Annual layers in the GrIS snow/firn were mapped using the University of Kansas' Center for Remote Sensing of Ice Sheets (CReSIS) ultra-wideband snow radar during OIB Arctic Campaigns from 2009 through 2012 (Leuschen, 2014). The snow radar operates over the frequency range from $\sim$ 2 to 6.5 GHz (Panzer et al., 2013; Rodriguez-Morales et al., 2014). The snow radar uses an FMCW design to provide a vertical-range resolution of $\sim$ 4 cm in snow/firn, capable of resolving annual layering, where preserved, to tens of meters in depth (Medley et al., 2013). OIB flights operate multiple instruments, including lidars and radars, spanning a range of frequencies (Koenig et al., 2010; Rodriguez-Morales et al., 2014). The snow radar was chosen for this study because its vertical resolution and penetration depth are optimized for detecting annual layers from the surface of the ice sheet. It is noted, however, that the CReSIS Accumulation Radar and Multichannel Coherent Radar Depth Sounder (MCoRDS) are also capable of detecting accumulation on decadal to multi-millennial timescales, respectively, using dated isochrones (e.g. Miège et al., 2013; MacGregor et al., 2016).

Accumulation rate and snow/firn density profiles were derived from the MAR RCM (v3.5.2; X. Fettweis, personal communication, 2015). MAR is a coupled surface–atmosphere model that simulates fluxes of mass and energy in the atmosphere and between the atmosphere and the surface in three dimensions and is forced at the lateral boundaries with climate reanalysis outputs (Gallée, 1997; Gallée and Schayes, 1994; Lefebre et al., 2003). It incorporates the atmospheric model of Gallée and Schayes (1994) and the Soil Ice Snow Vegetation Atmosphere Transfer scheme (SISVAT) land surface model, which includes the multi-layer Crocus snow model of Brun et al. (1992). The MAR v3.5.2 simulation used here has a horizontal resolution of 25 km and utilizes outputs from the European Center for Medium Range Weather Forecasting (ECMWF) ERA-Interim global atmospheric reanalysis at the lateral boundaries (Dee et al., 2011). Additional details are described by Fettweis (2007), with updates described by Fettweis et al. (2011, 2013) and Alexander et al. (2014). MAR has been validated with in situ data and remote sensing data over the GrIS, including data from weather stations (e.g. Lefebre et al., 2003; Fettweis et al., 2011), in situ and remotely sensed albedo data (Alexander et al., 2014) and ice-core accumulation rates (Colgan et al., 2015), and it has been used to model both past and future SMB (Fettweis et al., 2005, 2013). We use accumulation rates and density profiles simulated by MAR for the period during which the radar data were collected (2009 to 2012).

In MAR, the initial falling snow density (${\mathit{\rho }}_{\mathrm{s},\mathrm{0}})$ is parameterized as a function of the temperature in the first model layer ($T_{\text{air}})$ in ${}^{\circ }$C (at roughly 3 m above the surface) and 10 m wind speed ($V)$ in m s${}^{-\mathrm{1}}$. The parameterization differs depending on atmospheric temperature as follows.

If $T_{\text{air}}$ is greater than $-$5 ${}^{\circ }$C, then $${\mathit{\rho }}_{\mathrm{s},\mathrm{0}}=\max (200,109+\mathrm{6}{T}_{\text{air}}+26\sqrt{V}).$$

If $T_{\text{air}}$ is less than $-$5 ${}^{\circ }$C and $V$ > 6 m s${}^{-\mathrm{1}}$, the parameterization of Kotlyakov (1961) is used: $${\mathit{\rho }}_{\mathrm{s},\mathrm{0}}=\max (200,\mathrm{\hspace{0.17em}104}\sqrt{V}).$$

If $V$ < 6 m s${}^{-\mathrm{1}}$ the initial snow density is set to the fixed value of 200 kg m${}^{-\mathrm{3}}$.

