Inverse estimation of snow accumulation along a radar transect on Nordenskiöldbreen, Svalbard
Abstract
We present an inverse modeling approach to reconstruct annual accumulation patterns from ground-penetrating radar (GPR) data. A coupled surface energy balance-snow model simulates surface melt and the evolution of subsurface density, temperature, and water content. The inverse problem consists of iteratively calibrating accumulation, serving as input for the model, by finding a match between modeled and observed radar travel times. The inverse method is applied to a 16 km GPR transect on Nordenskiöldbreen, Svalbard, yielding annual accumulation patterns for 2007–2012. Accumulation patterns with a mean of 0.75 meter water equivalent (mwe) a^{−1}contain substantial spatial variability, with a mean annual standard deviation of 0.17 mwe a^{−1}, and show only partial consistency from year to year. In contrast to traditional methods, accounting for melt water percolation, refreezing, and runoff facilitates accurate accumulation reconstruction in areas with substantial melt. Additionally, accounting for horizontal density variability along the transect is shown to reduce spatial variability in reconstructed accumulation, whereas incorporating irreducible water storage lowers accumulation estimates. Correlating accumulation to terrain characteristics in the dominant wind direction indicates a strong preference of snow deposition on leeward slopes, whereas weaker correlations are found with terrain curvature. Sensitivity experiments reveal a nonlinear response of the mass balance to accumulation changes. The related negative impact of small-scale accumulation variability on the mean net mass balance is quantified, yielding a negligible impact in the accumulation zone and a negative impact of −0.09 mwe a^{−1}in the ablation area.
Key Points
- Inverse modeling beneficial for accumulation reconstruction from GPR
- Variability in accumulation in space linked to terrain parameters
- Small-scale accumulation variability lowers the mean mass balance
1 Introduction
Understanding the processes that determine variability of snow accumulation in space and time is of great importance for both mass balance modeling as well as interpretation of local mass balance measurements and ice core-derived accumulation rates. Spatial variability in accumulation not only induces local variations of the mass balance, it also affects the area-averaged mass balance due to the nonlinear response of the mass balance to changes in accumulation. In the context of this paper, the mass balance refers to the sum of accumulation (snow/rainfall), runoff, and sublimation/riming and hence accounts for subsurface storage and refreezing of melt water in the snowpack. Glacier mass balance models, requiring accumulation input, usually rely on interpolated local observations or low-resolution climate model output, thereby ignoring the role of small-scale accumulation variability.
Spatial variability in snow accumulation in mountainous regions is controlled by both local variability in precipitation as well as redistribution of the snow by wind. On a larger scale, precipitation variability has been linked to changes in altitude [e.g., Sevruk, 1997], whereas on a smaller scale variations in local vertical wind speeds associated with flow divergence/convergence become significant [e.g., Paulcke, 1938]. The uplift of already deposited snow by wind depends on the local wind speed and direction relative to the ground, which is known to vary significantly in space due to undulations in the terrain. Slopes directed toward the wind suffer from erosion, whereas deposition occurs in the lee of undulations [Liston and Sturm, 1998; Winstral et al., 2002]. Modeling efforts to simulate the redistribution of snow by wind with snow models indicate the necessity of detailed modeling of local wind fields and snow drift processes [Lehning et al., 2008; Mott et al., 2010], which is complex and computationally expensive. Other modeling attempts rely on parameterizations built on correlations between snow deposition and terrain-based parameters [Liston et al., 2007; Gascoin et al., 2013].
Observational data, providing constraints on spatial accumulation variability, are valuable for validation of snow model output, as well as to improve our understanding of processes involved in shaping the snow cover. While ice cores and stake measurements provide point estimates of accumulation, ground-penetrating radar (GPR) has been extensively used to reconstruct spatial accumulation variability along transects in Antarctica [e.g., Frezzotti et al., 2007; Verfaillie et al., 2012], Greenland [e.g., Dunse et al., 2008; Miège et al., 2013], the Alps [Machguth et al., 2006], and Svalbard [e.g., Pälli et al., 2002; Taurisano et al., 2007]. In the accumulation zone of a glacier rapid densification occurs near the surface in summer, related to melt-freeze cycles, vapor transport, and radiation transfer. This produces annual summer surface horizons with increased density in the firn pack [Colbeck, 1986]. These layers are associated with a high reflectivity of the GPR signal [Kohler et al., 1997]. In subsequent years, accretion on top of high-density layers as a result of refreezing of percolating melt water may further enhance the density contrast of internal reflection horizons (IRHs) [Dunse et al., 2008]. The two-way travel time (TWT) of the radar signal traveling between the surface and an IRH depends not only on the depth of the IRH but also on the relative dielectric permittivity of the penetrated medium. In a cold firn pack, the permittivity is determined by the density of the firn, and empirical relations have been developed relating TWTs to firn density and depth [e.g., Robin et al., 1969; Kovacs et al., 1995]. In temperate firn, the presence of water, stored in pore spaces in the firn, significantly increases the electric permittivity [e.g., Looyenga, 1965]. Hence, information on vertical densities and water content is required when converting TWTs in temperate firn into annual layer masses [Pettersson et al., 2004].
Inverse methods have regularly been applied in the field of glaciology and can effectively be used to indirectly obtain information about variables that are not well constrained by measurements, including basal slipperiness, basal topography, and/or ice viscosity [e.g., Gudmundsson and Raymond, 2008; Arthern and Gudmundsson, 2010; Gillet-Chaulet et al., 2012]. The concept of inverse estimation of accumulation from radar layering has previously been illustrated in an ice dynamical framework by Waddington et al. [2007] and Steen-Larsen et al. [2010].
