Estimating englacial radar attenuation using depth profiles of the returned power, central West Antarctica



[1] We analyzed depth patterns of geometrically corrected returned power Pc from within the ice of central West Antarctica to develop a proxy for englacial radar attenuation. The depth patterns of [Pc]dB (Pc in the decibel scale) at individual sites were first approximated with least squares vertical gradients of the local mean [Pc]dB for five depth ranges. Variations of these gradients along radar tracks show identifiable trends but have local anomalous features over distances less than ∼5–10 km that are caused by smaller reflection from tilted internal layers above steep beds and other factors. Consequently, extraction of an attenuation proxy from the returned power requires mitigation of reflectivity variations. Next, returned power was synthesized only from bright layers assembled over distances much wider than the local anomalous features. Individual data ensembles show a clear upper (not lower) cutoff in [Pc]dB. The cutoff power decreases with depth linearly between ∼500 and ∼1600 m, which defines the upper envelope gradient. With the aid of attenuation and reflectivity modeling, we concluded that the upper envelope gradient can be an attenuation proxy in the isothermal ice that is expected in the upper half or more of central West Antarctica. The estimated attenuation rate in the upper ∼1600 m varies 5 dB km−1 (one way), equivalent to lateral temperature variations of about 2°C or chemistry variations of up to a factor of 2. This range indicates that the attenuation variations can significantly affect delineation of bed wetness on the basis of contrasts in power returned from the bed.

1. Introduction

[2] Geometries of isochronous layers detected by radio echo sounding have been used as natural markers to decipher glacial evolution [e.g., Waddington et al., 2007]. Returned power analysis can potentially reveal additional information about ice characteristics. Bed-returned power has been used as a proxy for basal conditions; brighter and dimmer beds have been interpreted as wetter and drier beds, respectively [Bentley et al., 1998; Carter et al., 2007; Jacobel et al., 2009; Peters et al., 2005]. Retrieving bed conditions requires extracting path effects through the overburdened ice. Gades et al. [2000] used spatial variations of depth-averaged returned power from within ice as a proxy of englacial loss. Winebrenner et al. [2003] and Jacobel et al. [2009] estimated attenuation averaged over the full ice thickness using the path length dependence of the power returned from the bed. However, spatial uniformity of the attenuation has not been studied yet; attenuation is a function of ice temperature and chemistry [MacGregor et al., 2007] so that attenuation can vary spatially. Besides delineation of wet beds, anomalous attenuation within the ice sheets can provide remote-sensing means to examine macroscopic thermal and chemical structures of the ice, which significantly affect ice rheology.

[3] In this paper, we examine vertical (depth) variation of the returned power from within ice in the West Antarctic Ice Sheet (WAIS) divide area, which separates ice flow toward Ross and Amundsen embayments (Figure 1). This area is chosen because of availability of usable radar data, eventual availability of ground truth data from a full depth ice core, and thick, nearly isothermal ice, which makes interpretations of the depth profiles simpler. Linear least squares sense gradients of the geometrically corrected returned power are derived for different data ensembles, and their relevance to attenuation is assessed with the aid of attenuation and reflectivity modeling. Finally, we discuss corresponding implications for spatial variations of ice temperature and chemistry in the WAIS divide area. All symbols used in this paper are summarized in the notation section.

Figure 1.

(a) Map of the Antarctic Ice Sheet. (b) Regional and (c) local regional surface topography (contours) and ice thickness (background color) in the WAIS divide area. Gray rectangles in Figure 1a and 1b show the coverage of Figures 1b and 1c, respectively. In Figure 1b, the WAIS divide area is a saddle between ice domes over Executive Committee Range (E) and at the boundary between Ross (R)/Amundsen (A) and Weddell Sea catchments. Byrd ice core site (B) and WAIS divide core site (C) are also shown. Thick contours are drawn at 500 m intervals, and thin contours are drawn at 100 m. In Figure 1c, in the vicinity of the divide area, 10 m interval contours are drawn only between 1750 and 1900 m asl. Surface topography is from Liu et al. [1999], and ice thickness is a composite of a continental BEDMAP model [Lythe et al., 2001] and a regional model by Support Office for Aerogeophysical Research [Morse et al., 2002]. Gray curves show locations of four longitudinal profiles, each in south (Sx), central (Cx), and north (Nx) series and the other four cross-flow profiles (Ny). Dark and light gray scales are assigned to show approximate distances ξ from the divide location along each longitudinal profile. Candidates for ice core drilling sites A, B, and C are shown with solid circles; see Morse et al. [2002] for coordinates of these sites.

2. Background

2.1. WAIS Divide

[4] Different local subglacial environments associated with sediments, volcanism, and faults introduce inherent heterogeneity to the dynamics of the WAIS [Anandakrishnan et al., 1998; Bell et al., 1998; Blankenship et al., 2001]. Complex spatial and temporal patterns of retreat of the marine-based WAIS [Conway et al., 1999] ultimately affect the WAIS divide area, where a full depth ice core is currently being drilled. Continental-scale ice flow models [e.g., Huybrechts, 2002] show that at the Last Glacial Maximum (∼15 ka), this area could have been a saddle between two major ice domes, one associated with the Executive Committee Range and the other associated with the high bed between the Weddell and Ross/Amundsen catchments. The lack of major topographical structures of the bed directly beneath the current divide location (Figure 1) indicates that the current divide has little local topographic control and may have been migrating [Neumann et al., 2008].

2.2. Radar Data

[5] This study uses radar data collected in the austral summer of 2000 by the Support Office for Aerogeophysical Research (SOAR) at University of Texas. The radar is a 60 MHz, pulse-modulated (250 ns pulse width) system with an incoherent receiver recording returned power but not phase. Characteristics of this radar are described by Blankenship et al. [2001]. Radar calibration data were provided for this study by M. Peters (personal communication, 2005).

[6] We analyzed 12 longitudinal ∼140 km radar profiles (four of each of the Sx, Cx, and Nx series) and four ∼50 km profiles (Ny series) crossing the Nx series (Figure 1). Each recorded waveform is a mean of 2048 waveforms collected over ∼14 m along-track distance. We measure the location ξ along individual radar tracks from the WAIS divide toward Ross Sea (ξ positive) and Amundsen Sea (ξ negative). The four profiles in the Sx, Nx, and Ny series are mostly parallel to each other, and separations between profiles are roughly 5–7 km. Separations between Cx profiles become larger at downstream sites since the topography is convex and ice flow diverges.

2.3. Returned Power

[7] Returned power (echo intensity) P from within ice is a function of characteristics of the radar instrumentation S and of ice I along a round-trip propagation path to a target at depth z. It is also affected by geometrical spreading G. The relationship among these can be written as P = SI/G or in the decibel scale ([x]dB = 10log10x) [e.g., Bogorodskiy et al., 1985; Matsuoka et al., 2004]:

equation image

The term G can be derived from radar height h above the surface, target depth z below the surface, and depth mean relative permittivity ɛ from the surface to depth z:

equation image

Permittivity varies significantly in the top <100 m of ice due to densification, which causes only negligible variations in G from that for pure ice (<0.05 dB at depths greater than 50 m). Therefore, we apply a uniform permittivity ɛ (=3.2), equivalent to a propagation speed of 168 m μs−1.

