Innovative inversion approaches for buried objects detection and imaging

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International Dotoral Shool in Information

and Communiation Tehnology

DISI- University of Trento

Innovative Inversion Approahes for

Buried Objets Detetion and Imaging

Maro Salui

Tutor:

AndreaMassa, Professor

University of Trento

Advisor:

GiaomoOliveri, Aggregate Professor

Co-Advisor:

PaoloRoa, Aggregate Professor

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The study, development, and analysis of innovative inversion tehniques for the

detetionandimagingofburiedobjetsisaddressedinthisthesis. Theproposed

methodologiesarebasedontheuseofmirowaveradiationsandradartehniques

forsubsurfae prospeting,suh as,forexample, the GroundPenetrating Radar

(

GP R

). Morepreisely,thereonstrutionofshallowburiedobjetsisrstly

ad-dressedbyaneletromagnetiinversesatteringmethodbasedontheintegration

of the inexat Newton (

IN

) method with an interative multisaling approah.

Theperformanesofsuhaninversionapproahareanalyzedbothwhen

onsid-eringtheuseofaseond-orderBornapproximation(

SOBA

)andwhenexploiting

the full set of non-linear equations governing the sattering phenomena for the

buried senario. The presented methodologies are partiularly suitable for

ap-pliations suh as demining (e.g., for the detetion of unexploded ordnanes,

UXOs

, and improvisedexplosive devies,

IEDs

), for ivil engineering

applia-tions (e.g., for the investigationof possible strutural damages, voids, raks or

water inltrations in walls, pillars, bridges) as well as for biomedial imaging

(e.g., for earlyaner detetion).

Keywords

GroundPenetratingRadar(

GP R

),Inverse Sattering,MirowaveImaging,

Iter-ative Multi-SalingApproah, Inexat Newton, Conjugate Gradient, Frequeny

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[C1℄ F. Viani, M. Donelli, M. Salui, P. Roa, and A. Massa,

Oppor-tunisti exploitation of wireless infrastrutures for homeland

seu-rity, Pro. 2011 IEEE AP-S International Symposium, Spokane,

Washington, USA, pp. 3062-3065, July 3-8, 2011.

[C2℄ F. Viani, M. Salui, P. Roa, G. Oliveri, and A. Massa, A

multi-sensorWSNbakboneformuseummonitoringandsurveillane,

Eu-CAP 2012, Prague,Czeh Republi, Marh26-30, 2012.

[C3℄ A. Saverio, T. Cadili, G. Liborio, M. Russo, G. Oliveri, M. Salui,

and A. Massa, A GA-based strategy for the alibration of satellite

ommuniation phased array antennas, EuCAP 2013, Gothenburg,

Sweden, April 8-12, 2013.

[C4℄ G.Oliveri, M. Salui, A. Massa, D. Gonzalez-Ovejero, and C.

Cra-eye, Analytially designed planar aperiodi arrays in the presene

ofmutual oupling, EuCAP 2013,Gothenburg, Sweden,April8-12,

2013.

[C5℄ F. Viani, F. Robol, M. Salui, E. Giarola, S. De Vigili, M. Roa,

F. Boldrini, G. Benedetti, and A. Massa, WSN-based early alert

systemforpreventingwildlife-vehileollisionsinAlpsregions-From

thelaboratorytest tothereal-worldimplementation, EuCAP 2013,

Gothenburg, Sweden, April 8-12, 2013.

[C6℄ L.Mania, P.Roa, M.Salui,M. Carlin,and A. Massa,

Satter-ingdata inversionthrough intervalanalysisunderRytov

approxima-tion, EuCAP 2013, Gothenburg, Sweden, April8-12, 2013.

[C7℄ M.Salui, D. Sartori,N. Anselmi, A. Randazzo, G. Oliveri, and A.

Massa, Imaging buried objets within the seond-order Born

ap-proximation through a multiresolution-regularized inexat-Newton

method, 2013 International Symposium on Eletromagneti Theory

(EMTS 2013),Hiroshima, Japan,May 20-24, 2013 (Invited).

[C8℄ G.Oliveri, P.Roa,M.Salui, E.T. Bekele, D.H.Werner,and A.

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Buena Vista,Florida,USA, July 7-12, 2013.

[C9℄ M. Salui, P. Roa, G. Oliveri, and A. Massa, "An innovative

frequeny hopping multizoom inversion strategy for GPR

subsur-faeimaging,"15thInternational Conferene onGround Penetrating

Radar (GPR2014), Brussels, Belgium, June 30- July 04,2014.

[C10℄ M. Salui, G. Oliveri, A. Randazzo, M. Pastorino, and A. Massa,

"Multi-Fousing Proedure based on the Inexat-Newton Method

for Eletromagneti Subsurfae Prospeting,"European Geosienes

Union General Assembly (EGU2014), Vienna, Austria, April 27

-May 2, 2014.

[C11℄ M. Salui, L. Tenuti, C. Nardin, G. Oliveri, F. Viani, P. Roa,

andA.Massa,"Civilengineeringappliationsofgroundpenetrating

radar reent advanes the ELEDIA Researh Center," European

Geosienes Union General Assembly (EGU2014), Vienna, Austria,

April27- May 2,2014.

[C12℄ M.Salui,L.Tenuti, C.Nardin, M.Carlin,F.Viani,G.Oliveriand

A. Massa, "Gpr survey through a multi-resolution deterministi

ap-proah,"IEEEAP-SInternationalSymposium,Memphis,Tennessee,

USA,July 6-12, 2014.

[C13℄ L. Mania, G. Oliveri, E. Bekele, M. Carlin, M. Salui, C. Nardin,

E. Martini, S. Mai, and A. Massa, Optimized synthesis of

wave-manipulatingdevies within the system-by-design paradigm, IEEE

AP-S International Symposium, Memphis, Tennessee, USA, July

6-12,2014.

[C14℄ A.Massa, G.Oliveri, M. Salui,C. Nardin, and P. Roa, Dealing

withEMfuntionaloptimizationthroughnewgeneration

evolutionary-basedmethods, IEEE International Conferene on Numerial

Ele-tromagnetiModelingandOptimizationfor RF,Mirowaveand

Ter-ahertz Appliations, Pavia,Italy, May 14-16, 2014 (Invited).

[C15℄ M. Salui, G. Oliveri, M. Gregory, D. Werner, and A. Massa, A

frequeny-tunablemetamaterial-basedantennausingareongurable

AMCgroundplane, 8thEuropeanConfereneonAntennasand

Prop-agation (EuCAP 2014), The Hague, The Netherlands, Apr. 6-11,

2014.

[C16℄ L. Poli, M. Salui, P. Roa, and A. Massa, Adaptive GA-based

thinningstrategy for pattern nulling in sub-arrayed phased arrays,

8th European Conferene on Antennas and Propagation (EUCAP

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Massa, Multi-salingdeterministi imaging for GPRsurvey, IEEE

International Conferene on Antenna Measurements & Appliations

(IEEE CAMA 2014), (Antibes Juan-les-Pins, Frane), Nov. 16-19

2014.

[C18℄ G. Oliveri, P. Roa, M. Salui, and A. Massa, Eient synthesis

of omplex antenna devies through system-by-design, Pro. 2014

IEEE Symposium Series on Computational Intelligene (IEEE SSCI

2014),Orlando,Florida,USA, Deember 9-12, 2014 (Aepted).

[C19℄ G.Oliveri, P.Roa,M.Salui,andA.Massa, Evolutionof

nature-inspiredoptimizationfornewgenerationantennadesign, Pro. 2014

IEEE Symposium Series on Computational Intelligene (IEEE SSCI

2014),Orlando,Florida,USA, Deember 9-12, 2014 (Aepted).

[C20℄ G. Oliveri, P. Roa, D. H. Werner, E. Bekele, M. Salui, and A.

