Microwave Interrogating Signals and Acoustic Wave

Induced Reections

R. A. Albanese

Human Eectiveness Directorate

Biomechanisms and Modeling Branch

AFRL

Brooks AFB, TX 78235

H. T. Banks and J. K. Raye

Center for Research in Scientic Computation

North Carolina State University

Raleigh, NC 27695-8205

email: htbanks@eos.ncsu.edu

April 28, 2002

Abstract

Aclassofinverseproblemsisformulatedfortheestimationofmaterialdielectric

pa-rametersusingreectionsofpulsedmicrowaveinterrogatingsignalsfrommoving

acous-tic interfaces in the dielectric. A model foracoustic pressure-dependence of material

constitutiveparameters inMaxwell'sequationsispresentedandanalyzed. Theoretical

An increasing number of papers in electromagnetic material interrogation are oriented

to-ward nondestructive oreven noninvasive detection of material properties. Among these are

dielectric properties,such aspermittivityand conductivity, and/orgeometry,includingsize,

shape, and location of occlusions. For example, in subsurface damage detection via eddy

current techniques (see[7], [5], and [6]) the goalis toidentify regionsof lowor zero

conduc-tivitywhichrepresentvoids orbarrierstoelectricalconductivityasappearindelaminations,

corrosions,and noncontactingcracks. The subsurface orinternal structuralinterrogationwe

focus onhere depends less onthe conductivity of the material than onthe materialelectric

polarization. Such techniquesare thereforeapplicable innon-invasiveexamination oftissue,

whichis alowconductivedielectric, to detectsuch anomaliesasabnormalities incell

struc-ture and chemical composition as well as in underground detection of mines and bunkers

containingrebar concrete withhighly conductivecomponents.

There are several ways in which these polarization based interrogation techniques may be

attempted. One may discern material properties by observing electromagnetic microwave

impulsesafterthey havebeen transmittedthrough the target(see the discussions and

refer-ences in Chapter 1 of [4]and [1]). DiÆculties for this approach include large and/or highly

dissipative and dispersive material targets as well as the infeasibility of placing sensors on

the back side of the target.

A second approachentails the interrogation via reections froma target with a highly

con-ductive back boundary. Examples include use of a metal tipped catheter (this, however,

scarcely qualies as noninvasive!) in tissue regions that are accessible, an artery for

in-stance,ortargetswithmetalbacked surfaces,suchasaircraft,ormetalcontainingstructures

including rebar concrete and communication equipment. However widespread use of such

techniques is not feasible in many targets, especially in biomedical diagnostics involving in

vivo interrogation.

Athird possibility,treatedinthis paper,entailsuse ofatravelingacousticwaveasa

reect-ing virtual interface for propagating microwave impulses. It is rather well-accepted (e.g.,

see [22]) thatacousticpressure waveswillinteractwith electromagneticsignals inways that

oftenmimicinterfacialpartialreection/partial transmissionfor the electromagneticwaves.

In [4]the authorsinvestigated aspectsof elementary electromagnetic/acousticwave

interac-tion. The modelingpresented in[4]was rathernaive and not basedon any specicphysical

mechanisms. Moreover, a xed (standing) acoustic wave was employed as the reecting

in-terface. In this paper, we continue the investigations begun in [4] where a remote antenna

generates a pulsed microwave interrogating signal that reects from natural material

inter-faces. Here we attempt to develop both theory and computation to demonstrate that one

canusereections frombothboundaryinterfacesandtravelingacoustic-generated interfaces

tion2,wepresentsuchapolarizationmodelfoundedonacceptedmechanisms. Ourmodelis

based onthe Debyemodelfor orientationalpolarization. Wemodify this modelto

incorpo-ratepressure-dependence and allowfortheelectromagnetic/acousticinteraction thatresults

in the reections. Other approaches to modeling electromagnetic/acoustic interaction are

outlined in [10]. In Section 3, we develop a theoretical framework for the inverse problem

whichunderlies the interrogationtechnique. Finallyin Section4,weprovidesample inverse

problemcalculations tosupportthe feasibiltyof the technique.

Under certain assumptions (see also [4]), includingmaterial homogeneity in directions

per-pendicular to the direction of electromagnetic wave propagation and the use of a polarized

planar electromagnetic wave, we may model the interrogation technique using a one

di-mensional form of Maxwell's equations with temporally and spatially varying coeÆcients.

Provided that theseassumptions hold, Maxwell's equationscan bewritten

@

@z

E =

0 @

@t

H (1)

@

@z

H =

@

@t

D+E+J

s

(2)

where E and H are the electric and magnetic elds, D is the electric ux density, and J

s

is the source current density. The macroscopic polarizationP is introduced via the electric

ux density through the relation

D=

0

r

E+P (3)

where

r

representsthe eectsofinstantaneouspolarizationinthedielectric. Wediscussthe

form of P in great detail inSection 2. By taking the appropriate derivativesof (1)and (2)

and employingequation (3), we obtain

0

r

E+

0

P +

0

_

E E

00

=

0 _

J

s

: (4)

Here and throughoutwe use _

E to denote @

@t

E and E 0

to denote @

@z E:

We consider (4) in the domain 0 z 1 with air in the interval [0;z

1

] and a dielectric

material in the region (z

1

;1] (0 < z

1

< 1). We enforce an absorbing boundary condition

on the left (z = 0) and a perfectly conductive boundary condition on the right (z = 1).

Weleave the initialconditions for the electric eld ingeneral form,but we assume that the

polarizationand its rst time derivative are initiallyzero (this may be done withoutloss of

### 1

### z=0

### Dielectric

### Material

### Air

### z=1

### Boundary

### Absorbing

### Boundary

### Perfectly Conductive

### Interrogating

### E&M

### Waves

### Waves

### Pressure

### Acoustic

### z=z

Figure 1: Schematicof geometry

2 Polarization model

In this section,wepresent and motivateour model for pressure-dependent polarization. We

begin by discussingpolarizationingeneraland thenexplain howitpertainstoour problem.

2.1 Mechanisms of polarization

Electric polarizationis by denition the electric dipole moment per unit volume. The

for-mationoftheseelectricdipolescanbecausedby several mechanisms[2], [3]whichwebriey

summarize here.

Electronic polarization/ Optical polarization/ Induced polarization Anappliedeld

displaces the electron cloud center of an atom with respect to its nucleus. This

in-ducesadipolemoment. Electronicpolarizationisfoundinboth materialsthatpossess

moleculeswithlargedipolemoments(polarmaterials)andthosethatdonot(nonpolar

materials).

Atomic polarization/ Ionic polarization/ Molecular polarization An applied

elec-tric eld may displace the atoms in the molecules, changing the distance between

the atoms, and thus changing the dipole moment. Atomic polarizationonly occurs in

terialpossessespermanentdipolemomentsthatarerandomlyoriented. Whenaeldis

applied,these dipoles align themselves with the eld. Since orientational polarization

is reliant upon the existence of permanent dipole moments, it only is found in polar

materials.

Interfacial polarization The impurities and defects in crystal can impede the ow of

charge created by an appliedeld. The resultingcharge accumulation can result in a

dipolemoment. This typeof polarizationis found only incrystals.

Themultiplenamesforeachtypeofpolarizationcanbeconfusing,especiallywhencomparing

the research of dierent contributors. We attempt to refer to each mechanism by the rst

name given above. We point out that in addition to this terminology both atomic and

electronic polarizationare sometimes referred to asdistortional polarization[2].

