Radar power returned from ice-sheet beds has been widely accepted as an indicator of bed conditions. However, the bed returned power also depends on englacial attenuation, which is primarily a function of ice temperature. Here, using a one-dimensional attenuation model, it is demonstrated that, in most cases, variations in bed returned power are dominated by variations in englacial attenuation, rather than bed reflectivity. Both accumulation rate and geothermal flux anomalies can interfere with the interpretation. With the consequence, analytical radar algorithms that have been widely accepted likely yield false delineations of wet/dry beds. More careful consideration is needed when diagnosing bed conditions. Spatial patterns of shallow englacial radar reflectors can be used as a proxy for accumulation rates, which affect ice temperature and thus returned power. I argue that it is necessary to simultaneously interpret the returned power and englacial-reflector patterns to improve the bed diagnosis.
 Radar power returned from ice-sheet beds (a.k.a. bed-echo intensities) has been widely accepted as an indicator of bed conditions [e.g., Bingham and Siegert, 2007; Carter et al., 2009]. Many studies assume englacial attenuation rates to be uniform so that local attenuation is proportional to local ice thickness [e.g., Bentley et al., 1998; Rippin et al., 2004; Jacobel et al., 2009]. However, a recent study of central West Antarctica indicates that the depth-averaged attenuation rate in the upper half of the ice sheet varies horizontally by 5 dB km−1 (one way) along a 120-km-long radar transect [Matsuoka et al., 2010a]. The mean ice thickness in this area is roughly 3 km, indicating that the englacial attenuation can cause the bed returned power to vary by 30 dB. This variation is much larger than the contrast (10–15 dB) in Fresnel reflectivity between wet and dry beds [e.g., Peters et al., 2005]. Therefore, the observed variation in the attenuation rate can easily interfere with the delineation of wet ice-sheet beds on the basis of contrasts in the returned power. As such, the conventional belief that brighter and dimmer reflectors are an indication of wetter and dryer beds, respectively, should be reassessed.
 This paper presents the model results of englacial attenuation for realistic ranges of ice thickness, surface accumulation, and geothermal flux in Antarctica and in Greenland. These results are used to assess a conventional analytical algorithm for delineating wet beds on the basis of the returned power. I argue that inaccurate attenuation estimates obtained using this algorithm (or more generally the conventional belief of regional uniformity in the attenuation rate) constitute a critical pitfall in radar diagnosis of ice-sheet bed environment. Alternative strategies to mitigate the uncertainty are discussed.
 Radar-measured bed returned power Pbed is a function of instrumental characteristics, englacial attenuation to the bed Lbed, bed reflectivity Rbed, and a geometrical factor (spreading loss). The geometric factor Gbed associated with the bed echo can be assumed independent of permittivity, because a fraction of the firn layer (roughly 100 m) to the thickness of the entire ice column is small and anisotropy in the permittivity caused by non-uniform alignments of ice crystals (ice fabrics) is negligible. Therefore, Gbed can be determined with ice thickness and so it is straightforward to remove the effects of Gbed from Pbed. The geometrically corrected returned power from the bed can be written in the decibel scale as
for the case in which the instrumental characteristics are sufficiently stable [e.g., Jacobel et al., 2009; Matsuoka et al., 2010b]. The loss due to multiple reflections between englacial reflectors is negligible for radio waves [e.g., MacGregor et al., 2007]. Extinction of the returned power caused by ice-fabric-induced birefringence can be mitigated if radar polarization is chosen correctly [Matsuoka et al., 2009]. Therefore, the quality of as a proxy for bed reflectivity depends primarily on the accuracy of attenuation extraction.
 Attenuation has contributions from pure ice and soluble ions (acidity and sea-salt chloride), all of which have different temperature dependences [MacGregor et al., 2007]. In this paper, temperature dependence of the attenuation is approximated with that of the pure ice contribution to seek an approximate picture of the kind of behavior to be expected.
 The depth profiles of ice temperature and the basal melting rate are estimated for steady states using a one-dimensional heat flow model coupled with a kinematic ice flow model [Morse et al., 2002]. The kinematic model approximates the depth profiles of horizontal ice velocity with two piece-wise linear functions. The thickness of the softer bottom layer in the model is assumed to be 0.2H, which gives a good approximation at flank sites located at distances of 1H or greater from the divide. The estimated ice temperature T is then used to derive the local attenuation rate N (dB km−1 for one way): N = 0.914σpure exp[−E/k (1/T − 1/Tr)], where σpure = 7.2 μS/m, E = 0.55 eV, Tr = 251 K, and k is the Boltzmann constant [MacGregor et al., 2007, 2009; Matsuoka et al., 2010a, 2010b]. Finally, N is integrated along the two-way travel path to the bed so that the attenuation [Lbed]dB associated with the bed returned power is derived.
