Calculations of Arctic Ice‐Ocean Interface Photosynthetically Active Radiation (PAR) Transmittance Values

Sea ice algae play an important role in the Arctic Ocean ecosystem, driving primary production in the spring and sequestering carbon to the deep ocean. Up to 45% of Arctic Ocean primary production occurs in ice‐covered areas; photosynthetically active radiation (PAR) is fundamental to driving this production. Sea ice, and particularly snow, strongly scatter and reflect light, reducing the amount of PAR transmitted to the ice‐ocean interface. The effect that varying thicknesses of sea ice (0.2–3.5 m) and snow (0.01–1 m) have on the value of PAR transmittance at the ice‐ocean interface are considered for a Winter, Spring, and Summer scenario. When characteristic Arctic Ocean conditions (2 m sea ice and ∼0.2 m snowpack) are modeled, there is roughly a two‐fold difference in PAR transmittance at the ice‐ocean interface between the Winter (0.003) and Spring (0.007) scenarios and an order of magnitude difference with the Summer scenario (0.04). The modeled values correlate within one standard deviation of measured values and show good agreement with extended pan‐Arctic Ocean field campaign measurements. The results also indicate that simple exponential decay methods may lead to inaccurate results, and radiative‐transfer modeling is required to accurately predict PAR transmittance at the ice‐ocean interface. Therefore, this study offers a novel mathematical technique to predict the value of PAR transmittance at the ice‐ocean interface. Coupled with year‐round near‐real‐time sea ice and snow thickness remote sensing data, this technique may improve understanding of primary production and carbon budgets in the changing Arctic Ocean.


Introduction
Arctic sea ice covers a vast expanse of the Earth's surface (∼6.2-15.3 million km 2 , 1981-2010 median values) and plays a significant role in regulating the climate through the large value of the albedo of sea ice and snow (Fetterer et al., 2017).The Arctic Ocean also acts as a large biome, with primary production yielding up to ∼682 Tg C year 1 ; 30%-45% of the total primary production occurs in ice-covered areas (e.g., Popova et al., 2010;Sakshaug, 2004;Zhang et al., 2010).Sea ice algae are an important component of primary production in the Arctic Ocean and play a substantial role in driving both the Arctic food web in the spring and the Arctic carbon cycle by sequestering carbon to the deep ocean from the surface (Arrigo, 2014;Assmy et al., 2013;Boetius et al., 2013;Fernandez-Mendez et al., 2014;Glud et al., 2007;Hancke et al., 2018;Soreide et al., 2010).Photosynthetically active radiation (PAR, wavelength range 400-700 nm) is fundamental to this biological productivity beneath the sea ice, with 85% of spatial variability and 72% of interannual variability in primary production associated with under-ice PAR and the depth of winter mixing (Gradinger, 2009;Petrich et al., 2012;Popova et al., 2010).
The age, thickness, and extent of sea ice alongside snow depth in the Arctic Ocean are declining, leading to a dynamic change in primary production (Castellani et al., 2021;Thoman et al., 2020).Sea ice, and particularly snow, strongly attenuates downwelling shortwave radiation and reduces the amount of PAR available for biota at the ice-ocean interface and the water column below (Maykut and Untersteiner, 1971;Perovich, 1996).The reduction in sea ice and snow has resulted in an increased flux in shortwave radiation absorbed in all areas of the Arctic Ocean, from 146 TW in 1988to 209 TW in 2007, which has resulted in a 50% increase in primary production by ice algae and phytoplankton (Arndt & Nicolaus, 2014;Lim et al., 2022;Nicolaus et al., 2012;Zhang et al., 2010).The implications of increased PAR availability have been extensively studied (e.g., Castellani et al., 2021;Grenfell & Maykut, 1977;Light et al., 2008Light et al., , 2015;;Lim et al., 2022;Perovich et al., 1998a;Shiozaki et al., 2022;Wassman and Reigstad, 2011) and it has been postulated that there is sufficient PAR in much of the Arctic Ocean for algae to begin growing by February (Stroeve et al., 2021).However, the effects of increased PAR availability are complicated and multifarious, with decreasing sea ice potentially leading to an alteration of the productive regions of the Arctic Ocean owing to stratification changes, which could also affect benthic carbon sequestration (Wassman and Reigstad, 2011).Indeed, there may be increased primary production by ice algae in the central Arctic Ocean but a decrease in regions that are highly productive and presently weakly stratified, such as the Barents Sea, which may have implications for the subarctic fisheries, which are currently the richest fisheries on Earth (Arrigo and Dijken, 2011;Lund-Hansen et al., 2020a, 2020b;Randelhoff et al., 2019;Slagstad et al., 2011;Wassman and Reigstad, 2011).Thus, Wassman and Reigstad (2011) argue that it is important to improve existing biological models (e.g., Lund-Hansen et al., 2015, 2020a, 2020b;Petrich et al., 2012;Popova et al., 2010;Randelhoff et al., 2019;Sakshaug, 2004;Slagstad et al., 2011;Zhang et al., 2010) with information on how PAR is affected by low solar zenith angle (SZA), cloud cover, and, most importantly, variable sea ice thickness and snow depth.