In agiven material,polarizationcan be the result of one ormore of thesefour mechanisms.

We are primarily interested in materials that contain a high water-content, such as living

tissue or mud, so here we focus on polar liquids. The polarization in this class of liquids

tends to depend mostly on the orientation of permanent electric dipoles in the molecules

(orientational polarization) and the distortion of the molecules by an applied electric eld

(electronic and atomicpolarization)[16]. Withthis inmind, we focus onthese polarization

mechanismsin the remainder of our discussions.

Inthe presenceof mostappliedelectricelds,the polarizationofahigh water-content liquid

is both distortional and orientational. At high (optical) frequencies however, the electric

eld oscillates so rapidly that it does not hold any orientation long enough for the dipoles

to align with it. Thus, the orientational polarization is virtually insignicant [26]. This

implies that atsuÆciently high frequencies, the only contributionto the dielectric constant

oroptical index of refraction is fromelectricaldistortion[16].

Since thepolarizationof apolarliquidhas multiplemechanisms,we expect thata complete

model must incorporate them all. Orientational polarizationis suggestive of a mechanism

with an exponential decay factor, such as the one in the model proposed by Debye [2].

However, a system rarely conforms exactly to the modeldescribed by the Debye dispersion

equations due to the fact that the polarizational decay may not be represented accurately

by a mechanismwith one relaxation time[17]. On the otherhand, distortional polarization

causes charges to behave somewhat like linear harmonic oscillators; thusit is reasonable to

modelthemassuch(theLorentzmodelisanexample). Neitherofthesetypesofmodelsalone

willbe suÆcient to completely describe the polarization of a polarliquid. Nonetheless as a

rststep andtoillustrateourideas, webaseourmodelonthe Debyemodelfororientational

The Debye model [4] can be represented by the rst order ordinarydierentialequation

_

P +P =

0 (

s

1

)E; (5)

orby

P(t;z)= Z

t

0

g(t s;z)E(s;z) ds

with kernel

g(t)=exp

t

0 (

s

1 )

:

In these equations,

s

is the static relative permitivitty and

1

is the value of permittivity

for an extremely high ( innite) frequency eld. In this model, the value of the relative

permittivity

r

of (3) inthe dielectric is given by

1

; that is,

r

=1in[0;z

1

] and

r =

1 in

(z

1

;1]:The variable isthe relaxationtime of the dielectric.

In[2],Andersondescribesapotentialdoublewellformulationforanatomicmodelthatleads

to the Debye polarization model. In this model, the dielectric is made up of independent

noninteracting particles; each particle has two equilibriumpositions separated by a barrier

of high potential. One considers a charged particle with two equilibrium positions A and

B located a distance d from each other. Between them is a potential barrier W such that

W k

B

T where k

B

is the Boltzmann constant and T is the temperature. (See Figure2.)

If there is no electromagnetic eld present, one assumes that the particle oscillates about

either equilibrium, and on occassion, obtains enough energy to cross the potential barrier

and jumpintotheother well. Over time,for constant temperature, the particleisnear Aas

oftenas near B and the probability of nding the particlenear a given wellis 1

## W

## B

## A

## d

## V

## eEd

## A

## W

## B

## d

## E

## V

Figure 2: Potentialdouble wellmodelwith and withoutan appliedeld

When an electric eld E is applied in the direction from A to B, the potentials at each

equilibriumare no longerequal, for instance V

A >V

B , and

V

A V

B

=edE;

where e is the charge of the particle. (See Figure 2.) A result from Boltzmann statistics

implies that the probability that a particle has potential V is proportional to exp ( V

k

B T

),

so that now it is more likely to nd the particle near equilibrium B. As before, a particle

can jump from one equilibrium to the other if it acquires enough energy. For a potential

barrier W, the probability that a particle can cross this barrier in the direction from B to

A is proportional to exp ( W

k

B T

), with proportionality constant w

0

2

, the assumed frequency

of oscillationdue to thermal agitation of the particle about the equilibrium. Likewise, the

probability that the particle can cross the barrier in the direction from A to B is given by

w

0

2 exp (

W edE

k

B T

). UsingtheseprobabilitiesandthefactthatthetotalnumberN =N

A +N

B

of particles is constant, one can derive [2] (see also page 387 of [15]) a linear rst order

dierentialequation todescribe the dierenceN

B

(t) N

A

(t) innumberof particlesinwells

B and A at any time t

d

dt (N

B

(t) N

A (t))=

w

0

exp

W

k

B T

(N

B

(t) N

A (t))+

ed

2k

B T

NE

:

The polarization P(t) due to the applied electromagnetic eld is proportional to N

B (t)

N

A

(t): Byrelating with w0

exp

W

k

B T

and

s

1 with

w0

exp

W

k

B T

ed

2k

B T

N,one thus

pressure-dependent. One approach to understanding this pressure-dependence is to extend

the aboveargumentsand consider the polarizationfromanon-equilibriumthermodynamics

perspective. A discussion of this nature isgiven in [20].

Wehowever take a dierent approach toincorporatingpressure-dependence into the Debye

model. We present the model here and provide motivation in Section 2.3. We begin by

assumingthatthe material-dependentparametersinthe dierentialequation(5)dependon

pressure, i.e.,

(p) _

P +P =

0 (

s

(p)

1

(p))E =

0

((p) (p))E:

Wesuppose asa rstapproximationthat each ofthe pressure-dependent parameterscan be

represented as amean value plus aperturbationthat is proportionaltothe pressure

(p) =

0

+~=

0 +

p

(p) =

0

+~ =

0 +

p

(p) =

0 +

~

=

0 +

p:

Then the equation

(p) _

P +P =

0

((p) (p))E

can be written

(

0 +

p)

_

P +P =

0 (

0

0 +(

)p)E: (6)

Werecallthatthepolarizationtermin(4)involvessecond-ordertimederivatives. Toexpress

(6)in compatible form,we take the time derivativeof both sides to obtain

P =

(1+

_ p)

(

0 +

p)

_

P +

0 (

0

0 +(

)p)

(

0 +

p)

_

E+

0 (

)p_

(

0 +

p)

E (7)

with

_

P =

1

(

0 +

p)

P +

0 (

0

0 +(

)p)

(

0 +

p)

From here,we can use (8) in(7) andthen replaceP in (4)with the expression given by (7).

Additionally,weshouldnotethattherelation

r =

1 in(z

1

;1]becomes

r =

1

(p)=

0 +

p

in (z

1 ;1]:

2.3 Motivation for pressure-dependence of polarization

Thepolarizationdescribed byboththeoriginalandpressure-dependentDebyemodelsisdue

tothealignmentofpermanentdipolemomentswiththeappliedeld. Thistendency toalign

is inhibited by the presence of centrifugalor gyroscopic forces. These forces are caused by

molecularrotationsand collisions[26]. Thepressure inapolarizablemediuminuences and

isinuencedbytheshortrangeparticleinteractioninthemedium,includingthesemolecular

rotationsand collisions. This interaction between particlesmayserveto inhibitorfacilitate

the alignment of dipolemoments with the applied eld,resulting inthe modication of the

orientationalpolarization[14]. Figure3 depicts this schematically.