Massa, Design and synthesis of innovative metamaterial-enhaned

arrays, 2013 IEEE Antennasand Propagation Soiety International

Symposium, (Orlando, USA), pp. 972-973,July 7-13, 2013.

[C21℄ M.Carlin, M. Salui, L.Tenuti, P. Roa, and A. Massa, Eient

radome optimization through the system-by-design methodology,

9th European Conferene on Antennas and Propagation (EUCAP

2015),Lisbon, Portugal,April 12-17,2015 (Submitted).

[C22℄ L. Tenuti, M. Salui, L. Poli, G. Oliveri, and A. Massa,

Physial-information exploitation in inverse sattering approahes for GPR

survey, 9thEuropeanConfereneonAntennasandPropagation

(EU-CAP2015), Lisbon, Portugal,April12-17, 2015 (Submitted).

[C23℄ F. Viani,M.Salui,F. Robol, E.Giarola,and A. Massa, WSNsas

enablingtoolfornext generationsmartsystems, AttiXIX Riunione

Nazionale di Elettromagnetismo (XIX RiNEm), Roma, pp. 393-396,

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[R1℄ F.Viani,M.Salui,F.Robol,G.Oliveri,andA.Massa,Designofa

UHFRFID/GPS fratalantenna forlogistismanagement, Journal

of Eletromagneti Waves and Appliations, vol. 26, no. 4, pp.

480-492, 2012.

[R2℄ F. Viani, M. Salui, F. Robol, and A. Massa, Multiband F

ra-talZigBee/WLANAntenna forUbiquitous Wireless Environments,

Journal of Eletromagneti Waves and Appliations,vol. 26, no.

11-12, pp. 1554-1562, 2012.

[R3℄ L. Poli, P. Roa, M. Salui, and A. Massa, "Reongurable

thin-ningforthe adaptiveontroloflineararrays,"IEEE Transationson

Antennas and Propagation,vol. 61, no. 10,pp. 5068-5077, 2013.

[R4℄ B. Majone, F. Viani, E. Filippi, A. Bellin, A. Massa, G. Toller, F.

RobolandM.Salui,"Wirelesssensornetworkdeploymentfor

mon-itoringsoil moisture dynamis at the eld sale," Proedia

Environ-mentalSienes, vol. 19,pp. 426-435, 2013.

[R5℄ M. Salui, G. Oliveri, A. Randazzo, M. Pastorino, and A. Massa,

Eletromagnetisubsurfae prospeting by amulti-fousinginexat

Newtonmethodwithintheseond-order Bornapproximation,

Jour-nal of Optial Soiety of Ameria A, vol. 31, no. 6, pp. 1167-1179,

2014.

[R6℄ T. Moriyama, G. Oliveri, M. Salui, and T. Takenaka, A

multi-salingforward-bakward timesteppingmethodfor mirowave

imag-ing, IEICE Eletronis Express, vol. 11, no. 16, id. 20140569, pp.

1-10,Aug. 2014. doi:10.1587/elex.11.20140578.

[R7℄ T. Moriyama, E. Giarola, M. Salui, and G. Oliveri, On the

ra-diation properties of ADS thinned dipole arrays, IEICE

Eletron-is Express, vol. 11, no. 16, id. 20140578, pp. 1-12, Aug. 2014.

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Eletromagneti subsurfae prospeting by a fully nonlinear

multi-fousinginexatNewtonmethod,JournalofOptialSoietyof

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1 Introdution 1

2 Inverse Sattering Equations for the Subsurfae Problem 5

2.1 Geometry of the Problem . . . 6

2.2 Mathematial Formulation . . . 6

3 Multi-FousingInexatNewtonMethodwithintheSeond-Order Born Approximation 13 3.1 Introdution . . . 14

3.2 Problem Formulation . . . 14

3.3 Reonstrution Method . . . 15

3.4 Numerial Assessment . . . 18

3.4.1 Calibrationof the

IMSA

−

IN

−

SOBA

. . . 18

3.4.2 HomogeneousSquare and L-shaped Cylinders . . . 24

3.4.3 O-shaped Cylinder . . . 24

3.4.4 InhomogeneousCylinders . . . 28

3.4.5 Square Cylinder withstrong ondutivity . . . 30

3.5 Disussions . . . 32

4 Eletromagneti Subsurfae Prospeting by a Fully Nonlinear Multi-fousing Inexat Newton Method 33 4.1 Introdution and motivation . . . 34

4.2 Mathematial formulation . . . 35

4.3 Numerial Results . . . 37

4.3.1 Calibration . . . 39

4.3.2 Eets of Noise . . . 42

4.3.3 Eets of the Dieletri Properties of the Target . . . 47

4.3.4 Reonstrution of Targets with Small Details. . . 50

4.3.5 Reonstrution of Targets with Higher Condutivity . . . . 51

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5.3 FHMF-CG Inversion Proedure . . . 58

5.4 Numerialand ExperimentalValidation. . . 62

5.4.1 Rationaleand Figuresof Merit . . . 62

5.4.2 NumerialValidation . . . 63

5.4.3 Experimental Validation . . . 75

6 Conlusions 81 6.1 ComparisonBetween Dierent Approahes . . . 82

6.2 FinalRemarks. . . 89

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3.1 Performane Assessment (O-Shaped Satterer

ℓ

≈

λ

b

2

,

SNR

∈

[10

,

30]

dB) - Error values and omputationalindexes. . . 26

4.1 Performane vs. Noise (Square Satterer -

L

= 0

.

32

λ

b

, (

x

c

=

−

0

.

16

λ

b

,

y

c

=

−

0

.

58

λ

b

),

ε

r

= 5

.

5

,

σ

= 0

.

01

S/m [

τ

= 1

.

5

℄,

ε

rB

=

4.0,

σ

B

=

0.01 S/m)- Totalnumberof performed outer

iter-ations,naltnessvalues,andreonstrutionerrorsfortheBARE

and the IMSA (

s

=

S

= 4

) IN approahes. Totalexeution time

on a PC with Intel(R) Core(TM)2 CPU 6600 2.40GHz, 2GB

RAM. . . 43

4.2 Performane vs. Target Condutivity (Square Satterer -

L

=

0

.

32

λ

b

, (

x

c

=

−

0

.

16

λ

b

,

y

c

=

−

0

.

58

λ

b

),

ε

r

= 5

.

5

,

σ

= 0

.

1

S/m

[

τ

= 1

.

5

−

j

5

.

39

℄,

ε

rB

=

4.0,

σ

B

=

0.01 S/m) - Reonstrution errors for the bare IN and the IMSA-IN (at step

s

=

S

= 4

)

approahes. . . 52

5.1 IllustrativeExample[Hollowsquare prole,

ε

rB

= 4

.

0

,

σ

B

= 10

−3

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2.1 Geometry of a subsurfae imaging problem. (a) Cross-borehole

and (b) half spae setup.

λ

b

is the wavelength inthe bakground

material. . . 6

3.1 Geometry of the problem and imagingsetup. . . 15

3.2 Sensitivity Analysis (HomogeneousSquare Satterer -

ℓ

≈

λ

b

3

,

τ

=

1

.

5

,

SNR

= 20

dB) - Behaviour of the integral error

Ξ

tot

versus

η

and

W

when

Q

=

Q

∗

,

S

=

S

∗

(a), and versus

K

when

η

=

η

∗

,

W

=

W

∗

,and

S

=

S

∗

(b). Plotofthetotal,internal,andexternal

error as a funtionof

S

when

Q

=

Q

∗

,

η

=

η

∗

,and

W

=

W

∗

(). . 19

ℓ

≈

λ

b

3

,

τ

=

1

.

5

,

SNR

= 20

dB,

S

=

S

∗

) - Atual (a) and retrieved (b)()

ontrast proles when (b)

Q

=

Q

∗

,

W

=

W

∗

,

η

= 10

−

4

; ()

K

=

K

∗

,

W

= 40

,

η

=

η

∗

. . . 21

ℓ

≈

λ

b

3

,

τ

=

1

.

