)
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 7
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
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 8
Homogenization : forward problem
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)
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 9
Homogenization : forward problem
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)
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 10
Homogenization : forward problem
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)
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 11
Homogenization : forward problem
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 ...
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 12
Homogenization : forward problem
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
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 13
Homogenization : forward problem
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.
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 14
Homogenization : forward problem
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.
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 15
Homogenization : forward problem
Two non periodic scale homogenization : workflow
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.
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 16
Homogenization : forward problem
Two non periodic scale homogenization : workflow
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)
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 17
Homogenization : forward problem
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%.
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 18
Homogenization : forward problem
A 2-D random example
The source is an explosion (1.5Hz of central frequency, 3.6Hz.λmin ' 0.8km.)
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 19
Homogenization : forward problem
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
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 20
Homogenization : forward problem
A 2-D random example
Order 0 homogenized model
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 21
Homogenization : forward problem
A 2-D random example
Order 0 homogenized model
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 21
Homogenization : forward problem
A 2-D random example
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)
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 22
Homogenization : forward problem
A 2-D random example : source at the center of the square
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 23
Homogenization : forward problem
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 :
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 24
Homogenization : forward problem
A 2-D random example : source at the center of the square
-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
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 25
Homogenization : forward problem
Marmousi2 example
Homogenization allows for simple meshes, even for complex media.
Mesh for the original model Mesh for the homogenized model
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 26
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
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 27
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.
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 28
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.
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 29
Homogenization : inverse problem
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)
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 30
Homogenization : inverse problem
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
)
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 31
Homogenization : inverse problem
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
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 32
Homogenization : inverse problem
Residual homogenization in layered media
3000 3500 4000 4500 5000time (s)
0
premtest1homogenized
3000 3500 4000 4500 5000time (s)
0
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 33
Homogenization : inverse problem
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
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 34
Homogenization : inverse problem
Full waveform inversion in layered media
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
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 35
Homogenization : inverse problem
Full waveform inversion in layered media
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 ;
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 36
Homogenization : inverse problem
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
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 37
Homogenization : inverse problem
Full waveform inversion in layered media
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.
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 38
Homogenization : inverse problem
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
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 39
Homogenization : inverse problem
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
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 40
Homogenization : inverse problem
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
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 41
Homogenization : inverse problem
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
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 42
Homogenization : inverse problem
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
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 43
Homogenization : inverse problem
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 ?
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 44
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
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 45
Homogenization : Topography
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)
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 46
Homogenization : Topography
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
Y. Capdeville , P. Cupillard , J.J. Marigo , P. Cance , L. Guillot , E. Stutzmann , J.P. MontagnerMesoImage 2013, 13/12/2013 47
Homogenization : Topography
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