In this study, we present a novel inverse approach to extract spatial and temporal accumulation variability from GPR data. A coupled surface energy balance-snowpack model computes surface melt and subsequent percolation, storage, refreezing, and runoff of water in order to simulate the subsurface evolution of density, temperature, and water content. Accumulation, serving as input for the coupled model, is iteratively adjusted to find a match between modeled and observed TWTs of selected IRHs. The method is applied to GPR data obtained along a transect on Nordenskiöldbreen, Svalbard, to reconstruct annual accumulation patterns for the period 2007–2012. Reconstructed accumulation along the transect forms the basis for a further analysis of the pattern in relation to terrain characteristics and wind, as well as a discussion and quantification of the negative impact of short-scale variations in accumulation on the overall mass balance. In comparison to traditional methods, accounting for variability in density and water content along the transect is shown to have a substantial impact on reconstructed accumulation. Additionally, accounting for postdepositional processes (melt percolation, refreezing, and runoff), which leads to discrepancies between annual accumulation (i.e., snow mass falling at the surface) and annual layer mass (i.e., mass stored between two IRHs in the firn), is shown to significantly improve accumulation estimates in areas with substantial surface melt.
Our main objective in this work is to reconstruct and analyze spatiotemporal accumulation variability on an Arctic glacier from GPR data. A brief description of the study area is given in section 2. In sections 3 and 4, we present the GPR data, the coupled model, and discuss the inverse approach. Output of sensitivity experiments is discussed in section 5. We present, validate, and discuss reconstructed accumulation in section 6.
2 Study Area
Nordenskiöldbreen is a tidewater glacier in central Svalbard, connected to a large ice plateau, Lomonosovfonna (Figure 1). The glacier covers an altitudinal range from sea level to ∼1200 m above sea level (asl) and flows into the Adolfbukta fjord, where the glacier front is actively calving along part of its width. Continuous GPS data reveal annual mean ice velocities up to ∼60 m a^{−1}in steeper parts and decreasing velocities toward the ice plateau [den Ouden et al., 2010]. Annual mean temperatures, rising at a rate of 0.25 K per decade at Svalbard Airport since 1912 [Førland et al., 2011] have amplified surface ablation and explain the observed ongoing retreat of the glacier tongue since the end of the Little Ice Age around 1900 A.D. [Rachlewicz et al., 2007].
In May 1997, a 120 m deep ice core was drilled on top of the Lomonosovfonna ice plateau, providing a record of accumulation back to 1598 A.D. [Pohjola et al., 2002; Van Pelt et al., 2013]. A previous study by Pälli et al. [2002], using low-frequency GPR data from the upper accumulation zone (above 1000 m asl, Figure 1) on Nordenskiöldbreen, discusses long-term trends in accumulation between 1963 and 1997. In contrast to Pälli et al. [2002], the GPR transect in this study covers not only the upper part of the accumulation area (above ∼1000 m asl) but also lower parts of the accumulation zone and ablation area, subject to significant refreezing and runoff.
3 Data Treatment
3.1 GPR
On 17 April 2012 a GPR profile was gathered along a 16 km transect of Nordenskiöldbreen using a Malå ProEx impulse GPR system with 500 MHz shielded antennae (Figure 2). The signal sampling frequency was set to 19.5 GHz and the traces were collected every 0.3 s, and each trace was stacked eightfold. The antennae were dragged ∼5 m behind a snowmobile at a speed of ∼10 km h^{−1} giving a trace spacing of roughly 0.8 m. The vertical resolution is theoretically less than the Nyquist frequency λ/4, where λis the wavelength of the radar signal. In our case, the theoretical range resolution is ∼0.1 m. However, the radar wavelength depends on the velocity of the signal and thus will vary with depth due to density variations. The GPR transect is located along the main flow line, spanning from 1198 to 550 m asl (Figure 1) and is gently sloping with a mean and maximum slope of 2.4° and 5.7°, respectively. The profile was positioned using kinematic carrier-phase differential GPS mounted on the snowmobile. A fixed base station was located at Terrierfjellet, a mountain just south of the transect, creating a maximum baseline of 14 km. The position of the GPR antennae was determined by interpolation along the traveled trajectory. The horizontal and vertical accuracy of the antenna position is estimated to be less than 0.1 m (including instrumental errors of ∼20 mm and trajectory interpolation errors).
Processing of the GPR profile includes band-pass filtering to remove high-frequency noise, applying a spherical and exponential compensation gain function to compensate for attenuation and geometrical spreading of the radar signal and application of a normal move out to compensate for antenna offset [Jol, 2009]. Continuous layers of high reflectivity (IRHs) were manually picked. The accuracy of picking of the IRHs TWT was estimated at 1.1×10^{−9} s based on repeated picking of the IRHs. Starting from the highest point of the transect, five IRHs could be traced with confidence continuously for the first 7300 m of the GPR transect. For the remaining part of the profile (in the lower accumulation and ablation area), excessive signal diffusion and/or absence of a firn pack enabled selection of only the uppermost IRH.
The continuity of the IRHs across the profile in the GPR radargram (Figure 2) indicates they are likely isochrones [Spikes et al., 2004; Eisen et al., 2006; Miège et al., 2013]. Based on the distinct and clear character of the IRHs, we assume that they are annual rather than multiannual or intraannual. In comparable climatic conditions in southeast Greenland, Miège et al. [2013] found annual IRHs in the higher part of a transect and fainting of some annual horizons at lower elevation. Diffusion and eventually fainting of IRHs following excessive melt [Dunse et al., 2008; Brown et al., 2011] is apparent beyond ∼5000 m but is unlikely to cause absence of IRHs higher up. Validation of the annual character of the upper two IRHs is discussed in section 6.3.
