The depth of a supraglacial lake can be approximated as (after Philpot, 1989) $$z=[\ln \left(A_d,-,R_{\mathrm{\infty }}\right)-\ln (R_{\mathrm{lake}}-R_{\mathrm{\infty }}\left)\right]/g,$$ where $z$ is lake depth in meters, $A_d$ is the lake bottom albedo, $R_{\mathrm{\infty }}$ is the reflectance of optically deep water, $R_{\mathrm{lake}}$ is the reflectance of a lake pixel, and $g$ is related to the losses in upward and downward travel through the water column (units: m${}^{-\mathrm{1}})$. Based upon a description of the processes that take place as light enters, passes through, and exits a lake, this method has been used successfully in both Greenland and Antarctica (e.g., Banwell et al., 2014; Sneed and Hamilton, 2007). It is physically based and therefore easy to adjust when measurements of lake water and lake bed properties are available. However, this method assumes that lake water has little to no dissolved or suspended matter and would be severely impacted by surface waves (wind-driven ripples, choppy waves, etc.). Additionally, it requires that the lake bottoms have low slopes and a homogeneous albedo (Sneed and Hamilton, 2011). While most of these assumptions hold for supraglacial lakes in Greenland (Sneed and Hamilton, 2011), lake bottoms are known to be too inhomogeneous to support the approach generally. In addition, optically deep water (i.e., deep lakes or ocean where the upwelling radiance originates from the water column without any bottom signal contribution) is not always available in inland Landsat scenes. The effects of these shortcomings on supraglacial lake depth retrievals have not been quantified.

In this study, for application to OLI imagery, $R_{\mathrm{\infty }}$ was obtained from dark ocean or lake water in the scene, following Sneed and Hamilton (2007, 2011). If no coast was available in the scene containing the lake, $R_{\mathrm{\infty }}$ was obtained from another scene further along the path (with an implicit assumption of similar atmospheric conditions). The parameter $g$ was calculated following earlier studies (Smith and Baker, 1981; Sneed and Hamilton, 2007) but with an updated absorption coefficient from Pope and Fry (1997, Table 3); for more details, see the Supplement.

$A_d$ was obtained from the reflectance immediately outside identified lake areas. However, in order to test this approximation for $A_d$, we also solve for lake bottom albedo rather than assuming it to be the same as the surrounding ice. We use spectral mixture analysis (Lillesand et al., 2007) to define a fractional coverage of ice ($r_{\mathrm{i}})$ and cryoconite ($r_{\mathrm{c}}=\mathrm{\hspace{0.17em}1}-r_{\mathrm{i}})$ in each lake bottom pixel. To create a determinable equation after introducing this new unknown ($r_{\mathrm{i}})$, we use reflectances from two OLI spectral bands (indicated with subscripts 1 and 2, below) and derive end-member reflectances for ice ($R_{\mathrm{i}\mathrm{1}}$ or $R_{\mathrm{i}\mathrm{2}})$ and cryoconite ($R_{\mathrm{c}\mathrm{1}}$ or $R_{\mathrm{c}\mathrm{2}})$ using glacier reflectance spectra from Pope and Rees (2014b) in conjunction with OLI spectral response functions in both bands (Barsi et al., 2014). We input these parameters into Eq. (1) and then combine the expressions by equating lake depth, thus obtaining $$[{\displaystyle \frac{r_{\mathrm{i}}\left(R_{\mathrm{i}\mathrm{1}},-,R_{\mathrm{c}\mathrm{1}}\right)+R_{\mathrm{c}\mathrm{1}}-R_{\mathrm{\infty }\mathrm{1}}}{R_{w\mathrm{1}}-R_{\mathrm{\infty }\mathrm{1}}}}{]}^{g_{\mathrm{2}}}=[{\displaystyle \frac{r_{\mathrm{i}}\left(R_{\mathrm{i}\mathrm{2}},-,R_{\mathrm{c}\mathrm{2}}\right)+R_{\mathrm{c}\mathrm{2}}-R_{\mathrm{\infty }\mathrm{2}}}{R_{w\mathrm{2}}-R_{\mathrm{\infty }\mathrm{2}}}}{]}^{g_{\mathrm{1}}}.$$ After Eq. (2) is solved for $r_{\mathrm{i}}$, the bottom albedo for one OLI spectral band can be calculated and subsequently used to compute lake depth: $$\begin{array}{}& & A_{d\mathrm{1}}=r_{\mathrm{i}}{R}_{i\mathrm{1}}+\left(\mathrm{1},-,r_{\mathrm{i}}\right)R_{\mathrm{c}\mathrm{1}}& & z={\displaystyle \frac{\log \left(R_{\mathrm{lake}\mathrm{1}},-,R_{\mathrm{\infty }\mathrm{1}}\right)-\log (A_{d\mathrm{1}}-R_{\mathrm{\infty }\mathrm{1}})}{-g_{\mathrm{1}}}},\end{array}$$ where $R_{\mathrm{lake}\mathrm{1}}$ is water leaving reflectance (as in Eq. 1) for the first band in the pair used and $z$ is lake depth.