[8] In the stratified ice of the WAIS divide area, ice characteristics I can be approximated with reflectivity R of the targeted internal layer, dielectric attenuation L integrated along the propagation path, and signal level reduction B caused by crystal-alignment-fabric-induced birefringence:

equation image

Loss of the radar power caused by multiple reflections is insignificant [MacGregor et al., 2007].

[9] Figure 2 shows a typical depth variation of the retuned power [P]dB (radar waveform). The returned power near the surface is masked primarily by off-nadir scattering from the ice sheet surface. Depths at which the shallowest englacial signal appears depend on surface conditions (e.g., smooth firn or open crevasses) and instrumental conditions (e.g., aircraft height, antenna beam width, and pulse width). In the data that we analyzed, this depth is mostly uniform at ∼150 m, and the first distinct trackable layer appears between 150 and 200 m. In this paper, we examine depth dependences of the returned power below this depth, where [I]dB is not affected by near-surface processes such as scattering at the ice sheet surface and refraction.

Figure 2.

A typical form of returned power versus depth (ξ = 60 km along Sx4 (Figure 4)). Three bars (labeled DCL, deepest coninuous layer; SN, shallowest noise; and DS, deepest signal) show depth ranges between z1 (200 m) and z2, over which the local mean gradients are derived (section 3.2). Solid circles show five bright layers that were tracked over long distances and used to examine [Pbrightc]dB (section 3.4). Noise floor is at −106.9 dBm.

2.4. Factors Controlling Returned Power

2.4.1. Attenuation

[10] Following standard textbooks [e.g., Ulaby et al., 1986], attenuation L integrated over the two-way travel path between depths z1 and z2 is written as

equation image

N(z) represents a local attenuation rate per unit path length at depth z accounting for a single trip. The factor of 2 accounts for a round trip.

[11] Local attenuation rate N (in the unit of dB km−1, one way) is proportional to ice conductivity σ that has linear dependences on acidity C1 and salinity C2 and associated three Arrhenius-type dependences on ice temperature T including the pure ice effect [MacGregor et al., 2007]:

equation image

where i is pure ice (i = 0), acidity (i = 1), and salinity (i = 2); c is the speed of light in vacuum; ɛ0 is the permittivity of free space; σi0 is the pure ice conductivity (i = 0; μS m−1) or molar conductivities (i = 1, 2; (μS m−1) (μmol L−1)−1) at the reference temperature Tr = 251 K; Ci (μmol L−1) are concentrations (C0 = 1, C1 is acidity, and C2 is salinity); Ei are activation energies corresponding to individual components; and k is the Boltzmann constant. The terms σi0 and Ei are frequency independent over HF and VHF ranges. Note that acidity and salinity here are “radar effective” values depending on in situ conditions of soluble ions (see MacGregor et al. [2007] for details). Gudmandsen [1971] used a simpler temperature dependence for conductivity with a single Arrhenius form to represent natural ice from north Greenland. It does not include effects of chemical constituents explicitly, but the parameterization is made to account for them implicitly.

[12] Figure 3 shows relationships between the ice temperature and the local attenuation rate N based on parameterizations by Gudmandsen [1971] and MacGregor et al. [2007] of the terms in equation (5). The latter is derived using the depth mean acidity and salinity at Siple Dome, West Antarctica. Hereafter, we call them Gudmandsen- and Siple Dome–type parameterizations. MacGregor et al. [2007, 2009] demonstrated that L(z) is more affected by depth mean chemistry than large Ci peaks usually associated with volcanic eruptions since such Ci peaks are localized in narrow depth ranges. The sensitivity of attenuation rate to ice temperature dN/dT (slope of Figure 3) is 1–1.5 dB km−1 °C−1 at −20°C with some dependence on chemistry and 0.5 dB km−1 °C−1 at −30°C with negligible effects of chemistry.

Figure 3.

Ice temperature dependence of local attenuation rate N accounting for a single trip derived from Gudmandsen (solid line) and Siple Dome (dashed line) parameterizations [Gudmandsen, 1971; MacGregor et al., 2007]. The former does not explicitly include effects of chemical constituents. The latter uses depth mean acidity of 1.3 μmol L−1 and salinity of 4.2 μmol L−1 obtained at Siple Dome, West Antarctica.

2.4.2. Reflectivity

[13] There are three known mechanisms of radio wave reflection within the ice sheets. In the upper hundreds of meters, contrasts in density (affecting permittivity) are important. At greater depth, contrasts in acidity (affecting conductivity) and fabric (affecting permittivity) are predominant [Fujita et al., 1999].

[14] The Fresnel (specular) reflectivity Rδɛ for a permittivity contrast δɛ caused by density or fabric is frequency- and temperature-independent [Fujita et al., 2000]. The Fresnel reflectivity [Rδσ]dB for a conductivity contrast δσ can be written as [Paren, 1981]

equation image

where f (MHz) is the radar frequency, and acidity and salinity contrasts are δCi (μmol L−1). The inverse frequency dependence causes acidity contrasts to be expressed in reflectivity most strongly at low frequency.

[15] At depths greater than several hundred meters, both acidity and fabric contrasts, but not density contrasts, cause reflections. The frequency at which relative importance of permittivity (fabric) and conductivity (acidity) switches (crossover frequency) can vary depending on the ice properties. The crossover frequency is somewhere between 60 and 179 MHz in East Antarctica [Fujita et al., 1999; Matsuoka et al., 2003]. This is consistent with an expectation from the ice properties and typical contrasts of acidity and ice fabrics found in ice cores [Fujita and Mae, 1994]. The crossover frequency is higher if ice is warmer and/or includes more acidity but lower if contrasts in ice fabric are more developed.

[16] The Fresnel reflectivities Rδσ and Rδɛ are an approximation for smooth and flat interface. When these conditions are not satisfied, the in situ (radar-measured) reflectivity R is proportional to the Fresnel reflectivity but smaller. We discuss these effects in the context of the data interpretations.

3. Quantifying Vertical Variation of Returned Power

[17] In this section, radar data collected in the WAIS divide area are examined in order to establish a radar proxy of englacial attenuation. We first discuss general features of the power returned from within the ice (section 3.1). Next, we linearly approximate the vertical profiles of the returned power in the decibel scale with two distinct methods, using the local mean of [Pc]dB (sections 3.3 and 3.4) and using only bright layers with broader horizontal averaging (sections 3.3 and 3.4). Modeling presented in section 4 infers that the second better accounts for horizontal variations in internal reflectivity.