3.2 Subsurface Data
Snow pits were dug at multiple sites near the GPR transect (Figure 1) in April 2011 (S4 and S6) and April 2012 (S1, S2, S3, and S5), providing vertical temperature and density profiles. A 12 m shallow firn core was drilled on 13 April 2012 at the start of the transect (S6). On 20 April 2012, temperatures in the borehole were recorded at 10 depths, with a spacing of 1.0 to 2.0 m. A density distribution in the firn core was obtained by weighing the 86 core segments with known volume.
4 Model and Inverse Method
4.1 Coupled Model
We use a coupled surface energy balance-snowpack model to simulate surface melt and subsequent percolation, storage, refreezing, and runoff of melt water. Here we give only a brief description of relevant model features. For a more elaborate overview of both the surface energy balance model and the multilayer snow model, as well as the meteorological input, the reader is referred to Van Pelt et al. [2012, and references therein]. Meteorological input comes from interpolated output of the regional atmospheric climate model RACMO [Ettema et al., 2010] (air temperature, pressure, and specific humidity) and data from a weather station at Svalbard Airport (precipitation and cloud cover), all downscaled to the 3 h model resolution [Van Pelt et al., 2012].
As water percolates in the snow/firn, a small amount of water (up to a few percent of the total layer mass) is held by capillary forces and is referred to as irreducible water [Schneider and Jansson, 2004]. Additionally, when percolating water reaches impermeable ice, it accumulates on top of the ice, filling the remaining pore space, hence forming a slush layer. Slush water is set to runoff gradually, whereas in case bare ice is exposed at the surface, runoff occurs instantaneously [Reijmer and Hock, 2008]. The subsurface model tracks the depth of annual summer surfaces, which are defined when the surface height reaches a minimum during the melt season. Summer surface depths are hence modeled independently of the vertical density profile. A limitation of the firn model in this work and other studies is the lack of accounting for variability in vertical propagation speed of the percolating water. In the model, vertical water transport is instantaneous and fluxes are controlled by the rate of refreezing and capillary space in different layers. Implementing more realistic vertical water transport velocities requires modeling microscale processes like piping and impedance of water flow on top of ice lenses, enabling lateral transport of water [e.g., Pfeffer and Humphrey, 1998; Bell et al., 2008; Wever et al., 2013], which is beyond the scope of this work. Alternative models to describe vertical transport of percolating water have been discussed by Mitterer et al. [2011a].
The subsurface model contains a total of 140 vertical layers, with a spacing in the upper 10 m of 10 cm and a spacing of 1.0 m in the lower 40 m of the vertical grid. In the event of mass addition or removal at the surface, due to melt, accumulation, or sublimation/riming, vertical model layers shift accordingly, effectively describing advection of mass and energy (section 3.2). In addition to calibration of model parameters discussed in Van Pelt et al. [2012], the fresh snow density ρ_{f} is chosen equivalent to the mean observed snow density in the upper 20 cm of all snow pits. Additionally, the fresh snow albedo α_{f}, affecting the amount of surface melt and subsurface refreezing, was selected such that vertical mean densities and temperatures match best with the observed mean of the shallow firn core, yielding α_{f}=0.885.
4.2 Simulated Radar Travel Times
From modeled densities and water content of individual model layers, bulk annual layer densities and water content between modeled summer surfaces are computed. Bulk densities and water content are used in equations 4–6 to obtain integrated simulated radar travel times of IRHs. By using bulk values we ignore the effect of microscale density and water content variability on integrated radar travel times of annual layers, which introduces a minor error in modeled travel times in wet firn. In the context of this work with values of θ_{w}<5%, the associated uncertainty appears to be of negligible magnitude.
4.3 Iterative Inverse Approach
The coupled model is used as a forward model in an inverse approach, aiming to find a match between modeled and observed TWTs of the selected annual IRHs. This is done by iteratively adjusting surface accumulation, serving as input for the coupled model. The GPR transect is projected onto the distributed model grid by averaging of all GPR data points falling within a 40×40 m model grid cell. This results in a total of 495 points along the transect.
A backward reconstruction is applied, which implies reconstructing accumulation associated with the most recently formed layer first and successive reconstruction of older layers hence effectively going back in time. By doing so, future accumulation and melting/refreezing is already known when reconstructing accumulation associated with an older layer. Reconstructed annual accumulation does not only depend on what happens in subsequent years but is also slightly affected by the history of the firn pack (e.g., through heat exchange with deeper layers). Proper spin-up of the firn pack is hence desired to initialize firn properties. Therefore, the coupled model is run since 1989 till the start of the reconstruction experiment, forced with accumulation time series from Svalbard Airport at sea level (mean 0.19 mwe a^{−1}), increasing linearly with altitude to 0.61 mwe a^{−1}above 850 m asl (equivalent to the mean annual accumulation found by Pälli et al. [2002] for 1963–1999). The relative insensitivity of reconstructed accumulation to spin-up accumulation (A_{init}) is quantified and discussed in section 5.
A time series of accumulation (in meters water equivalent) is taken directly from the Svalbard Airport record, resulting in a time vector of 12-hourly values. The model internally distinguishes between snowfall and rainfall, as discussed in Van Pelt et al. [2012]. The iterative inverse method aims to find scaling factors of the accumulation time series for the periods between the formation of consecutive IRHs. The scaling procedure is performed independently for all grid points along the transect.
