Ice samples

To investigate the real part of permittivity as a function crystal orientation, ten specimens of single crystal collected from Mendenhall Glacier, Alaska, were used as shown in Table 1. Six of them were used to measure ϵ^′ _‖c . Four of them were used to measure ϵ^′ _⊥c . The orientation of the c axis in each single crystal specimen was determined using a universal stage, which measures the orinetation of the c axis of ice by applying optical birefringence of the ice crystal. A report by Reference LangwayLangway (1958) gave details of the equipment and the technique to determine orientation of the c axis. We confirmed that the orientation of the c axis determined with the universal stage was the same as that determined by X-ray analysis in our laboratory. The procedures to cut the ice specimens and how it was placed in the waveguide are described in the next section. All ice specimens were bubble-free and transparent. Their densities are shown in Table 1. Average value of the density is 913 kgm⁻³. The angle between the c axis of the samples and the applied electric field vector is also shown in Table 1. Here, the angle is defined as the angle between two axes in the plane which contains them. Concentration of total impurity ions in ice was of the order of 0.1 ppm (Reference Fujita, Shiraishi and MaeFujita and others, 1992). Such very low impurity does not influence the real value of permittivity.

Experimental procedures

Measurements were made at temperatures between about −30°C and melting point. 9.7 GHz was used for all measurements. Experimental procedures are described as the following three steps: (1) preparation of the ice sample; (2) setting up the sample to the dielectric measurement system; (3) measurement and calculation. These procedures and precautions are described below.

The specimen prepared for each measurement was a rectangular prism with cross-section 10.9 × 22.9 mm² and length 30.0 mm, cut from an original ice block of single crystal with diameter of the order of 10 cm. The transverse dimension of the rectangular specimen was the same as the inner dimension of the rectangular waveguide. When each specimen was cut from the original single crystal ice block, the orientation of its c axis was investigated first. The error of this investigation is ±5°. Next, each rectangular ice specimen was cut out with a band saw. Then the c axis of each specimen was made to orient parallel to the shortest or longest side of the rectangular prism so that the c axis was parallel or perpendicular to the electric field when placed in the waveguide. After making each rectangular prism, the orientation of the c axis in each rectangular prism was measured again for confirmation. Then the angle between the c axis of each specimen and the electric field vector, when placed in the waveguide, was calculated. This is shown in Table 1. Finally, the surface of each prism was finished with a microtome, which smoothed the surface precisely. Once a specimen was prepared, it was inserted into 30 mm long waveguide whose standard was WRJ-10. The waveguide was made from brass with inner wall coated in silver. One side was short-circuited by the metal plate and the other side was terminated with a flange.

In inserting the sample into the waveguide, care was taken to eliminate the small air gap between the inner wall of the waveguide and the ice sample because it causes serious error in the measured result of permittivity. However, such a small air gap does not affect the orientation of the c axis in the waveguide, because the ice specimen was cut very precisely. To eliminate the small air gap, each ice sample was inserted into the waveguide in a bath filled with ion-exchanged water at melting point: the specimen and the water which filled the gap were frozen together. This procedure ensured that permittivity measured was that of the specimen itself and the negligible layer of ice which filled the gap, at most a few per cent of the whole ice measured. After freezing, ice outside the waveguide was completely removed and the surface of the specimen at the flange was cut to a plane with roughness less than 0.1mm. Next the 30 mm long waveguide containing the sample was connected to the waveguide which comes from slotted line, and was set in a freezer in which temperature could be varied between −70°C and 25°C (Fig. 2). In connecting the waveguides, a polyethylene sheet was put between them to prevent vapour leaking from a small slit in the slotted line. The influence of the polyethylene on the experimental results was calibrated and corrected in calculation.

Measurements were carried out varying temperatures at a rate of 10°C (h⁻¹). When permittivity was measured with varying temperature at a slower rate, (e.g. when temperature was fixed for a few hours), no difference was found in the experimental results. In addition, no difference was found in either lowering or raising the temperature. To obtain the permittivity, the input impedance of the empty waveguide was first measured, then the input impedance of the specimen-filled waveguide. The input impedance of the specimen-filled waveguide varies with temperature, not only by the temeprature-dependence of permittivity in ice, but also by the thermal expansion of the waveguide and temperature dependence of resistivity at the inner wall of the waveguide. Their influences were corrected by measurements of the input impedance of the empty waveguide at various temperatures. Estimated error of the measurement was at most ±0.01.

Anisotropy of ϵ_∞

Reference EvansEvans (1965) showed that most of ϵ_∞ measured in earlier studies is expressed by

Fig. 4. ϵ^′ _‖c and ϵ^′ _⊥c compared with the results of ॉ_∞ in the earlier studies.

Taking into account the error term of Equation (equ.3), the real parts measured in this study are included in ϵ_∞, given by Equation (equ.4). Reference Johari and CharetteJohari and Charette (1975) measured the complex permittivity of ice at 35 and 60 MHz above −25°C. As shown in Figure Fig. 4, the real part at 35 MHz is slightly larger than our result, especially at higher temperatures above −7°C, but that at 60 MHz is very similar to ϵ^′ _‖c . Reference Mätzler and Wegmüller.Mätzler and Wegmüller (1987) measured the complex permittivity at frequencies between 2.4 and 9.6 GHz at temperatures above −30°C using the cavity-resonator method. They assumed ice to be isotropic and concluded that the real part is expressed as:

Equation (equ.5) is shown in Figure 4. ॉ_∞, of Equation (equ.5) is completely equal to ϵ^′ _‖c . Then we can conclude that the real parts obtained in this study agree approximately with these earlier results. It is not clear why ॉ_∞ of Equation (equ.5) is completely equal to ϵ^′ _‖c , although the former is assumed to be isotropic. Possible causes of this strange coincidence are as follows: (1) there are small systematic errors in the results; (2) the assumption that ice was isotropic was not correct in Equation (equ.5). Even if the former is the case, and even if the small systematic error exists in the results of this study, Δॉ′ in Equation (equ.3) is not influenced by such error at all.

