JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 100, NO. C7, PAGES 13,699-13,711, JULY 15, 1995
Boundson the complex permittivity of sea ice
K. Golden
Departmentof Mathematics,Universityof Utah, Salt Lake City
Abstract.An analyticmethodfor obtainingboundson effectivepropertiesof
composites
is appliedto the complexpermittivity•* of seaice. The seaice is
assumed
to be a two-component
randommediumconsisting
of pureice of permittivity
½•andbrineof permittivity•2. The methodexploitsthepropertiesof e* as an analytic
functionof the ratio El/e2. Two typesof boundson •* are obtained. The first bound
R1 is a regionin the complexe* planewhich assumesonly that the relativevolume
fractionsPl andP2 - i -Pl of the ice andbrine are known.The regionR1 is bounded
by circulararcsand •* for any microgeometry
with the givenvolumefractionsmust
lie insideit. In additionto the volumefractions,the secondboundR2 assumesthat the
seaice is statistically
isotropicwithinthe horizontalplane. The regionR2 is again
boundedby circulararcsandlies insideR1. Built intothe methodis a systematic
way
of obtainingtighterboundson •* by incorporatinginformationaboutthe correlation
functionsof the brine inclusions.The boundingmethoddevelopedhere, which does
not assumeany specificgeometryfor the brine inclusions,offers an alternativeto the
classicalmixing formulaapproachadoptedpreviouslyin the studyof seaice. In these
mixingformulas,specificassumptions
are madeaboutthe inclusiongeometry,which
are simplynot satisfiedby the sea ice undermany conditions.The boundsR1 and R2
are comparedwith experimentaldata obtainedfrom artificiallygrown seaice at the
frequencies4.8 and 9.5 GHz. Excellentagreer0entwith the data is achieved.
wide variety of possible microstructuresand the high
dielectric contrast of the components,it is, in'general,
The remote sensingof sea ice generatesmany inter- quite difficult to accurately predict the effective comesting technical and theoretical problems. A central plex permittivity of sea ice. Nevertheless,many models
questionin the application of remote sensingtechniques have been proposed to describe the dielectric behavto studying sea ice is how the physical microstructure ior of sea ice, suchas thoseof Addison[1969],Hoekof the ice determines its scattering and effective elec- stra and Capillino[1971],$togryn[1985],Tinga et al.
tromagnetic properties. This question plays a funda- [1973],Vant et al. [1978],GoldenandAckley[1981],Simental role in correctlyinterpretingscatteringdata and hvolaandKong[1988].Typically,the seaiceis assumed
images from sea ice. The determinatio.n of the electro- to consistof a hostmedium(pure ice) containingspherical or ellipsoidalinclusions(brine or air). Various efmagnetic
properties
frommicrostructural
information
is particularly interesting in the case of sea ice, which fective medium theories,suchas the coherentpotential
is a complex random medium consistingof pure ice approximation, have been used to derive "mixing formulas" for the effective permittivity e* of the system.
with brine and air inclusions.
The details of the microstructure,namely the geometryand relative volume Each formula is judged accordingto how well it predicts
of the inclusions,dependstronglyon the temperatureof experimental data.
While mixing formulas for e* are certainly useful,
the ice, the conditionsunder which the ice was grown,
and the history of the sample under consideration, as their applicability to the full range of microstructures
discussed
by Weeksand Ackley[1982]. Owing to the presentedby sea ice is limited. For example, as sea ice
1.
Introduction
warms up to aroun.
d -5øC, the brine pocketstend to
Paper number 94JC03007.
coalesceor "percolate" and form a connected matrix
of brine, in which case the sea ice becomesa porous
medium. This processrepresentsan important stage
in the evolution of a sea ice sheet, as it precedesthe
0148-0227/95/94JC-03007 $05.00
phe•nomenon
known as "brine rejection," where some
Copyright 1995 by the American GeophysicalUnion.
13,699
13,700
GOLDEN:
COMPLEX
PERMITTIVITY
of the brine flows out of the sea ice, due to its porosity. The electromagneticpropertiesof the ice following
the rejection are dramatically changed. Clearly the assumptionthat the brine (or air) componentof seaice is
contained in individual ellipsoidal inclusions,is simply
not satisfied when the brine forms a connected matrix.
We must therefore questionthe validity of thesemixing
formulas
for warm
ice.
OF SEA ICE
ample,seeKozlov[1989],Golden[1990],Golden[1992],
Golden[1994],BerlyandandGolden[1994],Bruno[1991]
and K. Golden [Statisticalmechanicsof conducting
phasetransitions,submittedto J. Math. Phys., 1995].
It is interestingto note in this contextthat the connectednessproperties of a particular component dominate
the behavior of many technologicallyimportant materials, including doped semiconductors,porous media
(e.g., sandstones),piezocomposites,
thermistors,solid
rocket propellants, and cermets.