3.1. Echogram

[18] Figure 4a shows an echogram along profile Sx4 closest to the ice core site. Except for subglacial mountains at ξ ≈ −40 km, the bed is mostly flat and ice thicknesses are roughly ∼3000 m (ξ < −50 km) and ∼3400 m (ξ > −20 km). The returned power gradually decreases down to the noise floor (∼107 dBm (Figures 2 and 4a)). This noise floor is not the top of an echo-free zone [e.g., Drews et al., 2009; Fujita et al., 1999] since the returned power decreases smoothly to that floor. The thick bottom layer with insignificant returned power (e.g., ξ ≈ −35, −55, and −65 km) is not related to the echo-free zone. Any actual echo-free zone, if present, must have a top deeper than the noise floor.

Figure 4.

Radar data along profile Sx4 closest to the ice-coring site. Vertical gray bars show the location of the WAIS divide ice core site and the site of the radar waveform shown in Figure 2. (a) Echogram. The bed identified with a time-differentiated echogram is marked with white asterisks. Deepest continuous layer (DCL) is marked at selected locations with crosses. Short interruption of the data at ξ ≈ 0 km is caused by off-nadir pointing of the radar, when the aircraft banked while turning its heading (Figure 1c). (b) Local mean gradients 〈d[Pc]dB/dz〉 (equation (8)) and (c) their residuals from the regional characteristic curve shown in Figure 5. These values are shown with color stripes; the colors assigned in Figures 4b and 4c represent values on logarithmic and linear scales, respectively. The ordinate shows the lower limits z2 of the depth ranges over which the vertical gradients are calculated (Figure 2). Vertical gradients to fixed depths (z2 = {1000 m, 1500 m}) are shown in both Figures 2b and 2c.

3.2. Vertical Gradient of the Local Mean Returned Power

[19] To further characterize the depth patterns of the returned power, we examined geometrically corrected returned power

equation image

where G is given by equation (2). From equations (1), (3), and (7), the vertical gradient of [Pc]dB can be approximated with

equation image

where notation image represents the mean of x(z) between z1 and z2 so that image = {x(z2) − x(z1)}/{z2z1}. Here, we neglect 〈d[B]dB/dz〉 since B is expected to be depth independent (Appendix A) and 〈d[S]dB/dz〉 since S is sufficiently constant during measurement at one location (Appendix B). When the reflectivity has no vertical gradient (〈d[R]dB/dz〉 ≈ 0), the radar-derived vertical gradient 〈d[Pc]dB/dz〉 determines the path-integrated attenuation gradient 〈d[L]dB/dz〉, which is related to the depth-averaged attenuation rate image (equation (4)):

equation image

Equations (5), (8), and (9) show that if depth variations in reflectivity are minimal, we can hope to interpret the radar-derived vertical gradient in the geometrically corrected returned power in terms of depth-averaged attenuation rates and associated thermal and chemical variations in the ice.

[20] The vertical gradients of local mean, geometrically corrected returned power were derived as linear least squares regression of [Pc]dB from the radar data over five depth ranges. This derivation is not identical with the definition of the vertical gradient (equation (8)) but practical since gradients derived only with returned power at two depths are more sensitive to reflectivities at these depths (Figure 2). All depth ranges share the same upper limit z1 but have variable deeper limit z2. It is impractical to vary both upper and lower limits since the gradient is affected by the presence or absence of bright layers near the depth limits. The lower limits z2 are defined as 1000 m, 1500 m, the deepest continuous layer (DCL), depths of the shallowest noise (SN), and depths of the deepest signal (DS) (Figure 2). More layers can be picked beneath the DCL, but they are interrupted and not continuous over the entire length (∼140 km) of the profile. SN and DS are defined as shallowest and deepest depths at which the returned power reaches the noise level (Figure 2 and Appendix B). Depths of DCL, SN, and DS are defined relative to local returned power in order to partially homogenize the returned power near the lower limit so that the gradient is less sensitive to the presence or absence of bright layers near the lower limit. The upper limit z1 is set at 200 m so that the analysis does not require accurate understanding of the near-surface processes.

[21] The ordinate of Figures 4b and 4c shows the variable lower depth limits along profile Sx4. DCL depth increases toward the Amundsen Sea, but its shape is affected by subglacial hills at ξ ≈ −40 km. DS and SN depths show patterns totally different from DCL depths. Large DS depths at ξ ≈ 60 km is caused by the presence of the distinct bright layer, reported by Jacobel and Welch [2005] and Neumann et al. [2008].

[22] We interpreted rapidly varying components of the vertical gradients along the profile as noise and removed them by low-pass filtering (see paragraph 26). We used a cutoff of 1 km for this filter, which corresponds to roughly one third to one of the local ice thickness. Color stripes in Figure 4b show the radar-derived vertical gradients. One prominent and consistent feature is that the gradient is more negative when calculated over a shorter, shallower depth range. This feature is found regardless of depth range. Figure 5a shows the vertical gradients along all 16 profiles in terms of variable deepest limits (z2 = {DCL, SN, DS}). Above ∼500 m depth, the gradients become rapidly less negative with increasing depth. The variation is weaker beneath ∼500 m. We characterize the dependence of the vertical gradient on the lower-limit depth z2 in the entire study area with a single curve represented by a least squares polynomial seventh-order fit to all of the data. Figure 5b shows residuals from this characteristic curve. Residuals are close to evenly distributed around the characteristic curve at depths between ∼500 and ∼2000 m. The characteristic curve and residuals show anomalous structure at greater depths due to the high-order polynomial fit and declining data volume where we do not attempt analysis.

Figure 5.

Regional characteristic curve of the depth dependence of the local mean gradients 〈d[Pc]dB/dz〉. (a) Values of the gradients are plotted against the lower limit z2 of individual depth ranges. Numerous DCL and SN values are hidden behind SN and DS, respectively. The gray curve shows the regional characteristic curve derived as a seventh-order polynomial fit to the observations. (b) Residuals from the regional characteristic curve. Negative residuals mean that geometrically corrected returned power decreases more rapidly than the regional characteristic. These residuals are plotted in Figure 4c and mapped in Figure 6.

[23] Figure 4c shows the residuals of the vertical gradient from the regional characteristic curve (Figure 5b), as well as the gradients for the fixed lower limits (z2 = {1000 m, 1500 m}). More negative gradient residual means that [Pc]dB decreases more rapidly with depth than the regional trend. Vertical gradients for fixed depth ranges have similar lateral patterns as those for variable depth ranges. The patterns of residuals for different depth ranges correspond well with each other. This agreement can be found both on tens of kilometers scale (e.g., negative deviations at 10 < ξ < 30 km and −35 < ξ < −25 km) and also on several kilometers scale (e.g., ξ ≈ −55 and −63 km). Extremely negative residuals (<−6.4 dB km−1; twice the standard deviation) are commonly found for the three depth ranges at locations where z2 for DS (and thus z2 for DCL and SN as well) are small. The locations of such extremely negative residuals are consistent with weak echo locations (Figure 4a). However, not all weak echo locations have such negative residuals (e.g., ξ ≈ −50 km).