After snow falls to the surface, snow compaction in MAR is described according to the scheme of Brun et al. (1989), where the compaction of a layer (d$\mathit{\delta }z/\mathrm{d}t$) of thickness $\mathit{\delta }z$ is given by $${\displaystyle \frac{\mathrm{d}\mathit{\delta }z}{\mathrm{d}t}}={\displaystyle \frac{-\mathit{\sigma }\mathit{\delta }z}{C{\mathit{\rho }}_{\text{dry}}}}0.25e^{(-23\mathit{\rho }-0.1|T_{\mathrm{s}}\left|\right)},$$ where $\mathit{\rho }$ is the dry-snow density (g cm${}^{-\mathrm{3}})$, $T_{\mathrm{s}}$ is the snow temperature (degrees Celsius) of the layer, $\mathit{\sigma }$ is the vertical stress from the snow above (kg m${}^{-\mathrm{1}}$ s${}^{-\mathrm{1}})$ and $C$ is a function of snow grain size and snowpack liquid water content.

The SUrface Mass balance and snow depth on sea ice working group (SUMup) dataset (July 2015 release) compiles publicly available accumulation-rate, snow depth and density measurements over both sea ice and ice sheets (Koenig et al., 2013). We use two subsets of these data. First, to characterize density across the GrIS, we extract the snow/firn density measurements ranging in depth from the snow surface to 15 m (the depth to which MAR predicts firn density), which contains over 1500 measurements from snow pits and cores at 62 sites. At each site, the number of measurements ranges in number between 8 and 170 and maximum depths range from 1 to 15 m. (Koenig et al., 2014; Koenig and SUMup, 2015; Miège et al., 2013; Mosley-Thompson et al., 2001; Hawley et al., 2014; Baker, 2012) (Fig. 1). Second, to compare radar-derived and in situ accumulation rates, we consider only accumulation-rate measurements within 5 km of OIB snow radar data, a criterion that includes 11 cores from the SUMup dataset (Mosley-Thompson et al., 2001). To expand this comparison, an additional dataset of 71 cores was included (J. McConnell, personal communication, 2015), providing 23 additional cores within 5 km of OIB snow radar data (Fig. 1).

Because we seek to derive accumulation rates from near-surface radars across large portions of the ice sheet, we require firn density profiles that cover the entire GrIS. Modeled snow/firn density profiles from MAR were investigated for use. However, a preliminary comparison of the SUMup-measured density profiles to MAR-estimated density profiles showed that MAR simulated density values in the top 1 m of snow/firn were lower (0.280 $\pm$ 0.040 g cm${}^{-\mathrm{3}})$ than observed (0.338 $\pm$ 0.039 g cm${}^{-\mathrm{3}})$ (Fig. 2). The comparison of measured and modeled density was simultaneous in time. Specifically, the MAR density profile output on the day of the measurement was used in this comparison. We consider it beyond the scope of this study to investigate and explain why MAR underestimates near-surface density. Here we assume that the firn density in the top 1 m is 0.338 g cm${}^{-\mathrm{3}}$. Below 1 m, the model and observed densities are similar (4 % mean difference with the model generally overestimating measured density slightly), so the spatially varying modeled density profiles are used for 30 April of each year. Hence, a hybrid measured–modeled density profile is used to determine accumulation rates from the snow radar data (Fig. 2).

Our assigned uncertainty in the top meter is the relative standard deviation in observed density (12 %), which we assume is due to the natural variability in surface density. This uncertainty is higher than the assumed mean measurement uncertainty of 2–5 % (Proksch et al., 2016) and smaller than the mean difference between the modeled and observed values within the top meter (16 %). No spatial bias is evident between the mean model used here in the top 1 m of snow/firn and the observed density.

The radar travel time is converted to depth ($z)$ using the snow/firn density profile and the dielectric mixing model of Looyenga (1965). Errors in radar-derived depth come from two primary sources: (1) the dielectric mixing model chosen and (2) layer picking. The choice of the dielectric mixing model maximizes potential error at a density of $\sim$ 0.300 g cm${}^{-\mathrm{3}}$. The maximum possible difference in depth over 15 m is 3 % assuming a constant density of 0.320 g cm${}^{-\mathrm{3}}$ and < 1 % assuming a constant density of 0.600 g cm${}^{-\mathrm{3}}$ (Wiesmann and Matzler, 1999; Gubler and Hiller, 1984; Schneebeli et al., 1998; Looyenga, 1965; Tiuri et al., 1984). The second source of error occurs during manual adjustment of the picked layers (Sect. 4.3.4) and is estimated to be a maximum of $\pm$3 range bins, or $\sim$ 8 cm. Given the relative standard deviation in accumulation rate, the range bin error contributes a mean uncertainty of 7 % with a range of 4 to 24 %. Lower accumulation rates have a higher relative error from layer picking.

