Two Layer Foldy-Lax Scattering Model - casis.llnl.gov 14.99 cm n 2 = 3; h 2 ... Neither the United...
Transcript of Two Layer Foldy-Lax Scattering Model - casis.llnl.gov 14.99 cm n 2 = 3; h 2 ... Neither the United...
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