On 15 June 2017, we used the Ocean Optics setup to collect spectra from three bright and one dark bare ice surfaces (Gege et al., 2019) that were missing the typical surface scattering layer (Fig. 1a, b). We therefore assume that their optical properties are comparable to pond bottoms. Illumination was diffuse and stable which was indicated by the negligible standard deviation of the 30 spectra contained in one measurement (Fig. 1c).

On 2 July 2017 between 00:35 and 01:18 LST, we performed 12 nadir measurements of a bare ice surface, likewise missing a surface scattering layer (Fig. 2a), under clear sky conditions and a mean solar zenith angle of 74.89∘ with the Ibsen setup (Gege and König, 2019). Here we use the average spectrum. The large standard deviation may be attributed to surface metamorphism during the measurement (Fig. 2b).

On 10 June 2017, we collected 49 melt pond spectra (Gege et al., 2019) and corresponding pond depths in three melt ponds. Two of the ponds had a bright blue color, while the third one was very dark, which is also apparent in Fig. 3. The pond site was located in a ridged area, and ice thickness measurements from 14 June 2017 showed that ice thickness was ≥0.9 m below the bright ponds and ≤0.5 m below the dark pond, which indicates that the bright ice is older. We presume that the dark ice may have been a refrozen lead. However, no ice cores were analyzed to determine the respective ice types.

The bottoms of the bright ponds were mostly smooth and solid but also featured a few cracks and highly scattering areas that were very porous. The dark pond bottom was more heterogeneous and featured cracks and areas that were porous and riddled with holes (Fig. 4).

At each pond, we referenced the Ocean Optics spectrometers using the Spectralon^® panel before data acquisition. We performed spectral measurements from the edge of the pond or waded through the pond avoiding shading. We did not observe any wind-induced disturbances of the water surface and waited for the water surface to settle before performing measurements inside the ponds. All measurements were performed under clear sky conditions between 12:23 LST and 14:43 LST and corresponding solar zenith angles between 58.90 and 61.04∘. Directly after each spectral measurement, we used a folding ruler to measure pond depth at the same location. Depths ranged between 6 and 25 cm with an average of 17.60 cm. Figure 5 illustrates the melt pond spectra and corresponding pond depths.

Even though the spectra appear smooth at first view, the hardly visible amount of noise in the data becomes relevant for calculating derivatives. To smooth the spectra, we therefore resampled all spectra to a 1 nm spectral sampling by linear interpolation and then applied a running average filter with a width of 5 nm.

We used the Ocean Optics bare ice spectra from overcast sky conditions (Sect. 2.1.1) as pond bottom reflectance.

Analyses of optical properties of water samples showed only negligible amounts of chlorophyll a, colored dissolved organic matter and total suspended matter. Moreover, Podgorny and Grenfell (1996) report that the signal of scattering in meltwater is overwhelmed by the scattering in the bottom ice. We therefore defined a pure water column without additional absorbing or scattering water constituents and computed remote sensing reflectance in shallow water above the water surface according to Eq. (2.20b) in Gege (2015): Rrssh(λ)=(1-σ)(1-σL-)nw2⋅Rrssh-(λ)1-ρu⋅Q⋅Rrssh-(λ)3+Rrssurf(λ), where σ, σL- and ρu are the reflection factors for Ed and upwelling radiance (Lu-) and irradiance just below the water surface. σ and ρu are 0.03 and 0.54, respectively, while σL- is calculated from the viewing angle (0∘ for a nadir-directed sensor). nw is the refractive index of water (≈1.33), and Q is a measure of the anisotropy of the light field in water, approximated as 5 sr. Rrssh- is the remote sensing reflectance just below the water surface according to Albert and Mobley (2003): Rrssh-(λ)=Rrs-(λ)⋅[1-Ars,1⋅exp⁡{-(Kd(λ)+kuW(λ))⋅zB}]+Ars,2⋅Rrsb(λ)⋅exp⁡{-Kd(λ)4+kuB(λ))⋅zB}, where Ars,1 and Ars,2 are empirical constants and Kd, kuW, and kuB describe the attenuation of the water body with depth zB defined by its absorption and backscattering and the viewing and illumination geometry. The first part of Eq. () describes the contribution of the water body and the second part the contribution of the bottom. Rrs- is the remote sensing reflectance of deep water just below the water surface defined by the absorption and backscattering of the water body and the viewing and illumination geometry. Rrsb is the remote sensing reflectance of the bottom that is defined as the sum of the fractional radiances of all contributing bottom types defined by their albedos and under the assumption of isotropic reflection. Rrssurf in Eq. () is the ratio of radiance reflected by the water surface and Ed. We set Rrssurf to zero; thus, the last part of Eq. () can be ignored. We further used a solar zenith angle of 60∘, similar to the in situ measurements, and a viewing angle of 0∘ (nadir).