Another limitation of mixing formulas is that they
provide only a preciseprediction of what e* should be,
given, say, a value for the brine volume and perhaps
some information about the distribution of ellipsoidal 2. Description
of Theory and Methods
orientations,as considered
by SihvolaandKong[1988].
They do not provide any information on the range of
reasonable values for e*, which would be very useful,
given that in any experiment,as one measurese* for
a variety of samples, one obtains a scatter of values in
the e* plane, which may or may not coincidewith the
precisevalue predicted by the mixing formula.
We remark that one of the principal motivating factors behind this work is the goal of developingpractical electromagneticmethods of distinguishingsea ice
types, such as first-year from multiyear ice. In typical inversescatteringalgorithms, what is reconstructed
from scattered electromagneticfield data is the complex permittivity which is "seen" by the wave. When
Owingto the abovelimitationsin the mixingformula
approachwhichare causedby the wide varietyof relevant sea ice microstructures,
we begin here a new approachto stu(]yingthe effectivecomplexpermittivity
e* of seaice by obtainingboundson e* in the complex
plane. As a first stepin the development
of this approach,we considerthe seaice to be a two-component
randommediumconsisting
of pureice of complexpermittivity el in the volumefraction pl and brine of com-
plexpermittivitye2in the volumefractionp2 - 1- pl.
In this paperwewill neglectthe effectsof air inclusions,
whichis a reasonableassumptionin many situations,
suchas for first yearice whichhasnot yet undergone
the wavelengthis longerthan the microstructuralscale, brinerejection,whereair canreplacemuchof the brine.
which for sea ice would correspond to frequenciesof less To obtain the bounds,we apply a methodintroducedinthan 10 or 20 GHz, this reconstructed permittivity will
dependently
byBergman
[1978]andMilton[1979]
which
be e*, the effectivepermittivity. (Of course,e* itself exploitsthe propertiesof e* as an analyticfunctionof
can vary throughout any given sea ice sheetas the mi- the ratio el/e2. The methodwasdeveloped
furtherand
crostructural geometry, such as brine volume, changes appliedin varioussettingsby bothBergmanand Milthrough the sheet.) One of our principal goalshere is ton, and the complex bounds used here were obtained
to study in detail how the statistical properties of the
sea ice microstructure
are connected
independently
by Milton[1981]andBergman
[1980].A
to these observed
mathematicalformulationof the methodwasgivenby
GoldenandPapanicolaou
[1983]andsomeof its impliWe also remark that another motivation for our study cationswereexplored
by Milton andGolden[1985].
of sea ice is that from the point of view of the mathThe keystepin thisformulation
is establishing
an in-
values of e*.
ematical theory of composite materials, understanding
the electromagneticbehaviorof seaice pushesthe limits
of currently available mathematical techniques. From
this point of view, sea ice is an extremely interesting
example of a composite material, which combines in
one medium many of the challengesencounteredin the
development of the mathematical theory; it is a multicomponent composite,whosecomponentshave highly
contrasting properties characterizedby complexparameters and whosemicrostructure exhibits a wide range of
geometries. In particular, one of the most mathematically intriguing aspectsof the microstructureis the coalescingor percolation of the brine cells, which occurs
around the critical temperature Tc • -5øC, and results
in a microstructural transition from separated brine
pockets to a connectedbrine matrix. The influence of
the connectednessproperties of composite microstructures on bulk electromagneticbehavior has been a topic
of considerable
recent mathematical
interest.
For ex-
tegral representationfor e*, which is then usedto obtain
the bounds.As in the caseof typicalmixingformulas,
the boundsare derivedin the quasi-staticlimit, sothat
theyare validwhenthe wavelength
is longcompared
to
the scaleof the inhomogeneities-.
We nowdescribethe boundsmoreprecisely.For the
simplestboundit is assumed
that onlyex, es, p•, and
p2 are known.The boundconsists
of a lens-shaped
regionin the complexe* planein whichthe valuesof e* for
all possiblemixtures of the two materials in the given
volume fractions must lie. For the caseof sea ice, where
there is a wide variety of possible microstructures, the
bound is valid for all of them, given only the brine volume. Built into the method, though, is a systematic
way of obtaining tighter boundson e* if one knowsfurther statistical information about the geometrybeyond
simply the volume fractions. More precisely,there is a
hierarchyof increasinglytighter bounds,where the nth
order bound dependson knowledgeof the n-point cor-
GOLDEN:
relation function
of the mixture.
COMPLEX
PERMITTIVITY
OF SEA ICE
13,701
The bounds described
realization and not frequency.) For a two-component
aboveare first order and dependonly on one-point cor- medium with componentpermittivitiesel (ice) and e2
relation or volume fraction information.
For the case of (brine) we write
second-order
bounds it turns out that if the material
is
statistically isotropic, i.e., if e* does not depend on the
direction of polarization of the electric field, then one
can obtain the boundswithout any direct knowledgeof
the two-point correlation function.