Measuring PAR, and particularly under-ice PAR, in the Arctic Ocean is challenging, expensive, and prohibitive (e.g., Light et al., 2008Light et al., , 2015;;Matthes et al., 2020;Pavlov et al., 2017), resulting in the majority of research occurring in the summer under melting conditions (Perovich, 1998b).New autonomous vehicles such as biogeochemical-Argo floats and the 24 currently active Ice-Tethered Profilers can provide large spatiotemporal resolution under-ice data throughout the year from the Arctic Ocean (Ardyna et al., 2020;Berge et al., 2016;Biogeochemical-Argo Planning Group, 2016;Boles et al., 2020;Jayne et al., 2017;Krishfield et al., 2006;Laney et al., 2017;Roemmich et al., 2019;Toole et al., 2010).There have also been significant improvements in nearreal-time snow and ice thickness data from the Arctic Ocean through the dual-altimetry data provided by CryoSat-2 and ICESat-2 and with the Copernicus Polar Ice and Snow Topography Altimeter (CRISTAL) mission due to be launched in 2027 (e.g., Garnier et al., 2021;Landy et al., 2022;Tilling et al., 2018).There is also airborne laser, radar, and electromagnetic data from the Arctic Ocean through the Alfred Wegener Institute's IceBird program, in addition to 20 Snow Buoys distributed during the MOSAiC campaign for sea ice and snow monitoring (Jutila et al., 2022;Kern et al., 2020;Nicolaus et al., 2021).These improvements in Arctic Ocean monitoring techniques allow for more accurate modeling studies (e.g., Landy et al., 2022;Stroeve et al., 2021), which can be used with near-real-time and in-situ sea ice and snow data.Therefore, the study presented here offers a simple mathematical technique to estimate PAR transmittance (the fraction of incident light which is transmitted) at the ice-ocean interface, through different types and thicknesses of sea ice and snow, simply by measuring the downwelling PAR at the surface of the snow.Typical sea ice thicknesses found in the Arctic Ocean are not optically thick or quasi-infinite (King et al., 2005) for the propagation of visible light.In contrast to light propagation in snow, it is anticipated that a simple exponential decay of light through the ice will not be independent of the optical properties of the materials above and below the ice (i.e., the low albedo of the ocean).
Several excellent studies have modeled PAR transmittance through sea ice; however, they have only considered limited sea ice or snow conditions in a diffuse environment (e.g., Hill et al., 2018;Light et al., 2008Light et al., , 2015;;Pavlov et al., 2017;Perovich, 1990Perovich, , 1991;;Slagstad et al., 2011;Taskjelle et al., 2017;Verin et al., 2022;Zhang et al., 2010) or have applied a relatively simple exponential decay approach for light attenuation (e.g., Lim et al., 2022;Stroeve et al., 2021) that requires knowledge of the surface albedo to use.The study presented here uses a coupled atmosphere-sea ice/snow radiative-transfer model to calculate PAR transmittance for three different sea ice and snow scenarios: Winter, Spring, and Summer.It should be emphasized that the study presented here aims to use radiative-transfer modeling to produce an empirical relationship.The aim is not to demonstrate improvements in radiative-transfer modeling of sea ice requiring comprehensive comparisons with other detailed and specific modeling studies.Different sea ice thicknesses (0.2-3.5 m) and snow depths (0.01-1 m) typical for the Arctic Ocean are considered for each scenario (Kurtz & Harbeck, 2017).The radiativetransfer model results are verified by a comparison with field measurement data from across the Arctic Ocean at different times of the year.Finally, the effect that the solar zenith angle (60-84°) can have on PAR transmittance through sea ice and snow is also considered and compared with diffuse PAR transmittance for the Spring scenario.

