Introduction
Electrical resistivity measurements on ice shelves are very rare. Prior to the present work, only two such profiles had been completed, the first near Roosevelt Island (Reference HochsteinHochstein, 1967), the second on the McMurdo Ice Shelf (Reference Hochstein and RiskHochstein and Risk, 1967). But the McMurdo Ice Shelf is thin, has a regime quite different from that of the Ross Ice Shelf, and is modified by brine soaking within the ice column. Thus the usefulness of resistivity studies, strongly suggested by Hochstein’s earlier work, remained untested.
The measurements reported on here were made as part of the ongoing Ross Ice Shelf Geophysical and Glaciological Survey (RIGGS), a survey of the entire Ross Ice Shelf that includes measurements of ice thickness and surface strain-rates. The parent Ross Ice Shelf Project (RISP) will, from a program of drilling through the ice, yield well-determined values of density and temperature as a function of depth. Resistivity measurements have been undertaken as an opportunity to take advantage of these known parameters to learn more about the electrical characteristics of Antarctic firn and ice, and the effect of possible bottom melting or freezing on resistivity profiles. The initial resistivity profiles reported on in this paper were carried out where a number of other geophysical and glaciological measurements yielding reasonably accurate density-depth information were made, but not at a present (or probable future) drill site. Temperatures are thus unknown. The observations have therefore been compared with resistivities to be expected on the basis of calculated temperature-depth curves. The analysis leads to a good determination of the actual resistivity, rather broad limitations on acceptable values of the activation energy and the bottom melt/freeze rate, and a fair insight into the possible resolving power of resistivity profiling when appropriate temperature models are better known.
Field Measurements
The resistivity soundings were made near the RIGGS I base camp (station BC) in late December 1973 and January 1974. Measurements were made along two perpendicular lines, called Profile A and Profile B, with a common center; Profile A (Fig. 1) was taken along the direction to station J9 (the initial RISP drill site), roughly parallel to the direction of ice flow. Measurements were made using both Schlumberger and equatorial-dipole arrays (Fig. 2), the results of which should be nearly identical for a simple one-dimensional variation of resistivity in the ice, that is, a resistivity which varies only as a function of depth. The lines were extended to a maximum separation (a in Fig. 2) of approximately 600 m along each profile.
The current source was a bank of 45 V dry cells, producing a maximum of 810 V. 1.2 m copper rods were used for both current and potential electrodes at larger spacings; nails were used instead at short spacings to maintain a satisfactory ratio of spacing to electrode depth. Experiments were made with multiple electrodes in an attempt to reduce the electrode contact resistance, but these were not very successful. The best success in reducing contact resistance was gained by soaking the firn surrounding the electrode with salt water. Potentials were measured with a Keithley model 600A electrometer, having an input impedance of 1014 Ω. No absolute calibration of the electrometer was possible during the actual measurement, but comparisons with an electronic voltmeter and two cathode-ray oscilloscopes in the base-camp laboratory all showed agreement within 5%. Copper wire with a very high-resistance insulation, of the same type used for resistivity measurements on land, was employed.
Fig. 1. Map of the Ross Ice Shelf showing the locations of stations BC and J9 and the orientations of the resistivity profiles. The rectangular network carries grid coordinates ; circular arcs and radiating lines are geographic coordinates. The light dashed line marks the approximate grounding line between the ice shelf arid the West Antarctic ice sheet. Gaps in the grounding line indicate probable ice streams.
Fig. 2. Diagrams of the two types of array used in the resistivity measurements.
Both ground voltage and current were usually observed to decrease after the current circuit was closed, often by as much as an order of magnitude over a period of about 10 s. However, the ratio of voltage to current showed no consistent change with time, suggesting that polarization and induction effects were not seriously affecting the apparent resistivity values.
The principal difficulty in making resistivity measurements in such, a highly resistive medium as the ice is to maintain an adequate signal-to-noise ratio; noise levels must be minimized. Man-made electromagnetic signals from any kind of local radio broadcast (communications system, aircraft beacons, etc.) interfered seriously with the measurements and had to be stopped. The effect of natural micropulsations was highly variable. About one-third of the time the micropulsation level was negligible, about one-third of the time it caused noticeable drift of the zero point in the measurements, and about one-third of the time it was bad enough that measurements could not be made at all.
Data Reduction
The basic data comprised sets of current I and voltage difference V measurements, each set consisting of several (usually six) series of simultaneous readings of I and V as they decayed, the direction of current flow being reversed for each new series. For each arrangement of the electrodes a plot of V versus I was drawn, defining (ideally) a straight line, the slope of which was taken as proportional to the apparent resistivity (e.g. Fig. 3a). (Using slopes rather than mean V/I ratios gives the heaviest weight to the strongest signals, which presumably represent the largest signal : noise ratios.) Not all sets satisfactorily defined single straight lines. In a few cases, owing either to a non-zero background voltage in the ice or to imperfect zeroing of the electrometer, there was a separation between points corresponding to opposite directions of current flow. In that case, the data were accepted if they could be satisfactorily fitted by two parallel straight lines with numerically equal V-intercepts of opposite sign (Fig. 3b; Fig. 3c shows an example of an unacceptable set). Where I varied too little to define satisfactory regression lines (e.g. Fig. 3f), means of the V/I ratios were accepted measures of the apparent resistivities.