The second method we consider uses spectral band ratios to derive water depth. It has been used in shallow marine settings (e.g., Dierssen et al., 2003) and alluvial rivers (e.g., Legleiter and Overstreet, 2012) and has been adapted for use on the Greenland Ice Sheet (Legleiter et al., 2014). While the physically based method above is highly dependent on $A_d$ and $g$, earlier studies show that the spectral band-ratio method is expected to be more robust to variations in these parameters (Legleiter et al., 2009; Stumpf et al., 2003). This is because the method relies on relative behavior in two different wavelengths, as opposed to absolute optical behavior.

This spectral band-ratio method employs an empirically derived quadratic formula to relate lake depths to the ratio of the reflectance of two spectral bands ($R_{\mathrm{1}}$ and $R_{\mathrm{2}})$: $$\begin{array}{}& & z=a+bX+c{X}^{\mathrm{2}},& & X=\ln (R_{\mathrm{1}}/R_{\mathrm{2}}).\end{array}$$ This empirical method requires the derivation of calibrated coefficients (i.e., $a$, $b$, and $c)$, and coefficients vary depending on which sensors and bands are used (Legleiter et al., 2014). We calculate these coefficients using a known set of reflectances and depths (from in situ measurements, see Sects. 3.1 and 4.1).

In summer 2010, Tedesco and Steiner (2011) used a small remote-controlled boat equipped with a compact spectroradiometer and a small sonar to collect coincident lake-bottom reflectance and depth over one lake in West Greenland (Tedesco et al., 2015). We use 2226 unique sample points from that study to evaluate the performance of the remote sensing methods described above. Field spectra are convolved to account for the spectral response of the spaceborne sensors as follows: $$r_{\mathrm{nb}}={\displaystyle \frac{\underset{\mathrm{0}}{\overset{\mathrm{\infty }}{\int }}r\left(\mathit{\lambda }\right)R\left(\mathit{\lambda }\right)\mathrm{d}\mathit{\lambda }}{\underset{\mathrm{0}}{\overset{\mathrm{\infty }}{\int }}R\left(\mathit{\lambda }\right)\mathrm{d}\mathit{\lambda }}},$$ where $r_{\mathrm{nb}}$ is the narrowband reflectance, $r\left(\mathit{\lambda }\right)$ is the spectral reflectance, $R\left(\mathit{\lambda }\right)$ is the relative spectral response (Barsi et al., 2014), and $\mathit{\lambda }$ is the wavelength. In order to emulate sensor dynamic range and radiometric resolution, we impose minimum and maximum reflectances and round reflectance values to the appropriate precision (i.e., 8 or 12 bit; see Pope and Rees, 2014a). We then regress the convolved reflectances and in situ depth measurements to test the goodness of fit of the physically based relationship presented in Eq. (1) and the empirical method described in Eqs.(5) and (6).

Landsat 8 launched on 11 February 2013 and became operational on 30 May 2013 (Roy et al., 2014). OLI collects spectral data gridded at 30 m spatial resolution (15 m for panchromatic data). We calculate top-of-atmosphere (TOA) reflectance using calibration coefficients provided in the image metadata and a solar elevation cosine correction (USGS, 2013). Based on a sensitivity analysis of path radiance to water vapor and ozone using an atmospheric radiative transfer model (see Sect. 5), we do not atmospherically correct the images.