Reference Johari and CharetteJohari and Charette (1975) measured ϵ^′ _⊥c of artificial single-crystal ice and ॉ_poly of polycrystalline ice at 35 and 60 MHz. They could not detect any difference between ϵ^′ _⊥c and ॉ_poly. Since they did not observe the ice fabric of the polycrystalline ice, it is assumed that the ice fabric shows isotropy. In this case, ϵ^poly=ϵ^′ _⊥c +(ϵ^′ _‖c -ϵ^′ _⊥c )/3. Using Equation (equ.3), we obtain ϵ^poly-ϵ^′ _⊥c =0.012, which is approximately equal to their experimental error, 0.010. This may be the reason why they could not obtain the anisotropy.

Reference Humbel, Jona and ScherrerHumbel and others (1953) measured the complex permittivity of single crystal ice at low frequencies in the kHz region and found that ϵ^′ _‖c , was clearly larger than ϵ^′ _⊥c . Based on their result, anisotropy between static permittivities, ϵ^′ _s‖c and ϵ^′ _s‖c is 15% at −5°C and decreased with increasing frequency. In addition, Reference KawadaKawada (1978) obtained similar results. Although the anisotropy has not been detected at higher frequencies, we conclude now that ϵ^′ _‖c is larger than ϵ^′ _⊥c over the frequency range from MHz to GHz.

ॉ_∞ is determined by vibration of the water molecule in ice which represents the higher frequency dispersion in the infrared region (Fig. 1). The precise behaviour of this molecular vibration has not been investigated and anisotropy of the real part cannot be explained. Reference EhringhausEhringhaus (1917) measured the anisotropy of the real part at frequency of sodium-D line and found it to be 0.0037 or 0.2% at 1°C.

Reflection induced by dielectric anisotropy

It is well known that ice fabric changes with depth in the ice sheets.Reference Harrison Harrison (1973) proposed a hypothesis that variation of ice fabric was one of the major causes of internal reflections observed by RES in the ice sheets. Unfortunately, this has not been supported because the dielectric anisotropy was not considered to be so large as to produce internal reflections. In this study, however, we can obtain the large dielectric anisotropy Δॉ′ = 0.037.

When a microwave or a radio wave are normally incident on a plane boundary between two ice media, 1 and 2, characterized by permittivity ॉ′₁ and ॉ′₂ respectively, if the wave travels from ice medium 1 to ice medium 2, the power reflection coefficient PRC is given by

where Z₁ and Z₂ are the intrinsic impedance of the two ice media, respectively. If change in impedance is due to only the real part of the complex permittivity, PRC, given by Reference Paren and RobinParen and Robin (1975) is:

where δॉ′ is the difference in permittivity between two media (= ॉ′₁ − ॉ′₂). Equations (6) and (7) hold only for reflection at a single plane boundary, separating two media infinitely extended regions. Equation (equ.7) holds only when δϵ^′<<ϵ^′ ₁, ϵ^′ ₂. The present discussion satisfies this condition because maximum δϵ^′ is only 1.2% of ϵ^′ ₁ and ϵ^′ ₂. To discuss exactly, we should consider the influence of birefringence, i.e. reflection of the ordinary wave and the extraordinary wave in ice should be considered separately. However, since the purpose of this discussion is to estimate the order of the magnitude of reflection coefficient caused by the dielectric anisotropy, the effect of birefringence is ignored. When change in the real part of permittivity is due only to the contribution of dielectric anisotropy, Equation (equ.7) can be expressed as

Here, D_a is a coefficient which shows the degree of contribution of ∆ϵ^′ to change in permittivity between media 1 and 2. D_a is a number between 0 and 1.

Figure Fig. 5 shows the relation between D_a and PRC. In the extreme case, i.e. when we assume that

and is 1 and PRC is about −50 dB. If and medium 2 is isotropic, we obtain D_& = ⅓. Then PRC is about −60 dB. Moreover, even when D_a is as small as 0.03 ∼ 0.1, PRC is about −70∼ −80dB. Since PRC observed by RES in the ice sheets is about −70 ∼ −80 dB, it is concluded that the measured dielectric anisotropy can be one of the dominant causes of internal reflections. Because the temperature dependence of ∆ϵ^′ is negligibly small in the temperature range of the cryosphere, PRC is not influenced by temperature. Details of the relation between the ice-fabric variation and PRC are reported in another paper (Fujita and Mae, this volume).

Fig. 5. Power Reflection Coefficient (PRC) as a function of D_a.

Reference Ackley and KeliherAckley and Keliher (1979) computed a depth profile of PRC using the measured physical properties of core to the bedrock (324 m) taken at Cape Folger, East Antarctica, in order to compare with observed radio-echo reflections. They used the variation of density, bubble size and shape, and ice fabric as parameters. Adopting ∆ϵ^′ = 0.0037 as the dielectric anisotropy, they calculated PRC to be about −90 dB which was negligibly small compared with values deduced using other parameters (

−80dB). Applying our result, ∆ϵ^′ = 0.037, as the correct dielectric anisotropy, we obtain PRC ≅−70dB and, even in the case of Ackley and Keliher, the dielectric anisotropy becomes a dominant cause of the internal reflection.