A SECTION of S.P.R.I, radio-echo record is shown in Figure 1. It shows a depth profile for a 7 km length of path, at a depth of approximately 3 000 m. The echo from bedrock can be clearly distinguished as an almost continuous line close to the centre of the vertical range scale. Though the strength is steady, on the average, over the width of such a section, it is clear that it varies considerably over a distance of a few metres, or tens of metres. This phenomenon of “fading” has been recognized, and understood in principle, since the earliest experiments in radio-echo sounding. It arises from the interference between reflections from different facets of the rough reflecting surface. Though it was known that the appearance of the echoes would depend on the form of the bedrock surface, the variation was treated as a minor nuisance to trouble the depth sounder.

Fig. 1. A section of radio-echo record showing a normal fading bedrock echo.

However, it has been shown by Harrison (unpublished) and Reference BerryBerry (1973), among others, that the variations in the echo are related, through the echo sounder characteristics, to the statistical properties of the roughness of the bedrock surface. They showed that it is, in principle, possible to infer much about the surface from the behaviour of the radio echoes, and this paper describes the first attempts to make such inferences using S.P.R.I. echo-sounding technique.

Throughout the experiments, a radar with a carrier frequency of 60 MHz was used, giving a radio wavelength in ice of 2.8 m.

Preliminary work was carried out using the results of the 1971/72 S.P.R.I.-N.S.F. Antaretic radio-echo sounding programme. Though the primary purpose of the fieldwork was to record a simple depth profile of the ice, the photographic records are of such quality that some detailed aspects of the echoes may be observed, classified, and compared, area to area. Comparisons were essentially qualitative, since the records do not contain a quantitative measure of the received power, and a detailed statistical analysis was not possible. However, it was thought that significant differences between echoes should in some cases be detectable in a visual examination.

A qualitative measure of the echo strength could be obtained by examination of the exposed area of film corresponding to the bedrock echo. The degree of exposure is a function of the received power, and may be classified realistically into three classes; weak, medium, and strong. The results of a survey of the recorded echoes is mapped inFigure 2. The strength is highest in the cold, central regions of the ice sheet, as would be expected. The observed strengths are, however, somewhat higher than might be expected in particular in a region north of Vostok station, and somewhat outside the centre of outflow of ice, known as Dome “C”.

The possible significance of this feature only became apparent in the light of the major finding of this work on the Antarctic results, that is, the discovery and identification of lakes of liquid water underlying the central region of the east Antarctic ice sheet (Oswald and Robin, 1973).

Fig. 2. A map of the qualitative classification of echo strengths observed in east Antarctica for the area of the 1971 survey.

The sites of the observed lakes can be seen, indicated by diamond dois inFigure 3, and an example of a lake echo trace is shown inFigure 4.

We can now confirm, from the radio-echo records, the identification of the lakes as consisting of water, by inspecting the surface slopes of the lakes and of the overlying ice.

We consider the equilibrium of a fluid of density p_i, underlying a mass of ice of density p_i . If the ice has a surface gradient of α the condition for hydrostatic equilibrium in the fluid is that a gradient β exists in the ice-fluid interface such that:

or

In the majority of cases the surface slopes of the ice are so small that measurements are not possible. However, in two locations the surface slopes of 1: (1 000±500) and 1: (500±250) gave rise to gradients of — 1: (70±10) and —1: (50±10) respectively in the interface.

Entering these figures in the above formula, we arrive at the following values for p₁ using p_i = 0.92 Mg m^-3: p₁ = 0.99±0.04 Mg m^-3 or 1.01 ±0.05 Mg m ^-3.

Fig. 3. The diamonds mark the positions of subglacial lakes in east Antarctica.

These results can be seen to be very close to the density of water and, as the alternative possibilities are few, we may take this as confirmation of our identification of the lakes.

Though the very long continuity of these echoes was limited to a small number of sites, the other anomalous characteristics, of high strength, long, though irregular fading lengths, and short echo “tails”, occur frequently in the areas of high echo strength inFigure 2. An example of such an echo is shown inFigure 5.

Comparison with neighbouring bedrock echoes, and of the variation of strength, by region, with calculations based on temperature estimates, suggests an augmented reflection coefficient at the lower boundary of the ice.

The existence of the lakes establishes that some areas of the base of the ice sheet are at the pressure melting point. It seems unlikely that melting is confined to these specific localities where the topography is suitable for the formation of extensive lakes. We should expect it to occur over a much wider area, and the combination of the above echo characteristics suggests that water does indeed exist over the areas indicated.

