Introduction

The rate of net mass accumulation (A, in kg m⁻² а⁻¹) at the surface (or surface balance) for particular locations in the Antarctic and Greenland ice sheets is derived from firn emissivity (Reference ZwallyZwally, 1977). The derivations are based on a hyperbolic function of emissivity and are applicable to areas of dry-snow and upper-percolation facies (for a definition of terms refer to Reference BensonBenson (1962)). We limit the discussion to areas of dry-snow facies (Figs 1–3). The ice sheets are represented by circles centered at grid-point locations that correspond to the intersection of lines 100 km apart; the origin lines are set at longitudes of 0°–180° and 90° W–90° E for Antarctica, and 45° W and 135° W–45° E for Greenland (horizontal lines have “y” numbers, and vertical lines, referred to as columns, have “x” numbers). Graham Land and eastern Palmer Land in Antarctica, and ice caps which are not attached by ice in Greenland, are excluded from this study. The distribution of the areas of dry-snow facies is adopted with minor modifications from Reference GiovinettoGiovinetto (1964a) for Antarctica and from Reference BensonBenson (1962) for Greenland (Table 1).

Fig. 1. East Antarctica. Circles are centered on grid points 100 km apart, a. Conterminous ice sheet; full circles show locations where net ablation at the surface has been indicated in some compilations, b. Area delimited by the dry-snow line; full links indicate the location of points excluded from the sampled area.

Fig. 2. West Antarctica. Circles are centered on grid points 100 km apart. a. Conterminous ice sheet, b. Area delimited by the dry-snow line; full circles indicate the location of points excluded from the sampled area.

Fig. 3. Greenland. Circles are centered on grid points 100 km apart. a. Conterminous ice sheet. b. Area delimited by the dry-snow line; full circles indicate the location of points excluded from the sampled area.

Table 1. Areas of dry-snow facies. N, number of grid points, each centered on a nominal area of 10⁴ km ². EA, East Antarctica: WA, West Antarctica; G, Greenland

We first compare the derived accumulation obtained at stations (A _ds) for which there are reliable field measurements of accumulation (A _fs) to determine sets of coefficients of the function for each ice sheet, We then derive accumulation values for grid points 100 km apart (A _dg) and compare them with interpolated grid-point values obtained from the latest contoured compilations of held data (A _fg). The comparisons are an indirect assessment of the compilations. The A _dg values are produced by repeatable application of the emissivity method on the basis of evenly distributed remotely sensed data collected in 1973–79 and are means representative of periods of the order of 10 years ending in 1976 (Table 2). The A _fg values are obtained from isopleth patterns drawn following subjective interpolation and extrapolation criteria of unevenly distributed held data representative of periods ranging between orders of 1 and 100 years, although the bulk of the data is characterized by overlapping 10 year means distributed over about 45 years (roughly 1940–85).

Table 2. Approximate periods represented by bulk of each data set

Emissivity Method

Our method for deriving the accumulation rate from satellite measurements of brightness temperature (T _B, in K) and mean annual surface temperature (T _M, in K) at particular locations is based on equations given by Reference ZwallyZwally (1977). The T _B and T _M values are obtained by bilinear interpolation from 30 km × 30 km grid-square Nimbus-5 ESMR data (T _B 4 year mean, 1973–76 (Reference Zwally, Comiso, Parkinson, Campbell, Carsey and Gloersen.Zwally and others, 1983; Reference Parkinson, Comiso, Zwally, Cavalieri, Gloersen and CampbellParkinson and others.(1987)) and Nimbus-7 THIR data (T _B for 1979 (Reference ComisoComiso, 1994)).

The connection between derived values and microwave emissivity (E) is made by combining a solution of the radiative transfer equation with the grain-growth function given by Reference GowGow (1969). The parameter in both equations is the inverse gradient, 1/(dR ³/dz), of the grain-size (R) with depth (z). Reference ZwallyZwally (1977) gives E as a function of s = dR ³/dz from a simplified analytic solution of the radiative transfer equation (cf. Reference Comiso, Zwally and SabaComiso and others, 1982; Reference Van der Veen and JezekVan der Veen and Jezek, 1993). Replotting of the values in Figure 1 of Reference ZwallyZwally (1977) shows that 1/(dR ³/dz) is approximately a hyperbolic function of E,

(1)

Equation (34)of Reference ZwallyZwally (1977). which is derived from the grain-growth function given by Reference GowGow (l969), is

(2)

where k is a constant, p ₀ is the average firn density in the top 10 m, e is the activation energy of the growth process, R is the gas constant, and T is the physical temperature. A value of e/R equal to 5250 with k p ₀ = 10¹⁰ is obtained from fitting Equation (2) to the data of Reference GowGow (1969) on R(z) using T = T _M. Therefore, from Equation (2),

(3)

and

(4)

which is the desired equation, using E = Т _B/Т _M according to Equation (46) of Reference ZwallyZwally (1977).