3.3. Properties of Spatial Variability

[24] Figure 6 shows maps of the gradient residuals for the three depth ranges. In general, extremely negative (twice the standard deviation) residuals are found on the Amundsen Sea side along the Sx series, while extreme positive deviations are found on the Ross Sea side. Except for these sites with the extreme residuals, moderate residuals are found over the entire area. The gradient along four profiles in each track series varies consistently in terms of the along-track distance from the divide, which is typically found at ∼10 < ξ < ∼30 km along Sx profiles. Major regional patterns are consistent regardless of the depth ranges, but magnitudes of the residuals are different. These regionally consistent features indicate that the radar-derived vertical gradient is a significant parameter that can be used to examine ice properties.

Figure 6.

Maps of the local mean gradient residuals from the regional characteristic curve (Figure 5). The color bar is the same as in Figure 4c. Contours show surface elevations; their intervals are identical with Figure 1. The residuals are shown only when the lower limit of the depth ranges is less than 2000 m to avoid anomalous structures (Figure 5b).

[25] Strong spatial correlations between the extremely negative residuals and weak echo locations in the radargram above steep beds are observed. It suggests that smaller radar-measured reflection (backscattering) from tilted internal layers causes significant reflectivity gradients 〈d[R]dB/dz〉 so that the local mean returned power gradients 〈d[Pc]dB/dz〉 have extremely negative residuals (equation (8)). Such features appear over along-profile distances shorter than ∼5–10 km. Some major residuals commonly found for different depth ranges are not correlated with weak echo locations associated with the steep beds. That conflicts with an expectation that more tilted layers at greater depths cause more anomalous gradients for longer depth ranges, suggesting that such features could also be caused by other mechanisms.

[26] Reflection from tilted layers is evidence that the layers are not perfectly smooth at the scale of the radar wavelength (∼2.7 m in ice). When a radio wave reflects from a rough surface, reflections from multiple focal points interfere with each other, which cause variations (fading patterns) of the returned power along a profile [Berry, 1973; Neal, 1982]. When the spectral wavelength and magnitude of roughness follow a Gaussian distribution, the returned power maxima would appear at every ∼5 m of along-track distance, which are not detected in the radar data averaged over ∼14 m. More generally, rough layers probably do cause rapid along-track variations of the vertical gradients. Another possible cause of the rapidly varying components of the vertical gradients is interference of returned power from multiple layers within the vertical resolution of the radar (∼21 m, travel distance within half of a pulse width of 250 ns). These interference effects are more sensitive to separations between reflection layers, rather than bulk dielectric properties of ice. Therefore, we interpreted rapidly varying components as noise and smoothed it out by the filtering (Figures 4b, 4c, and 6).

[27] In conclusion, the vertical gradients 〈d[Pc]dB/dz〉 of the local mean returned power represent both regional and local ice characteristics. The gradient residuals from the regional characteristic curve are sensitive to anomalously small reflectivities localized at great depths mostly associated with steep beds, which introduce significant reflectivity gradients. Thus, the least squares vertical gradients of local mean values depend on both reflectivity and attenuation gradients (equation (8)). In section 3.4, we discuss a method to mitigate the effect of reflectivity variations associated with the analysis in order to get a better proxy for depth-averaged attenuation rate.

3.4. Returned Power From Bright Layers

[28] Figure 2 shows that the returned power varies locally more than 20 dB. For a small depth range (z2z1 ≈ 0), the geometric factor and path-integrated attenuation vary negligibly so that most of the local variability in the returned power is caused by local reflectivity variations (equation (3)). To mitigate effects of variable local reflectivity and to retrieve more reliable attenuation-related information, we now analyze reflections only from “bright” layers. Reflectivities [Rbright]dB associated with the individual bright layers may have a narrower range than the entire range of reflectivities used in the gradient analysis in sections 3.2 and 3.3, but path-integrated attenuation [L]dB to a given depth z is unrelated to whether or not the reflection layer at that depth z (not other layers between the surface and z) is locally bright. The use of bright means that layers are continuous more than tens of kilometers and able to be tracked with an in-house automatic routine [Matsuoka et al., 2009a]; 3–10 layers were picked for each profile. Definitions of bright layers are thus somewhat arbitrary, but the goal is to more narrowly restrict the effect of the reflectivity gradient 〈d[R]dB/dz〉 on the radar-derived vertical gradient of 〈d[Pc]dB/dz〉.

[29] Figure 7 shows [Pbrightc]dB from bright layers along profile Sx4. A range of [Pbrightc]dB at similar depths is roughly 20 dB, indicating that [Rbright] still locally vary roughly 20 dB. As we discussed in section 3.3, [Pbrightc]dB is also affected by fading and anomalously small reflectivities of tilted layers, which likely cause this variation. However, [Pbrightc]dB has a clear upper envelope.

Figure 7.

Depth variations of the geometrically corrected returned power [Pbrightc]dB from bright layers along profile Sx4. [Pcbright]dB (gray dots) have a clear upper envelope [Pmaxc]dB (circles) at a given depth, which decreases linearly with depth (line). The upper envelope gradient 〈d[Pmaxc]dB/dz〉 and its standard deviation for the entire profile Sx4 are also shown. The inclined dashed line shows increases of [Pc]dB due to [G]dB when [P]dB remains at the noise floor (equations (2) and (7)).

[30] To represent the upper envelope, we divided all [Pbrightc]dB values into 10 depth bins and defined the upper envelope value [Pmaxc]dB (shown with circles in Figure 7) in each depth bin; see Appendix C for details of the derivation. Along profile Sx4, the upper envelope slopes linearly with depth (in the decibel scale) between ∼400 and ∼1800 m. At shallower depths, the upper envelope is larger than the values expected from the linear trend between 400 and 1800 m (see section 4). At greater depths, returned power is close to the noise floor (approximately −107 dBm), and the upper envelope is more or less depth-independent.

3.5. Vertical Gradient of the Upper Envelope of Returned Power

[31] We approximated the vertical gradient of the upper envelope by the linear regression of [Pmaxc]dB, which has the property in analogy to equations (8) and (9):

equation image

The upper envelope gradient is a linear approximation of the depth variations in local maximum returned power (Figures 2 and 7). In contrast, the vertical gradient analyzed in sections 3.2 and 3.3 is a vertical gradient of local mean returned power. The derived upper envelope gradient and its standard deviation in the regression along the entire Sx4 (Figure 7) is −22.1 ± 1.2 dB km−1. For all 16 profiles, the gradient is −20.7 ± 0.4 dB km−1. If 〈d[Rmax]dB/dz〉 were actually negligible, these results would give 〈N〉 ≈ 11.1 ± 0.6 dB km−1 and 10.4 ± 0.2 dB km−1 along profile Sx4 and along all 16 profiles, respectively (equation (10)).

