Corresponding Author: W. Dierking, Climate Sciences Division, Alfred Wegener Institute for Polar and Marine Research, D-27570 Bremerhaven, Germany. (firstname.lastname@example.org)
 The sensitivity of radar measurements to snow accumulation rate is determined by the firn volume characteristics of the ice sheets. Here we present a new approach for calculating the volume backscattering of dry firn, which is combined with recently developed empirical parameterizations of firn grain size and density as functions of depth, surface temperature, and accumulation rate. To this end, dense medium radiative transfer theory is applied to calculate the volume scattering and absorption coefficients. The coefficients for the density transition between snow and dense firn are evaluated using polynomial interpolation. For testing the method, we used measured accumulation rates, Envisat ASAR C-band wide-swath mode images, and QuikSCAT Ku-band backscattering data from Dronning Maud Land, Antarctica. The comparison between measured and simulated backscattering coefficients shows that no tuning parameter is necessary to obtain the correct absolute level of scattering intensity. The robustness of accumulation rate retrievals depends on the consideration of technical and environmental factors. Due to the presence of sastrugi on the ice sheet surface the measured intensities are sensitive to the radar look direction. Wind compaction of snow and depth hoar formation change the depth-dependent snow density and grain size profiles. Theoretical simulations revealed that the backscattering coefficient at C-band is more sensitive to changes of accumulation rates than at Ku-band. Penetration depths can vary significantly, dependent on radar frequency and firn characteristics. This has to be taken into account when comparing accumulation rates from different locations.
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 Recent studies show the accelerating shrinkage of the Greenland and West-Antarctic ice sheets [Rignot et al., 2011]. If the observed trend in mass loss continues, the ice sheets will become the dominant contributors to sea level rise during the 21st century. The mass input by accumulation is one parameter required to deduce whether the ice sheet mass balance (i.e., the difference between mass gain and loss) is positive or negative. Mass gain is quantified by measurements of accumulation rates, which are the result of precipitation, sublimation, and wind erosion and redistribution of snow at a given location. In the field such measurements are carried out in snow pits or by using stakes, ultrasonic sounders or ground penetrating radar [Eisen et al., 2008]. Another possibility is the analysis of layers in firn cores. All methods are restricted to comparatively small regions on the ice sheets. Investigations using satellite data show that accumulation rates can also be derived from passive or active microwave sensors [e.g., Arthern et al., 2006; Flach et al., 2005; Munk et al., 2003; Drinkwater et al., 2001; Forster et al., 1999]. This approach, which is promising for interpolating between the spatially sparsely distributed field measurements, is the topic of this study.
 Over the ice sheets, the signal intensity received by a radiometer or radar depends on the size of ice grains in the firn, the firn volume structure (e. g. layering, presence of ice lenses and pipes) and on the moisture content [e.g., Rignot, 1995; Forster et al., 1999; Steffen et al., 2004; Oveisgharan and Zebker, 2007]. The basis of the retrieval algorithms from passive or active microwave data is the fact that relatively large snow grains develop under low accumulation rates and vice versa, and that the thickness of annual layers is proportional to accumulation rate. Most of the methods and studies described here, including our model, are restricted to the dry snow zone of the ice sheets in which the microwave signals are not affected by moisture or inclusions of solid ice bodies in the firn volume.
 Retrieval algorithms for accumulation rates on ice sheets are in some cases established on the basis of relatively simple empirical relationships, which are derived from comparisons of measured accumulation and microwave signatures [e.g., Drinkwater et al., 2001; Winebrenner et al., 2001; Rotschky et al., 2006]. More complex algorithms in which sensitivities to environmental conditions are explicitly considered consist of two steps: In the first step, theoretical or empirical relationships are provided that describe variations of grain size, density and temperature with depth as functions of accumulation rate and average annual surface temperature or location [Forster et al., 1999; Munk et al., 2003; Flach et al., 2005]. In the second step, microwave brightness temperatures or backscattering coefficients are calculated from the simulated firn parameter profiles. The results are used to generate look-up tables of theoretical microwave values versus annual mean surface temperature (in some cases geographical coordinates are used) and accumulation rate. The latter can then be retrieved for locations for which satellite data of the brightness temperature or backscattering coefficient are available and the average annual surface temperature is known. In this study, we applied the two-step approach. For simulating firn metamorphosis in the first step, we employ relationships that were recently developed using field data from Greenland and Antarctica [Linow, 2011]. The accumulation rate obtained in this way is an average value over the firn column from the surface down to a depth that depends on the frequency of the microwave sensor and on firn properties.
 In the case of passive microwave measurements the accumulation retrieval has to be based upon models of the layering structure if working with frequencies below approximately 7 GHz [Arthern et al., 2006]. West et al.  showed that the emission signature is controlled by coherent reflections between layer interfaces whereas particle scattering is negligible because of the small grain size. The situation is different for radar sensors that are operated in the backscattering mode (i.e., e. the received signal travels along the same path as the transmitted one). In this case, the signal intensity depends only on incoherent scattering (with possible exceptions at vertical incidence) [Hallikainen and Winebrenner, 1992]. Hence, for active sensors – and also for passive microwave radiometers at higher frequencies – scattering from ice grains is dominant. In order to explain the results of correlation measurements using interferometric synthetic aperture radar (InSAR) data from the ERS-1/2 tandem mission, Oveisgharan and Zebker  had to include incoherent scattering contributions from interfaces between layers. Their model is valid for a situation in which the uppermost annual layers include a distinct sub-layer of coarse-grained hoar that has formed during summer. The interfaces between the coarse-grained hoar and the fine-grained firn act as additional scattering sources. Using ERS-1 or -2 data [Forster et al., 1999; Munk et al., 2003] the consideration of surface or interface scattering is necessary because of the low incidence angle.
