JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 86, NO. C9, PAGES 8107-8116, SEPTEMBER 20, 1981
Modeling of AnisotropicElectromagnetic
Reflection
K.
M.
GOLDEN
From
AND
Sea Ice
S. F. ACKLEY
U.S. Army Cold RegionsResearchand EngineeringLaboratory,Hanover,New Hampshire03755
The contribution of brine layers to observedreflective anisotropyof sea ice at 100 MHz is quantitativelyassessed,
and a theoreticalexplanationfor observedreflectiveanisotropyis proposedin termsof
anisotropicelectricflux penetrationinto the brine layers.The seaice is assumedto be a stratifieddielectric consistingof pure ice containingellipsoidalconductinginclusions(brine layers)uniformly aligned
with their long axesperpendicularto the preferredcrystallographic
c axis direction.The asymmetrical
geometryof the brine layersis shownto producean anisotropyin the complexdielectricconstantof sea
ice. The contributionof theselayersto the reflectiveanisotropyis examinedwith a numericalmethodof
approximatingthe reflectedpowerof a radar pulseincidenton a slabof seaice. Mixture dielectricpermittivities are calculatedfor both the electricfields parallel and perpendicularto the c axis direction.
Thesepermittivitiesare then usedto calculatepowerreflectioncoefficientsat eachinterfacein the air/sea
ice/seawatersystem.Significantbottomreflection(R = 0.09) occurswhen the polarizationis parallel to
the c axis.However,whenthe polarizationis perpendicularto the c axis,the return is almostcompletely
extinguished(R < 0.003). This extinctionis due primarily to absorptivelossassociatedwith the conductinginclusionsand secondarilyto an impedancematchat the ice/water interfacethat resultsin transmission rather than reflection of the wave.
1.
low the cutofffrequencyfor parallelpolarizationin the guide.
INTRODUCTION
A fundamentalproblem with this theory is the lack of eviRecent studiesof arctic fast ice [Cherepanov,1971; Weeks
and Gow, 1979;Kovacsand Morey, 1978]have shownthat the dencefor tl•e existenceof brine celis•m parallel arrays of
crystallographicc axes(optic axes)of sea ice have not only a plates.We feel that it is morenaturaland predictivelyfruitful
to considerthe brine structureas consistingof individual, fipreferredhorizontal orientation,but also a preferredazimunite,
anisotropic
bodiesratherthancontinuous
plates(Figure
thal direction within the horizontal plane. Campbelland Or3).
ange [1974] and Kovacsand Morey [1978] have established
that the strengthof the return from impulseradar soundings
2. THEORY
(centerfrequencyat 100MHz) of seaice dependsstronglyon
We proposethat the asymmetricalgeometryof the brine indirection. Maximum reflection from the ice/water interface
clusions
causesan anisotropyin the penetration of the imoccurswhen the pulse is polarized so that the electric field
parallelsthe preferredc axis direction.Minimum bottom reflection occurswhen the polarization is perpendicular to the
preferredc axis direction.In the most extremecasesof this
anisotropy,the bottom signalappearsto be almostcompletely
extinguished(Figure 1).
Associatedwith a preferred c axis orientation within the
horizontal plane is anisotropicbrine structure.The ice/water
interfacewhere seaice growth takesplace consistsof dendritic
pingingelectricfield into the brine layers.This anisotropic
penetrationis associatedwith an anisotropyin the effective
complex dielectric constant of the sea ice, which determines
the power returned to the receiver.
To facilitate analysisof the anisotropy,we assumethat a
slabof seaiceof thickness
T is composed
of I horizontallayers
of uniformthickness
d, whichwe take as 10cm. Eachlayer i
consistsof pure ice of complexdielectricconstantel -- ei' +
plates
thatareelongated
perpendicular
tothecaxis
direction
je•"
-- 3.17
+ j 0.013
[Johari
andCharette,
1975]
which
surrounds
uniformly
distributed
andaligned
ellipsoidal
brin.e
inand extendinto the seawater below (Figure 2). As growth occurs, the sea water between the dendritic plates becomes
trapped and makes up the so-calledbrine layers of sea ice
[WeeksandAssur,1967].The brine layersnear the seaice bottom are elongatedin the plane perpendicularto the c axis direction and have their shortestdimensionparallel to the c axis
direction.
Thus maximum
bottom reflection occurs when the
electric field is nermal to the principal brine/ice interface
(normalpolarization),and minimum bottomreflectionoccurs
when the electric field is tangentialto the principal brine/ice
interface (tangential polarization).
