Introduction

Reference EvansEvans (1965) reviewed the dielectric properties of ice at temperatures from −0.1°C to −66°C and at frequencies from 10 Hz to 10⁵ MHz. At frequencies up to a few hundred megahertz, dielectric absorption is due to a relaxation process with a single, temperature-dependent relaxation frequency. At higher frequencies the dominant influence is infrared absorption, which shows no temperature dependence. In this paper we present new experimental data on dielectric absorption in ice, obtained from radio-echo studies at 30 to 440 MHz. These results, together with other published measurements in this band of frequencies are compared with the high frequency extreme of the relaxation spectrum and with the low frequency extreme of the infrared spectrum. Satisfactory agreement is obtained if account is taken of both the temperature and the purity of the ice.

Method

At very high frequencies (V.H.F.) dielectric absorption in ice is so low that the usual bridge methods of measurement are rather difficult. Instead we have estimated the absorption by measuring the attenuation of radio signals propagating through the ice. From electromagnetic theory the attenuation of plane radio waves of frequency f is 90

f tan δ decibels per kilometre path where the complex permittivity of ice is ϵ ^⋆ = ϵ′−jϵ″, the loss tangent is tan δ = ϵ″/ϵ′ and f is in MHz. Since for ice is nearly constant at V.H.F. we take f tan δ as a convenient measure of the dielectric absorption.

During 1963 the author carried out radio-echo-sounding experiments on the Brunt Ice Shelf, Antarctica (lat. 75° 31′ S., long. 26° 40′ W.) (Watford, unpublished). Soundings were mostly at 35 MHz but there are some results at 100 and at 440 MHz. Echo strengths from natural targets at the bottom of the ice shelf were measured using calibrated attenuators and from these measurements we estimate the mean dielectric absorption in the Brunt Ice Shelf. We must take account of other factors which may affect the signal strength of radio echoes. Estimates show that scattering and partial reflections within the ice do not contribute significantly to the signal attenuation but that the inverse square law, the directivity of the aerials and refraction into the snow do affect signal strengths. We may estimate these contributions to the total attenuation with sufficient precision, but an important factor which is rather uncertain is the nature of the radio-echo target at the bottom of the ice shelf. A radio-echosounding traverse across the Brunt Ice Shelf (Walford, unpublished) shows that at 35 MHz the ice–sea-water interface at the bottom of the ice shelf behaves like groups of targets approximately a hundred metres across. The groups are separated by horizontal distances of approximately one kilometre and between groups the echoes fade by at least 60 decibels. This is a remarkably large range; possibly heavy absorption of radio signals occurs only where salty or wet ice is present within the ice shelf and echoes are detected where the ice shelf contains only pure ice. We assume that the strongest radio echoes are reflected from non-absorbing targets about a hundred metres in diameter and we regard the resulting values of f tan δ as defining the maximum dielectric absorption for pure ice, probably of inland origin (see Reference Barclay and BruntBarclay, 1964).

Radio-echo sounding suffers from uncertainty in the path of propagation of the signals and in the nature of the echoing target. These uncertainties are avoided by the use of guided waves, and a separate series of values off f tan δ has been obtained from measurements using twin-wire transmission lines (Walford, unpublished). The lines consist of two parallel copper wires 20 cm apart initially mounted horizontally 2 m above the snow. A balanced feeder at one end of the lines transmits radio waves which are reflected at the other end by a large metal plate. We use a small magnetic probe to plot the resulting standing-wave pattern. These preliminary experiments show that the power losses along the lines due to radiation and to ohmic resistance are small and may be neglected in the second part of the experiment. Then the lines are lowered to the snow surface and allowed to bury naturally in drift snow to a depth of about one metre. Standing-wave patterns are again plotted and from the standing-wave ratios we calculate the dielectric absorption in snow. Over forty experiments of this type were carried out at frequencies between 30 and 300 MHz in snow of measured temperature and density.

The accuracy of the transmission-line experiments is limited in practice by a troublesome effect arising with the transmission lines buried in snow. A few metres of the line must be led into the air to permit standing wave measurements, and a little power is therefore reflected at the air–snow interface. The standing-wave pattern is modified by an unknown amount and is found to be irregular in shape due to the irregular snow surface nearby. With the equipment available in the field no experimental arrangement was found to eliminate these effects,which reduce the accuracy of the experiments, but the unwanted reflections could probably be avoided by measuring the attenuation of radio signals with an impedance bridge connected across the lines and allowed to bury with them.