 For the retrieval of accumulation rates over Greenland using the two-step approach both C-band [Forster et al., 1999; Munk et al., 2003; Oveisgharan and Zebker, 2007] and Ku-band [Flach et al., 2005] radar systems were used. Except in the work by Oveisgharan and Zebker , modeling was carried out using a radiative transfer approach assuming independent Rayleigh scattering, which is valid for volume fractions of scattering particles <0.03 [Winebrenner et al., 1989]. To account for near-field effects in a dense medium such as firn, Munk et al.  adjusted their values of the scattering albedo manually to match the observed backscattering coefficients. Flach et al.  included a manually adjustable “conversion factor” for scattering by dense firn. Forster et al.  used an optical equivalent snow grain radius, which corrects for the effects of near-field interactions and the grain size distribution in annual layers. They obtained theoretical levels of backscattering intensity that were biased compared to measurements by 0.5 and 1.5 dB, respectively. Oveisgharan and Zebker  employed a method based on the Born approximation presented by Mätzler , which is derived under the assumption that the ice particles scatter independently (free arranged particles). In this approach, the dense medium effect is accounted for by the decrease of the scattering amplitude due to the changing ambient medium as the density increases. Their model reproduces the observed levels of backscattering intensity and interferometric correlation. However, a point-to-point comparison between simulated and measured data is not provided.
 The objective of our study is to develop a method for simulation of the radar intensity as a function of firn parameters that can be applied for the high-density layers at larger depths and without any tuning factor. To this end we use the dense medium transfer theory (DMRT), which was derived from wave theory under the quasi-crystalline approximation with coherent potential (QCA-CP) [Wen et al., 1990]. The DMRT QCA-CP is valid if the volume fraction of the scattering particles is less than 0.25 [Tsang et al., 2007] corresponding to snow/firn densities of 0.23 g/cm3. Estimates for penetration depths of microwave signals into dry firn are provided, for example, by Munk et al. , Weber-Hoen and Zebker , Rott et al. , and Legresy and Remy . Values range from 12 to 60 m at C-band and approximately 6 to 7 m at Ku-band. Close to the surface, the density of firn is about 0.3 to 0.45 g/cm3, values of 0.55 g/cm3 may be reached already at depths of 5 m, and at a depth of 60 m, the density is between 0.7 and 0.85 g/cm3 [Oerter et al., 1999, Hörhold et al., 2011]. Hence it is necessary to find a method, which makes it possible to use the DMRT over the full range of firn densities. We achieve this by an approach we term scatter regime bridging. In the snow (density ≤ 0.23 g/cm3) and the ice regime (density ≥ 0.69 g/cm3, volume fraction of scattering air bubbles in the ice <0.25) DMRT can be directly applied for computing the radar intensity. For firn of densities between 0.23 and 0.69 g/cm3, the radar intensity is obtained by interpolating between the two regimes.
 Most of the studies dealing with modeling the radar response as a function of accumulation rate have been concentrating on Greenland. Here, the dry snow zone covers approximately 40 percent of the ice sheet. In Antarctica, almost the entire continent can be categorized as dry snow zone except for a narrow strip along the coast, For our study, we selected an area in Dronning Maud Land, Antarctica, for which accumulation rates were measured in the field at numerous locations.
 When dealing with the retrieval of snow accumulation rates using imaging radar or scatterometers, one has to consider variations of the backscattering coefficient that are related to firn characteristics generated under the influence of katabatic winds [e.g., Ashcraft and Long, 2005]. These variations are a function of the radar look direction and incidence angle and are dependent on the geographic location [e.g., Long and Drinkwater, 2000; Rott et al., 1994; Rotschky et al., 2006].
 The paper is organized as follows: The backscattering model and our approach for backscattering calculations are introduced in section 2. Firn parameters required as input are obtained from empirical relationships for depth dependent snow grain size and density. In section 3, the simulated backscattering coefficients are compared to satellite radar measurements at C- and Ku-band over Dronning Maud Land. Hereby, the influence of different factors such as azimuthal modulations and inaccuracies in radar parameters (local incidence angle, geolocation error, calibration) on deviations between model and measurements are considered, which impact the robustness of the accumulation retrieval. In section 4, we analyze the sensitivity of the radar backscattering to accumulation rates, grain radius, and mean annual temperature, dependent on radar frequency. Penetration depths and the age of the corresponding firn layer are calculated. The potential of using C- and Ku-band radar data for the retrieval of accumulation rates is critically assessed.
2. Volume Scattering Simulations
2.1. Backscattering Model
 Formally, the model we use is identical to the one described by Flach et al.  which consists of a number of horizontal firn layers, each one characterized by constant values for density, grain size, and temperature. In our implementation, each layer corresponds to half a year of accumulation. In this way, it is in principle possible to consider seasonal variations of firn parameters, e. g. the inclusion of coarse-grained hoar sub-layers or seasonal variations of density described, e.g., in Hörhold et al. . In the recent version of our snow metamorphosis model, however, seasonal variations were not taken into account. The choice of the number of layers corresponding to one year of accumulation is restricted by the ratio between layer thickness and snow particle size, which has to be larger than unity. It is furthermore assumed that the interface between two layers is smooth and does not contribute to scattering. Electromagnetic particle-interactions across interfaces and multiple-scattering are neglected. The total backscattering coefficient σ0 of N layers is
where σvj0 is the backscattering coefficient from the volume of layer j, Γi-1,i is the transmissivity between layers i-1 and i (Γ01 at the air-firn interface), θi is the incidence angle on the interface between layers i and i + 1, and L is the one-way loss factor given by
where κe is the extinction coefficient, d is the layer thickness, and θ′ is the refraction angle. Transmissivity Γ and backscattering coefficient σ0 depend on the polarization of the radar waves.
 The difference of our approach to the model used by Flach et al.  occurs in the calculation of the volume backscattering coefficient, which in a general form is given by Fung :
Here, ω = κs/κe is the scattering albedo, κs the volume scattering coefficient, and the extinction coefficient is κe = κs + κa, where the volume absorption coefficient is denoted by κa. Instead of assuming independent scattering, we compute albedo and extinction coefficient from expressions valid for a dense medium, which can be found in Wen et al. [1990, equations (2), (4), and (5)]. In the DMRT QCA-CP approach, the wave interactions between the fields of neighboring scatterers are considered. The theory is valid for small particles (assuming spheres with radius r, the validity criterion is 2πr/λ < 1, with λ the radar wavelength in snow).
