[PPT] - Filtering random medium effects for imaging Liliana Borcea PowerPoint Presentation

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Filtering random medium effects for imaging Liliana Borcea Computational and Applied Mathematics Rice University borcea@caam.rice.edu Collaborators: Fernando Gonz´ alez del Cueto, CAAM, Rice University George Papanicolaou, Mathematics, Stanford University Chrysoula Tsogka, Mathematics, University of Crete Support: ONR: N00014-05-1-0699 NSF: DMS-0604008; DMS-0354658

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Topic Problem: Image with arrays of sensors compactly supported re- flectors buried in heterogeneous, strongly backscattering media. Difficulty: The backscattered field can overwhelm the echoes from the reflectors that we wish to find.

Can we differentiate the coherent echoes from the incoherent,

backscattered field? Can we design filters that emphasize the coherent field needed in imaging?

We discuss such filters for finely layered media.
These media are a good case study because they give a worse

scenario in terms of backscattering effects. For example, wave localization is sure to occur.

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Formulation of the problem Data: time traces of pressure p(t,

xr, xs).

Source is fixed at

xs = (xs, z = 0).

Receivers at

xr = xs + (h, 0),

|h| ≤ a

2.

Image the support S of reflectivity ν(

x).

1 V 2(

x)∂2

t p(t,

x, xs) − ∆p(t, x, xs)

= −f(t) ∂z δ(

x − xs),

t > 0, p(t,

x, xs)

= 0, t ≤ 0, where

x = (x, z) and

1 V 2(

x) =

1 v2(z) + ν(

x)

v(z) is rough (fine layering + interfaces) scattering.

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Illustration motivated by exploration geophysics

10 20 30 40 50 60 70 80 5 10 15 20 25 30 35 40 10 20 30 40 50 60 70 80 5 depth speed

time xr 100 133 167 200 233 267 300 333 5 10 15 20 25 30 35 z (range) x (cross−range) 30 40 50 60 70 80 5 10 15 20 25 30 35 40

Frequeny band: 20−40Hz, v(z) fluctuates about 3km/s. Central wavelength is 100m, the reflectors are at depth 6km and the fine layering is at scale 2m. Imaging function: J (

ys) =

xr p(τ(

xr, ys, xs), xr, xs)

τ(

xr, ys, xs) = travel time computed with smooth part of v(z).

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Illustration of filtering the layer echoes

time xr 100 133 167 200 233 267 300 333 5 10 15 20 25 30 35 time xr 100 133 167 200 233 267 300 333 5 10 15 20 25 30 35

Recorded traces Filtered traces

z (range) x (crossrange) 30 40 50 60 70 80 5 10 15 20 25 30 35 40 z (range) x (crossrange) 30 40 50 60 70 80 5 10 15 20 25 30 35 40

Migration image Image with filtered data.

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Outline

Mathematical model of the data.
Introduce the filters, which use simple ideas from geophysics.
Explain through analysis why they work.
Present numerical results.

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Model of the medium

The sound speed v(z) has a (piecewise) smooth part c(z) and

a rough part supported in z < 0. For z ≥ 0 we have v(z) = co. 1 v2(z) = 1 c2(z)

1 + σµ

z

ℓ

,

−Lj < z < −Lj−1, j = 1, 2 . . .

Interfaces at z = −Lj, j = 1, 2 . . ., due to jump discontinuities
f c(z) or to isolated blips over depth intervals ∼ wavelength.
Fine layering modeled with random, stationary process µ.

The process µ has mean zero, no long range correlations and it is properly normalized. ℓ = correlation length and σ gives strength of fluctuations.

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Mathematical model of the data

We let: p(t,

xs + (h, 0), xs) D(t, h) .

The layer echoes to be annihilated are

Dlay(t, h) ∼

dω

2π

f(ω)
dK

ω

2π

2 R(ω, K)e−iωt+iωK·h where R(ω, K) is the reflection coefficient of the medium and

K = horizontal slowness vector of plane waves.

The echoes from the compact reflectors to be imaged are

modeled with the Born approximation. The transmitted field between the array surface and the compact reflectors is determined by T (ω, K) = transmission coefficient.

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The scattering series∗ R(ω, K) = R1 + · · · + T1 ˜ T1T2 ˜ R2T2 ˜ T1T1e−2iω(τ1+τ2) + . . . R1 T1 ˜ T1 T2 ˜ R2 T2 ˜ T1 T1 −L1 −L2

We use tilde for the strong interfaces.
Travel times of the plane waves between the interfaces

τj(K) =

−Lj

−Lj−1

dz

1

c2(z) − K2

vertical slowness

, j = 1, 2, . . .

9 ∗Series for transmission T (ω, K) between A and compact scatterers is similar.