Results

Figure 1 shows values of f tan δ calculated from the measured signal strengths of radio echoes through the ice shelf and from the experiments with buried transmission lines. The temperature profile through the ice shelf is not directly known but is assumed similar to that through the ice shelf at Maudheim. On this assumption we find that a suitable weighted mean for the Brunt Ice Shelf is −15°C. The temperature of the surface layers down to 2 m was measured during the transmission-line experiments and was found to vary from −18°C to −22°C at the depth of the lines and a mean value of −20°C is used in Figure 1. In each case the measurements on drift snow have been corrected by Weiner’s formula (see Reference EvansEvans, 1965; Reference WalfordWalford, unpublished) so that they are equivalent to solid ice: the value of the Formzahl was assumed to be 2 and the magnitude of the correction was never greater than a factor of 2. Figure 1 includes other published measurements of dielectric absorption in ice at frequencies from 100 kHz to 10⁴ MHz.

Fig. 1. f tan δ of ice versus frequency. Ice temperatures are marked in degrees Celsius below zero. Curve a: Expected dielectric absorption due to resonance absorption at infra-red wavelengths (Reference Cartwright and ErreraCartwright and Errera, 1936). Curves b: Expected dielectric absorption due to relaxation absorption at lower radio frequencies (Reference Auty and ColeAuty and Cole, 1952).

● Westphal (Reference RagleRagle and others, 1964). Laboratory measurements at V.H.F. on ice from the Ward Hunt Ice Shef. Ellesmere Island.

▼ Reference LambLamb (1946) and Reference Lamb and TurneyLamb and Turney (1949). Laboratory measurements at V.H.F. on non-annealed ice made from distilled water.

× Reference CummingCumming (1952). Laboratory measurements on ice made from distilled water, melt water or tap water.

■ Reference Von HippelVon Hippel (1954). Laboratory measurements on non-annealed ice made from conductivity water.

T: Reference WalfordWalford (unpublished). Field measurements by transmission line methods on snow of the Brunt Ice Shelf Antarctica, corrected for the density of the snow. Mean snow temperature −20°C.

R₁, R₂, R₃: Watford (unpublished). Field measurements by radio echo sounding at 35, 100 and 440 MHz respectively through the Brunt Ice Shelf, Antarctica, corrected for the density of the snow. Weighted mean temperature of the ice shelf −15°C

The new data are compared first with the low-frequency extreme of the infrared absorption spectrum. Infrared absorption in ice has been studied at wavelengths as long as 150 μm, equivalent to 2×10⁷ MHz (Reference Cartwright and ErreraCartwright and Errera, 1936). Curve A in Figure 1 is extrapolated from these measurements using the equation f tan δ = Af ² which is derived from elementary resonance theory (Reference Bleaney and BleaneyBleaney and Bleaney, 1957). We obtain the constant A and hence absolute values of dielectric absorption from the measurements of Cartwright and Errera. The extrapolation is correct to within a factor of two provided no peaks occur in the infra-red absorption spectrum of ice at wavelengths greater than 150 μm. The experimental data in Figure 1 becomes rather insensitive to temperature above a few hundred megahertz and appears to approach curve a asymptotically. The agreement is remarkably good bearing in mind that curve a is extrapolated over four decades of frequency. We conclude that above a few hundred megahertz dielectric absorption in ice is dominated by the infrared spectrum and probably no peaks occur in this spectrum below 2×10⁷ MHz.

We next compare the new experimental data in Figure 1 with the high-frequency extreme of the relaxation spectrum. The relaxation spectrum of pure polycrystalline ice is accurately described by Debye’s theory for polar dielectrics with a single relaxation frequency f _r. Measurements by various authors are generally in good agreement with the Debye equation:

The static value of the relative permittivity of ice ϵ _s increases smoothly from 92 to 124 as the temperature falls from 0 to −66°C and the high-frequency limiting value ϵ _∞, is 3.17. The relaxation frequency f _r varies with the absolute temperature T and is given in megahertz by the following equation (after Reference EvansEvans, 1965, p. 774)

An approximate form of the Debye equation useful for calculating dielectric absorption at the high-frequency extreme of the relaxation spectrum is

This gives f tan δ to within 5 per cent at frequencies greater than 25 f _r and it is used to plot the set of curves B in Figure 1. The curves agree qualitatively with the experimental data: between 0.1 and 100 MHz f tan δ varies rapidly with temperature and rather little with frequency. Quantitative agreement however is less satisfactory and there are discrepancies of as much as a factor of ten. These discrepancies probably arise from the effects of ionic impurities upon the dielectric relaxation process in ice.

The Effect of Impurities

According to the Debye theory, at frequencies above 25 f _r, dielectric absorption in ice is independent of frequency. Except for the effects of infrared absorption bands, the data in Figure 1 generally support this hypothesis, and so f tan δ is plotted as a function of the impurity content of the ice with temperature as the only parameter in Figure 2. Reference CookCook (1960), Westphal, and Kuroiwa (1954.) explicitly state the nature and concentration of major impurities in their samples but the present author has estimated the impurity concentrations for the remaining data in Figure 2 from the original papers. The impurity content of the Brunt Ice Shelf is not directly known and the values in Figure 2 were assigned by comparison with the Maudheim ice shelf (Reference Brocas and DelwicheBrocas and Delwiche, 1963; Walford, unpublished). The estimates of impurity concentrations are probably accurate only to a factor of three and no attempt has been made to take account of differences in chemical composition between various samples. Such differences may be rather small because four authors use samples of naturally occurring ice or snow, Reference CookCook (1960) uses ice of similar composition to natural sea ice and Reference Auty and ColeAuty and Cole (1952) use pure ice made from conductivity water.