[32] To see more detailed variations in this area, lateral variations of the upper envelope gradient are examined along radar tracks. Derivation of the upper envelope gradients requires radar data ensembles that have a statistically adequate number of data points at various depths. Also, one value of the upper envelope gradients must be derived over a certain distance, over which ice characteristics can be assumed similar but layer depths are slightly different. As we discussed in section 3.3, zones of anomalously small reflection are narrow (less than ∼5–10 km). We chose the ensemble window to be wider (20 km) so that 〈d[Pmaxc]dB/dz〉 values in data ensembles are less affected by anomalous weak echo locations. The center points of the 20 km windows are moved by 10 km so that the segment closest to the divide ranges −10 < ξ < 10 km and the second closest segments range from the divide to ±20 km. To increase the number of data points in one data ensemble, radar data from all four profiles in each Sx, Cx, and Nx series are combined so that one upper envelope gradient is derived over about a 20 km by ∼15–20 km area. Here we make a pragmatic assumption that ice characteristics in each regional series of the profiles coherently vary in terms of the along-flow distance from the current divide. On the basis of regional distribution of the local mean gradient 〈[Pc]dB/dz〉 (Figure 6), this assumption is acceptable. Data are excluded from the analysis of the upper envelope gradients if the depths are smaller than 500 m or larger than 2000 m since the former show a different trend and the latter have small signal-to-noise ratios (Figure 7). The derived upper envelope values (shown with circles in Figure 7) are located at depths shallower than 1600 m in most cases and up to 1800 m in thick ice areas. Thus, the upper envelope gradients are derived to depths less than ∼60% of the local ice thickness in most cases.

[33] The upper envelope gradients derived independently for neighboring 20 km long segments vary gradually at most sites (Figure 8). In general, errors in the linear regression are smaller than the along-track variations, suggesting that the upper envelope of [Pc]dB linearly decreases with depth and its depth dependence is consistent in an area of 20 km (6–10 ice thicknesses; along the tracks) by ∼15–20 km (across the tracks).

Figure 8.

Upper envelope gradients 〈d[Pmaxc]dB/dz〉 for individual 20 km long segments with their center positions at ξ = 20i km (red) and 20i + 10 km (blue), where i = 0, 1, 2, 3. Error bars show the standard deviation in the linear regression (Figure 7). Mean and standard deviation along each regional series are −21.0 ± 2 dB km−1 (Sx), −23.5 ± 3.1 dB km−1 (Cx), and −23.0 ± 1.8 dB km−1 (Nx); the means are shown with dotted lines.

[34] Regional patterns of the local mean gradient (Figure 6) and upper envelope gradient (Figure 8) are notably different; anomalously small values at weak echo locations are found only in the local mean gradient, not in the upper envelope gradient. We argue that this difference is primarily caused by the difference between 〈d[R]dB/dz〉 and 〈d[Rmax]dB/dz〉. Differences in spatial smoothing (1 km versus ∼20 km) make insignificant differences in 〈N〉 since depth-averaged ice temperature and chemistry probably vary smoothly. Therefore, we conclude that the upper envelope gradients are much less affected by local reflectivity variations. Equation (10) shows that when reflectivity gradients are minimal, the upper envelope gradient is more related to the depth-averaged attenuation rate.

4. Modeling Depth Profiles of the Returned Power

[35] Prominent features of the depth variations of the returned power are rapid decreases of [Pc]dB and [Pmaxc]dB at small depths (Figures 2, 5, and 7) and a linear depth dependence of [Pmaxc]dB ensembles over 20 km long segments at middepths (Figure 7). To interpret these features, we model depth dependence of path-integrated attenuation and reflectivity.

4.1. Attenuation

[36] Attenuation is a function primarily of ice temperature and secondarily of ice chemistry (section 2.4.1) so that we first model attenuation by accounting only for ice temperature using a temperature model (Appendix D), provided that the ice sheet is in a steady state. The temperature modeling shows that the WAIS divide area has thick, nearly isothermal ice (Figure 9). We characterize the depth of the lower limit of this isothermal ice using a depth ZTs+3 at which the ice temperature is higher than Ts by 3°C or ∼10% of the temperature difference between the surface (approximately −31°C) and the bottom (at the melting point [Neumann et al., 2008]). The majority of our study area has surface mass balances larger than 0.2 m yr−1, and ice thickness ranges between 2000 and 3000 m. ZTs+3 in such an area is greater than ∼0.5H. Some areas on the Amundsen Sea side have ice as thin as ∼1200 m, but surface mass balance is large (0.6–0.7 m yr−1) so that ZTs+3 is greater than 0.7H. Overall, the WAIS divide area has nearly isothermal ice in the top half or more of its thickness.

Figure 9.

Normalized depths ZTs+3 at which ice temperature is 3°C higher than the surface temperature Ts. This depth is used to represent the thickness of the isothermal ice; over the depth range to ZTs+3, ice temperature varies by ∼10% of the difference between the surface temperature (approximately −31°C) and the melting point at the bed. Geothermal flux is assumed to be (a) 60 mW m−2 and (b) 80 mW m−2. Ranges of H and b are chosen to cover the observed ranges of modern values in the WAIS divide area [Morse et al., 2002].

[37] These ice temperature profiles were used to estimate depth-averaged attenuation rate 〈N〉 (equation (5)). Results below are virtually independent of the choice of attenuation parameterizations (section 2.4.1). Figure 10 shows Z25dB at which local attenuation rate N(Z25dB) is larger than N(0.2H) by 25 dB km−1. Here, 0.2H is used to represent the isothermal ice. Local attenuation rates increase more rapidly at greater depths where ice temperature rapidly increases. However, depth-averaged attenuation rate 〈N〉 varies less rapidly. For any combinations of the ice thickness and surface mass balance shown in Figures 9 and 10, image differs less than 1.0 dB km−1 from 〈N00.2H. If the ice thickness and surface mass balance are limited to observed combinations in the WAIS divide area, image differs less than 0.25 dB km−1 from 〈N00.2H. This suggests that if chemistry is uniform with depth, attenuation is mostly depth-independent in the upper half or more of the ice thickness at any given location in our study area.

Figure 10.

Normalized depth Z25dB at which local attenuation rate N(z25dB) differs by 25 dB km−1 from the local attenuation rate in the isothermal ice N(0.2H). The ranges of H and b are identical with Figure 9. Geothermal flux is assumed to be 80 mW m−2; smaller geothermal flux gives larger Z25dB.