Radiative-Transfer Model
The Tropospheric Ultraviolet-Visible Radiation Model (TUV-snow) (Lee-Taylor & Madronich, 2002;Madronich and Flocke, 1998) is a coupled atmosphere-sea ice/snow radiative-transfer model.The model operates an Earth and Space Science 10.1029/2023EA002948 eight-stream DISORT algorithm (Stamnes et al., 1988) that computes PAR from the top of the atmosphere into sea ice and snow layers, producing accurate PAR intensity versus depth profiles, and is exhaustively described in Lee-Taylor and Madronich (2002).The model has been used previously in several studies (e.g., France et al., 2011;Hancke et al., 2018;King et al., 2005;Lamare et al., 2016;Marks and King, 2013;Marks et al., 2017Marks et al., , 2014;;Redmond Roche and King, 2022) focusing on albedo, e-folding depth and PAR in sea ice and snow and is comparable to similar publicly available models (e.g., Flanner et al., 2021).In this study, the TUV-snow model is used to calculate PAR from the ice-ocean interface for all three-sea ice and snow scenarios considered below.Additionally, the PAR at each sea ice and snow level is available at the online repository (Redmond Roche and King, 2023) for all 1,326 modeled runs considered in this study.
The sea ice and snow layers in the TUV-snow model are assumed to be horizontally homogeneous.The optical properties of each layer are controlled by: an asymmetry factor, g, which is 0.89 for layers of snow (Warren and Wiscombe, 1980) and 0.98 for ice layers (Mobley et al., 1998); a wavelength-dependent absorption cross-section, σ abs ; a much larger wavelength-independent scattering cross-section, σ scatt ; and the assigned sea ice or snow density (France et al., 2011;Hancke et al., 2018;King et al., 2005;Lamare et al., 2016;Lee-Taylor & Madronich, 2002;Marks andKing, 2013, 2014;Marks et al., 2017;Redmond Roche and King, 2022).The light scattering is controlled by the sea ice and snow matrix, and the absorption is controlled by a combination of the ice, black carbon, and the under-ice ocean layer.The TUV-snow model is separated into 201 layers comprising a 62-layer atmosphere from 90 km to the snow surface; a 20-layer snowpack with a total thickness ranging from 0.01 to 1 m to the sea ice surface; and a 119-layer sea ice with a total thickness ranging from 0.2 to 3.5 m to the iceocean interface.The total number of layers is selected as it allows for a balance between several thin millimeter scale layers at the ice-ocean, ice-snow, and snow-atmosphere interfaces without adversely affecting computational time.The typical structure and thickness of the individual layers are detailed in Table 2 of both Marks and King (2014) and Redmond Roche and King (2022), which have been emulated to provide consistency with these previous studies.The atmosphere in the TUV-snow model is assumed to be aerosol-free within a typical ozone concentration of 300 DU (Dobson units).The distance between the Earth and Sun is set to 1 au (astronomical unit).An under-ice ocean layer is assigned to the model with characteristic Arctic Ocean wavelength-dependent albedo values (<0.09) that are calculated using the COART model (Jin et al., 2006;Lamare, 2017;Seitz, 2011).

Sea Ice and Snow Optical Properties
The sea ice and snow absorption spectrum are taken from Warren and Brandt (2008).A low-background mass ratio of black carbon typical for the Arctic snow (10 ng g 1 ) (e.g., Jiao et al., 2014;Warren, 2019) is assigned to the snow layers; a typical sea ice black carbon mass ratio (5.5 ng g 1 ) is assigned to the sea ice layers, similar to previous studies (e.g., Redmond Roche and King, 2022).The black carbon mass absorption coefficient is computed using a Mie calculation with a refractive index of 1.0 for the snow medium and a refractive index of 1.33 for the sea ice medium.The wavelength dependent black carbon mass absorption cross-section is for a typical 0.13 μm diameter particle and is taken from Figure 3, panel d, in Dang et al. (2015).The light scattering is controlled by the sea ice and snow matrix, and the light absorption is controlled by a combination of the ice absorption spectra, the mass ratio of black carbon present, and the low albedo of the ocean below the ice.The sea ice density is fixed at 920 kg 1 m 3 (Zhang et al., 2020), and the snow density varies from 300 kg m 3 for the cold polar snow to 400 kg m 3 for the coastal windpack snow and 500 kg m 3 for the melting snow (Gerland et al., 1999;Grenfell & Maykut, 1977;Liston et al., 2020;Perovich, 1990Perovich, , 1996;;Stroeve et al., 2020;Timco and Frederking, 1996).In previous TUV-snow model work examining under-ice PAR (e.g., Hancke et al., 2018;King et al., 2005), sea ice freeboards were not considered.The study presented here advances the model by adopting a freeboard following a method used in previous studies (e.g., Alexandrov et al., 2010;Zhang et al., 2020), where the freeboard can be estimated by assuming the sea ice is in hydrostatic equilibrium: where F i is the sea ice freeboard, H i is the ice thickness, H sn is the snow thickness, ρ I is the sea ice density, ρ sn is the snow density, and ρ w is the seawater density.For the equation, the sea ice density is set to 920 kg 1 m 3 , and the snow density is set to 320 kg 1 m 3 for all three scenarios, to be consistent with Alexandrov et al. (2010) and Zhang et al. (2020).Therefore, the TUV-snow model utilizes a fundamental three-layer model that is comparable Earth and Space Science 10.1029/2023EA002948 to the literature (e.g., Briegleb & Light, 2007;Holland et al., 2012;Light et al., 2008Light et al., , 2015;;Stroeve et al., 2021), and comprises an interior sea ice layer, below a drained sea ice layer, and an overlying snowpack for positive freeboards.A two-layer model comprising an interior sea ice layer and an overlying snowpack is calculated for negative freeboards.