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The coherent and incoherent field D(t, h) = C(t, h)

coherent

+ N(t, h)

incoherent
The coherent field corresponds to pure transmission through

the random medium. It consists of the echoes from the deter- ministic structures (the interfaces and the compact scatterers). C(t, h) is modeled with the O’Doherty Anstey (ODA) theory: The fine layering effects on the coherent echoes are: pulse broad- ening, strong attenuation (exponential) and small random arrival time shifts. C(t, h) =

P

ϕP

     

t − [ arrival time

τP(h) + ǫδP(h)]

ǫ , h

     

.

N(t, h) is due to reflections in the random slabs.

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Coherent and incoherent intensity

The coherent echoes from the top scattering interfaces are

strong and dominant in the data.

The incoherent backscattered field is weaker, but the coherent

echoes from deep deterministic structures are also weak.

time xr 100 133 167 200 233 267 300 333 5 10 15 20 25 30 35 time xr 100 133 167 200 233 267 300 333 5 10 15 20 25 30 35

Recorded traces Filtered traces

The layer annihilator filters are designed to suppress the

primary echoes in C(t, h) scattered once at an interface.

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Definition of annihilator: Step 1

normal move-out
D(t, h) → D(Tc(h, z), h) using the roundtrip time to z < 0

Tc(h, z) = 2

z

1 − c2(s)K2

c

c(s) ds + hKc where h 2 =

z

c(s)Kc

1 − c2(s)K2

c

ds

Kc = d

dhTc(h, z) Example for c(z) = co Tco =

h2 + 4z2

co and

K[Tco] =

h c2

Tco(h, z) = cos θ1

co z

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Definition of annihilator

The annihilation step

D (Tc(h, z), h) − 1 |I(h)|

I(h)

D (Tc(h + ξ, z), h + ξeh) dξ where I(h) is an averaging interval around h and eh = h

h.

return to (t,h)
z = ζc(h, t)

with inverse of travel time map Tc (h, ζc(h, t)) = t If c(z) = co we have ζco = −1

2

c2

0t2 − h2

The filter is

QcD(t, h)= D (Tc(h, z), h)−

1 |I(h)|

I(h)

D (Tc(h + ξ, z), h + ξeh) dξ

z = ζc(h, t)

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Numerical results

10 20 30 40 50 60 70 80 2 3 4 5 6 depth sound speed

time xr 100 150 200 250 300 5 10 15 20 25 30 35 time xr 100 150 200 250 300 5 10 15 20 25 30 35

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Numerical results

10 20 30 40 50 60 70 80 1 2 3 4 5 depth sound speed

time xr 100 150 200 250 300 5 10 15 20 25 30 35 time xr 100 150 200 250 300 5 10 15 20 25 30 35

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Numerical results

10 20 30 40 50 60 70 80 1 2 3 4 5 depth sound speed (km/s)

time xr 133 167 200 233 267 300 333 5 10 15 20 25 30 35 time xr 133 167 200 233 267 300 333 5 10 15 20 25 30 35

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Why is the annihilation better than expected?

x z 5 10 15 20 25 30 35 40 50 55 60 65 70 75 80 x z 5 10 15 20 25 30 35 40 50 55 60 65 70 75 80

By design, the filter Qc annihilates the coherent echoes from

the strong interfaces. The numerics confirm this.

But all the examples show that the incoherent, backscattered

field is annihilated too. To explain the surprising effectiveness of Qc we need to go deeper in the fluctuation theory, beyond ODA.

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Analysis setup for the annihilation of incoherent echoes

Take time t ∈ (0, t⋆], with t⋆ < first coherent arrival time as

if we had just the random medium, up to depth Lt⋆.

The model of the data becomes

D(t, h) = N(t, h) = 1 2

dω

2π

f(ω)
dK

ω

2π

2 Rt⋆(ω, K)e−iωt+iωK·h, where Rt⋆(ω, K) = reflection coefficient of the medium in [−Lt⋆, 0].

It is easy to get that E {Rt⋆(ω, K)} = 0 so that E {N(t, h)} = 0.
Our goal is to show

E

[Qc N(t, h)]2

≪ E

N 2(t, h)
.

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The moments of the reflection coefficients

To study Rt⋆(ω, K), it is convenient to define it as

Rt⋆(ω, K) = lim

zր0 Rt⋆(ω, K, z),

where Rt⋆(ω, K, z) = reflection coefficient of the medium in [−Lt⋆, z].

We have the Ricatti equation

∂ ∂zRt⋆(ω, K, z) = −iωσµ(z/ℓ)c(K, z) 2c2(z)

e2iωτ(K,z) [Rt⋆(ω, K, z)]2 +

e−2iωτ(K,z) − 2Rt⋆(ω, K, z)

+ · · ·

for z > −Lt⋆ and the initial condition Rt⋆(ω, K, −Lt⋆) = 0. Here c(K, z) =

1

c2(z) − K2 and τ(K, z) =

z

−Lt⋆

dz′ c(K, z′).