After reconstructing accumulation since the formation of the first IRH (in 2011), accumulation reconstruction of the second layer (2010–2011) involves iteratively running the model between 1 July 2010 and 17 April 2012, with previously calibrated accumulation for 2011–2012 and iteratively updated scaled accumulation for 2010–2011. In the same spirit, annual accumulation associated with the third, fourth, and fifth annual layers has been reconstructed. Ultimately, we have scaled accumulation estimates for all points along the transect and every year for the period 2007–2012, as well as dates of summer surface formation per year. From this, we compute total accumulation between the formation of two IRHs. When discussing annual accumulation in the following sections, we refer to surface accumulation between the formation of two summer surfaces in consecutive years.
5 Sensitivity Experiments
All uncertainties in the inverse approach related to model initialization, uncertain modeling physics, inaccurate computation of TWTs from modeled subsurface properties, and uncertainty in layer picking contribute to the error in the reconstructed accumulation patterns. We perturb parameter settings and compare mean accumulation for a 1 year (2009–2010) and a 5 year period (2007–2012). Results are shown in Table 1.
Standard | ΔA | ΔA | |||
---|---|---|---|---|---|
Parameter | Unit | Value | +/− | (2007–2012) | (2009–2010) |
A_{init} | mwe a^{−1} | 0.45–0.61 | +0.10 | +0.001 | +0.002 |
−0.10 | −0.002 | −0.004 | |||
ε_{f} | - | equation 6 | +10% | −0.028 | −0.030 |
−10% | +0.033 | +0.035 | |||
m_{liq} | - | - | +10% | −0.002 | −0.005 |
−10% | +0.003 | +0.004 | |||
ρ_{f} | kg m^{−3} | 344 | +25 | +0.024 | +0.021 |
−25 | −0.027 | −0.028 | |||
t_{ref} | 1 July | −90 days | −0.000 | −0.004 | |
t_{2009} | ns | 20–69 | +1.1 | +0.000 | +0.055 |
−1.1 | −0.000 | −0.050 | |||
Δz | m | 0.1–1.0 | × 2 | −0.004 | −0.004 |
/ 2 | +0.002 | +0.002 |
- ^{a} ΔA is given in mwe a^{−1}. Perturbations of the initialization accumulation (Ainit), firn permittivity (ε_{f}f), maximum pore space for irreducible water storage (mliq), fresh snow density (ρ_{f}f), vertical grid spacing (Δz), fresh snow albedo (αf), TWT of the 2009 layer (t2009), and starting date (tref).
Perturbing the spin-up accumulation pattern (A_{init}) since 1989 by 0.10 mwe a^{−1}is shown to have only a minor impact on reconstructed accumulation, indicating a small effect of initial firn conditions. This motivates the use of the backward reconstruction (starting with the most recent layer) rather than going forward in time (starting with the oldest layer) introducing large errors related to unknown future accumulation and mass exchange between layers.
Conversion of simulated subsurface densities and summer surface depths into TWTs relies on an empirical relation developed by Kovacs et al. [1995], based on Antarctic firn measurements. Uncertainty in the associated firn permittivity ε_{f}(equation 6) directly affects computed TWTs and indirectly affects accumulation. We find that perturbing ε_{f} by 10% induces a substantial error of 0.030 mwe a^{−1}(3.9%) in accumulation (2007–2012). Although there is no indication that the mixing model, described by equations 5 and 6 is not applicable to the firn pack in this study, different mixing models exist, yielding slightly different relations between permittivity, firn density, and water content [Mitterer et al., 2011b].
Another empirical relation has been used to compute the maximum pore space in the firn pack (m_{liq}) available for storage of irreducible water [Schneider and Jansson, 2004]. A 10% perturbation of m_{liq}, affecting the water content distribution in the firn pack is shown to have a minor impact on reconstructed accumulation (2007–2012) of on average 0.002 mwe a^{−1}(0.3%).
The fresh snow density ρ_{f} directly affects the mean density of the firn pack and hence modeled summer surface depths. Perturbing ρ_{f} by 25 kg m^{−3}induces a significant change in reconstructed accumulation (2007–2012) of on average 0.025 mwe a^{−1}(3.4%). Modeled densities are validated against observations in section 6.3.
The chosen starting date (t_{ref}; set to 1 July) of the iterations influences the temporal accumulation distribution (rather than the total accumulation) between the formation of two IRHs. A sensitivity experiment with the starting date perturbed by 90 days illustrates the minor influence of the choice of starting date on reconstructed accumulation.
Uncertainty in layer picking was estimated at 1.1×10^{−9}(section 3.1). Perturbing the observed TWT of the IRH formed in 2009 (t_{2009}) results in a substantial error in accumulation of 0.053 mwe for 2009–2010. It should be noted that erroneous selection of an IRH is compensated for by an error of opposite sign for the accumulation in the previous year. Hence, the effect of inaccurate layer picking on the 5 year mean accumulation is likely to be small. Potential errors in dating of the IRHs are discussed in section 6.3.
Finally, sensitivity experiments with perturbed vertical grid resolution (Δz), multiplied and divided by a factor 2, are performed. Results show the relative insensitivity of reconstructed accumulation on Δz. Furthermore, convergence under grid refinement is apparent. These results indicate there is little improvement to be gained by a further increase of the vertical grid resolution.