An initial impetus for the present research was our dissatisfactionwith the model proposedby Kovacsand Morey
[1978] to explain the reflectiveanisotropy.In this model, the
brine structure near the bottom
of the ice is assumed to be-
have as a parallel plate waveguide,where 100 MHz is well beThis paper is not subjectto U.S. copyright.Publishedin 1981 by
the American GeophysicalUnion.
Paper number IC0767.
clusionsof complexdielectricconstante2= e2'+ je2" = 80 + j
1000 [I/ant, 1976]and relativevolume vt,(0,wherej = x/-2-1.
The geometryof the inclusionsin layer i is specifiedby the
principalaxesa(0, b(0, and c(0, and is characterized
by the
depolarizationfactorsn•,(0 definedby
abc
fo
ø•
n•,
= -•(k:+s)x/(a:
+ds
s)(b
2+s)(c
• +s)
(1)
wherek = a, b, c [Stratton,1941].Thesefactorsdependonly
on the axial ratiosof the ellipsoids.The a axisof an ellipsoid
is horizontaland perpendicularto the preferredcrystalc axis
direction,the b axisparallelsthisdirection,andthe ellipsoidal
c axis is vertical, with a > c >> b near the sea ice bottom. The
sea ice lies above a homogeneous
half-spaceof sea water of
complexdielectricconstantew= 80 + j 773 [Stratton,1941],
andbelowa homogeneous
half-spaceof air of complexdielectric constante• = 1 + j0.
8107
We will nowexaminethedielectric
behaviorof brinelayers
8108
GOLDEN AND ACKLEY: MODEL OF E-M REFLECTION FROM SEA ICE
'.r
..
..
. .-
...t:'
..
..
iZ'ce.-::Bot.'tom.
Fig. 1. Maximum and minimum bottomreflectionsspacedat 90ø [from Kovacsand Morey, 1978].
to understandthe observedreflectiveanisotropy.We will examMe the behaviorof the radar pulseby only consideringdielectricand reflectivepropertiesof the centerfrequencyof the
plane wave pulse.
AnisotropicElectric Flux Penetration
The electricfieldE:z,,insideoneof the ellipsoidalinclusions
in the bottomlayer/, whensubjectedto an initially uniform
electricfield E directedalongoneof the principalaxesk = a,
b, c, is given by [Tinga et al., 1973]
1
Sea
Water
Fig. 2. Dendriticplatesat ice/waterinterface,elongatedper-
E:,--'e,+n,(/)(1
- vt,(/))(e:
- e,)E
(2)
and is independent
of positioninsidethe ellipsoid.This posi-
dependsprimarily on the conditionthat
pendicularto the c axisdirection,extendinginto the seawaterbelow. tional independence
Seawater
trapped
between
theplates
becomes
thebrine
layers
[froma <<3,,whichissatisfied
forthewavelength
3,-- 1.7m at 100
WeeksandGow,1979].
MHz in pure ice.
GOLDEN AND ACKLEY: MODEL OF E-M
REFLECTION FROM SEA ICE
8109
.....
.
............
;:.:.::'.."".
....
',•.}}.-':.;;;'"-•;'•*•::•-::--"
..
-.,
....... .•..•,.:
......
.... .:.,.--;::t..:;.,
.}::.'?'•'
....'.......
. :::•::
.......
;......
...::.:
:..::";;::;':...::;...
,.;::...;...........?:::...?
;;::-.;-'::':*
;'':;'¾.;
':':t*/;'"-.
.........
......
.:,½:':"'
:.
....
;-'......
7* .:,..,:...:.
.
: -.........
-. ,:.
:.-.?.--.
:.:.::-
..::
:f'•::..::.
.....
•,:½
.......
,.. ....
:...... ,:,,.,;;:.
:;,'..
..;:..,.*'
.-.
.......
....--:
:.-,.,;./;;;.'.'
.
............
:..:;:
.:.,...:• .:,.<:
...... ..f ::
-. '-"
..:....-:•. .:
..:,,.-......,-
•-
;•:•:'-'-
...::.:..,.
:-
;.-:,.::•.:
L;'*'.
...::,,.,
...,:%.:,.......-:-......
".-'-:.-:.z:•..
...........
'* .....
-::c:;
.......,;.... ,•.::.:..:•:(.::
./:..
.................
::::;'..'•;;%-.