In the present work we apply the first order and
isotropic second order bounds to sea ice with given
salinities and temperatures. These bounds are com-
e(x, co)- elXl(X, co)+ e•X•(x, co),
(1)
where X•' is the characteristicfunction of medium j -1, 2, which equals one for all realizationscoe f• having
mediumj at x, and equalszero otherwise.Let E(x, co)
and D(x, co)be the stationary randomelectricand displacementfields, respectively,satisfying
paredwith the experimentaldata of Arconeet al. [1986]
O(x,
for artificially grown sea ice at 4.8 and 9.5 GHz. Given
a sample of sea ice at temperature T and of salinity
$, we estimate the brine volume p•. using the equation
-
(2)
V. D(x, co)- 0
(3)
of Frankensteinand Garner [1967]. Then for eachfre-
V x E(x, co)- 0
(4)
quency, 4.8 and 9.5 GHz, and for a given temperature
T, we find the complex permittivity of the brine e2 us-
<
(s)
ing the calculations
of $togrynandDesargant[1985].In
>=
where ek is a unit vector in the kth direction, for some
k = 1,...,d.
In (5), angle bracketsmean ensemble
general, we obtain excellent agreementwith the experaverageoverf• or spatialaverage
overall of IRa.
imental data. Small discrepanciesbetween the bounds
In view of the local constitutivelaw (2) we definethe
and the real data are inevitable, though, and may be
effective complex dielectric tensor e* via
explained with uncertainties in the brine volumes, the
unaccountedfor presenceof air, or scattering effects
<D>
= e* <E>,
k-1,...,d,
(6)
when the wavelength is larger but comparable to the
or,
e•k = < Di >, i,k-1,...d,
(7)
typical brine length scale.
*
3. Bounds on the Effective Permittivity
We now present the bounds on e* discussedabove. In
this section we first formulate the effective permittiv-
ity problem, briefly describethe method for obtaining
the bounds, which is called the analytic continuation
method, and then state the bounds. A more detailed
summary of the method is given in the appendix. We
also refer the interested reader to Golden and Papanico-
where the right side of (7) dependson k via (2) and
(5). For simplicity we focuson one diagonalcoefficient
e* -
* Noting that through the homogeneityprop•kk'
erty e*(Ael,Ae2) = Ae*(el,e2), e* dependsonly on the
ratio h = el/e2, we may divide (7) by e2 and define
6'
= -
62
= <
+
>,
61
= -.
62
(8)
The two main propertiesof re(h) are that it is •nalytic
off the interval (-•, 0] in the h plane•nd that it m•ps
laou[1983],Golden[1986]and Bergman[1982]for more the upper half plane to the upper half plane, i.e.,
detail.
Let us begin by formulating the effectivecomplexpermittivity problem for a two-componentrandom medium
Ira(re(h)) > 0,
Ira(h) > 0,
(9)
in all of I•a, d-dimensional
Euclideanspace,with trans- where Im denotes imaginary part.
lation invariant statistics. Formulating the problem in
this way is for mathematical simplicity but is analogousto assumingthat the brine statistics are the same
throughouta given sampleof seaice and that the sample is large comparedto the typical brine length scale,
which is reasonable for comparison with data taken for
samplesby Arconeet al. [1986]. For real sea ice the
brine statistics and structure will change with depth
and might differ from floe to floe. However, for a relatively small depth interval of, say, 10 cm, and in any
givenfloe the assumptionshouldbe reasonablywell satisfied. Now let e(x, co)be a (spatially) stationaryrandom field in x • I• a and co• f•, wheref• is the set of
all realizationsof our random medium. (We are dealing here with an ensemblef• of random media, indexed
by the parameter co, which representsone particular
Such functions
which are analytic off some segment of the real axis
and obey (9) are called Herglotz functions.
As mentionedin section2, the key step in the analytic
continuation method is obtaining an integral representation for e*. For this purpose it is more convenient to
look at the function
F(s)=l-m(h),
s=l/(1-h),
(10)
whichis analytic off [0,1] in the s plane. It was then
provenby Goldenand Papanicolaou[1983] that F(s)
has the following representation
F(s)-
f0•aU(•)
(XX)
where/• is a positivemeasureon [0,1]. (Very loosely
13,702
GOLDEN: COMPLEX PERMITTIVITY
OF SEA ICE
speaking,we may think of dtt(z) as g(z)dz, for some and C(a) getstransformedto the circulararc
densityg(z) whichis allowedto have "6 function"coma
1-a
ponents.) One of the most important featuresof (11)
- --+•
,
0<a<l,
is that it separatesthe parameter information in s 1/(1- e•/e2) from informationabout the geometryof
C*(a)
the mixture, which is all contained in it, through dependenceon X•. In fact, statistical information about
thegeometry
is inputinto(11)viathemoments
of/•,
- -
(17)
so that a plays the role of Pl, the volume Ëaction of
medium 1. In fact, the expressions
in (16) and (17)
are simply the complexpermittivities of laminates(or
layers) of e• and 62 in the relative volume fractions a
and 1- • with the applied field parallel and perpendicular, respectively, to the lamination direction. The arc
and line segmentare parameterizedby the relative volUme fraction a of 61. Becausethere are actual geome-
•01
.
