Effects of Birefringence Within Ice Sheets on Obliquely Propagating Radio Waves

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 47, NO. 5, MAY 2009 1429

Effects of Birefringence Within Ice Sheets onObliquely Propagating Radio Waves

Kenichi Matsuoka, Larry Wilen, Shawn P. Hurley, and Charles F. Raymond

Abstract—In this paper, effects of birefringence on radio wavesobliquely propagating though polar ice sheets are examined tofacilitate interpretations of bistatic and side-looking radar data.A formalism applicable for arbitrary radar configurations is de-veloped to predict the returned power from within and beneaththe ice sheets that have arbitrary alignments of ice crystals (icefabrics). We applied this formalism to a range of ice fabrics foundin ice cores and assessed the effects of birefringence in terms ofray-path configurations, ice fabrics, and radar frequency. Pre-dicted frequency dependence of the bed return power replicatesprominent features observed at Greenland NGRIP ice-core site.Results show that birefringence in ice of 1 km or more thicknesswith strong (weak) fabric can reduce the power returned from thebed 2 dB or more at frequencies higher than 200 MHz (20 MHz)as compared to isotropic ice. This suggests that quantitative in-terpretation of the power returned from the bed requires carefulassessment of birefringence almost everywhere over the ice sheets.Application of this formalism also suggests a radar-frequencyrange usable for attenuation measurements, possible effects offabric on synthetic aperture radar processing, and a feasibility ofremote sensing of ice fabric.

Index Terms—Arctic regions, birefringence, ice, radarpolarimetry.

I. INTRODUCTION

RADIO-WAVE remote sensing has revealed spatial varia-tions of bed-echo intensities in polar ice sheets relevant

to the motion of ice over the base. The location of brighterand dimmer bed echoes can identify regional differences in bedcharacters [6], [8], [20]. However, mapping of the bed condi-tions and quantitative interpretation of the bed return powerrequires accurate evaluations of other factors that may affectthe bed-echo intensity in addition to the bed conditions. Twosignificant factors that modify bed-echo intensity are dielectricattenuation [18], [29], [36] and birefringence [12], [23]. Dielec-tric attenuation within ice is affected by both ice temperatureand chemical constituents [9], [29]. Birefringence has been

Manuscript received February 13, 2008; revised June 29, 2008. First pub-lished November 18, 2008; current version published April 24, 2009. Thework of K. Matsuoka and C. F. Raymond was supported by the NationalScience Foundation under Grant ANT-0440847. The work of L. Wilen wassupported by the National Science Foundation under Grants OPP-0135989 andANT-0439805.

K. Matsuoka and C. F. Raymond are with the Department of Earth andSpace Sciences, University of Washington, Seattle, WA 98195 USA (e-mail:[email protected]).

L. Wilen is with the Department of Physics and Astronomy, Ohio University,Athens, OH 45701 USA.

S. P. Hurley was with the Department of Physics and Astronomy, OhioUniversity, Athens, OH 45701 USA. He is now with Liquid Crystal Institute,Kent State University, Kent, OH 44242 USA.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TGRS.2008.2005201

Fig. 1. Variations of the echo intensity for monostatic copolarized antennaconfigurations (Fig. 2). c3 is assumed vertical so that c1 and c2 are in thehorizontal plane (Section II-A). The ordinate shows the copolarized echointensity relative to the echo intensity when the phase difference δ betweenthe two principal wave components equals 0◦ (8), so that the intensity ispolarization-independent as found in isotropic ice. (a) 90◦-interval periodic(45◦ symmetric) variations in the echo intensity caused by birefringence.The echo intensities are illustrated for five δ with 0.2π intervals. (b) 90◦-interval periodic variations shown in the panel a are modified if attenuationand/or scattering is polarization-dependent so amplitudes of two principal wavecomponents are different (Section V-A). Three curves are drawn for amplituderatios of the two principal wave components of 0.85, 0.90, and 0.95.

detected by radar in both land ice and shelf ice and is known tobe caused by anisotropic alignments of ice crystals (ice fabrics)[13], [19], [23], [27], [57]. Birefringence has been studiedextensively for the case when a principal axis of the fabric isaligned with the radio-wave propagation path (Fig. 1). Thisconfiguration holds for monostatic radar measurements usingtransmitter and receiver close to each other, yielding verticalpropagation path through the ice, since in many cases, the fabrichas a near-vertical principal axis. However, little attention hasbeen paid to effects from birefringence when a principal axisof the fabric is not aligned with the radio-wave propagationpath. This configuration happens in nonvertical ice fabrics, inbistatic-radar measurements (Fig. 2), and in side-looking radarmeasurements. Recently, the bistatic-radar configurations havebeen used to measure attenuation [40], [56]. Side-looking radaris first-time applied to map bed topography and conditions overa swath in Greenland [1], [41] and has a large potential to mapbed conditions with finer lateral resolutions than conventionalgrid surveys using nadir-looking radar.

Here, we examine birefringence caused by ice fabric withan emphasis on bistatic-radar measurements. Birefringence forobliquely propagating radio wave is formulated with a stan-dard wave-propagation theory adapted for ice-sheet sounding

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1430 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 47, NO. 5, MAY 2009

Fig. 2. Geometry of bistatic-radar measurements shown in (a) oblique viewand (b and c) top views. (a) The transmitting (Tx) and receiving (Rx) antennasare separated by 2s. Thickness of the ice below the midpoint (closed circle)is d. These give the propagation polar angle θw = tan−1(s/d). (b) Cross-firecopolarized antenna configuration. Angle ϕw shows the azimuth of the lineconnecting Tx and Rx measured from x1-axis. Antenna (polarization) azimuthϕa is also measured from x1-axis ϕa = 90◦ + ϕw . (c) End-fire copolarizedantenna configuration. ϕa = ϕw .

(Sections II and III). This model is applied to a range of icefabrics found in polar ice cores to assess effects of birefringencein terms of ray-path configurations, ice fabrics, and radar fre-quency (Section IV). The results indicate that quantitative inter-pretation of the bed return power requires careful assessmentsof ice-fabric-origin birefringence almost everywhere over theice sheets. Other factors than birefringence that modify the bedreturn power were also examined (Section V). Using the modelresults, we discuss radar settings best suited for characterizationof the bed and ice fabrics (Section VI).

II. DIELECTRIC ANISOTROPY IN ICE SHEETS

Mechanical and dielectric anisotropy of ice-sheet ice resultsfrom characteristic (preferred) patterns of ice-crystal orienta-tion (e.g., [2], [3], and [7]). Anisotropic fabrics are producedduring deformation, which, in turn, influence further deforma-tion. Crystal-alignment patterns (ice fabrics) in the ice sheetsare therefore variable, depending on strain history.

During deformation, the crystal c-axes rotate away from theextension axis and cluster along the compression axis. Single-pole patterns generally occur under uniaxial compression orsimple shear, whereas girdle patterns generally occur underuniaxial extension. Between these two extremes are a morecommon elongated single-pole (or weak vertical girdle) pat-terns. More complicated fabrics are found in ice shelves, deepice, and ice caps close to the coast, where ice is typicallywarmer and strained in a more complicated manner. Sincesingle ice crystals are dielectrically anisotropic [16], [34], thebulk ice on the scale of radar wavelength (deci to tens ofmeters) involving many crystals (grain size approximately in

millimeters) can be birefringent depending on the statistics ofcrystal orientation and volume.

A. Bulk Dielectric Properties Derived From Ice-Core Data

Crystal orientation and volume are obtained by thin-section(submillimeter thick) analysis of ice cores (e.g., [4], [11],[47], and [51]). We follow a standard method described in[30] to represent the statistics of c-axes orientation relevant torefractive index. A symmetric second-rank fabric tensor C canbe defined by

C =1V

N∑j=1

υ(j)c(j) ⊗ c(j) (1)

where ⊗ denote the outer product. υ(j) and c(j) are volume andc-axis orientation vector of the jth ice crystal; V (=

∑Nj=1 v(j))

is the volume of the total crystal ensemble. Real principal val-ues of C are denoted as χ1, χ2, and χ3 in ascending order withcorresponding orthogonal principal directions c1, c2, and c3,respectively. The principal values have the property χ1 + χ2 +χ3 = 1. These principal values can be used to categorize icefabric patterns. For example, χ1 = χ2 = χ3 = 1/3 for randomice fabric; χ1 = χ2 = 0 and χ3 = 1 for a perfect single-polefabric; χ1 = 0 and χ2 = χ3 = 0.5 for a perfect girdle fabric;and χ1 < χ2 � χ3 for more general elongated fabrics in theplane that includes c2 and c3.