## +

## -

### Orientational

### polarization

### Molecular

### collisions & rotations

### perturbations

### Pressure

### Applied E field

### Direction of

### applied E field

Figure 3: Pressure-dependence of orientationalpolarization

Thisinteractionmakessense intuitively;however, wewanttobetterunderstandthe

interac-tion mechanism. Specically, we want to consider each polarizationparameter individually

s

the presence ofaconstantappliedelectriceld. In 1850and 1879respectively, Clausiusand

Mossitti independently discovered that for any given material,the quantity

s

0 1

=

s

0 +2

is proportional to the material density (page 155, [18]; page 140, [25]). Pressure variations

in the material cause changes in its density. These changes are reected in the static

per-mittivityof the materialdue tothe law ofClausius and Mossitti. So the staticpermittivity

can be expected to depend onpressure.

The pressure-dependence of

s

does not necessarily suggest the pressure-dependence of

1 ;

the permittivity of a material under a very high frequency electric eld. However in the

interest of generality, we allow for the possibility that

1

is pressure-dependent.

Pressure-independent behavior of

1

is just aspecial case of our model(see Section2.1) with

=0

so that

1

(p)=

0 :

Lastly we examine the feasibility of the pressure-dependence of the relaxation parameter

. To do so we consider a dipolar liquid which consists of freely moving molecules. If

an individual dipole changes its orientation, the nearby dipoles shift to compensate and

produce a new equilibrium position. Their collective motion can be viewed as a viscous

frictional damping force that acts on the original dipole. When an electric eld is applied,

itsforcecausesthe dipoletoalignitselfwith theeld. Therateofalignmentdependsonthe

amount of friction. However since the dipole is subject to the eects of Brownian motion,

this rate alsodepends onthermal uctuations. Taking this into account Debye derived the

followingexpression for the relaxation(page 73,[2])

=

2k

B T

where is the frictional constant. Dipoles arranged in smaller groups are less apt to resist

reorientation [13]. This leads to diminished frictional eects. Variations in pressure likely

alter the cohesion of dipole groupings and thus aect the friction. A specic example of

this relationship isgiven in(page 63,[21]) for hard sphereuids. In this case, the frictional

viscosity constant isgiven by

= k

B T

mD

where m isthe particlemass and D isthe self-diusioncoeÆcient. The self-diusion

D= 1

2 R

k

B T

m

2

p

k

B T

1

1

where R is the hard sphere diameter, p is the pressure, and is the liquid density. Clearly

in this examplethe relaxationparameter ispressure-dependent.

3 Theoretical results

In order to consider (4) from a theoretical perspective, it is convenient to write it in

vari-ational form. We formulate the Gelfand triple V ,! H ,! V

, where H = L 2

(0;1) and

V = H 1

R

(0;1) f 2 H 1

(0;1) : (1) = 0g. We let < ; > denote the usual L 2

inner

product. Thenwemaywrite (4)asavariationalformofMaxwell'sequationinsecondorder

form

<a

E(t);>

V

;V

+<b _

E(t);>+<e

P(t);>

+c _

E(t;0)(0)+

1

(E(t);)=< F(t); >

V

;V

; t2[0;T];

(9)

for all 2V. The sesquilinearform

1

is dened by

1

(; )=c 2

< 0

; 0

>;

where c 2

= 1

0

0

> 0 is constant and the parameter functions a;b; and e are determined

by the geometry, conductivity, and instantaneous polarization of the dielectric. Since the

absorbing boundarycondition _

E cE 0

=0 atz =0isanaturalcondition,we incorporateit

into the variational formulation of the equation. The superconductive boundary condition

atz =1 however is essential and isimposed in the denition of V.

MotivatedbythepolarizationmodeldescribedinSection2,wemayrewrite(9)asthegeneral

variationalform

<a

E(t);>

V

;V

+<b _

E(t);>+<hE(t);>

+< R

t

0

G(t;s;)E(s;)ds;>

+c _

E(t;0)(0)+

1

(E(t);)=<F(t);>

V

;V

; 2V; t 2[0;T];

E(0;z)=E

0 (z)

_

E(0;z)=E

1 (z);

For the pressure-dependent Debye polarizationmodelgiven inSection 2,wehave

a(t;z) = 1+(

1 1)I

(z

1 ;1)

=1+(

0 +

p(t;z) 1)I

(z

1 ;1)

b(t;z) = 0 + 1 0 0( 0 0 +( )p(t;z) ) ( 0 + p(t;z)) I (z1;1)

h(t;z) = 1 0 0( )p(t;z)_

( 0 + p(t;z)) ( 1+ _ p(t;z)) 0( 0 0 +( )p(t;z) ) ( 0+p(t;z)) 2 I (z1;1)

G(t;s;z) = 1

0

(1+p(t;z)) _ 0(

0 0 +( )p(s;z)) ( 0 + p(t;z)) 2 ( 0 + p(s;z)) exp R t s d 0+p(;z) I (z 1 ;1) c 2 = 1 0 0

F(t;z) = 1 0 _ J s (t) 1

(; ) = c 2 < 0 ; 0 >: (11) (HereI

istheindicatororcharacteristicfunctionforaset .) Wenotethatthesesquilinear

form

1

: V V ! R is V-continuous and V-elliptic,so that there exist positive constants

c

1 ;c

2

such that the following inequalitieshold.

1

(; )=c 2

< 0

; 0

>c 2 j 0 j H j 0 j H c 1 jj V j j V 1

(;) =c 2 < 0 ; 0 >=c 2 j 0 j 2 H c 2 jj 2 V :

In [9], the well-posedness of (10) is considered for general coeÆcient, kernel, and forcing

functions underthe following assumptions

A1)The coeÆcient aalong with itsderivativesa_ and a are inL 1

(0;T;L 1

[0;1]); and forall

z 2[0;1],a(z)a

0

; forsome 1a

0 >0.

A2) The coeÆcient b and its time derivative _

b are in L 1

(0;T;L 1

[0;1]) and b(t;z) 0 for

all(t;z)2[0;T][0;1]:

A3) The coeÆcient h isin L 1

(0;T;L 1

[0;1]):

A4) The kernel function G is inL 1

([0;T][0;T];L 1

[0;1]):

A5) The sequilinearform

1

is given by

1

(; )=c 2

< 0

; 0

A6) The forcing functionF is inH (0;T;V ):

The result is the following theorem. (See [9] fora detailedproof.)

Theorem 1: Under assumptions A1)-A6), the system (10) possesses aunique solution and

(E; _

E)depends continuously on initialdata (E

0 ;E

1

) and forcing function F from

(E

0 ;E

1

;F)2V HH 1

(0;T;V

) to (E; _

E)2L 2

(0;T;V)L 2

(0;T;H).

Wemay verify that assumptions A1)-A6)hold for (11) { see [9].

3.1 Estimation of parameters in the general variational

form

The previous well-posedness result provides a framework in which to formulate parameter

estimation problems. Asdescribed inSection1, the generalMaxwellsystem treatedby this

result arises from a class electromagnetic interrogation problems. The crux of these

prob-lems isestimation ofcertainparameter values, namelydielectric constantsand conductivity

coeÆcients, forthe materialunder interrogation. The estimationproblemtypicallyinvolves

ndingthe parameter values thatprovidethe best tbetween the modeland data collected

from the actual system, using, for example, a least squares criterion. These parameter

estimates may then beused to characterize the material.

