Comparison of Pond Depth and Ice Thickness Retrieval Algorithms for Summer Arctic Sea Ice

Abstract

In order to satisfy the demand of key sea ice parameters, including melt pond depth H_p and underlying ice thickness H_i, in studies of Arctic sea ice change in summer, four algorithms of retrieving H_p and H_i were compared and validated by using optical data of melt ponds from field observations. The Malinka18 algorithm stood out as the most accurate algorithm for the retrieval of H_p. For the retrieval of H_i, Malinka18 and Zhang21 algorithms could also provide reasonable results and both can be applied under clear and overcast sky conditions, while retrievals under clear sky conditions are more accurate. The retrieval results of H_i for Lu18 agreed better with field measurements for thin ice (H_i < 1 m) than that for thick ice, but those results of H_p were not satisfactory. The König20 algorithm was only suitable for clear sky conditions, and underestimated H_p, while showing a good agreement with H_p < 0.15 m. For Arctic applications, Malinka18 and Zhang21 algorithms provided a basis and reference for the satellite optical data such as WoeldView2 to retrieve H_p and H_i. Malimka18 also showed the ability to retrieve H_i, except for the Lu18 algorithm if pond color captured by helicopters and unmanned aerial vehicles were available. This study identifies the optimal algorithm for retrieval of H_p and H_i under different conditions, which have the potential to provide necessary data for numerical simulations of Arctic sea ice changes in summer.

1. Introduction

Melt ponds are the most distinctive characteristics of Arctic sea ice in summer. The albedo of pond is significantly smaller than that of snow and ice, which will lead to more solar energy flux into the ice and ocean beneath it. This causes the melt pond to melt further and then reduce the albedo [1,2]. Such an albedo feedback mechanism is one of the reasons for the rapid decrease in Arctic sea ice in summer. Furthermore, the desalination process of sea ice can also be affected by melt ponds. Ice porosity will increase during the melting of sea ice. Then, the coalesced brine channels give rise to downward flushing of surface pond water into the ocean due to gravity and reduce the salinity of Arctic sea ice in summer [3]. These facts stress the importance of getting reliable observations and simulations on melt ponds.

Most recent measurements on melt ponds concentrated on the horizontal scale, such as melt pond fraction and size [4,5,6], whereas with the development of numerical models, more and more studies demonstrated that H_p and H_i are the key factors that affect the optical properties of melt pond. For example, Lu et al. [7] confirmed that the melt pond albedo increases mainly with H_i for thin ice (H_i < 1 m), while it depends on H_p and H_i for ice thickening, and finally chiefly on H_p for H_i > 3 m. The transmittance was dependent only on H_i. Flocco et al. [8] indicated that refreezing pond is essential for the growth of sea ice in the autumn and early winter. The thicker the underlying ice, the longer it will take for the melt pond to freeze completely. The greater the depth of the melt pond, the slower the growth rate of sea ice is during the freezing period. Compared to thick bare ice, melt ponds greatly reduce the surface albedo, and two to three times more solar energy is absorbed by ponds [9]. As the deepening of the melt pond occurs, the solar energy absorbed by a pond is extremely larger than that by underlying ice and ocean beneath ice [10]. These studies put forward requirements on the measurements of pond depth and ice thickness. While it is difficult to achieve in situ data on large spatial scales due to severe environmental conditions in the Arctic Ocean. Satellite remote sensing is also difficult to determine H_p, because melt ponds are relatively shallow. In addition, changes in pond depth are related to ice thickness [11], and remote sensing of the sea ice thickness in summer has always been gaped [12]. Cryosat-2 and ICESat can only offer the Arctic sea ice thickness from October to next April, which is related to the accuracy of open lead identification, the model parameters such as sea ice density, ocean coverage and snow thickness and density, and the uncertainty of laser or microwave penetration depth in the melting ice [13,14]. Farrel et al. [15] found the ability of ICESat-2 to estimate H_p, but ICESat-2 retrievals of maximum H_p were deeper than those typical in situ measurements [16].

