Two Layer Foldy-Lax Scattering Model

Sean K. Lehman, Ph.D.

Lawrence Livermore National Laboratory

May 21, 2014

x (meters)

z (m

eter

s)

Point Source

λ1

λ2

n1 = 1; λ1 = 14.99 cm

n2 = 3; λ2 = 5.00 cm

−1 0 1

−1

−0.5

0

0.5

1

x (meters)

z (m

eter

s)

Point Scatterer

λ1

λ2

n1 = 1; λ1 = 14.99 cm

n2 = 3; λ2 = 5.00 cm

−1 0 1

−1

−0.5

0

0.5

1

LLNL-PRES-653640This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract

DE-AC52-07NA27344.

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model

Administrivia Incantations to Keep the C-Students at Bay

DisclaimerThis document was prepared as an account of work sponsored by an agency of the United States government.Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employeesmakes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy,completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its usewould not infringe privately owned rights. Reference herein to any specific commercial product, process, or serviceby trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement,recommendation, or favoring by the United States government or Lawrence Livermore National Security, LLC. Theviews and opinions of authors expressed herein do not necessarily state or reflect those of the United Statesgovernment or Lawrence Livermore National Security, LLC, and shall not be used for advertising or productendorsement purposes.

AuspicesThis work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore NationalLaboratory under Contract DE-AC52-07NA27344.

“Scary incantations in the wrong hands...”— From the Collide song Falling Up

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model

Foldy-Lax Theory

Foldy-Lax theory provides a model of (frequency domain) wavescattering for a collection of point scatterers

I M. Lax, “Multiple Scattering of Waves”, Reviews of ModernPhysics, October 1951, 23(4), pp287–310

I L. L. Foldy, “The Multiple Scattering of Waves: I. GeneralTheory of Isotropic Scattering by Randomly DistributedScatterers”, Pysical Review, February 1945, 67(3 & 4),pp107–119

I M. Lax, “The Multiple Scattering of Waves: II. The EffectiveField in Dense Systems”, Physical Review, February 1952,85(4), pp621–629

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model

Foldy-Lax Theory in a Nutshell – I

The integral form of the Helmholtz equation

ψscat(r,Rsrcn , ω) = k2

0 (ω)

∫dr′ G (r, r′, ω) o(r′) ψtot(r′,Rsrc

n , ω),

specialized to point scatterers is

Scatterer location,Scattering amplitude,

Refractiveindex

ψscat(r,Rsrcn , ω) = k2

0 (ω)∑j

τj G (r,Xj , ω) ψtot(Xj ,Rsrcn , ω)

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model

Foldy-Lax Theory in a Nutshell – II

The total field as observed at the scatterers must be known:

ψscat(r,Rsrcn , ω) = k2

0 (ω)∑j

τj G (r,Xj , ω) ψtot(Xj ,Rsrcn , ω)

The total field due to source n is given by

ψtot(r,Rsrcn , ω) = ψinc(r,Rsrc

n , ω) +

k20 (ω)

∑j

τj G (r,Xj , ω) ψtot(Xj ,Rsrcn , ω)

Evaluate this at r = Xj (excluding the self-scattering terms wherethe Green function diverges):

ψtot(Xj ,Rsrcn , ω) = ψinc(Xj ,R

srcn , ω) +

k20 (ω)

∑j ′ 6=j

τj ′ G (Xj ,Xj ′ , ω) ψtot(Xj ′ ,Rsrcn , ω)

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model

Foldy-Lax Theory in a Nutshell – III

We cast this into the form of a matrix equation,

ψtot(Xj ,Rsrcn , ω)−

k20 (ω)

∑j ′ 6=j

τj ′ G (Xj ,Xj ′ , ω) ψtot(Xj ′ ,Rsrcn , ω) = ψinc(Xj ,R

srcn , ω),

[I − k2

0 (ω)G(ω)T]︸ ︷︷ ︸

A

ψ̄totn (ω)︸ ︷︷ ︸x

= ψ̄incn (ω)︸ ︷︷ ︸b

solve

ψ̄totn (ω) =

[I − k2

0 (ω)G(ω)T]−1

ψ̄incn (ω)

and back substitute,

ψscat(r,Rsrcn , ω) = k2

0 (ω)∑j

τj G (r,Xj , ω) ψtot(Xj ,Rsrcn , ω)

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model

Aside on Stability

For a non-diverging solution, require

k20 ‖G (ω)T‖2 < 1

This bounds the scattering amplitudes,

|τ | < 4π

(J − 1) k20 (ω)

where Rmin is the minimum scatterer separation and J is thenumber of scatterers.

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model

The Green Function

Nothing has been said of the Green function.

ψ̄totn (ω) =

[I − k2

0 (ω) G(ω) T]−1

ψ̄incn (ω)

ψscat(r,Rsrcn , ω) = k2

0 (ω)∑j

τj G (r,Xj , ω) ψtot(Xj ,Rsrcn , ω)

Consider a two planar layer environment:

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model

Sommerfeld Integral

Deriving the Green function, G (r,Xj , ω), for a two planar layermedium requires solving a Sommerfeld integral such as,

G (r⊥, z) =i

2(2π)2

∫dk⊥

1

γ0(k⊥, ω)e iγ0(k⊥,ω)z e ik⊥·r⊥

where

γ0(k⊥, ω) =√k2

0 (ω)− k2⊥

which is difficult.