The water-equivalent accumulation rate $\dot{b}$ in m w.e. a${}^{-\mathrm{1}}$ at along-track location $x$ is $$\dot{b}\left(x\right)={\displaystyle \frac{z\left(x\right)\mathit{\rho }\left(x\right)}{a\left(x\right){\mathit{\rho }}_{\mathrm{w}}}},$$ where $z$ is the depth of layer in m, and $\mathit{\rho }$ is average snow/firn density to depth $z$ in kg m${}^{-\mathrm{3}}$. Hence, the numerator is the mass in kg m${}^{-\mathrm{2}}$ to depth $z$, $a$ is age of the layer in years from the date of radar data collection and ${\mathit{\rho }}_{\mathrm{w}}$ is the density of water in kg m${}^{-\mathrm{3}}$ (e.g. Medley et al., 2013; Das et al., 2015). Depth $z$ is calculated using the radar two-way travel time (TWT), the snow/firn density ($\mathit{\rho })$ and the Looyenga (1965) dielectric mixing relationship as follows: $$z={\displaystyle \frac{\mathrm{TWT}c}{\mathrm{2}{\left(\frac{\mathit{\rho }}{{\mathit{\rho }}_i},\left({{\mathit{\epsilon }}^{\prime }}_i^{\mathrm{1}/\mathrm{3}},-,\mathrm{1}\right),+,\mathrm{1}\right)}^{\mathrm{3}/\mathrm{2}}}},$$ where TWT is the travel time to the dated layer in sec, $c$ is the speed of light in m s${}^{-\mathrm{1}}$, ${\mathit{\rho }}_i$ is ice density in kg m${}^{-\mathrm{3}}$ and ${{\mathit{\epsilon }}^{\prime }}_i$ is the dielectric permittivity of pure ice. Combining these two equations gives $$\dot{b}\left(x\right)={\displaystyle \frac{\text{TWT}\left(x\right)\mathit{\rho }\left(x\right)c}{\mathrm{2}a\left(x\right){\mathit{\rho }}_w{\left(\frac{\mathit{\rho }\left(x\right)}{{\mathit{\rho }}_i},\left({{\mathit{\epsilon }}^{\prime }}_i^{\mathrm{1}/\mathrm{3}},-,\mathrm{1}\right),+,\mathrm{1}\right)}^{\mathrm{3}/\mathrm{2}}}}.$$ The cumulative mean snow/firn density ($\mathit{\rho })$ is determined by the density profile described in Sect. 4.1. The layers are picked in the radar data using a semiautomated approach described in Sect. 4.3.

Layer ages are determined by assuming spatially continuous layers are annually resolved and dated accordingly from the year the radar data were collected. The radar data were collected during springtime (April–May) and the surface is assumed to be 30 April to align with the modeled accumulation rate, which was processed to monthly values. Subsurface picked layers are assumed to be 1 July $\pm$ 1 month, so the first layer represents 10 months and each subsequent layer is 12 months. Peaks in radar reflectivity are, assuming ice with no impurities, caused by the largest change in snow density. In the ablation and percolation zone, the peak in density difference occurs in the summer between the snow layer and ice or the snow/firn layer and the high-density melt/crust layer, respectively (e.g. Nghiem et al., 2005). In the dry-snow zone, the peak density contrast also occurs in the summer between the summer hoar layer and the denser snow/firn layer (e.g. Alley et al., 1990). While melt/crust and hoar layers can form at other times, it is assumed they will be smaller density contrasts and, therefore, cause a smaller radar reflection than the dominant layers which occur near 1 July. We assume picking errors are randomly distributed in space due to the heterogeneous nature of snow/firn. Density peaks have been shown to vary in depth along ice-core transects, likely due to small-scale microstructure differences (e.g. Machguth et al., 2016).