"::::':"'"'"'"'"":'*•'½?':
F"??-'
.....
"
'*'"
- '-* *" ' ..
ß '½....,..,*
'" '"'
....
•:S:•:..:.....•
-.' ..":'•.".
'"' ::t.f'":.:".
Fig. 3. Photomicrographof a thin sectionof sea ice illustratingbrine pocket shapes[from Weeksand Assur, 1969].
We assumethe fields are of the form E(t) = Eo e-j•' and
E2,(t)= E2,oe-j(•'+s)withno spatialdependence,
whereto=
2w x 108tad/s, and 8 is the phasedifferencebetweenthe
fields.From a polynomialfit of actualarcticseaice data,we
set to(/) -- 0.29. We take the following values as the dimensionsof a bottom layer inclusion:b -- 0.1 mm [Andersonand
Weeks,1958],c = 5b = 0.5 mm [Kovacsand Morey, 1978],and
a = 3 mm [Weeksand Assur, 1967].We evaluate (2) for the
two casesof interest,k = a and k = b, where rta(1)-• 9.67 x
10-3 and no(1)= 8.39 x 10-l.
Case 1: Normal Polarization,E = En (parallel to c axis in
Figure 2). By using the above parameters
IEbl
I/rnl
Dielectric
Behavior
The power flowing from the impinging electromagnetic
wave into a brine layer, measuredin W/m 3, is givenby
1
W=•-geE2,o
2
(5)
where g: is the conductivity of brine. Then the ratio of the
power lossesfor the two polarizationsis
W,= (1/2)g2(4.01
x 10-')2Eo2V
+ (1/2)g,Eo2V/v•,(1)
= 2200
(1/2)g2(5.26x 10-3)2Eo2V
+ (l/2)g,Eo:•V/v•,(1)
(6)
5.26 x 10-3
(3)
with 8 -- 85.1o. This small ratio impliesthat the electricfield is
wherewe assumethat g, is the conductivityof pure ice, V is
the volumeof an ellipsoidalbrine layer, and that the amplitude of the wavefield in ice is equalto Eo. Thus we expect
essentially
excluded
from
thebrine
layer
during
normal
polar-much
more
power
tobeabsorbed
bybottom
seaicefortanization.
Theexclusion
arises
physically
from
thebuildup
of gential
polarization
than
fornormal
polarization.
Significant
charge
(polarization
and
free)
at
the
brine/ice
interface
that
conduction
currents
are
set
up
in
the
brine
layers
during
tancreatesa large depolarizing field inside the inclusion and termMatesthe normal field in the ice matrix [Goldenand A ckley,
1980].
Case 2: TangentialPolarization, E = E, (perpendicularto c
axis in Figure2). In thiscase,the moduliof the fieldssatisfy
legal-4.01
x 10-1
IE,I
(4)
with 8 = 61.8ø. Now, nearly half the applied field strengthhas
penetratedinto the inclusion.This penetrationarisesphysically becausethe field is directed along the brine/ice interface causingadjacent polarization chargesto cancel and free
chargesto travel rather than accumulate.
gentialpolarizationsothat thereis a largeimaginarymixture
dielectricconstant(e," a W,) with high attenuationand little
signalreturnedto the receiver.For normalpolarizationthere
is a small imaginarymixture dielectricconstant(en"a Wn)
with significantsignalreturn.Note that we are considering
primarily conductioneffects,since 100 MHz is below the resonant frequency for water dipole rotations.
In additionto anisotropic
absorption
of the wave,the asymroetryof the brineinclusions
causesanisotropyin the real part
of the effective dielectric constant of the sea ice. Horizontal
elongationof the brine layersperpendicularto the preferredc
axisdirectionrendersthe dipolemomentsof the brine layers
larger for tangentialpolarizationthan for normal polariza-
8110
GOLDENAND ACKLEY:MODELOF E-M REFLECTION
FROM SEAICE
neckingandsubsequent
freezing-out
of the brinelayers,we
linearly decreasethe a/b and c/b ratios from a: b: c -- 30:1: $
in thebottomlayerto a' b'c -- 1' 1'0.5 in thetoplayer.The
vertical dimensionc is decreasedto a value lessthan a -- b at
the top to indicatethat the longerdimensions
of brine inclusions
therehavea tendencyto be horizontal,dueto freez-
ingprocesses.
Themodelweuseisa rough,first-order
approximationdesigned
to reflectonly the grossfeaturesof brine
structure:
markedanisotropy
near the bottomand isotropy
near the top.