Different values of scattering cross-section are assigned to each of the layers for the three scenarios and are summarized in Table 1.In this study, three scenarios are considered based on the time of year and temperature: Winter (<<0°C), Spring (<0°C), and Summer (≥0°C), and are comparable to Phases 1, 2, and 3 from Verin et al. ( 2022), respectively.The values of the interior and drained sea ice scattering cross-sections are consistent with the vertical scattering profiles presented in Figure 8 by Light et al. (2015) of snow-covered multi-year and first-year sea ice, obtained from ICESCAPE cruises in summer 2010 and 2011 and Barrow Alaska in April 2012 (Light et al., 2008).The scattering cross-sections of the interior sea ice is set to 0.03 m 2 kg 1 , and the drained sea ice is set to 0.3 m 2 kg 1 for all scenarios.Significant variations in baer sea ice scattering cross-sections are possible, particularly between multi-year and first-year sea ice (e.g., France et al., 2011;Grenfell & Maykut, 1977;King et al., 2005;Lamare et al., 2016;Marks andKing, 2013, 2014;Perovich, 1990Perovich, , 1996;;Redmond Roche and King, 2022), therefore, snow-covered sea ice with a drained layer is selected to better represent the Arctic Ocean where baer sea ice is uncommon.The values of scattering cross-sections for the snow are taken from previous radiative-transfer studies (e.g., France et al., 2011;King et al., 2005;Lamare et al., 2016;Marks andKing, 2013, 2014) and are typical for cold polar (20 m 2 kg 1 ; 0.1-0.5 mm), wind-packed (7.5 m 2 kg 1 ; 0.5-2 mm) and melting snow (1.25 m 2 kg 1 ; 2-5 mm).These values of scattering cross-sections have been recently confirmed with measurements by Verin et al. ( 2022), where they are described as Phase 1 "fresh snow," Phase 2 "windslab," and Phase 3 "melt form" snowpacks, respectively, based on the scattering cross-section values derived from specific surface area measurements.

PAR Transmittance Calculations
The downwelling PAR is calculated at each of the 201 layers in the TUV-snow model using the equation: where I(λ) is the downwelling monochromatic irradiance and λ is the wavelength.The "flat plate" PAR is considered in this study owing to it being the typical method instruments measure PAR in the field; however, scalar values of PAR may be up to 1.8-4.2times greater as the photosynthetic tissue in algae acts as scalar collectors of light (Galindo et al., 2017;Kirk, 2011;Pavlov et al., 2017).The PAR transmittance through the snow and sea ice is calculated as the ratio of PAR at the ice-ocean interface, PAR base , relative to the PAR at the snow surface, PAR surf (i.e., PAR base /PAR surf ).The value of PAR at the ice-ocean interface may depend on the solar zenith angle, therefore the study presented here will consider two regimes: diffuse sky conditions and clear sky conditions.Thin snowpacks (5, 10, and 20 cm thick) are considered at increasing solar zenith angles (45, 53, 60, 66, 72, 78, and 84°), spaced equally in cos Θ, and consistent with the possible values of solar zenith angle for the Arctic latitudes (Woolf, 1968).Ultimately, this study will provide algorithms to predict PAR transmittance values at the ice-ocean interface depending on the time of year and temperature and the type and thickness of sea ice and snow.

PAR Transmittance
The TUV-snow model was used to calculate the PAR transmittance through different sea ice and snow types.