We computed linear mixtures of the two measured bottom albedos in 25 % steps (100 % dark, 0 % bright; 75 % dark, 25 % bright; …; 0 % dark, 100 % bright). Using this setup, we generated a spectral lookup table (LUT) by increasing pond depth from 0 to 100 cm in intervals of 1 cm, which is adequate for the great majority of melt ponds on Arctic sea ice. The final LUT contains 505 spectra (Fig. 6).

According to the Beer–Lambert law, the extinction of light at a certain wavelength in a medium is described by an exponential function. Here we assume that multiple scattering in meltwater and (multiple) reflections at the pond surface, bottom and sidewalls can be neglected to approximate the radiative transfer. Figure 7a illustrates the exponential decrease in Rrs with water depth at 700 nm for the five different bottom type mixtures. To linearize the effect, we computed the logarithm of the spectra (Fig. 7b). Lastly, we computed the first derivative of the logarithmized spectra (Fig. 7c) for each band by applying a Savitzky–Golay filter using a second-order polynomial fit on a 9 nm window (The Scipy community, 2019b).

We then computed Pearson's correlation coefficient (r) as (The Scipy community, 2019c) 5r(x,y)=∑i=0n-1(xi-x´)(yi-y´)∑i=0n-1(xi-x´)2∑i=0n-1(yi-y´)2, where xi and x´ are the depth of the ith sample and the average depth, yi and ý are the slope of the logarithmized reflectance at a certain wavelength of the ith sample and the average slope of the logarithmized reflectance at a certain wavelength, and n is the number of samples.

The orange curve in Fig. 8 illustrates the wavelength-dependent correlation coefficients of the slope of the logarithmized spectra and pond depths in the LUT. We observe an almost perfect negative correlation in bands between 700 and 750 nm. We performed the same processing for the simulated spectra as for the in situ pond spectra. The blue curve in Fig. 8 illustrates the wavelength-dependent correlation coefficients of measured pond depth and the slope of the logarithmized in situ spectra. We likewise observe strong negative correlations in the wavelength region around 700 nm.

To investigate the similarity of the dark and bright ice spectra, we normalized both bottom spectra at 710 nm and found a high spectral similarity between ∼590 and ∼800 nm (Fig. 9). Consequently, the slope of the logarithmized spectra is widely independent from the chosen bottom albedo in this wavelength region. Assuming that this also applies to ice spectra recorded under clear sky conditions, we used the Ibsen bare ice measurement to develop a model for clear sky conditions accordingly.

Due to the strong negative correlation in the simulated as well as in the measured data, we chose the slope of the logarithmized spectrum at 710 nm (r=-1.0 and -0.86 for simulated and in situ data, respectively) to develop a simple linear model. We used scikit-learn's LinearRegression function (Pedregosa et al., 2011) to fit a linear model to the simulated data with the Ibsen bare ice spectrum as bottom albedo using the method of ordinary least squares.