The brinevolumevb(0is variedaccording
to polynomial
fitsof datacalculated
fromsalinityandtemperature
profiles
obtainedfrom arcticfastice by W. F. Weeksand A. J. Gow
(personalcommunication,1978). Data are consideredfrom
CapeKrusenstern,
Barrow(ChukchiSea),andHarrisonBay
(Figures5-7). Brinevolumeswerecalculated
by usingthe
equationof Frankenstein
andGarner[givenin WeeksandAssur, 1967],
O, +0.532
$,[-49.185
b.
vb(O
--i-•
(8)
whereS, is thesalinityof layeri expressed
in pansperthousands,
and8,isthetemperature
of layeri expressed
in degrees
Celsius.
Fig.4. Brinelayers(topview)nearthebottomof seaice(a) begin
to 'neck'withdecreasing
temperature
furtherup in the ice sheet(s)
andfreezeoutintocylinders
(c) andellipiticalcylinders
(d) [fromAn-
Salinity(%.)
02•4
6
8
I0
Temperature(•½)
12
-I0
-8
-6
-4
-2
0
L1' I ' ' ' ' ' 'J
dersonand Weeks,1958].
zo
tion. Thus, the real part of the dielectricconstantof oriented
sea ice is higher for tangential polarization than for normal
polarization[Goldenand Ackley, 1980].
3.
To illustrate
NUMERICAL
MODEL
how the above effects determine
the nature of
the reflectiveanisotropy,we calculatepower reflectioncoefficients(powerreceivedfrom a giveninterface/incidentpower)
for each of the n -- I + 1 interfaces of the aft/ice/sea water
i OOL-•---L--L--L--L--L--L•.--L•
system,and for each polarization.
Layer DielectricConstants
Brine Volume
The effective complex dielectric constantof layer i when
subjectedto an initially uniform electricfield directedalong
the k axesof the ellipsoidsis [Tinga et al., 1973]
e•,(0= •, +
-
0
[
•
ac)
(7)
The formula is evaluatedby usingthe appropriatevaluesof •l
and •2, alreadyquoted.(See Pant [1976]for a discussionof the
applicabilityof (7) to time varyingfields.)The brinevolume
0.050
ß;
'
I,
'
0.1O0
[
I
0.150
'
I
0.200
'
[
/ u.
IZ'
=('.'Xllt•)Z•-17.70xl
4
+(1.70x10-'iZ
+4.24x1F
ß
and depolarizationfactorsare varied for eachlayer to account
for the change in the physical propertiesof the sea ice with
depth.
Immediately above the ice/water interface, the ice is relatively warm and containsa high concentrationof elongated
brine layers(Figure 4a). Further up the.temperaturedecreases
and, consequently,
the brine layersbeginto 'neck' (Figure 4b)
and changeinto rows of closelyspacedcylinders(Figure 4c)
and finally into elliptical cylinders(Figure 4d) lAndcroonand
Weeks, 1958]. Above the transition zone the brine structure Fig. 5. Temperature,salinity, and brine volume profilesfor Cape
tends to have a more random
orientation.
To simulate
the
Krusenstern.
GOLDEN AND ACKLEY: MODEL OF E-M REFLECTION FROM SEA ICE
Interfacial PowerReflectionCoefficients
8111
Salinity(%0)
We will now calculate interfacial power reflection coeffi-
0 II lllJl
cientsR•(O (powerreflectedby interface//powerincidenton
20
interfaceO at each interfacei for the two polarizationsof the
incident wave [Ward, 1967]. The characteristicbulk impedanceZ•,(Oof layer i for polarizationalongthe k axesof the ellipsoidsis defined by
Zk(0
---¾•,(0
Temperature
(*C)
.
• I
I
'
40
(9)
where¾•,(0is the propagationconstantof layer i definedby
•(0 -- •/•,o%'(0
+ jt•0 gJO
(•0)
g•,(Ois the conductivityof layer i for polarizationk, and • =
4•r x 10-7 henry/m is the permeabilityof free space.Sincewe
Salinity(%0)
2 4 6
8 I0 12 -20
-16
-12
-8
-4
LiI'I 'l'll ,'.
20
•1 Ill _
Temperature
(øC)
0
,
Brine Volume
0
--
40
I .o.
•oo.
aoa3o
•_.•
ui,(Z)
=(3.30x10-w)Z'-
--
.•60
•
--
80
O.4O
40
-
..