In practice, the experimental data is compared with nite dimensional numerical

approxi-mations to the model. In this section, we examine the relationship between the parameter

estimation problems for the originalsystem (10) and for acorrespondingnite dimensional

system. We suppose that the coeÆcients and sesquilinear form in both (10) and its nite

dimensionalapproximationdepend onaparameterq ina set Q. If the exactsolutionto the

originalsystem (10)were accessible, we would consider the problemof minimizingthe least

squares cost functional

J(q;w)= N

t

X

i=1

jOE(t

i

;q) w

i j

2

(12)

over q 2 Q where w = fw

i g

Nt

i=1

is a set of observations taken at times t

i

; Q is a set of

admissible parameters, and O is an observation operator. The form of O depends on the

particular application and set of observations. For example, if w

i

is a measurement of the

electric eld taken at a spatial point z at time t

i

, then the operator O entails evaluations

of the function E(t

i

;;q)at apointin space. Sincewecannot obtain aclosed formsolution

to (10), we use the solution E N

(t;q) to a nite dimensional approximating system. The

solutionE N

(t;q) liesin V N

<a(q)

E (t);>

V

;V

+<b(q) _

E (t);>+<h(q)E (t); >

+< R

t

0

G(t;s;;q)E N

(s;) ds;>

+c _

E N

(t;0)(0)+

1 (q)(E

N

(t);)=< F(t); >

V

;V

E N

(0;z)=P N

E

0 (z)

_

E N

(0;z)=P N

E

1 (z)

(13)

for all 2 V N

: In particular, we dene the piecewise linear basis elements f N

j g

N 1

j=0 with

nodalvalues N

j

(k=N)=Æ

kj

; k =0;1;:::;N; and let V N

= span f N

0 ;

N

1

;:::; N

N 1

g V:

Then we dene P N

to be the quasi-L 2

(0;1) projection (see [23], [4], [12]) of V

onto V N

dened by

<P N

v

;v N

>

N =<v

;v N

>

V

;V

for v

2V

and for allv N

2V N

where

<w N

;v N

>

N

Z

1

0 I

N

(w N

v N

)(z) dz

and I N

is the nodalvalue linear interpolationoperatorfor V N

. It is shown in [23] that the

operator P N

iswell-dened and satises

jP N

j

H

K

0

1 jj

H

for 2H

jP N

j

V

K

0

2 jj

V

for 2V:

(14)

As expected, the corresponding cost functional forthe nite dimensionalsystem is

J N

(q;w)= Nt

X

i=1

OE N

(t

i

;q) w

i

2

: (15)

Againthe form of the operator O is chosen to correspond to the typeof data collected.

In [9], we established the well-posedness of (10) with solutions E in L 2

(0;T;V) and _

E 2

L 2

(0;T;H); where V = H 1

R

(0;1) and H = L 2

(0;1); and we also veried that a unique

solutionto (13) exists. Theseresults hold provided that, foreach q2Q; Assumptions

A1)-A6) are satised and the sesquilinear form

1

is V-continuous and V-elliptic. Moreover in

[24], we show that the solution E of (10) has the enhanced regularity E 2 H 3

(0;T;V

H (0;T;H)\ H (0;T;V) under consistency conditions for the initial conditions and the

followingassumptions:

A7) The secondtime derivative,

b ; of b is inL 1

(0;T;L 1

[0;1]):

A8) The rst and second time derivatives, _

h and

h; of h are in L 1

(0;T;L 1

[0;1]):

A9) The rst and second derivatives with respect to the rst temporal variable, d

dt

G and

d 2

dt 2

G; of the kernel function G are in L 1

((0;T)(0;T);L 1

[0;1]):

A10) The forcing function F is in H 2

(0;T;V

) and is of the form F(t;z) = ~g(t)Æ(z) with

~

g(t)2H 2

(0;T) and g(0)~ = _

~

g(0) =0. (This assumption replaces A6).)

(Verication of these assumptions A7)-A9) for the pressure-dependent Debye polarization

modelare alsogiven in[24].)

Wenowmakethe followingassumptionsabout theset ofadmissibleparametersQ,the state

space V N

, and the projection operatorP N

.

B1) The nite dimensional set Q lies in a metric space ~

Q with a metric ~

d and is compact

with respect to this metric.

B2) The nite dimensional subspaces V N

are subsets of V.

B3) Foreach 2V, j P N

j

V

!0as N !1.

B4) Foreach 2H, j P N

j

H

!0as N !1.

Vericationsof B3)and B4)for ourparticular P N

are given in[23]. Wenowmakeafurther

assumption on the sesquilinear form

1

. We assume that

1 =

1

(q) is dened on Q and

satises

H1)

j

1 (q

1

)(; )

1 (q

2

)(; )j ~

d(q

1 ;q

2 )jj

V j j

V

for q

1 ;q

2

2Q where depends onlyon Q.

Fortheelectromagneticsysteminconsiderationinthispaper,AssumptionH1)isunnecessary

since

1

is independent of q. However for the purpose of establishing a moregeneral result,

we donot assumehere that our sesquilinear formis parameter independent.

Furthermore we makethe following assumption about our coeÆcients.

A11)The coeÆcientsdepend continuously on q so that as ~

d(q;q N

i) ja(q) a(q )j

L 1

!0

ii) jb(q) b(q N

)j

L 1 !0

iii) jh(q) h(q N

)j

L 1

!0

iv) jG(q) G(q N

)j

L 1

!0:

The above continuity along with the compactnessof Q implies that the images a(Q);b(Q);

h(Q); andG(Q) are compact. ThuseachcoeÆcientcan bebounded independentlyof q. We

assume throughoutthat all bounds onour coeÆcientsdo not depend onq.

By solving the parameter estimationproblems related to(13), (15) we obtain asequence of

estimates fq N

g. We wish to demonstrate that under certain conditions this sequence (or a

subsequence) converges to the estimate corresponding to the problem related to (10), (12).

In order to do this, we state the following claim, which can be found (along with a proof)

as Theorem 5.1in [11].

Theorem 2: To obtain convergence of at least a subsequence of fq N

g to a solution qof

minimizing(12)subjectto(10),itsuÆces,underassumptionB1),toarguethatforarbitrary

sequences fq N

g in Qwith q N

!q inQ, we have

OE N

(t;q N

)!OE(t;q):

In [11], the operator O is general enough to include functions that map functions f such

thatf :T !V tothe spaceofobservations W;whereT isanappropriatelychosen(see [11]

and [8]) subset of [0;T] that contains the times of observation and V is a space containing

E(t;): In the numerical examples presented in this paper, the observations correspond to

the values of the electric eld at the point z = 0 at various times, i.e., fE(t

i

;0)g; thus the

operatorO involvespointwise evaluationofE atmanypointsintime and onespecic point

in space.

We suppose that V N

and P N

satisfy B2)-B4), the sesquilinear form

1

satises H1), the

coeÆcients satisfy assumptions A1)-A11) and we letq N

2 Q be arbitary such that q N

! q

in Q. Our primary goalis toshow that asN !1

E N

(t;0;q N

)!E(t;0;q) (16)

for each t 2 [0;T]: However, here we verify a more general result. We show that for each

E (t;q ) ! E(t;q) in the V norm

_

E N

(t;q N

) ! _

E(t;q) in the H norm

(17)

as N !1;where E N

; _

E N

are the solutionsto (13) and E; _

E are the solutionsto (10). We

note that we may evaluate these functions pointwise in t due to the enhanced regularity of

solutions. Moveover, using the equivalence of norms, we see that (17) implies (16) and we

have the result weneed for our computations.

Wepoint out that fora sequence q N

=q forall N;the desired result impliesconvergence of

thenitedimensionalapproximationtothetruesolution. Thisisimportantwhenconsidering

numericalapproximationsto the solution.