Following [24], we define tensor m related to refractiveindex as

m = mr⊥cI + ΔmC (2)

where I is the unit isotropic tensor and Δm is the single-crystal birefringence defined by mr

‖c − mr⊥c, where mr

⊥c

(∼√

3.152) and mr‖c(∼

√3.189) are the real parts of the

refraction index measured perpendicular to and along c-axis.These values are the mean of laboratory-measured values atthree frequencies at −21 ◦C [16], [34]. mr

‖c is about 1.1% larger

than mr⊥c, yielding Δm = 1.04 × 10−3. m⊥c and m‖c depend

on ice temperature, but Δm does not depend significantlyon radio-wave frequency in the HF–UHF ranges or on icetemperature for the range of temperature found in ice-sheetinteriors. Equation (2) is valid when each ice grain is reasonablyclose to being spherical [24]. The principal directions of mare aligned with those of C. Principal values are mi = (1 −χi)mr

⊥c + χimr‖c (i = 1, 2, 3). We define these to be the prin-

cipal refractive indexes in the standard terminology (e.g., [39,p. 237]). For example, m1 =m2 =m3 =mr

⊥c+1/3Δm=2/3mr

⊥c + 1/3mr‖c for random (isotropic) fabric, m1 = m2 =

mr⊥c and m3 = mr

‖c for a perfect single-pole fabric, and m1 =m3 = (mr

⊥c + mr‖c)/2 and m2 = mr

⊥c for a vertical-girdle fab-ric. Note that mi for the single pole matches a single crystal asrequired.

Propagation oblique to principal directions of C or equiva-lently m is standardly analyzed on the basis of the represen-tation quadric of the relative dielectric impermeability B. Bis defined to have the same principal directions as C and mwith principal values m−2

1 , m−22 , and m−2

3 that are reciprocals

MATSUOKA et al.: EFFECTS OF BIREFRINGENCE WITHIN ICE SHEETS 1431

TABLE ISTATISTICS OF ICE FABRICS MEASURED WITH POLAR ICE CORES

of the dielectric permittivity. In a principal coordinate systemm aligned with c1, c2, and c3, the representation quadric isgiven by

xc21

m21

+xc2

2

m22

+xc2

3

m23

= 1 (3)

where xc1, xc

2, and xc3 are coordinates along c1, c2, and c3. The

representation quadric is an ellipsoid known as the indicatrix.The three principal radii of the indicatrix are the principalindexes of refraction m1, m2, and m3. Equations (1)–(3) fullydefine the indicatrix based on a measured fabric.

B. Ice Fabric Measured by Ice Cores

Table I summarizes fabric characteristics determined in vari-ous polar ice cores from thin-section data. θ represents polar an-gle of c3 measured from the core axis. Because the azimuths ofice cores (or of the thin sections relative to the core) are almostalways unavailable, we assumed that c1 and c2 are consistentthroughout the sampled depth range. This implicitly assumesthat principal orientations of the strain configuration have notvaried at the ice-core site or upstream from it. We assume aconstant volume [υ(j) = V/N in (1)] for crystals to derive prin-cipal values at each depth. Under these assumptions, all valuesin Table I are derived as algebraic means weighted by a normal-ized number density of sampling depths. Thus, Table I does notgive complete information about the ice fabrics. Nevertheless,

ice cores are the most direct sources of detailed informationabout ice fabrics, and therefore, we rely on them to pin down alikely range for fabrics used in our calculations (Section IV).

GRIP, GISP2, and Dome Fuji ice cores are from centralparts of the polar ice sheets, where horizontal shear strain ispresumably very small. χ2 − χ1 and θ from these sites rangebetween 0.03 and 0.05 and between 3◦ and 9◦, respectively. Inaddition, χ3 is roughly 0.7. Antarctic Dome C ice core shows acomparable range in the top 1500 m [52]. Data from the deepest8% of the ice thickness at GRIP and GISP2 show quite differentfeatures from those at shallower depths. Excluding these datafrom the deepest 8%, χ2 − χ1 and θ at GRIP are 0.031 and 4.8◦,respectively, and χ2 − χ1 and θ at GISP2 are 0.025 and 6.0◦,respectively. Siple and Taylor Dome ice cores from the RossEmbayment in Antarctica were drilled in the vicinity of the ice-flow divide which is more or less ridge rather than symmetricdome. Due to this topography and large accumulation relativeto the ice thickness, ice cores show larger χ2 − χ1 and θ thaninland ice domes.

As ice moves downstream, it experiences significant hori-zontal extension and/or compression along the flow path andmay develop larger χ2 − χ1. Mizuho ice core was retrievedabout 750 km away from Dome Fuji, and ice at the core siteflows convergently into a fast-flowing glacier. χ2 − χ1 andθ from Mizuho are 0.18 and 9.0◦, respectively. These largevalues are obtained from the top 36% of the ice. Similar fabricpatterns (vertical girdle) to that at Mizuho are also found in


more inland regions, Vostok [26] and NGRIP [51]. At NGRIP,vertical-girdle fabrics appear at approximately 36% of thesampled ice thickness, while random or strong single-polefabrics appear at other depths. Therefore, depth mean χ2−χ1 ofNGRIP shown in Table I is similar to that from inland domes.

In many cases, c3 is aligned close to vertical. Therefore,a vertically propagating radio wave in the monostatic radarconfiguration is affected by birefringence ∼Δm(χ2 − χ1),which indicates a range of 3%–20% of Δm. In bistatic-radarmeasurements, apparent anisotropy in the refractive index canpotentially be much larger. A range of fabrics reviewed herewill be used to examine realistic birefringent signatures inSection IV.

III. OBLIQUE RADIO-WAVE PROPAGATION

THROUGH ICE SHEETS

Oblique radio-wave propagation can be described in a wayanalogous to the propagation of light through a single biaxialcrystal, as outlined in most mineral optics and electromagneticstextbooks (e.g., [37] and [46]). However, with the best of ourknowledge, a detail prescription dedicated for realistic condi-tions in ice is not given yet. Here, we briefly describe a casesimplified slightly from the general formalism of biaxial opticsin anticipation of applications to ice, where the anisotropy ofmacroscopic refraction index is quite small. In general, onemust consider differences between the normal of the wavefrontand the propagation direction, but these will be assumed to benegligible [28].

Radio-wave propagating in an arbitrary direction with arbi-trary polarization can be decomposed into two orthogonallypolarized components. These components travel at two speedsdefined by the two distinct refractive indexes determined by ref-erence to a corresponding central section ellipse of the indica-trix. The central section ellipse is formed by a plane through thecenter of the indicatrix perpendicular to the wavefront normal.The polarization of these wave components are aligned withthe orientations p

1and p

2of the principal radii of the central

section ellipse, and the corresponding refractive indexes m1 andm2 are given by the principal radii. Because of the differentwave speeds of the two wave components, the polarizationrotates along the propagation path. The primary calculationalproblem is to find the total of the summed incremental rota-tions along the propagation path between the linearly polarizedtransmitting and receiving antennas.

We modeled the propagation process with separate coordi-nates for the downward (single primed x′

1−x′2−x′

3) and upward(double primed x′′

1−x′′2−x′′

3) propagation paths that x′3 and

x′′3 align with the corresponding propagation directions. These

frames are defined in relation to the space-fixed x1−x2−x3 co-ordinates (Appendix I). As follows, we first discuss propagationin a single layer under the single-primed coordinates.

Orientations p′1

and p′2

of the local principal radii of the cen-tral section ellipse of the indicatrix can be expressed in terms ofthe x′

1−x′2−x′

3 coordinate system by p′Ti

= R(ϕ′)x′Ti where

R(ϕ′) =(

cos ϕ′ sinϕ′

− sin ϕ′ cos ϕ′

)(4)

and ϕ′ is the angle between p′i

and x′i(i = 1, 2).

Transmission of the radio wave decomposed into two polar-izations parallel to p′

1and p′

2can be described by a transmis-

sion matrix T

T =(

T1 00 T2

)(5a)

T1 = exp [2πim′1kl′f/c] (5b)

T2 = exp [2πim′2kl′f/c] (5c)

where l′ is path length, f is radar frequency, c is the speed oflight in vacuum, and k(= c

√ε0μ0) is the propagation constant.

m′1 and m′

2 are principal radii of the central section ellipse sothat they are function of angle θ′ between c3 and x′

3. However,they do not depend on ϕ′. In most cases, ice layers are locallyparallel to the ice-sheet surface above it so that l′ can beapproximated as d/ cos θw, where θw is determined by the radarconfiguration and d is the layer thickness (Fig. 2). Therefore, Tis independent of ϕ′ and can be written as T (θ′).