5

,

SNR

= 20

dB,

Q

=

Q

∗

,

W

=

W

∗

,

η

=

η

∗

) - Behaviourof

Φ

and

ζ

versus the

IMSA

−

IN

iterationnumber (a). Plot of the retrieved ontrast proles when (b)

S

= 1

, ()

S

= 2

, (d)

S

= 3

,

(e)

S

= 4 =

S

∗

. . . 23

3.5 Performane Assessment (

τ

= 1

.

5

,

SNR

∈

[10

,

40]

dB) - Be-haviourofthe

Ξ

tot

asafuntionof

SNR

whendealingwithSquare

or L-Shaped targets (a). Plot of the ontrast proles retrieved

by(b)()

BARE

−

IN

−

SOBA

and(d)(e)

IMSA

−

IN

−

SOBA

when

SNR

= 10

dB.(b)(d)Square satterer;()(e)L-Shaped

satterer. . . 25

ℓ

≈

λ

b

2

,

SNR

∈

[10

,

30]

dB) -Behaviourof the

Ξ

tot

asafuntion of

τ

obtained by

BARE

−

IN

−

SOBA

and

IMSA

−

IN

−

SOBA

. . . 26

ℓ

≈

λ

b

2

,

SNR

=

20

dB)-Plotoftheontrastprolesretrievedby(a)()(e)

BARE

−

IN

−

SOBA

and (b)(d)(f)

IMSA

−

IN

−

SOBA

when (a)(b)

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3.8 Performane Assessment (Inhomogeneous Satterers,

SNR

= 20

dB)-Plotof the atual(a)(b) and retrieved ()-(f)ontrast

pro-les by ()(d)

BARE

−

IN

−

SOBA

and (e)(f)

IMSA

−

IN

−

SOBA

for (a)()(e) Double-L and (b)(d)(f) Conentri

tar-gets. . . 29

3.9 Performane Assessment (Homogeneous Square Satterer -

ℓ

≈

λ

b

3

,

R {

τ

}

= 1

.

5

,

SNR

∈

[10

,

30]

dB) - Behaviour of the

Ξ

tot

asa funtion of

σ

c

(a). Plot of the real (b)() and imaginary (d)(e)

parts of the ontrast proles retrieved by (b)(d)

BARE

−

IN

−

SOBA

and ()(e)

IMSA

−

IN

−

SOBA

when

SNR

= 20

dB.. . 31

4.1 Cross-boreholeimaging onguration. . . 35

4.2 Calibration (Square Satterer -

L

= 0

.

32

λ

b

, (

x

c

=

−

0

.

16

λ

b

,

y

c

=

−

0

.

58

λ

b

),

ε

r

= 5

.

5

,

σ

= 0

.

01

S/m [

τ

= 1

.

5

℄,

ε

rB

=

4.0,

σ

B

=

0.01 S/m,

SNR

= 20

dB) - Atual target used for the algorithm

ali-bration. . . 39

L

= 0

.

32

λ

b

, (

x

c

=

−

0

.

16

λ

b

,

y

c

=

−

0

.

58

λ

b

),

ε

r

= 5

.

5

,

σ

= 0

.

01

S/m [

τ

= 1

.

5

℄,

ε

rB

=

4.0,

σ

B

=

0.01

S/m,

SNR

= 20

dB) - Total reonstrution error vs.

α

(

α

∈

[0

.

1

,

0

.

9]

) fordierent values of

Q

inthe range

Q

∈

[10

,

100]

. . . . 40

L

= 0

.

32

λ

b

, (

x

c

=

−

0

.

16

λ

b

,

y

c

=

−

0

.

58

λ

b

),

ε

r

= 5

.

5

,

σ

= 0

.

01

S/m [

τ

= 1

.

5

℄,

ε

rB

=

4.0,

σ

B

=

0.01

S/m,

SNR

= 20

dB) -Best tness value for dierent

(

Q, α

)

pairs. 40

L

= 0

.

32

λ

b

, (

x

c

=

−

0

.

16

λ

b

,

y

c

=

−

0

.

58

λ

b

),

ε

r

= 5

.

5

,

σ

= 0

.

01

S/m [

τ

= 1

.

5

℄,

ε

rB

=

4.0,

σ

B

=

0.01

S/m,

SNR

= 20

dB)-(a)Realand(b)imaginarypartsofthe

re-onstruteddistributionoftheontrastfuntionwhen

Q

=

Q

opt

=

50

and

α

=

α

opt

= 0

.

9

. . . 41

L

= 0

.

32

λ

b

, (

x

c

=

−

0

.

16

λ

b

,

y

c

=

−

0

.

58

λ

b

),

ε

r

= 5

.

5

,

σ

= 0

.

01

S/m [

τ

= 1

.

5

℄,

ε

rB

=

4.0,

σ

B

=

0.01 S/m,

SNR

= 20

dB) - Reonstrution errors for dierent values

of

∆

ε

rB

. . . 42

L

= 0

.

32

λ

b

, (

x

c

=

−

0

.

16

λ

b

,

y

c

=

−

0

.

58

λ

b

),

ε

r

= 5

.

5

,

σ

= 0

.

01

S/m [

τ

= 1

.

5

℄,

ε

rB

=

4.0,

σ

B

=

0.01 S/m,

SNR

=

10 dB) - Reonstruted

dis-tributions of the ontrast funtion (real part) when using (a)()

IMSA-IN and (b)(d) IN under (a)(b) full-nonlinear and ()(d)

approximate onditions (SOBA). . . 44

L

= 0

.

32

λ

b

, (

x

c

=

−

0

.

16

λ

b

,

y

c

=

−

0

.

58

λ

b

),

ε

r

= 5

.

5

,

σ

= 0

.

01

S/m [

τ

= 1

.

5

℄,

ε

rB

=

4.0,

σ

B

=

0.01 S/m,

SNR

=

10 dB) - Reonstruted

dis-tributions of the ontrast funtion (imaginary part) when using

(a)() IMSA-IN and (b)(d) IN under (a)(b) full-nonlinear and

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L

= 0

.

32

λ

b

, (

x

c

=

−

0

.

16

λ

b

,

y

c

=

−

0

.

58

λ

b

),

ε

r

= 5

.

5

,

σ

= 0

.

01

S/m [

τ

= 1

.

5

℄,

ε

rB

=

4.0,

σ

B

=

0.01 S/m,

SNR

= 10

dB) - Fitness (a) and

re-onstrution errors (b) versus outer iterations index,

i

. () Error

index values at eah fousingstep

s

(

s

= 1

, ..., S

). . . 46

4.10 Performane vs. Target Permittivity (Hollow Cylinder -

L

ext

=

0

.

48

λ

b

,

L

int

= 0

.

16

λ

b

,

(

x

c

= 0

.

08

λ

b

, y

c

=

−

0

.

48

λ

b

)

,

σ

= 0

.

01

S/m,

ε

rB

=

4.0,

σ

B

=

0.01 S/m,

SNR

= 20

dB) - Reonstrution

errors for dierentvalues of

τ

. . . 47

4.11 Performane vs. Target Permittivity (Hollow Cylinder -

L

ext

=

0

.

48

λ

b

,

L

int

= 0

.

16

λ

b

,

(

x

c

= 0

.

08

λ

b

, y

c

=

−

0

.

48

λ

b

)

,

ε

r

= 6

.

2

,

σ

= 0

.

01

S/m [

τ

= 2

.

2

℄,

ε

rB

=

4.0,

σ

B

=

0.01 S/m,

SNR

= 10

dB)

- Reonstruted distribution of the ontrast funtion. (a) Atual

ongurationand(b)realand(d)imaginarypartsprovidedbythe

IMSA-IN strategy and () real and (e) imaginary parts obtained

by the BARE-IN. . . 48

4.12 Performane vs. Target Sales (E-Shaped Satterer -

ε

r

= 5

.