Figure 1 illustrates the PAR transmittance at the ice-ocean interface as a two-dimensional surface, displayed in REDMOND ROCHE AND KING three dimensions, for the Winter, Spring, and Summer scenarios under varying thicknesses of sea ice (0.2-3.5 m) and overlying snow (0.01-1 m).The sea ice is assumed to be in hydrostatic equilibrium (e.g., Alexandrov et al., 2010;Zhang et al., 2020), and both positive freeboards comprising a three-layer structure, and negative freeboards comprising a two-layer structure, are considered for completeness and to explore a large range of sea ice and snow physical parameter space.The downwelling irradiance at the snow surface is diffuse owing to a cloud layer assigned to a 1 km altitude in the model.Differences in the downwelling PAR at the snow surface are present between the three scenarios owing to the changing albedo of snow and sea ice that occurs as there is a change in the thickness or optical properties of the underlying snow or sea ice.However, the PAR transmittance value ( PAR base PAR surf ) is relative to the downwelling PAR at the snow surface and is therefore invariant to changes in the amount of downwelling irradiance in diffuse light conditions (King et al., 2005).
The change in the value of PAR transmittance forms a smooth surface as a function of sea ice and snow thickness across the three seasonal scenarios.As the sea ice thickness increases from 0.2 to 3.5 m and the snow thickness remains unchanged, there is a relative decrease in PAR transmittance values to ∼18% of the original value for the Winter, Spring, and Summer scenarios.As the snow thickness increases from 0.01 to 1 m and the sea ice thickness remains unchanged, there is a relative decrease in PAR transmittance values to ∼0.021%, ∼0.076%, and ∼1.5% of the original value for the Winter, Spring, and Summer scenarios, respectively.Therefore, the PAR transmittance surface has a significantly steeper gradient as the snow thickness increases compared to an increase in the sea ice thickness.The PAR transmittance shows signs of complex behavior, and a weak inflection point on the surface appears present as the sea ice reaches 1 m thickness; however, as the sea ice and snow increase in thickness, the PAR transmittance value decreases monotonically.
The study presented here examines a broad range of sea ice and snow conditions resulting in the PAR transmittance value varying over several orders of magnitude.The smallest PAR transmittance values of 4.37E 6, 2.83E 5, and 0.0013 for the Winter, Spring, and Summer scenarios, respectively, occurred when the maximum sea ice (3.5 m) and snow (1 m) thicknesses were examined.Conversely, the largest PAR transmittance values of 0.16, 0.27, and 0.57 for the Winter, Spring, and Summer scenarios, respectively, occurred when the minimum sea ice (0.2 m) and snow (0.01 m) thicknesses were examined.The Winter scenario has the largest scattering cross-section values, resulting in it being the most efficient scenario for extinguishing PAR with depth, whereas the Summer scenario has the smallest scattering cross-section values, resulting in higher PAR transmittance to the ice-ocean interface, as is commonly the case in the Arctic Ocean (e.g., Nicolaus et al., 2013;Oziel et al., 2019).The Arctic Ocean typically has ∼2 m thick sea ice with a characteristic snow cover between October and April of ∼0.2 m (Kwok, 2018;Kwok et al., 2020;Lee et al., 2021;Rostosky et al., 2018); when these conditions are considered for each scenario, there is roughly a two-fold difference in PAR transmittance values between the Winter (0.003) and Spring (0.007) and an order of magnitude difference with the Summer (0.04).Lund-Hansen et al. (2020a, 2020b) recorded maximum snow surface PAR at Station Nord, Greenland (81°N), as ∼120 μmol m 2 s 1 in mid-March (Winter), ∼700 μmol m 2 s 1 in late-April (Spring), and ∼1,200 μmol m 2 s 1 in mid-June (Summer).Utilizing the three scenarios presented in this study, these surface values of PAR would equate to maximum values of PAR at the ice-ocean interface under typical Arctic Ocean conditions of ∼0.36, ∼4.9, and ∼48 μmol m 2 s 1 , for the Winter, Spring, and Summer scenarios, respectively.