We choose two study areas for applying OLI imagery (see Fig. 1). One site located in northwestern Greenland (including Sverdrup Gletsjer, Dietrichson Gletsjer, Sermersuaq, and Kjer Gletsjer, on Melville Bay; 56.2966–58.7186${}^{\circ }$ W, 74.9685–75.7808${}^{\circ }$ N) is an area with a high concentration of lakes and was imaged four times by OLI throughout summer 2013. A larger region farther to the south is examined using all available OLI scenes collected over the Sermeq Kujalleq region in West Greenland in 2014. For a list of all OLI scenes used in this study, see Table S2 in the Supplement.

Using the calculated TOA reflectances, we define supraglacial lake extent using the ratio between the blue and red bands (Banwell et al., 2014; Box and Ski, 2007). However, since OLI bands are slightly different from those of past sensors, we could not use published thresholds for extent. We set the threshold for this ratio at 1.5 (versus 1.05–1.25 for ETM$+$ in Banwell et al., 2014) based upon visual comparison with the imagery. We then visually inspected and manually adjusted the threshold mask to remove coastal water areas (i.e., not on the ice sheet) and clouds. Although Leeson et al. (2013) describe such thresholding as too coarse for low-resolution imagery (i.e., MODIS), they do acknowledge its utility for higher-resolution imagery (ASTER, Landsat, etc.). We remove regions four pixels or smaller (i.e., small lakes likely comprised solely of mixed pixels) or less than two pixels wide (i.e., linear features likely to be channels, not lakes) from the lake mask.

We interpolate the lake mask using a nearest neighbor algorithm in order to apply the physically based method to the higher-resolution panchromatic band. Where both panchromatic and spectral bands were used together, we bilinearly interpolate the panchromatic image to 30 m resolution.

We use submeter ($\sim$ 0.5 m pixel${}^{-\mathrm{1}})$ stereo imagery from WorldView-1 and WorldView-2 to create DEMs of lake areas both before filling and after drainage. Similar validation for ASTER has been carried out with airborne LiDAR from before lake drainage (Georgiou et al., 2009) and for estimating lake drainage volumes (Stevens et al., 2015). We generate the high-resolution WorldView DEMs using the open-source NASA Ames Stereo Pipeline tool (Moratto et al., 2010; Shean et al., 2016). For both the Sermeq Kujalleq and northwestern sites, we use DEMs from 6 different days, for a total of 12 DEMs (see Table S2).

WorldView-1 image data have a geolocation accuracy of better than 4.0 m horizontal 90 % circular error of probability and WorldView-2 better than 3.5 m (DigitalGlobe, 2014). Thus, the imagery and DEMs are more precisely positioned than the 15–30 m OLI.

The vertical accuracy of the derived DEM products is less than 5.0 m 90 % vertical error of probability with submeter relative vertical precision (Mitchell, 2010). Differencing a WorldView DEM with an Airborne Topographic Mapper LiDAR profile over a pronounced basin in northeastern Greenland provided a standard deviation over the spread of elevations of 0.25 m. Considered conservatively, differencing one WorldView DEM with a second DEM collected 1 year later provided a standard deviation of 0.58 m for the elevation differences (Willis et al., 2015). Stacks of 13 and 17 overlapping WorldView-1 and WorldView-2 DEMs over Summit Station and Tracy Glacier, Greenland, provide absolute vertical accuracy estimates of $\sim$ 2.0–3.0 m relative to airborne LiDAR measurements ($\sim$ 10 cm accuracy). After removing absolute horizontal and vertical offsets from all DEMs, the relative vertical accuracy ($\mathrm{1}\mathit{\sigma }$) for the stack was $\sim$ 15–30 cm (Shean et al., 2016).

We resample the DEMs to the same grid as OLI imagery using cubic interpolation. The OLI and WorldView acquisitions are from different dates; although lake basins do ablate during the summer, this should not have significant impact on the results presented here, because most supraglacial lakes in Greenland remain fixed over bedrock-controlled surface depressions (Lampkin and VanderBerg, 2011). Using the lake mask, we identify a shoreline for a given date (see Sect. 3.2), which is then used to derive lake depth. We remove outliers of impossibly shallow (i.e., negative depth) or deep (> 65 m) values as errors in the DEM. In addition, we remove lakes having a standard deviation in lake elevation along the shoreline of larger than 1.5 m. These steps also mitigated any potential bias caused by temporal offset between DEM and spectral depth measurements. After filtering, over 250 000 pixels (30 m) in total remained for spectral lake depth validation over 6 days in 2013 and 6 days in 2014.

