Corresponding author: A. F. Banwell, Scott Polar Research Institute, University of Cambridge, Cambridge CB2 1ER, UK. (firstname.lastname@example.org)
 During more intense melt years, supraglacial lakes on the Greenland Ice Sheet are found to fill more rapidly and to drain earlier in the melt season. They also develop at higher latitudes and elevations. Since the rapid drainage of lakes has been shown to enhance basal sliding and since the lake volume can play an important role in determining if and when it drains, understanding and modeling the processes which control lake filling rates forms a key development in successfully modeling the possible impacts of lake drainage events on ice motion. We have developed a surface water routing and lake filling model and applied it to a 100 km2area of the Paakitsoq region, west Greenland. The model takes the time series of calculated runoff (melt minus refreezing) over the area and calculates flow paths and water velocities over the snow-/ice-covered surface, routing the water into topographic depressions which can fill to form lakes. Runoff is calculated from a distributed, surface energy-balance model coupled to a subsurface model, which calculates changes in temperature, density and water content in the snow, firn and upper ice layers, and hence refreezing and therefore net runoff. The model is calibrated against field measurements of a filling lake in our study area during June 2011 and can be used to calculate the filling rate of the instrumented lake with a high degree of accuracy. The filling rate of the instrumented/modeled lake depends on melt and routing from the immediate lake catchment and from overflowing lakes in upstream catchments.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 Because of the important role of supraglacial lakes in the hydrology of the GrIS surface as well as the bed, the routing of meltwater to surface depressions, and thereby the filling, overflowing, and ultimately drainage of surface lakes by hydrofracture should be included in any future high-resolution model of ice dynamics applied to the ice sheet margin. To this end, we report here on the development and calibration of a water routing and lake filling model which uses the calculated surface runoff from an existing distributed, high spatial resolution model of surface mass balance (SMB) [Rye et al., 2010; Banwell et al., 2012] to calculate supraglacial lake filling rates over a 100 km2 area of the Paakitsoq region of the west central GrIS. Although the dynamics of the ice sheet as a whole are likely to be most influenced by the cumulative effect of multiple lake drainage events across much greater spatial and temporal ranges than explored in this study, this work is an important intermediate step that is necessary before more simplified models of the same process can be developed over larger spatial scales and longer temporal scales. Further work will be undertaken in due course to validate the model against other independent data sets and to investigate the filling rates of lakes over a larger area and in future climate warming scenarios.
2. Study Site and Data
 Our study site, the “Ponting” area, is a 10 km × 10 km area located within the ∼2300 km2 Paakitsoq region in west central Greenland, northeast of Jakobshavn Isbrae (Figure 1). This area was chosen as a field site because satellite imagery showed the consistent filling and drainage of lakes in this region on an annual timescale at the time of our planned field season. The Ponting area is ∼15 km inland from the ice sheet margin with ice elevations between ∼750 m and ∼980 m above sea level (masl). In this study we focus on “Lake Ponting,” located centrally within our model domain (69.589°N, −49.783°E, 962 masl) (Figure 2), ∼10 km north of the JAR 1 Greenland Climate Network (GC-Net) automatic weather station [Steffen and Box, 2001]. We focus on the time period 9 to 30 June 2011, during which we have field measurements of snow depth and lake level, which we use for model calibration.
 The SMB model used to calculate melt/runoff (section 3.1) is driven with a full range of meteorological variables from the JAR 1 GC-Net station [Steffen and Box, 2001]. The SMB model is calibrated using average daily snow surface height measurements against four poles drilled into the ice (located with 100 m of “Camp,” Figure 2) from 9 to 29 June 2011, during which time no precipitation fell.