This input is obtainedby equatingan expansionof (11) tries (laminates) which att•n these complexbounds,
in powersof 1/S, the coefficientsof which are the mo- the bounds are s•d to be optimal, in other words, with
ments ;tn, to a correspondingperturbation expansion no •sumptions at •l about the geometry, these are the
arounda hbmogeneous
medium(s = ½x:or el = 62)
best bounds one can obt•n.
For real parameters with
of anotherrepresentation
for F(s) involvinga resolvent e• • e•, R0 collapsesto the interval
formulafor E. This procedure,whichis describedin
the appendix, yields
;UO-- Pl,
(13)
If thevolume
fractions
pl andp• - 1- pl areknown,
then (13) must be satisfied.Let R• be the corresponding region, which is bounded by circular arcs. In the F
plane one of these arcs is parameterized by
if only the volume fractions are known, and
PiP2
;ti-
C•(z)- pl
(14)
d
8--Z
if the material is statistically isotropic, where d is the
O• z • p•
(19)
wherez is a real parameter•d C(z) is fractionallinear
in z, like C(a) above. To exhibit the other arc, it is
dimensionof the system as in 6 and (7). In general, convenient to considerthe auxiliary function
knowledgeof the (n q- 1)-point correlationfunction of
e•
1-sF(s)
the meditiinallowscalculationof ;t;• (in principle).
Boundson e* or F(s) are obtainedby fixing s in (11),
varyingover admissiblemeasures;t (or admissiblegeometries),suchas thosethat satisfyonly (13), and find- whichis a Herglotzfunctionlike F(s), analyticoff [0,1].
ing the correspondingrange of valuesof F(s). This Then in the E plane, we can parameterize the other
-
_
procedureis described
in the appendix.We nowgive circular boundary of R• by
formulas for the boundaries of these regions. In addition to the first- and second-order
½•(z)- P2
bounds described in
0 < 2,< Pl
(21)
the Introduction, there are zeroth order boundswhich
assumeno knowledgeof the geometry,not eventhe vol- In the 6
* plane,R• hasverticesplel +p262 and (p•/61 +
ume fractions. They exploit only the conditionsthat
p2/62)-1 and is boundedby the arcs
0 _< ;to _< i and F(s1) _< 1. In the F plane, let
R0 be the regioncontainingthe rangeof valuesof F(s).
C1(/•) -- 62+ p•
Then R0 is boundedby a circular arc C(a) and a line
61 -- 62
segmentL(a),
(22)
C(a)-
a
s-(1-a)'
L(a)---a
0<a<i
s'
-
'
(15)
62 -- 61
pl)
(23)
Th6 arc C(a) is circular becauseas a function of a,
C(a) is a fractionallinear transformation
of a, which where/•is a realparameter
andC•(/•) and• ([:•)are
maps the class of circles and lines to itself. When s is
again fractional linear in /•. Again, these bounds are
complex,then the line segment[0,1] in the a planegets optimal and are attained by a number of geometries,
mapped to an arc of a circle. If we transform R0 to
including coated elliptical cylindersconsideredby Mil-
the e* plane, then its verticesare 61 and ca, L(a) gets ton [1981].When 61and 62are real, R1 collapses
to the
interval
transformed to the line segment
(16)
(p•/e• + p2/62)-• _<e* _<p•el + p2e2,
(24)
GOLDEN:
which are the classical arithmetic
COMPLEX
PERMITTIVITY
and harmonic mean
bounds.
If the material is further assumed to be statistically
OF SEA ICE
13,703
found experimentally. Compared to the brine, the complex permittivity of the pure ice e• is basically independent of temperature and frequency, as found by Miitzler
isotropic,
i.e., eik-- e 5ik,then(14)mustbesatisfied
as and Wegmiiller[1987],and we take e• = 3.15+ i0.002.
,
*
In Figure 1 we plot the bound R0 with R• contained
well. The corresponding
regionR2, again, has circular
arcs for boundaries. In the F plane one of these arcs is inside, with the frequency, $,T, and p2 values shown.