The results (Table 1) of depth–reflectance regressions for all methods are shown in Fig. 2. We base the bands tested here using in situ data upon those identified in the literature (e.g., Box and Ski, 2007; Sneed and Hamilton, 2007; Tedesco and Steiner, 2011), as well as the OLI's new coastal band and the significantly narrowed panchromatic band (0.500–0.680 $\mathrm{\mu }$m, at 15 m spatial resolution). ETM$+$ high and low gain results are virtually indistinguishable, and so only low gain results are shown here. For each regression, we use the correlation coefficient ($r)$ and the root mean square error (RMSE, relative to sonar depths) to assess the performance of each method. The results of the physically based method show that the OLI blue and coastal bands do not perform well relative to other bands (RMSE of 3.10 and 11.03 m, respectively; $r$ of 0.29 and 0.05, respectively). The OLI Band 3 (green, 0.525–0.600 $\mathrm{\mu }$m; 0.78 m, $r=$ 0.78) performs as well as legacy ETM$+$'s Band 2 (green, 0.525–0.605 $\mathrm{\mu }$m; 0.77 m, $r=$ 0.79). Finally, both OLI Band 4 (red, 0.640–0.670 $\mathrm{\mu }$m) and Band 8 (panchromatic, 0.500–0.680 $\mathrm{\mu }$m) outperform their analogous ETM$+$ bands (RMSE of 0.28 and 0.63 m, respectively; $r$ of 0.96 and 0.84, respectively).

Red light attenuates more strongly in water than green or blue light. So, for the same lake depth, there will be a larger (and easier to measure) change in net reflectance for red wavelengths than shorter wavelengths. However, the rapid attenuation of red light means that only shallower lakes may be measured in this band. The maximum in situ lake depth measurement is $\sim$ 5 m, well within the red light limit, but deeper lakes may exist in the overall study area. We address this issue below by using many Landsat scenes and WorldView DEMs.

We investigate the two-band physically based method (where $A_d$ was calculated) with a range of emulated OLI bands (see Table 1). We find similarly high correlation coefficients ($r=$ 0.94) to the regression method. Nevertheless, only the combination of blue and green bands had an RMSE below 1 m. This method appears to slightly overestimate lake depths. We investigate the reasons for this with the OLI and WorldView data below.

Applying the empirical method using field data (see Table 1, Fig. 2) indicates that the more continuous bands of the ETM$+$ outperform the narrower (less spectrally auto-correlated) bands of the OLI when estimating lake depths. However, the addition of the coastal band should allow the OLI still to perform quite well ($r$ > 0.92, RMSE < 0.38), in particular when paired with the green or panchromatic bands.

Our analysis shows that supraglacial lake depth retrievals using OLI are as good as or better than ETM$+$ retrievals. We identify the best methods for OLI (identified with asterisks in Table 1) based on the highest correlation coefficients and lowest RMSEs. We then apply these methods to OLI data and validate them with WorldView stereo DEMs.

In the northwestern Greenland study area, we identified 694 lakes on 2 July 2013 with a total area of 27.2 km${}^{\mathrm{2}}$, 1259 lakes totaling 43.7 km${}^{\mathrm{2}}$ on 18 July 2013, 955 lakes totaling 38.8 km${}^{\mathrm{2}}$ on 3 August 2013, and 274 lakes totaling 8.6 km${}^{\mathrm{2}}$ on 19 August 2013. We calculate lake depths with all previously discussed methods, as well as an average between the two best single-band depth estimates. Total lake volume in the study area increased in early July, stayed almost constant as lake growth areas moved higher in elevation over the following three weeks, and then decreased again toward the end of August as cooler conditions prevailed (see Fig. 3). While all methods show the same pattern of surface water storage, the total water volumes derived with the different methods differ by over a factor of 2.

For both of our case study regions, northwestern and southwestern Greenland, we difference all overlapping areas of OLI-derived lake depths and WorldView-derived DEMs. The statistics of this comparison are shown in Fig. 4. As seen in the northwestern Greenland case study, the results are divided into two groups. OLI-derived depths using Band 3, Bands 2 and 3, a ratio of Bands 1 and 3, and a ratio of Bands 1 and 8 all considerably overestimate lake depth relative to the DEMs. However, the physically based single-band method for the red band (OLI Band 4) only slightly underestimates lake depth ($-$0.1 $\pm$ 1.7 m), while the panchromatic band (OLI Band 8) slightly overestimates lake depth (0.1 $\pm$ 1.4 m).