5

,

σ

= 0

.

01

S/m [

τ

= 1

.

5

℄,

ε

rB

=

4.0,

σ

B

=

0.01 S/m,

SNR

=

20dB)

- Reonstruted distribution of the ontrast funtion (real part).

(a)Atualongurationandreonstrutionswith(b)(d)IMSA-IN

and ()(e)IN under(b)()full-nonlinearand(d)(e)approximate

onditions (SOBA). . . 49

4.13 Performane vs. Target Sales (E-Shaped Satterer -

ε

r

= 5

.

5

,

σ

= 0

.

01

S/m [

τ

= 1

.

5

℄,

ε

rB

=

4.0,

σ

B

=

0.01 S/m,

SNR

=

20

dB) - Reonstruted distribution of the ontrast funtion

(imag-inary part) with (a)() IMSA-IN and (b)(d) IN under (a)(b)

full-nonlinearand ()(d) approximate onditions (SOBA). . . 50

4.14 Performane vs. Target Condutivity (Square Satterer -

L

=

0

.

32

λ

b

,

(

x

c

=

−

0

.

16

λ

b

, y

c

=

−

0

.

58

λ

b

)

,

ε

r

= 5

.

5

,

σ

= 0

.

1

S/m

[

τ

= 1

.

5

−

j

5

.

39

℄,

ε

rB

=

4.0,

σ

B

=

0.01 S/m,

SNR

=

10 dB) -Reonstruted distributionof the ontrast funtion. (a)Realand

(b) imaginary parts provided by the IMSA-IN strategy and ()

real and (d)imaginary parts obtained with the bare IN. . . 51

5.1 Problem denition - Geometry of the problem (a), plot of the exitationsignal in(b)timedomain(i.e.,

χ

(

t

)

)and ()frequeny domain(i.e.,

X

(

f

)

),andofatypialGPR trae

u

e

v

(

r

m

, t

)

(d)and assoiated sattered eld

e

s

v

(

r

m

, t

)

(e). . . 56 5.2 FHMF-CGInversionProedure -Flowhartofthe GPR

prospet-ing method. . . 59 5.3 IllustrativeExample[Hollowsquare prole,

ε

rB

= 4

.

0

,

σ

B

= 10

−3

(20)

5.4 IllustrativeExample[Hollowsquare prole,

ε

rB

= 4

.

0

,

σ

B

= 10

−3

S/m, Noiseless data℄ Dieletri proles retrieved by (a)()(e)(g) FH-CG and(b)(d)(f)(h)FHMF-CG when(a)(b)

q

= 2

(

f

2

= 300

MHz), (a)(b)

q

= 3

(

f

3

= 400

MHz), (a)(b)

q

= 4

(

f

4

= 500

MHz),(a)(b)

q

= 5

(

f

5

= 600

MHz). . . 65 5.5 PerformaneAssessment[Hollowsquareprole,

ε

rB

= 4

.

0

,

σ

B

=

10

−3

S/m,

τ

= 1

.

0

℄ Behaviour of the integral error vs. the

SNR

(a),and dieletri proles retrieved by (b)(d) FH-CG and ()(e) FHMF-CG when (b)()

SNR

= 30

dB, (d)(e)

SNR

= 10

dB. . . 66 5.6 Performane Assessment [Square prole,

ε

rB

= 4

.

0

,

σ

B

= 10

−3

S/m℄ Behaviour of the integral error vs.

τ

(a), and dieletri proles retrieved by (b)(d) FH-CG and ()(e) FHMF-CG when (b)()

τ

= 1

.

0

,(d)(e)

τ

= 2

.

2

when

SNR

= 30

dB. . . 68 5.7 PerformaneAssessment [Two-bar prole,

ε

rB

= 4

.

0

,

σ

B

= 10

−3

S/m,

τ

= 1

.

4

℄ Behaviourof the total integral error versus

M

and

SNR

for (a)FH-CG and (b)FHMF-CG. . . 70

5.8 PerformaneAssessment [Two-bar prole,

ε

rB

= 4

.

0

,

σ

B

= 10

−3

S/m,

τ

= 1

.

4

,

SNR

= 20

dB℄ Atual (a) and dieletri proles retrieved by (b)(d) FH-CG and ()(e) FHMF-CG when (b)()

M

= 19

,(d)(e)

M

= 76

. . . 71

5.9 Comparative Assessment [Cirle prole [96℄,

ε

rB

= 9

.

0

,

σ

B

=

10

−2

S/m,

ε

= 9

.

05

,

σ

= 0

.

0

,

k

=

K

= 3

℄ Real (a)() and imag-inary parts (b)(d) of the atual (a)(b) and FHMF-CG retrieved prolewhen

SNR

= 50

dB()(d),and (e) behaviourof the inte-gralerrorvs. the

SNR

. . . 73

5.10 Comparative Assessment [Large square prole [87℄,

ε

rB

= 9

.

0

,

σ

B

= 10

−2

S/m,

τ

= 3

.

0

,

k

=

K

= 6

℄(a)Behaviouroftheintegral

error vs. the

SNR

and (b) atual and () FHMF-CG retrieved proleswhen

SNR

= 50

dB. . . 74 5.11 ExperimentalValidation -Dataset [97℄-Photooftheexperimental

setup (ourtesy of Prof. M. Guy) (a), geometry of the problem (b),and full measured radargram available in[97℄ (). . . 76

5.12 Experimental Validation - Dataset [97℄ [

V

= 21

℄ Real (a) and imaginaryparts (b) of the FHMF-CG retrieved prole. . . 77

5.13 ExperimentalValidation - Dataset [97℄-Real(a)()(e)and imag-inary parts (b)(d)(f) of the FHMF-CG retrieved proles when (a)(b)

V

= 5

,()(d)

V

= 11

, and (e)(f)

V

= 41

.. . . 78

6.1 Comparative Assessment (Square Satterer at Dierent Depths

-L

= 0

.

16

m

, (

x

c

= 0

.

0

m

,

ε

r

= 5

.

5

,

σ

= 0

.

01

S/m [

τ

= 1

.

5

℄,

ε

rB

=

4.0,

σ

B

=

0.01S/m,

SNR

= 20

dB) -Loationof the

illumi-nating soures and of the measurement points for the (a)

(21)

-L

= 0

.

16

m

, (

x

c

= 0

.

0

m

,

ε

r

= 5

.

5

,

σ

= 0

.

01

S/m [

τ

= 1

.

5

℄,

ε

rB

=

4.0,

σ

B

=

0.01 S/m,

SNR

= 20

dB) -Atual targetused for

the omparison for (a) top (

y

c

=

−

0

.

16

m

), (b) intermediate (

y

c

=

−

0

.

4

m

)and () bottom (

y

c

=

−

0

.

64

m

)ongurations. . . 83 6.3 Comparative Assessment (Square Satterer at Dierent Depths

-L

= 0

.

16

m

, (

x

c

= 0

.

0

m

,

ε

r

= 5

.

5

,

σ

= 0

.

01

S/m [

τ

= 1

.

5

℄,

ε

rB

=

4.0,

σ

B

=

0.01 S/m,

SNR

= 20

dB) - Final reonstrution

obtainedbythe

IMSA

−

IN

methodwhenonsideringa(a)(b)() ross-boreholeand (d)(e)(f) anhalf spae setup.. . . 85

-L

= 0

.

16

m

, (

x

c

= 0

.

0

m

,

ε

r

= 5

.

5

,

σ

= 0

.

01

S/m [

τ

= 1

.