 The surface routing and lake filling model (SRLF model) (section 3.2) is calibrated against measured lake level data. Two HOBO pressure sensors were installed in Lake Ponting's lake basin on 13 June 2011. The lake drained via hydrofracture on 19 June 2011, giving 6 days of lake filling data. One sensor was firmly secured to an aluminum pole drilled into the ice at a height of ∼0.5 m above the ice surface. The second sensor was loosely attached to the pole so that it rested on the ice but could slide down the pole as the bottom of the lake melted, while remaining close to the pole. This setup allowed for the ablation rate at the bottom of the lake to be calculated from the difference between the time series of lake depth recorded by the two sensors [Tedesco et al., 2012]. Both sensors recorded pressure every five minutes on an internal data logger and were recovered following lake drainage. The sensors have a water level accuracy of 0.5 cm and a resolution of 0.21 cm. Pressure data were corrected for altitude and for barometric pressure changes using data from a third sensor located <1 km from the lake. Measured lake depth data were converted to volume using a depth-volume relationship derived from a digital elevation model (DEM) of the area (see below). A handheld GPS allowed us to measure the horizontal position of the sensors to an accuracy of ±5 m. Comparing this position with the 100 m resolution DEM suggested that the sensors were installed in a DEM cell that was 0.7 m higher than the lowest elevation DEM cell of the lake basin; we therefore add 0.7 m to all the pressure sensor data so they are relative to the deepest part of the lake, and we define the depth-volume relationship for this lowest DEM cell.
 In addition to converting lake depth to volume, a surface DEM is also required by the SMB model to spatially distribute meteorological data and compute topographic shading, slope angles and aspects (section 3.1). The DEM is subsequently used by the SRLF model to route the modeled meltwater across the snow/ice surface to topographic lows to form lakes (section 3.2). We use the ASTER Global Digital Elevation Model (GDEM) which has a nominal grid size of 30 m (http://asterweb.jpl.nasa.gov/gdem.asp). We checked the original GDEM for obvious artifacts in the area and none were found. The GDEM quality files for the Paakitsoq region show ASTER stacking numbers here lie between 8 and 12, yielding an accuracy of ±18.2 m (SD) <500 m elevation, and ±13.8 m (SD) >500 m [MacFerrin, 2011]. The original data were smoothed with a 6 × 6 cell median filter to remove small-scale noise then resampled to a 100 m resolution using bilinear interpolation.
3.1. Surface Mass Balance Model
 We model hourly melt and runoff using the high-resolution SMB model described byRye et al. , and subsequently developed by Banwell et al. . The SMB model consists of three coupled components: (1) an energy balance component that calculates the energy exchange between the glacier surface and the atmosphere; (2) an accumulation routine; and (3) a subsurface component, which simulates changes in temperature, density and water content in the snow, firn and upper ice layers, and hence refreezing and net runoff. Here, we describe the SMB model only briefly, concentrating on the adaptations that have been made to the model presented more fully by Rye et al.  and Banwell et al. .
 The SMB model uses a range of meteorological variables from the JAR1 GC-Net station notably: incoming shortwave radiation (diffuse and direct), air temperature, relative humidity, and wind speed readings at a nominal height of 2 m above the ice surface. As incoming longwave radiation data are not available for the Paakitsoq region, these data were calculated using parameterizations based on the work ofKonzelmann et al.  following Banwell et al. .
 Following Banwell et al. , the mass balance year runs from 1 October to 30 September. We are particularly interested in model output during the times of our measurements, notably June 2011. The model also requires a year of meteorological data used repetitively for 5 years for spin-up purposes (see below). Due to instrument failure at JAR 1 from 1 January 2011 to 17 May 2011, and the commencement of our modeling work in late July 2011, we only have meteorological data for 2011 from 18 May to 25 July 2011. We therefore synthesize a mass balance year from the following JAR 1 measurements: 1 October to 31 December 2010; 1 January to 17 May 2010; 18 May to 25 July 2011; 26 July to 30 September 2010. The model is run for 5 years for spin-up purposes, then during the sixth year, we generate output for the three week period commencing on 9 June 2011 for input to the SRLF model. Values for fresh snow albedo (0.80) and ice albedo (0.45) are set based on average measured albedo values at JAR 1 for the year of climate data (2010/2011) used by the SMB model. These values are very similar to those set for fresh snow albedo (0.82) and ice albedo (0.48) in the study byBanwell et al.  which calibrated the same SEB model for a larger 450 km2 subsection of the Paakitsoq region for two mass balance years (2000/2001 and 2004/2005).