The fact that high-angle reflections are not frequently seen in association with these echoes suggests that any channels are incised downwards into the rock, with relatively flat upper surfaces. This would agree with the form of a network of pools and “Nye channels”, or possibly Reference WeertmanWeertman’s (1972) “modified sheet”. In this case the distinction between the two, with only very low flow rates involved, is probably small.

Fig. 4. A section showing the echo from a sub-ice lake.

Fig. 5. The echo from a position where the ice is thought to be at the pressure-melting point, without the formation of a definite “lake”.

It has been possible, from the foregoing argument, to make some more or less tentative deductions concerning the nature of the base of the Antarctic ice sheet. However, if we wish to perform a more detailed analysis, and to be able to distinguish more subtle differences in the bottom surface than that between water and generalized rock, a quantitative measure of the instantaneous echo power is necessary.

The Devon Island expedition of 1973 presented an opportunity to make precise and detailed recordings of the echo waveform, as a pilot study for large-scale work in the Antarctic and elsewhere. The equipment already provided a display of the envelope of the received waveform, in the form of an “A-scope” trace. An 8 mm ciné camera was used to photograph this display at pre-determined intervals along the sounding track, giving a complete record of the signal envelope, accurate to ± 1 dB over a range of about 60 dB. One such photograph can be seen inFigure 6.

Fig. 6. Fig. 6. An example of an “A-scope” trace. The horizontal axis covers about 800 m of ice depth: the vertical axis is a logarithmic scale of echo strength, with a range of some 60 dB. The trace is initiated at the left-hand end of the scale, and the peak near the centre represents the bottom echo.

The soundings in Devon Island covered the lines shown inFigure 7. These include a fairly dense net close to the camp, and three radiating traverses, two running towards the north, and one towards the west. We wished to obtain a statistical description of the small-scale bedrock roughness in these areas, and also to locate the geological boundary which is thought to pass beneath the ice in a north-south direction across the island.

A-scope recordings were made at 50 m intervals over almost the entire survey path. In addition, recordings were made at I m intervals over distances of several hundred metres at several widely spaced sites over the area of the survey. They are lettered inFigure 7. At each of these sites the ice thickness was effectively uniform, and varied from site to site between 400 and 800 m. The closely spaced records give an almost continuous record of the echo waveform, and the large number of them allows us to make a detailed statistical analysis.

A total of some 25 000 A-scope recordings were made. In order to reduce the stored data to manageable proportions for analysis, it is necessary to select the components of the echo which will most satisfactorily yield the required information. Harrison’s general wave treatment has been used extensively in the reduction and analysis, since his terminology is closely linked with S.P.R.I. radio-echo practice. His assumptions are those of Kirchhoff diffraction theory, mainly concerning the amplitude and degree of “spikiness” of the reflecting surface roughness. The imposed restrictions are not important in the present case.

It seems reasonable to direct our inquiry initially towards the most accessible aspect of the received signal, namely the maximum observed level of power in the echo. If we can establish that much of the information obtainable about the reflecting surface is represented in the behaviour of the maximum level, we shall have gone a long way towards reducing the complexity, and also the tedium, of the analysis.

Fig. 7. The inset map shows the eastern end of Devon Island, the ice cap covering most of the area to the east of the dotted line. The positions of the survey lines, and of the nine sets of closely spaced strength measurements, can be seen at larger scale in the main part of the figure. Contours are in feet; 5000 ft = 1520 m.

The validity of the maximum level in this sense depends largely on the extent to which the rough reflecting surface scatters power away from the “specular reflection” ray paths.

We describe the roughness in terms of the parameters a, T , ø ₀, and ß, representing respectively the r.m.s. height of the deviations from smoothness, the horizontal autocorrelation length of the deviations, the phase change introduced by a in the reflection of the radio waves (φ₀ = 4πa/λ), and the r.m.s. slope existing in the surface, (β = 2a/T).

In the case where both φ₀ > I, and β > (p/Z_o)^½ >, (where p is the spatial half-length of the transmitted pulse, and z₀ is the distance from sounder to bedrock), the mean value of the received power will be reduced by a factor

below that expected from a smooth surface. Otherwise we may take this mean to be within at most 2 dB of the expected value.

We may obtain a rough estimate of ß by observing the fall-off of received power with angle of reflection (as determined by the extension of the delay beyond that of the “first return”).