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Scaling and the asymptotic regime

The reference length scale is Lt⋆.
Introduce ǫ ≪ 1, where ǫ =pulse width / t⋆.
Assume separation of scales ℓ ≪ λo ≪ Lt⋆ and model them as

ℓ λo ∼ λo Lt⋆ ∼ ǫ ≪ 1, σ = O(1). Array aperture ≫ λo.

Thus ω ω

ǫ in the Ricatti equation for Rǫ t⋆(ω, K, z) = Rt⋆

ω

ǫ , K, z

.
Furthermore, ℓ = (ǫ/σ)2l = O(ǫ2) and the random driving

σ ǫ µ

z

(ǫ/σ)2l

takes the canonical form of white noise as ǫ → 0.

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The white noise limit∗ ǫ → 0 We have rapid decorrelation of Rǫ(ω, K, z) over ω and K and E

  Rǫ

t⋆

ω + ǫ˜

ω 2 , K + ǫ ˜ K 2 , z

Rǫ

t⋆

ω − ǫ˜

ω 2 , K − ǫ ˜ K 2 , z

   →

ds
dχ

W1 (ω, K, s, χ, z)exp

i˜

ω(s − Kχ) − iω ˜ Kχ

where

∂Wn ∂z + 2n

  

1 c(z)

1 − K2c2(z)

∂Wn ∂s + c(z)K

1 − K2c2(z)

∂Wn ∂χ

   =

n2 Lloc(ω, K, z)

Wn+1 + Wn−1 − 2Wn
,

n ≥ 0, with initial condition Wn(ω, K, s, χ, −L) = 1o(n)δ(s)δ(χ).

21 ∗Fouque, Garnier, Papanicolaou, Solna - Springer 2007

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The result lim

ǫ→0 E

|N(t, h)|2

= 1 4(2π)3

dω ω2|

f(ω)|2

dK K

h W1(ω, K, t, h, 0).

The annihilation works if we subtract averages of the traces

after move-out in an interval |Iǫ(h)| = ǫ|I(h)| ∼ λo around h. lim

ǫ→0 E

|Qc N(t, h)|2

= 1 2(2π)3

∞

−∞dω ω2|

f(ω)|2

dK K

h W1(ω, K, t, h, 0) 1 |I(h)|

I(h) dξ{1 − cos [ωξ(Kc − K)]}
The annihilation depends on support of W1 in K. In the case

c(z) = co, W1 is a Dirac δ(K − Kco) so we have lim

ǫ→0 E

|QcoN(t, h)|2

= 0.

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The probabilistic representation of W1

The solution W1(ω, K, t, h, 0) is given by

W1 = E

    10 (N(0)) δ   t − 2

−L

1 − c2(z)K2

c(z) N(η(z))dz − Kh

  

δ

  h − 2

−L

N(η(z))c(z)K

1 − c2(z)K2dz

  

N(η(−L)) = 1

    

N(η(z)) = Markov jump process on positive integers, with

absorbing state N = 0 and η(z) =

z

ds Lloc(ω, K, z) The jump times have exponential density 2N2e−2N2. The jumps are to N + 1 or N − 1 with equal probability.

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Illustration of subtraction after normal move-out of two traces at h = 5λo and h′ = 7.5λo

10 20 30 40 50 60 70 80 2 3 4 5 depth sound speed

50 100 150 200 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 time raw Dc(t,h,h’) 50 100 150 200 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 time raw Dc(t,h,h’) 50 100 150 200 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 time raw Dc(t,h,h’)

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Illustration of subtraction after normal move-out of two traces at h = 5λo and h′ = 20λo and h′ = 7.5λo

50 100 150 200 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 time raw Dc(t,h,h’) 50 100 150 200 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 time raw Dc(t,h,h’)

If we subtract traces at offsets |h′ − h| > O(λo), there is no annihilation.

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Velocity estimation

10 20 30 40 50 60 2 2.5 3 3.5 4 4.5 5 depth sound speed true medium estimation 10 20 30 40 50 60 1.5 2 2.5 3 3.5 4 4.5 depth sound speed true medium estimation 5 10 15 20 25 30 35 40 0.5 1 1.5 2 2.5 depth sound speed true medium estimation 5 10 15 20 25 30 35 40 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 depth sound speed true medium estimation

We estimate c(z) by minimizing the energy of the filtered traces

Q˜

cD(t, h) over the trial speed ˜

c(z).

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Conclusions

We have shown that it is possible to filter out the random

medium effects for imaging in strongly backscattering media.

The approach described here relies heavily on the layered struc-

ture of the medium.

It uses common ideas in exploration geophysics such as nor-

mal move-out, gather flattening, differential semblance velocity estimation.

We have seen that although these methods are based on the

single scattering approximation, the filters perform surprinsingly well in random media, where multiples dominate.

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Results with isotropic clutter

z x 30 40 50 60 70 80 5 10 15 20 25 30 35 40 z x 30 40 50 60 70 80 5 10 15 20 25 30 35 40 z (range) x (cross−range) 30 40 50 60 70 80 5 10 15 20 25 30 35 40