5

℄,

ε

rB

=

4.0,

σ

B

=

0.01 S/m,

SNR

= 20

dB) - Final reonstrution

(22)

(23)

Introdution

Inreentyears,therehasbeenagrowinginterestinthedevelopmentof

imag-ingsystems basedon theuse ofmirowaveradiations[1℄-[5℄. Dueto the

ompa-rable values of the inident wavelength and objet linear dimensions, the

phys-ial phenomenon involved in these systems is the sattering of eletromagneti

waves. Approahes based on mirowaves an be protably employed in several

diagnostisenarios,suhasnondestrutivetestingandevaluation(NDT/NDE)

of materialsin ivilengineering [6℄-[9℄, medialimaging for breast aner

dete-tion [10℄-[12℄, shallowinvestigationof Earth's subsurfae [13℄ as wellasretrieval

ofeletromagneti andgeometrial harateristisof satterersburiedunder the

air-soilinterfae [14℄[18℄.

Oneof thekey instrumentsforsubsurfae monitoringand imagingistheground

penetrating radar (

GP R

) [13℄[19℄ whih an be used, for example, for verifying

thestruturalstabilityofonrete struturesandfor rak detetioninside

ina-essiblematerials. Althoughverygoodresultshavebeenobtainedbyusing

GP R

,

thesolutionofinversesatteringproblemsfor burieddetetionisstilla

halleng-ing issue, espeially onsidering the need for fast and aurate apparatuses for

illuminatingthetargetundertest and measuringthesattered radiation,aswell

asfor eientproedures toretrievethe geometrialand dieletri properties of

objets buried under ground with an aeptable level of resolution. In

parti-ular, onerning the inversion proedures, it seems that further researhes are

requiredin orderto overomethe limitationsarising fromthe wellknown issues

ofnon-linearity andill-posedness haraterizingthebasieletromagneti

formu-lation[5℄. Thenon-linearity isdiretlylinked tothe dependene ofthe unknown

total eld inside the investigation area on the satterer properties [20℄, while

the ill-posedness auses the solutionto be extremely sensitive to noise aeting

available data for the inversion. Moreover, the availablemeasured data are

lim-itedandpratialmeasurementsarearriedoutfromlimitedtransmitter-reeiver

positions,resultinginlimiteddatadiversity [20℄. Forthese reasons, eient

reg-ularization tehniques [21℄-[23℄ apable to mitigate the above mentioned issues

(24)

and seond-order [27℄-[29℄ Bornapproximations.

In this ontext, it has also been proved that deterministi inversion proedures

[30℄-[32℄ an provide very aurate reonstrution results, although they suer

fromastrongdependeneontheinitializationphase. Ontheotherhand,the use

of stohasti tehniques has also been proposed [33℄-[38℄. Stohasti approahes

an eiently overome the above limitation, but they exhibit a signiantly

higheromputationalost [41℄[42℄.

Amongdeterministiapproahes, inexat-Newton(

IN

)methods[28℄[29℄[43℄-[49℄

havebeen proven tobeeetiveaslinearizationandregularizationtoolsfor

solv-ing inverse-sattering problems, both numerially and experimentally [44℄.

Ba-sially,these methodsprovidealinearizationof theimagingequationsby means

of a Newton's expansion through the Fréhet derivative, and solve them in an

approximate way [29℄. However, the appliation of suh an approah has been

mainly limited to the free-spae senario, while a more omplex formulation is

neededwhendealingwithsubsurfaeprospeting[50℄. The

IN

methodhasbeen

preliminary applied to retrieve buried objets in [28℄ within the seond-order

Born approximation (

SOBA

) [27℄. By exploiting suh a seond-order

approx-imation, a signiant redution of the omputational burden an be ahieved,

thankstoaredutionoftheproblemunknowns(thedieletriparameters),sine

the internal total eletrield is writtenas the sum of the known inident eld

andtheinternallinearizedsattered eld(whihisalsoexpressed intermsofthe

transmitted eld)[29℄.

It must be also notied that multi-resolution approahes [51℄-[53℄ have been

proven tobevery eetive inreduing the amount of loalminima arisingfrom

the non-linearity of the free-spae inverse-sattering problem, bringing a

bet-ter exploitation of the available information from olleted data and yielding

both auratereonstrutions andhigh omputationaleieny. Thesynergeti

integration of a diret regularization tehnique, suh as the

IN

method, and

the iterative multi-salingapproah (

IMSA

) [54℄ has been shown to eetively

takleboththe non-linearity and theill-posedness/ill-onditioningof mirowave

imaging problems by exploiting the best properties of the two strategies and

mutuallyoveroming theirintrinsilimitationsintomographiimaging[48℄-[47℄.

Asamatteroffatthe exploitationofsuhanapproahleadstoastrong

simpli-ationof the problem,thanks itsapabilityto enfore a higherresolution only

inthe so-alledregions-of-interest(

RoIs

)[54℄.

Moreover, a signiant advantage in using

GP R

as the subsurfae prospeting

tool isrepresented by the availability of wide-band measurements [59℄, overing

awiderangeofthe mirowaveradiationspetrum. In fat,pulsed

GP R

systems

are based on the transmission of short eletromagneti pulses in time-domain

[59℄, whihpenetrate insidethe host mediumandare partially reeted towards

the reeiving antennas eah time a disontinuity of the dieletri harateristis

(25)

sat-teringapproahes anbefurtherextended byintroduingadditionalinformation

omingfromtheintrinsifrequenydiversityoftheolleteddata. Insuhaway,

the exploitation of wide-band

GP R

measurements requires the development of

multi-frequeny tehniques whih are able to protably exploit the information

assoiatedto dierent omponents of the measured spetrum.

Following the above onsiderations, this thesis presents two eient

single-frequenytehniquesbasedontheintegrationoftheinexat-Newton(

IN

)method

with a multifousing tehnique, and then a multi-frequeny approah whih is

abletoeetivelyexploitthefrequeny diversityof

GP R

measurementsthrough

aFrequeny-Hopping(

F H

) sheme.

Thesis outline

The thesis is organized as follows. Firstly, the basi equation governing in

fre-queny domainthe satteringphenomenainsubsurfae problems areintrodued

inChapter 2. Then,asingle-frequenyapproahbasedonthe

IN

methodunder

the seond order Born approximation (

SOBA

) is presented in Chapter 3. An

improved version of this tehnique, treating the full non-linear inverse

satter-ing problem is shown in Chapter 4, extending to strong satterers the imaging

apabilities of the rst approximated approah. Finally, Chapter 5 presents an

innovativemirowaveinverse-satteringnestedapproahombiningaF

requeny-Hopping(

F H

)proedureandaMulti-Fousing(

MF

)tehniquefordealingwith

multi-frequeny GPR measurements. Finally,a omparison amongthe dierent

presented tehniques is given and some nal onlusions are drawn in Chapter

(26)

(27)

Inverse Sattering Equations for

the Subsurfae Problem

In this hapter, the basi equations mathematially modeling the subsurfae

inverse sattering problem in frequeny domain are presented. More preisely,

the two equationsompletely desribing the elds measured within and outside

the buried investigationdomainare referred toas state and data equations.

It is shown that the problem of retrieving the eletromagneti harateristis

of unknown objets buried below the interfae in a half spae senario an be

reformulatedastheminimizationofasuitableostfuntion. Suhaostfuntion

aounts for both the mismath between the measured and omputed sattered

eldoveragivenobservationdomainandforthemismathbetweenthemeasured

(28)

2.1 Geometry of the Problem

Letusonsider aset of ylindrialsatterers buriedina homogeneous, isotropi

andnon-magnetihalfspae medium[Fig. 2.1℄. Theuppermedium(i.e.,

y >

0

)

is supposed to be air, with dieletri properties equal to those of the vauum

(

ε

0

= 8

.

85

×

10

−

12

Farad/m,

µ

0

= 1

.