[38] To explore possible effects of chemistry, we derived 〈N0H of isothermal ice at −31°C using ice core chemistry at Siple Dome (H = 1 km), West Antarctica [MacGregor et al., 2007]. The Siple Dome ice core is used since chemistry records in our depth range are not yet available in the WAIS divide area. When a full depth (1 km) profile of the chemistry is used, 〈N0H is 6.9 dB km−1 for the isothermal ice at −31°C. When a depth mean value for chemistry is used, 〈N0H is 7.4 dB km−1. Chemistry effects on attenuation at the WAIS divide may be less than at Siple Dome since the Siple Dome ice core chemistry used here includes the climate change during the past 100 kyr [Brook et al., 2005], but the radar data that we analyzed spanned only a shorter period (2.3–10 ka at the WAIS divide core site [Neumann et al., 2008]).

4.2. Reflectivity

[39] When ice is isothermal, depth patterns of the acidity origin reflectivity are determined only by the acidity contrasts (equation (6)). As a preliminary approach, we assess a depth profile of the acidity origin conductivity at a uniform ice temperature of −31°C using Siple Dome ice core chemistry data. Radar-effective acidity (section 2.4.1) was derived at ∼0.2 m intervals using ion chromatography data [MacGregor et al., 2007]. We used 10 m running mean acidities as local background acidities so that the differences between the measured acidity and the running mean give local acidity contrasts and thus reflectivity at ∼4600 depths.

[40] Mean and standard deviation of the estimated acidity contrasts δC1 are 1.23 and 0.84 μmol L−1, respectively. Accounting for all estimated acidity contrasts, regardless of their magnitude, gives the reflectivity gradient 〈d[R]dB/dz〉 of −1.6 dB km−1. Nevertheless, an insignificant reflectivity gradient is predicted at the 99% confident level by accounting for acidity contrasts larger than 2, 4, and 6 standard deviations. Such large acidity contrasts are identified at 255, 40, and 7 depths, respectively, while only 3–10 layers along each profile are used to derive [Rmax]dB. This suggests that 〈d[Rmax]dB/dz〉 is virtually negligible but 〈d[R]dB/dz〉 is significant.

[41] At shallow depths, density contrasts (not acidity) are the predominant reflection mechanism (section 2.4.1). We estimated depth variations of the density contrasts in isothermal ice and then depth profiles of the density origin reflectivity (Appendix E). The estimated reflectivity values are −65 and −74 dB at 200 and 500 m, respectively, giving 〈d[R]dB/dz200 m500 m ≈ −27.8 dB km−1.

[42] Magnitudes of the estimated density origin reflectivities at 200 and 500 m are identical with those of estimated acidity origin reflectivity at the 18th and 289th largest acidity contrasts in the Siple Dome ice core. This implies that the density contrasts are the most significant reflection mechanism to depths of less than 500 m, which is consistent with the dependence of the vertical gradient on the lower-limit depth (Figure 5).

5. Estimates of Attenuation at WAIS Divide

[43] Modeling reflectivity shows that the predominant reflection mechanism changes at several hundred meters. This result explains the distinct depth dependences of the upper envelope gradient above and below ∼500 m (Figure 7) and the stronger depth dependence of the local mean gradient at small depths (Figure 5). All local mean gradients were derived from a fixed upper depth (200 m) to various lower depths so that the local mean gradients to greater depths are affected by large gradients of density origin reflectivity close to the upper limit in the same way.

[44] The upper envelope of returned power ensembles is unlikely affected by fading and anomalously small reflectivities of tilted layers. Therefore, they can be interpreted in terms of Fresnel reflectivity derived from ice core data. Our assessment using the Siple Dome ice core spanning the past 100 kyr shows that depth variations of reflectivity are expected to be negligible in isothermal ice, yielding nearly linear depth trend in [Pc]dB. This assessment is consistent with the observed trend in the upper envelope. Therefore, we conclude that in the isothermal ice, the upper envelope gradients can be a much better proxy for depth-averaged attenuation rates than local mean gradients. This conclusion may be invalid in areas where temporal acidity variations are significant. However, we argue that temporal chemistry variations are not a serious problem for this kind of analysis since this method is applicable only to depths where ice is nearly isothermal. In such regions, accumulation rates are large compared to the ice thickness so that the depth range for the analysis covers a short time period.

[45] The upper envelope gradients for 20 km long windows range between −17 and −27 dB km−1, and its regional mean value is −20.8 dB km−1. The range gives 〈N∼500 m∼1600 m 8.5 and 13.5 dB km−1, respectively, when reflectivity gradients are negligible (equation (10)).

[46] For comparison, depth-averaged attenuation rate 〈N0H for the full ice thickness was estimated to be 20 dB km−1 at Siple Dome [MacGregor et al., 2007] and 8 dB km−1 at Vostok [MacGregor et al., 2009], using ice core data and equations (4) and (5). Radar-based estimates of 〈N0H at various sites in Antarctica and Greenland range between 6 and 27 dB km−1 (see summaries of measured values by Matsuoka et al. [2009b] and Jacobel et al. [2009]). Our estimate in the upper half shows that englacial attenuation in the WAIS divide area is somewhat larger than in other areas.

[47] Variation of attenuation in the upper half thickness over this area is estimated to be 16 dB for a round trip through the upper 1600 m. That is comparable with an expected reflectivity contrast between wet and dry beds [Peters et al., 2005], indicating a need for accurate extraction of englacial attenuation variations before dependable detection of bed wetness can be made on the basis of contrasts in power returned from the bed.

[48] The upper envelope gradients can be used as an attenuation proxy in areas where the accumulation to ice thickness ratio is relatively large so that ice temperature is relatively uniform to middepths and thus the path-integrated attenuation can be approximated as linear (in the decibel scale) with reasonable accuracy. Figure 9 shows that ice is nearly isothermal in the top half if accumulation to ice thickness ratio is larger than 10−4 yr−1 and lateral motion of ice is minimal. Figure 10 gives guidance how deep radar data can be used to retrieve attenuation with the linear approximation. We note that it should be carefully assessed whether the segment length (20 km in this study) and the definition of the upper cutoff (excluding 97th–99th percentiles or 70th–90th percentiles) satisfy the conditions mentioned in section 3.4 and in Appendix C; we choose these numbers a posteriori. If ice is not isothermal, nonlinear approximation of the depth patterns of the returned power is necessary, which is a next step of this kind of methodological development.

6. Interpretations of the Regional Attenuation Variations

[49] The attenuation proxy 〈d[Pmaxc]dB/dz〉 allows us to explore temperature and chemistry distributions within the ice. Using Siple Dome and Gudmandsen parameterizations (section 2.4.1), the regional mean attenuation proxy can be interpreted as a regional mean ice temperature of −26.1°C ± 0.3°C and −26.9°C ± 0.3°C, respectively. Regardless of the choice of attenuation parameterizations, a range of estimated local variations in the ice temperature is 2°C–3°C. An alternative interpretation is that the observed range of the attenuation is caused by variations in the chemistry under small ice temperature variations fully represented by orographic lapse rates of the current ice sheet surface. For chemistry, we examined three cases by varying only acidity (C2 = 0 in equation (5)), only salinity (C1 = 0), and both acidity and salinity, but their ratio is fixed as at Siple Dome (C1/C2 = 0.31). All these cases require that the depth mean acidity (and/or salinity) varies by a factor of 2 in the WAIS divide area.