Clearly, knowingthe brine volume(R•) yieldsa tremenparameterized by
dous improvement over the R0 bound, which only assumesknowledgeof the values of e• and e2. In Figure
2 we have plotted two different regionsR• for the same
p• (s - z)
C2(z)
- s(s-z-p•/d)'O_•
z_•(d-1)/d.(25)
sampleof seaicewith $ = 50/00,to showhowchanging
In the E planethe other arc is parameterizedby
p2(sz) 1)/d]'O•_z•_l/d. (26)
(•2(z)
- s[s-zp,(dWhen e• and e2 are real with e• •_ e2, R2 collapsesto
the interval
/( 1 p•)
e* < e2+ p•
-•
/( •1--1•2 + •pe)
el +P2
e2-- el
+
•
•1
--
(27)
the temperature shifts the region in the e* plane. Alternatively, we can view Figure 2 as showinghow changing
the brine volume can dramatically affect the location
and size of R•. In Figure 3 we have plotted the smaller
R• with T = -11øC from Figure 2 with R2 inside of
it. For the calculation of R2 here and in subsequent
figures,whoseboundariesare given by (25) and (26),
we have used d -- 2, since as the wave propagatesvertically down through a slab which is not too thick, the
properties are reasonably uniform in this direction. In
this case the electric field only has componentswithin
the horizontal plane, so that the system is effectively
[1962].The real boundsin (27) are attainableby coated two-dimensional. The assumption of isotropy in this
spheregeometries,but the complexarcs (25) and (26) configuration, which is the configuration used by Atwhich are the famous bounds of Hashin
and $htrikman
are not generallyattainable,as notedby Milton [1981] coneet al. [1986],is then an assumptionaboutisotropy
within the horizontal plane, which is a topic considered
and Bergman[1982].
by Cherepanov
[1971],Kovacsand Morey [1978],Weeks
and Gow[1980]and Goldenand Ackley[1981].(We remark that since the scaleson the Re(e*) and Ira(e*)
4. Comparison With Experimental Data
axes are not the same, the circular arcs which form the
In this section we considerplots of the bounds R0, R•,
and R2 and compare them with the experimental data
of Arconeet al. [1986].For any givensampleof seaice
boundaries of the regions may not necessarilyappear
circular.)
In Figures 4a, 4b, and 4c we begin the comparison
of our bounds
with
the measured
values
of e* for ar-
of salinity $ and temperature T the brine volume p2
may be estimated using the equation of Frankenstein tificially grown sea ice of Arcone et al. [1986]. The
data points in Figure 4a were taken from Arcone et al.
and Garner [1967],
+ o.53' .
p•_$(49.185
To determine e2, the complex permittivity of the brine.
we use the empirically determined formulas of $togryn
[1986],Figure 8. They representa rangeof p• values
from 0.031 to 0.041, and salinityvaluesfrom 3.8O/oo
to 4.4O/oo. To constructthe bound,we havechosen
the midpointvaluesof p• = 0.036 and S = 4.1 ø/00,
and then inverted (28) to obtain T = -6øC. The data
points in Figure 4b range in p2 values from 0.018 to
and Desargant[1985],whichare basedon a Debye-type 0.023, with the same salinity range as in Figure 4a, and
relaxation equation,
we obtain the bound through the same midpoint procedure. In Figure 4c, the data have been taken from the
slab measurements
of Arconeet al. [1986],Fige.•--e•+ I -i27rf•+i2•e0--•
'
(29) 85-4
ure 17, with a range in temperature from -12.5øC to
which has only one resonance,ignoring a possiblespread -15.5øC, so that we choose the midpoint T = -14øC,
in relaxation times. In (29), es and e• are the limit- with S = 2.4ø/00,to constructthe bound. •Ve chose
ing static and high frequencyvaluesof the real part of this particular slab because there was little difference
e2,Re(e2), r is the relaxationtime, f is the frequency,cr between the measured salinity before and after the exis the ionic conductivity of the dissolvedsalts, e0 is the periments were done, so that we were fairly confident
permittivityof freespace,andi - (-1)•/2. The ionic that the p• value given by (28) would be representative
conductivity is assumedto be independent of frequency. of what was in the sample. By comparisonof the data
Then e2 for brine in equilibrium with sea ice is deter- with R•, there appears to be some evidence of a small
mined by the four real parameters e•, es,r, and or,each degreeof anisotropy, which agreeswith what was found
of which depends only on temperature, and all were directly by Arcone et al. [1986]. However,thesedis-
13,704
GOLDEN'
COMPLEX
PERMITTIVITY
OF SEA ICE
4O
f
35
3O
-
4.8 GHz
$ -
50/00
T
-2øC
-
25
2O
15
10
p2=0.12
0
10
20
30
40
50
60
70
Re(e*)
Figure 1. Boundson complexpermittivitye* for seaice. The largeregionR0 assumes
nothing
about the microstructure.The smallregion-R1insideassumes
a brinevolumeof p•. - 0.12; f is
frequency, $ is salinity, and T is temperature.
crepanciesmay also be accounted for by uncertainties perature. The data shownin our figuresare a representative set of the data from slabs 85-1, 85-2, and 85-3.