Radio-wave propagation through a single layer can be fullydescribed as WoutR(−ϕ′)T (θ′)R(ϕ′)WT

in , where Win/out arecomponents of polarization vectors of the radio wave enteringand leaving this layer under the single-primed coordinates.Sequential applications of this term allow one to describeradio-wave propagation through multiple layers. Equations (4)and (5) are valid for the upward propagation described in thedouble-primed coordinate system by replacing single-primedvalues by the double-primed values.

To model a round-trip propagation through the ice sheetfor a bistatic configuration, we make pragmatic simplificationsdiscussed as follows. When the radio wave is reflected from atarget beneath a single layer, the resultant complex amplitudeAMP can be written as

AMP=WrR(−ϕ′′)T (θ′′)R(ϕ′′)SR(−ϕ′)T (θ′)R(ϕ′)WTt (6)

where Wt and Wr are components of transmitting and receivingradar-polarization vectors, respectively. In the x−y−z coordi-nate system (Appendix I), Wi = (0, 1, 0) and (1, 0, 0) when theantennas are in the cross-fire [Fig. 2(b)] and end-fire settings[Fig. 2(c)], respectively. Appropriate alternative expressions forWi would allow us to examine radio-wave propagation forarbitrary polarizations. S represents scattering characteristicswhich must account for the single-primed to the double-primedframes. Similarly earlier, (6) can be adapted to multilayeredice by multiplying the term RTR with appropriate parametersdetermined by the fabric at individual layers. For side-lookingradar, the terms RTR for the downward and upward propaga-tions would be identical. A similar matrix expression was usedto model monostatic radar measurements [19].

We prescribed single- and double-primed angles θ andϕ as well as m1 and m2 for two illustrative special cases,uniaxial and biaxial indicatrices (Appendices II-A and II-B).The uniaxial indicatrix represents single-pole fabrics but withprincipal directions in arbitrary directions (χ3=1; c3 = x3).This situation might arise from single-pole fabric thatdeveloped from principally uniaxial compression but wherethe compression axis was not perfectly vertical. The biaxial


Fig. 3. Azimuthal angles (a), (e) ϕ′ and (c), (g) ϕ′′ and apparent anisotropy (b), (f) Δmuniaxial(θ′) and (d), (h) Δmuniaxial(θ

′′) of perfect single-pole fabrics(χ3 = 1; χ1 = χ2 = 0). The upper and lower rows show results for the fabrics with the tilted c3 by θ = 5◦ and 10◦, respectively. An identical gray scale isused. [Panels (b) and (f)] Δmuniaxial(θ

′) and [panels (d) and (h)] Δmuniaxial(θ′′) are standardized (see text): (a) θ = 5◦, (b) θ = 5◦, (c) θ = 5◦, (d) θ = 5◦,

(e) θ = 10◦, (f) θ = 10◦, (g) θ = 10◦, (h) θ = 10◦.

indicatrix represents vertical-girdle fabrics (χ1 =χ2 =a;χ3 =1− 2a; a < 1/3; c3 = x3) that developed under uniaxialtension near ice ridge or onset of fast flow. The refraction-indextensor m has three distinct principal values but has one axisaligned vertically. These two fabric types are typically found inthe majority of the ice sheets (Section II-B). The formalism forvertical-girdle fabric will be generalized later (Appendix II-C)so that any fabric can be considered with this model.

IV. EFFECTS OF BIREFRINGENCE MODELED FOR A

REALISTIC RANGE OF ICE FABRICS

Formalisms for birefringence associated with two distinctice fabric patterns (Appendix II) allow us to evaluate birefrin-gence in terms of radar frequency, antenna configurations, icethickness, and ice fabrics. We simulate birefringence for severalfabric patterns. Measured fabrics from polar ice cores (Table I)are used to approximate a range of the fabric; angle θ betweenc3 and x3 is approximated by an angle between c3 and the coreaxis (Table I). Although the measured fabrics never perfectlyagree with idealized fabric types discussed in this paper sincemost ice cores are not geographically oriented, detail calcula-tions are unfeasible at present. In this section, all results areobtained for a single-layered ice sheet with depth-independentfabrics and with paralleled upper and lower boundaries. Thesetreatments, however, provide a picture of the kind of behavior tobe expected and a route to practical analysis for more detailedfabrics measured in future ice cores. In Section VI-C, a morerealistic approach including depth-variable fabrics is adapted tointerpret radar data collected at NGRIP.

For the case of internal layers within the polar ice sheets, ascattering matrix depends on radar frequency [17] and polariza-tion [32]. For the case of ice-sheet bed, scattering characteristics

are less known. In this paper, we examine the effects of bire-fringence and approximate the scattering process as isotropicperfect (specular) reflection. With this approximation, S = −Iwhich has the same components for any coordinate system. Weignore reflection loss at the ice-sheet surface and at internallayers and attenuation during the round trip; we discuss possibleviolations of these assumptions in Section V.

For all calculations, the radio-wave polar angle θw (Fig. 2)is varied between 0◦ and 40◦. For reference, in recent bistaticattenuation measurements, θw was 8.7◦ in [40] and varied from2.9◦ to 29◦ in [56]. With no loss of generality, we take c3 of thetilted single-pole fabric to be in the x2−x3 plane (ϕw = 90◦

in Fig. 2). Results are independent of W , as long as Wr = Wt

(copolarized). For all results, we set mr‖c =

√3.189 and mr

⊥c =√3.152, which correspond to ice at −21 ◦C. Model results

presented here agree with results known for the monostaticsetting at θw = 0◦ [Fig. 1(a)].

A. Principal Axes of the Fabric in Relation to the ObliquelyPropagating Coordinates

We first examine azimuthal angle ϕ′ and ϕ′′ in terms of ϕw

and θw. Because we assumed |S1|= |S2| and |T1|= |T2| (5,6), the radio-echo intensity for cases when 45◦ < ϕ′ < 90◦ isthe same as for 90◦ − ϕ′; i.e., ϕ′ = 45◦ is a symmetry axis.Therefore, we always choose to map ϕ′ and ϕ′′ into the rangefrom 0◦ to 45◦. We also show results only for 0◦ < ϕw < 90◦

here, because we restricted fabrics to either single pole orvertical girdle so that ϕ′ and ϕ′′ are symmetric about ϕw =0◦

and 90◦.Azimuths ϕ′ and ϕ′′ of the transmitted radio wave in the

downward and upward frames for the single-pole fabrics(χ3 =1;χ1 =χ2 =0) are shown in Fig. 3. When c3 is vertical


Fig. 4. (Left column) Azimuthal angles ϕ′ and (right column) normal-ized Δmbiaxial for four vertical-girdle ice fabrics. χ3 is consistent (0.7)for all panels, but χ2 − χ1 varies. For vertical-girdle fabrics, ϕ′ = ϕ′′.Contour legends and gray scale are identical with Fig. 3: (a) χ2 −χ1 = 0.025, (b) χ2 − χ1 = 0.025, (c) χ2 − χ1 = 0.05, (d) χ2 − χ1 =0.05, (e) χ2 − χ1 = 0.075, (f) χ2 − χ1 = 0.075, (g) χ2 − χ1 = 0.1,(h) χ2 − χ1 = 0.1.

(θ=0◦), ϕ′ and ϕ′′ are zero for any combinations of θw andϕw. ϕ′ and ϕ′′ are both small when c3 is nearly vertical(θ <∼ several degrees) and the propagation polar angle θw islarge (>10◦). However, they become distinct for more tiltedsingle-pole fabric. In contrast to the monostatic case, ϕ′ and ϕ′′

can be nonzero, when the polarization plane includes the twominor principal axes c1 or c2 (ϕw =0◦ or 90◦). There aredramatic changes in the principal-axis orientation for relativelysmall changes in the propagation direction when θ=θw

and ϕw ∼ 90◦. This corresponds to the situation when thepropagation direction is aligned with the so-called optic axis ofthe index ellipsoid [37, p. 55]. Optic axes are directions alongwhich the radio wave is not split into two wave components.For this direction, the intersection with the ellipsoid is almostcircular, and small changes in direction can result in largedifferences in the orientation of the ellipse of intersection. Aswe will show in the next section, differences in the refraction

index are very small when this happens, consistent with thealmost circular ellipse.