Optical data have enough spatial resolution to distinguish shallow melt ponds, and different algorithms addressing this issue have also been proposed [17,18,19,20,21]. The reflectance or the albedo of melt ponds is a common input parameter for these algorithms. However, these algorithms are developed for different purposes and under different in situ conditions. It is difficult to compare directly with each other and determine the optimal value. Thus, in this study, these algorithms were evaluated in detail and validated using field observations to determine the best estimation algorithm. The applicability of these algorithms under the clear and overcast sky, using optical sensors onboard satellites and unmanned aerial vehicles (UAVs), was also discussed.

2. Methods and Data

The four algorithms are briefly summarized below, and some necessary improvements to suit the objectives of this study are also included. Optical data of melt ponds obtained during in situ measurements are presented as validations to the retrieval algorithms.

2.1. Methods

2.1.1. Algorithm of Lu18

Lu et al. [19] (LU18 hereafter) showed that the pond color was influenced by H_p, H_i, incident solar radiation level and optical properties of ice, while H_p and H_i were the major factors among all. Theoretically, the color signal measured by digital camera is the radiance measured at three wavelength bands (red, green and blue). The radiance reflected by a surface into a sensor is defined by the bidirectional reflectance distribution function (BRDF) of the surface, which is a geometrical description of the surface albedo. LU18 took a two-stream radiative transfer model (TEA) to calculate the pond surface albedo, and then the colorimetric method was used to convert the upwelling spectrum into the color in three RGB color space [7,19]. Then, the relationship between pond color and both H_p and H_i was established. The retrievals of H_i and H_p by using the inverse function were verified with the measurements but did not find a clear relationship. However, a clear relationship was found for thin ice (H_i < 1 m), with correlation coefficient R = 0.82 and root-mean-square error ε = 0.156 m.

In order to evaluate the application of the algorithm in the clear sky, we took the Delta-Eddington model (BL model) to replace the TEA model and considered the solar zenith angle and calculated the color of a melt pond [22]. The underlying ice was divided into one “drained layer (DL)” and four “interior layers (INT)” in the BL model. The scattering coefficient of DL was 70 m⁻¹, and 20 m⁻¹ for INT. The absorption coefficient of sea ice and water of the model was obtained from Perovich [23]. The scattering coefficient for pond water and ocean was negligible. The refractive indices of air, ice and water were 1.0, 1.31 and 1.33, respectively. The asymmetry parameter g was 0.94 for sea ice [22]. The BL model can also calculate different sky conditions by changing the ratio of direct and diffuse incident irradiance. The pond optical data collected on the clear sky corresponded to the range of solar zenith angle of 50–70°, and only a few data corresponded to 70–80°. We mainly considered the situation of 50–70° and found that RGB intensity and albedo did not change much with the solar zenith angle. We took the clear sky in August with solar disk visible from Grenfell and Perovich [24] as the incident solar irradiance, and the solar zenith angle set to 60°.

2.1.2. Algorithm of Malinka18

Malinka et al. [18] (MA18 hereafter) indicated that pond surface reflectance is mainly dependent on the pond bottom albedo α_b under different sky conditions, which in turn determines both H_i and inherent optical properties of ice, namely scattering and absorption. The refractive index and absorption coefficient of ice were from Warrant and Brandt [25]. So, the melt pond surface albedo α at direct and diffuse incident solar radiation can be computed through structural formula containing α_b and H_p. These three parameters, H_i, H_p and transport scattering coefficient σ_t, which were varied in analytical formulas to provide the best fit of the measured and modeled spectra in the range of 350–1300 nm at various sky conditions. MA18 also confirmed that the effect of the background (e.g., white ice or snow) was negligible. The retrievals of H_i were found to be in good agreement with the in situ measured values from three fields (R = 0.75 and ε = 0.215 m). However, the retrieval of H_p was not so good as H_i since pond surface albedo was less affected by H_p (R = 0.79 and ε = 0.582 m), and some points retrieved were different from the measured values by more than two times. All parameters for MA18 were as follows. The refractive index of water was taken from Damion and Masumura [26] and Kedenburg et al. [27]. The Fresnel reflective coefficient was calculated through the formula in Malinka et al. [28]. Data from Segelstein [29] provided the water absorption and the spectral scattering coefficient [30]: σ_w(λ) = 0.0017 × (550/λ)^4.3. All the parameters were changed with spectral, and g was included in σ_t (σ_t = σ(1 − g)).