The emissivity method requires accurate determination of Т _B and Т _M. A change of 1 K in either Т _B or Т _M would result, on average, in a difference of 5% in the derived accumulation rate. Thus, the reliability of any comparison between derived and field accumulation is enhanced if Т _M is determined from the same 30 km × 30 km grid-square THIR database format used to determine Т _B.

Areal and Temporal Variabilities

The only 12 consecutive monthly mean series of surface temperature processed from the THIR database is for 1979 (Reference ComisoComiso, 1994). Although this is a short period. whatever uncertainties are introduced by the areal and temporal variability of surface temperature particular to that year, the value of Т _M determined for 1979 is more representative of actual firn temperature to use with 1973–76 Т _B data than the use of 10 m firn temperatures measured at different times between 1949 and 1990 (Table 2).

The areal and temporal variabilities of accumulation (e.g. Reference GiovinettoGiovinetto, 1964b) contribute to uncertainty in the comparisons between A _ds and A _fs , and between A _dg and A _fg. The contribution to uncertainty by the areal variability cannot be avoided, i.e., A _ds , A _dg are values derived for each point from bilinear interpolations of average values of Т _в and Т _M centered on sensor-grid areas of 900 km², whereas A _fs values are determined from field measurements at a point, and A _fg values are interpolated for a point.

The contribution to uncertainty by the temporal variability is less critical than it may appear because the optical depth of firn (thickness of top strata contributing approximately two-thirds of the total external emission) is approximately 5 m (Reference ZwallyZwally, 1977). Using the firn-density summaries of Reference KojimaKojima (1964) and the accumulation map of Reference Giovinetto and BentleyGiovinetto and Bentley (1985), we estimate that in Antarctica the derived values are means representative of periods with an overall range between 1940 and 1976 (Table 2). In Greenland, from the density profiles shown by Reference BensonBenson (1962) and the accumulation map of Reference Ohmura and Reeh.Ohmura and Reeh (1991), we estimate that the derived values are means representative of periods with an overall range between 1956 and 1975. The overall range of the periods represented in the A _fs data is between 1940 and 1990, and in the A _fgdata between 1940 and 1985. Thus, the time overlap between the derived and field data sets is > 1/2 and < 3/4.

Comparison with Field Data

Initial values of the coefficients a ₀ and a ₁ are chosen by curve fitting (А ехр(5250/T _M)) to the hyperbolic function of T _B/T _M in Equation (4), using A _ds and A _fs values. Preliminary studies of the emissivity method were made using held data from 357 stations in Antarctica and 89 stations in Greenland; the sources of А _fs data are listed elsewhere (paper in preparation by H. J. Zwally and others). The Antarctic A _fs data sets selected were those in which accumulation was obtained by the radioactive- or stable-isotope method, or by the stake-network method; if obtained by the stratigraphic method, only those sets for which the rate could be substantiated by one of the other methods were selected. The A _fs data sets for Greenland do include sets in which the stratigraphic method was the only one used to obtain the rate, i.e., all methods were considered equally reliable.

In the areas of dry-snow facies, values of £ range from 0.63 to 0.88 in the Antarctic data, and from 0.67 to 0.81 in the Greenland data. In the latter data set. the emissivity approaches its asymptotic value of 0.95 Equation (4)) when included in the fit as the third adjustable coefficient. The value of 0.95 is also consistent with the asymptotic value in the solutions of the radiative transfer equations (Reference ZwallyZwally, 1977; Reference Comiso, Zwally and SabaComiso and others, 1982). Thus, the asymptotic value of E is fixed at 0.95, and the coefficients a ₀ and a ₁ are then adjusted to optimize the linear correlation between A _ds values as derived from Equation (4) and A _fs values to give a linear slope of unity and intercept of zero.

The preliminary values for the coefficients were a ₀ = −5 and a ₁ = 6.5 for Antarctica, and a ₀ = −32 and a ₁ = 15.5 for Greenland. The results are summarized in Table 3. The regression coefficient (r) is 0.82 for Antarctica and 0.89 for Greenland. The root mean square (rms) of the residuals is relatively large for Antarctica (90 kg m−² a⁻¹) and relatively small for Greenland (36 kg m⁻² a⁻¹).