Wehaveestablished previouslythat thesolutionof (10)satisesE(t)2V and _

E(t)2H for

eacht. Since

jE N

(t;q N

) E(t;q)j

V jE

N

(t;q N

) P N

E(t;q)j

V +jP

N

E(t;q) E(t;q)j

V

and B3) guarantees jP N

E(t;q) E(t;q)j

V

!0 asN !1, weneed only showthat

jE N

(t;q N

) P N

E(t;q)j

V

!0 asN !1

for eacht 2[0;T]. In the same way, itsuÆces toshow that

j _

E N

(t;q N

) P N

_

E(t;q)j

H

!0 as N !1

for eacht 2[0;T] toobtain the second result.

Welet E N

=E N

(t;q N

);E =E(t;q); and N

E N

(t;q N

) P N

E(t;q):

Subtracting (10) from(13), we havefor 2V N

<a(q N

)

E N

a(q)

E;>+<b(q N

) _

E N

b(q) _

E;>

+<h(q N

)E N

h(q)E; >

+< R

t

0 (G(q

N

)E N

G(q)E) ds; >+c( _

E N

(t;0) _

E(t;0))(0)

+

1 (q

N

)(E N

;)

1

Weadd and subtract P E and its derivatives and rearrange terms toobtain <a(q N )( E N P N

E);>+

1 (q N )(E N P N E;) +c( _ E N

(t;0) P N

_

E(t;0))(0)

=<a(q)

E; > <a(q N

)P N

E; >+

1 (q)(E;) 1 (q N )(P N E;) +c( _

E(t;0) P N

_

E(t;0))(0)+<b(q) _ E b(q N ) _ E N

; >

+<h(q)E h(q N

)E N

;>+< R t 0 G(q)E G(q N )E N

ds; >:

Wechoose the test function = _ N 2V N so that <a(q N )( E N P N E); _ N >+ 1 (q N )(E N P N E; _ N ) +c( _ E N

(t;0) P N _ E(t;0)) _ N

(t;0)=

<a(q) E; _ N

> <a(q N )P N E; _ N >+ 1 (q)(E; _ N ) 1 (q N )(P N E; _ N ) +c( _

E(t;0) P N _ E(t;0)) _ N

(t;0)+<b(q) _ E b(q N ) _ E N ; _ N >

+<h(q)E h(q N )E N ; _ N

>+< R t 0 (G(q)E G(q N )E N ) ds; _ N >:

Wenote that

2<a(q N )( E N P N E); _ N >+2 1 (q N )(E N P N E; _ N ) = d dt <a(q N ) _ N ; _ N >+ 1 (q N )( N ; N )

<a(q_ N ) _ N ; _ N >:

1 2 d dt <a(q N ) _ N ; _ N >+ 1 (q N )( N ; N

) +c( _ N (t;0)) 2 = <a(q) E; _ N

> <a(q N )P N E; _ N

>+<a (q_ N ) _ N ; _ N > + 1 (q)(E; _ N ) 1 (q N )(P N E; _ N ) +c( _

E(t;0) P N _ E(t;0)) _ N

(t;0)+ <b(q) _ E b(q N ) _ E N ; _ N >

+<h(q)E h(q N )E N ; _ N

>+< R t 0 (G(q)E G(q N )E N ) ds; _ N >:

Integration with respect tot yields

<a(q N ) _ N (t); _ N

(t)>+

1 (q N )( N (t); N (t)) +2 R t 0 c( _ N (;0)) 2 d= 2 R t 0 n <a(q) E; _ N

> <a(q N )P N E; _ N

>+<a (q_ N ) _ N ; _ N > + 1 (q)(E; _ N ) 1 (q N )(P N E; _ N ) +c( _

E(;0) P N _ E(;0)) _ N

(;0)+<b(q) _ E b(q N ) _ E N ; _ N >

+<h(q)E h(q N )E N ; _ N

>+< R 0 G(q)E G(q N )E N ds; _ N > o d

+<a(q N ) _ N (0); _ N

(0)>+

1 (q N )( N (0); N (0)):

Wenow use the denition of N

to obtain

N

(0)=E N

(0) P N

E(0)=E N (0) P N E 0 =0 _ N (0)= _ E N (0) P N _ E(0)= _ E N (0) P N E 1 =0:

<a(q ) _

(t); _

(t)>+

1

(q )( (t); (t))

+2 R t 0 c( _ N (;0)) 2 d= 2 R t 0 n <a(q) E; _ N

> <a(q N )P N E; _ N

>+<a (q_ N ) _ N ; _ N > + 1 (q)(E; _ N ) 1 (q N )(P N E; _ N ) +c( _

E(;0) P N _ E(;0)) _ N

(;0)+<b(q) _ E b(q N ) _ E N ; _ N >

+<h(q)E h(q N )E N ; _ N

>+< R 0 G(q)E G(q N )E N ds; _ N > o d: (18)

In order toboundthe rightside of (18), we derive the followingestimates:

Estimate 1:

R

t

0

2<a(q)

E; _

N

> 2<a(q N )P N E; _ N

>+<a(q_ N ) _ N ; _ N > d = R t 0

2<(a(q) a(q N )) E; _ N

>+2<a(q N )( E P N E); _ N >

+<a(q_ N ) _ N ; _ N > d R t 0 j(a(q) a(q N )) Ej 2 H +ja(q N )( E P N E)j 2 H + 1 2

(5+ja(q_ N )j 2 H )j _ N j 2 H d ja(q) a(q N )j 2 L 1 R t 0 j Ej 2 H

d+ja(q N )j 2 L 1 R t 0 j E P N Ej 2 H d + 1 2

(5+ja(q_ N )j 2 L 1 ) R t 0 j _ N j 2 H d: Estimate 2: 2 R t 0 1 (q)(E; _ N ) 1 (q N )(P N E; _ N ) d = 2 R t 0 1 (q N )(P N _ E; N ) 1 (q)( _ E; N

) d+2(

= 2 t 0 1 (q N )(P N _ E _ E; N )+ 1 (q N )( _ E; N ) 1 (q)( _ E; N )d +2 1 (q)(E(t); N (t)) 1 (q N )(E(t); N (t)) + 1 (q N )(E(t) P N E(t); N (t)) R t 0 c 2 1 jP N _ E _ Ej 2 V + 2 ( ~ d(q;q N )) 2 j _ Ej 2 V +2j N j 2 V d + c 2 1 jP N E(t) E(t)j 2 V + 2 ( ~ d(q;q N )) 2 jE(t)j 2 V +2j N (t)j 2 V

where >0 is arbitrary.