 Due to the low intensity contrast at the interfaces between single layers, interface-volume reflections can be neglected for dry firn. We tested this for a model of three firn layers with only small differences in their dielectric constants and found that the results of equation (1) compare perfectly with the full radiative transfer solution. The contribution of multiple scattering depends on the radar frequency and the firn grain size. According to scatter modeling results presented by Tsang et al. , multiple scattering can raise the radar signal by a few dB if the scattering albedo is close to one. Only very few of the test sites that we investigated revealed values of the albedo larger than 0.7 at C-band and 0.8 at Ku-band. In these cases, the accumulation rates were very low (<0.03 m w.e./a), and the layers with such high scattering albedo were located at larger depths (>15 m at C-band, >5 m at Ku-band). For a radar frequency of 17.5 GHz, a grain radius of 0.7 mm, and a snow depth of 1 m, Tsang et al.  obtained backscattering coefficients that were 4–6 dB higher if multiple scattering was included. Considering their results and our measurement conditions we estimate that the contribution of multiple scattering is 1–4 dB, with lower values at C-band and at sites with higher accumulation rates.
 At certain locations in Greenland and Antarctica, formation of depth hoar layers occurs at the onset of the summer season [Alley et al., 1990; Rott et al., 1993]. Depth hoar is characterized by comparatively large grain sizes (approximately 2–5 mm) and low densities (0.1–0.3 g/cm3). Hence, it causes a comparatively high dielectric contrast to the adjacent fine-grained and dense winter layers. This means that scattering contributions may occur from the interfaces between summer and winter layers, as was discussed by Oveisgharan and Zebker . At depths of about 5 m, the hoar layers collapse. For deeper layers, the dielectric contrast is hence reduced, but because of the large grain size and density, the volume scattering coefficient of hoar at those depths is relatively high. However, it is yet not clear whether the occurrence of depth hoar layers is a more localized or wide spread phenomenon. In the snow metamorphosis model that we use here, winter-summer variations of firn properties are not considered.
2.2. Firn Profiles in the Dry Snow Zone
 Grain size profiles were measured employing X-ray microfocus computer tomography on 12 m long firn cores from six test sites in Greenland and Antarctica [Hörhold, 2010]. On the basis of these data, an empirical relationship for calculating the effective firn grain size as a function of depth, dependent on annual mean surface temperature and accumulation rate, was derived [Linow, 2011]. The effective grain size is determined from the total surface area of particles in a unit volume of the firn sample. For the derivation of an empirical equation relating density profiles to annual mean surface temperature and accumulation rate, density profiles down to depths between 100 and 200 m were available from eleven locations in Greenland and Dronning Maud Land. The empirical relationships for density and grain size profiles used throughout this study are given in the appendix. Seasonal variations of grain size and density could not be clearly identified in the measured profiles and are consequently not included in the empirical relationships. We used LST (Land Surface Temperature) data from MODIS (Moderate-resolution Imaging Spectroradiometer) on the Aqua and Terra satellites to derive the mean annual surface temperature and seasonal temperature variations. MODIS surface temperatures are given as 8-day averages with a spatial resolution of 1 km. Temperature propagation into the firn is modeled as an exponentially decaying oscillation reflecting seasonal variations of the mean surface temperature [see, e.g., Flach et al., 2005]. Details about the simulation of snow metamorphosis can be found in Linow .
2.3. Volume Scattering and Absorption
 Volume absorption (κa) and scattering (κs) coefficients are dependent on the dielectric properties of the firn (and hence on temperature), as well as on density and radar wavelength. In addition, the volume scattering coefficient is affected by the size of the scattering particles. Since we assume that the scatterers can be approximated by spheres, effects of radar polarization are neglected. Both coefficients are independent of radar incidence angle and firn layer thickness.
 The starting point of our approach is to calculate κa and κs of snow (ice particles in a background of air) at ice volume fractions between 0.05 and 0.25 and of ice with air bubbles at air volume fractions in the same range (corresponding to ice volume fractions between 1.0 and 0.75). At a certain stage, a transition takes place from scattering ice particles in a background of air to scattering air inclusions in a background of ice. The basic idea of our approach is to cover this critical range by interpolating between the theoretical curves of κa and κs obtained in the “snow” and the “ice”-regime (“scatter regime bridging”) as is shown in Figure 1. When calculating the backscattering response in practice, the irregularly shaped ice grains in the “snow” regime and the air pockets in the “ice” regime are approximated by spheres, using the effective radius derived from the total ice grain and air bubble surface area in a given volume [Linow, 2011]. In the transition zone with volume fractions ν between 0.4 and 0.6, this approximation cannot be applied. It would require a sudden switch from spherical ice grains with interstices of air to spherical air bubbles embedded in ice. We assume that (a) the absorption in the ice is a continuous function of ν, no matter whether the ice is forming the scattering particles or the host medium, (b) that the size distributions of ice particles and the interstices filled with air are similar at ν around 0.5, and (c) that the volume scattering coefficient is a continuous function of ν. Because of the bridging between the “snow” and the “ice” regime, we avoid the need to separate explicitly scatterer and host.
 The absorption coefficient κa increases continuously as a function of volume fraction. The gradient dκa/dνi in the example of Figure 1 is strongest at volume fractions between 0.6 and 0.8. The volume scattering coefficient κs reveals a more complicated shape with two local maxima. We investigated how the width of the interpolation zone influences the calculated backscattering coefficient resulting from equation (1). Between interpolation intervals [0.25, 0.75] and [0.35, 0.65] we obtained a difference of <0.2 dB at C-band and <0.6 dB at Ku-band. The difference is larger when the scattering contributions from layers with volume fractions inside the interpolation zone have a stronger weight in the total sum on the right-hand side of equation (1). Since the penetration depth at Ku-band is smaller than at C-band, the firn layers located in the interpolation zone dominate the backscattered signal. The volume scattering coefficient is very sensitive to the particle radius, but variations due to temperature can be neglected in the range from −10°C to −50°C, as our numerical simulations showed. The volume absorption coefficient is independent of particle size, but very sensitive to temperature variations.