 Subglacial reflectivity depends on the dielectric contrast between ice and the materials beneath the ice, as well as the roughness of the interface [e.g., Peters et al., 2005]. If the interface is smooth and flat, the reflectivity can be represented by Fresnel reflectivity. A Fresnel reflectivity for a subglacial lake surface and a dry bed is −3.3 dB and −19 dB, respectively, so the reflectivity contrast between dry and wet beds is less than ∼15 dB (Text S1 of the auxiliary material). The wetter bed has the larger reflectivity, but horizontal movement of the subglacial water hampers more realistic estimates of bed reflectivity in terms of basal melting rate using this one-dimensional model. In the present study, I assign a simple step function so that [Rbed]dB = 0 dB when the bed is melting, and [Rbed]dB = −10 dB when the bed is not melting.
3. Estimated Bed Returned Power
 The depth-averaged attenuation rate 〈N〉 (= [Lbed]dB/(2H)), the basal melting rate, and the geometrically corrected bed returned power are estimated in terms of accumulation rate A (Figure 1), geothermal flux G (Figure 2), and ice thickness H (Figure 3). In each case, other parameters are kept unchanged.
Figure 1 shows the predicted features in terms of surface accumulation rate between 0.02 m a−1 and 0.3 m a−1. This range is found, roughly speaking, in the Ross embayment, the Filchner-Ronne ice shelf, and inland Antarctica (surface elevation > ∼1,500 m a.s.l.), where the accumulation rate varies between 0.1 and 0.2 m a−1 over the horizontal distance of 50 to 400 km [van de Berg et al., 2006]. In Greenland, this range of mass balance is found inland where the surface elevation is higher than ∼2,000 m a.s.l. [Box et al., 2004]. The depth-averaged attenuation rate 〈N〉 decreases monotonically as accumulation increases, regardless of the bed condition (melting or not melting). The attenuation varies less significantly when the accumulation is larger (i.e., ice is colder to greater depths) because of the Arrhenius-type temperature dependence of the attenuation. Figure 1c shows that significant (>10 dB) variations in the bed returned power can be caused by attenuation variations, even if the bed is uniform. The 10-dB step change of the returned power (caused by the reflectivity change) at the boundary of the wet and dry beds can be compensated for when the accumulation increases by 0.03 m a−1 or decreases by 0.05 m a−1 for this specific case. Compensation of the 10-dB step requires larger variations in the surface accumulation when the ice is thicker and/or the geothermal flux is larger: ±0.07 m a−1 (H = 3 km), ±0.09 m a−1 (H = 3.5 km) and ±0.12 m a−1 (H = 4 km), when the geothermal flux is 50 mW m−2.
Figure 2 shows the predicted features in terms of the geothermal flux. In previous studies, the mean (± standard deviation) of geothermal flux over the Antarctic ice sheet has been estimated to be 62 ± 18 [Shapiro and Ritzwoller, 2004] and 65 ± 13 mW m−2 [Fox-Maule et al., 2005] with the tuning proposed by Pattyn . A wider range of geothermal flux (20–135 mW m−2) is predicted over the Greenland ice sheet [Greve, 2005] but its spatial distribution is not as well known as that of the Antarctic ice sheet. When the bed is not melting, 〈N〉 increases monotonically as the geothermal flux increases. When the bed is melting, 〈N〉 remains uniform regardless of geothermal flux, because although a larger geothermal flux melts more ice, the ice temperature remains unchanged. The 10-dB step change at the wet/dry boundary can be compensated for if the geothermal flux decreases by 20 mW m−2 for this specific case. When the ice is thicker, less geothermal flux is required to compensate for the contrast between the wet and dry beds (e.g., 15 mW m−2, when H = 3 km and A = 0.3 m a−1).
 These modeling results indicate that the typical assumption of regional uniformity of the attenuation rate is invalid for quite simple, but likely, glaciological settings.