Fig. 3. Sample plots of potential difference (V) versus current (I). The individual measurements are identified by S for Schlumberger array, D for dipole array, subscripts A and B denoting the profile direction, and two numbers in the form a/b indicating the separations a and b. Open and solid circles denote current flow in opposite directions; small numbers indicate the number of readings plotted at the same point on the graph. Note that the scales are not the same for each plot.
The scatter exhibited by the various sets differed considerably (e.g. Fig. 3a and 3d) ; sets were still accepted so long as a regression line was well defined, since even scattered data yield standard errors which are small on the compressed scale of an apparent resistivity plot (e.g. Fig. 4). As electrode separations increased; the current tended to decay more slowly, sometimes becoming nearly constant (Fig. 3e). At the largest separations the currents were typically of equal magnitude in opposite directions, but the voltages in the two directions were often decidedly different (Fig. 3f). The electric field strengths corresponding to the imbalances in measured voltage were in the range of a few tens to a hundred millivolts per kilometer, reasonable values for natural telluric potentials.
Fig. 4. Plot of all apparent resistivity data points. Profile A. Error bars show standard deviations from linear fits to plots like those in Figure 3. Heavier bars indicate a Schlumberger array, lighter bars a dipole array. The solid curve is calculated from Model 1 (see Table II).
Apparent resistivitiesFootnote *,pa, were calculated from the mean values of V/I according to the general formula
wherein K has different values, K s and K d respectively, for the Schlumberger and dipole arrays.
The appropriate second-order K-factors for substitution in Equation (1) are
and
(Reference Keller and FrischknechtKeller and Frischknecht, 1966 ; K s is given by their equation (75), and K d is readily derivable from their equation (72)). The spacings a and b have somewhat different meanings in the two cases (Fig. 2 and next paragraph).
In the numerical modeling described below, apparent resistivities have been calculated using the equation appropriate to the Schlumberger array
where (∂U/∂r) a is the horizontal gradient of the normalized potential (see Appendix) at distance r = a from a single electrode, and a is the separation (half-spacing) of the current electrodes (Fig. 2). Equation (2) is valid to the second order in a2/b2, where b is the potential-electrode separation (i.e. the length of the potential dipole). For the equatorial dipole array, the apparent resistivity is again proportional to the potential gradient, but a second-order correcting term is required :
where b now represents the (identical) lengths of both the potential dipole and the current dipole, and a is the distance between their mid-points (Fig. 2). To avoid recalculating model apparent resistivities for the dipole array using Equation (3), thus requiring separate model fitting for the Schlumberger and dipole data, we have included the second-order factor from Equation (3) in the K factor, giving (to second order)
When Kd’ is used in place of K d in Equation (1), the dipole data as well as the Schlumberger data can be compared directly to models calculated from Equation (2).
Apparent resistivities along Profiles A and B are shown in Figures 4 and 5 ; Schlumberger-array and dipole-array data along each profile have been plotted together. (The solid curves refer to a model discussed later.) A large scatter in observed values at the shorter distances is obvious; less obvious but still clear (especially in Figure 4) is a tendency for the dipole-array points to fall below the others at intermediate distances. Both of these characteristics can be explained if there are local inhomogeneities in the firn, such as ice lenses, wind crusts, or radiation crusts, that provide short conductive paths. These would naturally become more disturbing as the electrode separation was decreased, and, being local, would result in apparent resistivities that were not reproducible with electrodes in slightly different locations. It would follow further that the largest apparent resistivities at a particular separation probably best represent the bulk resistivity of the firn.
Fig. 5. Same as l-’igure 4, Profite B.
Assuming that near-surface inhomogeneities are indeed distorting the results, we have considered, for modeling purposes, only data for which the current-electrode spacing was greater than a certain minimum. That minimum was chosen, on the basis of consistency of the data, to be 10 m for Profile A and 8 m for Profile B. Thus, all measurements at plotted separations less than the minimum and, in addition, dipole measurements at separations less than 100 m on Profile A and 60 m on Profile B, were excluded from succeeding figures (except Figure 16 in the discussion of anisotropy). (Another reason not to use data at separations less than 8 or 10 m is that they are affected by seasonal warming in the upper few meters of the firn, which causes the observed decrease in p a toward very small separations. Analysis of those data, which should yield information about the activation energy in the firn, will be carried out separately.) Right at 10 m on Profile A, only the higher of two points that differed by 50% was retained.