 The model requires a spin-up period of 5 years in order for the surface mass balance and the subsurface temperature and density profiles to attain equilibrium [Banwell et al., 2012]. Another key aim of the spin-up period is to produce a snowpack thickness on 9 June 2011 (of the main (sixth year) model run) equal to that measured at “Camp” (Figure 2) on the same date. In order to achieve this we prescribe the model with precipitation at a constant rate per hour, in meters water equivalent (mwe), on one random day per week from 1 October to 30 April inclusive (called “winter precipitation” hereafter). We prescribe winter precipitation in this way, rather than, for example, having it fall at lower rates continuously throughout the 7 months since it is more realistic and will allow more realistic subsurface temperature and density profiles to evolve over the winter. During model calibration (see below) we establish the total amount of winter precipitation that is required for the modeled snowpack thickness to match the measured snow thickness on 9 June 2011. Following Banwell et al. , the threshold temperature for precipitation falling as either snow or rain is set at 2°C.
 The subsurface model is described fully by Rye et al. , but briefly, it calculates temperature, density, and water content on a one-dimensional vertical grid extending at least 25 m from the surface into the ice sheet. Meltwater generated by the surface energy balance component percolates through the grid, with refreezing occurring where the temperature is below 0°C and the density is less than that of ice. The cell below receives any residual meltwater if either of these conditions is not met, or if there is excess meltwater after refreezing. Meltwater percolates until it reaches the impermeable snow/ice interface where superimposed ice may be formed. If the rate at which meltwater reaches this interface exceeds the rate of superimposed ice formation, then excess water will form runoff [Rye et al., 2010]. The hourly runoff in mmwe calculated by the SMB for each model grid cell is used as input to the SRLF model (section 3.2).
3.1.1. SMB Model Calibration Method
 The SMB model is calibrated using measured daily snow surface height data measured at Camp (Figure 2). We calibrate the SMB model based on (1) measured snowpack thickness on 9 June; (2) day on which superimposed ice becomes exposed; (3) total surface height decrease from 9 June until superimposed ice is exposed; and (4) average rate of snow surface height decrease over this time period. To best match these four criteria, we choose the most suitable values for two model parameters which we do not have suitable observations to constrain (1) total winter precipitation and (2) initial snow density (i.e., the density of snow which has just fallen onto the ice sheet surface). For total winter precipitation, we parameterize for a range of values from 0.36–0.44 m, and for initial snow density we parameterize for a range of values between 300 kg m−3 and 400 kg m−3 (Table 1). Initial sensitivity tests involving a much wider range of values indicated that these ranges gave the best match between measured and modeled snow depth on 9 June and are also consistent with suggested ranges of values in the literature [Bassford, 2002; Bales et al., 2009; Burgess et al., 2010; Cuffey and Paterson, 2010; Rye et al., 2010]. We appreciate that the range of values for initial snow density appears to be high, but as snow densification due to settling, compression and the action of the wind is not accounted for by the model, but is instead driven by melting and refreezing alone, a relatively high value for snow density is expected to be established during model calibration [Bassford, 2002; Wright, 2005; Rye et al., 2010]. For example, Banwell et al.  established that the best value for initial snow density for both 2000/2001 and 2004/2005 was 400 kg m−3. All combinations of parameter values at the given intervals in Table 1 are used for individual model runs. The measured and modeled ablation curves are plotted and compared qualitatively. The modeled curves that clearly show a bad match with measured data are immediately discarded. To determine the highest quantitative match, root mean square errors (RMSEs) between the measured and the modeled curves showing good visual matches with measurements are calculated.