We found that the solar zenith angle affects the slope and y intercept of the linear model. Because the model should be applicable to a wide range of solar zenith angles, we implemented a second model to derive the slope and y intercept of the linear model for various solar zenith angles. We used WASI to generate spectral libraries for different solar zenith angles (0, 15, 30, 45, 60, 75, 90∘) and found that the resulting change in slope and y intercept can each be described by an s-shaped curve. We used SciPy's optimize.curve_fit function (The Scipy community, 2019a) to fit generalized logistic functions (Richards, 1959) into the data. Using these functions, the model's slope and y intercept can be computed for different solar zenith angles (Fig. 10).

The model is 6z=aθsun+bθsun∂log⁡Rrs(λ)∂λλ=710nm, where z is the predicted pond depth and θsun is the solar zenith angle. a and b are offset and slope as follows: 7aθsun=-20.6+0.790.8+5.8exp⁡-0.13⋅θsun12(cm) and bθsun=-1619.88+94743.64255.3+7855exp⁡-1.3⋅θsun119.9(cm). We further computed the coefficient of determination (R2) as recommended by Kvålseth (1985) as 9R2(y,y^)=1-∑i=0n-1(yi-y^i)2∑i=0n-1(yi-y´)2, where yi and y^i are the true (simulated) and predicted values of the ith sample, n is the number of samples, and y´i=1n∑i=0n-1yi (Pedregosa et al., 2011; scikit-learn developers, 2018). In addition, we also computed the root mean square error (RMSE) as 10RMSE(y,y^)=1n∑i=0n-1(yi-y^i)2 and the normalized RMSE (nRMSE) as 11nRMSE(y,y^)=RMSE(y,y^)y´⋅100. For the model described above, we obtained a perfect correlation (r=1.0; probability value p=8.9×10-172), an R2 of 1.0 and an RMSE of 0.56 cm (nRMSE=1 %) on the simulated training data.

Measurements of albedo have a long tradition in Arctic research (e.g., Grenfell, 2004; Nicolaus et al., 2010; Perovich, 2002; Perovich and Polashenski, 2012) because albedo is an important quantity in climate models and can be measured with a single irradiance detector. In this study, we conducted measurements of Rrs because our model should be applicable to remote sensing data, and the quantity measured in optical remote sensing is radiance. It is only appropriate to derive an accurate radiance directly from the albedo of a Lambertian surface. This assumption, however, is not valid for specular water surfaces and may easily introduce errors. Morassutti and Ledrew (1996) identified changing Ed as the main error affecting reflectance data recording. To tackle this issue, we used a combination of two spectrometers described in Sect. 2.1.

Field spectroscopy is influenced by external factors and the measurement design itself. In contrast to ruler measurements, the spectrometer acquires information of an area. To ease comparison and limit the influence of spatial heterogeneities, we used a fore optic with a 1∘ field of view to minimize the footprint (∼1 cm at a height of 60 cm). However, holding the instruments perfectly still for a period of several seconds is challenging, and even small changes in the position result in changes in the viewing angle, which increases the footprint of a measurement. For future campaigns, we therefore recommend using a gimbal to minimize the influence of roll and pitch of the handheld spectrometer setup. Another issue might have been reflections of the black spectrometer housings on the water surface possibly contributing to the offset between modeled and measured data.

Different refraction indices of wet and dry surfaces may cause part of the observed offset. Furthermore, using bottom albedos obtained from dry surfaces in WASI introduce a systematic offset. However, it remains unclear if the ice surface used to compute the spectral library was wet or dry.

Some of the scattering may be introduced by reflectances at the water surface, which we did not consider in the LUT computation because the necessary values for the parametrization are unknown. Another influence may be the different solar zenith angles between bare ice and pond measurements. The potential influence of the mentioned factors may be worth further examination to refine the model.

Measuring the depth of a pond may appear trivial, but the bottom of a pond is frequently not flat and solid but can be slushy or riddled with holes. In addition, performing two measurements with a spectrometer and a folding ruler at the exact same location is difficult. We therefore recommend using a laser pointer at the end of the pole for orientation. These uncertainties explain some of the scattering in Fig. 11. Interpretation of field photographs of the pond bottoms, however, did not indicate any systematic errors associated with pond bottom characteristics.