Three Schlumberger-array points on Profile A at relatively large separations (150 m, 200 m, and 350 m) were rejected simply because they showed discordantly high apparent resistivities compared with the rest of the data (Fig. 4), despite perfectly satisfactory V versus I plots. We have no specific explanation to offer for these discrepancies, but all three had much shorter voltage-electrode spacings than other, apparently satisfactory measurements at the same separation, suggesting that signal levels were below some instrumental noise level. No such problem arose with the Schlumberger-array measurements on Profile B or with dipole-array measurements on either profile. The selected points are listed in Table I and shown in Figures 6 and 7.
Table I. Accepted values of apparent resistivity standard deviations quoted are from linear fits to plots of V against I such as those in Figure 3.
Fig. 6. Selected data, Profile A, together milk models showing the effect of changing bH (Models 1, 6 and 7).
Fig. 7. - Same as Figure 6, Jar Profile B.
Three characteristic zones can be seen in the apparent resistivity data (and models, discussed below) at distances greater than 10 m. First, from 10 m to 100 m, is a region of rapidly decreasing ρ a, corresponding to the strong effect of the increasing density with depth in the upper 50 m of the ice shelf on the actual resistivity in the ice (Fig. 13). The effect of temperature in this zone is secondary, because the temperature change over this top 10% of the ice thickness is at most only a few degrees (Fig. 11). The marked decrease in slope at around 100 m is a direct result of the marked decrease in the density-depth gradient at the firn-ice boundary. In the next zone, from 100 m to about 500 m, the primary factor is the temperature gradient between depths of 50 m and 250 m in the ice. At distances greater than 500 m, in the third zone, apparent resistivities fall off increasingly more rapidly with distance owing to the increasingly important effect of the highly conductive sea-water beneath the ice shelf.
Analysis
For the ease of computation, Reference HochsteinHochstein (1967) fitted his apparent resistivity curve on the shelf ice with a sequence of layers of constant resistivity. A sequence of layers, however, does not well represent the real resistivity variation to be expected in an ice shelf. Because of the strong dependence of resistivity on density and temperature, and the continuous variation of each with depth, a model incorporating a continuous resistivity-depth function p(
) is much to be preferred. We have therefore developed a computer program to calculate apparent resistivities on an ice shelf conforming to such a model. The relevant theory is laid out in the Appendix. The net result is a program which yields curves of apparent resistivity as a function of the vertical gradient d(ln p)/d
and the surface resistivity p
o ≡ p(0), the latter appearing only as a simple factor so that p
a/p
o is independent of p
o. This means that changes of p
o do not change the shape of any of the model curves of p
a or p(
) shown in the figures of this paper, which all employ logarithmic scales, but merely raise or tower them.
The effect of temperature on the resistivity of solid ice is assumed to be in accordance with the Arrhcnius function:
where E is the activation energy, k Boltzmann’s constant (8.62 × 10−5 eV K−1 = 1.38 × 10−23 J K−1), and T the absolute temperature. The corresponding term in d(ln p)/dz is — (E/kT 2)(dT/dz). For specification of the temperature-depth function wc employ the analysis ofReference Crary Crary (1961[a]) for a steady-state ice shelf of thickness H:
where
α is the thermal diffusivity, b the accumulation rate in thickness of ice, and the subscripts o and H refer to the surface and the base of the ice, respectively. (Equation (4) docs not take into account the effect on advection of lower values of the density and diffusivity in the firn layers; inclusion would cause a modification of model ρ a curves similar to that from doubling b 0, which is negligible for the purposes of this paper; see the Discussion, below.)