Estimate 3:

2c( _

E(;0) P N _ E(;0)) _ N (;0) c 2 j _

E(;0) P N

_

E(;0)j 2 +j _ N (;0)j 2 c 2 K 1 j _ E P N _ Ej 2 V +j _ N (;0)j 2 :

(Here we use the fact that jj 2

V

is equivalent to j 0 j 2 H +j(0)j 2

so that jj 2 V ~ K(j 0 j 2 H + j(0)j 2 ) ~ Kj(0)j 2 :) Estimate 4: 2 R t 0 <b(q) _ E b(q N ) _ E N ; _ N > d = 2 R t 0 f<b(q)( _ E P N _ E); _ N

>+<(b(q) b(q N ))P N _ E; _ N >

+<b(q N ) _ N ; _ N >gd R t 0 jb(q)( _ E P N _ E)j 2 H +jb(q N ) _ N j 2 H

+j(b(q) b(q N ))P N _ Ej 2 H +3j _ N j 2 H d jb(q)j 2 L 1 R t 0 j _ E P N _ Ej 2 H

2 R

t

0

<h(q)E h(q N )E N ; _ N > d = 2 R t 0

f<h(q)(E P N

E); _

N

>+<(h(q) h(q N ))P N E; _ N >

+<h(q N ) N ; _ N >g d R t 0 jh(q)(E P N E)j 2 H +jh(q N ) N j 2 H +3j _ N j 2 H +j(h(q) h(q N ))P N Ej 2 H d jh(q)j 2 L 1 R t 0 jE P N Ej 2 H

d+jh(q N )j 2 L 1 R t 0 j N j 2 H

d+3 R t 0 j _ N j 2 H d +jh(q) h(q N )j 2 L 1 R t 0 jP N Ej 2 H d: Estimate 6: 2 R t 0 < R 0 (G(q)E G(q N )E N )ds; _ N > d = 2 R t 0 < R 0 G(q)(E P N E)ds; _ N

>+< R 0 (G(q) G(q N ))P N E ds; _ N > +< R 0 G(q N ) N ds; _ N > d R t 0 fj R 0 G(q)(E P N E) dsj 2 H +j R 0 (G(q) G(q N ))P N E dsj 2 H +j R 0 G(q N ) N dsj 2 H +3j _ N j 2 H g d = R t 0 R 1 0 j R 0 G(q)(E P N E) dsj 2 dz d + R t 0 R 1 0 j R 0 (G(q) G(q N ))P N E dsj 2 dz d + R t 0 R 1 0 j R 0 G(q N ) N dsj 2

dz d+3 R t 0 j _ N j 2 H d jG(q)j 2 L 1 R t 0 R 1 0 j R 0 (E P N E) dsj 2

dz d+jG(q) G(q N )j 2 L 1 R t 0 R 1 0 j R 0 P N E dsj 2 dz d +jG(q N )j 2 L 1 R t 0 R 1 0 j R 0 N dsj 2

jG(q)j 2 L 1T t 0 1 0 jE P N Ej 2 L 2 (0;)

dz d+jG(q) G(q N )j 2 L 1 T t 0 1 0 jP N Ej 2 L 2 (0;) dz d +jG(q N )j 2 L 1T R t 0 R 1 0 j N j 2 L 2 (0;)

dz d+3 R t 0 j _ N j 2 H d jG(q)j 2 L 1T 2 R t 0 jE P N Ej 2 H

d+jG(q) G(q N )j 2 L 1 T 2 R t 0 jP N Ej 2 H d +jG(q N )j 2 L 1T 2 R t 0 j N j 2 H

d+3 R t 0 j _ N j 2 H d:

Using these estimates, Assumption H1), the V-continuity and V-ellipticity of

1

; and the

fact that jj 2

H jj

2

V

, we may rewrite(18) as

j p a(q N ) _ N (t)j 2 H +c 2 j N (t)j 2 V +2 R t 0 cj _ N (;0)j 2 d Æ N 1

(t)+Æ N

2

(t)+2j N (t)j 2 V + R t 0 n j _ N (;0)j 2 +( 23 2 +jb(q N )j 2 L 1 + 1 2 ja(q_

N )j 2 L 1)j _ N j 2 H

Æ N

2

(t) = ja(q) a(q N )j 2 L 1 t 0 j Ej 2 H d + 2 ~ d(q;q N ) Z t 0 j _ Ej 2 V

d+jb(q) b(q N )j 2 L 1 Z t 0 jP N _ Ej 2 H d

+ jh(q) h(q N )j 2 L 1 +T 2

jG(q) G(q N )j 2 L 1 Z t 0 jP N Ej 2 H d+ 2 ~ d(q;q N )jE(t)j 2 V :

Since >0 isarbitrary, wemay choose ittobesuchthat 1>c

2

2>0. Furthermore, the

wave speed csatises2c >>1. Wethen use Assumptions A1)-A4)toclaim that thereexist

constants

1 ;

2

>1and 1a

0

>0, independent of q,such that

a 0 j _ N (t)j 2 H +(c 2 2)j N (t)j 2 V + R t 0 (2c 1)j _ N (;0)j 2 d Æ N 1

(t)+Æ N 2 (t)+ R t 0 1 j _ N j 2 H + 2 j N j 2 V d:

Finally recallingthe bounds on

1 ;

2 ;a

0

; and c

2

2,we may rewrite the inequalityas

a 0 j _ N (t)j 2 H +(c 2 2)j N (t)j 2 V Æ N 1

(t)+Æ N 2 (t)+ Z t 0 2 c 2 2 1 j _ N j 2 H + 1 2 a 0 j N j 2 V d Æ N 1

(t)+Æ N 2 (t)+ 1 2 a 0 (c 2 2) Z t 0 a 0 j _ N j 2 H +(c 2 2)j N j 2 V d:

Inordertoapply Gronwall'sinequalitytoobtainuniformconvergence of N

and _

N

intas

N !1; we must establish the uniform convergence of Æ N

1

and Æ N

2

: We have from B3) and

B4) that j E(t) P N E(t)j H ; j _ E(t) P N _ E(t)j V

; and jE(t) P N

E(t)j

V

converge to zero as

N ! 1 for each t: Since this convergence is dominated and fE(t)g

t2[0;T]

is compact in V,

we havethat

Æ N

1

!0 uniformlyin t asN !1:

Moreover, theboundednessof E; _

E;and

E given bythe enhanced regularity resultsand the

Æ

2

!0uniformly int as N !1and q !q in ~

Q:

Then we may apply Gronwall'sinequality toconclude that

sup

t2[0;T] j

N

(t)j 2

V

! 0 asN !1

sup

t2[0;T] j

_

N

(t)j 2

H

! 0 asN !1

whichis suÆcient toprovethe desiredresult.

3.2 Estimation of parametersin thesystem withpressure-dependent

Debye polarization

The general system (10) is formulated to accomodate systems arising from a variety of

electromagnetic interrogation problems. We are concerned here with a particular system

that incorporates apressure-dependent modelforDebye polarization. We demonstratethat

this system satises Assumptions A11), B1)-B3) and H1) and thus that the results of the

previoussectionapply. (Wenotethat verications ofAssumptionsA1)-A10) aregiven in[9]

and [24].)

The system we wish to consider is given by (10) with the parameter-dependent coeÆcients,

kernel and forcing functions,and sesquilinearform(11).