Table 1. Ranges From Which Parameter Values Were Chosen for SMB Model Calibration
Initial snow density (kg m−3)
Total winter precipitation (m)
3.2. Surface Routing and Lake Filling Model
 Lakes form in topographic hollows on the ice sheet surface. The rate at which they fill (and hence the water volume within any given lake at any given time) is controlled by the size of the supraglacial catchment which supplies the lake, the rate of water production within the catchment, and the rate of water flow within the catchment. We calculate the rate of water production using the SMB model (section 3.1), but the location and size of lakes, their catchment areas, and water routing within and between the catchments are controlled by the surface topography. The SRLF model we have developed consists of two main components. The first component takes a DEM of the surface and analyses the DEM to identify the topographic hollows which can contain lakes, the catchment areas which feed each lake, and the topological routing of water between catchments if the water level in any given lake reaches the overflow. The second component of the model calculates the time delay between melt production and that water entering the lake by calculating the route taken by water, and the water flow velocity, within each catchment in order to calculate input hydrographs for each lake.
3.2.1. Lake and Catchment Identification Algorithm
 The identification of watersheds (and hence catchment areas), flow accumulation (upstream areas) and flow directions over a DEM are common operations within the hydrological sciences. However, most algorithms in common use rely on the artificial filling of surface depressions within the DEM (see Arnold for a review). As we are specifically concerned with calculating the time-dependent filling of these depressions with water to form lakes, we need another approach. Thus, we use the lake and catchment identification algorithm (LCIA) developed byArnold  for calculating lake and catchment extent as this does not require the artificial filling of surface sinks. Full details of the algorithm are given by Arnold , but briefly, the algorithm begins by calculating the surface slope, and from this the direction of steepest descent, for each cell within the DEM. Any cell with no lower neighbor is defined as a sink, and becomes a potential nucleus for a lake. The algorithm then searches the flow direction matrix to find the set of DEM cells which ultimately flow into each sink cell. This identifies a series of separate catchment areas feeding each sink cell. The algorithm then searches each catchment boundary for the lowest possible DEM cell over which water would pour as the sink (and any surrounding cells lower than their neighbors) flood with water. The maximum areal extent of any lake within each catchment can be calculated, as cells within the catchment lower than the level of the outflow would flood with water as the lake fills. The DEM also allows the lake hypsometry (depth/volume/area) to be calculated. This process also allows the connectivity between catchments to be identified, as the location of the outlet cell, and the DEM cell into which water would pour, are known.
Figure 3 shows the calculated catchments for each lake basin in the Ponting area, the overflow point for each catchment, the maximum area that each of the lakes can reach before overflow, and the topological links between each catchment if overflow occurs.
3.2.2. Flow Delay Algorithm
 The flow direction matrix calculated by the LCIA allows the water flow path from any given DEM cell to the sink cell (or lake) to be calculated. However, this information by itself does not allow any flow delay to be calculated. Thus, we link the LCIA with the flow delay algorithm (FDA) initially developed by Arnold et al. . The FDA uses the flow direction matrix, and the surface slope matrix, together with assumptions about the physical processes controlling water flow, to calculate a flow delay time between each DEM cell and its sink cell. Over a glacier surface, water is assumed to move across snow-covered cells by Darcian flow in a saturated layer at the base of a seasonal snowpack [Colbeck, 1978], or flow across “bare” ice cells in a supraglacial stream, governed by the Manning's equation [Arnold et al., 1998, equations 2 and 3]. Thus, for every DEM cell, a “travel time” for water to cross the cell can be calculated; this time depends on the slope of the cell, whether the cell is ice or snow (which governs the physical processes assumed to control the flow), and the parameter values which govern the water flow. For snow-covered cells, the parameters are the snow porosity and permeability; for ice cells the parameters are the assumed channel geometry and roughness. By integrating the travel times downslope along the calculated flow path, a total delay time from any given cell to its sink can be calculated. This travel time will vary as the snow cover across a catchment is lost over the melt season, and as any lake within the catchment expands, as this effectively shortens the path the water follows.