_
-
,oo,2o-[, :
•
140
160--,I
, , i , I ' I , I , ! , I I I I.
0.050
Brine Volume
0.100
ß
•)
I
-
øø
0.150
0.200
0 'oI i I t I l
O/
VbiZ)
=(l'17xlO'7)Z$(l'82xlO'qZ2
l
aO
ß+
(7.23x10+
1.74x10
-2
/
i
I
I
I
i
!
,
Fig. 7. Temperature,salinity,and brine volumeprofilesfor Harrison
Bay.
•o
assumethat the lossesin seaice at 100 MHz are primarily due
to conductioneffects,i.e., ek" (0 ----gk(0/eoW,(11) becomes
¾k(O
-- •oJeoP•k(O
(11)
IOO
The complexamplitudereflectioncoefficientre(Oat an interface i dependson the mismatchof the bulk impedanceZ/•(i -
12o
Zk(O of layer i below the interface:
1) of layeri- 1 abovetheinterfaceandthebulkimpedance
r•(O =
160
,
I
I
I
Fig. 6. Temperature,
salinityandbrinevolumeprofilesfor Barrow
(ChukchiSea).
z,,(•- i)- z•(o
zjii)+ z,,(O
(12)
On substitutionof (10) and (12) this becomes
rk(O---e•x/-•+
x/e•(i
- 1)
(13)
8112
GOLDEN AND ACKLEY: MODEL OF E-M
REFLECTION
FROM SEA ICE
The interfacialpowerreflectioncoefficientRff(0 is obtained plane wave traveling one way through a lossylayer i of thicknessd is decreasedby a factor e-"k(%where a•,(0 is defined by
yon Hippel [1954] as
from the complex amplitude coefficientthrough
Rff(O -- rk(0* rn(O
(14)
where the asteriskdenotesthe complex conjugate.Note that
this expressiondependson frequencyonly through the dis-
--•--(41+ (•"(0/•'(0)
2
(19)
persivenatureof the dielectricconstants
of the constituents
of
the system,seawater, ice, brine, and air.
where X is the free spacewavelength.One-way power, there-
Bulk PowerReflectionCoefficients
power transmissioncoefficientfor layer i. We de•e the attenuated power reflectioncoefficientby
fore, decreases
by a factor(e-•(ø• 2, whichmay be termeda
Supposeour ice sheetconsists
of onelossless
layer of thicknessd. Let a plane wave pulseof temporaldurationAt traveling throughair be normallyincidenton interface1. Further
= H e-4a•(øR&•(O
(20)
assumethat At << d/V•, where V• is the wave velocity in ice.
The primarypulsereturnedto the air from interface2 hasthe
followingbulk power reflectioncoefficient(power returnedto
air by the interface/powerincidenton ice slab,without atten-
for m • 2, 3, ..., n, and
uation):
form=
-- [1 -
x
R•(m) = Rff(m)
(21)
1.
(15) Beam Spread
x [l -
To obta• realistic reflection coe•cients, we est•ate
A secondarypulsereflectedback into the air by interface2
beam
sprea•g effectswith the F•is transmissionformffia [Kraus,
1953],which givesthe ratio of the power • the load of a receiv•g antennaP, to the power radiated by a transmitt•g antenna P, • terms of the distance between the antennas r, the
wavelength•, and the d•ectivities of the transmitt•g and receiv•g antenna, D, and D,,
has the bulk power reflectioncoefficient
•(2) = [• - •'(•)] x •'(2) x •'(•) x •'(2) x [•- •'(•)]
(16)
There are infinitely many of these reflectedpulses.In our
modelwe neglectmultiplereflectioneffectsbeyondfirstorder
and consideronly primary reflections,so that the bulk power
reflectioncoefficientof interfacern for polarizationk is given
P'=D•,
(22)
by
The directivity can be expressedas
i=l
fff(O,
•) •
for rn -- 2, 3, ..., n, and
R•fi(rn)-- Rff(m)
wheref(O,•) is the norma•ed powerpatternand •
(18)
AttenuatedPowerReflectionCoefficients
The attenuatedpowerreflectioncoefficient
R•(O for interfacei (powerreturnedto air by interface//powerincidenton
interface0 is calculated
fromRff(0 by includinglosses
from
travel throughsea ice. The amplitudeof a monochromatic
Sea
Tce
Rs (i).-R'T(i)
4e
D=
RS(i)
2.