The dependence of p on density is less well defined than its dependence on temperature. Following Reference Glen and ParenGlen and Paren (1975), we have tried two different equations that give the relative permittivity ∊ of a mixture of two different components of relative permittivities ∊1, and ∊2: Looyenga’s equation (Reference LooyengaLooyenga, 1965)
and Böttchcr’s equation (Reference BöttcherBöttcher, 1952)
In each, v is the volume proportion of the second dielectric. Both are based on the modification of solutions of Laplace’s equation brought about by the introduction of small bodies of one permittivity into a medium of another permittivity. They differ in the permittivity contrast between the bodies and the surrounding medium ; Looyenga’s analysis appears better because of a differential approach that keeps the two permittivities always nearly equal during the gradual admixture. Glen and Paren point out the likelihood that these equations are at least mathematically valid for complex permittivities. We have extrapolated further to assume their applicability when the imaginary part of ∊, (ω∊0p)−1, where ∊0 is the permittivity of free space, becomes very large with respect to the real part, i.e. when the frequency cu becomes very small. This leads, in the limit, to the case of d.c. conduction. Even though conduction currents replace displacement currents, Laplace’s equation still holds in a grossly homogeneous medium if there is no free charge accumulation. In the firn layers, however, p is a strong function of
so ∇·E is not zero (see Appendix, Equation (A2)) and Laplace’s equation is not valid. Looyenga’s and Böttcher’s equations thus do not lie on a firm physical foundation. Nevertheless, lacking a better theory, we try them on an empirical basis. Equations (5) and (6), when applied to an air-ice mixture, reduce simply to
ρfirn = ρice/v3,
for Böttcher’s equation, where v is now the ratio of firn density to ice density. The corresponding contributions to
The effect of pressure on the resistivity of the solid ice is negligible in the ice shelf. At the base of an ice shelf 500 m thick the pressure is less than 50 bars (5 MN m−2), corresponding to a resistivity change of less than 1% (Reference Chan, Chan, Davidson and Whalley.Chan and others, 1965). However,Reference Hochstein Hochstein (1967) found that an empirical relationship based on pressure appeared to fit measured resistivities on firn cores fairly well down to the sampling depth limit of 15 m, and he suggested the extrapolation of that relationship to greater depth. We have, therefore, tried his model
where p is the hydrostatic pressure, as a substitute for a direct density dependence. The corresponding contribution to d(In p)/d
is (— 0.35/p) dp/d
Application To Ross Ice Shelf Measurements
To apply the analysis to the field measurements, the variation of density with depth is needed. Measurements at station BC extend only to a depth of 10 m ; for greater depths in the firn, the density was calculated from the variation of seismic compressional-wave velocity with depth, using a velocity-density relationship developed by Reference KohnenKohnen (1972). The wave velocities were in turn determined from seismic refraction shooting along two profiles that coincided with the resistivity profiles. The velocity-depth functions that were found (Reference RobertsonRobertson, unpublished) were slightly different in the two directions, the velocities (and hence the calculated densities) being higher along Profile A for depths less than 50 m (Fig. 8). The velocity differences are large enough to be real, indicating some seismic anisotropy in the firn.
Fig. 8. Density-depth plats calculated from seismic refraction shooting (lines) and density measurements on ice cores from station J9 (circles).
We have adopted densities calculated along Profile B, because anisotropy is more likely to raise the velocity above the isotropic value than to lower it, and because Profile B yielded densities that were closer to actual density measurements (Reference LangwayLangway, 1975; also shown in Fig. 8) in a hole drilled in November 1974 at station J9 (Fig. 1). Had densities been calculated from seismic velocities along Profile A rather than Profile B, model apparent resistivities would have been about 10% higher between roughly 30 and 100 in, the difference diminishing rapidly outside that range. Such a change would not significantly alter the model fits.
Use of Crary’s equation (Equation (4)) for estimating the temperature distribution required numerical values for several parameters. The ice thickness H was taken to be 493±10 m from radar soundings by J. W. Clough (Reference RobertsonRobertson, unpublished); we took the surface balance rate b0 to be 0.08 m year−1 as determined by Reference Clausen and DansgaardClausen and Dansgaard (in press), from identification of radioactive fallout horizons; the basal temperature T H was assumed to be —2°C, the freezing point of sea-water, and the thermal diffusivity α was taken to be 1.2 × 10−6 m2 S−1 (Reference CraryCrary, 1961 [a]). Of these quantities, the only one about which there is great enough uncertainty to be of possible significance is b 0; the values found throughout the region byReference Clausen and Dansgaard Clausen and Dansgaard (in press) were only half as great as those determined from pit studies on I.G.Y. traverses (Reference Crary, Crary, Robinson, Bennett and BoydCrary and others, 1962). This uncertainty turns out to be of minor, if not completely negligible importance (see Discussion, below).
We treat the activation energy E and the basal balance rate bH , as parameters to be varied in attempts to fit the observed apparent resistivity curve.
Model Fitting
As an initial model, we chose E = 0.25 eV (24 kj mol−1) in accordance with measurements on ice-sheet ice from “Byrd” station and elsewhere (Reference Fitzgerald and ParenFitzgerald and Paren, 1975; Reference Glen and ParenGlen and Paren, 1975), bH = 0, and Looyenga’s equation for the density effect (Model 1; for a list of all models considered in this paper see Table II). p 0 was varied to provide a best fit of the model to the observations at separations greater than 100 m corresponding to conduction principally in solid ice. The fit to both profiles at those separations is very good (middle of the three curves in Figs 6 and 7) showing that the chosen values of E and bH are completely in accord with the data. The value of p 0 however, is 12% higher for Profile A than for Profile B. Since changes in p0 do not change the shape of the pa curves, we may combine the results of Profile B with those of Profile A simply by shifting them uniformly upward by 12% (Figs 10, 12 14, and 15) for study of the average characteristics of the ice. We will consider later the question of the apparent anisotropy.
Table II. Apparent resistivity models The figure numbers quoted in the last three columns indicate where plots resulting from the models can be found.