6 Results and Discussion
6.1 Annual Accumulation Patterns 2007–2012
Application of the inverse approach results in annual accumulation patterns for 2007–2012. Figure 4 shows reconstructed accumulation patterns per year (Figures 4a–4e) and a composite pattern (Figure 4f). For x>7300 m, where x is the distance along transect, only one IRH could reasonably be identified, thereby restricting accumulation reconstruction to the uppermost layer. Comparing annual patterns reveals distinct year-to-year variability in the mean, horizontal gradient, and standard deviation. Reconstructed annual mean values range from 0.63 to 0.88 mwe a^{−1} with a 2007–2012 average of 0.75 mwe a^{−1}. We find substantial year-to-year discrepancies in the degree of spatial variability with standard deviations ranging from 0.12 (2008–2009) to 0.23 mwe a^{−1} (2009–2010), corresponding to 15 to 30% of annual mean accumulation. Despite altitude decreasing with x, spanning an altitudinal range of ∼1200–550 m asl, we find weak trends in both the 2011–2012 pattern and the 2007–2012 mean. Horizontal gradients of the annual patterns vary substantially in both magnitude and sign with values ranging from −0.08 (2010–2011) to +0.02 mwe a^{−1}km^{−1}(2007–2008).
Continuous GPS velocities obtained at multiple sites along the transect (Figure 1, S1–S5) [den Ouden et al., 2010] for the period 2010–2012 are included in Figure 4f. When interpreting consistency of annual accumulation patterns, one should keep in mind that mass is transported down the transect, which leads to a horizontal shift with depth of variability along the transect [Arcone et al., 2005]. Given observed velocities up to 58 m a^{−1}, we expect 2007-2008 accumulation may have shifted by up to 200-300 m in 2012. The dashed blue line in Figure 4 coincides with peaks in the composite accumulation pattern and illustrates that a phase shift is apparent in the lower part of the accumulation area. As a result of this phase shift, systematic variability in the composite pattern is partly suppressed in regions with high-flow velocities. Additionally, dynamic thinning/thickening of accumulation layers may occur in areas where ice flow diverges/converges, yielding some uncertainty in modeled summer surface depths of older layers. This effect is estimated to be small for the short period under consideration.
Part of the transect covered by Pälli et al. [2002] nearly overlaps with the transect in this study (Figure 1). Pälli et al. [2002] presented accumulation patterns for the periods 1986–1999 and 1963–1999 of which the lowest 4 km are shown in Figure 4f. Reasonable agreement is found between spatial patterns. The mean in our study (0.75 mwe a^{−1}) is higher than the averages of 0.71 and 0.54 mwe a^{−1}for 1986–1999 and 1963–1999 found by Pälli et al. [2002]. This could indicate significantly higher accumulation rates since 2007 relative to the long-term mean. However, in addition to uncertainty in our approach, these discrepancies could also arise from substantial systematic errors (up to 35%) involved in dating the 1986 and 1963 layers in Pälli et al. [2002].
6.2 Vertical Profiles of Density, Temperature, and Water Content
Figure 5 shows simulated subsurface density, temperature, and water content distributions along the transect at the end of the reconstruction (17 April 2012). Depths of the modeled summer surfaces are indicated.
Subsurface temperatures at on April 2012 in Figure 5a show a temperate firn pack for 2000<x<8000 m at depths greater than around 5 m. This can be ascribed to substantial refreezing of percolating melt water, while cold wave penetration during the 2011–2012 winter season induced nontemperate near-surface conditions. In the lower accumulation area/ablation area (x>8000 m) and the highest part of the accumulation area (x<2000 m), heat release by melt water refreezing is less prominent and cold subsurface conditions prevail.
Subsurface densities in Figure 5b reveal strong horizontal variability along the transect, which is clearly correlated with variability in depth of the summer surfaces and hence accumulation. Although enhanced gravitational densification of snow for higher accumulation is accounted for, this does not prevent densities at a certain depth below the surface to decrease with accumulation, as near-surface low-density snow is buried at a faster rate [Ligtenberg et al., 2011]. Nevertheless, when comparing simulated bulk density and snow depth of the fresh snowpack over the course of a winter season, we find bulk density increasing with accumulation as a thicker and older snowpack has had more time to settle. The latter is in line with findings by Jonas et al. [2009].
The simulated subsurface water content in Figure 5c shows the presence of small amounts of irreducible water (volume fraction up to 3%) in the firn pack below depths of ∼5 m for 3000<x<8000 m. The water has been stored in the firn pack during previous melt seasons and has gradually been refrozen from above during the 2011–2012 winter season. For x>8000 m, firn depth is too limited to accommodate deep storage of irreducible water, while for x<3000 m, the limited amount of melt water does not percolate deep enough to survive winter refreezing. The local storage capacity of irreducible water depends on the firn porosity and is hence determined by the firn density [Schneider and Jansson, 2004].
6.3 Validation
As a means of validation, we compare simulated subsurface profiles of density and temperature to firn observations along the transect. Observational data comprise density and temperature profiles from multiple snow pits along the transect in April 2011 and 2012, as well as a 12 m shallow firn core, drilled at the start of the transect on 13 April 2012 (Figure 1). We compare model output and observations at identical points in time.
6.3.1 Shallow Firn Core Profiles
Shallow firn core densities and temperatures are compared to modeled values in Figure 6 and show similar patterns. After calibration of the fresh snow albedo α_{f} remaining discrepancies of the vertically averaged density and temperature are reduced to 5 kg m^{−3} and 0.2 K, respectively. Associated RMSE values for the density and temperature profiles are 86 kg m^{−3} and 0.53 K, respectively. Whereas modeled temperature variability agrees well with observations, the lacking ability of the model to simulate variability in surface density and vertical transport rates of percolating water leads to a lack of modeled microscale density variability. For example, high-density layers arising from local vertical water flow impedance, lateral water flow, and preferential refreezing are hence absent in the model output (see also section 4.1). We believe reconstructed annual accumulation is only slightly affected by the lack of modeled microscale vertical density variability as mainly the bulk annual layer density affects the TWT of the radar signal rather than microscale density variability within an annual accumulation layer. Some uncertainty in reconstructed accumulation arises from uncertainty in the modeled depth and concentrations of stored liquid water in the firn pack at the time of radar observations.