ϕ′(= ϕ′′) for vertical-girdle fabrics are shown in Fig. 4 leftpanels. In these cases, c3 is vertical so that the polar angle θof c3 is zero, and c1 and c2 are taken to align with x1 and x2

directions, respectively. ϕ′ reaches 45◦ at (ϕw, θw) = (45◦, 0◦)as known in the monostatic radar configuration [Fig. 1(a)].Combinations of ϕw and θw that make ϕ′ ∼ 45◦ constitute astrip toward ϕ = 0◦. ϕ′ varies by approximately 45◦ when ϕw

varies only several degrees at the end of this strip near ϕw = 0◦.At this end, θw = 12◦, 17◦, 21◦, and 24◦ for χ2 − χ1 = 0.025,0.05, 0.075, and 0.1, respectively, for the constant χ3 = 0.7.These angles correspond to a half of the so-called optic anglefor the biaxial media; x′

3 and x′′3 with these θw align the optical

axes so that p1

and p2

are in the optical plane [37, p. 77].As found in the single pole, the difference in the refractionindex (right column in Fig. 4) is small as expected for a nearlycircular ellipse.

B. Refraction Index at Two Principal Polarizations

Refraction indexes m1 and m2 along p1

and p2

eachfor uniaxial and biaxial indicatrices give apparent anisotropyin the refraction index Δmuniaxial(θ′), Δmuniaxial(θ′′), andΔmbiaxial in the plane perpendicular to the propagation axissuch that

Δmuniaxial(θ′) = m1 − m2. (7)

m1 for uniaxial indicatrix is always consistent with m⊥c, whilem2 for uniaxial indicatrix and m1 and m2 for biaxial indicatrixdepend on θ′ (Appendix II).

Figs. 3 and 4 show Δmuni/biaxial/ cos θw divided by Δm ofthe single crystal (2) for tilted single-pole and vertical-girdlefabrics, respectively. The term cos θw compensates for thedifference of propagation-path lengths in terms of separationbetween the transmitter and receiver. With the standardization,contours in Figs. 3 and 4 specify polar and azimuthal angles thatgive the same phase difference for a given ice thickness. Unlikeϕ′ and ϕ′′, Δmuniaxial(θ′) and Δmuniaxial(θ′′) are not symmet-ric about ϕw = 0◦ (Fig. 3). They are symmetric about ±90◦

however. We show results only for 0◦ < ϕw < 90◦ becauseΔmuniaxial(θ′) for downward propagation for cases −180◦ <ϕw < 0◦ is consistent with Δmuniaxial(θ′′) for upward prop-agation for cases 0◦ < ϕw < 180◦ (or vice versa). Note thatΔmbiaxial is symmetric about both ϕw = 0◦ and 90◦. Thesecharacteristics are caused by symmetry of the fabric pattern.

Table I shows that χ2 − χ1 ranges up to about 0.2 in the polarice sheets, so anisotropy for the monostatic radar measurementsis less than 0.2 Δm. However, anisotropy can be much largerthan that when the polar angle θw is large.

C. Phase Difference Over Full Path

The phase difference δ between two principal wave compo-nents during a one-way trip can be defined as

δ/d = 2πΔmuni/biaxial/λ cos θw. (8)


Fig. 5. Frequency-depth combinations which give the phase difference δ =π/2 during a one-way trip through ice in terms of standardized Δmuni/biaxial.For standardization, see text. Legend for contours are consistent with those inFigs. 3 and 4. δ reaches π/2, for instance, when 70-MHz radio wave propagatesthrough 1000 m of ice with Δmuni/biaxial ∼ 0.1Δm cos θw .

The phase difference is given in [19] for monostatic radarmeasurements in terms of principal values of the ice-fabrictensor. Here, it is divided by cos θw to account for a longer pathlength (l′ = l′′ = d/ cos θw) for the oblique propagation. Fig. 5shows depth and frequency combinations that give the phasedifference π/2 for one-way travel. These combinations are ofparticular interest because the echo extinction happens whenthe phase difference reaches π for a round trip (δ is different fordownward and upward propagations through single-pole fab-rics). Contours in Fig. 5 represent Δmuni/biaxial/Δm cos θw,which is equal to (χ2 − χ1)/ cos θw when the propagation axisaligns along c3.

Table I suggests that θ and χ2 − χ1 observed in the center ofthe polar ice sheets are less than about 5◦ and 0.05, respectively.These ice fabrics give Δmuni/biaxial less than 0.1 Δm cos θw

when θw is less than about 20◦ [Figs. 3(b) and (d) and 4(b) and(d)]. Fig. 5 shows that, for these ice fabrics, phase differencereaches π/2 when 20 and 60 MHz radar is used for approx-imately 3600- and 1200-m-thick ice, respectively. Airborneradar is typically operated at frequencies higher than 60 MHzin Antarctica and Greenland. This suggests that airborne datacollected over Antarctica and Greenland potentially includesignal drops caused by birefringence.

D. Echo-Intensity Variations

Fig. 6 shows echograms in the ϕw − θw domain for four icefabric patterns in rows and for four different sets df in columns,where d and f stand for depth and radar frequency, respectively.df may range between 1 × 109 m/s (equal to 1-km ice thicknessand 1-MHz radar frequency) and 4 × 1012 m/s (equal to 4 kmand 1 GHz). df corresponding to these four columns are shownwith thick curves in Fig. 7. Monostatic measurements (θw =0◦) give the echogram symmetric about ϕw = 45◦. However,when the polar angle θw is nonzero (oblique propagation), thissymmetry no longer holds.

As df increases, the echogram shows a more complicatedpattern. To find out df that cause significant (or insignificant)

echo variations in the radar data induced by birefringence, weexamined the echo-intensity variations for realistic conditions.In radar measurements, a typical bandwidth is roughly 10% ofthe center frequency [21], [31], [38]. To account for the effectof the bandwidth, radar frequency is perturbed by ±5% at eachvalue of df . The echo averaged over this range of df is definedas the echo at a given θw and φw and the center df is defined asdfbandwidth. This averaging is not made for Fig. 6. dfbandwidth

is initially set at 4 × 1012 m/s and decreased until variations inthe echo intensities over a given ϕw − θw domain become lessthan 2 dB. This maximum dfbandwidth that gives less than 2-dBvariations is defined as dfmax.

Thin curves in Fig. 7 shows dfmax for six ice fabric patterns.To derive dfmax, polar angle θw varies between 0◦ and 40◦.However, a narrower range of θw does not assure insignificantvariations in the echo intensity for a given dfbandwidth, sincea signal drop at a given azimuth happens at a polar angle θw

mostly less than 10◦ (Fig. 6). Fig. 7 shows that 100 MHzor higher radar data include birefringent-origin echo-intensityvariations more than 2 dB, even if ice fabric is quite gentle(contour d; χ2 − χ1 = 0.025). Significant signal drops are notevenly distributed in terms of θw (Fig. 6) so we further exam-ined dfmax for partial azimuthal ranges. Table II shows dfmax

for four 30◦-width azimuthal ranges (0◦–15◦ is equivalent to±15◦ around 0◦). As the rightmost column of Table II indicates,there is the “window free from echo extinction due to birefrin-gence ” at which the effect of birefringence is insignificant atmuch higher dfband than other azimuths. The azimuthal rangeof this window is ±15◦ (75◦ < ϕw < 105◦) from the planeincluding c2 (vertical-girdle fabrics) or the plane includingnonvertical c3 and x3 (tilted single-pole fabrics).

V. APPLICABILITY OF THE MODEL

TO ACTUAL ICE SHEETS

In addition to ice fabric, directional refractive indexes canbe produced by density variations at shallow depths if direc-tional rough surface such as sastrugi and dunes is preservedwithin the ice. However, even if such a directional densitystructure exists, it becomes less significant with increasingdepth, while anisotropy in the fabric develops with increasingdepth. Thus, most of bulk birefringence associated with bedechoes is presumably caused by ice fabric rather than possibledirectional density at shallow depths for most sites in polar icesheets.

All results showed in Section IV are derived with an as-sumption that only the phase of the two principal radio-wavecomponents are changed by the ice fabric. Amplitudes of thewave components are modified by dielectric attenuation withinice, transmission loss at the air/ice interface and within ice, andscattering loss at the bed. If these effects are isotropic, relativeamplitudes of the two wave components remain the same, so theformalism presented here can be used straightforwardly. If theseeffects are anisotropic, then the formalism must be slightlymodified: Attenuation effects can be prescribed if m1 and m2

(5) includes the imaginary part. Scattering can be considered byextra matrices in (6). Here, we assess these possible effects thatmay require slight modification to the formalism.