26

×

10

−

6

Henry/m and

σ

0

= 0

S/m). The lossy lower half spae of bakground relative permittivity

ε

rB

and bakground

ondutivity

σ

B

S/m, ontains a set of satterers loated within the known

in-vestigationdomain

D

inv

[Fig. 2.1℄and desribedbydisontinuous(wrtthe

bak-ground) prolesof permittivity

ε

r

(

r

)

and ondutivity

σ

(

r

)

,where the position

vetor

r

denotes a point inthe transverse plane (i.e.,

r

= (

x, y

)

).

-2.5

-2

-1.5

-1

-0.5

0

0.5

-1.5

-1

-0.5

0

0.5

1

1.5

y/

λ

b

x/

λ

b

Measurement points

Source locations

D

inv

Free Space

Soil

ε

rB

,

σ

B

,

µ

0

ε

0

,

µ

0

τ

(x,y)

τ

(x,y)

-2.5

-2

-1.5

-1

-0.5

0

0.5

-1.5

-1

-0.5

0

0.5

1

1.5

y/

λ

b

x/

λ

b

Measurement points

Source locations

D

inv

Free Space

Soil

ε

rB

,

σ

B

,

µ

0

ε

0

,

µ

0

τ

(x,y)

τ

(x,y)

(a) (b)

Figure2.1: Geometryof asubsurfae imagingproblem. (a) Cross-borehole and

(b) halfspae setup.

λ

b

is the wavelength inthe bakground material.

2.2 Mathematial Formulation

Inthefollowing,weassumethattheunknownburiedtargetsareilluminatedbya

setof

V

inidentmonohromatiwavesproduedbyasetofinnitelineurrents

oriented along the

z

axis,whihan bearranged inboth halfspae[Fig. 2.1(a)℄

or ross-borehole[Fig. 2.1(b)℄ setup 1

. Given that, the generated inident waves

are of transverse magneti (

T M

)type, suh that

E

inc

(

v

)

(

r

) =

E

inc

(

v

)

(

r

)

b

z

, v

= 1

, ..., V.

(2.1)

1

Hybrid ongurationsan exist, too, where the soures of em waves are displaed both

(29)

SUBSURFACE PROBLEM

Moreover, weassumethatforeah

v

-thilluminationthelongitudinal

ompo-nent of the sattered eletri eld vetor is olleted at

M

measurement points

loated atposition

r

(

v

)

m

,

m

= 1

, ..., M

dening the observation domain

D

obs

.

Following the lassialinverse sattering approah [5℄, the problemof retrieving

the shape, the position and the eletromagneti harateristis of the targets

buried within

D

inv

is formulated as the problem of reonstruting the so-alled

ontrast funtion,dened as

τ

(

r

) =

ε

eq

(

r

)

−

ε

B,eq

ε

0

(2.2)

where

ε

eq

(

r

) =

ε

0

ε

r

(

r

)

−

j

σ

(

r

)

ω

(2.3)

and

ε

B,eq

=

ε

0

ε

rB

−

j

σ

B

ω

.

(2.4)

Given (2.3) and (2.4),it iseasy toverify that the real part ofthe ontrast is

given by

ℜ {

τ

(

r

)

}

=

ε

r

(

r

)

−

ε

rB

(2.5)

while the imaginary part depends on the frequeny via the angular speed

ω

= 2

πf

as

ℑ {

τ

(

r

)

}

=

σ

B

−

σ

(

r

)

ωε

0

.

(2.6)

Denoting with

υ

(

j

)

the ross-setion of the

j

-th target buried within

D

inv

(

j

=

1

, ..., J

, being

J

the total number of satterers),wethen have

τ

(

r

) =







0

r

∈

/

P

J

j

=1

υ

(

j

)

τ

(

r

)

r

∈

P

J

j

=1

υ

(

j

)

(2.7)

sine outside the support of the

J

buried targets the equivalent permittivity

and the ondutivity is that of the bakground medium(i.e.,

ε

eq

(

r

) =

ε

B,eq

and

σ

(

r

) =

σ

B

) and nodisontinuityan be observed by the propagatingimpinging

waves.

As a matter of fat, the total eld measured at position

r

when the

J

targets

are buried inside the investigation domain an be deomposed as the sum of

twoseparate ontributions, represented by theinidenteldand bythe so-alled

sattered eld, respetively

E

(

tot

v

)

(

r

) =

E

(

inc

v

)

(

r

) +

E

(

scatt

v

)

(

r

)

, v

= 1

, ..., V.

(2.8)

Given the ylindrial symmetry of the problem [Fig. 2.1℄ and the isotropi

harateristis of the medium at hand, also the total eld and the sattered

eld result

z

-oriented (i.e.,

E

(

v

)

tot

(

r

) =

E

(

v

)

tot

(

r

)

b

z

and

E

(

v

)

scatt

(

r

) =

E

(

v

)

(30)

v

= 1

, ..., V

). If on the one hand the inident eld

E

(

v

)

inc

(

r

)

is referred to the

half spae senario when no objets are loated below the interfae [Fig. 2.1℄,

onthe otherhand the sattered eld isthe ontributiontothe total eld dueto

the presene of satterers buried within

D

inv

. More preisely, the total eld is

ompletelydesribed by means of the followingset of Maxwellequations [59℄











▽ ×

E

(

tot

v

)

(

r

) =

−

jωµ

0

H

(

tot

v

)

(

r

)

▽ ×

H

(

tot

v

)

(

r

) =

jωε

eq

(

r

)

E

(

tot

v

)

(

r

) +

I

0

δ x

−

x

(

v

)

δ y

−

y

(

v

)

b

z

▽ ·

ε

eq

(

r

)

E

(

tot

v

)

(

r

) = 0

▽ ·

µ

0

H

(

tot

v

)

(

r

) = 0

(2.9)

where

H

(

v

)

tot

(

r

)

isthe total magneti eld atposition

r

H

tot

(

v

)

(

r

) =

H

tot,x

(

v

)

(

r

)

x

b

+

H

(

v

)

tot,y

(

r

)

y

b

(2.10)

and the impressed urrent for the

v

-th illumination is expressed in expliit

formas

J

0

(

r

) =

I

0

δ x

−

x

(

v

)

δ y

−

y

(

v

)

b

z

(2.11)

where

I

0

is the amplitude of the urrent owing along an innite

z

-oriented line loated at position

x

(

v

)

, y

(

v

)

. In (2.9), the divergene of

ε

eq

(

r

)

E

(

v

)

tot

is set

tonull (i.e.,

ε

eq

(

r

)

E

(

v

)

tot

is solenoidal) sine itis easilyveried that

▽ ·

ε

eq

(

r

)

E

(

tot

v

)

=

∂

∂z

n

ε

eq

(

x, y

)

E

tot

(

v

)

(

x, y

)

o

= 0

.

(2.12)

Similarly, in absene of targets within

D

inv

, the inident eld satises the

following set of equations [59℄











▽ ×

E

(

inc

v

)

(

r

) =

−

jωµ

0

H

(

inc

v

)

(

r

)

▽ ×

H

(

inc

v

)

(

r

) =

jωε

hs

E

inc

(

v

)

(

r

) +

I

0

δ x

−

x

(

v

)

δ y

−

y

(

v

)

b

z

▽ ·

ε

hs

E

(

inc

v

)

(

r

) = 0

▽ ·

µ

0

H

(

inc

v

)

(

r

) = 0

(2.13)

where

H

(

v

)

inc

(

r

)

isthe inidentmagneti eldat position

r

(31)

SUBSURFACE PROBLEM

and

ε

hs

is a piee-wise onstant funtion dening the (possibly omplex)

dieletri permittivityof the halfspae senario athand

ε

hs

=







ε

0

y >

0

ε

B,eq

y <

0

.