At separations less than 100 m the model curves reflect principally the effect of density variations. The fit to Profile A appears satisfactory, but that to Profile B and to the combined profiles is not so good, particularly if it is true that the higher apparent resistivities at a particular distance are to be preferred.
Choice of model of density dependence
We next tried Böttcher’s equation for p(v) and Hochstein’s equation for p(p) (Models 2 and 3; Figs 9 and 10). Model 3 is entirely wrong in shape, with too small a resistivity gradient above 50 m and too large a gradient below. Clearly, the resistivity is a function of density, as expected, rather than of pressure. Hochstein’s relation has therefore not been considered further.
Model 2, matched to Model 1 at distances greater than 200 m, has the right shape but insufficient range in pa. The reason is that d(ln p)/d
is steeper for Model I in the critical depth range of 20–100 m. It appears that Looyenga’s equation fits significantly better than Böttcher’s. Even a large bottom-freeze rate (bH
> 0), which would greatly increase the temperature gradient in the upper part of the shelf (e.g. bH
= + 0.5 m year−1, Fig. 11), would not increase ρa
enough at small separations (cf. curves for bH
= o m year−1 and bH
= 0.5 m year−1 in Figs 6 or 7) to bring Model 2 into agreement with the observed apparent resistivities.
Fig. 9. Resistivity versus depth in the ice shelf according to three different models of resistivity as a function of density (Models 1, 2 and 3).
Fig. 10. Apparent resistivity curves following from the three models in Figure 9, compared with combined data from Profiles A and B.
Fig. 11. Calculated temperatures versus depth assuming b0 = o.o8 m/year and bH as indicated (Models 1, 6 and 7).
One other factor must be considered, however. The activation energy appropriate to the firn is probably much higher than that for ice.Reference Kopp Kopp (1962) found values of 0.7 eV (67 kJ mol−1) for compressed fresh snow, around 0.9 eV (87 kJ mol−1) for compressed granular snow and from 0.8 eV (77 kJ mol−1) to as high as 1.4 eV (135 kJ mol−1) for samples from depths of 10 to 40 m in the Greenland ice sheet; his observations for compressed snow have recently been confirmed byReference Fitzgerald, Fitzgerald, Glen and Paren. Fitzgerald and others (in press). Kopp associates the high activation energies with conduction in the quasi-fluid film which exists on the outer surface of the ice grains. To take this factor into account, we have tried a simple model in which the activation energy is taken to be 1.0 eV (96 kJ mol−1), corresponding to grain-surface conduction, down to the firn-ice boundary at a depth of 40 m, and 0.25 eV (24 kJ mol−1), corresponding to bulk conduction, at greater depth. The result (Fig. 12) is an increased slope in the calculated apparent resistivities at separations less than 100 m that brings the “Looyenga model” (Model 4) into remarkably close agreement with the observations, while still leaving the “Böttcher model” (Model 5) substantially too low. Only if there is a temperature gradient in the upper 50 m of the ice large even in comparison with that calculated for a 0.5 m year−1 bottom accumulation rate, could the “Böttcher model” be brought into agreement with the observations. Further consideration of this problem will have to await resistivity measurements at a drill-hole site where the temperature variation with depth is known.
Fig. 12. Apparent resistivity curves for Models 4 and 5, showing the effect of taking ∊ = 1,0 eV (96 kJ mot−1) near the surface. Data are combined from Profiles A and B.
Despite the good agreement with the observations, it should be borne in mind that the adoption of Looyenga’s relation for d.c. conduction in the firn layers docs not rest on a firm physical foundation, particularly if conduction along grain surfaces is indeed the dominant mechanism, and that close agreement, therefore, is not necessarily to be expected. Thus in examining the effects of changing various parameters, care must be taken not to be influenced by the goodness of fit between models and observations at distances less than 100 m. Partly to emphasize that fact, the remaining modeling in this paper is done on the basis of values of E that are constant throughout the thickness of the ice shelf.
Changes of bH, E, and b0
The rather large values bH
= ±0.5 m year−1 (Models 6 and 7) yield the temperature profiles, p(
) functions, and apparent resistivity curves shown, together with corresponding curves for bH
= o, in Figures 11, 13, 6, and 7. The three apparent resistivity curves have been matched, arbitrarily, at a distance of 200 m, which also has the effect of providing about the best fit for each curve. Although the differences between models are slight, the curve for bH
= 0 docs appear to fit both profiles (Figs 6 and 7) significantly better than those for rapid rates of bottom freeze or bottom melt. If other assumptions are correct, the resistivity measurements thus indicate that the bottom balance rate is numerically less than 0.5 m year−1.
Fig. 13. Resistivity versus depth for Models 1, 6, 7 and 8.
Fig. 14. Models of apparent resistivity showing the effects of changing ∊ and b0 (Models 8 and 9). Data are combined from Profiles A and B.