For this system, the set of admissible parameters Q is a subset of R 7

, where seven is the

number of parameters tobe estimated (inaddition tothe six polarizationparameters from

Section 2, one is often interested in estimating the conductivity coeÆcient ). Here we

consider q 2 Q R 7

where q =(;

0 ;

0 ;

0 ;

;

;

): We choose the admissible set Q to

insurethat ourDebyecoeÆcients arewell-dened. First,because ofthe physicalmeaningof

these parameters, the values of ;

0 ;

0 and

0

must be positive. Then for a given pressure

wave pwith p;p_ 2L 1

(0;T;L 1

(0;1))and axed Æ>0;we admit onlyvalues of

;

; and

suchthat

0 +

p(t;z);

0 +

p(t;z);and

0 +

p(t;z)aregreaterthanÆ forallz 2[0;1]

and t 2[0;T]. In additionto these requirements,we assume that the admissibleparameter

set is closed and bounded inR 7

:

We recall that q N

! q in the standard Euclidean metric is equivalent to the convergence

of each component of q N

. Moreover, any closed and bounded sets Q in R 7

are compact

and satisfy the conditions of B1). The conditions B2)-B4) are satised by V N

; which is

in this case the set of nite dimensionallinear piecewise basis elements,and the projection

operator P N

. To verify H1), we note that

1

(q)(; ) = c 2

< 0

; 0

> is independent of q

and j

1 (q

1

)(; )

1 (q

2

)(; )j=0 forany q

1 ;q

WenextverifyA11)i)-iv)forthe coeÆcientsinourmodel. Wenotethat asq !qwe have,

for agiven p;p_ 2L 1

;

j N

j!0 (19)

N 0 + N p 0 + p L 1

!0 (20)

N 0 N 0 +( N N )p 0 0 +( )p L 1

!0 (21)

N 0 + N p 0 p L 1

!0 (22)

( N N )p_

(

)p_

L 1

!0 (23)

and N _ p _ p L 1

!0: (24)

We use (20) directly to claim that ja(q N

) a(q)j

L

1 ! 0 whenever q N

! q; thus A11)i)

holds.

In demonstratingthat A11)ii)holds, we observe that

jb(q N ) b(q)j L 1 1 0 N + sup t2[0;T] sup z2[z 1 ;1] N 0 N 0 +( N N )p(t;z) N 0 + N p(t;z) ! 0 0 +( )p(t;z) 0 + p(t;z) :

Thenwemayapplyequations(19),(21),and(22)and thequotientruleoflimitstoconclude

that jb(q N

) b(q)j

L 1

!0 asq N

!q and A11)ii)is satised.

jh(q N

) h(q)j

L

1 sup

t2[0;T] sup z2[z1;1] ( N N

)p(t;_ z)

N 0 + N p(t;z) (

)p(t;_ z)

0 + p(t;z) + (1+ _ p(t;z))( 0 0 +( )p(t;z)) ( 0 + p(t;z)) 2 (1+ N _ p(t;z))( N 0 N 0 +( N N )p(t;z)) ( N 0 + N p(t;z)) 2 :

From equations (23), (22), (24), and (21) and the product and quotient rules of limits, we

may conclude that A11)iii)holds.

To showthat A11)iv) holds, weargue that

jG(q N ) G(q)j L 1 sup (t;s)2[0;T][0;T] sup z2[z1;1] exp Z t s d N 0 + N

p(;z) (1+ N _ p(s;z))( N 0 N 0 +( N N )p(s;z)) ( N 0 + N p(t;z)) 2 ( N 0 + N p(s;z)) 2 exp Z t s d 0 + p(;z) (1+ _ p(s;z))( 0 0 +( )p(s;z)) ( 0 + p(t;z)) 2 ( 0 + p(s;z)) 2 :

Wenotethat equation(22) coupledwiththe quotientrule forlimitsallowsustoassert that

exp Z t s d N 0 + N p(;z) !exp Z t s d 0 + p(;z) (25) as q N

! q. Thus, we may use equations (24), (21), (22), and (25) with the quotient and

product rules for limitstoverify that

jG(q N

) G(q)j

L 1 !0

as q N

!q.

Wehavethusveriedthatthe theoryestablishedinSection3.1canbeappliedtothe system

Intheprevioussection,wedemonstratedthata(sub)sequenceofminimizersfq N

gofthecost

functionals(15)convergestoaminimizerqof(12). Inthis sectionwepresentcomputational

results for the problem of nding q N

for a xed N. (We attemptto choose N large so that

q N

is close toq.)

Thisproblemisequivalenttoourmainobjective,estimatingthepolarizationand

conductiv-ityparametersofadielectricbycomparingnumericalsolutionsofthemodelwith

experimen-tal data. We recall from Section 2 that our polarization modelhas six material-dependent

parameters;inthissectionwex andconsider onlythesesix variables

0 ;

0 ;

0 ;

;

;and

inthe equation

(

0 +

p)

_

P +P =

0 (

0

0 +(

)p)E:

Wewanttotestthefeasibiltyofestimatingthemfromexperimentaldata. Atthistime,wedo

notyethavedatafromexperiments(an experimentaldevicetoobtainsuch dataiscurrently

beingconstructed); insteadwecreatesimulateddatafromourcomputations. Thesimulated

data consists of the boundary data from a numerical approximation to the solution of the

system with added noise. (See [9] for sample numerical solutions to the forward problem.)

Wecompute this approximate solutionwith xed parametersvalues. Thesevalues are then

thoughtofasour\unknown" truematerialparameters. Thegoalistoestimatethesevalues.

Weappraiseour abilitytosolvetheproblemby comparingthe estimates withthe truexed

values. Ifwecannotaccurately approximatethe parametervaluesinthis context, wecannot

expect tobe able toestimate them inan experimentalsetting.

We let q generically denote the set of parameters we wish to estimate in the examples

presented below; these may include the mean values in the polarization model,

0 ;

0 ; and

0

and/or the coeÆcients of pressure in the polarization model,

;

; and

: We let q

denote the true values of the corrsponding \unknown" parameters. We leave the values of

allother parameters xed.

There are two sets of electromagnetic reections that reach the boundary. The rst, after

the initialsignal,arethe reectionsfromthe air/dielectricinterface andthe secondarefrom

the virtual interface produced by the acoustic pressure wave. (Figure 4 depicts each set of

reectionsseparately.) Insomescenarios,using datathatcontainsonlyoneset ofreections

may be advantageous. For example, one may use the data from the initialsignal and the

reections from the air/dielectric interface (i.e., the rst section of data in Figure 4) to

renethe initialparameter estimatesand then use theserenementswiththe data fromthe

acoustic interface reections to obtain nal estimates. In another approach, one may use

justthe datafromthe acousticinterfacereections toestimatetheparameters. Inany case,

J(q)=

i2I (E

i

data

E(t

i ;0;q))

2

:

whereI correspondstoanappropriatelychosendataset. (Sincehereweconsiderexclusively

the nite dimensional system for axed N, wedrop the N for ease of notation.)

### 0

### 1000

### 2000

### 3000

### 4000

### 5000

### 6000

### 7000

### 8000

### 9000

### −200

### −150

### −100

### −50

### 0

### 50

### 100

### 150

### 200

### 1

### 1.2

### 1.4

### 1.6

### 1.8

### 2

### 2.2

### 2.4

### 2.6

### x 10

### 4

### −200

### −150

### −100

### −50

### 0

### 50

### 100

### 150

### 200

Figure 4: The two sets of reections that reachthe boundary

WeusedaNelderMeadoptimizationroutine[19]tondthe parametervaluesthatminimize

the cost function. This optimizationmethodis agradient-free, simplexsearchmethod. The

tolerance for the dierencebetween subsequent function evaluations. We choose the initial

estimates to have varying levelsof error in relationto the true parameter values. For these

computations,we set the terminationtolerance at 1e-09.

As already noted, we created simulated data to test our algorithms. The data set without

noise is simplyobservations atthe boundary taken from a forward simulation of the model

using the parameterset q

. The data sets with errorwere created by addinganappropriate

amount of normally distributedrelative randomnoise to the originaldata set. The random

noise was generated by the MATLAB command randn which creates normally distributed

noise with mean 0 and variance 1 and was scaled and shifted appropriately. Because the

noise is relative, the magnitude of noise is greater in the intervals of data that contain the

initialinterrogating impulseand the reections.