Combining these two best-performing bands, the resulting spectral and DEM-derived lake depths are in close agreement, showing a difference of 0.0 $\pm$ 1.6 m. We infer that the optimal method for estimating supraglacial lake depth with OLI is to take an average of the physically based (see Eq. 1) depths as derived from the red and panchromatic channels (bold in Table 1). It is likely that the spread in depths is the result of a combination of factors including temporal offset between DEM and spectral data collection, image coregistration, and atmospheric effects, as well as uncertainties inherent in the lake depth retrievals. Despite meter-scale uncertainties (1.6 m) at the pixel level, the mean lake depth derived from these methods agrees well.

We apply the lake depth algorithm (i.e., average of single-band depths from OLI red and panchromatic bands) to 34 OLI images from the summer of 2014 over the Sermeq Kujalleq area (see Figs. 1 and 5). The total meltwater storage in supraglacial lakes peaked near 3 km${}^{\mathrm{3}}$ across the entire study area in mid-July 2014. There are many shallow lakes (0.3 to 1.5 m depth) and many lakes with depths of 2.5 to 4 m. Few lakes exceed 5.5 m depth (see Fig. 6a). The preponderance of shallow lake pixels reflects the fact that the observed lakes have low surface slopes at their edges.

If the water stored in supraglacial lakes in row 12 of path 008 in mid-July were spread across the whole 25 246 km${}^{\mathrm{2}}$ of ice in the scene, it would have an average depth of almost 3 cm. In other scenes, calculations provide average depths of 0.5 to 1.5 cm. Our maximum observed value is almost as high as the volume in supraglacial streams measured by Smith et al. (2015), reinforcing the potentially daily turnover of a well-connected surface system they observed. Indeed, Tedesco et al. (2012) observe bare ice melt rates next to supraglacial lakes in West Greenland of $\sim$ 2.5–3 cm day${}^{-\mathrm{1}}$, similar to those observed by van den Broeke et al. (2011). This implies that lakes are storing on the order of 1 day's worth of melt (or less), indicating daily or subdaily residence times, depending on connectivity.

The Sermeq Kujalleq data set also provides a time series that shows lake growth and drainage/freezing (see Fig. 5a). There are many factors that contribute to lake growth and drainage, including temperature, insolation, albedo, topography, and ice dynamics. These complex drivers are related to the more easily quantified mean elevation and latitude of each scene. For example, isolating the coastal scenes shows the delayed onset of melt and earlier shutdown in the north compared to the south (see Fig. 5b).

To further refine our investigation of geographic factors associated with lake depth over the summer season, we examine single swaths of OLI imagery through time. Path 008 (in the WRS-2 reference scheme; Irons et al., 2012), which transects the lower Sermeq Kujalleq, shows a strong influence both of elevation and latitude in rates of lake growth and water storage (Fig. 5c). Isolating Path 006, however, conflates the effects of elevation and latitude on surface meltwater storage, but because we have more temporal coverage (see Fig. 5d) we see the decline of total lake volume as summer progresses toward autumn. Again, higher latitude and elevation delay melt onset (i.e., Path 006, Row 012). For 006/013 and 006/014, it is likely that the reduced ice sheet area within 006/014 is the explanation for the reduced meltwater volume. Rates of increase and decay of lake volume are similar for this pair.

The distribution of lake depths (by pixel) with elevation is shown in Fig. 6b. Lakes are distributed from $\sim$ 300 to $\sim$ 2100 m elevation. Maximum lake depths occur at about 1200 m a.s.l. At lower elevations, lake depths recorded by our method vary significantly, likely due to rapid lake growth and drainage across a range of dates at lower elevations, versus the higher elevation maximum depths mostly derived from an OLI image on 30 July 2014. From 1200 to 2100 m, measured lake depths decline steadily with less variation. This likely reflects a combination of factors, including the variations in induced surface topography of the ice sheet as it flows over undulating bedrock (Lampkin and VanderBerg, 2011). At higher elevations, slow flow leads to low-amplitude ice surface topography thus shallower depressions, and there is also less available meltwater to fill ice-surface depressions. In addition, while lakes are less likely to variably fill and drain at higher elevations, there was also reduced imagery available from $\sim$ 30 July 2014 onwards. Therefore, the more consistent maximum depths at higher elevations are a combination of incomplete temporal coverage and elevation. Further down, more melt and higher amplitude topography from faster ice flow facilitate lake formation. However, below 1200 m, increased ablation begins to reduce this topography. In addition, the volume of melt available will determine whether depressions are large enough to hold lakes or instead drain via connecting supraglacial channels. The melt volume and therefore the relationship between lakes and channels will thus vary both seasonally and with elevation (Lampkin and VanderBerg, 2014).