./4
= 6.83
fafsmOao
Rs(n)
R
'r(i)/
i
.
rt
Water
is a dif-
ferential elementof solid angle. For s•p•city we assumethat
the transmitt•g/receiv•g ante•a • our study radiates energy unifo•ly over a cone subtend•g a total polar angle of
90ø. The directivity D = D, = D, then becomes
for m -- 1. Figure 8 givesa schematicof the situation.
Air
(23)
Time•"
Fig. 8. Schematicdiagramfor bulk powerreflectioncoefficientcalculation[adaptedfrom Petrin, 1979].
(24)
GOLDEN AND ACKLEY:
MODEL OF E-M
The final power reflectioncoefficientsRk(0 are obtained by
multiplyingeachRk'•(0 by the beamspreadfactorS(0
a(0 -- s(0 a/(0
(25)
where
s(0 =
m I
4•r
X2hi
REFLECTION
FROM
SEA ICE
8113
and h is the distance from the antenna to interface i. Since re-
flectionmeasurementsare often donewith the antennaresting
on the ice (but not coupledto the ice), we normalizethe S(0
so that S(1) = 1. Thus, there is no spreadingof the beam for
the surfacereflection,although it spreadsout into the ice.
The above-describednumerical method for obtaining reflection profiles was carried out on the computer at Dartmouth College in Hanover, New Hampshire.
(26)
E 40
i
20
4080-
•...•_.•
I:'n
0.0:•0
-
0.060
Power Received/Power
0.090
O.120
60
Transmitted
Fig. 9a
•
o
80
20
IOO
40-
120 -
60
140 -
8t3-
160
Et
_
0
0.025
0.050
Power Received/Power
IO0
0.075
O. I00
Transmitted
Fig. 9c
120
140 --
ti
Et
160
0
I
0.025
Power
0.050
Received/Power
0.075
,
O0
Transmitted
Fig. 9b
Fig. 9. Profilesof the R•,(0for (a) CapeKrusenstern,
(b) Barrow,and(c) HarrisonBay.Et denotestangentialpolarization, En denotesnormal polarization.
8114
GOLDEN AND ACKLEY: MODEL OF E-M
0
Dielectric
2
i
'
I
'
I
Dielectric
Constant
:5
,
I
REFLECTION FROM SEA ICE
4
5
!
'
0
•
I
20
'
I
40
'
I
Constant
60
i
I
80
'
I
i00
i
I
120
'
I
I
•o-•
20-
I
I
40-
4o-I
i
i
I
•
I
'"
i
ioo--I
i00-
i
J
120-
120-
140-
140
160-
160
i
Fig. 10a
I
,
I
,
I
i
I
I
I
,
Fig. 10b
Fig. 10. Mixturedielectric
constant
profiles
at HarrisonBayfor(a) normalpolarization
and(b) tangential
polarization.
4.
of thetransmitted
signalis returnedto thereceiverby the ice/
waterinterfacefor normalpolarization,whilethe tangentially
RESULTS
AnisotropicBottomReflections
polarizedsignalisvirtuallyextinguished
(R,(n)< 0.003).Very
Our model exhibits marked anisotropyof bottom refleclittle,if any,signalis returnedfromwithintheicesheet.Since
tions from sea ice at 100 MHz. Figure 9 givesthe calculated
we haveassumed
that the top layer is isotropicin the horizonreflectionprofilesfor the threesites.On the average,about 9%
tal plane,surfacereflections
are isotropic.
Power
Transmission
0.2
I
20
0.4
,
I
Coefficient
0.6
,
I
0.8
I
I
1.0
'
I
AnisotropicComplexDielectricConstants
In our model, reflectiveanisotropycan only existwith anisotropicdielectric constants.Figure 10 showsthe calculated
dielectricconstantprofilesfor HarrisonBay.The anisotropyis
mostpronouncedin the lower regionswherethe brine volume
and a/b ratio are high (i.e., where the brine layersare numerous and well defined). Figure 11 showsthe one-way power
60
transmission
coefficients
e-2da(o
for HarrisonBay.The waveis
highly attenuatedin the lower regionsduring tangentialpolarizationbecauseof the very largecontributionof e", as sug-
gestedpreviously.The ratio of the imaginarydielectricconstantsfor the bottom layer at Harrison Bay e/'/en" -- 47.3/
0.027 = 1800 is roughly in agreementwith the ratio of the
IO0
power lossesin (6), the calculationof which wasbasedon normal exclusionand tangential penetrationof electricflux.