6.3.2 Snow Pit Profiles
Next we compare simulated vertical temperatures, densities, snow mass, and snow depths with values observed in snow pits, dug in April 2011 and 2012. Modeled and observed values are compared in Table 2. We find small discrepancies between modeled and observed bulk densities (on average 3.4%). Much larger discrepancies are found between modeled (area-averaged) and observed (pointwise) snow depths of on average 13.6%, thereby contributing heavily to discrepancies in snow mass of on average 15.1%. It should be stressed here that modeled area-averaged values of a 40×40 m model grid cell are compared to pointwise snow pit measurements. Additionally, snow pits were not dug exactly along the radar transect (average discrepancies in the order of tens of meters). This is relevant, because over distances of tens of meters, snow properties may vary substantially. In Greenland Dunse et al. [2008] found a standard deviation of spatial accumulation variations on a 100×100 m grid of 15%, whereas on two plots of identical size Maurer [2006] computed a standard deviation of snow depth of on average 25%. Based on the above, we hypothesize that a substantial part of the found snow mass discrepancies (Table 2) can be attributed to the pointwise character of the snow pit observations and odd positioning of the pits relative to the transect.
Snow Depth | Density | Snow Mass | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Location | Date | Mod. | Obs. | Δ(%) | Mod. | Obs. | Δ(%) | Mod. | Obs. | Δ(%) |
S4 | 1 April 2011 | 2.26 | 2.50 | 10.1 | 370 | 361 | 2.3 | 0.84 | 0.90 | 7.8 |
S6 | 4 April 2011 | 1.78 | 1.60 | 10.7 | 367 | 363 | 1.2 | 0.65 | 0.58 | 11.8 |
S1 | 13 April 2012 | 1.56 | 1.50 | 3.9 | 410 | 397 | 3.1 | 0.64 | 0.60 | 7.0 |
S2 | 15 April 2012 | 2.27 | 1.85 | 20.4 | 391 | 404 | 3.3 | 0.89 | 0.71 | 17.1 |
S3 | 14 April 2012 | 1.88 | 2.25 | 17.9 | 378 | 400 | 5.7 | 0.71 | 0.90 | 23.6 |
S5 | 14 April 2012 | 1.53 | 1.85 | 18.4 | 376 | 393 | 4.5 | 0.58 | 0.73 | 23.3 |
All pits | 1.88 | 1.93 | 13.6 | 382 | 386 | 3.4 | 0.72 | 0.74 | 15.1 |
A comparison of the snow pit density and temperature profiles and model output is shown in Figure 7. Modeled bulk densities agree well with observations (difference 4 kg m^{−3}, Table 2). Nevertheless, the model is unable to replicate the microscale density variations (RMSE averaged over all pits of 45 kg m^{−3}), resulting from the complex interaction of the surface layer with the atmosphere through e.g. wind compaction, moisture exchange and precipitation type [Colbeck, 1986]. As mentioned, a lack of modeled microscale density variability is of minor importance for the accumulation reconstruction. Comparing snow pit temperatures with modeled profiles (Figure 7b) reveals that the model generally captures vertical variability in subsurface temperatures reasonably well. On average, vertical mean temperatures per pit differ by 0.9 K, with a RMSE averaged over all pits of 1.4 K.
In order to validate the annual character of the upper two selected IRHs, we compare mean snow mass for all pits in 2011 and in 2012 to modeled mean values for the 2 years. In case selected continuous IRHs would bound multiannual layers, mean modeled values would overestimate snow mass by a factor 2 or more. Alternatively, selection of multiple IRHs per annual layer would result in a severe underestimation of mean modeled snow mass along the transect. For the snow pits in 2011, we find an average modeled and observed snow mass of 0.75 and 0.74 mwe, respectively, whereas for 2012, a mean snow mass of 0.71 (modeled) and 0.74 mwe (observed) is found. From this we conclude that the upper two IRHs likely represent summer surfaces and bound annual layers. Means to confirm the annual character of the deeper layers are lacking.
6.4 Advantages of the Inverse Method
The use of a coupled model in an inverse method to infer snow accumulation has several benefits over traditional methods. In contrast to traditional methods, we (1) account for mass exchange between layers due to melt water percolation and refreezing, (2) explicitly model densities along the transect, and (3) account for water content in the firn. Next, we quantify the impact of these factors on reconstructed accumulation.
With the inverse method, we can distinguish between accumulation and mass stored in annual layers, i.e., between two IRHs. These may differ as accumulated snow may melt, percolate in the snow/firn, refreeze in deeper layers, or runoff. Traditional methods assume layer mass equals accumulation, thereby disregarding vertical mass transport in the firn. In areas with substantial melt percolation, this may introduce large errors in reconstructed accumulation. Figure 8a compares annual mean accumulation and layer mass for the period 2007–2012 along the upper part of the transect. Layer mass is lower than accumulation for the entire transect, due to refreezing below the 2007 IRH and/or runoff of percolating melt water. The water flux through the oldest IRH is therefore a measure for the discrepancy between surface accumulation and mean annual layer mass. Discrepancies increase toward lower altitudes as melt percolation and runoff intensifies. The above illustrates the use of the inverse method in areas with substantial melt. Along this part of the transect, using a traditional approach leads to errors ranging from 0.010 to 0.161 mwe a^{−1} (mean 0.060 mwe a^{−1}).