Finally, Figures 5a, 5b, 6a, and 6b were created to These three slabs were chosenbecausetheir salinity val-
in the brine volume, which is discussedbelow.
be able to directly comparewith Arconeet al. [1986, ues(beforethe experiment)wereall closeto eachother,
Arconeet
Figures16 and 17], in which the real and imaginary with $ = 4.8,5.5 and 5.4 ø/00,respectively.
parts of e* are plotted separately as a function of tern- al. [1986,Figures16 and 17] plot data for ice whichis
4.5
f
4
-
4.8 GHz
$ -
5%0
3.5
3
Im(e*)
2.5
2
1.5
T
=
-11øC
P2
--
0.025
0.5
O3
4
5
6
7
Re(e*)
T
=
-2øC
P2
-
0.12
8
9
10
11
Figure 2. Bounds on e* for sea ice with different brine volumesp2 or different temperatures T.
GOLDEN: COMPLEX PERMITTIVITY OF SEA ICE
13,705
•.2
f = 4.8GHz
//•
S = 5ø/•0
T
=-11
C
•
0.8-
0.6-
0.4-
0.2-
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4
4.1
4.2
Figure 3. Boundson e* for givenbrine volume(outer boundR•) and assumingisotropywithin
the horizontalplane (inner bound R2).
warmed and subsequently cooled down. We only used
boundsfor that valueof T and with $ = 5 ø/00. For
the "warming" data, as once the ice has been warmed
a given value of T, the R2 bound gives a maximum
above -5øC, brine leaks out and air replaces some of
and minimum value for both Re(e*) and Im(e*) which
the brine, which is reflectedin the "cooling"data. For
any given value of T, to create the upper and lower
bounds shown in our figures, we first computed the R2
corresponds to the coordinates of the vertices of R2.
1.8
I
f
1.6
s
T
1.4
p2
=
I
The upper and lower bounds for any given T shown
in Figure 5a, for example, correspond to these maxi-
I
I
I
I
I
I
4.75 GHz
4.1
0/00
•/•
0.
1.2
Im(•*)
0.8
0.6
0.4
•
I
I
03.2 3.4 3.6 3.8
I
4
I
I
I
I
4.2 4.4 4.6 4.8
5
Re
Figure 4a. Comparisonof experimental data on e* with/i•l for given brine volumeP2 and with
bound R2 which assumesisotropy as well.
13,706
GOLDEN'
1.8
I
I
f
1.6
1.4
COMPLEX
=
I
I
OF SEA ICE
I
I
I
I
I
4.75GHz
-
S = 4.10/00
_
T
=
PERMITTIVITY
-
-11øC
P2= 0.0205
-
1.2
Im(e*)
1
0.8
0.6
0.4
0.2
03.2 3.4 3.6 3.8
4
4.2 4.4 4.6 4.8
5
Re
Figure 4b. Same as Figure 4a, except for temperature of -11øC and p2 of 0.0205.
mum and minimum valuesof Re(e*) obtainedfrom the
verticesof R2. Note that by consideringRe(e*) and
sidesparallel to the axes) with two verticesmatching
Ira(e*) separately,it is easierfor data pointsto lie inside the bounds,comparedwith plotting the pointsin
ity through plotting Re(e*) and Ira(e*) separately,we
found that we couldobtain goodfit with the data by
using/{2 insteadof R1. We remark that the assumption
of isotropyin the horizontalplane, which is implicit in
constructingR2, is fairly reasonable,as seenin the previous figures. Furthermore, sincethe sea ice was artifi-
the complex plane. More precisely, points which lie
inside the boundsrepresentedin Figures 5a, 5b, 6a,
and 6b, if plotted in the complexe* plan.
e, wouldonly
be requiredto lie in the corresponding
rectangle(with
1.8
i
i
f
:1.t51.4-
i
-
o.s GHz
S -
2.4 0/00
T
=
-14øC
P2
--
0.01
the two vertices of/{2.
Because of this added flexibil-
i
1.21-
Im(e*)
0.80.6-
0'41
0.2
I
I
03.2 3.4 3.6 3.8
I
4
I
I
I
I
4.2 4.4 4.6 4.8
5
Re (e*)
Figure 4c. SameasFigure4a, exceptfor frequency
of 9.5 GHz,salinityof 2.40/00,temperature
of-14øC,
and P2 of 0.01.
GOLDEN:
10
COMPLEX
PERMITTIVITY
i
0
I
-5
OF SEA ICE
i
13,707
I
I
-10
f
-
4.8 GHz
S
-
5 ø/oo
I
-15
I
-20
T
Figure 5a. Comparisonof experimentaldata on e* with upper and lower boundson Re(e*)
obtained from R2, as a function of temperature, at 4.8 GHz.