2.1.3. Algorithm of König20

The dependence of pond reflectance on its depth is apparent only in the red range and only for light ponds with high albedos of the underlying ice [7,31]. König and Oppelt [20] (KO20 hereafter) used this effect and developed a linear model to derive melt pond depth. The Water Color Simulator (WASI) can simulate the way pond water changes the reflected spectra of bare ice. Strong negative correlations appeared between pond depth and slope of log-scaled spectra for in situ measurements and simulated by WASI in the 700–750 nm (the Pearson’s correlation coefficient r = −0.86 and −1.0, respectively). Additionally, the spectra of dark and bright ice were quite similar between 590 and 800 nm. So, the slope of the logarithmized spectrum at 710 nm could be chosen to develop a simple linear model. The solar zenith angle affected the slope and intercept of this linear model under clear sky conditions. The retrieved H_p and measured in situ of the solar zenith angle between 58.9° and 61° exhibited a powerful correlation (R = 0.89 and ε = 0.028 m) under the clear sky. In Figure 11 of König and Oppelt [20], validations data of pond depth were quite shallow within the range of 6–25 cm and solar zenith angles around 60°.

Under the assumption of a Lambertian bidirectional reflectance distribution, the remote sensing reflectance, namely inputs of KO20, was somewhat consistent with the surface albedo. In order to extend the algorithm to a wider range, the albedo was used to retrieve the pond depth, and more field measurements on melt pond optics summarized in

Table 1. The eight datasets including different sky conditions.

Source	Location	Measure Time	Size	Hp	Hi	Measured Data	Albedo Wavelength	Sky Condition
Perovich [35]	Near 71°N, 156°W	1995.6.9	-	0.06 m	1.6 m	Albedo and color	400–1000 nm	Clear sky
Perovich et al. [36]	75°N, 142°W to 80°N, 162°W	1998.6–1998.8	<35 m	0–0.5 m	-	Albedo	399–1000 nm	Clear and overcast sky
Perovich et al. [37] Malinka et al. [18]	75°N, 142°W to 80°N, 162°W	1998.6–1998.8	-	0–0.5 m	0–1.2 m	Albedo	400–1000 nm	Overcast sky
Polashenski et al. [38]	Near 71°N, 156°W	2008.6 and 2009.6	<20 m	0–0.3 m	-	Albedo and color	350–1300 nm	Clear and overcast sky
Light et al. [39]	67°N, 150°W to 75°N, 175°W	2010.6 and 2011.7	<20 m	0–0.37 m	0.49–1.5 m	Albedo	350–1300 nm	Clear and overcast sky
Istomina et al. [32]	84°N, 31°E to 82°N, 129°E	2012.8	-	0–0.5 m	0.4–3 m	Albedo and color	350–1300 nm	Clear and overcast sky
Wang et al. [40]	76°N, 167°W to 83°N, 180°W	2016.8	-	0–0.3 m	0.5–1 m	Albedo and color	350–950 nm	Overcast sky
Cao et al. [41]	79°N, 156°W to 85°N, 170°W	2018.8	-	0–0.3 m	1–1.5 m	Albedo and color	350–920 nm	Overcast sky

were used to verify the model. The bare ice spectra from overcast sky conditions served as pond bottom reflectance. The forward model of the WASI can generate libraries of melt pond albedo, which is different from the remote sensing reflectance of KO20. However, the other parameters in WASI were the same as KO20. Figure 1a is the spectral look up table (LUT), which shows the linear mixtures of the two measured bottom albedos in 25% steps and is quite similar to the results in Figure 6 of König and Oppelt [20]. Every bottom-type mixture contains 101 spectral, namely the pond depth ranging from 0 to 100 cm. The blue curve in Figure 1b is the wavelength-dependent correlation coefficient of pond depth and the slope of the logarithmized spectral corresponding to LUT. The orange curve in Figure 1b is r for the in situ measurements [32].