[50] Statistics (Student's t tests) show that all Nx, Cx, and Sx profiles have linear trends in the attenuation proxy from low on Amundsen side to high on Ross side, with confidence levels of 80%, 70%, and 55%, respectively (Figure 8). Over the depth range that we studied, depth variation of ice temperature is dominated by accumulation. The observed trend cannot be produced if the regional accumulation pattern has been maintained, regardless of the magnitudes of the accumulation. Snow pit work near the divide shows that soluble ions are deposited wet [Kreutz et al., 2000]. If soluble ions have steadily deposited either wet or dry, their concentrations would be independent of or inversely proportional to the accumulation, respectively, neither of which explain the observed attenuation patterns.

[51] The attenuation proxy near the divide (−10 < ξ < 10 km) on the Nx profiles is significantly larger than the divide sites on Cx and Sx profiles, while no significant difference is found between the Cx and Sx divide sites. A possible interpretation is that ice temperature 80–180 km northwest along the divide from the core site is ∼2°C warmer than at the core site. An alternative interpretation is that acidity is larger by a factor of ∼1.4 with the other effects held constant. This along-divide pattern qualitatively matches the 40 year averaged chemistry revealed by three ice cores drilled near these divide sites. Two ice cores near the Sx3 and Cx4 divides (00-1 and RISD-A cores, respectively) have comparable mean concentrations for excess sulfate, a major contributor to the acidity, which are much larger than those from the ice core 50 km north of the Nx4 divide (00-4 core) [Dixon et al., 2005]. This is opposite from what is expected based on attenuation.

[52] These examinations indicate that the observed attenuation proxy patterns cannot be readily explained with steady state conditions in the WAIS divide area. They suggest a complicated, rather than steady or simple, glaciological evolution in this area.

7. Synthesis

[53] To estimate englacial radar attenuation, we analyzed depth patterns of geometrically corrected returned power [Pc]dB in the decibel scale collected with 60 MHz airborne radar in central West Antarctica. The depth patterns of [Pc]dB are related to depth-averaged attenuation and depth profiles of reflectivity; a better attenuation proxy requires correspondingly better mitigation of reflectivity variations. Least squares (local mean) gradients of [Pc]dB along radar tracks have extremely anomalous values over steep beds, which are caused by small reflectivity from tilted internal layers. The local mean gradients are greatly affected by the reflectivity variations to represent attenuation. To mitigate this feature, we examined depth patterns of [Pc]dB in data ensembles over a distance much wider than these features. These ensembles still have large local variations of [Pc]dB at given depths but have a clear upper envelope. The upper envelope decreases linearly with depth between ∼500 and ∼1600 m, where ice is expected to be isothermal. Our estimates using Siple Dome ice core data show that significant acid origin reflectivity has no depth dependence in the isothermal ice. Therefore, we conclude that the upper envelope gradient could be a proxy of depth-averaged attenuation rates in the isothermal ice.

[54] Estimated depth-averaged attenuation rates for individual 20 km by 15–20 km segments vary ±2.5 dB km−1 around the regional mean value of 11.3 dB km−1 for a single trip to ∼1600 m. Regional patterns of the estimated attenuation suggest that there are distinct temperature variations larger than expected from the orographic effects. Alternatively, the attenuation patterns can be interpreted in terms of chemistry, indicating more complicated features than expected from steadily wet (or dry) depositions under the modern accumulation pattern.

[55] The range of the estimated attenuation rate yields a lateral variation of 16 dB in the returned power. The attenuation in the upper 1600 m is already comparable with bed reflectivity variations of ∼15 dB caused by frozen-unfrozen alternations of bedrock [Peters et al., 2005]. This strongly suggests that extracting attenuation precisely is critical to delineate wetter and dryer bed in this area.

[56] Isochronous radar layers have been used as an established natural marker to examine glacial history, but ice flow models used for the interpretation usually assume simple ice rheology and bed properties. Assessing spatial variability in attenuation ultimately provides remote-sensing means of thermal and chemical characteristics of the ice and of bed conditions, which can constrain ice flow models more realistically. This study demonstrates that radar profiling data, which already cover a significant area over Antarctica and Greenland, can provide attenuation proxies and ultimately complement understanding of current subglacial conditions and glacial history.

Appendix A:: Effects of Ice Fabric

[57] The term B in equation (3) represents signal level reduction due to the interference of two principal wave components in anisotropic (birefringent) ice. It is measured relative to isotropic ice as a function of a phase lag ϕ between the two wave components and an azimuthal angle between the (vertical) radar polarization plane and principal axes of the fabric (in the horizontal plane). When the polarization plane has an angle of ±30° from one of the principal axes, B is 2 dB (ϕ = π/2) and 6 dB (ϕ = π). Depths at which ϕ reaches such significant values depend on anisotropy of the ice fabric and radar frequency. A depth mean fabric anisotropy measured with the NGRIP ice core drilled on a topographical ridge in Greenland [Wang et al., 2002] shows that ϕ of the 60 MHz radio wave reaches π/2 at a depth of 2270 m (73% of the ice thickness). The depth at which ϕ reaches π/2 would be 2270 m (75% of the ice thickness) at Dome Fuji, but ϕ for 60 MHz radio wave never reaches π/2 at GRIP and GISP2 (see statistics of the ice fabric as given by Matsuoka et al. [2009b]). All the radar profiling analyzed in our study follows maximum topographic gradient, i.e., approximate ice flow direction. Usually one of the principal axes of the ice fabric has a small angle with the flow vector. Therefore, on the basis of the estimates of small ϕ and small angle between the radar polarization and the principal axes, we assume that B is depth-independent so that 〈d[B]/dz〉 is negligible for our analysis. This assumption will be assessed using the future ice core.

Appendix B:: System Stability Requirements for Returned Power Analysis

[58] The radar instrumental factor S can be written as a sum of a long-term mean S0 and a term δS that represents temporal variations. When the depth pattern of a single waveform is analyzed, the property in equation (8) requires uniform δS over the delay time in which the radio wave completes the two-way travel (e.g., a round trip in 4200 m thick ice takes 50 μs). In practice, hundreds of waveforms are stacked to reduce thermal (random) noise, and then one averaged waveform is recorded. The stability requirement for this case is that the mean δS(t) over n waveforms 〈δS(t)〉n is uniform in terms of the delay time.

[59] Usually, transmitting power, amplifier characteristics, and receiver sensitivities are measured by stacking hundreds of waveforms so that uncertainties in the calibration data represent 〈δS(t)〉n. Therefore, uncertainties in the radar specifications can be used as a measure of (in)stability of δS required for our analysis. The one exception is the noise floor level that varies with instantaneous environment of the electrical noise. However, possible fluctuations of the noise floor do not affect our analysis since we defined the deepest depth (DS) of radar signals as a margin (2 dB) above the noise floor level (approximately −107 dBm). This margin can be smaller, but it does not benefit our subsequent analysis since such deep bright echoes are caused by extremely large reflectivity (section 3.3). In this study, uncertainty in [P]dB is within ±0.6–1.7 dB (M. Peters, personal communication, 2005).