For a surface with a Gaussian vertical distribution, the dependence of received power P_r on angle of reflection is given by Harrison as:

where θ is the inclination of the reflection to the vertical, P_t the transmitted power, G the antenna gain, k = 2π /λ the wave number of the radiation, and r₀ the distance from transmitter to reflecting point.

Table I shows these estimates of β for the nine sets of data mentioned above. They are consistently somewhat less than the critical value (p/z₀)^½ for each set. The difference is small, however, and we should bear in mind the possibility of a small reduction in power. It will be possible later to check these estimates against those derived from the autocorrelation functions of the received power.

The quantities z_o p_α, λ, refer to the ice depth, one-half the transmitted pulse length, and the radio wavelength in ice. In this case, p = 23 m, λ = 2.8 m. They define the various critical values, with which those quantities derived from the observations are to be compared.

The derived quantities are marked with asterisks.

β_I is the estimate of the r.m.s. slope derived from the decrease of received power with angle of reflection.

τP is the observed autocorrelation length of the received power.

β₂ is the estimate of the r.m.s. slope derived from the observed autocorrelation length.

T is the autocorrelation length of the bedrock roughness.

ø _o is the phase change introduced by reflection from a surface with r.m.s. vertical displacement a.

The angle (p/z_o) ½ defines the edge of the area illuminated for Harrison’s “first return”, when the centre of the pulse has just arrived at the surface.

The mean observed strengths are plotted against the depth of ice at each site inFigure 8. The strength is expressed in terms of the implied attenuation by absorption in ice, and by losses on reflection, after correction for that due to geometrical spreading of the wavefronts.

Seven of the points can be seen to lie close to a straight line. Their correlation coefficient is 0.97, and a linear regression analysis results in a line of gradient 4.8 dB/100 m. Remembering the two-way passage through the ice, this corresponds to absorption of 2.4 dB/100 m, averaged over ice between depths of 400 and 800 m. An assumption of uniform temperature (and therefore absorption) throughout the ice gives rise to an intercept at zero depth of — (4.8 ± 1.4) dB. This is the implied reflection coefficient at the bedrock surface. The numerical similarity of gradient and intercept is, of course, entirely fortuitous, and has no physical significance.

The isothermal assumption is, of course, a gross oversimplification. This figure for absorption implies, from Westpha’s (Reference Robin, Robin, Evans and BaileyRobin and others, 1969) figures, a mean absorption temperature of —12 º C for the ice between 400 and 800 m depth. The temperature near the surface is known to be about —23° C in this area. We can arrive at an improved estimate of the absorption, though with less apparent precision, by using a mean absorption temperature of — 20° C between the surface and 400 m depth. This corresponds to absorption of 1.6 dB, and gives a figure of — 11 dB for the reflection coefficient. ’This is within the expected range of — 10 to —20 dB, but is still somewhat higher than might be expected, since the bedrock in this area is thought to be granitic with a reflection coefficient of about — 15 dB. However, the uncertainties in the temperature distribution, and in the dependence of absorption on temperature probably include a range of at least ±4 dB in the estimate.

Fig. 8. Echo strength versus ice depth on the Devon Island ice cap. The power is expressed in terms of the total attenuation due to absorption in ice (A_i) and reflection losses at the bedrock. The dotted lines represent the standard error arising from the linear regression.

Of the nine points plotted inFigure 8, we have so far mentioned only seven, which we grouped owing to their high degree of correlation. There are two further points which do not conform to this line, and which we are treating as separate. They represent the sets of measurements made at the outward end of the western traverse, and are separated from their expected positions on the regression line by some 15 dB. Some of the discrepancy may be accounted for in terms of absorption, when we remember that the surface of the ice will be somewhat warmer at the lower altitude of these sites than that at the higher eastern sites.

Equating the change in the ice temperature to that in the mean air temperature for the drop in altitude of 500 m, we may expect a temperature change of between 4 and 6° C. The corresponding rise in absorption should be between 0.4 and 0.7 dB/100 m, giving a total rise, for a two-way passage through 600 m of ice, of between 5 and 9 dB. We still have a discrepancy of between 6 and 10 dB, which can only be satisfactorily explained in terms of the reflection loss. The large uncertainty in the value of the reflection coefficient applies mainly to the absolute value. The relative values for the two areas are, we believe, different by 8±2 dB, which is highly significant in view of the standard error of estimate of ± 1.4 dB derived from the seven eastern points.