 Using the calculated delay times for each time step, each hourly melt increment from the SMB is added to the appropriate sink (or lake) cell(s) at the appropriate time step. As the model run progresses, distinct input hydrographs for each sink are produced. The total accumulated volume of water within each lake at a given can then be calculated, and from this (and the calculated lake hypsometry), the time step at which the lake overflows its rim can be calculated; lakes effectively fill with water from the original sink cell “upward” by successive flooding of the next lowest DEM cell(s) within the catchment until the water depth reaches the level of the calculated outflow cell. Once a lake is full, any further water inputs are passed into the downstream catchment, as calculated by the LCIA. In this way, water can flow in a series of “cascades” from its initial source cell, through a series of full lakes, until it either reaches a lake which is yet to overflow, or until it reaches the edge of the DEM domain.
 Preliminary model runs showed that water tended to be delayed in the catchment for too long and did not fill the instrumented lake fast enough when compared with measurements. Field observations suggested that once the snowpack had thinned to a threshold thickness, water starts to flow quickly in the form of slush flows or in channels incised into the saturated snowpack. Consequently, the algorithm was modified by introducing a parameter of threshold snowpack thickness at which flow switches from Darcian to channelized, rather than assuming that flow switching occurs once all the snow in a DEM cell has melted. This is a simple representation of the physical processes occurring on the surface of the GrIS that is closer to our observations, and allows more water to reach the lake basin more quickly. This threshold snow thickness parameter was tuned during model calibration (see below).
3.2.3. SRLF Model Calibration Method
 The hourly runoff per model grid cell from the calibrated SMB model is used to drive the SRLF model. This model is calibrated through comparison of the modeled and measured lake filling data (i.e., the cumulative volume over time) for Lake Ponting. Through tuning model parameters, we aim to best match (1) the onset of meltwater arrival in the lake; (2) the initial lake filling rate before any overflow occurs; (3) the timing of the diurnal cycles in the initial filling rate; and (4) the lake filling rate once potential inflow from overflowing upstream lake(s) has occurred.
 We do not have suitable observations to constrain values for the three parameters which control supraglacial water flow velocity in the snowpack and so we perform multiple model runs and compare the results with measurements in order to identify suitable values. The three parameters are (1) snow permeability; (2) effective snow porosity; and (3) the threshold snowpack thickness (which we call z) for the Darcian to channelized flow switching. However, as snow permeability and effective snow porosity appear as a simple numerator and denominator, respectively, in Colbeck's  equation [Arnold et al., 1998, equation 2], we combine them into one parameter, k, given by k (m2) = snow permeability (m2)/effective snow porosity. Suitable ranges of values for snow permeability and effective snow porosity are given in Table 2. These ranges fall within those used by Marsh , Arnold et al. , and Willis et al.  and give us a range of k values from 1.6 × 10−8 m2 to 4.8 × 10−8 m2. For z, initial sensitivity tests encompassing a range of values from 0.00 to 0.40 mwe indicated the most appropriate range of values to be between 0.20 and 0.30 mwe (Table 2). Initial sensitivity tests indicated that the model was quite insensitive to Manning's roughness (n) and hydraulic radius (R). We therefore use constant values of R = 0.035 m and n = 0.05 m−1/3 s [e.g., Arnold et al., 1998; Willis et al., 2002].