As we can see from Figure 1b, the simulated and measured pond surface albedo show a strong negative correlation in the range of 600–750 nm with the results similar to KO20. The slope of the logarithmized spectral albedo at 710 nm (r = −0.94 and −1.0 for in situ measurements and simulated data by WASI) can be chosen to develop the linear model. The linear model can be developed with the melt bare ice spectral albedo in Arctic summer from Malinka et al. [33] as the bottom albedo in WASI for overcast sky conditions:

$$H_p=-1.0456-1100.5{\left[\frac{\partial log\alpha }{\partial \lambda }\right]}_{\lambda =710nm}$$

(1)

For clear skies, we still choose the slope of the logarithmized spectrum at 710 nm to develop the models since the data we used in Table 1 came from eight field campaigns and only a few data were reported as clear. The Rayleigh atmosphere with the Arctic background aerosol was assumed under clear sky conditions [34]. In this assumption, the solar zenith angle was determined from pond reflectance in the band 1250–1300 nm (perfectly) or 850–900 nm according to MA18. For accuracy, we used WASI to general LUT for every solar zenith angle and found the corresponding model.

2.1.4. Algorithm of Legleiter14 and Zhang21

Legleiter et al. [17] (LE14 hereafter) estimated melt ponds depth on Greenland glacier by using the ratio of the surface reflectance of two bands. An optimal band ratio analysis (OBRA) algorithm was employed to find the maximum correlation coefficient between the pond depth and the reflectance of two bands. A linear model was then developed with R = 0.96 and ε = 0.470 m in the yellow-orange region of the spectrum for H_p in the range of 0.31 to 10.45 m. Furthermore, LE14 also convolved the in situ measured spectrum with the spectral response functions of satellites such as WoeldView2 (WV2) and the correlation between convolved spectrum and H_p was still high (R = 0.96).

In order to establish the general relationship between H_p and pond surface albedo and then retrieve H_p, and take the changes of Arctic summer sea ice thickness into account, Zhang et al. [21] (ZH21 hereafter) employed the radiative transfer models to supplement the LE14 algorithm. The maximum r appeared when λ₁ = 359 nm and λ₂ = 605 nm with the relationship for H_p = 1.49X − 0.02 and X = ln [α(λ₁)/α(λ₂)]. ZH21 also discussed the influence of the sky condition and found that the effects of sky conditions on the relationship between H_p and X can be neglected. In the case of underlying ice thickness, the r between H_i and X was good for λ₁ = 447 nm and λ₂ = 470 nm and H_i = 225.34X + 0.20.

2.2. Data

Eight datasets of in situ measurements on melt pond optics were summarized in Table 1. They were applied for the evaluation of the above algorithms under different sea ice and sky conditions.

From late May to mid-August, melt ponds can go through four stages: pond formation, pond drainage, pond evolution and pond refreezing. The melt rate may be 2.25 cm day⁻¹ from 17 July to 14 August [18,37]. Optical measurements on melt ponds with open water surface were the idealized validation data for this study. However, during in situ observations, some melt ponds were covered by snow, which were removed as a result of great difference between the model and the measured data. There were also some melt ponds with ice lid on the surface. However, the influence of ice lid varying from 0 to 3 cm on pond albedo and transmittance can be neglected [42]. Therefore, these ponds could still be used for model verification. Measuring of the pond depth remained at the same location despite changes in the pond depth based on field reference and pond horizontal extent was generally less than 35 m according to available records. Measurements on melt pond color were relatively rare. To provide enough field validations, the measurements on pond albedo were converted to RGB color according to LU18, as an additional supplement to the absence of pond color measurements.

3. Results

3.1. Retrievals of Pond Depth

Sky conditions pose an important impact on the optical properties of melt ponds, we therefore separately discussed the retrievals of pond depth under overcast sky and clear sky conditions. All the algorithms treated the pond water and underlying ice as parallel layers having only finite depth. So, we did not consider the pond size as retrieving H_p.

A comparison between retrievals and measurements of H_p under overcast sky conditions was shown in Figure 2. The above models’ retrievals of H_p were in the range of 0–0.6 m, which were in accord with the measured range. Among all, MA18 algorithm had the best retrieval results, with R = 0.76 and ε = 0.120 m (Figure 2b). It was firstly because this algorithm used the spectral albedo and had the most data points among all. Secondly, all the parameters for the MA18 algorithm were functions of spectral, more accurate than other algorithms. The retrieval results for H_p of MA18 algorithm were mostly within the range of ±2ε.