5. Discussion
cially grown in an outdoor pool, where there is no welldeveloped,long-termcurrent direction(whichis usually
present when the c axis orientation of sea ice is highly
In comparing our bounds with the data of Arcone
anisotropic,
asstudiedby Cherepanov
[1971],Weeksand et al. [1986],we havefoundti/at perhapsthe single
Gow[1980]and Goldenand Ackley[1981]),we should most important quantity which determines the dielecnot expect marked anisotropy in the data from Arcone
tric properties of sea ice is the brine volume p2, and the
regionsR• and Rs are very sensitiveto this value. Un-
et al. [1986].
I
3.5
f
-
4.8 GHz
$ = 50/00
2.5
Im(e*)
1.5
0.5
0
-0.5
0
-5
-10
-15
-20
Figure 5b. Comparisonof experimentaldata on e* with upper and lower boundson Ira(e*)
obtained from R2, as a function of temperature, at 4.8 GHz.
13,7o8
GOLDEN:
lO
COMPLEX
PERMITTIVITY
I
I
OF SEA ICE
I
I
f
__
9.5 GHz
ø/oo
6
•
....
0
I, ,
-5
,I,
-10
I ,, ,
-15
I
-20
T (oc)
Figure fla.
Comparisonof experimentaldata on •* with upper and lower boundson Re(e*)
obtained from R2, as a function of temperature, at 9.$ GHz.
fortunately, the brine volume is quite difficult to measure experimentally. In comparingour boundswith the
data, it appearedthat the largest discrepancies
occurred
when there was the most uncertainty in the calculated
value of the brine volume. As Arcone et. al. note, brine
wasleakingoutof theslabsofseaiceuntiltheirtemperaturedroppedbelow-24øC, and resumedagainwith T
between -5 ø and -6øC. Even though the leaked brine
was collected so that accurate salinity measurements
could be made, under such circumstancesthe equation
quenciesof 4.8 and 9.5 GHz we have found no evidence
of any systematicoffsetof the experimentalvalueswhen
comparedwith the bounds. We concludethen that at
these frequencies,scattering lossesare minimal. In current work with S. Ackley and V. Lytle, we are comparing the quasi-staticboundswith higher-frequencydata,
overa rangeof 26.5-40.0 GHz, wherethe validity of the
quasi-static assumptionis uncertain. This work will be
reported elsewhere.
Finally, as mentionedin section4, we did not use the
of Frankensteinand Garner [1967],whichis basedon "cooling"data of Arconeet al. [1986],as air has retheoretical considerations,cannot be expected to provide a very accurate prediction of the actual brine volume in the sample. In fact, we noted that adjusting the
brine volume in the bound R1 could usually capture
any scatter of data points associatedwith a particular
S,T pair. This observationleads one to suspectthat
the brine volume of a given sample of sea ice could be
deducedby measuringe* at a number of different frequenciesand checkingwhich brine volume yields the
best correspondencebetween the bounds and the data.
placed some of the brine. The effect this has on e* is
that it apparently lowersthe measuredvaluesof both
Re(e*) and Ira(c*). In orderto obtainboundsto model
this situation, as well as multiyear ice which generally
has more air volume than does first year ice, we must
consider the sea ice to be a three-component random
medium consistingof pure ice, brine, and air. However, the mathematical theory of bounding e* for three
or more componentsis quite a bit more involved than
for two, and involvesthe theory of severalcomplexvariablesin which there arise new mathematical phenomena
not encounteredwith functionsof a singlecomplexvariable. In fact, the generaltheory of complexboundswas
restricted to two componentsuntil the work of Golden
As mentioned in section 2, the bounds are derived
in the quasi-static or infinite wavelengthlimit and are
valid when the wavelengthA is large comparedwith the
typical brine length scale/3,which is of the order of millimeters. When A becomescomparableto •, we might andPapanicolaou
[1985],andthe firstsetof comprehenexpect that scatteringeffectsbecomemore significant. sive boundson e* in the multicomponent casewas found
In this case we would expect that measured values of by Golden[1986]. Subsequently,
Milton [1987a,b]imIm(e*) shouldbe higherthan thosepredictedby the proved Golden's boundsand significantlyextendedthe
bounds due to scattering losses. For the current fre- bounding theory. In subsequentwork we would like to
GOLDEN: COMPLEX PERMITTIVITY
OF SEA ICE
I
13,709
I
3.5
I
f
-
$ -
9.5 GHz
50/00
2.5
1.5
0.5
-0.5
0
-5
-10
T (øC)
-15
-20
Figure6b. Comparison
ofexperimental
dataone* withupperandlowerbounds
onIra(e*)
obtainedfrom/•2, as a functionof temperature.at 9.5 GHz.
'applythe Golden-Milton boundsto three-component
Now we describehow (A3) can be usedto inject ge-
sea ice.
ometrical information into the measure 12. First, for
Appendix: The Analytic Continuation
medium(s=ooorh=l),
I.sl> 1, (A1) can be expanded
abouta homogeneous
Method
Here we describe in more detail following Golden
F(s)- 12-2ø
TM
s + '•'+...,
12n
z'•d12(
z)ß (A4)
to a Taylorseries
expansion
ofm(h) [1986],howthe boundsR0, R1, and R2 are obtained. Thisisequivalent
kVebeginwith the fundamentalrepresentation
formula •*/•2(el/•2) aroundh = 1. Equating(A4) to the same
presented in section 2,
expansionof (A3) yields
F(s) -
f0•
,
(A1)
12,•= (-1) '• < X•[(Fx:)'•e•]ße• >.