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model

A Fast Sommerfeld Integral Solver

I A. Hochman, “A Numerical Methodology for EfficientEvaluation of 2D Sommerfeld Integrals in the DielectricHalf-Space”, IEEE Trans. Antennas and Propagation,February 2010, 58(2), pp413–431.

I MATLAB function, “Numerically-DeterminedSteepest-Descent Path (NDSDP)”: ndsdp()

I http://web.mit.edu/hochman/www/

I Put the ndsdp() function into the planar layer Green function

I Put the Green function into the Foldy-Lax solver

The result is a fast-ish MATLAB two-layer Foldy-Lax solver: It mustperform two ndsdp() calls for each scatterer for each frequency.

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model

Numerically-Determined Steepest-Descent Path

For each frequency, Numerically-Determined Steepest-DescentPath (ndsdp.m) solves,

x (meters)

z (m

eter

s)

Point Source

λ1

λ2

n1 = 1; λ1 = 14.99 cm

n2 = 3; λ2 = 5.00 cm

−1 0 1

−1

−0.5

0

0.5

1

x (meters)

z (m

eter

s)

Point Scatterer

λ1

λ2

n1 = 1; λ1 = 14.99 cm

n2 = 3; λ2 = 5.00 cm

−1 0 1

−1

−0.5

0

0.5

1

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model

Examples

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model

Simulated Verification

−1.5 −1 −0.5 0 0.5 1 1.5

−1

−0.5

0

0.5

1

x (meters)

z (m

eter

s)

Geometry

n1 = 1; λ1 = 14.99 cm

n2 = 3.4; λ2 = 4.41 cm

λ1

λ2

0 20 40 60 80

Pluse length (cm)

Pulse

λ1

0 2 4 6 8 10 12Frequency (GHz)

Pulse Spectrum

−1.5 −1 −0.5 0 0.5 1 1.5

0

5

10

15

20

25

30

35

40

x (meters)

time

(ns)

Multimonostatic Data Matrix

x (meters)

z (m

eter

s)

Multimonostatic Migration

λ1 λ2

−0.5 0 0.5

−0.5

0

0.5

1

1.5

x (meters)

z (m

eter

s)

Multistatic Migration

λ1 λ2

−0.5 0 0.5

−0.5

0

0.5

1

1.5

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model

Aluminum/Copper Layer

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model

Aluminum/Copper Layer

−10 0 10

−10

−5

0

5

x (mm)

z (m

m)

Geometry

n1 = 1; λ1 = 1.21 cm

n2 = 1.2; λ2 = 1.01 cm

λ1

λ2

0 2 4 6Pluse length (mm)

Pulse

λ1

0 10 20 30Frequency (MHz)

Pulse Spectrum

x (meters)

time

(μs)

Multimonostatic Data Matrix

−10 0 10

−2

0

2

4

6

8

10

12

−10 0 10

−10

−5

0

5

x (mm)

z (m

m)

Migration of Simulated Data

c0 = 6046.5 m/s

c = 5038.8 m/sn2 = 1.2

Transceiver Location (mm)

Ran

ge B

ased

upo

n Al

umin

um V

eloc

ity (m

m)

Multimonostatic ResponseAluminum Velocity = 6046.5 m/s; Copper Velocity = 5038.8 m/s

31 multistatic transducerswith 1 mm separation

−15 −10 −5 0 5 10 15

6

8

10

12

14

16

18

20

22

24

26

−15 −10 −5 0 5 10 15

−12

−10

−8

−6

−4

−2

0

2

4

6

8

c0 = 6046.5 m/s

c = 5038.8 m/sn2 = 1.2

x (mm)

z (m

m)

Migration

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model

Buried Resolution Phantom

0 20 40 60 80 100 120

−10−5

05

x (cm)

z (c

m)

Geometry

n2 = 3.5; λ2 = 3.43 cmλ2

λ1n1 = 1; λ1 = 11.99 cm

0 20 40 60

−0.50

0.5

Pluse length (mm)

Pulse

λ1

0 5 10 15

0.0050.01

0.0150.02

Frequency (GHz)

Pulse Spectrum

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model

Buried Resolution Phantom

Object Field Time Series

0 0.5 1

2

4

6

Background Field Time Series

0 0.5 1

2

4

6

Scattered Field Time Series

0 0.5 1

2

4

6

0 0.5 1

0

5

10

x (meters)

z (m

eter

s)

Migration

0 0.2 0.4 0.6 0.8 1 1.2

00.020.040.06

x (meters)

z (m

eter

s)

0 0.2 0.4 0.6 0.8 1 1.2

00.020.040.06

x (meters)

z (m

eter

s)

0 0.2 0.4 0.6 0.8 1 1.2

00.020.040.06

Simulated MMDM

Multimonostatic Migration of Simulated Data

Multistatic Migration of Simulated Data

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model

Summary

I Foldy-Lax is a full point scatterer scattering theory with theonly concession being that of no self-scattering.

I The theory is derived independently of the Green function.

I A Sommerfeld integral solver is used to create a two-layerGreen function for a Foldy-Lax scattering code.

Sean K. Lehman, Ph.D. Two Layer Foldy-Lax Scattering Model