 The increase of the volume scattering coefficient for smaller volume fractions between 0 and an upper limit of 0.1 to 0.2, and the subsequent decrease for volume fractions up to 0.4 is a typical characteristic of volume scattering from dense media that was observed in laboratory experiments [e.g., Wen et al., 1990] and in numerical simulations of snow scattering [e.g., Tsang et al., 2007]. For larger volume fractions, however, the results of the DMRT-QCA deviate from observations. Kendra et al.  described measurements with a truck-mounted scatterometer over artificial snow, for which the backscattering coefficient predicted by the dense medium theory was considerably smaller than the observed value at a snow density of 0.5 g/cm3. This is in qualitative agreement with our approach since the values of κs on the interpolated segments are larger than the ones that are predicted by DMRT-QCA at 0.25 < ν < 0.5. Tse et al.  compared solutions of DMRT-QCA and the Numerical Maxwell Model of three-dimensional simulations (NMM3D) and found that the scattering coefficient resulting from DMRT-QCA simulation was too low for ν between 0.3 and 0.45.
 For a firn column that was simulated assuming an accumulation rate of 0.1 m w.e./a and a mean annual temperature of −21°C, the volume absorption and scattering coefficients are shown in Figure 2. The shape of κa close to the surface is influenced by the temperature profile at the time of data acquisition. A broad peak characterized the curve of κs, which means that many layers at moderate depths contribute almost equally to the scattering return. In computations of the backscattering coefficient σ0 according to equation (1) the depth of maximum volume scattering is influenced mainly by the profile of absorption and, to a minor degree, by the transmissivity between the firn layers.
 When evaluating the volume extinction from a column of firn it has to be considered that during the process of snow metamorphosis, larger grains are growing at the expense of smaller ones. This means that the number of ice particles per unit volume decreases as a function of depth (see Figure 3). In our model we calculate the number of particles in the upper layers from firn density and ice particle radius at a given depth, which are output parameters of the empirical firn parameter equations mentioned above. We assume that the number of ice particles and air voids must remain constant at larger depths at which particles are in the stage of densest packing. For randomly distributes spheres this stage occurs at ν ≈ 0.55 [Anderson and Benson, 1963]. By measuring density of firn cores, Hörhold et al.  found maximum packing densities between 0.5 and 0.6 g/cm3 corresponding to 0.55 < ν < 0.65. We examined measured density and grain size profiles and found that the number density of ice particles did not decrease further for volume fractions larger than about 0.5–0.55 (see Figure 3).
 Since we deal with a huge number of particles in a sufficiently large volume, we assume that the number of interstices filled with air is equal to the number of ice particles at ν = 0.55. The number density of air bubbles in pure ice, which is our modeling approach in the “ice regime” of the firn, is then kept constant over the range of 0.55 < ν < 1.0. This means that the radii of the air bubbles must decrease with increasing ν, which is realistic since the bubbles in dense firn are compressed at larger depths.
 Depth dependent profiles of volume scattering and absorption coefficients κs and κa were derived from given firn parameter and temperature profiles. To this end, κs and κa-values were calculated at ν = 0.05, 0.10… 0.95. Values of κs(ν) are provided for discrete radii r between 0.1 mm and 2.7 mm (validity of Rayleigh approximation) with a spacing of 0.1 mm, values of κa(ν) for discrete temperatures T between −50°C and −5°C with a spacing of 5°C. For each pair (r, ν) and (T, ν) from the firn parameter profiles, the corresponding κs and κa- values were determined by bilinear interpolation on the (r, ν) and (T, ν) planes, respectively, using the calculated grids of κs(ri, νj) and κa(Tk, νj). The very weak sensitivity of κs to temperature was neglected. Volume scattering and absorption coefficients were then fed into equations (2) and (3).
 The dielectric constant for each layer was calculated using the equations for dry snow given by Tiuri et al.  for ν < 0.5, and a two-phase-mixture equation for spherical air inclusions in case of larger volume fractions [Ulaby et al., 1986]. Subsequently, the transmissivities and the refraction angle were determined for each layer. Finally, the backscattering coefficient for the whole firn column was evaluated using equation (1). The number of layers, N, was limited by the penetration depth into the firn which is defined by
where di is thickness of layer i, θi is the transmission angle at the upper interface of layer i, and N is the number of layers for which the sum given by (4) equals one. We also carried out test simulations with limits of 1.2, 1.5 and 2 times the penetration depth. Compared to results in which the number of layers was fixed by the single penetration depth, we found deviations of 0.3, 0.5 and 0.65 dB, respectively. Since we had to consider that - because of the accumulation of round-off errors- the results are biased toward larger values if too many layers are included, we determined the limiting value of N according to (4). The numerical uncertainty in our results is below 1 dB. Using C-band data, Munk et al.  found that beyond 100 annual layers, values of the calculated backscattering intensity did not vary as additional layers were included. As will be shown later, penetration depths at C-band are in most cases larger than the depth corresponding to a firn layer age of 100 years.
3. Comparison With Measurements
3.1. Field and Satellite Data
 For testing the model, we used field data of snow accumulation measured at more than 600 locations in Dronning Maud Land (Figure 4). A detailed description of the data can be found in Rotschky et al. . For each site with known accumulation and mean annual temperature, particle size, density, and temperature profiles were calculated as a function of depth (first step of modeling, see above). The theoretical backscattering coefficients (equation (1)) were then evaluated using these profiles down to the penetration depth of the radar signal (second step of modeling). As the penetration depth (equation (4)) depends on the firn properties, the number of years spanning the firn column included in the computation of the backscattering coefficient differs between field sites. For each field site, we determined measured backscattering coefficients for comparison with the theoretical results. We used images from ESA's Envisat ASAR sensor acquired in wide-swath mode (WSM) at HH-polarization and with an incidence angle range between 15 and 43 degrees, and QuikSCAT scatterometer data from 2006 that were obtained at HH-polarization (46 degree nominal incidence angle). Details for the ASAR images are given in Table 1.
 In our analysis, we focus on two test sites: (1) the Kottas Traverse, which is characterized by a wide range of measured accumulation rates of up to 0.5 m w.e./a, and (2) the Amundsenisen Plateau and the ice ascent north of it with accumulation rates of up to 0.15 and 0.5 m w.e./a, respectively [Rotschky et al., 2006; Oerter et al., 1999].