(2.15)

Given that, it follows that the sattered eld satises the following set of

equa-tions [59℄











▽ ×

E

(

scatt

v

)

(

r

) =

−

jωµ

0

H

(

scatt

v

)

(

r

)

▽ ×

H

(

scatt

v

)

(

r

) =

jωε

hs

E

scatt

(

v

)

(

r

) +

jω

∆

ε

(

r

)

E

(

v

)

tot

(

r

)

▽ ·

ε

hs

E

(

scatt

v

)

(

r

) = 0

▽ ·

µ

0

H

(

scatt

v

)

(

r

) = 0

(2.16)

where

H

(

v

)

scatt

(

r

)

is the sattered magneti eld atposition

r

H

(

scatt

v

)

(

r

) =

H

scatt,x

(

v

)

(

r

)

b

x

+

H

scatt,y

(

v

)

(

r

)

b

y

(2.17)

and

∆

ε

(

r

)

modelsthedisontinuitybetween thedieletripermittivityofthe

satterers and the surrounding homogeneousmedium

∆

ε

(

r

) =

ε

eq

(

r

)

−

ε

hs

.

(2.18)

Bylookingat(2.16)weanobservethatthesatteredeldisduetoanequivalent

soure that models the presene of the unknown satterers inside

D

inv

, dened

as[59℄

J

eq

(

r

) =

jω

∆

ε

(

r

)

E

(

tot

v

)

(

r

)

.

(2.19)

Byre-arranging(2.16)andimposingtheontinuityofthetangentialomponents

ofboththeeletriandmagnetieldsattheinterfae(i.e.,at

y

= 0

),eventually

[59℄ the

z

-omponent ofthe sattered eldfor pointsloated belowthe interfae

[i.e.,

y <

0

, Fig. 2.1℄ an beomputed as

E

scatt

(

v

)

(

r

) =

k

2

B

Z

D

inv

τ

(

r

′

)

E

tot

(

v

)

(

r

′

)

G

buried

(

r

,

r

′

)

d

r

′

(2.20)

while the sattered eld for points loated above the interfae [i.e.,

y >

0

,

Fig. 2.1℄ isexpressed as

E

scatt

(

v

)

(

r

) =

k

B

2

Z

D

inv

(32)

In (2.20) the integral Green's funtion

G

buried

(

r

,

r

′

)

relates points below the

interfae to points belowthe interfae (i.e.,

y <

0

and

y

′

<

₀

) and, aording to

Eq. (4.42)in [59℄ and tothe denition of the ontrast funtion given in (2.2),it

isdened as

G

buried

(

r

,

r

′

) =

G

buried

(

x, y, x

′

, y

′

) =

−

j

4

π

ε

0

ε

B,eq

R

+

∞

−∞

exp

(

−

ju

(

x

′

−

x

))

k

yB

[

exp

(

−

jk

yB

|

y

−

y

′

|

) +

µ

0

k

yB

−

µ

0

k

y

0

µ

0

k

yB

+

µ

0

k

y

0

exp

(

−

jk

yB

(

y

′

+

y

))

i

du

(2.22)

while the funtion

G

half

−

space

(

r

,

r

′

)

linkspointsbelow the interfae topoints

above the interfae (i.e.,

y >

0

and

y

′

<

₀

) and is dened as

G

half

−

space

(

r

,

r

′

) =

G

half

−

space

(

x, y, x

′

, y

′

) =

−

jµ

0

2

π

ε

0

ε

B,eq

R

+

∞

−∞

exp

(

−

jk

yB

y

′

)

exp

(

jk

y

0

y

)

exp

(

−

ju

(

x

′

−

x

))

µ

0

k

yB

+

µ

0

k

y

0

du.

(2.23)

In (2.22) and (2.23)

k

y

0

and

k

yB

are funtions of the integration variable

u

and are dened asfollows







k

2

y

0

=

k

y

2

0

(

u

) =

k

0

2

−

u

2

y >

0

k

2

yB

=

k

2

B

(

u

) =

k

B

2

−

u

2

y <

0

(2.24)

where

k

0

=

ω

√

ε

0

µ

0

and

k

B

=

ω

√

ε

B,eq

µ

0

are the wave-number infree-spae and in the lossy bakground medium, respetively. Finally, the following salar

integralequationsanberetrieved, mathematiallymodelingthe buried

satter-ingproblem

E

inc

(

v

)

(

r

) =

E

tot

(

v

)

(

r

)

−

k

2

B

R

D

inv

τ

(

r

′

)

E

(

v

)

tot

(

r

′

)

G

int

(

r

,

r

′

)

d

r

′

r

∈

D

inv

(2.25)

E

scatt

(

v

)

(

r

) =

k

2

B

R

D

inv

τ

(

r

′

)

E

(

v

)

tot

(

r

′

)

G

ext

(

r

,

r

′

)

d

r

′

r

∈

D

obs

(2.26)

inwhih

D

obs

∈

/

D

inv

istheobservationdomain,wherebothsouresand measure-ment points are supposed tobe loated [Fig. 2.1℄. The former integralequation

is alled state equation, while the latter is the so-alled data equation, and

both need to be solved in a numerial way in order to retrieve the unknown

ontrast funtion

τ

(

r

)

and the unknown total eld inside

D

inv

. Clearly, sine

G

int

(

r

,

r

′

)

(ommonly known as the internal Green's funtion) relates points

whih are loated inside

D

inv

, it will always oinide with

G

buried

(

r

,

r

′

)

(33)

SUBSURFACE PROBLEM

otherhand,

G

ext

(

r

,

r

′

)

(ommonlyknown astheexternal Green'sfuntion)

re-lates points inside

D

inv

to points outside it (i.e., belonging to the observation

domain

D

obs

∈

/

D

inv

). Then, if a half spae setup is onsidered, where mea-surement points are loated above the interfae (i.e.,

y

(

v

)

m

>

0

, for

v

= 1

, ..., V

and

m

= 1

, .., M

[Fig. 2.1(a)℄), we will have that

G

ext

(

r

,

r

′

) =

G

half

−

space

(

r

,

r

′

)

.

Otherwise,ifaross-boreholesetupisonsidered,wheremeasurementpointsare

loated below the interfae (i.e.,

y

(

v

)

m

<

0

, for

v

= 1

, ..., V

and

m

= 1

, .., M

[Fig.

2.1(b)℄), wewill have that

G

ext

(

r

,

r

′

) =

G

buried

(

r

,

r

′

)

.

In order to solve the inverse sattering problem, both the unknowns and

the state and data equations need to be disretized. A ommon hoie is to

use retangularbasisfuntions[56℄partitioningthe investigationdomaininto

N

subdomains

τ

(

r

) =

P

N

n

=1

τ

(

r

n

)

ψ

n

(

r

)

E

tot

(

v

)

(

r

) =

P

N

n

=1

E

tot

(

r

n

)

ψ

n

(

r

)

(2.27)

resultingin the followingvetor of unknowns

Θ

=

n

τ

(

r

n

) ;

E

tot

(

v

)

(

r

n

) ;

n

= 1

, ..., N

;

v

= 1

, ..., V

o

.

(2.28)

Given that, the disretized form ofthe state equation (2.25) beomes

E

inc

(

v

)

(

r

n

) =

E

tot

(

v

)

(

r

n

)

−

k

B

2

P

p

=1

τ

(

r

p

)

E

(

v

)

tot

(

r

p

)

G

int

(

r

n

,

r

p

)

r

n

,

r

p

∈

D

inv

(2.29)

while the data equation in(2.26) beomes

E

scatt

(

v

)

r

(

m

v

)

=

k

2

B

P

N

n

=1

τ

(

r

n

)

E

(

v

)

tot

(

r

n

)

G

ext

r

(

m

v

)

,

r

n

r

(

m

v

)

∈

D

obs

,

r

n

∈

D

inv

.