The correlation of the ZH21 algorithm was quite similar to that of MA18 with R = 0.71, ε = 0.091 m (Figure 2d). This was because ZH21 used two models for comparison and verification, taking the model to calculate the r of all pond depths and X of 350–1000 nm albedo, and then selected the optimal band. On the contrary, the performance of the KO20 algorithm for the overcast sky condition was not as good as ZH21 and MA18 (Figure 2c), similar to the conclusions of König and Oppelt [20]. Reflection of clouds on the surface contributed a lot to the reflectance measurements under overcast sky conditions. Changes in the optical path of the incident light in water also played a part in such cases.

The relationship between the retrieved and measured H_p was not clear for the LU18 algorithm (Figure 2a), implying that the connection between H_p and pond color was relatively weak. Firstly, the pond color, in some instances, can be considered as a kind of spectral albedo corresponding to specific wavelengths, since the albedo determined poorly by pond depth and was mainly dependent on pond bottom albedo, especially for the ice thinner than 1.5 m [7,33,43]. Secondly, LU18 just used pond color to retrieve H_p, which was significantly insufficient compared to MA18. Thirdly, only the red intensity of melt ponds for thick ice (H_i > 1.5 m) highly depended on H_p while on H_i for thin ice (H_i < 1.5 m). The green and blue intensities were specifically dependent on H_i according to LU18.

The comparisons under clear sky conditions were mostly consistent with the results under overcast skies. It can be seen from Figure 3 that the MA18 algorithm also had the highest correlation (R = 0.93) and the smallest root-mean-square error (ε = 0.079 m) among all. The retrieval results for H_p under clear sky conditions were clearer than overcast sky conditions for the ZH21 algorithm (Figure 2d and Figure 3d). This can be explained that the irradiance reflected by the atmosphere under clear sky conditions is smaller than that under overcast sky conditions [44]. For the clear sky, the correlation between the measured and estimated values for the KO20 algorithm was still not clear (Figure 3c), and the estimations were significantly lower than the measurements especially when pond depth was larger than 0.15 m. However, a clearer correlation appeared when the pond depth was lower than 0.15 m (R = 0.71, ε = 0.026 m). It was because the dependence of reflectance on the pond depth was only apparent in the red range and only for light ponds with high albedos of the underlying ice. As the depth of the pond increases, the change in spectral albedo becomes gradually insignificant, especially for the dark pond [7]. Therefore, this algorithm using the log-scaled spectral albedo at 710 nm is currently only suitable for shallow ponds. The results for LU18 under clear sky conditions (R = 0.40, ε = 0.277 m) were basically consistent with the results under overcast sky conditions (Figure 3a), still depicting that the correlation between pond color and H_p was loose.

3.2. Retrievals of Underlying Ice Thickness

There were few field measurements on underlying ice thickness compared to pond depth data, and most of them were conducted under overcast sky during Arctic summer. Thus, the retrievals of ice thickness were only validated under overcast sky conditions, as shown in Figure 4.

The retrieved ice thickness of the MA18 algorithm was more reasonable than others with R = 0.79, ε = 0.507 m (Figure 4b). The ZH21 algorithm could also be used to retrieve ice thickness (Figure 4c), while the correlation coefficient was slightly lower than the MA18 algorithm (R = 0.75). The uncertainty of H_i retrievals for LU18 was the greatest among all, with R = 0.70, ε = 0.976 m (Figure 4a). However, the retrieval results for thin ice (H_i < 1.5 m) were better for LU18, and all of the retrievals were within the range of ±ε.

A possible reason was that the TEA model assumed that the melt ponds and underlying ice were homogenous and with constant inherent optical properties. Additionally, melt ponds and underlying ice had fairly flat bottoms and horizontal variations can be neglected. The thin first-year ice (FYI) is more applicable to these assumptions than thick multiyear ice (MYI), because melt ponds on FYI are shallower and larger than those on MYI owing to the rougher topography of MYI generally [45]. So, the contrasts at the boundary between ponded and bare ice had a larger impact on the measurements of MYI [46]. Another reason was that the pond depth on MYI is deeper than FYI, and greater depth creates greater irradiance absorbed by the pond water, especially when at large wavelengths. This in turn affected the reflected irradiance [10]. However, the results for the LU18 algorithm can still be encouraging. The effects of global warming are taking a toll as the snow and ice now melt more rapidly than in the past, and the ice thickness is continuously thinning. Lindsay and Schweiger [47] showed that the annual mean sea ice thickness over the Arctic Basin declined 0.58 ± 0.07 m decade⁻¹ between 2003 and 2012. The annual mean ice thickness had decreased from 3.59 m to 1.25 m over the period 1975–2012, and in September, the mean ice thickness declined from 3.01 m to 0.44 m. Rationally, the retrievals of thin ice (H_i < 1 m) are critical for Arctic sea ice in summer.