4. Assessment of Analytical Algorithms
 Now traditional approaches determining bed reflectivity from radar data are applied to the modeled bed returned power in order to assess their validity. The algorithm to extract attenuation from the radar-observed returned power involves linearly approximating the relationship between geometrically corrected returned power and ice thickness H in a data ensemble (Figure 4). Ice-thickness derivative of equation (1) gives:
If the bed reflectivity has no depth dependence, represents the depth-averaged attenuation rate 〈N〉 averaged over the area where the data ensemble was collected. For the analysis using equation (2), the radar data should be collected over a range of the ice thickness so that variations in are exhibited by the variations in [Lbed]dB (Text S2). Another approach is to give the regional-mean attenuation rate a priori. In each case, anomalies in the measured returned power from the prediction obtained using the regional-mean attenuation rate (derived with equation (2)) have been accepted as an indicator of bed reflectivity and bed wetness.
 The depth-averaged attenuation rates 〈N〉 are modeled for the case in which only the ice thickness varies, while the other factors are kept uniform (Figure 3b). The modeled depth-averaged attenuation rates have a maximum at the wet/dry boundary and vary by roughly 5 dB km−1 for ice thicknesses ranging between 1 and 4 km. The algorithm described above is applied to the modeled bed returned power (Figure 4). For the 3-km thickness range, is largest when the bed is not melting, whereas is nearly zero when the basal melting rate is small. A more complication is that is sensitive to a choice of the thickness ranges over which the algorithm is applied for (Figure 3b). Therefore, analysis using the traditional approach is non rigorous (Figure 3c).
5. Discussion and Conclusions
 These modeling results indicate that the typical assumption of a regionally uniform attenuation rate is invalid in most cases and that currently accepted radar algorithms likely yield false estimates of the anomalous bed reflectivity. I argue that more careful consideration should be given to the interpretation of bed returned power, which requires accurate estimates of ice temperature and chemistry.
 Radar-observed isochronous reflectors (Figures 1a, 2a, and 3a) have constrained ice-flow models in order to infer past accumulation rate and ice-sheet evolution [e.g., Nereson et al., 1998; Waddington et al., 2007]. Thus far, bed diagnosis on the basis of the returned power has been performed independent of such modeling. However, reflector patterns and the returned power from within and beneath the ice sheets should be interpreted interdependently in order to infer glaciological conditions that satisfy all aspects of the radar data, with the aid of thermo-mechanical modeling of the ice flow. Depth variations of radar-measured englacial returned power can be used to constrain the attenuation rate, if the ice is nearly isothermal [Matsuoka et al., 2010a]. Knowing the ice temperature allows this method to be adapted to non-isothermal ice, so that the attenuation can be estimated using englacial reflectors (except at depths close to the bed). Modeling isochrones also helps to extrapolate the acidity and sea-salt chloride concentrations available at ice core sites to a wider area and eventually to account for spatial variations in the chemistry effects on the attenuation rate [Carter et al., 2009; MacGregor, 2008].
 Poor knowledge of geothermal flux more seriously hampers the interpretation of radar data, because the radar-reflector patterns remain virtually unchanged when the basal melting rate is on the order of 10−3 m a−1 or less (Figure 2a). Pattyn  conducted ensemble experiments using a range of geothermal flux and surface accumulation datasets and concluded that geothermal flux is the most significant source of bed prediction uncertainties. The determination of geothermal flux by means other than ice-penetrating radar is critical for assessing basal conditions.
 Although enormous radar datasets are being prepared, rigorous analytical models to interpret these radar data are not yet well established. Tracking englacial reflectors is labor intensive, but is necessary not only for examining the accumulation and flow histories but also for constraining the attenuation and ultimately diagnosing the current bed conditions.
 In conclusion, inaccurate estimates of the englacial attenuation are a serious pitfall in diagnosing bed conditions on the basis of the returned power. The development of better models is critical in order to simultaneously interpret various properties of the radar data.
 The author would like to thank E. D. Waddington and T. A. Neumann for providing the numerical codes of the ice temperature model, and H. Conway, J. A. MacGregor, B. Kulessa, R. G. Bingham and two anonymous reviewers for their helpful comments. This study was supported by US National Science Foundation grants OPP-0338151 and ANT-0538674 through the University of Washington and by Center for Ice, Climate, and Ecosystems (ICE) at the Norwegian Polar Institute. Eric Rignot thanks two anonymous reviewers.