 Before retrieving backscattering coefficients σ0 from the Envisat ASAR WS (wide swath) images, we applied an averaging filter at a window size of 2 × 2 pixels. Hence, every value of σ0 is from an effective area of about 225 × 225 m2. In our model simulations of the backscattering coefficient σ0 we used -for each measurement point- the respective local radar incidence angle to avoid incidence angle normalization. The reason is that gradients dσ0/dθ can vary considerably between different locations [e.g., Rott et al., 1994; Rotschky et al., 2006]. The backscattering coefficients on the Amundsenisen Plateau and the northward ice ascent were retrieved from the ASAR image acquired in February 2009. A part of the Kottas traverse is also covered by the eastern part of this image but it is seen under small incidence angles of around 20 deg. In such a case, scattering contributions from surface and layer interface undulations are relatively strong. Consequently, we obtained differences of 2 to 6 dB between simulations and backscattering coefficients. In the SAR images from summer 2008 and winter 2009, the Kottas Traverse is located in areas of larger incidence angles, and the surface scattering contribution is lower (see Table 1).
3.2. Inaccuracies and Variations of the Measured Data
 The accuracy of the observed σ0-values depends on errors in radar intensity calibration and geolocation, effects of the scalloping phenomenon in ASAR WSM images, speckle, noise level, and uncertainties in the local incidence angle due to errors in the used digital elevation model. The calibration accuracies for different ASAR image products are listed in the Envisat ASAR Monthly Report (http://earth.eo.esa.int/pcs/envisat/asar/public_reports/). Between December 2008 and June 2009, the deviation between the measured and the nominal radar cross section of reference transponders was between 0.13 ± 0.75 and 0.11 ± 0.63 for the WSM mode, the absolute calibration accuracy was hence better than 1 dB. The NESZ (noise-equivalent sigma zero) of the WSM images is between −21 dB at smaller and −26 dB at larger incidence angles. All our measured σ0-values are clearly above the noise floor. Considering the equivalent number of looks (ENL) for ASAR WS-mode (about 13 at the resolution of the original data) and the spatial averaging (resulting in ENL ≈ 39), 95 percent of the measured σ0-values cluster in the ±2 dB range around the mean value [Oliver and Quegan, 1998, p. 106; Ulaby et al., 1982, pp. 486–487].
 Scalloping is a modulation of the image radar intensity in range direction that occasionally occurs in burst-mode SAR data products such as the ASAR WSM [Hawkins and Vachon, 2003]. In the data we had available, sharp boundaries between adjacent stripes forming the WSM image could hardly be recognized. They were more pronounced in darker areas of those images acquired on the ascending orbit, where the intensity contrast between both sides of the boundary was found to be between 0.3 and 1.8 dB. Hence, variations of σ0 due to scalloping may influence the accumulation retrieval in some cases. Areas along our profiles, however, were not visibly affected.
 The geolocation error was found to be below the WSM sample spacing of 75 m [Schubert et al., 2008]. Uncertainties arising from errors in the retrieval of the local incidence angle are difficult to estimate since we have no detailed information on the accuracy of the used RAMP DEM [Liu et al., 2001] in terms of elevation gradients in our test region. In general it can be assumed that these uncertainties can be significant at steep slopes but negligible over horizontal planes. Large parts of the Kottas Traverse and most measurement sites on Amundsenisen Plateau are on rather flat terrain, which means that errors in local incidence retrieval can be neglected there.
 A potentially important contribution to the observed deviations between modeled and measured data is the azimuth modulation of σ0 that can be observed over certain areas on Antarctica and Greenland when measuring at varying radar look directions [e.g., Long and Drinkwater, 2000; Ashcraft and Long, 2005]. It was found that this anisotropy in scattering is mainly caused by sastrugi, which are wind-formed erosion features with a preferred orientation. It is furthermore assumed that the undulating snow surfaces of former years are preserved down to deeper snow layers [Ashcraft and Long, 2006]. For the region in which our test sites are located, Rotschky et al.  analyzed azimuth variations of σ0 using ERS-2 scatterometer data (which are VV-polarized). The amplitudes of these variations are between 0.5 and 3 dB. For more detailed investigations, we applied a tool provided by D. Long from Brigham Young University (BYU), USA, to compute the azimuth modulation dependent on the radar look direction. (see ftp://www.scp.byu.edu/outgoing/data/ts/azmod/ScatterometerAzimuthModulation.doc). The tool uses data from the Advanced Scatterometer (ASCAT) aboard the EUMETSAT METOP satellite, which is operated at VV-polarization using six side-looking antennae yielding high angular resolution. In the algorithm on which the tool is based the sensitivity of the azimuthal modulation to the incidence angle is neglected. Backscattering coefficients are normalized to 40 degrees. Also provided is the gradient dσ0/dθ, which is assumed to be linear. Observations, however, reveal that the gradient decreases at larger incidence angles [see, e.g., Oveisgharan and Zebker, 2007, Figure 4; Forster et al., 1999, Figure 7].
 To get an estimate of the modulation amplitude for our test sites, we used the BYU algorithm to estimate the expected minimum and maximum backscattering intensity for the local ASAR incidence angles along the Kottas traverse (indicated by the symbols in Figure 5). The results revealed differences between minimum and maximum backscattering coefficient from 3.0 to 6.5 dB. The values for the ascending path are higher compared to the descending path because of the lower ASAR incidence angles (see Figure 7). In our case the data from the ascending path are close to the possible minimum, which means that the radar look angle is oriented roughly in the katabatic wind direction. The look angle at the descending pass is rotated by about 80 degrees (largely across wind) and data are close to the possible maximum. Since the ASAR data are HH-polarized, it is not possible to compare ASCAT and ASAR data directly. However, on the basis of results reported in the field of ocean wind scatterometry [e.g., Horstmann et al., 2000; Mouche et al., 2005] we assume that the modulation depth (amplitude) is of similar magnitude for both VV and HH-polarization. Hence, the intensity difference between our ASAR ascending and descending images (Figure 5) can be explained -at least partly- by the effect of azimuthal modulation.
 Since the simulated results for the volume backscattering coefficient depend on the accumulation rates measured at our test sites, their accuracy and variations have to be analyzed as well. The effect of snow compaction is difficult to consider when stakes are used for the measurement of accumulation [Takahashi and Kameda, 2007]. It is realistic to assume that the accumulation rates given in Rotschky et al.  are slightly smaller than the “true” ones by approximately up to 10 percent.