Table 3. Summary statistics of Station data (A _ds = f(A _fs)). A, Antarctica; G, Greenland; EA, East Antarctica; WA, West Antarctica

The scatter pattern in the Antarctic data indicated that at least two sets of coefficients a ₀ and a ₁ for Equation (4) are required for Antarctica; one set where the rate of accumulation is low and firn is exposed to large temperature gradients near the surface for several years (e.g. the plateau area in East Antarctica where A _fg 100 kg m⁻² a⁻¹ or approximately 50% of the area of the ice sheet); another where snow grains drift at relatively high speed and for relatively long distance before final deposition (e.g. the interior of West Antarctica and the southern area of major ice shelves where A _fg < 200 kg m⁻² a⁻¹). The 100 and 200 kg m⁻² a⁻¹ isopleths enclose nearly 50% and 80%, respectively, of the area of the ice sheet (Reference Giovinetto. and Bull.Giovinetto and Bull, 1987).

The data also showed widespread scatter where A _dg, A _fg > 200 kgm⁻² a⁻¹, i.e. in that area of Antarctica where most of the accumulation data were obtained by the stratigraphic method, and where sporadic melting (Reference Zwally and Fiegles.Zwally and Fiegles, 1994) may affect emissivity. Therefore, the field data for Antarctica selected to determine the coefficients were first reduced to include only accumulation data obtained by radioactive- or stable-isotope method, and split between East and West Antarctica (Figs 4a and 5a). The data consist of 82 stations in East Antarctica (Reference Picciottn, Crozaz and de BreuckPicciotto and others, 1971; Reference Vinogradov and Lorius.Vinogradov and Lorius, 1972; Reference Young, Pourchet., Kotlyakov, Korolev and DyugerovYoung and others, 1982) and 69 stations in West Antarctica including the areas of dry-snow facies in the Ross and Filchner–Ronne ice shelves (Reference Clausen, Dansgaard, Nielsen and CloughClausen and others, 1979; Reference Reinwarth, Graf. and KohnenReinwarth and Graf, 1985; Reference Whillans. and BindschadlerWhillans and Bindschadler, 1988; personal communication from I. M. Whillans, 1993).

Fig. 4. East Antarctica. Scattergrams of accumulation values (in kg m ⁻² a ⁻¹) derived from emissivity (A _ds, A _dg) and determined from field data (A _fs, A _fg). a. Station data. b. Sampled grid-point data. c. Grid-point data after exclusions.

Fig. 5. West Antarctica. Scattergrams of accumulation values (in kg m ⁻² a ⁻¹) derived from emissivity (A _ds , A _dg) and determined from field data (A _fs, A _fg). a. Station data. b. Sampled grid-point data. c. Grid-point data after exclusions.

The data sets listed above were used to determine the values for the coefficients; these are a ₀ = −20 and a ₁ = 14.75 for East Antarctica, and a ₀ = −40 and a ₁ = 14.5 for West Antarctica (Table 3).

The scalier pattern for Greenland is rather uniform across the whole range of accumulation (Fig. 5a), and the coefficients are adopted as determined in the preliminary analysis (Table 3) on the basis of field data for 89 stations (Reference Koch. and Wegener.Koch and Wegener, 1930; Reference PatersonPaterson, 1955; Reference Hamilton.Hamilton, 1956; U.S. Army Transportation Board, 1960; Reference LangwayLangway, 1961; Reference BensonBenson, 1962; Reference Mock and AlfordMock and Alford, 1964; Reference QuervainQuervain, 1969; Reference Clausen, Gundestrup, Johnsen, Bindschadler and Zwally.Clausen and others, 1988).

The regression coefficients for East and West Antarctica are 0.79 and 0.82, respectively. In each case, the rms of the residuals is smaller than for the larger data set (N357): 16 kg m⁻² a⁻¹ for East Antarctica and 37 kg m⁻² a⁻¹ for West Antarctica. This is partly because there is a better correlation between A _ds and A _fs values, and partly because the A _fs data do not exceed values of 125 kg m⁻² a⁻¹ in East Antarctica or 220 kg m⁻² a⁻¹ in West Antarctica. Therefore, we will tentatively limit the comparison between A _dg and A _fg values to areas where A _fg 101 kg m⁻² a⁻¹ in East Antarctica, and A _fg < 201 kg m⁻² a⁻¹ in West Antarctica.