We seefrom Figure 10b that the real part of the mixture
120
dielectricconstantof the bottomlayer for tangentialpolarization at Harrison Bay has become quite large (109), in fact
larger than that of brine itself (80). The finite extent of the
140
brine layerscausesthem to behaveas macroscopic
dipoles.
Free and polarizationchargecan build up at the endsof the
brine layers,renderingthe seaice highly polarizable,more so
160
than pure brine with no interfaces.
The combinationof the large real and imaginaryparts of
the
mixture dielectric constantsfor tangential polarization
Fig. 11. One-waypowertransmission
coefficient
profilesat Harrison
Bay.Lowtransmissions
throughthelayersindicatehighattenuation. causesthe bulk impedanceof the bottom seaice layer to be
GOLDEN AND ACKLEY.' MODEL OF E-M REFLECTION FROM SEA ICE
2O
2O
4O
40-
60-
60-
80-
80-
8115
.
IOO
I00-
120
120 -
140
140 -
160
ß
I
0
I
0.025
0.050
I
I
--
160
-I•.•.......•..
•
!
0.075
0.100
PowerReceived/Power
Transmitted
0
I
[•
I
]
-
.
0.0•5
0.050
0.075
0.1•:•
PowerReceived/Power
Transmitted
Fig. 12a
Fig. 12b
Fig. 12. Profilesof the/•k(0 at HarrisonBayfor (a) a'b'c = 20:1'5 in thebottomlayerand(b) a:b'c = 50' 1'$ in the
bottom layer.
similarto thatof thewaterbelow.Consequently,
afterdown- thisbumpin explaining
apparent
travel-time
isotropy
[Kovacs
ward propagation,
the remainingenergyof the tangentially andMorey,1979].We leavethisaspectof the reflectiveanipolarizedwavetendsto betransmitted
ratherthanreflected
at sotropyopento moreexhaustive
experimentation.
the ice/waterinterface.The normallypolarizedwaveencoun-
tersa largeimpedance
mismatch
andis reflected.
Sensitivityof Parameters
Figure 12givesreflectionprofilesfor bottomlayerbrinege-
5. DISCUSSION
We have assumed extreme
sea ice. Effective
axial
order in the microstructure
ratios and orientations
of brine
of
in-
ometrysPi•cified
by (1) a:b:c -- 20:1:5 and(2) a:b:c --
clusionsat a given depth or region within the sea ice are actually distributed about some mean. However, the standard
tangential electricfield so that attenuation decreases.An in- deviations from preferred bottom c axis azimuth given by
creasein a allowsgreaterpenetrationof a tangentialelectric Weeksand Gow [1979] are between5ø and 15ø, indicating a
field so that attenuation increases. An increase in c has a simirelatively high degreeof ordering. Our purposewas to show
lar effect to a decrease in a.
that if one views brine inclusionsas finite, anisotropicconducting bodiesthat are small comparedto wavelength,then
Internal Reflections
the observedreflectiveanisotropyfollows naturally from the
It is very interestingto note the appearanceof an internal behavior of the electromagneticfield around and in the bodreflection
located
at a depthof about2T/3 fortangential
po- ies.To illustratebettermechanismsdeterminingreflectiveanilarization(Figures9a, 9c, and 12a).In the modelthisbump in sotropy,we have taken a very simplifiedmodel of seaice. Exthe reflectionprofilearisesfrom a superposition
of two effects. amination of these limiting casesmay render the observed
Large brine volume and axial ratio gradients combine to data more understandable.
For instance, our analysis focused on a limiting case of
createfairly significantinterfacialreflectioncoefficientswhich
increasewith depth. When the ice slab is 1ossynear the ice/ brine structure, namely brine layers (Figures 4a and 4b).
water interface,the bottom reflectioncoefficientsare greatly Throughout much of the ice sheet,however,brine structure
decreaseddueto attenuation,leavinga bumphigherin the re- takesthe form of verticalcigar-shapedinclusions(Figures4c
50:1:5.
A decrease in a allows more effective exclusion of a
flection
profile.Therehasbeenspeculation
ontheexistence
of and4d).Nevertheless,
therewill stillbeanisotropy
in electric
8116
GOLDEN AND ACKLEY: MODEL OF E-M
flux penetration,as long as the distancebetweenadjacentinclusionsis lessthan the distancebetweenadjacentrowsof inclusions.The reflective anisotropy associatedwith these ordered'cigars'is weakerthan that associated
with brine layers.