Hochstein’s (1967) analysis of resistivities near Roosevelt Island led to an estimate of £ between 0.3 and 0.5 eV (29-48 kJ mol−1) ; we have tried E = 0.4 eV (38 kJ mol−1) (Model 8) with the results shown in Figures 13 and 14. Model curves are again matched at 200 m. The fit for E = 0.25 eV (24 kJ mol−1) is clearly superior to that for E = 0.4 eV (38 kJ mol−1) if the bottom balance rate is small.
Because of the discrepant estimates of surface balance rates mentioned above, an apparent resistivity curve has been calculated for a model in which b0 has been doubled, to 0.16 m/year (Model 9). The change in the calculated resistivities (Fig. 14) is imperceptible at distances greater than 200 m, and trivial at all distances.
Since E and bH are both related to the temperature effect on the resistivity, one might expect to be able to vary both of them in such a way as to maintain a fit to the data. This is indeed the case, as seen from Models 10 and 11 (Fig. 15). The decreased temperature gradient in the upper half of the shelf that results from increasing the bottom melt rate can be compensated for by increasing the activation energy; conversely, the effect of bottom freezing with the attendant increase in the upper temperature gradient can be balanced by a smaller activation energy. In the light of other evidence, however, the most likely combination is E close to 0.25 and bH close to 0.
Fig. 15. Models of apparent resistivity showing simultaneous changes in ∊ mid bH (Models 10 and 11),
Absolute Resistivity
Up to this point we have been discussing the shape of the apparent resistivity curves, i.e. resistivity ratios relative to an unspecified p0. We now wish to consider the actual magnitude of the resistivity in the solid ice. There is no simple correspondence between a measured apparent resistivity and the actual resistivity at any particular depth—pa at a particular distance depends on p over a whole range of depths. Thus to estimate p at a particular depth we must assume a specific model. (Note that the model chosen need not correspond to the correct melt/freeze rate or activation energy; it need only yield the correct resistivities.)
The best fit to the observations comes from Model 4 (Fig. 12), which indicates an actual resistivity at a depth of loom of 7.4 × 104 Ω m according to Profile A, and 6.6 × 104 Ω m according to Profile B. (100 m was chosen for the depth simply as a round value within the range where the actual resistivities strongly affect the apparent resistivity at the separation (200 m) at which models and observations were matched.) We can get an idea of the uncertainty in the resistivity by considering Models 1, 10, and 11 (Figs 6 or 7 and 15), which yield apparent resistivity curves very similar to each other but falling significantly below Model 4. The resistivities at 100 m according to these models lie within ±1% of 6.3 × 104 Ω m (for Profile A), 15% less than the estimate from Model 4. In view of the substantially better fit of Model 4 to the observed data, the author believes that the corresponding average resistivity, 7 × 104 Ω m, should be in error by less than 10%.
The temperature at 100 m depth, assuming steady-state with bH = o, is —23°C. The corresponding average conductivity, 1.4 × 10−5 Ω−1 m−1 is just about twice the values found byReference Hochstein Hochstein (1967) from his resistivity profiles on Roosevelt Island and immediately adjacent on the Ross Ice Shelf, and by Reference Fitzgerald and ParenFitzgerald and Paren (1975) from laboratory measurements on samples from the deep core at “Byrd” station, although well within the range of other measurements from cold, polar glaciers and ice sheets (Reference Glen and ParenGlen and Paren, 1975).
The accuracy of the resistivity measurements on Antarctic ice both in the field and in the laboratory is certainly great enough that a factor of two must have physical significance, i.e. that the ice at station BC is truly about twice as conductive as the ice at “Byrd” station to the east and the ice on and near Roosevelt Island to the north. There is no reason to suppose that the snow from which the ice at station BC formed was inherently more conductive than that around Roosevelt Island or “Byrd” station (since the ice at 100 m depth is about 1 000 years old, it presumably fell on the surface near the Siple Coast grounding line about 200 km upstream). Indeed, one might instead expect ionic impurities derived from the Ross Sea (Reference LangwayLangway and others, 1974) in the snow at Roosevelt Island to increase conductivities there, but there is no evidence of such an effect. The conductivity difference therefore probably reflects some difference in the densification or strain history of the ice in the different locations. Although it is difficult to say just what the significant difference in history might be until the basic cause for the remarkably high conductivity of polar ice in general is better known, resistivity profiles might nevertheless be useful in tracing flow lines in ice shelves, where movement rates are great enough that ice at a few hundred meters depth originated a significant distance up-stream.
