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: [email protected]
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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