While comparing Figure 4 with Figure 2, it was found that the retrievals of H_i seem to be more reasonable than that of H_p under overcast sky conditions, especially for the results of LU18 algorithm. This can be explained by the fact that only the red intensity of the pond color depends on H_p as H_i > 1.5 m, and the color depends on ice thickness under other situations [19]. For MA18 algorithm, the correlation coefficient of retrieved H_p (R = 0.76) was slightly lower than that of retrieved H_i (R = 0.79). Moreover, the retrievals of pond depth contained more uncertainties because, for example, some values can differ more than 2 times from the measurements (Figure 2b). The results for the ZH21 algorithm were quite similar to MA18. The reasons for the better performance of these algorithms as estimating ice thickness were as follows: firstly, the reflected upwelling irradiance or the RGB intensity came primarily from the scattering of underlying ice. Therefore, the pond surface albedo or the color was associated more with the underlying ice and the corresponding scattering coefficient than melt pond water. Secondly, the underlying sea ice was not flat, especially at the pond–ice interface [41]. All retrievals of H_i and H_p were average values, while in situ measurements always gave a random value taken at a fixed location. The underlying ice thickness was relatively large compared to pond depth, so the measurement error had a greater impact on H_p than on H_i. Finally, pond water could have some impurities that increase the absorption coefficient, and the additional absorption would result in larger retrievals of H_p than the measurements.

4. Discussions on Application

4.1. The Application for the Satellite Optical Data

These algorithms are potential retrieval algorithms for satellite remote sensing. The wavelength bands of satellite optical sensors should be investigated because they have different spectral response functions. For example, the WV2 contains seven channels: coastal (400–450 nm), blue (450–510 nm), green (510–580 nm), yellow (585–625 nm), red (630–690 nm), red edge (704–745 nm) and near infrared (770–895 nm). The satellite images of WV2 have higher spatial resolution and more spectral channels relative to ASTER, MODIS and Landsat 7. Similar to LE14, we convolved albedo to the sensor bands for WV2 to get the relationship between pond depth and spectral response functions using the ZH21 and MA18 algorithms.

It was clear from Figure 5a that the convolved albedo corresponding to the spectral response functions of WV2 pointed to the general relationship between H_p and X, and so did H_i and X from Figure 6a. However, r was lower than the results of spectral albedo for all H_p and H_i, with r = 0.93 for H_p and r = 0.86 for H_i. At the same time, the R for retrievals of H_p in Figure 5b was slightly lower, and the uncertainties were a bit greater than the results using the spectral albedo as inputs in Figure 2d, similar to that in H_i (Figure 6b). Firstly, it can be explained by the fact that some bands of albedo with a lower correlation to pond depth were also included in the convolution calculation as the satellite channels were wide, which increased the uncertainty. Secondly, the WV2 signal did not fully cover the 350–1000 nm band which thereby lost some important information. For example, the correlation coefficient was the largest for the spectral albedo at λ₁ = 359 nm, but the satellite channels did not include this band.

The r for H_i was good only when λ₁ and λ₂ were both small. If comparing r between H_p and H_i from Figure 5a and Figure 6a, we found that r for H_p was slighter higher than H_i. This could be firstly explained by the fact that only the solar irradiance in 350–600 nm can transmit melt ponds and underlying ice [7], and then the scattering characteristics of the sea ice will lead to the backscattering in these bands, namely upwelling irradiance on the pond surface, which was affected by the ice thickness and scattering coefficient of ice. Secondly, according to ZH21, r was large between H_i and X only when both λ₁ and λ₂ were small for spectral albedo. The blue channel (450–510 nm) for WV2 contained some band that had a low correlation between H_i and X. When comparing the retrieval results of H_p and H_i from Figure 5b and Figure 6b, H_i was more reasonable than H_p, thus the results were consistent with Section 3. LE14 and ZH21 also showed that broader bands can be used to retrieve H_p. If the influence of different satellites was ignored, ZH21 calculated the relationship between the ration X* of the convolved albedo with a step of 50 nm and H_p. The maximum r appeared when λ₁* = 355–404 nm and λ₂* = 564–613 nm. R was 0.60 between measurements of H_p and retrievals by using X*, which was slightly higher than the results by using the spectral response functions of WV2.