Fig. 6. Echograms in terms of azimuth and polar angles of the radio-wave propagation for four ice fabrics and for four products of ice thickness (d) andradar frequency (f). Rows correspond to different fabrics; top two rows are for single-pole fabrics with different tilts, and bottom two rows are for vertical-girdle fabrics with two χ2 − χ1 but the constant χ3(= 0.7). Columns correspond to four different df given by thick contours shown in Fig. 7. From left toright, df = 0.1, 0.5, 1, and 2 × 1012 m/s. Identical gray scale in decibels [panel (i)] is used for all panels to show echo intensity relative to the isotropicice: (a) θ = 5◦, (b) θ = 5◦, (c) θ = 5◦, (d) θ = 5◦, (e) θ = 10◦, (f) θ = 10◦, (g) θ = 10◦, (h) θ = 10◦, (i) χ2 − χ1 = 0.025, (j) χ2 − χ1 = 0.025,(k) χ2 − χ1 = 0.025, (l) χ2 − χ1 = 0.025, (m) χ2 − χ1 = 0.1, (n) χ2 − χ1 = 0.1, (o) χ2 − χ1 = 0.1, (p) χ2 − χ1 = 0.1.

A. Attenuation Within Ice

Dielectric attenuation is proportional to ice conductivity.Bulk conductivity of ice having contributions from pure ice,acidity, and salinity [35] is virtually frequency-independent upto about 300 MHz [18]. Anisotropy in pure-ice conductivity isabout 20 (±6)% at frequencies less than 1 MHz [18] but withinexperimental errors (about 13%) and thus insignificant in the30-GHz range [34]. No measurements of the anisotropy in pure-ice conductivity are available between these two frequency

ranges. In addition, there is no evidence of anisotropy in theacidity and salinity contributions. Therefore, in our estimates asfollows, we assume 20% anisotropy in the pure-ice conductivitycomponent. Pure ice contributes less than 69% of the bulkconductivity, if ice temperature ranges between −10 ◦C and−40 ◦C and the acidity is more than 2 μmol/l. If ice includesmore acid and sea salt, the relative contribution of the pure iceto the bulk conductivity is smaller, and thus, the anisotropy inbulk-ice conductivity is smaller than the estimate as follows.


Fig. 7. Relationships between depth d and radar frequency f that give thesame phase difference δ. Four thick contours labeled 0.1, 0.5, 1, and 2 ×1012 m/s represent df that give echograms shown in Fig. 6 from left to rightin this order. Contours labeled with letters give dfmax for the full range of ϕw

(0◦–90◦) for different ice fabrics: single-pole fabrics with (a) θw = 1◦, (b) 5◦,and (c) 10◦; vertical-girdle fabrics with different (d) χ2 − χ1 = 0.025, (e) 0.1,and (f) 0.2 but with the constant χ3 = 0.7.

TABLE IIdfmax (= ×1012 m/s) AT WHICH THE ECHO INTENSITY VARIES 2 dB

Anisotropy of the bulk conductivity can be written asαΔχ(l′ + l′′) decibels, where α represents a ratio of the pure-ice component to the other contributions from acidity and salin-ity to the bulk conductivity. l′ = l′′ = d/ cos θw if ice layersare parallel to the ice-sheet surface. α varies exponentiallywith temperature and linearly with acidity and salinity [29].Δχ = χ2 − χ1 = Δmuni/biaxial/Δm gives an anisotropy ofice-fabric tensor in the central section ellipse. cos θw accountsfor a path-length variation caused by oblique propagation. Forice with acidity of 2 μmol/l, α for a round trip is 5.87 ×10−3 dB/m at −10 ◦C, 2.06 × 10−3 dB/m at −20 ◦C, and6.64 × 10−4 dB/m at −30 ◦C (see [18] for details of conduc-tivity). For 3000-m-thick ice at −20 ◦C, slightly anisotropicice (Δχ = 0.01) yields |T2|/|T1| 0.99, and more anisotropicice (Δχ = 0.1) yields the ratio 0.93. This ratio is smallerif ice temperature is higher because of different temperaturedependences of molar conductivity and pure-ice conductivity.Nevertheless, this ratio is no less than 0.87 (α for −10 ◦C and2 μmol/l of acidity, Δχ/ cos θw = 0.1 and d = 2000 m).

Fig. 1(b) shows the echo intensity for a possible range of|T2|/|T1| when the radio wave propagates along c3 (monosta-tic). The radio wave traveling along c2(ϕw = 90◦) is attenuatedmore than that along with c1. When the ratio decreases to0.9, the azimuth of the signal drops shifted by about 5◦, andecho intensities at ϕw = 90◦ is no more than 1 dB smallerthan the echo at ϕw = 0◦. Therefore, amplitudes of the twowave components can differ less than 1 dB except for extreme

cases such as ice thicker than ∼2000 m has a depth-meantemperature higher than −10 ◦C and is highly anisotropic butincludes very low chemical constituents. Overall, attenuationanisotropy does not modify the results shown in the previoussection. Attenuation can be included in the theory if (5b) and(5c) account for imaginary part of the refraction index m1

and m2.

B. Scattering Within Ice

Scattering within ice can be caused by acidity, density,and ice-fabric nonuniformities [17]. Among these scatteringcauses, the ice-fabric nonuniformities are the most likelycause of anisotropic scattering and transmission losses forthe two principal radio-wave components. Anisotropy in thescattered power found by radar survey in Antarctica is no morethan 10 ∼ 15 dB [32], [33]. For typical backscattering coeffi-cients of −60 ∼ −80 dB, the scattering anisotropy makes thetransmission-wave magnitudes different but only by −60 dBand thus negligible. Multiple reflections at the internal layersare also negligible [29].

C. Scattering at the Upper and Lower Interfaces of Ice

Fresnel reflectivity at the air/ice interface is −5.5 dB, whichgives transmission loss 1.5 dB. As long as this loss is isotropic,our analysis is not affected regardless of the magnitude of theloss. If the surface has directional roughness in the scale ofwavelength (1.7 m in ice at 100 MHz), the loss at the surfacemay be frequency- and polarimetric-dependent. Modeling sucheffects is beyond the scope of this paper; we leave it forfurther work.

It is also uncertain whether scattering at the bed can alterthe phase and amplitudes of each wave component. Scatteringcharacteristics of bed of icebergs [43], ice shelves, and icesheet nearby the grounding line [49] have been examined, butscattering from more inland ice-sheet bed is less constrained.Fresnel reflectivity varies 20% in terms of the incident anglebetween 0◦ and 45◦. However, it is eventually a function of theincident angle regardless of dielectric properties of the bed (i.e.,wet or dry bed) so that this effect can be taken into accountusing geometry of the radar configurations [56].

VI. IMPLICATIONS OF RADAR DATA

A. Effects of Ice Fabric on Estimating EnglacialAttenuation and Bed Reflectivity

An algorithm proposed by Winebrenner et al. [56] estimatesdepth-integrated attenuation within ice from variations in theecho intensities collected with different antenna separations(i.e., path lengths) using bistatic-radar configurations at aconstant azimuth. It works regardless of the bed reflectivity(i.e., wet or dry bed) but does not account for birefringence.Correct extraction of attenuation and birefringence from thebed-echo intensity is essential to estimate bed reflectivity.Here, we discuss how the fabric affects the Winebrenneralgorithm.


It is required to vary the path length more than the attenuationlength La (e folding, i.e., ∼1/2.7 = −4.3 dB) for good accu-racy [29]. La estimated with various methods are 500–700 mat South Pole (ice thickness d: 2800 m [5]), 120–170 m at SipleDome (1000 m [56]), 170 m at Greenland NGRIP (3070 m[40]), and about 250 m at Ross Ice Shelf (roughly 1000 m[6], [44]). These ranges of the attenuation lengths and icethicknesses require θw to vary from 0◦ to 13◦ ∼ 27◦.

When the fabric is unknown and thus ϕw is arbitrary, bistatic-radar measurements aiming to derive attenuation should bedone with HF or very low VHF radio wave (Fig. 7).Winebrenner et al. [56] used 3 MHz at Siple Dome, WestAntarctica, and birefringence is negligible in their radar data.Unlike monostatic measurements, echo intensity observed withthe bistatic measurements is asymmetric about ϕw = 45◦

(Fig. 6). dfmax for |ϕw| ≤ 15◦ from the plane containing c2 is atleast one order higher than dfmax for the entire azimuthal rangeof 90◦ (Table II). Through this window free from extinction dueto birefringence, the echo intensity varies no more than 2 dBat any radar frequency less than 890 MHz in most of icesheets and 210 MHz in the area with strongly developed fabric(Table II, case c) as long as radar bandwidth is 10% of thecenter frequency or wider. Therefore, if this azimuthal windowis identified, VHF or higher frequency radars can be used tomeasure attenuation and thus bed reflectivity with insignificanteffects of birefringence.