 As the accumulation rates retrieved from satellite data are in general averages over a long time period, they may deviate from the ones measured in the field. This occurs if the rates of recent years, determined from the upper firn layers, differ from the ones obtained at larger depths, which date longer back. The accumulation data we used for comparison are from annual stake readings and were collected over a 5-year-period from 1997 to 2001. The annual readings revealed partly considerable variations from the 5-year-mean, ranging from 40 to 300 percent. In case of such large annual variations one would need measurements over a longer time period to reduce the statistical error of the mean value. The effect of varying accumulation rates on the radar signal can also be studied with our model, but this is beyond the scope of this paper.
 Finally, we note that the snow characteristics are changed when heavy vehicles move along the measurement traverses. In fact, the Kottas traverse can be identified in some of our ASAR WSM-images because of its relatively high radar intensity. The intensity contrast between the traverse and the adjacent area is larger when the backscattering coefficient of the latter is low. We found a maximum contrast of 0.5 dB which may have to be taken into account if σ0-values are taken from pixels located directly on the position of the traverse.
 A comparison between the backscattering coefficients σ0 obtained from Envisat ASAR WSM and the results of numerical scattering simulations using the method described in the preceding sections is shown for the Kottas traverse in Figure 6. Uncertainties due to calibration and speckle are less than 3 dB. The neglect of multiple scattering in the model simulations may account for uncertainties of the same order (see section 2.1). Calibration errors result in a constant offset, speckle because of the large ENL in a cluster of intensity values equally distributed around the line of perfect correspondence.
 If an azimuthal surface modulation exists, it causes an offset of the data cluster to the right in Figure 6, i.e., the measured values of σ0 are larger than the simulated ones. The reason is that the calculated values represent the volume scattering contribution and must hence be at or below the minimum level of the measured σ0-values. From Figure 6 it can be deduced that the backscattered intensity measured along the ascending path is lower than in case of the descending one. Since the simulated intensities are from volume scattering, the data clusters should be located below the line of perfect correspondence. We investigated this in more detail. The result is shown in Figure 7.
 In Figure 7, accumulation rates along the Kottas Traverse are shown together with simulated values of σ0 and data from ASAR. The accumulation rates were obtained from stake-readings between 1997 and 2001 [Rotschky et al., 2006], In rare cases of very low accumulation rates, simulations could not be carried out since the size of the scattering particles in a number of the deeper layers was too large, i.e., the Rayleigh-criterion was not valid. The radar intensities along the traverse were determined for the ascending and descending orbit and for austral summer (December 2008) and winter (May and June 2009). In general, summer and winter curves are in good correspondence, larger differences up to 1 dB over small distances are only observed occasionally. We attribute the difference between the measured σ0-values of the ascending and descending orbit to the effect of azimuthal modulation, which is smaller over the profile segment between 72.4°S and 73°S, and larger south of it. According to Rotschky et al.  at least a part of the latter profile segment is characterized by large sastrugi and redistribution of snow due to wind erosion. The southern part of the profile is viewed at smaller incidence angles than the northern part, which means that surface scattering contributions are more pronounced.
 In the northern part of the profile, approximately from 72.2°S to 73°S, the simulated data compare well with the σ0–values measured on the ascending orbit. The data from the descending orbit are higher. Further south, the model curve is at the same intensity level as the data from the descending pass, whereas the values belonging to the ascending pass are considerably lower. This is reflected in Figure 6, which is easier to interpret with Figure 7 at hand.
 Along the northernmost profile section shown in Figure 7, the deviations between model and measurements increase considerably. We interpret this profile section as transition to the percolation zone, which is characterized by large values of σ0 due to the existence of strong scatterers in the firn, such as coarse-grained material and ice lenses. Rotschky et al.  determined the boundary between percolation and dry snow zone from the differences of σ0-values between ESCAT C-band and NSCAT Ku-band data acquired in 1997. The corresponding latitude along the Kottas Traverse was 71.9°S, which is farther north than our value. However, on the basis of microwave radiometer data from January 2004 it was deduced that melting occurred also further south [Welker, 2007].
 The results visualized in Figures 6 and 7 reveal that the volume scattering model together with the parameterizations for snow metamorphosis predicts the correct level of backscattering intensity without any tuning parameters. However, we have to keep in mind that the volume scattering level determines the observable minimum σ0-value. Hence, the theoretical curve should rather follow (or be lower than) the measured data from the ascending orbit, which is not the case south of 73°S. Since we could not find any evidence of the scalloping effect along the traverse, we conclude that our parameterization of snow metamorphosis needs to be modified over the profile segment between 73°S and 74.2°S which is characterized by wind erosion (see above). Due to the impact of strong wind, snow crystals may be broken into smaller pieces. The result is the deposition of a fine-grained, compacted snow layer, which results in a lower backscattering coefficient. In Figure 10b, section 4 below, examples for the sensitivity to changes in the grain size profile, dependent on accumulation rate, are presented. The accumulation rate over the southern profile segment varies between 0.1 and 0.2 m w.e./a. According to Figure 10b a reduction of the average upper grain size by a factor >2 is required to lower the simulated curve to the level of the backscattering coefficients of the ascending orbit. This is not unrealistic since one of the authors (W. Rack) observed snow grain sizes much smaller than 0.5 mm in wind compacted firn.
 In Figure 8 (top), backscattering coefficients σ0 at C-band obtained over Amundsenisen Plateau and the ice ascent are shown in comparison to modeled values (black dots). This area is relatively flat, and the azimuthal modulation is relatively low [Rotschky et al., 2006]. The mean annual surface temperature on the Plateau is between −43°C and −49°C, whereas it is between −19°C and −27°C along the Kottas Traverse and −36°C to −20°C on the ice ascent. Since the loss factor (imaginary part of the dielectric constant) of freshwater ice is very low, in particular at lower temperatures, and its measurement is difficult [see, e.g., Ulaby et al., 1986, chapter E-3], simulations were also carried out with the imaginary part larger by 10 percent compared to the standard model. In this case, the effect was only marginal (not shown). However, it cannot be excluded that one has to assume an even higher uncertainty for the loss factor of freshwater ice.