Table 2. Ranges From Which Parameter Values Were Chosen for Calibration of the SRLF Model
k (snow permeability/effective snow porosity) (m2)
1.6 × 10−8 to 4.8 × 10−8
0.8 × 10−8
z (threshold snowpack thickness for Darcian to channelized flow switching) (mwe)
 When the measured lake filling data for Lake Ponting are analyzed (Figure 5), we hypothesize that the dramatic increase in lake filling rate at ∼12:30 local time (LT) on 18 June is due to the overflowing of one or both of the upstream lakes, Lakes X and Y (Figure 3). However, although this was the time at which the magnitude of meltwater discharge into Lake Ponting increased significantly, it is also possible that water started to overflow into Lake Ponting at a lower rate from the upstream lake(s) a few days earlier before a large, high capacity channel was able to form. We calibrate the SRLF model by performing model runs in order to quantitatively establish which combination of parameter values produces the best match between the early period of measured and modeled data. During these runs we focus on the 3.7 km2 catchment area of Lake Ponting, without allowing for any inflow from the upstream Lakes X and Y. This enables us to best match the onset of meltwater arrival in the lake basin and the initial filling rate of Lake Ponting before the catchment area supplying Lake Ponting increases due to meltwater inflow from overflowing upstream lakes. Once the combination of parameter values for k and z which give the lowest RMSE between the early modeled and measured data has been determined, these values are used in subsequent model runs allowing for overflow from the upstream catchment(s), enabling us to observe the effect that inflow from the overflow of Lakes X and Y has on the filling rate of Lake Ponting.
4. Model Calibration
4.1. SMB Model Calibration
 Measured snow depth was 0.99 m on 9 June. We measured a total snow height decrease of 0.64 m between 9 and 25 June when superimposed ice was exposed, suggesting that 0.35 m of superimposed ice formed at this site. Fifteen SMB runs covering the full range of parameter values in Table 1 were undertaken. A selection of graphs of modeled ablation using different parameter value combinations are plotted alongside the measured snow ablation graph in Figure 4. The model captures the magnitude of the total snow height decrease and the average ablation rate between 9 and 25 June. The model does not capture the minor variations in measured ablation rate during this time period likely due to local snow conditions at the measuring stakes. The model run which produces the best match with the measured data has the parameter values of initial snow density 350 kg m−3 and winter precipitation 0.42 m (Figure 4). The calculated RMSE is 0.022 m. This is low in comparison to the RMSE of 0.049 m calculated when the parameter values of initial snow density 400 kg m−3 and winter precipitation 0.44 m are used instead. Importantly, the slightly lower density value of 350 kg m−3 (compared to 400 kg m−3) more accurately captures the slight increase in ablation rate on 13 June. These values are therefore set as parameter values in subsequent SMB model runs.
4.2. SRLF Model Calibration
 Using hourly melt input per DEM cell from the calibrated SMB model, the SRLF Model is run 25 times in order to explore the full range of parameter values for z and k (Table 2). Graphs comparing the measured lake filling curve to various modeled lake filling curves are produced. By visually inspecting the graphs, it appears that the best fit between the modeled and measured data produced when parameter values of z = 0.275 mwe and k = 4.0 × 10−8 are used. As examples, Figure 5a shows the measured lake filling curve alongside modeled lake filling curves for various k values given a z value of 0.275 mwe, and Figure 5b shows modeled lake filling curves for various z values given a k value of 4.0 × 10−8. However, this match is only good up until 16:00 LT on 16 June. Up until this point, (marked by a vertical dashed line in Figures 5a and 5b), the model run using these parameter values produces an RMSE between the measured and modeled data of 1.7 × 104 m3 (i.e., 4% of the cumulative lake volume at 16:00 LT on 16 June). Beyond this point, the gradient of the measured lake filling data starts to show a slight increase and deviates from the almost linear gradient of the modeled lake filling graph. These two parameter values are therefore used in subsequent model runs.
5. Results and Discussion
 As previously mentioned, our measured volume data for Lake Ponting indicates a sudden rise in filling rate at 12:30 LT on 18 June (Figure 5). Furthermore, field observations on 19 June (a few hours after Lake Ponting had drained by hydrofracture), showed a large incised channel, containing a river routing water into the Lake Ponting basin coming from the direction of Lakes X and Y (Figures 2 and 3). This channel had not been visible during our first visit to the basin on 13 June. We hypothesize, therefore, that the sudden increase in lake filling rate at 12:30 LT on 18 June was due to the initiation of, or sudden rise in, meltwater inflow from one or more upstream lakes. Our measurements and field observations do not allow us to determine whether the river was routing water just from Lake X, or whether Lake Y was also overflowing in to Lake X so that Lake Ponting was receiving water from both upstream catchments as well as its own after 18 June 2011.