Neither monostatic radar measurements nor current ice coresidentify c1 and c2 axes. However, regional ice-sheet topographyprovides reasonable constraints in some cases. For instance,uniaxial extension typically found along ice ridge causes thevertical-girdle fabric with the vertical plane perpendicular tothe extension axis [14], [33]. Similar patterns were found inupstream of outlet glaciers [15] and over floating ice [26].If no such supplemental information is available, then multi-polarimetric measurements with the monostatic configuration[32] must be made first so that signal drops at ϕw = 45◦

are identified. Then, the bistatic-radar measurements must bemade at two azimuths ±15◦ from each of c1(ϕw = 0◦) andc2(ϕw = 90◦) to find out azimuths of the window free fromextinction due to birefringence. Measurements at two azimuthsare necessary, since monostatic radar data are incapable todistinguish c1 and c2.

B. Effects of Birefringence on SAR Processing

Synthetic aperture radar (SAR) processing has been em-ployed to enhance spatial resolution of the ice-thickness mea-surements and to remove clutter. Unfocused SAR algorithmswithout compensating phase difference have been successfullyapplied to ice-sheet study [22], [25], [42], [44]. Unfocused SARcan be used when the phase difference is less than 45◦ over thesynthetic aperture length [48].

Relative phase variations in the radio wave can be inducedas follows: 1) a phase difference of the wave components withdifferent polar angles associated with the aperture length and2) a phase difference between the two principal wave compo-nents for a given polar angle. Since polar-angle dependenceof Δmuni/biaxial is small (Figs. 3 and 4), the former phase

Fig. 8. Frequency dependence of the bed echo at NGRIP for a polar angleof 8.7◦. (a) Simulations using ice-core data from NGRIP [51]. Gray scale isidentical with Fig. 6. Dash and solid lines show azimuths of 55◦ and 65◦,respectively, to facilitate a comparison with the radar-observed data. (b) Bed-echo intensity measured with (gray) bistatic radar [40] and simulated for (dash)ϕw = 55◦ and (solid) 65◦.

difference can be no larger than 0.6◦ for radar configurationsused for Greenland (aperture length of ∼130 m and flight heightof ∼500 m [22]). Thus, it is totally negligible; effects of bire-fringence for SAR processing can be assessed with theoreticalframework for monostatic radar [19]. The latter is certainlybeyond 45◦ for many cases. Δmuni/biaxial that causes 45◦ phasedifference over a round trip can be given by contour labels inFig. 5 divided by four. If the radar frequency is higher than100 MHz and ice is thicker than several hundred meters, whichhappen in most radar measurements, δ reaches 45◦. Althoughfurther assessments of the SAR algorithms are beyond the scopeof this paper, birefringence might induce significant errors inthe SAR processing to determine basal conditions of the polarice sheets. It is more problematic in deep outlet glaciers fromGreenland and Antarctic ice sheets, since fabric there is likelyto be strongly developed.

C. Detection of Ice Fabric Using Bistatic Radar

Ice-flow studies suggest that variations in viscosity causedby ice fabric is necessary to explain observed surface ice-flow speed, geometry of radar-detected englacial layers, andstratigraphy in ice cores [45], [50]. Recent radar measurementsin Antarctica characterized regional distribution of the icefabrics [19], [32], [33], but all of them used monostatic radarconfigurations. As we demonstrated in Section IV-D, bistaticconfigurations are more capable than monostatic configurationsto characterize ice fabrics; the bistatic-radar data can determineazimuths of c1 and c2, but monostatic radar is incapable to doso because of symmetry about ϕw = 45◦ [Fig. 1(a)]. Here, toassess feasibility of remote sensing of ice fabrics, we examinebistatic-radar data collected at NGRIP in Greenland [40] usingice-fabric data from the ice core [51].

Paden et al. [40] made a wideband-frequency (110–500 MHz) bistatic measurement for a given antenna separation


(θw = 8.7◦). Bed-echo intensity derived with deconvolutionof system parameters shows a linear decrease by 8 dB overthe frequency range and more rapid variations in terms of thefrequency [Fig. 8(b)]. They interpreted that the linear trendincludes frequency dependence of the attenuation (3 dB) andbed reflectivity (5 dB). The frequency variation of the echointensity is simulated using the fabrics measured at 133 depthswith the NGRIP ice core (Table I). We assumed that c3 isvertical, and c1 and c2 are aligned to given orientations (i.e.,north and east, respectively) through the entire ice so thatformalism for the vertical-girdle fabric can be applied. Bedis assumed to be isotropic perfect Fresnel plane (S = −I).Unknown azimuths of the ice-core-derived fabric in a fixedframe x1−x2−x3 yields eventually unknown ϕw. Thus, wesimulated the echo for a frequency range between 100 and500 MHz and for an azimuthal range between 0◦ and 90◦.

Fig. 8(a) shows radio echo in terms of radar frequency andpolarization azimuth. Frequency modulations between 100 and500 MHz cause zero to three major signal drops and zero totwo minor signal drops depending on the azimuth. Fig. 8(b)compares the simulated results with the radar observation [40].Although azimuths of the radar polarization is unknown, it isapparent that the simulations with ϕw = 0◦–40◦ and 70◦–90◦

do not match with the radar observation. Rather, simulationswith ϕw ∼ 60◦ replicate prominent features of the echo suchas major signal drops around 150, ∼300, and 470 MHz.Since vertical c3 and constant azimuths of c1 and c2 areassumed, we think that this level of agreements between theradar data and simulations is encouraging. We argue that somefeatures in the radar data are caused by birefringence. Distinctfrequency variations demonstrated here suggest that windband-frequency measurements can provide a reasonable proxy todecipher in situ ice fabric and assess fabric effects associatedwith radar echoes from interior and bed of the ice sheets.Likely causes of residual in the observed data that cannotbe explained with the birefringence model are frequency de-pendence of the bed reflectivity induced by wavelength-scaleroughness of the bed, and assumptions in the model that prin-cipal orientations of the fabrics are constant over the entire icecolumn.

VII. CONCLUSION AND OUTLOOK

Birefringence for radio waves propagating obliquely withinice sheets is examined in terms of the ice-fabric and radar con-figurations. The birefringence was formulated for two specialcases: tilted single-pole fabric and vertical-girdle fabric. Thelatter is generalized for arbitrary fabric (Appendix II-C). Radio-wave propagation though a single layer is described in relationto a reference frame fixed in space so that this formalism can bestraightforwardly applied to multilayered media.

Computations are made for an approximate range of icefabrics found in polar ice cores. The results provide importantconsequences in bistatic and side-looking radar measurements.When the bed-echo intensity is collected with various antennaseparations to examine englacial attenuation, radar frequencyless than 20 MHz must be used for 1000-m-thick ice (df =2 × 1010) if ice fabric is well developed (Table II, fabric-type f).

If ice thickness is doubled, appropriate radar frequency must be10 MHz or less. Otherwise, ice fabrics can cause significantreduction of the echo intensity so the derivation of the attenua-tion becomes problematic. In inland parts of polar ice sheetswhere ice fabric is less developed, this frequency range canbe wider. Unlike monostatic radar measurements, the echo isasymmetric about 45◦ off the principal axes of the fabric. Whenazimuthal angle ϕw ranges ±15◦ from the plane including c2

(vertical-girdle fabrics) or the plane including nonvertical c3

and the vertical axis (tilted single-pole fabrics), the fabric effecton the bed-echo intensity is minimal. Measurements at theseazimuths make birefringence negligible for most cases. Thiswindow free from extinction due to birefringence can be usedfor attenuation and bed-reflectivity measurements using thebistatic radar. To identify this window, the ice-fabric axes mustbe inferred from bistatic measurements at multiple azimuthsor other sources such as ice-sheet topography, ice cores, ormonostatic polarimetric-radar measurements.