The depths returned by the empirical (band-ratio) method considerably overestimate lake depths relative to the WorldView DEMs. The method is entirely dependent upon the calibration of the input parameters (i.e., $a$, $b$, and $c)$. The parameters used in this study are in turn based solely upon extrapolation from in situ measurements at a single lake. Therefore, it is possible that the lake used for calibration is not representative of lakes in our study region. Legleiter et al. (2014) note that the coefficients for the empirical method may be scale dependent, and values calculated from field data may not be appropriate for the 30 m OLI pixels. Indeed, other work (Moussavi et al., 2016) both calibrates and validates spectrally derived depths with WorldView DEMs to show that the band-ratio/empirical method and single-band/physically based method perform similarly well. The use of a ratio of coastal and green reflectances performed well for lake depth retrieval using WorldView-2 imagery (Legleiter et al., 2014). Therefore the band-ratio method may, with better parameters, produce results consistent with the physically based single-band approaches.

The physically based depth retrievals show a large spread in total water volume returns. Physically based depth retrievals rely on accurate bottom albedos ($A_d)$ and water absorption coefficient ($g)$. While $A_d$ is derived from the imagery, $g$ is always calculated for each spectral band based on laboratory measurements and is therefore consistent across all OLI scenes. Comparison of laboratory-measured $g$ with those derived from in situ data (see Table 1) shows that when the laboratory-measured $g$ is higher than that obtained from regressing in situ data, lake depths are overestimated and vice versa. For example, OLI Band 3 (green) shows a 70 % difference in directly measured and regressed $g$, and it overestimates lake depths by a mean of 2.4 $\pm$ 2.1 m relative to WorldView DEMs. By contrast, Band 4 (red) and Band 8 (panchromatic) have very small differences between measured and regressed $g$ ($-$0.06 and 0.06 %, respectively) and yield accurate lake depth estimates ($-$0.1 $\pm$ 1.7 m and 0.1 $\pm$ 1.4 m, respectively) relative to WorldView DEMs.

Water absorption properties also vary with wavelength. For example, poor performance in blue and coastal bands is related to very low absorption. Red wavelengths attenuate relatively quickly in water, and this is described by a relatively high $g$ (0.7507 m${}^{-\mathrm{1}})$ compared to green (0.1413 m${}^{-\mathrm{1}})$. This high $g$ for red light makes it less sensitive to errors in $g$ than green wavelengths. Lake depth estimates using a red channel are also less sensitive to $A_d$ than with a green channel (Tedesco and Steiner, 2011), again due to the high absorption for longer wavelengths. Ultimately, as long as the sensor radiometry is able to measure the return from deep-water pixels, longer wavelengths (i.e., red) can return generally more accurate lake depths because they are less sensitive to the input parameters.

To evaluate other studies in the literature and compare them with our results, we apply the same methods we use (i.e., lab-measured absorption/scattering parameters and appropriate spectral response functions) to calculate $g$s for ETM$+$ bands (see Table 1). Tedesco and Steiner (2011) studied the accuracy of ETM$+$'s green band for lake depth estimation. They tested different multipliers of the diffuse attenuation coefficient for downwelling light to get the water absorption coefficient $g$. They showed that for ETM$+$'s green band, sonar and spectral depths correlated better when a larger multiplier was used. This is broadly consistent with the 70 % offset between observed and theoretical values that we observe (Table 1). They also find that this offset “cannot be easily explained, aside from a possible chlorophyll concentration in the water, currently considered to be unlikely”. Morriss et al. (2013) used ETM$+$'s red band and extracted a higher value of $g$ (0.86 m${}^{-\mathrm{1}})$; this is very close to the regressed value we observe of 0.83 m${}^{-\mathrm{1}}$ (see Table 1), and so we expect their depth estimates to be slightly overestimated.