(AS)
It is throughthis momentformulathat correlationin-
where
formationabout the geometryis injectedinto the inte-
r-l-e*/e2,
s--1/(1-e•/e2),
(A2)
gral representation.
If all the moments12,•are known,
thenthe measure12is uniquelydetermined.Then (A1)
and 12is a positivemeasureon [0,1]. An alternative providesthe 'analyticcontinuationof the power series
operatorrepresentation
for F(s) can be obtainedfrom in (A4) to its full domain of analyticity,which is the
full complexs planeexcluding[0,1].
manipulationof (2)-(5),
F(s) -< ?(•[(s+ Fx•)-•e•] ße• >,
(A3)
Boundson e* are obtainedfrom (A1) as follows.First
we describehow R0 is obtained. By (A5), 120equalsp•,
the volume fraction
of medium
1. Then
whereF = V(-A)-iV .. In the Hilbertspace
0 •_ 120_<1.
(A6)
with y1 in the inner product,FX1 is a boundedselfadjointoperatorof normlessthan or equalto one. The
integralrepresentation
(A1) is the spectralrepresenta- For s • C•[0, 1], whereC is the complexplane,F(s, 12)
tionof the resolvent
(s+Fx1) -•, where12is the spectral in (A1) is a linearfunctionalfromthe set M of positive
measureof the family of projectionsof FX1, which pro- measuresof mass_< 1 on [0,1] into C. Now M is a
videsa proof of (A1). We then seeexplicitly how 12 compact, convexset of measures.The extreme points
dependson the geometrythroughX•. Another proof (or boundary)of the rangeof valuesof F(s, 12)in C are
attained by one-point (Dirac) measuresaSa,0 < a <
of (A1) exploitsthe Herglotzpropertyof F(s).
13,710
GOLDEN: COMPLEX PERMITTIVITY
1, 0 _• a _• 1, since they are the extreme points of M.
OF SEA ICE
The functionFx(s) is, again,a Herglotzfunctionwhich
Forthesemeasures
(whichareso-called"5functions"),has the representation
F has the form
F(s)-•,
0_•a_• 1,
$--a
0•_a•_ 1.
F1($
) --j•01
d•l(z)
(A7)
8-2:
(A16)
Under (A14), Fx is knownonly to first order
Thecondition
F(s- 1) •_1 determines
theallowed.
regionR0 in the F plane.It is the imageof the triangle
- p2
Fi(s)
/pld+...
in (a• a) space,definedby a +a • 1, 0 _• a '5•I, 9 • a •
$
1, underthe mapping(A7)i wh{chis described
in (1.5).
,
(A17)
The boundsin (15) •e optimal and can be :•:t•ined whichforcesthe zerothmoment/z01
of •1 to be
by laminates of e• and e2 aligned i•erpendicu•arand
parallel to the applied field. The arcsare traced out as
the volume
fraction
I• - p2/pld.
(A18)
varies.
If the volumefractionsp• and p2 - I -p• are fixed Then applying the same procedure as for R• yields the
bound R2, where the boundaries are circular arcs pa-
as well as s, then •0 is fixed, with
rameterizedby (25) and (26). We remark that higher•0 - P•.
(A8)
Then applyinga similarextremalanaJysis
as above
order correlation information can be convenientlyincor-
poratedby iterating (A15), as doneby Golden[1986].
shows
thatthevalues
ofF lieinside
thecircleparametrized
Acknowledgments.
Theauthorwouldliketoihank
by
Cl(z)
pl
(A9)
On the other hand,
Eric Bair for preparing the graphs and for helpful suggestionsin comparing the bounds with the data. In
addition,. the author is grateful to Steve Ackley, Steve
Arcone, Tony Gow, Vicky Lytle, Jin Kong, and Graeme
Milton for helpfuldiscussions
on seaice and generaldielectric theory. Finally, the author expresseshis thanks
E($)- I - el/e*
1-sF($)
= $[1
- F(s)] (A10)
to Art Jordan and Reza Malek-Madani
at ONR for their
is alsoa Herglotzfunctionlike F(s) and hasa represen- encouragementand support of this work. This research
tation
=
wassupported
byONRGrantN000149310141
andNSF
/0a-z)
Grant
DMS-93073244.
$--Z
Expansionof E(s) yields v0 - p•, so that the valuesof
References
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-oc _• z •_ 2.
(A12)
In the e* plane the intersection Rx of these two regions is bounded by two circular arcs correspondingto
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