Model and Field Data Comparison
The accuracy and similarity of the modeled PAR transmittance values to field measurements were assessed by undertaking a comparison with several studies (Hancke et al., 2018;Leu et al., 2010;Light et al., 2008Light et al., , 2015;;Lund-Hansen et al., 2015;Matthes et al., 2019;Nicolaus et al., 2013;Perovich et al., 1993Perovich et al., , 1998b) ) spanning the Arctic Ocean; from 89.5°North, the Wandel Sea, Beaufort Sea, Svalbard, Arctic Archipelago, and Baffin Bay.The measurements occurred between March and September and have been modeled according to the sea ice and snow conditions designated by the Winter, Spring, and Summer scenarios in Table 1.The Winter, Spring or Summer scenario was selected based on the temperature and time of year of the measurements and are comparable to phases 1, 2, and 3 of Verin et al. ( 2022): Winter << 0°C, Spring < 0°C, and Summer ≥ 0°C.The comparison between the measured and modeled results is shown in Figure 2 (a); all sea ice and snow details are listed in Table A1, and Figure 2 (b) employs a Bland-Altman plot (Bland & Altman, 1986) to compare the difference between the modeled values and the measured values (i.e., the log measured and log modeled average (x-axis) versus log measured log modeled (y-axis)).On average, the modeled values are 144% of the measured values, indicating a slight overprediction from the values of the small field data set.All values from the three scenarios lie within one standard deviation of the differences, equating to up to 17% underestimations and a factor of two overestimations.The asymmetry in the standard deviations is owing to the overall overestimation of the modeled values, which is particularly apparent for points 4, 5, 6, and 7 from the Spring scenario and points 11 and 12 from the Summer scenario.
In general, the majority of measurements are typical of the transitional period between Winter and the onset of melt (Perovich et al., 1998b); three Winter scenario measurements from March and April (Campbell et al., 2016;Leu et al., 2010;Nicolaus et al., 2013), five Spring scenario measurements from May (Hancke et al., 2018;Matthes et al., 2019;Nicolaus et al., 2013;Perovich et al., 1993Perovich et al., , 1998b)), and five Summer scenario measurements from June to September (Light et al., 2008(Light et al., , 2015;;Lund-Hansen et al., 2015;Matthes et al., 2019;Nicolaus et al., 2013) were identified in the literature.Ideally, more measurements of PAR transmittance from throughout the year and particularly from the Winter would be available for a more robust statistical comparison with the  A1.modeled results.Where sea ice freeboards have been reported there is generally a good agreement with our predicted freeboards: Nicolaus et al. ( 2013) report a small positive freeboard for points 1, 5, and 11, and we predict 6.5-13.5 cm positive freeboards, respectively; Matthes et al. (2019) report 4 ± 4 cm and 9 ± 3 cm positive freeboards and we predict 5.8 and 9 cm positive freeboards, respectively; and Lund-Hansen et al. ( 2015) report a 10.2 cm positive freeboard and we also predict a 10.2 cm positive freeboard.There is a poorer correlation between the positive freeboards reported by Light et al. (2008) of 38 cm and Light et al. (2015) of 33 cm, with our predicted values of 21.2 and 15.1 cm, respectively.However, Light et al. (2008) do report that this was an anomalously large freeboard relative to the average height of 14 cm.
Figure 2 confirms that the three-layer structure used for the positive freeboards works well with the three scenarios described (points 1-7, and 9-13).Figure 2 also confirms that the two-layer model used for negative freeboards also works well for point 8, even with an extreme 95 cm snow cover on 1.1 m of sea ice (Hancke et al., 2018).Therefore, the TUV-snow model can accurately predict the PAR transmittance values at the ice-ocean interface, and the three scenarios replicate the different seasons of the year well.
The model was additionally assessed against 415 measured values of PAR from three extended field campaigns in Barrow, Alaska (Nicolaus et al., 2013), 82-89.5°N in the Arctic Ocean (Lund-Hansen et al., 2015), and near Qikiqtarjuaq Island, Baffin Bay (Oziel et al., 2019), and the results are presented in Figure 3.The modeled values generally show a good agreement with the measured values for the three scenarios, with all modeled values being within one order of magnitude of the measured values.When error bars are added for each scenario, based on the variability in light scattering cross sections in snow described in the literature (e.g., France et al., 2011;King et al., 2005;Lamare et al., 2016;Marks andKing, 2013, 2014), the vast majority of the measured values fall within the scattering-based uncertainties.Like Figure 2, the modeled values generally overestimate PAR across the three scenarios.Compared to the measured values from Nicolaus et al. (2013), there is a slight overestimation for all three scenarios; this may be owing to the fact that this was a land-fast sea ice site with large mineral particulate matter loading (up to 24 mg m 2 ) present in the ice.Compared to the measured values from Oziel et al. (2019), there is a slight overestimation in the Spring and Summer scenarios compared to 2015 measurements and for the Summer scenario compared to 2016 measurements.Sea ice algae were recorded developing in the bottom 3 cm of the ice in both years, with higher concentrations and much higher nitrate and nitrite concentrations throughout 2015.The ice algae concentrations peaked in June in both years (>20 mg m 2 ), and this may have led to the slight overestimations seen in both modeled Summer scenarios, whilst the slight overestimation for the Spring scenario compared to 2015 may be owing to increased activity at this time.The difference in the modeled values of PAR relative to the measured values may arise from a lack of increased absorption from the mineral particles or chlorophyll being incorporated into the model.The modeled values of PAR are more consistent compared to Lund-Hansen et al. (2015), and this may be owing to the fact that these were oceanic samples with fewer pollutants and consistently low chl-a (<0.6 mg m 2 ) in the sea ice.