Comparison of Grid-Point Data

In this section we compare A _dg and A _fg values for grid-point locations in Antarctica and Greenland. In Antarctica, the areas of dry-snow facies are sampled by 1152 grid-point locations for which A _fg values are interpolated from the contoured compilation of field data by Reference Giovinetto and BentleyGiovinetto and Bentley (1985). It should he noted, however, that seven locations for which A _fg values are negative (Fig. 4a, lines 26–29, columns 34–36) lie in the northern region of the Lambert Glacier basin, an area for which net ablation at the surface has been inferred from Landsat imagery (Reference McIntyreMcIntyre, 1985). If the rate of accumulation in those locations is negative, most or all of the ablation in that region is due to snow deflation and sublimation and for the purposes of this study should be included in the dry-snow data set.

Derived values are obtained for 613 locations in East Antarctica where A _fg < 101 kg m⁻²a⁻¹ (Fig. 4b; Table 4). The correlation is weak (r = 0.51, rms = 51), and the mean of A _dg is 27% larger than that of A _fg values. There are 19 locations where (A _dg – A _fg) > 200 kg m⁻²a⁻¹ (Fig. 1b). These are either in mountain areas or where Nimbus-5 ESMR data have occasionally shown differences of 30 K in T _B over the annual mean, which indicates melting (Reference Zwally and Fiegles.Zwally and Fiegles, 1994).

Table 4. Summary statistics of grid data

The A _fg values for the seven locations in the northern region of the Lambert Glacier basin range between 0 and −75 kg m⁻²a⁻¹. A _dg values for the same locations range between 43 and 232 kg m⁻²a⁻¹, supporting an earlier estimate of positive accumulation in the region (Reference Allison, Young and Medhurst.Allison and others, 1985).

If we discount the 19 locations in mountain areas or where occasional melting affects emissivity, and the seven locations in the Lambert Glacier basin, the reduced data set (N = 587; Fig. 4c; Table 4) shows a moderate correlation (r = 0.67) and a much smaller scatter (rms = 27). This improvement in the statistics, and the difference between the means (A _dg = 55 kg m⁻²a⁻¹, A _fg = 49 kg m⁻²a⁻¹), suggests that the compilation would lead to underestimates of accumulation of 12%. rather (ban the 27% mentioned above.

Derived values are obtained for 136 locations in West Antarctica where A _fg 201 kg m⁻²a⁻¹ (Fig. 5b; Table 4). The correlation is weak (r = 0.44. rms = 123), and the mean of A _dg is 55% larger than that of A _fg values. There are nine locations where (A _dg – A _fg) > 400 kg m⁻²a⁻¹ (Fig. 2b). These are in mountain areas or near grounding lines, or where Nimbus-5 ESMR data indicate occasional melting (Reference Zwally and Fiegles.Zwally and Fiegles, 1994). There are three other locations in this group, one (Fig. 2a, line 18, column 11) lying approximately 200 km east-northeast of Byrd Station, and two (Fig. 2a, lines 27 and 28, both on column 9) lying on the Ronne Ice Shelf, approximately 120 km southeast of the Hauberg Mountains. Several locations in the Amundsen Sea sector and in the southwest of the Ronne Ice Shelf show A _dg values that are larger than A _fg values, suggesting that the drawing of isopleths on the basis of extrapolation of field data (as of 1985 there were only a few stratigraphic observations in those areas) leads to underestimates of accumulation in those regions. Other issues have been discussed elsewhere (Reference Giovinetto, Bentley and BullGiovinetto and others, 1989).

If the nine locations mentioned above are discounted, the reduced data set (N = 127; Fig. 5c; Table 4) shows an even weaker correlation (r = 0.41), but, of relevance to our discussion, a significantly smaller scatter (rms = 88). Despite the weaker correlation (which, like all others mentioned in this paper, is significant at the 0.99 confidence level), the difference between the means (A _dg = 195 kg m⁻² а⁻¹, A _fg = 140 kg m⁻² а⁻¹) suggests that the compilation would lead to underestimates of accumulation of 39%, rather than the 55% mentioned above.

In Greenland, the area delimited by the dry-snow line is sampled by 54 grid-point locations, the A _fg values for which are interpolated from the contoured compilation of field data by Reference Ohmura and Reeh.Ohmura and Reeh (1991) (Fig. 3b). Derived values are obtained for these locations (Table 4; Fig. 6b). The correlation is robust (r = 0.89, rms = 43), and the mean of A _dg is 4% larger than that of A _fg values, a practically insignificant difference because it is within the error of determination of accumulation at particular sites.