In order to quantify the impact of accounting for density variability along the transect, we calculated the 2007–2012 mean layer mass that would be obtained from observed TWTs by assuming a vertical density profile equivalent to the mean for x<7300 m, thereby disregarding horizontal density variability. This is equivalent to traditional methods using a composite density record to directly convert TWTs into layer mass (using a conversion relation as in equation 6). The effect of ignoring horizontal density variability on reconstructed accumulation is shown in Figure 8b. Accounting for density variability along the transect is shown to substantially reduce spatial variability in reconstructed accumulation. Reduced variability is related to the dependence of densities at depth on accumulation (Figure 5b), which causes modeled TWTs to change more rapidly with changes in accumulation. This effectively reduces spatial variability in reconstructed accumulation by on average 19%.
Although the firn water content may only comprise a small fraction of the total firn volume (Figure 5c), the impact of firn water on the effective permittivity can be substantial due to the large relative permittivity of water. Irreducible water storage reduces the velocity of the radar signal, thereby increasing TWTs. In order to quantify the impact of neglecting water on reconstructed layer mass, we first calculate mean layer mass from observed TWTs and the modeled density distribution when assuming a zero water content (as in traditional approaches). These results are then compared with modeled layer mass in which the effect of irreducible water on TWTs is taken into account. Figure 8c illustrates that neglecting irreducible water storage leads to an overestimation of reconstructed layer mass. Accounting for water increases modeled TWTs, which is compensated by a lower accumulation rate. In case of high accumulation rates, annual layers extend deeper in the firn pack and reconstructed layer masses experience a larger influence of firn water. We find discrepancies up to 0.045 mwe a^{−1} and an average difference of 0.010 mwe a^{−1}. Note that the GPR data were gathered in spring, when the firn water content exhibits a seasonal minimum. More substantial errors can hence be expected when collecting data in other seasons.
6.5 Spatial Variability Versus Terrain Parameters
Previous work has demonstrated that the interaction of wind and terrain features controls the distribution of snow deposition. Snow redistribution by wind is known to erode snow on the windward side of ridges, whereas increased deposition occurs in the lee [e.g., Liston and Sturm, 1998; Winstral et al., 2002]. Additionally, reduced wind velocities on the leeward side of undulations may result in preferential precipitation [Lehning et al., 2008]. Collected differential GPS measurements of surface elevation along the GPR transect allow us to link accumulation variability to surface topography.
Before quantifying correlations between terrain features and accumulation, it is important to know the prevailing wind direction relative to the transect direction. For that purpose, observational and simulated wind data are employed. Wind velocity and direction at ∼4 m above the surface are recorded every 10 min at the AWS on Nordenskiöldbreen (Figure 1). The wind rose and histogram of all observations between March 2009 and August 2012 are shown in Figures 9e and 9f and show a clear prevalence of winds blowing from the northeast. Recent modeling of wind fields at a 2.7 km resolution using the atmospheric model WRF [Claremar et al., 2012] resulted in wind roses at points near the AWS and the ice summit (Figure 1, W1 and W2) for one summer (June–August 2009) and winter season (December 2009 to February 2010). In agreement with the AWS data, modeled wind roses at W1 in Figures 9c and 9d reveal prevailing winds from the northeast. A similarly consistent simulated pattern is found on the ice plateau at W2 (Figures 9a and 9b). This implies that prevailing winds are aligned with the downslope direction of the transect for x>4400 m, whereas for x<4400 m winds blow in the downslope direction of the transect with a horizontal inclination of ∼60°.
Table 3 presents correlations of normalized accumulation and terrain curvature, slope, and SI for the full transect (2011–2012), x<4400 m and 4400<x<7300 m (2007–2012 and 2011–2012). For the full transect (2011–2012), we find highly significant correlations between accumulation and slope and SI. The strong anticorrelation between accumulation and slope (r=−0.58) demonstrates preferred snow deposition on steep slopes directed away from the dominant wind direction. We find a much weaker correlation between accumulation and curvature (r=0.18), which confirms preferential deposition on the upwind side of concave surface depressions rather than in the concavities itself. For 4400<x<7300 m, normalized accumulation for 2011–2012 anticorrelates strongly with slope (r=−0.64), whereas the 2007–2012 pattern correlates strongly with curvature (r=0.53). The latter may well be unphysical and can be ascribed to the aforementioned effect of substantial ice velocities up to 58 m a^{−1}, inducing an unrealistic horizontal shift in reconstructed 5 year mean accumulation (see Figure 10a). As by definition a 90° phase shift is apparent between curvature and slope, the phase error in 2007–2012 accumulation is likely to enhance the correlation with curvature and reduce the correlation with slope. High correlations are found between accumulation and the sheltering index SI, indicating its use as a predictor of accumulation variability. Nevertheless, correlations of 2011–2012 accumulation with slope and SI are of comparable magnitude. For x<4400 m, we find low correlations between accumulation and slope, curvature and SI, which can be ascribed to the inclination of the wind relative to the transect and the lack of surface height variations on the ice plateau.
A11-12 | A07-12 | A11-12 | A07-12 | A11-12 | |
---|---|---|---|---|---|
(Full Transect) | (4400<x<7300) | (4400<x<7300) | (x<4400) | (x<4400) | |
Curvature | +0.18 | +0.53 | +0.13 | −0.13 | +0.01 |
Slope | −0.58 | −0.32 | −0.64 | −0.05 | −0.17 |
SI (100 m) | +0.54 | +0.45 | +0.64 | +0.08 | +0.18 |
SI (250 m) | +0.54 | +0.58 | +0.65 | +0.06 | +0.07 |
SI (500 m) | +0.54 | +0.57 | +0.61 | +0.00 | +0.00 |
SI (1000 m | +0.53 | +0.52 | +0.58 | +0.00 | −0.24 |
- ^{a} Correlations significant at a 99% confidence level are marked in bold.