 Having calibrated the SRLF model for the Lake Ponting basin only, without allowing for potential inflow from overflowing upstream lakes, we now rerun the model using the optimal parameter set for the whole 10 km × 10 km Lake Ponting area, allowing for lake overflow from basin to basin. The output from this model run is shown together with that from the original run without overflow and the measured lake volume data in Figure 6. We successfully model (1) the onset of meltwater arrival in the lake; (2) the initial lake filling rate before any overflow occurs; (3) the timing of the diurnal cycles in the initial filling rate; and (4) to an extent, the lake filling rate once overflow from upstream Lakes X and Y has occurred. The main discrepancies between the modeled and measured lake volume data are related to the timings in the overflow of the two upstream lakes: X and Y.
 The graph of modeled Lake Ponting volume allowing for Lakes X and Y overflow starts to deviate from the graph of modeled lake volume without upstream lake overflow at 16:00 LT on 16 June (Figure 6). This is when the modeled Lake X volume reached a maximum depth of ∼1.5 m and started to overflow into Lake Ponting. As the graph of measured lake volume also deviates from the graph of modeled lake volume without overflow at this time, it is likely that Lake X also started to overflow at this time in reality. The modeled Lake Y volume reached a maximum depth of 3.8 m and started to overflow into Lake X (and on to Lake Ponting) at 16:00 LT on 17 June. This is just over 24 h before the time (12:30 LT on 18 June) when we infer Lake Y to have overflowed into Lake X (and on to Lake Ponting) from the measured data. After this time, the modeled rate of filling of Lake Ponting is slightly less than the measured rate. This supports our hypothesis that water from both Lakes X and Y was necessary in order to produce the measured increase in the filling rate of Lake Ponting after 12:30 LT on 18 June. However, the measured rate may be slightly higher than the modeled filling rate due to channel incision into ice/snow, thereby allowing a higher discharge of water to flow from Lake Y, into Lake X, and on into Lake Ponting. This process is not accounted for by the model.
 Thus, it is likely that Lake X overflowed into Lake Ponting at 14:00 LT on 16 June (creating a slight rise in Lake Ponting's filling rate), before Lake Y then overflowed into Lake X, and on into Lake Ponting, at 12:30 LT on 18 June (creating a much more significant rise in Lake Ponting's filling rate). In our model, Lake X first overflows into Lake Ponting in agreement with the measured data, but then Lake Y overflows into Lake X on 17 June. Thus, there are discrepancies between our model and measurements, not in the magnitude of Lake Ponting volume increase after the overflowing of Lake X and Y, but in the timings of the overflow of Lake Y.
 In order to provide additional evidence to help constrain the timings of lake overflows, we consult the first available Landsat image after the 18 June 2011, from 3 July 2011 (Figure 2b). When this image is compared to the 17 June Landsat image (Figure 2a), Lake X appears to have completely drained by 3 July, whereas Lake Y is now significantly smaller (estimated to be 0.08 km2 on 3 July compared to ∼0.39 km2by 17 June). We can therefore infer that in reality, it was likely that Lake Y did not drain via hydrofracture, but instead probably overflowed into Lake X until the lake level reached the height of the catchment overflow point. Consistent with both measured and modeled lake filling data, we therefore infer that the rapid increase in Lake Ponting's filling rate on 18 June was due to both the overflow of Lake X and Lake Y (with the overflow of Lake Y likely occurring after the overflow of Lake X). Other sources of time-coincident satellite imagery at appropriate spatial resolutions needed to examine these lake drainage dynamics were not available.