The difference in the apparent resistivities along the two perpendicular profiles is also presumably a real physical phenomenon, but its significance is not clear. When the apparent resistivities from Profiles A and B are plotted together as measured (Fig. 16), rather than with an adjustment to match at a separation of 200 m, no consistent difference at separations less than 30 m appears, whereas an apparent anisotropy seems to be fully developed at distances greater than that. At a separation of 30 m the current flows almost entirely above the firn/ice boundary, so it is unlikely that any true crystalline anisotropy associated with a strongly developed preferred orientation of c-axes is responsible. It is probably significant in this regard that the field measurements at 30 m on Profile B were sufficiently disturbed to preclude any valid determination of the apparent resistivity (according to the criteria given above), and that the measurements at 35 m were still not of the highest quality. (No such difficulty was observed at 35 m on Profile A; unfortunately, measurement was not made at 30 m.) This suggests that there is some kind of conductivity anomaly at a depth of a few tens of meters or less—perhaps a healed crevasse. (The nearest present-day crevasses in a generally up-stream direction are about 70 km (300 years) away, corresponding to a depth of burial of about 40 m.)
Fig. 16. Apparent resistivity data points es measured on the two profiles, without adjustment for anisotropy. The highest value at each separation from each profile has been plotted. Profile A : open circles; Profile B: solid circles.
Discussion
Resolution of temperatures and basal melt/freeze rate
Model fitting led to the conclusion that the bottom balance rate lies between +0.5 m year−1 and -0.5 m year−1—the uncertainty is too large for the result to have much glaciological significance. However, the resolving power of the apparent resistivity models with respect to dT/d
and hence to bH
would be greatly improved if the models could be matched at 10 m, or even 100 m, instead of 200 m. This should be possible when resistivity measurements are completed at a site of known T(
), so that the value of the activation energy can be confirmed and the relationship between resistivity and density better determined. (Such measurements are planned for the RISP drill site in 1976-77.) Thereafter, it may be possible in other localities to determine T in the upper half of the ice shelf to within a degree or two, and thus estimate bH
with a resolution of perhaps ±0.1 m year−1; both are capabilities that would be of distinct glaciological value.
For two reasons, however, there is no part of the apparent resistivity curve that reflects at all sensitively the temperature gradient in the lower part of the shelf. In the first place, resistivities in that part of the shelf are largely masked by the effect of the sea-water, and in the second, the rapid decrease in apparent resistivity with distance causes a rapid decrease in potential difference between measuring electrodes as their separation is increased thus lowering the signal into the noise level. Even if the range of measurements could be increased by some means of enhancing the signal : noise ratio, the first problem remains ; it seems unlikely, therefore, that resistivity profiles can ever be used to examine directly the temperature variations in the lower half of an ice shelf. Only a major change in the resistivity of the ice, either upward or downward, might be detectable. An order-of-magnitude decrease near the bottom, such as could result from the freeze-on of saline ice, might be detected, whereas a large increase in the resistivity at the base of the shelf, even in a layer only ten or a few tens of meters thick should be readily detectable. (An increase in apparent resistivity at distances greater than 600 m does indeed appear on preliminary results of a resistivity profile south of Roosevelt Island, but the data have not yet been analyzed in detail.)
Steady-state assumption
The assumption was made in the analysis that the ice shelf is in steady-state. If it is not, and there is good reason to believe it may not be (Reference ThomasThomas, 1976), then Crary’s equation for the englacial temperatures (Equation (4)) is no longer valid. In that case, analysis of apparent resistivity curves by the procedure we have outlined here can still proceed, leading to a steady-state temperature profile that approximates the actual, transient profile. Alternatively, the calculation of temperatures in our computer program could be modified in accordance with any specific model of transient dynamic behavior of the ice shelf. The resistivity data could thus be very useful in distinguishing between dynamic models that imply different temperature-depth curves, particularly if the bottom balance rate were known independently from ice-thickness, movement rate, and surface-strain measurements.
Acknowledgements
Field measurements were made principally by B. K. Sternberg and by S. S. Brandwein. Other members of the geophysical field party, who aided in various ways, were J. W. Clough, J. D. Robertson, L. R. Whiting, and T. M. Kolich. Computer programming for the numerical analysis was carried out by A. B. Schubert. The author is grateful to R. H. Thomas for providing various data in advance of publication, and for useful discussions on the ice-shelf regime. The research was supported by National Science Foundation grant GV-36963.
Appendix
Theory of Apparent Resistivity
The differential equation governing the electrical potential from a single, d.c. point source of current in a conducting medium may be very simply developed. Ohm’s law gives
where E is the electric field, p is the electrical resistivity (in general a function of all coordinates) and j is the current density. Conservation of electric charge requires, for steady state
∇·j = o.
whence, from Equation (AI)
Let the electrical potential u be defined bv
then Equation (Aa) becomes
which can be rewritten
For application to an ice shelf, we assume that p varies only with depth z. We also adopt cylindrical coordinates. appropriate to the point source. Since we are interested in voltage : current ratios, it is convenient to define a normalized potential U = 2πu/I, where I is the input current. By symmetry. U will be a function of z and the radial distance r only. Then, from Equation (A4)
Next, consider the boundary conditions appropriate to the ice shelf. The geometrical arrangement is shown as (a) in Figure Al ; the requirements are that there be no vertical component of current at the surface, whence ∂U/∂z at z = o; that the potential and its normal derivative be continuous at - z = H; that (by symmetry) ∂U/∂z at z = o; and that U → o as r → ∞.