Appendix C:: Determination of the Upper Envelope

[60] To characterize the upper envelope of local returned power, we divided all [Pcbright]dB values into 10 depth bins that include the same number of measured [Pcbright]dB values. We define the upper envelope [Pmaxc]dB as the mean of [Pcbright]dB values within the largest 1%–3% (99th–97th percentiles) of the data. The mean depth of these [Pcbright]dB values is assigned as the representative depth of [Pmaxc]dB in that depth bin (circles in Figure 7). Depths of bright layers change gradually, so the data points tend to cluster rather than be evenly distributed. For this reason, the data are divided into bins that have the same number of data points, rather than into bins with uniform depth ranges. The data above the 99th percentile (largest 1%) are rejected to exclude layers with extremely large reflectivity; the objective of this analysis is to mitigate effects of reflectivity variations. One example of the extremely large reflectivity is the distinctly bright layer reported by Jacobel and Welch [2005] and Neumann et al. [2008] (Figure 4a); we estimated that this layer has a reflectivity ∼9 dB larger than a more normal layer found immediately (460 m) above it by extracting estimated attenuation over a round trip between these two layers (section 4.1).

[61] When lateral variations in the upper envelope are derived, the number of data points in each ensemble may be small. When the number of data points is less than 100, [Pmaxc]dB values are defined as the mean of the 70th–90th percentile values, not the 97th–99th percentile values, to avoid statistics of small numbers. Note that these rejection criteria were established a posteriori, but results are robust for changes in these numbers.

Appendix D:: Ice Temperature Model

[62] We use a one-dimensional heat flow model coupled with a kinematic ice flow model to estimate depth profiles of ice temperature and then returned power. The kinematic model approximates the depth profiles of horizontal ice velocity with two piecewise linear functions [Morse et al., 2002]. For this study, geothermal flux q is varied between 60 to 80 mW m−2; this range includes 69 mW m−2 at Siple Dome [Engelhardt, 2004] and 75 mW m−2 at Byrd [Gow et al., 1968]. Surface mass balance b is varied between 0.2 and 0.7 m yr−1 (ice equivalent), covering the range of the modern surface mass balance in this area [Morse et al., 2002]. Surface temperature Ts is fixed at −31°C, the mean annual temperature at the divide (∼1820 m above sea level (asl)) upstream of the core site, estimated from Byrd Station (1500 m asl), with a lapse rate of 10°C km−1 [Morse et al., 2002]. The expected range of Ts associated with surface elevation variations (∼150 m) in this area has virtually no effect on the depth patterns of the temperature estimated in section 4.1. The thickness of the bottom softer layer in the model is assumed to be 0.2H, where H is ice thickness. This gives a good approximation at flank sites 1H or more from the divide [Neumann et al., 2008]. Effects of horizontal advection are not taken into account, and all boundary conditions are kept steady. This treatment is acceptable since we seek an approximate picture of the kind of behavior to be expected.

Appendix E:: Density Origin Reflectivity

[63] The ice density ρi(z) at depths was derived using an empirical depth-density relation, [Schytt, 1958] with the exponential coefficient (2.75 × 10−2 m−1) from Byrd Station [Paterson, 1994] and surface density ρs ranging from 300 to 400 kg m−3. Next, the density ρb(z) of bubbly ice at depths was estimated using the ideal gas equation, provided that ice is isothermal (section 4.1). A density variation of 1 kg m−3 found at 1000 m in the Byrd ice core [Clough, 1977] is taken to derive depth variations of ρb(z). These allow us to derive (1) density contrasts equation imageequation image at depths, (2) associated permittivity contrasts using the Looyenga dielectric mixing formula [Looyenga, 1965], and (3) Fresnel reflectivity [Paren, 1981].


values in the decibel scale, i.e., [x]dB = 10log10(x).


mean of x(z) between depths z1 and z2.


signal level reduction relative to the isotropic ice due to birefringence (equation (3)).


acidity (i = 1) or salinity (i = 2) for attenuation modeling (equation (5)).


activation energies for pure ice (i = 0), acidity (i = 1), and salinity (i = 2) components (equation (5)).


geometric spreading (equation (2)).


ice thickness.


effects of ice characteristics on a round-trip propagation of radiowaves (equation (3)).


dielectric attenuation integrated along the propagation path (equation (4)).


local attenuation rate at depth z (equation (5)), dB km−1 for one way.


returned power measured with radar (equation (1)).


geometrically corrected returned power (equation (7)).


geometrically corrected power returned from a bright layer (equation (10)).


geometrically corrected power representing the upper envelope of the depth profiles of the power returned from bright layers.


in situ (radar-measured) reflectivity.


reflectivity associated with a bright layer.


reflectivity associated with the upper envelope (equation (10)).


Fresnel reflectivity due to a conductivity (δσ) or permittivity (δɛ) contrast.


effects of radar instruments (equation (1)).

S0, δS

long-term mean (S0) and temporal (δS) components of S.


ice temperature.


reference ice temperature for the attenuation model = 251 K (equations (5) and (6)).


temperature at the ice sheet surface.


depth at which modeled temperature is 3°C higher than the upper surface temperature Ts.


depth at which modeled local attenuation rate N(z) is 25 dB higher than N(0.2H).


surface mass balance.


speed of light in vacuum.


radar frequency.


radar (airplane) height above the surface.


Boltzman constant.


geothermal flux.


target (reflector) depth.

z1, z2

upper (z1) and lower (z2) boundaries of depth ranges for the derivation of vertical gradients.


acidity (i = 1) or salinity (i = 2) contrast at a reflection surface (equation (6)).


relative permittivity of ice (=3.2).


permittivity of the free space.

ρi, ρb, ρs

density of normal ice (i), bubbly ice (b), and firn density at the surface (s).


phase difference between two wave components propagating in the birefringent ice.




conductivity contrast at a reflection surface.


pure ice conductivity (i = 0) and molar conductivities for acidity (i = 1) and salinity (i = 2) contributions.


distance along individual radar tracks measured from the current flow divide location.


[64] We acknowledge SOAR for conducting radar surveys, Matthew Peters for providing calibration data, and Tom Neumann and Ed Waddington for providing computational codes of the temperature model. Katie Liu developed a GUI-based layer picking tool and tracked internal layers as a part of her undergraduate summer research program at the University of Washington, which was partly supported by Washington NASA Space grant. Comments from reviewers and the Associate Editor significantly improved the manuscript. This work was funded by U.S. National Science Foundation grants ANT-0338151, OPP-9726113, and OPP-9726500.