Banwell et al. (2014) and Arnold et al. (2014) also used ETM$+$'s green band with a $g$ of 0.1954 m${}^{-\mathrm{1}}$, $\sim$ 40 % higher than our regressed value of 0.14 m${}^{-\mathrm{1}}$, leading to depths overestimated by $\sim$ 30 %. Because the comparisons of Greenland and Antarctic lakes (Banwell et al., 2014) are based on relative depths, their conclusions are likely still valid. Arnold et al. (2014) concluded that their model under-predicted water depths, which could in reality mean that their model is behaving correctly but their validation data (i.e., ETM$+$ lake depths) were biased.

Using the same process as for Landsat sensors, we calculated $g$s for ASTER, MODIS, and WorldView-2 bands (see Table S1). Sneed and Hamilton (2007, 2011) used ASTER's green band for lake depth estimation ($g=$ 0.1180 m${}^{-\mathrm{1}})$. This is $\sim$ 20 % smaller than the regressed value of 0.15 m${}^{-\mathrm{1}}$ (see Table S1). They will therefore have likely underestimated lake depth (Sneed and Hamilton, 2007).

For all three studies, the regressed $g$s are much closer to the updated lab-based $g$s (see Sect. 2.1 and the Supplement) than those used in the studies. Adoption of the new $g$s presented here in Tables 1 and S1 would therefore likely lead to improved lake depth estimates.

For all sensors, wavelengths, and input parameters, an important consideration for reflectance-derived lake depth is the atmospheric correction used to prepare the multispectral imagery. All imagery is processed to TOA reflectance, which means that there is some extraneous path radiance remnant in the data. Therefore, TOA values will slightly overestimate the true reflectance. This offset will not be the same between bands and will influence the retrieved lake depths as discussed below.

The single-band physically based model requires that the reflectance of optically deep water be derived for each scene separately. Effectively, this shifts the exponential decay curve of light in lake water but does not change its shape. Therefore, as long as path radiance is assumed to be homogeneous across the 185 km wide OLI scene, TOA reflectance is sufficient for lake depth estimation. To test this assumption, the MODTRAN radiative transfer model (Berk et al., 2005) was used to simulate path radiance on a day for which OLI data were used in northwestern Greenland (18 July 2013) to investigate variations associated with variable water vapor and ozone across a Landsat scene. According to MODIS retrievals (accurate to 30 DU; Borbas et al., 2011), ozone variability within a Landsat scene is on the order of approximately $\pm$50 DU, which translates to a path radiance of $\pm$1.6 % in the red channel. For lake depth, this can propagate to a $\sim$ 20 % error in lake depth. Much of this error appears largely random for a given point in time and space. Thus, while it decreases confidence in individual lake depth retrievals, averaged water volume retrieval should not be biased. For water vapor there was a 0.3 % change in path radiance between the minimum and maximum Landsat scene values, making it a small contributor to overall error. Between days, however, path radiance effects due to water vapor may vary by an order of magnitude more.

For the multiple-band methods, the differential change in path radiance has larger effects. Sensitivity tests showed that a 3 % change in path radiance for one or both bands changed water volumes on the order of 10–30 %. Therefore, a more rigorous atmospheric correction is necessary in order to apply multi-band lake depth algorithms. Still, for the study here, because validation is conducted across 12 non-consecutive days in both spring and autumn, we do not expect atmospheric conditions to bias our conclusions.

There are additional limitations to our method. As discussed above, OLI lake depth estimates (average single-band estimates from red and panchromatic bands) are robust for regional averages but not single pixels. In addition, the threshold used to identify lake extent may need to be adjusted for different regions and scenes (e.g., Banwell et al., 2014; Box and Ski, 2007). Lake depth retrievals are also sensitive to variations in ice albedo, as well as to the presence of ice lids on the surface of supraglacial lakes, which can be common both in early and late summer. Cloud cover and Landsat's 16-day revisit time also limit the conclusions that can be drawn from OLI lake depths. Many studies have used daily MODIS data to identify and track supraglacial lakes (e.g., Liang et al., 2012; Selmes et al., 2011; Sundal et al., 2011). Fusing the higher temporal resolution of MODIS (or additional sensors such as ESA's upcoming Sentinel-2) and higher spatial resolution of OLI, along with more in situ calibration and validation data, should lead to unique insights to supraglacial water storage.