Fig. A1. Diagram of the coordinate system and boundary values for the solution of the electrical potential equation.
The solution, call it Ui in the simple case of constant resistivity pi in the ice and pw in the water is well known (e.g. Reference SundeSunde, 1949, section 2.5):
Where Fo is the zero-order Bessel function, and μi = ρi— ρw)/(ρi+ρw;)
When p varies arbitrarily with z, an explicit solution cannot be written; some sort of approximate solution must then be sought for any particular pz). The usual approach is to approximate pz). by a series of layers, or by some simple function for which an explicit solution can be found. The approach taken here is different—to find a numerical solution to the differential equation (A5) with the exact pz). desired.
For numerical analysis, two of the boundary conditions must be expressed differently. Since we have no analytical solution to Equation (A5), we know neither the potential nor its normal derivative at z = H. Fortunately, the resistivity of sea-water (≈1/3 Ω m) is orders of magnitude less than that of glacier ice (→ 104 Ω m), so that U is very small at z = H. For analytical purposes it would suffice to take pw). - — o, hence U = o, at Z = H, but this is not convenient numerically. Instead we assume that U(r, II) = [Ui(r,H)] pi = p0 = U0(r, H)], i.e. that U differs insignificantly from the very small value it would have in a shelf with constant resistivity equal to the actual resistivity po at z = 0. (The proportional difference between Ua(r, H) and U(r, H) may be large, but that does not matter since both are very small compared to U(t, o).) To find U0(r, H) in a form useful for numerical calculations, expand the term [I + μi exp (—2λH)] −1 in Equation (A6) and use the identity
to obtain
Then
where in μ0 = [μ i]ρ i = ρ 0
A similar situation obtains for large values of r. Since we cannot apply a numerical condition at r = ∞, we take instead as the boundary r = rI, where r, → H so that the effect of the shelf is small. Again, setting U = o is numerically inconvenient; we choose instead an approximation to the real potential distribution which can be used not only as a boundary condition at r — rI, but also as an initial value for the numerical iterations at all points.
The assumption is that the electric field at any point can be approximated by the field that would exist in the ice if the resistivity throughout were equal to its actual value at that point. Then for the corresponding potential, call it U = o, we have, from Equation (A3),
To build up a U, field in the ice shelf, we start with the approximation (8) and integrate upwards:
Since the series in Equation (A7) is alternating with diminishing terms, it converges uniformly and can be differentiated term by term, whence, using Equation (A9),
The boundary condition at r = ri is simply U(ru z) = U,(r1 z)-
The final step in comparing theory with measurements is to calculate the apparent resistivity pa . For a Schlumberger array, in which the distance between potential electrodes is small compared to that between current electrodes,
where a is the mean distance from potential to current electrodes (Fig. 2).
For numerical calculations we musl consider the source as having finite size ; wc define the potential on a small rectangular notch of dimensions ra and ; Z a. The simplest source potential is that, call it Us, for a semi-infinite
we therefore specify U = Us(r, z) along r = ra, o < z < za and z = za , o < r < ra .
The boundary conditions, including the source function, may be summarized (Fig. Alb):
The problem is now well posed, and solutions may be obtained by finite differences, subject to the specification of d(ln p)/dz. For applying finite differences with a constant grid size, changes of independent variables are convenient : let g =1n r (r in meters) and y= z2/3;. The first provides the logarithmic horizontal scale standard in resistivity work, while at the same time allowing the grid spacing in r to increase appropriately with distance The second provides a much more limited scale expansion since the temperature, and heure p, may be: expected to change rapidly with depth in the lower part of the shelf, too great a scale modification (such as a logarithmic scale) would be inadvisable. The transformation chosen yields grid point intervals in z of one or two meters near the surface, and about io m at the base of the ice.
Boundary condition (e) must now be modified since g -→ — ∞ as r → o; we replace r = o by r = r0, where r0 is small compared with the minimum separation employed in the actual measurements. The other boundary conditions follow by simple substitution.
With these transformations, the differential equation (A5) becomes
The quantity d(ln p)/dz is left untraiisformed to indicate that it is calculated as a function of z.
The numerical calculations leading to the models presented in this paper were carried out on a 50 × 75 grid, with r0, = 0.1 m, r1 = 10 km, and grid spacing δg = In 0.1 and δy = H2/3/75. The source notch size was taken to be 2 × 2 grid intervals, i.e. ra = exp (2 δg) and
a = (2 δy).