 The spatial resolution of the QuikSCAT data is 25 km. Therefore several sites of accumulation rate measurements and corresponding simulated σ0–values are included in one resolution cell over the Kottas Traverse, which explains the linear clusters of the open-square data marks in Figure 8 (bottom). Simulations were only carried out for cases in which the Rayleigh criterion was valid.
 The sensitivity of the C-band radar backscattering coefficient to accumulation rate and temperature is depicted in Figure 9. The measured backscattering coefficients reveal a significant spread. This is caused by local variations in snow properties yet not taken into account in our parameterizations. Another reason is the influence of processes that modify snow surface and volume structure but are only weakly or not at all linked to the accumulation rate, such as depth hoar formation or snow compaction by wind. The observed variations of σ0 are also a consequence of the azimuth modulation as discussed above.
4. Application of Modeling Results
 Sensitivities of the backscattering coefficient as a function of accumulation rate, mean annual surface temperature, and incidence angle can be deduced from Figure 10a. The simulated values are determined both by the backscattering model and the empirical firn parameter equations. The gradient dσ0/dA (A is accumulation rate) is largest at low accumulation rates and decreases toward larger accumulation rates. The decrease in sensitivity of the backscattering coefficient relative to accumulation rate was also found in other studies [e.g., Forster et al., 1999; Rotschky et al., 2006]. Considering these results and the achievable accuracy of radiometric calibration we judge that accumulation retrievals using C-band radar are realistic up to values of about 0.2 m w.e./a.
 Regarding sensitivities at different frequencies, mean annual air temperatures, and incidence angles we obtained the following results:
 1. The sensitivity of σ0 to accumulation rate is smaller at Ku-band than at C-band. The reason is that scattering at Ku-band is close to the limit from Rayleigh to Mie scattering and hence less sensitive to changes of the particle size.
 2. Lower temperatures are better suited for the retrieval of accumulation rates (Figure 10a).
 3. The simulated sensitivity to the incidence angle is very low, which is a typical characteristic of volume scattering models. At smaller incidence angles, gradients dσ0/dθ obtained from measurements are larger than the results obtained by using our model but they decrease if θ increases [e.g., Forster et al., 1999, Figure 7]. The cited figure indicates that at incidence angles larger than 35 to 40 deg, volume scattering may be dominant. As already noted earlier, the gradient depends also on location as was shown in different studies [Rott et al., 1994; Rotschky et al., 2006].
 It has to be considered that the radar waves are shorter and the incidence angle is lower within the firn volume. For example, the wavelength at C-band decreases from 5.7 cm to 4.2 cm in snow with a density of 0.4 g/cm3 at a temperature of −25°C. For an incidence angle θ0 of 30 deg at the surface, the transmission angle at the first snow-snow interface of our model varies between 19.2 and 18.8 deg for accumulation rates between 0.01 and 0.4 m w.e./a (assuming a mean annual temperature of −25°C). For θ0 = 40 deg, the corresponding transmission angles range from 30 to 29.4 deg. This means that scattering contributions from the interfaces may have to be taken into account as suggested by Oveisgharan and Zebker . There is no information on interface roughness available and it can be questioned whether it is possible to measure it directly. We conclude that accumulation rates should be retrieved using backscattering coefficients at C-band at larger incidence angles (≥35 deg). However, in upcoming studies the character of volume and potential interface scattering has to be further investigated.
 The sensitivity of the backscattering to the near-surface grain radius (obtained as an average over the uppermost 20 firn layers) depends on the accumulation rate and is weaker at lower accumulation rates (Figure 10b). In the parameterization of grain growth presented by Linow , the differences between the grain radii calculated for two different accumulation rates decrease with depth. At lower accumulation rates, for which the thickness of annual layers is smaller, the radar penetrates deeper into the firn. Therefore, it is less affected by the larger differences of grain size in the upper firn layers.
 Penetration depths of radar waves were calculated using equation (4). They depend on the volume scattering and absorption coefficient, and to a minor degree also on the incidence angle. For example, at a mean annual temperature of −25°C and an accumulation rate of 0.1 (0.2) m w.e./a, the penetration depths at C-band and incidence angles of 25 and 45 deg are 34.5 (37.5) m and 32.5 (35.6) m, respectively. Since the absorption is strongly linked with the imaginary part of the dielectric constant ε, the calculations of penetration depth are affected by the accuracy of the model or empirical equation used for computing ε. The age of the layer at the penetration depth is a function of the density profile and the accumulation rate. Along the Kottas Traverse penetration depths vary between 15 m and 40 m (Figure 11). Smaller depths are observed at sites of low accumulation rates because of larger scattering losses related to the presence of larger snow grains. Since firn layers are very thin in such a case, the age of the firn at the penetration depth is relatively high (more than 300 years). At sites of larger accumulation rates, the corresponding age may be as low as 50 years. However, since the correspondence between backscattering model and measured data decreases at accumulation rates >0.2 m w.e./a, the calculated penetration depths and layer ages have to be regarded as rough estimates in this range. The largest contribution to the backscattered signal comes from depths between 9 to 11 m. It is noted that penetration depth and age are not directly linked. The former depends strongly on temperature and grain size. Between 73 and 74.5 deg south the penetration depth increases, which can be explained by a decrease in mean annual temperature (Figure 11). The accumulation rate does not reveal any corresponding trend.
 Results for test site 2 are depicted in Figure 12. At the ice ascent between 71°S and 72.8°S, the calculated penetration depths range from 23 to 50 m (corresponding to layer ages of 20 to 260 years) at C-band, and from 8 to 13 m (5 to 50 years) at Ku-band. On the plateau, where the average annual temperature is less than −40°C, the corresponding values are between 30 and 80 m (230 to 1100 years) at C-band and between 11 and 18 m (40 to 200 years) at Ku-band. The decrease in temperature from −20° to −45°C causes an increase of penetration depths from the ice ascent to the plateau. Grain sizes and scattering loss in the direction of propagation are larger and penetration depths smaller at latitudes <72.5°S, where accumulation rates are low. This means that the temperature effect causing an increase in penetration depth is dominating and only weakly counterbalanced by the low accumulation rates. In comparison to Figure 11 it has to be noted that grain sizes at low accumulation rates are considerably smaller at the lower temperatures on the Plateau than at the higher temperatures along the Kottas traverse. This explains the larger penetration depth on the Plateau.