For the MA18 algorithm, the retrieval results of H_p and H_i taking the convolved albedo were shown in Figure 7. The performance for this algorithm had the same characteristics as the ZH21 algorithm containing more uncertainties than the results of spectral albedo, and retrieval results for H_i were clearer than H_p. This was because the corresponding convolved albedo was from the seven spectral channels of WV2 using satellite optical data and shallower than the wavelength band of 350–1300 nm using the spectral albedo. H_i was more dependent on albedo than H_p. Zege et al. [48] developed the Melt Pond Detector (MPD) algorithm to retrieve the melt pond fraction (MPF) and the spectral albedo of Arctic summer ice from MERIS (Medium Resolution Imaging Spectrometer) on ENVISAT (Environmental Satellite). The optical thickness of the water layer in the pond, namely H_p, was also a parameter in the MPD algorithm. A closed numerical experiment confirmed that the R between retrieved and true H_p was 0.35. However, the experiment was conducted under MPF = 0.4, where the retrieval error was close to maximal. Additionally, the data used to verify the MPD algorithm were insufficient, which was also a factor limiting the accuracy of the MPD algorithm.

Here, we are just testing the feasibility of WV2 in using satellite optical data to retrieve H_p and H_i. There will be more satellites and channels with the development of technology in the future. With the increase in satellite channels, the ZH21 and MA18 algorithms will be more accurate.

4.2. The Application for In Situ Optical Data

In addition to satellite remote sensing, the hyperspectral instruments onboard UAVs can record the geometry of melt ponds and measure surface reflectance remotely. Such airborne hyperspectral imagery has the characteristics of many wavelength bands and high spectral resolution. For example, the wavelength range of the airborne hyperspectral AISA Eagle Ⅱ is 400–1000 nm, with the spectral resolution of 3.3 nm and the spectral sampling interval of 9 nm. As in LE14, the pond surface albedo can be obtained by multiplying remote sensing reflectance, which is calculated as the ratio of upwelling radiance to downwelling irradiance by π under the assumption of a Lambertian BRDF. On the other hand, similar to Zege et al. [48], the MPD algorithm can retrieve related parameters about melt ponds and the albedo can further be calculated. Pond depth and underlying ice thickness can thus be estimated according to the ZH21 and MA18 algorithms. The KO20 algorithm can also be employed to obtain pond depth under clear sky conditions, and the LU18 algorithm is a good choice to obtain ice thickness, especially for H_i < 1 m.

Apart from hyperspectral imagery, melt pond color captured by helicopters and UAVs is an easier way to retrieve pond depth and ice thickness. The MA18 algorithm can also be used in this situation by using the RGB intensity of melt ponds, which is the quantities describing the reflected radiance by the surface. Additionally, the principle is similar to that of LU18. The colorimetric method is employed here to convert the spectrum into RGB, and then, the connection between the pond color and the 3D vectors (H_i, H_p, and σ_t) can be established. The results of MA18 by using pond color were shown on Figure 8. It was clear that the relationship between simulated and measured H_p was less correlated (R = 0.71), while the retrieval results for H_i were quite reasonable. The reason can be that the dependence of pond color on pond depth was less important than that on ice thickness. Compared to Figure 2a, the correlation coefficient of the MA18 algorithm was obviously larger than LU18 when retrieving H_p (Figure 8a). The ε, contrarily, was quite smaller than LU18. Even though the correlation coefficient of MA18 (R = 0.71) was similar to LU18 (R = 0.70) for retrieval H_i, ε (ε = 0.473) was two times smaller than LU18 (ε = 0.976), which can be seen in Figure 4a and Figure 8b. Thus, when we retrieve H_p and H_i using pond color, the MA18 algorithm will be more reasonable than LU18. It was because all the parameters in MA18 algorithm were wavelength dependent, such as the Fresnel reflection coefficient and refractive index between water and air, while LU18 used a constant. Taking the parameter settings similar to MA18 will increase the accuracy of the algorithm but will increase the computation consumption as well. Compared to Figure 7, we perceived that retrieval results of H_p and H_i using pond color were mostly similar to the results using convolved albedo for MA18 algorithm. The input parameters were only red–green–blue intensities for Figure 8, which was less than the seven spectral channels for Figure 7. The other parameters in MA18 algorithm were convolved with the spectral channels for Figure 7, while were all spectral dependent for Figure 8, which was more correct.