Solar Zenith Angle
PAR transmittance for typical Arctic solar zenith angles (Woolf, 1968) between 45 and 84°was examined with the TUV-snow model for clear sky conditions.The results were compared with the diffuse light conditions used throughout this study and are shown in Figure 4 and detailed in Table A2.A 1 m thick sea ice under the Spring scenario conditions was considered with thin (5 and 10 cm) snow cover, and with the modal Arctic Ocean firstyear ice snow cover (∼20 cm) in the model (Rostosky et al., 2018).As the solar zenith angle increased, the value of PAR at the snow surface significantly decreased from 1,685 μmol m 2 s 1 at 45°to 179 μmol m 2 s 1 at 84°.The amount of light at the ice-ocean interface decreased correspondingly, from 86.8 to 6.2 μmol m 2 s 1 , when a 5 cm snowpack was present.Under diffuse conditions, the value of PAR transmittance at the ice-ocean interface decreased with increasing snow cover, from 0.051 with a 5 cm snowpack to 0.025 with a 10 cm snowpack to 0.01 with a 20 cm snowpack.Figure 4 indicates that as the solar zenith angle increases from 45 to 84°, the points move further away from the line of equality, and there is an increasing difference in the value of PAR relative to the diffuse value.The poorest agreement is at 78-84°for the three snowpacks representing an empirical reduction factor of ∼0.68 from the diffuse value.When diffuse conditions and an 84°solar zenith angle are modeled, this reduction factor is demonstrated with the relative value of PAR transmittance being 0.052 for a 5 cm snowpack, 0.026 for a 10 cm snowpack, and 0.01 for a 20 cm snowpack.The relative values of PAR transmittance under clear sky conditions are 0.035 for a 5 cm snowpack, 0.017 for a 10 cm snowpack, and 0.007 for a 20 cm snowpack.As these are relative values of PAR transmittance, it does not matter what the solar zenith angle is for diffuse conditions.There is effectively no change from the diffuse conditions at a 45°solar zenith angle; however, as the solar zenith angle increases, the empirical reduction factor also increases: ∼0.94 at 53°, ∼0.87 at 60°, ∼0.8 at 66°, ∼0.74 at 72°, ∼0.69 at 78°, and ∼0.68 at 84°.The empirical reduction factors at different solar zenith angles between the Winter, Spring, and Summer scenarios are effectively the same and are shown in Table A2.When trendlines are fit through the modeled clear sky solar zenith angle, the empirical reduction factor relative to the diffuse conditions can be calculated using the following equation: where the coefficients α, β, χ, δ, and ε are presented in Table 2.
Therefore, the algorithm presented in this study is primarily for use in diffuse conditions, however, a worst-case error factor of ∼0.68 should be expected in clear sky conditions at solar zenith angles 78-84°.As solar zenith angles increase (i.e., approaching 90°), the tendency for snow to forward scatter light will appear to enhance light attenuation with depth faster than an exponential, as the light will be scattered out of the snow at glancing angles.
A more detailed description of this effect is given in Warren (1982) and King et al. (2005).

PAR Transmittance Equation
The value of PAR transmittance for the three scenarios considered in this study span several orders of magnitude for the Winter (5 orders), Spring (4 orders), and Summer (2 orders) scenarios.The value of PAR transmittance for a given snow thickness at the ice-ocean interface for each scenario was calculated using the equation: where α, β, and γ are the empirical constants determined from a Levenberg-Marquardt fitting procedure (Moré, 1978), and t is the ice thickness in centimeters.All coefficient values are given at a 1, 2, 5, or 10-cm snowpack resolution for the snow and are presented in  (Bland & Altman, 1986) was repeated utilizing Equation 4and the empirical constants presented in Tables 3-5.On average Equation 4gives values that  A2.  3 were also calculated using Equation 4and the empirical constants.As the measured snow depths from literature (and shown in Table A1) do not precisely correspond with the snowpack resolutions presented in Tables 3-5, these uncertainties may be anticipated, amongst other uncertainties discussed in Section 4.1, when using Equation 4. Therefore, there is no significant difference in using Equation 4as opposed to the TUVsnow model to predict the PAR transmittance value at the ice-ocean interface.