Wenext presentsampleresultsforspecicparameterestimationproblems. Werst consider

the problem of estimating q

= [

0 ;

0 ;

= 1

p

0

0

0

] = [78:2;5:5;0:10545728042059] from

data with varying levels ofnoise. Here

0

isso small thatit isadvantageousto estimate

;

a scaled function of

0

; an estimated value of

0

may be computed from an estimation of

: We use the data containing the initialsignal and the reections from the air/dielectric

interface to rene the initialparameter estimates q

0

and the data containing the reections

fromthe acousticinterface toobtain nalestimates. Wepresent the results inthe following

0

q

0

=1:0q

[78.2, 5.5, 0.10545728042059]

q

0

=0:95q

[78.190631, 5.499999, 0.105462]

q

0

=1:05q

[78.197841, 5.500000, 0.105458]

q

0

=0:9q

[77.128609, 5.499937, 0.107040]

q

0

=1:1q

[70.763992, 5.499504, 0.117540]

Final estimatefor data with 1% noise

q

0

=0:95q

[78.210472, 5.499997, 0.105444]

q

0

=1:05q

[77.260485, 5.499940, 0.106852]

q

0

=0:9q

[78.198944, 5.499998, 0.105461]

q

0

=1:1q

[78.482467, 5.500010, 0.105052]

Final estimatefor data with 5% noise

q

0

=0:95q

[74.876211, 5.499764, 0.110559]

q

0

=1:05q

[77.975797, 5.499939, 0.105722]

q

0

=0:9q

[78.337467, 5.500010, 0.105186]

q

0

=1:1q

[78.405972, 5.499987, 0.105175]

Table 1: Parameter estimation results for

q

=[

0 ;

0 ;

= 1

p

00

0

]=[78:2;5:5;0:10545728042059]

Theseresultsillustratethatitispossibletorecoveraccurateapproximationsof

0 ;

0

;and

inthepresence ofnoiseandwitherrorupto10%intheinitialestimates. Afewoftheresults

are unexpected, for instance the ability to approximate the values better in the presence of

5% noisewithaninitialguess with-10%errorthan withaninitialguesswith -5%error. We

suspect these anomaliesare due tothe simplex search natureof the optimization routine.

Theresultsinthetablesclearlyindicatethatwecanrecoverq

withoutmucherror. However

it is oftenillustrativeto compare the solutionscalculated with the estimates with the

solu-tions calculatedwith q

. Todothis, we plotthe absolutevalue of the errorfor the solutions

computed atthe boundary, i.e.,

jE(t

i

;0;q) E(t

i ;0;q

)j

where q is the nal estimate (given in the table). As an example, Figure 5 depicts this

errorfor the estimationproblemwith5% noise andaninitialguesswith -10%error. We see

that overall the magnitude of error is smalland that, as expected, the most error occurs in

### 0

### 0.5

### 1

### 1.5

### 2

### 2.5

### x 10

### 4

### 0

### 0.5

### 1

### 1.5

### 2

### 2.5

### 3

### 3.5

### 4

### x 10

### −3

Figure 5: jE(t

i

;0;q) E(t

i ;0;q

)j vs t

i {

Absolute error forthe parameter estimation problem

with 5% noise and an initialguess with -10% error.

Wenext consider the estimation of q

=[

;

;

]=[46:92;1:65;1:581139e 09] from the

previousdatasets. SincetheseparametersarethecoeÆcientsofpressure,they areirrelevant

and undeterminable until the electromagnetic/acousticinteraction occurs. Thuswe include

0

q

0

=0:99q

[46.450800, 1.633500, 3.16174e-09]

q

0

=1:01q

[46.92, 1.65, 1.58114e-09]

q

0

=0:95q

[ 44.574000, 1.567500, 5.52700e-09]

q

0

=1:05q

[49.266000, 1.732500, 0]

q

0

=0:9q

[42.228000, 1.485000, 0]

q

0

=1:1q

[51.61200, 1.815000, 0]

Final estimatefor data with 1% noise

q

0

=0:99q

[46.450800, 1.633500, 3.17078e-09]

q

0

=1:01q

[47.389200, 1.666500, -7.2435e-10]

q

0

=0:95q

[44.574000, 1.567500, 5.53493e-09]

q

0

=1:05q

[49.266000, 1.732500, -1.064074e-08]

q

0

=0:9q

[42.228000, 1.485000, 6.01095e-09]

q

0

=1:1q

[51.612000, 1.814500, -1.712928]

Final estimatefor data with 5% noise

q

0

=0:99q

[46.450800, 1.633500, 3.22126e-09]

q

0

=1:01q

[47.389200, 1.666500, -6.6658e-10]

q

0

=0:95q

[44.574000, 1.567500, 5.56261e-09]

q

0

=1:05q

[49.266000, 1.732500, -1.061871e-08]

q

0

=0:9q

[42.228000, 1.485000, 6.02759e-09]

q

0

=1:1q

[51.612000, 1.815000, -1.712024e-08]

Table 2: Parameter estimation results for

q

=[

;

;

]=[46:92;1:65;1:581139e 09]

Weareabletoobtainreasonableestimatesfortheparameters,especially

and

:Wehave

diÆculty estimating the value of

, most likely because the value is small. In general, the

estimation error increases with the error in the initialguess. However, increasing the noise

level inthe data doesnot signicantlyeect the estimation accuracy.

We note that we obtain better estimates for the mean values (

0 ;

0

; and

0

) than for the

pressure coeÆcients (

;

; and

). This is understandable, as the mean values are more

inuential in the system dynamics. They are also more important in identifying and

char-acterizing the material.

In an electromagnetic interrogation parameter estimation problem, an estimate is

suÆ-cient if it can be used to classify the material. We consider the results for estimating

q

=[

;

;

]=[46:92;1:65;1:581139e 09]usingdata with5% relativenormalnoise and

aninitialguess with-10% error. After solvingthe parameter estimationproblem,weobtain

the material under interrogation, we are successful in our attempt to solve the estimation

problem. On the other hand, if the characteristic material parameters fall within the

(hy-pothetical) range 45<

<47, 1:6 <

<1:7, and 1e 09<

<2e 09; we are unable

to characterize the materialwith our estimates and our attempt is unsuccessful. Ranges of

these parameter values for dierent materials have not been experimentally determined, so

we haveno concrete measure as yet to assess our ability tosolve the problem.

5 Concluding remarks

Wehave presented theoreticaland computationalresults foranew classof inverse problems

arisinginnondestructiveinterrogationof materials. Ourfocus isonreectionsfromacoustic

pressure wavesthatare movingthrough dielectricmaterialtargets. Adetailedatomicbased

model for acoustic-dependence of dielectric parameters in a Debye material was given and

this was incorporated into a theoretical framework for both forward solutions and least

squares parameter estimation.

Computational ndings suggest that primary parameters (relaxation, static permittivity,

etc.) willbereadily identiablewhileaspects of the nonlinear dependence onpressure (rst

order coeÆcients)may beascertained if appropriate data is available.

Our eorts on this methodology are continuing. An experimental device (similar to that

depicted in Figure 7.1 of [4]) iscurrently under construction. Datafrom this device willbe

used to test and validate the methodsdeveloped in this paper.

6 Acknowledgements

This research was supported in part by the Air Force OÆce of Scientic Research under

grantsAFOSR-F49620-01-1-0026,and AFOSR-F49620-98-1-0430andinpart through a

De-partment of Education GAANN Fellowship toJ. K.Raye under Grant P200A70707.

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