 There are two possible explanations for the source of the errors between the timings of the measured and modeled lake overflows. First, as already mentioned, these errors may be due to the model's inability to simulate the process of opening and growth of overflow channels from lakes. Second, the discrepancies between the timings of the measured and modeled lake overflows may be due to inaccuracies in the ASTER GDEM. As a test of this latter hypothesis, we carry out additional model runs with altered DEM topography and find that if the average depth of Lake Y was 0.87 m deeper (giving Lake Y an extra 270,000 m3 of volume in addition to its current maximum volume of 820,000 m3), Lake Y would overflow at 12:30 LT on 18 June, in agreement with the measured data. This modeled curve is plotted in Figure 6 (purple line).
6. Summary and Conclusions
 Using measured snow surface height data, we successfully calibrate a distributed, high-resolution surface mass balance (SMB) model for a small (100 km2) area of the larger (2300 km2) Paakitsoq region of west central Greenland for June 2011. Key SMB model parameter values are (1) total winter precipitation and (2) initial snow density. Values for 2011 are within the ranges found when the model was calibrated against longer time series of measured surface height and albedo over the mass balance years 2000/2001 and 2004/2005 [Banwell et al., 2012].
 We model the routing of this runoff across snow-/ice-covered cells to topographic depressions which fill to form supraglacial lakes, which can overflow into their downstream catchment(s) once full. This surface routing/lake filling (SRLF) model is calibrated using measured volume data from supraglacial Lake Ponting. The key SRLF model parameters are (1) snow permeability, (2) effective snow porosity, and (3) threshold snowpack thickness (z) for the Darcian/channelized flow switching.
 We successfully model (1) the onset of meltwater arrival in the lake, (2) the initial lake filling rate before any overflow occurs, and (3) the timing of the diurnal cycles in the initial filling rate; and (4) to an extent, the lake filling rate once overflow from upstream Lakes X and Y has occurred. Our modeled data also confirm that the rapid rise in the measured filling rate of Lake Ponting was due to the overflow of upstream Lake Y, which flowed into Lake X (which had already overflowed), then on into Lake Ponting, so that Lake Ponting suddenly received water from both upstream catchments once Lake Y overflowed its basin. There are discrepancies of around a day or so between the timings of the modeled and measured lake overflows. These discrepancies could be explained by the model's inability to simulate the process of opening and growth of overflow channels and consequent changes in water velocity, and/or inaccuracies in the DEM which could alter calculated lake volumes and hence the timing of overflow events.
 By linking a surface hydrology model with melt input from a calibrated mass balance model we show that we are able to model the filling rate of a supraglacial lake in an area of the Paakitsoq region in west Greenland with high accuracy. We also demonstrate that water inflow from overflowing lakes in surrounding catchments can play a key role in increasing the filling rate of a lake, and we are able to model the timings of these overflow events with relatively high accuracy given the quality of the available surface topographic data sets. As the rapid drainage of some of these supraglacial lakes by hydrofracture is thought to play a key role in the establishment of a hydraulic connection between the surface and subglacial drainage systems, advancing our ability to accurately model the temporal and spatial variability of lake volumes will ultimately improve our ability to predict changes in ice sheet dynamics, mass balance and sea level contributions.
 This work was funded by a UK Natural Environment Research Council (NERC) Doctoral Training Grant to A.F.B. (LCAG/133) (CASE Studentship with GEUS). Additional financial support was provided by the U.S. National Science Foundation under grants ARC-0907834 and ANT-0944248 awarded to Douglas MacAyeal, who we also thank for discussions during the final stages of the research. M.T. was supported by the NASA Cryosphere Program and NSF (grant ANS 0909388). World Wildlife Fund (WWF), St. Catharine's College (Cambridge), the Scandinavian Studies Fund, and the B.B. Roberts Fund are all acknowledged for financially supporting part of the fieldwork activities. We are grateful to Konrad Steffen and coworkers for the GC-Net data. We thank three anonymous referees and the Associate Editor, Martin Truffer, whose insightful comments helped us improve on an earlier draft of the paper.