6.6 Impact of Spatial Variability on Net Mass Balance
The response of the mass balance to a change in accumulation is nonlinear, due to feedbacks of accumulation and snow depth with the surface albedo and subsurface properties. Effectively, these interactions amplify the sensitivity of the mass balance to changes in accumulation. As a result of the nonlinear dependence of the mass balance on accumulation, the degree of small-scale accumulation variability influences the area-averaged mass balance.
In order to quantify the effect of small-scale accumulation variability on the net mass balance, we first extend the model transect toward the calving front (Figure 1, dashed line). Temporal accumulation variability, once again, comes from the Svalbard Airport record. We define a smooth-scaled reference accumulation pattern (A_{ref}), which is linearly increasing with altitude and maximizes above 850 m asl at 0.75 mwe a^{−1} (reconstructed mean in Figure 4). We perform multiple model experiments over the period 2007–2012 with perturbed accumulation (−0.6<A−A_{ref}<0.6mwea^{−1}) and divide the simulated annual mean mass balance into 50 m height bins. Resulting height profiles are shown in Figure 11a. In the accumulation zone, the mass balance responds approximately linearly to changes in accumulation, whereas in the ablation area the mass balance is most sensitive to negative accumulation perturbations. For very negative accumulation perturbations in the ablation area, annual accumulation may become zero, and the mass balance is insensitive to further reductions.
Figure 11b shows the mass balance versus altitude for a range of accumulation perturbations. The contour map consists of vertical lines, adopted from output in Figure 11a and is completed by means of spring-metaphor interpolation between these lines. In subsequent steps, we (1) subtract the reference mass balance in Figure 11b, (2) set the standard deviation σof accumulation variability, (3) apply a normal probability density function around A_{ref}, and (4) integrate over dA for every altitude. This results in a height profile of the net mass balance effect of spatial accumulation variability. We set σto the mean of the annual standard deviations of reconstructed accumulation for 2007–2012 of 0.17 mwe a^{−1}. Figure 12 shows the net mass balance effect of small-scale spatial variability with a standard deviation of 1, 2, and 3σ and clearly illustrates the net negative mass balance effect of accumulation variability in the ablation area. For σ=0.17 mwe a^{−1}, a mean mass balance effect of −0.043 mwe a^{−1}is found. In the ablation area, the relatively strong nonlinear response of the mass balance to accumulation changes induces a mean net mass balance reduction of 0.090 mwe a^{−1}. Perturbations of accumulation have a major impact on the mass balance if changes from ablation to accumulation area or vice versa occur. This affects nonlinearity in the net mass balance and explains extrema found around 400 and 500 m asl in Figure 12. The decline of the net mass balance effect below ∼100 m asl can be ascribed to the aforementioned absence of accumulation for very negative perturbations. In the accumulation zone, accumulation variability hardly affects the net mass balance, since the surface albedo is rather insensitive to accumulation changes.
7 Conclusions
A novel approach is presented and applied to extract spatial and temporal accumulation variability from GPR data. The method uses a coupled surface energy balance-snow model in an inverse approach and aims to find a match between simulated and observed TWTs of buried summer surfaces by iteratively adjusting accumulation, serving as input for the coupled model. The coupled model computes surface melt and subsequent percolation, storage, refreezing, and runoff of water in order to simulate the subsurface evolution of density, temperature, and water content. The inverse approach is applied to a 16 km GPR transect on Nordenskiöldbreen, Svalbard, to reconstruct annual accumulation patterns between 2007 and 2012. The use of a model in the inverse determination of accumulation rates is shown to provide substantial improvements over traditional methods. With this work, we hope to set an example how inverse modeling can help to increase the accuracy of accumulation reconstruction from GPR data.
Reconstructed annual accumulation patterns for 2007–2012 with a mean of 0.75 mwe a^{−1} show substantial spatial variability (on average 0.17 mwe a^{−1}), and patterns are only partly consistent from year to year. Compared to traditional methods, the inverse modeling approach (1) accounts for postdepositional processes (percolation, refreezing, and runoff) and avoids substantial errors in estimated accumulation in areas with substantial melt; (2) accounts for density variability along the transect, reducing spatial variability in accumulation by on average 19%; and (3) accounts for the effect of irreducible water on reconstructed accumulation. The latter effect is of minor influence here, as GPR data were collected during a seasonal minimum in firn water content. Relating spatial accumulation variability to terrain parameters shows generally strong correlations with slope and weaker correlations with curvature, thereby indicating preferential deposition of snow on steep slopes at the upwind side of concavities rather than in concavities itself. Finally, the nonlinear response of the mass balance to perturbations of accumulation induces a net negative mass balance effect as a result of small-scale variability in accumulation of on average −0.09 mwe a^{−1} in the ablation area. In the accumulation area, small-scale accumulation variability has a negligible impact on the mass balance.
Acknowledgments
This project is funded through the Royal Netherlands Academy of Arts and Sciences (KNAW) professorship of J. Oerlemans. R. Pettersson, V. A. Pohjola, B. Claremar, and S. Marchenko acknowledge funds from the Swedish Science Council, ESF-SvalGlac, the Nordic Centre of Excellence SVALI, and the Norwegian Polar Institute for support with the field observations. This publication is contribution 26 of the Nordic Centre of Excellence SVALI (Stability and Variations of Arctic Land Ice), funded by the Nordic Top-level Research Initiative (TRI). We finally thank the Editor B. Hubbard, Associate Editor M. Schneebeli, and the anonymous reviewers for their constructive comments, which have helped to substantially improve the quality of the manuscript.