To establish radar as a tool to detect ice fabric, a next stepis to examine radar data with azimuth-controlled ice-core data.Detection of azimuth of each ice-core piece is now feasible witha new drill for West Antarctic Ice Sheet Divide Ice Core Projectled by the U.S. Antarctic Program. Automatic fabric analyzerscan be used to measure ice fabric of this ice core at numerousdepths. Sonic logging can guide depths of the thin-sectionanalysis. Modeling of the polarimetric signatures using ice-coredata would eventually facilitate radar data to reveal fabric overwide areas. It is also necessary to examine sensitivities of theecho on various fabric and radar parameters to find out the bestfield strategy for characterizing ice fabrics. Because current icefabric is related to past ice-strain history, remote sensing of theice fabrics facilitates not only deciphering current ice viscosityrelated to ice fabrics but also constraining ice-sheet history ininland regions where other geological evidence is unavailable.

APPENDIX IBISTATIC COORDINATES

The transmitted radio wave is reflected at the bed below themidpoint between the transmitter and receiver, provided thatthe bed is reasonably flat over this separation and that the firnthickness is uniform. The radar configurations are expressedwith reference to a rectangular coordinate system x1−x2−x3

fixed in space; for simplicity, we choose x1-axis headed north,x2-axis headed east, and x3-axis headed downward vertically(right-handed system). The distance between the transmitterand receiver is 2s (Fig. 2). The azimuthal angle between themis ϕw measured from x1-axis toward x2-axis. Components ofa unit vector in the propagation direction of the radio wavecan be written as (sin θw cos ϕw, sin θw sin ϕw, cos θw) in thenonprimed coordinate where the polar angle θw is given bytan−1(±s/d) for downward and upward propagations. Pathlengths l′ is then defined as d/ cos θw, where d is the layerthickness measured along x3-axis (5), if layers are parallel tothe ice-sheet surface.

Single- and double-primed coordinates are used to describedownward and upward propagations, respectively. x′

3 and x′′3

coincide with the incident and reflected propagation paths,


respectively. The nonprimed x3 aligns vertically so that x′1

and x′′1 lie in the plane of incidence defined by x3 and

x′3. The nonprimed and single-primed frames are defined as

right-handed coordinates, while the double-primed frame isintentionally defined as a left-handed coordinate so that x′

2 andx′′

2 coincide. The single- and double-primed frames coincidewith the nonprimed frame when transmitting and receivingantennas are next to each other (monostatic). The single- anddouble-primed frames are related to the space-fixed x1−x2−x3

coordinate as follows:

x′1 = cos θw cos ϕwx1 + cos θw sinϕwx2 − sin θwx3 (9a)

x′2 = − sin ϕwx1 + cos ϕwx2 (9b)

x′3 = sin θw cos ϕwx1 + sin θw sinϕwx2 + cos θwx3 (9c)

x′′1 = cos θw cos ϕwx1 + cos θw sinϕwx2 + sin θwx3 (9d)

x′′2 =x′

2 = − sin θwx1 + cos ϕwx2 (9e)

x′′3 = sin ϕw cos ϕwx1 + sin θw sin ϕwx2 − cos θwx3. (9f)

APPENDIX IIDERIVATIONS OF FABRIC- AND

COORDINATE-DEPENDENT PARAMETERS

Prescriptions of values of single- and double-primed pa-rameters are derived for single-pole fabric (Appendix II-A),vertical-girdle fabric (Appendix II-B), and more general cases(Appendix II-C). These parameters are rotational angle ϕ′ aboutthe propagation axis that determines directions of p′

i, polar

angle θ′ between xc3 and x′

3, and refraction indexes m′i along p′

i.

A. Single-Pole Ice Fabric (Uniaxial Indicatrix)

For this uniaxial case, one needs only determine the principalorientation (i.e., Euler angles). The Euler angle ϕ′ in the x′

1−x′2

plane determines the two principal polarization azimuths of thetransmitted radio wave, while the polar Euler angle θ′ deter-mines the refraction indexes for each of the two polarizations.The same is true for the upward radio-wave substituting ϕ′′ andθ′′. The manipulations turn out to be mathematically almostidentical to those derived previously for use in an automatedtechnique for ice-fabric analysis [53], [55]. Results for themonostatic limit of this case but with an arbitrary angle betweentransmitting and receiving antennas were worked out in detailpreviously [27] and will be used as a check on our results below(see Appendix III).

Suppose the principal values of the refraction-index ten-sor are m1 = m2 = m⊥c and m3 = m‖c. Components ofthe most major principal vector xc

3 can be written as xc3 =

(sin θ cos ϕ, sin θ sinϕ, cos θ)T under the nonprimed coordi-nates fixed in space. Here, c-axes are represented by polar θand azimuthal ϕ angles measured from x1 to x2 axes and fromx3 axis, respectively. Note that, in our formalism, xc

3 is definedwith respect to the fixed frame which defines x3 to be in thedownward direction, but this is opposite to the usual conven-tion that is used for ice fabrics derived from thin sections. If

the transmission antenna is located at a position specified byazimuthal angle ϕw and polar angle θw (Fig. 2), components ofxc

3 in the single-primed coordinate system (denoted by xc′3 ) is

found by first rotating about the x3-axis by the angle ϕw (whichsimply shifts the azimuth of xc

3 by ϕw) and then rotating theresult about the x2-axis by the angle θw

xc′

3 =

⎛⎝ cos θw 0 − sin θw

0 1 0sin θw 0 cos θw

⎞⎠

⎛⎝ sin θ cos(ϕ − ϕw)

sin θ sin(ϕ − ϕw)cos θ

⎞⎠ (10)

xc′3 can also be written as (sin θ′ cos ϕ′, sin θ′ sinϕ′, cos θ′)T.

These equations yield θ′ and ϕ′ as

θ′= cos−1(cos θw cos θ + sin θw sin θ cos(ϕ − ϕw)) (11a)

ϕ′= tan−1

(sin θ sin(ϕ−ϕw)

cos θw sin θ cos(ϕ−ϕw)−sin θw cos θ

). (11b)

According to the standard results of uniaxial crystal optics [37],a wave component having polarization p

1will have refraction

index

m1 =m‖cm⊥c√

m2‖c cos2 θ′ + m2

⊥c sin2 θ′(12)

and a wave component having polarization p2

will have refrac-tion index m2 = m⊥c, independent of θ′.

The double-primed variables θ′′ and ϕ′′ represent values inthe frame of the direction of the reflected radio wave and can befound in a similar way to that of the downward propagation butwith θw replaced by −θw as follows:

θ′′ = cos−1 (cos θw cos θ − sin θw sin θ cos(ϕ − ϕw)) (13a)

ϕ′′= tan−1

(sin θ sin(ϕ−ϕw)

cos θw sin θ cos(ϕ−ϕw)+sin θw cos θ

). (13b)

B. Vertical-Girdle Ice Fabric (Biaxial Indicatrix)

Vertical-girdle fabric has one axis of the ellipsoid that isvertical. With no loss of generality, the xc

1- and xc2-axes may

be assumed to be aligned with the other two axes of theellipsoid, i.e., xc

1 = x1 and xc2 = x2. For this case, the general

prescription described in Section III must be implemented.Because the axes of the ellipsoid are aligned with the fixednonprimed frame, the equation for the ellipsoid is given by(x1/m1)2 + (x2/m2)2 + (x3/m3)2 = 1, where m1, m2, andm3 are the principal values of the refraction-index tensor asdetermined by (2). This equation can also be written in a matrixform as follows:

(x1 x2 x3 )

⎛⎝ m−2

1 0 00 m−2

2 00 0 m−2

3

⎞⎠

⎛⎝ x1

x2

x3

⎞⎠ = 1. (14)

In the single-primed downward frame, the ellipsoid can befound by applying the appropriate rotation matrices for the


coordinate transformation

( x′1 x′

2 x′3 )

×

⎛⎝ cos θw 0 − sin θw

0 1 0sin θw 0 cos θw

⎞⎠

⎛⎝ cos ϕw sinϕw 0

− sin ϕw cos ϕw 00 0 1

⎞⎠

×

⎛⎝ m−2

1 0 00 m−2

2 00 0 m−2

3

⎞⎠

⎛⎝ cos ϕw − sin ϕw 0

sinϕw cos ϕw 00 0 1

⎞⎠

×

⎛⎝ cos θw 0 sin θw

0 1 0− sin θw 0 cos θw

⎞⎠

⎛⎝ x′

1

x′2

x′3

⎞⎠ = 1. (15)

The ellipse of intersection between the ellipsoid and the x′1−x′

2

plane is found by taking x′3 = 0. The equation for the ellipse

can be written as

A′x′21 + B′x′

1x′2 + C ′x′2

2 = 1 (16)

where A′, B′, and C ′ are given in terms of m1, m2, m3, ϕw,and θw

A′ =(

cos2 ϕw

m21

+sin2 ϕw

m22

)cos2 θw +

1m2

3

sin2 θw (17a)

B′ = −(

1m2

1

− 1m2

2

)cos θw sin(2ϕw) (17b)

C ′ =sin2 ϕw

m21

+cos2 ϕw

m22

. (17c)

Once A′, B′, and C ′ are found, an Euler angle ϕ′ that deter-mine p

1and p

2in the single-primed frame and corresponding

refraction indexes m′1 and m′

2 can be found as

ϕ′ =12

tan−1

(B′

A′ − C ′

)(18a)

m′1 =

√2

A′ + C ′ +√

B′2 + (A′ − C ′)2(18b)

m′2 =

√2

A′ + C ′ −√

B′2 + (A′ − C ′)2. (18c)

The same development applies to the upward propagation, butθw is replaced by −θw. Because of the symmetry of the biaxialindicatrix, all m′′

1, m′′2, and ϕ′′ are the same as their single-

primed values [(18)].