 Penetration depths reported in the literature vary. Rott et al.  obtained values of 18 to 22 m at C-band based on field measurements of transmissivity and on satellite radiometer measurements at a site for which the 10 m firn temperature was at −25°C and the accumulation rate 0.35 m w.e./a. Using our model, we obtained about 32 m (northernmost part of the Kottas traverse at approximately 72.1°S; see Figures 7 and 11). Weber-Hoen and Zebker  reported values of about 25 to 40 m in the dry snow zone of Northern Greenland, for which mean annual temperatures are around −30°C and accumulation rates between 0.15 and 0.22 m w.e./a [Oveisgharan and Zebker, 2007]. This is in good agreement with our results from the Kottas Traverse (73°S–74.2°S).
 In general, average accumulation rates for Dronning Maud Land are regarded stable over several decades except over the coastal area north of approximately 72°S [Richardson-Näslund, 2004]. When retrieving average accumulation rates from radar data, the locally varying penetration depth has to be considered: larger accumulation rates are linked to shorter time intervals which means that changes of accumulation rate are easier to detect when comparing measurements acquired over several years.
 A new approach to model the volume backscattering from dry polar firn is presented, which is based on dense-medium theory. The backscattered intensity is calculated for snow of densities <0.23 g/cm3 (assuming ice grains in a host medium of air) and for ice of densities >0.69 g/cm3 (modeled as air bubbles in a background medium of ice) and interpolated between the snow and ice regime. We emphasize the fact that in contrast to the former studies by, e.g., Munk et al.  and Flach et al. , no tuning parameter is needed to obtain the correct absolute level of backscattering intensities as our analysis of data from Envisat ASAR (C-band) and QuikSCAT (Ku-band) acquired over Dronning Maud Land, Antarctica, revealed. The model compares well to measured data at accumulation rates up to 0.25 m w.e./a. At sites of higher rates and over the percolation zone the deviations are large. Our model, which is based on Rayleigh-scattering, needs to be supplemented by Mie-scattering to extend its range of validity.
 The measured σ0–values show comparatively large variations over short spatial scales, which decrease the accuracy of accumulation rate retrievals and are not yet reproduced by the scattering simulations. For robust retrievals, variations caused by azimuthal modulations of the backscattering coefficient have to be considered, which are related to wind-formed erosion surface features such as sastrugi. Also the parameterizations for simulating snow metamorphosis have to be extended to take into account specific environmental conditions such as wind compaction and depth hoar formation.
 The model can be used to compare the expected benefit of C- and Ku-band radars for accumulation rate retrievals. We found that the sensitivity to changes of accumulation rate is larger at C-band than at Ku-band. Modeling results indicate a considerable variability of the penetration depth caused by differences in local accumulation rates and mean annual temperatures. This means that time intervals over which the average accumulation rate is determined range from less than 10 years at Ku-band and high accumulation rate to more than 1000 years at C-band in areas of low accumulation rates and low mean annual temperatures. If this time interval exceeds decades, recent changes of snow accumulation can hardly or even not at all be recognized by comparing accumulation rates retrieved from radar data acquired with only a few years' difference. It has still to be investigated whether a combination of different radar frequencies or of radar and passive microwave data can solve this problem. In this context, also the additional use of radar altimetry with high vertical resolution may be beneficial. Hawley et al.  reported on airborne radar altimeter measurements from Greenland which revealed that internal annual layers can be imaged in the upper meters of dry firn. Whether space-borne altimeter will be capable to detect annual layers remains to be seen.
 To establish a robust scheme for the retrieval of accumulation rates from SAR data, the requirements are as follows:
 1. Zones of accumulation rates below and above 0.2 m w.e./a have to be separated. For the latter, a retrieval on the basis of active (and passive) microwave data cannot be applied because of the very low sensitivity of σ0 to higher accumulation rates. Considering the fact that accumulation rates are low over vast areas of Antarctica, this is not a serious restriction. In Greenland, accumulation rates below 0.2 m w.e./a occur only in the northeastern part of the dry snow zone and are larger elsewhere [Munk et al., 2003].
 2. The properties of the azimuthal modulation have to be specified further in terms of their sensitivity to the incidence angle and to polarization. The results have to be used to remove the surface contribution from the measured radar backscattering coefficient.
 3. More comparisons between in situ accumulation and radar data are required for different regions of Antarctica. Additional measurements of snow parameters (density, grain size) as a function of depth and dependent on environmental conditions (wind, temperatures) are needed to improve the model of snow metamorphosis, also taking spatial and temporal variations of weather conditions into account.
 4. The radar calibration accuracy has to be significantly better than 1 dB.
 5. Along with maps of snow accumulation, also the depth and the corresponding time period over which accumulation was determined have to be provided.
 The last item is important regarding the detection of recent (decadal) changes of accumulation over the Antarctic ice sheet. From our results we conclude that key areas for change detection should be characterized by a comparatively large accumulation rate and high mean annual temperature. In such an area the penetration depth of the radar signal is smaller and the period over which the accumulation rate is retrieved is shorter. However, it also has to be taken into account that the sensitivity of the measured radar intensity is higher at smaller accumulation rates and lower temperatures.
 For ease of reference, we provide here the empirical relationships for the simulation of snow metamorphosis. They were derived by Linow  based on field data from Greenland and Antarctica (see section 2.2 above).
 The vertical density profile as a function of mean annual surface temperature T [°C] and annual accumulation rate A [m w.e./a] is calculated from
Snow grain growth is obtained from
where r0 is the initial radius, K is growth rate, and t is time. The initial radius is given by
with T in [°C] and A in [m w.e./a].
 The growth rate is a function of mean annual surface temperature T [K] and seasonal temperature amplitude at the surface, ΔT [K]:
 ASAR images were provided by ESA (ALOS-ADEN, ID 3741, and Cryosat-2 Cal/Val, ID 4512). S.L. was funded by the German Science Foundation (SPP-1257: Mass Transport and Mass Distribution in the Earth System). Our study was also supported by the Royal Society of New Zealand and by the International Bureau, Federal Ministry of Education and Research, Germany (project NZL 10/024).