Fig. 2. f tan δ versus impurity content of ice. The temperature of each sample is marked in degrees Celsius below zero and smooth curves are drawn through measurements at similar temperatures by different authors.

A: Reference Auty and ColeAuty and Cale (1952). Laboratory measurements on relaxation spectrum of annealed ice made from conductivity water. High frequency limiting values of f tan δ plotted against estimated impurity content.

W: Westphal (Reference RagleRagle and others. 1964). Laboratory measurements at V.H.F. on two types of ice from the Ward Hunt Ice Shelf Ellesmere Island, f tan δ plotted against measured impurity content.

C: Reference CookCook (1960). Laboratory measurements at V.H.F. on arti_ficial samples of salty ice of similar composition to natural sea ice. f tan δ plotted against measured impurity content.

K: Reference KuroiwaKuroiwa [1956]. Laboratory measurements on the relaxation spectrum of natural snow, density 0.6g/cm³ collected at Sapporo, Japan. High frequency limiting value of f tan δ plotted against the measured impurity content.

M: Reference Watt and MaxwellWatt and Maxwell (1960). Field measurements on the relaxation spectrum of ice, Athabasca Glacier, Canada. High frequency limiting values of f tan δ plotted against the estimated impurity content.

R–T: Walford (unpublished). Field measurements at V.H.F. by radio-echo and transmission-line techniques on the Brunt Ice Shelf, Antarctica f tan δ plotted against the impurity content characteristic of Antarctic shelf ice (Reference Brocas and DelwicheBrocas and Delwiche, 1963). Temperature range −15 to −20°C

Despite uncertainties it is clear from Figure 2 that dielectric absorption in ice generally increases with impurity content. Over most of the range of temperatures and concentrations the absorption increases linearly with the impurity content. The measurements at V.H.F. by Westphal (personal communication and in Reference RagleRagle and others, 1964), Reference CookCook (1960) and the present author agree well with data extrapolated from measurements on the relaxation spectrum of ice by Reference Auty and ColeAuty and Cole (1952), Reference Watt and MaxwellWatt and Maxwell (1960) and Reference KuroiwaKuroiwa [1956].

Two laboratory studies have been made of the relaxation spectrum of ice as affected by temperature and impurity content. Brill and Camp studied ice doped with controlled amounts of ammonium fluoride (Reference BrillBrill, 1957; Reference Brill and CampBrill and Camp, 1961; Reference CampCamp, 1963) and Reference GränicherGränicher and others (1957) studied dielectric absorption and ice doped with hydrogen fluoride. They discuss their results using solid-state defect theory and account for dielectric absorption in terms of diffusion of charged defects through an ice crystal under an electric field (Reference GränicherGränicher and others, 1957; Reference GränicherGränicher, 1963; Reference JaccardJaccard, 1965). The two different laboratory studies use different doping chemicals but agree on the magnitude of the effects to within a factor of two. They show that as ice becomes more impure the static dielectric constant increases and the dielectric relaxation process takes place at higher frequencies, being now described by a spectrum of relaxation frequencies instead of a single one. If we attempt to compare the Iaboratory results with those in Figure 2 we find that satisfactory agreement is obtained for rather pure ice but that above a hundred parts per million of impurity discrepancies of a factor of ten occur. This is not surprising and may be due partly to the different chemical impurities present and partly to using the extrapolation equation f tan δ = (ϵ _s /ϵ _∞−1)f _r which applies strictly only to ice with a single relaxation frequency.

Summary

Present measurements support the view (Reference EvansEvans, 1965) that dielectric absorption at V.H.F. is caused partly by infra-red absorption bands and partly by the radio-frequency relaxation process. Relaxation is dependent upon temperature and impurity content and future experiments should include measurement of the ionic impurity content of the ice. There are few measurements on the V.H.F. spectrum or on the relaxation spectrum of natural, undisturbed ice or snow. There are no systematic studies of the effect upon the V.H.F. dielectric absorption spectrum of controlled amounts of known ionic impurities.

The measurements presented in this paper were carried out at Halley Bay, Antarctica, in 1963 when the author was a member of the British Antarctic Survey. He thanks the personnel of the station for their interest and assistance in the field work. He is also very much indebted to Dr. G. de Q. Robin and Dr, S. Evans for supervision and encouragement during his postgraduate work at the Scott Polar Research Institute, Cambridge.