To calculate the total uncertainty on the radar-derived accumulation rate, the largest error is assumed for density (12 %) and age (10 %) and the error for the mean accumulation rate is assumed for layer picking (7 %). Equation (3) shows that the density profile is used for calculating both depth and water equivalent. The derivative of Eq. (3) determines the correlated error between depth and density. Assuming uncorrelated and normally distributed errors between density, age and layer picking the mean accumulation-rate uncertainty is 14 %, with a range of 13 % for the highest accumulation rates and 27 % for the lowest accumulation rates. This relative uncertainty is similar to previous studies by Medley et al. (2013) and Das et al. (2015) for radar-derived accumulation rates.

A semiautomated layer detection algorithm is developed to process the large amounts of OIB snow radar data (> 10${}^{\mathrm{4}}$ km a${}^{-\mathrm{1}})$, analogous to the challenges faced by MacGregor et al. (2015) for analysis of very-high-frequency radar sounder data. While a fully automated method is ultimately desirable, we have found that it is necessary to manually check every automated pick, making adjustments as needed by an experienced analyst, to distinguish between spatially discontinuous radar reflections, caused by the natural heterogeneity of firn microstructure, and spatially consistent annual layers. Our algorithm processes the OIB snow radar data in four steps outlined below.

Annual radar-derived accumulation rates and their uncertainties were calculated for all 2009–2012 OIB radar data that contained detected layers (Fig. 4). The increase in coverage from 2009 to 2012 is related to an increasing number of OIB flights over the GrIS and adjustments to the snow radar antenna and operations that improved overall data quality. These accumulation-rate patterns are consistent with observed and modeled large-scale spatial patterns for the GrIS: high accumulation rates in the southeast coastal sector and lower accumulation rates in the northeast (Fig. 5). Year-to-year variability in the accumulation rate is also evident, even at the ice sheet scale; e.g., in the southeast accumulation rates were lower in 2010 than in 2011.

The radar-derived accumulation rate in Fig. 4 represents only the first layer detected by the snow radar, or approximately the annual accumulation rate from the year prior to data collection. For simplicity, we refer to this quantity as the annual accumulation rate, but we caution that it does not strictly represent the calendar year. The values shown in Fig. 4 represent only 10 months of accumulation, based on our assumption that the radar layers date to 1 July (Sect. 4.2) and that the data collection date is 30 April for all OIB data, which may differ from the actual flight date by up to a month. When comparing the first layer of radar-derived accumulation to modeled estimates from MAR (Fig. 5) or other accumulation measurements, these timing differences must be considered. Although the first layer represents only a partial year, all deeper layers represent a full year, from 1 July to 30 June. We simultaneously compare the time represented by the layer to MAR estimates of accumulation.

Figure 6 shows the number of detected layers, or previous years, discernable in the OIB radar data. For the majority of the GrIS, one to three annual layers are discernable. OIB flight lines are clustered in the ablation/percolation zones of the GrIS, where radar penetration depths are reduced by the increased density, englacial water and layering structure of the firn column (Fig. 3). In the GrIS interior, where dry-snow conditions allow deeper radar penetration, annual layering going back over 2 decades is detectable (Fig. 3). Figure 7 shows a histogram of depths for the first layer detected for years 2009 through 2012; 63 % are within the top 1 m of snow.

Crossover points were assessed to determine the internal consistency of the radar-derived accumulation rates (Figs. 8 and 9). While no consistent spatial pattern is found in the crossover errors, the largest discrepancies were found in 2011 and 2012 in the northwest and southeast (Fig. 8). Other inconsistencies are likely due to snow deposition occurring between flights in the southeast and incorrectly picked layers that were either sub- or multiannual in the northwest. Figure 9 shows a scatter plot of crossover points. There are relatively few outliers, and those that are outlying are generally offset by a factor of 2, suggesting an error in layer detection/dating rather than a radar-system error. Crossover differences per year, including the mean, standard deviation and maximum, are given in Table 1. These differences are comparable (mean of 0.04 m w.e. a${}^{-\mathrm{1}}$ or 4 range bins) to our inferred relative uncertainty of 14 %, emphasizing the overall validity of our methodology.