We deduce that, in the course of the traverse to the west of the camp, the geological boundary mentioned above was crossed, and that the radio waves were being reflected from a different type of rock in the western area. We make no attempt to identify the types of rock involved, owing to the uncertainty in the absolute value of the reflection coefficient. The transition is, however, in the sense of a change from higher to lower permittivity when travelling from west to east.

It is difficult to see how the accuracy of the absolute values can be improved without extensive measurements of real temperature profiles and more detailed understanding of the phenomenon of absorption.

We next examine the extent of variation of the received power over short distances. We define the “normalized variance” of the instantaneous power at a fixed delay time as:

where the angular brackets denote the arithmetic mean.

The value of vp’ depends on that of φ_o and of T. Reference Bramley and YoungBramley and Young (1967) have calculated vp¹ , for the first return, as a function of 1/T² for various values of φ_o , and his results are shown inFigure 9. The value exceeds unity only under the conditions that φ_o>I, and T is of the order of (az_o)½. This refers to focusing of reflections near the receiver.

Fig. 9. The normalized variance of the received power, plotted as a function of I/T²,for increasing values of φ_o between 0.2 and 4.

We have calculated the normalized variances for the nine sets of closely-spaced observations, and the values are shown in Table I. They cover a range from close to unity up to about four, with four sets at about 2.5.

In all cases it would appear that φ₀ > I, and that in the cases of sets A, B, D, F and TI, the horizontal correlation length T is in the region of (az₀)½.

The calculation of the variance has enabled us to place a lower limit on the vertical scale of the roughness, such that a > 0.25 m. We have an estimate for T²2a, which can give us values for T and a, when combined with values for the slopes, 2a/X.

Our third independent direction of approach for deriving the bedrock characteristics is through examination of the horizontal autocorrelation functions of the received signals. Though Berry (1973) states categorically that the “spatial fading ... is not a potential source of extra information about the form of the rough surface”, Harrison deduces that in many cases the horizontal autocorrelation length of the echo power is proportional to that of the surface roughness. We resolve this conflict as follows: Berry speaks of the rate of fading as a function of arbitrary delay lime, and states correctly that, in the case of very long-tailed echoes, it is uniquely defined by the delay considered. However, for short-tailed echoes with a pronounced maximum, delay times are not easy to measure accurately. The fading rate of particular features of the echo such as the maximum is a sensitive indicator of the angular spectrum of received power, which can then be related to that of the surface roughness. In S.P.R.I. experience, the condition for very long-tailed echoes is only frequently fulfilled in the case of reflections from heavily crevassed ice surfaces. It represents an extreme case, where the r.m.s. slope ß considerably exceeds the angle (p/z₀)^½ . Put in this context, Berry’s statement can be seen to agree with Harrison’s treatment of this case for the geometrically defined “first return”.

Fig. 10. Autocorrelation functions for nine sets of echo strength measurements at 1 m intervals.

The autocorrelation functions of received power have been calculated for the nine data sets, and are shown inFigure 10, up to a separation of 100 m. The “autocorrelation length” τP is defined as that separation of points for which the correlation coefficient falls below I/e. Values of τ_P obtained from these calculated functions are shown in Table I, and can be seen uniformly to exceed the minimum values λ₀ ^½2^3/2πp^½ defined for the first return, and also shown in Table I. We may, as a result, be more confident of our previous measurement of the mean received power.

We may now derive a second estimate for T|φ₀ , which we believe to be an improvement on that derived previously from the fall-off of power with angle. This estimate depends on a large number of measurements of the well-defined maximum level, rather than of the comparatively ill-defined position at which the power has fallen off by a fixed large factor (in that case 40 dB).

Harrison shows that, if ø₀ > 1, and τ_P for the first return exceeds its minimum value, then

These new values for T/ø₀ and β are given in Table I. The difference indicates that returns at angles between 25° and 40° to the vertical are of higher intensity than would be predicted by observation only of the first returns. We tentatively suggest that this is due either to the effect of refraction of power into the ice at about this angle, or to the presence of small morainal boulders at the base of the ice on an otherwise gently undulating surface.

The autocorrelation lengths for the two western sets of observations are shorter than for the other seven. This tends to support our previous suggestion that the western bedrock is of different type from that in the east.

Estimates of a and T are shown in Table I. Those derived for set B present an unresolved problem, in that a surface with such a small scale of vertical relief could not be expected to give rise to the observed value of variance, namely 2.9. However, the use of the condition for vP’≈ as I only gives an order-of-magnitude estimate for T, a and φ₀ , and we are reminded not to place too much weight on the actual figures derived by this method.