Fig. 6. Greenland. Scattergrams of accumulation values (in kg m ⁻² a ⁻¹) derived from emissivity (A _ds, A _dg) and determined from field data (A _fs, A _fg). a. Station data. b. Sampled grid-point data. c. Grid-point data after exclusions.

There are three locations for which the contribution to the rms matrix is largest (Fig. 2b). All three lie close to the delineation of the dry-snow line. We are currently examining Nimbus-5 ESMR and Nimbus-7 SMMR data for indications of melting. Assuming that the locations lie below the dry-snow line and are discounted, the improvement in the statistics for the reduced data set (N = 51, Table 4; Fig. 6c) is quite remarkable (r = 0.95, rms = 28). The difference between the means of the smaller data sets (A _dg = 190.7 kg m⁻² a⁻¹, A _fg = 200.0 kg m⁻² a⁻¹) suggest that the compilation of field data would lead to an overestimate of 5% instead of the 4% mentioned above.

Discussion

The compilation from which the A _fg values were interpolated for Antarctica (Reference Giovinetto and BentleyGiovinetto and Berttley, 1985) was based on A _fs data for the whole ice sheet (approximately 1500 stations and stake-network segments distributed along traverse routes! with the exception of the contours drawn in the northern Lambert Glacier basin as described above). The fact that in both East and West Antarctica the mean of A _dg is larger than that of A _fg suggests that the subjective criteria used in the interpolation and extrapolation of isopleths results in an underestimate.

The A _fs data used to select the coefficients are heavily weighted by the data of Reference Picciottn, Crozaz and de BreuckPicciotto and others (1971) for western Dronning Maud Land in East Antarctica, and of Reference Whillans. and BindschadlerWhillans and Bindschadler (1988) and I. M. Whillaus (personal communication, 1993) for western Marie Byrd Land. It will be of interest to compare how the coefficients determined for Last and West Antarctica would change when the results of field data being collected on, or analyzed for, the northern Amery Ice Shelf drainage system (including the northern area of the Lambert Glacier basin) and the southwestern Ronne Ice Shelf drainage system, are released. Preliminary maps of residuals (A _dg – A _fg) show distinct nucleation of positive anomalies in those areas.

The compilation from which the A _fg values were interpolated for Greenland (Reference Ohmura and Reeh.Ohmura and Reeh, 1991) was based on data for 251 stations distributed along traverse routes in the accumulation zone and precipitation data for 35 meteorological stations, mostly located along the coast, which provided ancillary information for the interpolation and extrapolation criteria used to draw the isopleths. The A _fs data used to determine the coefficients do not show any particular bias. Preliminary maps of residuals (A _dg – A _fg) show near-zero anomalies in both the north and south of the area of dry-snow facies; they also show generally slightly negative anomalies to the west and slightly positive anomalies to the east of the central ridge.

Conclusion

The method of determining accumulation using a hyperbolic function of firn emissivity appears to be reliable for the areas of dry-snow facies, provided that there are adequate and reliable field data to determine the coefficients in the function. The derived accumulation for Antarctica suggests that estimates based on the contoured compilation of Reference Giovinetto and BentleyGiovinetto and Bentley (1985) would lead to underestimates, particularly in West Antarctica. The determinations of the sets of coefficients for Last and West Antarctica need refinement because the distribution of reliable held data available at present for either ice sheet shows regional bias. The derived accumulation for Greenland suggests that the compilation of Reference Ohmura and Reeh.Ohmura and Reeh (1991) may lead to over-estimates by a small margin that presently is within the error of determination of accumulation at particular locations.

A more accurate facies zonation (e.g. Reference Fahnestock, Bindschadler, Kwok and Jezek.Fahnestock and others, 1993; Reference Zwally and Fiegles.Zwally and Fiegles, 1994) is needed in both Antarctica and Greenland. The change in the areal distribution of the accumulation rate suggested by the derived values indicates that the full area coverage of brightness temperature and surface temperature data available from passive-microwave satellite observations would help remove inaccuracies inherent in the interpolation and extrapolation over long distances of held data. The full use of passive-microwave databases (25 km × 25 km and 30 km × 30 km grid-squares) rather than the 100 km grid approach used in this paper should allow the refinement of estimates of the accumulation rate for large parts of the ice sheets as well as of particular drainage systems.