6.
CONCLUSIONS
The reflectiveanisotropyobservedin sea ice can be explainedby consideration
of the geometricalasymmetries
inherent in the brine structureof sea ice with a high azimuthal
order of c axis orientation. Anisotropic electric flux penetration into the brine layersdeterminesanisotropyin attentuation of the wave. Axial ratios of 30:1: 5 for brine layers in
the bottomlayer of seaice givea reasonablelevelof powerre-
REFLECTION
FROM SEA ICE
REFERENCES
Anderson,D. L., and W. F. Weeks, A theoreticalanalysisof sea ice
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Campbell,K. J., and A. S. Orange,The electricalanisotropyof seaice
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Cherepanov,N. V., Spatial arrangementof sea ice crystal structure,
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Golden, K. M., and S. F. Ackley, Modeling of anisotropicelectromagnetic reflectionfrom sea ice, Rep. 80-23, Cold Reg. Res. and
Eng. Lab., U.S. Army Corps of Eng., Hanover, N.H., 1980.
Johari, G. P., and P. A. Charette,The permittivity and attenuationin
polycrystallineand single-crystalice Ih at 35 and 60 MHz, J. Glaciol., 14, 293-303, 1975.
Kovacs,A., and R. M. Morey, Radar anisotropyof seaice due to preferred azimuthal orientation of the horizontal c-axesof ice crystals,
J. Geophys.Res., 83, 6037-6046, 1978.
Kovacs,A., and R. M. Morey, Anisotropicpropertiesof seaice in the
50-150 MHz range, J. Geophys.Res.,84, 5749-5759, 1979.
turned to the ice surface in the direction normal to the brine
Kraus, J.P., Electromagnetics,
McGraw-Hill, New York, 1953.
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to the
layer stratified media, Master's thesis,Univ. of New Hampshire,
c axis).
Variation of brine volume and geometry profiles during
model calculationswas found to changesubstantiallythe re-
Durham, 1979.
Stratton, J. A., ElectromagneticTheory, McGraw-Hill, New York,
1941.
Tinga, W. R., W. A. G. Voss, and D. F. Blossey,Generalized approach to multiphasedielectricmixture theory, J. Appl. Phys.,44,
flection profiles.If further experimentalwork relating ob3897-3902, 1973.
servedreflectionprofilesto seaice microstructurewere done,
radar soundingof seaice couldbe usedas a nondestructive Vant, M. R., A combinedempirical and theoreticalstudy of the diepropertiesof seaice over the frequencyrange 100 MHz to 40
'spectroscopic'
tool givinginformationon the microstructure lectric
GHz, Ph.D. thesis, Carleton Univ., Ottawa, 1976.
variationsthat ultimately control the strengthand other im- yon Hippel, A. R., Dielectricsand Waves,MIT Press, Cambridge,
portantproperties
of seaice.At present,microstructure
inforMass., 1954.
theoryfor geophysical
applicationsin
mation can only be obtainedby coringa smallsampleof ice. Ward, S. H., Electromagnetic
Mining Geophysics,
Societyof ExplorationGeophysicists,
Tulsa,
A methodologysuchas radar soundingthat couldextendmiOkla. 1967.
crostructureinformation to line profile and area measure- Weeks, W. F., and A. Assur, The MechanicalPropertiesof Sea Ice,
ments would be valuable in future scientificand engineering
CRREL Monogr., II-C3, AD, Cold Reg. Res. and Eng. Lab., U.S.
investigationsof sea ice.
Army Corps of Eng., Hanover, N.H., 1967.
Weeks, W. F., and A. Assur, Fracture of lake and sea ice, Res. Rep.
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Hanover, N.H., 1969.
Acknowledgments.
The authors would like to thank Wilford
Weeksfor supplyingsalinityand temperaturedata and for reviewing Weeks,W. F., and A. J. Gow, Crystalalignmentsin the fast ice of arctic Alaska, Rep. 79-22 Cold Reg. Res. and Eng. Lab., U.S. Army
the manuscript,AnthonyGow for supplyingdata and generalinfor-
mation on seaice structure,and StevenArcone for review of the man-
uscriptandconsultations
on electromagnetic
wavepropagation.
This
researchwassponsored
by National ScienceFoundationgrantDPP77-24528and this supportis alsogratefullyacknowledged.
Corps of Eng., Hanover, N.H.,
1979.
(ReceivedJanuary 12, 1981;
acceptedApril 27, 1981.)