5. Conclusions

This paper compared four different algorithms for retrieving melt pond depth H_p and underlying ice thickness H_i proposed in recent years and the results were summarized in

Table 2. Retrieval results and applications for the above four algorithms.

Algorithm	Input Parameters	Output Parameters	Overcast Sky Accuracies	Clear Sky Accuracies	Application	Illumination Condition	Instrument Setups
LU18	Melt pond color (RGB)	Hp and Hi	Hp: R = 0.27 ε = 0.295 m Hi: R = 0.70 ε = 0.976 m	Hp: R = 0.40 ε = 0.277 m	In situ measure of optical data	Clear and overcast sky conditions	Digital camera
MA18	Melt pond surface spectral albedo	Hp and Hi	Hp: R = 0.76 ε = 0.120 m Hi: R = 0.79 ε = 0.507 m	Hp: R = 0.93 ε = 0.079 m	Satellite optical data In situ measure of optical data	Clear and overcast sky conditions	Spectrometers
KO20	The slope of the log-scaled albedo at 710 nm and the solar zenith angle	Hp	Hp: R = 0.40 ε = 0.163 m	Hp: R = 0.71 (0 < Hp < 0.15 m) ε = 0.026 m	In situ measure of optical data	Clear sky conditions	Spectrometers
ZH21	Melt pond surface spectral albedo	Hp and Hi	Hp: R = 0.71 ε = 0.091 m Hi: R = 0.75 ε = 0.551 m	Hp: R = 0.81 ε = 0.101 m	Satellite optical data In situ measure of optical data	Clear and overcast sky conditions	Spectrometers

. The main conclusions were drawn as follows.

(1) The pond depth retrievals by the MA18 algorithm have the highest correlation coefficient with observations among all. The correlation of ZH21 algorithm is slightly lower than the MA18 algorithm. The LU18 algorithm is not suitable to determine pond depth. The KO20 algorithm underestimates pond depth but performs well when H_p < 0.15 m (R = 0.71) under clear skies. (2) For the retrieval of H_i, both MA18 and ZH21 algorithms can give reasonable results. Retrievals using the pond color for LU18 overestimate the ice thickness, but the LU18 algorithm provides more reasonable results for FYI than for thick MYI. (3) Considering the application of the four algorithms to different sky conditions, MA18 and ZH21 algorithms are more accurate in the clear sky condition than in the overcast sky, due to the decrease in irradiance from the atmosphere. KO20 algorithm is only reasonable for the clear sky. (4) Considering the ease of implementation of the four algorithms, retrievals using pond color is the easiest algorithm of all, while more additional information is required to improve the accuracy of LU18 and MA18. (5) MA18 is the most accurate for retrieving H_i and H_p, but requires pond surface spectral albedo (350–1300 nm) and all the parameters in the algorithm are wavelength dependent.

In current research studies on melt ponds, both H_i and H_p are the key parameters in many models, but there are very few in situ observation data and satellite remote sensing is also limited. This paper compares the four algorithms, providing the difference in applicability of optical data from satellite and in situ measurements, and feasibility for offering key parameters of summer sea ice changes. However, the results also show that the accuracy of the four algorithms can be further improved by adding additional information about melt ponds. It makes a request on the accumulation of in situ measurements on summer Arctic sea ice. Moreover, the retrieval algorithms of pond depth and ice thickness in this paper provide a theoretical basis for the investigations on the three-dimensional morphology of the melt pond, which dominates the shape and size of melt ponds, and further improve our understanding on the evolution process of sea ice surface melting. Both are future research plans.