The along-track radar-derived accumulation rates were gridded to the MAR grid for comparison. The mean-local, radar-derived accumulation rate was used when gridding. Because OIB flight lines are not spatially homogenous, each MAR grid cell represents a different number of radar-derived values, so grid cells are not sampled equally. With this discrepancy noted, this gridding method is still the most straightforward approach for this comparison. Figure 10 shows the difference between the radar-derived and MAR accumulation rates. The mean difference for all years is low (0.02 m w.e. a${}^{-\mathrm{1}})$. Table 1 shows the annual variability of the mean difference, which is low for every year except 2010, when large differences are seen over the southeastern coastal region of the GrIS (Fig. 10).

Figure 10 shows that MAR generally reconstructs accumulation rates well in the GrIS interior (consistent with the comparison with ice-core estimates presented by Colgan et al., 2015) but has larger differences around the periphery, especially in the southeast and northwest in particular years. In the southeast, MAR generally has higher accumulation rates, except in 2011 when there is a mixed pattern of agreement and higher accumulation rates. Higher values in the southeast are not surprising and are likely due to the large changes in surface topography that are not resolved by the relatively coarse model grid (Burgess et al., 2010). In 2011, the northwestern coastal region of the GrIS was well sampled by OIB and MAR has lower accumulation rates there in contrast to 2010, when the area was sampled and had higher values. The origin of this anomaly in the northwest is less clear, but it may be related to forcing at the lateral boundaries of MAR that may not capture a relatively small storm track into this region.

Figure 11 shows a scatter plot of the radar-derived and MAR-estimated accumulation rates. These values are not well correlated (Pearson correlation coefficient $r^{\mathrm{2}}=0.2$) and have large root-mean-square errors (RMSEs) (0.24 m. w.e. a${}^{-\mathrm{1}})$, emphasizing that further improvements in accumulation-rate modeling and measurements are needed, particularly over the southeastern and northwestern GrIS.

Between 2009 and 2012, OIB flew within 5 km of 34 core locations but only two locations, NEEM and Camp Century (Fig. 1), were coincident in time with the layers we detected. Each of these locations has two cores, providing annual accumulation rates and a measure of spatial variability. Figure 12 compares the radar-derived to core measured accumulation rates. At NEEM, the two ice cores and radar data are nearly co-located, within 0.6 km of each other. The radar-derived accumulation rates are self-consistent between 2011 and 2012 and agree well with the ice cores (RMSE of 0.06 m w.e. a${}^{-\mathrm{1}})$. For comparison, the two NEEM cores have an RMSE of 0.05 m w.e. a${}^{-\mathrm{1}}$ for the period of overlap. A timing discrepancy arises with this comparison because the ice cores, with higher dating resolution from isotopic and chemical analysis, are dated and reported for calendar years, whereas the radar-derived accumulation is assumed 1 July–30 June (Sect. 4.2). This mismatch in the measurement is likely evident in Fig. 12 by the differences in the annual peaks between the cores and radar-derived accumulation having similar means yet differing magnitudes from year to year.

Near Camp Century, the cores and radar data are farther apart from each other. The radar data are located within 4.4 km of the Camp Century core and the GITS core is located $\sim$ 8.2 km from the Camp Century core. These separations are likely responsible for the poorer agreement at this site of radar-derived accumulation rate to the Camp Century core (RMSE 0.10 m w.e. a${}^{-\mathrm{1}})$ and the larger difference (RMSE 0.07 m w.e. a${}^{-\mathrm{1}})$ in accumulation rate between the two cores for the period of overlap. At Camp Century, and throughout much of northern Greenland, two older, continuous layers were dated from the interior of the ice sheet and spatially traced. These layers, dated July 2000 and July 2001, could not be dated with the Camp Century data alone – hence the temporal gaps in annual accumulation rate at this location. While it is more difficult to analyze the results at Camp Century, with only three overlapping points and no continuous annual time series of radar-derived accumulation rates, our estimates are within the expected variability and capture the long-term mean value.