Solving theinversesattering problemis thenreformulatedasthe estimation

oftheunknownoeients

Θ

viathe minimizationofthefollowingostfuntion

Φ

n

Θ

ˆ

o

=

β

data

Φ

data

n

ˆ

Θ

o

+

β

state

Φ

state

n

ˆ

Θ

o

(2.30)

where

β

data

and

β

state

areonstantweights. In(2.30)thedataterm

Φ

data

n

ˆ

Θ

o

quantiesthe mismathbetweenthe known satteredeldolleted at

M

points

belongingto

D

obs

tothe satteredeldomputedfortheretrievedversionsofthe

unknowns (i.e.,

ˆ

Θ

=

n

τ

ˆ

(

r

n

) ; ˆ

E

tot

(

v

)

(

r

n

) ;

n

= 1

, ..., N

;

v

= 1

, ..., V

o

) aording to

(34)

Φ

data

n

ˆ

Θ

o

=

P

V

v

=1

P

M

m

=1

E

scatt

(

v

)

r

(

m

v

)

−

E

ˆ

scatt

(

v

)

r

(

m

v

)

2

P

V

v

=1

P

M

m

=1

E

scatt

(

v

)

r

(

m

v

)

2

(2.31)

where

E

ˆ

(

v

)

scatt

r

(

m

v

)

is the omputed sattered eld for the

m

-th probe under

the

v

-th illumination. Similarly,the state term of the ost funtion dened in

(2.30)measuresthedierenebetweentheknown inidenteldinside

D

inv

tothe

retrieved inident eld omputed aording (2.25) on the basis of the estimated

ˆ

Θ

Φ

state

n

ˆ

Θ

o

=

P

V

v

=1

P

N

n

=1

E

inc

(

v

)

(

r

n

)

−

E

ˆ

inc

(

v

)

(

r

n

)

2

P

V

v

=1

P

N

n

=1

E

inc

(

v

)

(

r

n

)

2

(2.32)

where

E

ˆ

(

v

)

inc

(

r

n

)

is the omputed sattered eld for the

n

-th point in

D

inv

(35)

Multi-Fousing Inexat Newton

Method within the Seond-Order

Born Approximation

In this hapter,the reonstrution of a shallow buriedobjet isaddressed byan

eletromagneti inverse sattering method based on ombining dierent

imag-ing modalities. In partiular, the proposed approah integrates the

inexat-Newton method with an iterative multi-saling approah. Moreover, the use of

the seond-order Born approximation (

SOBA

) is exploited. A numerial

val-idation is provided onerning the potentialities arising by ombining the

reg-ularization apabilities of the inexat-Newton method and the eetiveness of

the multi-fousingstrategytomitigatethe non-linearityandill-posednessof the

inversionproblem. Comparisonswith the standard "bare"approah interms of

(36)

3.1 Introdution

The aim of this hapter is to reformulate the integrated

IMSA

−

IN

inver-sion tehnique [48℄ inorder todeal with subsurfae imagingand toevaluate the

eetiveness of suh an approah when the seond-order Born approximation

(

SOBA

) is applied. Moreover, a diret omparison in terms of auray,

ro-bustness against dierent onditions and noise levels, as well as omputational

eieny is given when diretly omparingthe proposed

IMSA

−

IN

−

SOBA

approah with its standard bare implementation (

BARE

−

IN

−

SOBA

), as desribed in[28℄.

Towardsthisend,setion3.2providesthebasimathematialformulationusedto

modeltheburiedproblemunderthe

SOBA

. InSet. 3.3theombined

IMSA

−

IN

−

SOBA

is desribed. An in-depth numerial validationisthen provided in

Set. 3.4 in order to analyze the performane of the proposed approah and to

demonstrate its eetiveness and advantages over the

BARE

−

IN

−

SOBA

, under monohromati transverse magneti (

T M

) illumination onditions in a

ross-borehole setup similar to that used in [37℄. Finally, some onlusions are

drawn (Set. 3.5).

3.2 Problem Formulation

Letusonsideraylindrialsattererburiedinahomogeneoushalfspaemedium.

A ross-borehole measurement ongurationis assumed [Fig. 3.1℄. Let

τ

(

r

)

de-note the ontrast funtion insidethe inspeted area

D

inv

,as dened inequation

(2.2). The uppermedium is supposed tobe air, with dieletri properties equal

tothoseofthevauumandthepositionvetor

r

denotesapointinthetransverse

plane,i.e.,

r

= (

x, y

)

.

The target, whose ross setion is inluded in the inspeted area

D

inv

is

illu-minated by

V

inident waves, whih are produed by a set of innite line

ur-rents. They generate inident waves of transverse magneti type, suh that

E

(

inc

v

)

(

r

) =

E

inc

(

v

)

(

r

)

ˆ

z

,

v

= 1

, . . . , V

. Due to the ylindrialgeometry,the sattered and total elds results tobe

z

-polarized,too.

The basi equation for this inverse problem is therefore the following salar

in-tegral one

E

scatt

(

v

)

(

r

) =

E

(

v

)

tot

(

r

)

−

E

(

v

)

inc

(

r

) =

k

2

B

Z

D

inv

τ

(

r

′

)

E

tot

(

v

)

(

r

′

)

G

buried

(

r

,

r

′

)

d

r

′

,

(3.1)

whih is a nonlinear ill-posed Lippman-Shwinger equation, whose kernel is the

Green's funtion for the half spae [55℄ with denition given inequation (2.22).

In equation (3.1),

E

(

v

)

tot

and

E

(

v

)

scatt

are the

z

-omponents of the total and

sat-tered eletri elds (for the

v

-th illumination), respetively. Suh equation is

(37)

WITHIN THESECOND-ORDER BORN APPROXIMATION

-2.5

-2

-1.5

-1

-0.5

0

0.5

-1.5

-1

-0.5

0

0.5

1

1.5

y/

λ

b

x/

λ

b

Measurement points

Source locations

D

inv

Figure3.1: Geometry of the problemand imagingsetup.

E

scatt

(

v

)

(

r

)

∼

=

F

(

v

)

B

1

τ

(

r

) +

k

B

2

Z

D

inv

τ

(

r

′

)

F

B

(

v

1

)

τ

(

r

′

)

G

buried

(

r

,

r

′

)

d

r

′

=

F

B

(

v

2

)

(

τ

) (

r

)

,

(3.2)

where

F

(

v

)

B

1

denotes the rst order Born operatordened as

F

B

(

v

1

)

τ

(

r

) =

k

B

2

Z

D

inv

τ

(

r

′

)

E

inc

(

v

)

(

r

′

)

G

buried

(

r

,

r

′

)

d

r

′

.

(3.3)

Consequently, sine the ontrast funtion is independent of

v

, the inverse

sat-teringproblemanbeformulatedasthesolutionofthefollowingsetofequations

with respet tothe unknown

τ

F

B

2

(

τ

) =







F

B

(1)

2

(

τ

)

.

F

B

(

V

2

)

(

τ

)







=







E

scatt

(1)

.

E

scatt

(

V

)







=

E

scatt

(3.4)

Thedisreteounterpartsoftheaboveequationsanbeobtainedbypartitioning

them in square subdomainsin order to obtain pixelatedimages of the retrieved

distributionsof the dieletri parametersinside the inspeted area.

3.3 Reonstrution Method

Inordertosolveequation(3.4), aninner/outeriterativeshemebasedonan

IN

method is applied [28℄. The operator equation (3.4) is iteratively linearized by

usingtheFrehét derivativeof theoperator

F

B

2

. Thisstep leadstothe following linear operatorequation

(38)

where

τ

i

isthe ontrast funtionatthe

i

-thiterationand

F

′

τ

i

denotesthe Frehét

derivativeof the operator

F

B

2

at

τ

i

. Asdetailed in[29℄,

F

′

τ

i

isgiven by

F

τ

′

i

u

=







F

τ

′

i

(1)

u

.

F

τ

′

i

(

V

)

u







(3.6)

where

F

τ

′

i

(

v

)

u

(

r

) =

F

(

v

)

B

1

u

(

r

) +

k

2

B

R

D

inv

τ

i

(

r

′

)

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