C. More General Ice Fabrics

In the most general case, the fabric will result in a biaxialindicatrix ellipsoid that has principal axes with none of themcoincident with the vertical axis. To account for this, an extrarotation matrix must be introduced into (14) and (15) thatrepresent the biaxial ellipsoid in relation to the space-fixed(nonprimed) frame. If the fabric information is known a priori,then the rotation matrix may be constructed in a standard wayusing appropriate direction cosines between fabric principaldirections and the fixed frame. For inverse-modeling purposes,however, it may be convenient to construct rotation matricesfrom Euler angles to examine the effect of rotating the index

ellipsoid in various well-defined fashions. To rotate the indexellipsoid to an arbitrary orientation using three Euler anglesφi, θi, and ψi, rotation matrix Mtilt and its inverse M−1

tilt areapplied left and right of the diagnostic matrix with elements ofm−2

i (i = 1, 2, 3) in (15)

Mtilt =

⎛⎝ cos φi − sin φi 0

sin φi cos φi 00 0 1

⎞⎠

⎛⎝ 1 0 0

0 cos θi − sin θi

0 sin θi cos θi

⎞⎠

×

⎛⎝ cos ψi − sin ψi 0

sin ψi cos ψi 00 0 1

⎞⎠ . (19)

APPENDIX IIICOMPARISON WITH A PREVIOUS FORMALISM

It is useful to show that our general result reduces to theexpressions given in [27, eq. (1)–(3)] for the monostatic uni-axial case. Comparison of our Fig. 2 with Fig. 1 as well asother definitions in the respective papers allows one to makethe following notational correspondences: (Liu et al. [27], thispaper) = (along flow direction, x1-axis), (across flow direc-tion, x2-axis), (θ, ϕ), (α,ϕw), (Vθ, c/m1), (V0, c/m2), (β, θ),(h, d), (ω, 2πc/λ), where c is the speed of light in vacuumand d is ice thickness [l′ = d when θw = 0 in (5)] and m2 =m⊥c for single-pole fabrics. Furthermore, we note that, for themonostatic case, using our notation, we can take θw = 0, andhence, θ′′ = θ′ = θ, and ϕ′′ = ϕ′ = ϕ − ϕw.

We have implicitly assumed the “end-fire” configuration inmaking these correspondences. We may now use our model toderive the appropriate result with only one minor modification.In our formulation, the transmitting and receiving antennasmay be either parallel (copolarized) or perpendicular (cross-polarized), but Liu et al. [27] allow the transmitting antennato have an arbitrary orientation with respect to the receivingantenna, which is aligned either along or across the flow direc-tion. This may be allowed for in our formalism by modifyingthe rotation matrix in (6) to be R(ϕ). Then, if Wi in (10) aretaken as (1, 0, 0), the result should reduce to that for the along-flow case, and if Wi are taken as (0, 1, 0), the result shouldreduce to that for the across-flow case.

We now write down expression for the received amplitudesfor the two cases

AMPalong flow = (1 0)D(

10

)(20a)

AMPacross flow = (0 1)D(

10

)(20b)

where

D =(

cos(ϕ − ϕw) − sin(ϕ − ϕw)sin(ϕ − ϕw) cos(ϕ − ϕw)

)

×(

exp(4πim1d/λ) 00 exp(4πim2d/λ)

)

×(

cos ϕ sin ϕ− sin ϕ cos ϕ

). (21)


In these expressions, we assumed that S in our formalism wasequal to the identity matrix and then also substituted the identityfor the resulting combination R(ϕ′)R(−ϕ′′). We also combinedthe transmission (T ) terms for the downward and upward waveswhich results in an extra factor of two in the exponential. De-rived AMPalong flow AMPacross flow are squared to get power,yielding expressions identical to [27, eq. (1) and (2)] once thenotation difference are taken into account.

ACKNOWLEDGMENT

The authors would like to thank N. Azuma, S. Fujita, andT. Thorsteinsson for providing digital ice-fabric data fromDome Fuji, Mizuho, and NGRIP ice cores. They would also liketo thank J. Paden and P. Gogineni for providing the radar datafrom NGRIP. They would also like to thank NSIDC for distrib-uting the ice-fabric data for GRIP and GISP2. They would alsolike to thank two anonymous reviewers for providing thoughtfulreviews.

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Kenichi Matsuoka received the B.E. degree inphysics and the M.S. and Ph.D. degrees in earthenvironmental sciences from Hokkaido Univer-sity, Sapporo, Japan, in 1995, 1997, and 2002,respectively.

After three-year postdoctoral training both inJapan and the U.S., he has been with the Departmentof Earth and Space Sciences, University of Washing-ton, Seattle, since 2005, as a Research Assistant Pro-fessor. Since then, as a field-oriented Glaciologist, hehas led five glaciology projects funded by the U.S.

National Science Foundation, both in the Antarctic and Arctic regions. He hasused airborne and ground-based ice-penetrating radar as the primary tool toelucidate processes that control glacial evolution, decipher glacial history, andassess ongoing and future behavior of polar regions.

Dr. Matsuoka is a member of American Geophysics Union and InternationalGlaciological Society.

Larry Wilen received the B.S. degree in physicsfrom the University of California at Los Angeles,Los Angeles, in 1980 and the Ph.D. degree in physicsfrom Princeton University, Princeton, NJ, in 1986.

In 1996, following a postdoctoral research posi-tions at Technion, Israel, and at the University ofWashington, Seattle, he was with Ohio University,Athens, where he was an Assistant Professor withthe Department of Physics and Astronomy and wasappointed to Associate Professor in 2001. In 2005,he was with Unilever, Trumbull, CT, as a Research

Scientist/Group Leader. He is currently an Affiliate Associate Professor withOhio University and maintains active collaborations in academia at Yale andOhio Universities. His major research interests at Ohio University includedfabric and texture analysis in ice cores and their relation to paleoclimate,thermoacoustic prime movers and refrigerators, and surface and interfacialmelting of ice and associated environmental impacts.

Shawn P. Hurley received the B.S. degree in electri-cal engineering and the B.S. degree in engineeringphysics from Ohio University, Athens. He is cur-rently working toward the Ph.D. degree in chemicalphysics at the Kent State University Liquid CrystalInstitute, Kent, OH.

He has worked as a Summer Intern forCorning Inc.

Mr. Hurley is a member of the Society for Infor-mation Display, The International Society for OpticalEngineering, and Sigma Xi. He is an Engineer Intern.

Charles F. Raymond received the A.B. degree inphysics and mathematics from the University ofCalifornia at Berkeley, Berkeley, in 1961 and thePh.D. degree in geophysics from the California In-stitute of Technology, Pasadena, in 1969.

In 1969, he joined the Geophysics Programas an Assistant Professor with the University ofWashington, Seattle, where he was appointed to As-sociate Professor in 1973 and then Professor in 1979.His primary research has concerned the motion ofglaciers and ice sheets. Recent work has focused

on ice streams draining the marine-based West Antarctic Ice Sheet and theirpotential influence on rapid sea-level change. One avenue of investigation hasbeen mapping the disturbance of internal layers using ice-penetrating radar toilluminate the history of past motion. He is currently a Professor Emeritus withthe Department of Earth and Space Sciences, University of Washington.

Dr. Raymond is a member of American Geophysical Union and InternationalGlaciological Society.

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Effects of Birefringence Within Ice Sheets on Obliquely Propagating Radio Waves

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