Discussion
The Discussion section is separated into an analysis of how varying thicknesses of snow and sea ice affect PAR transmittance at the ice-ocean interface, how PAR decays as a function of depth in snow and sea ice, and the potential limitations and uncertainties of the study.

Predicting PAR Transmittance
The study presented here examines how varying thicknesses of different types of sea ice and snow representing winter, spring, and summer conditions in the Arctic Ocean affect PAR transmittance to the ice-ocean interface.The change in the optical properties of the sea ice, and particularly the snowpack through snow metamorphosis, throughout the year can result in a substantial increase in PAR transmittance with little or no change in the thickness of the sea ice and snow layers (Hancke et al., 2018).Conversely, a small increase in snow thickness (from 5 to 10 cm) can occur throughout the year (e.g., Verin et al., 2022) and cause a decrease in PAR transmittance values to ∼47% for the Winter scenario, ∼51% for the Spring scenario, and ∼63% for the Summer scenario, relative to the original values.Changes in the thickness of sea ice take longer to occur (e.g., Nicolaus et al., 2013) and are often less important on PAR transmittance than changes in the snow thickness, with a 10 cm increase in sea ice thickness resulting in a relative decrease in PAR transmittance to ∼96% of the original values for the Winter, Spring, and Summer scenarios.However, sea ice does play a significant role in modifying PAR transmittance, and first-year sea ice can be up to three times more transmitting than multi-year sea ice (Nicolaus et al., 2012); this significance is muted here owing to the fact that the values of the overlying snowpacks are more efficient at extinguishing PAR, and the sea ice optical properties are fixed according to Light et al. (2015).Ultimately, the large difference in scattering cross-section between the Winter, Spring, and Summer scenarios results in the PAR transmittance values at the ice-ocean interface having a two-fold difference between the Winter and Spring scenarios and an order of magnitude difference with the Summer scenario when characteristic Arctic Ocean conditions (Kwok, 2018;Kwok et al., 2020;Lee et al., 2021;Rostosky et al., 2018) are considered.
The modeled values of PAR transmittance show a good agreement with the measured values taken from the literature (Hancke et al., 2018;Leu et al., 2010;Light et al., 2008Light et al., , 2015;;Lund-Hansen et al., 2015;Matthes et al., 2019;Nicolaus et al., 2013;Perovich et al., 1993Perovich et al., , 1998b)).All three scenarios give a relatively accurate approximation within one standard deviation of the differences between the modeled and measured values of PAR transmittance at the ice-ocean interface.The three scenarios also showed a good agreement with the extended field campaigns (Lund-Hansen et al., 2015;Nicolaus et al., 2013;Oziel et al., 2019), although in general there was an overestimation in the modeled values of PAR.Whilst the vast majority of the modeled points do fall within the scattering-based uncertainties, the model will likely overestimate values of PAR if used in land-fast ice with high sediment inclusions (>20 mg m 2 ) or in sea ice with high algal concentrations (>20 mg m 2 ).A better understanding of how different algal concentrations affect PAR transmittance would enhance the accuracy and usability of the model on a pan-Arctic scale.It is possible that the PAR measurements were taken in clear sky conditions with high solar zenith angles, which could have decreased the transmittance by up to a factor of ∼0.68.Unfortunately, this information is rarely detailed in the literature.
There is also limited PAR data from the Winter in the literature (e.g., Campbell et al., 2016;Leu et al., 2010;Nicolaus et al., 2013) owing to the difficulty of operating in the Arctic Ocean at this time (e.g., Light et al., 2008Light et al., , 2015;;Matthes et al., 2020;Pavlov et al., 2017), and it is anticipated with the advent of new technologies (e.g., Ardyna et al., 2020;Berge et al., 2016;Biogeochemical-Argo Planning Group, 2016;Boles et al., 2020;Jayne et al., 2017;Krishfield et al., 2006;Laney et al., 2017;Roemmich et al., 2019;Toole et al., 2010) there will be an increase in data for this time of the year and the model parameterization can be enhanced.Fundamentally, there is a burgeoning interest in better understanding the availability of PAR in the Arctic Ocean in the winter and what the implications may be for primary production and calculating carbon budgets, particularly after Hancke et al. ( 2018) identified primary production commencing at a PAR transmittance value of 0.0003 (Vancoppenolle et al., 2013).Therefore, the TUV-snow model, coupled with improving year-round nearreal-time sea ice and snow thickness remote sensing data (e.g., Garnier et al., 2021;Jutila et al., 2022;Kern et al., 2020;Landy et al., 2022;Nicolaus et al., 2021;Tilling et al., 2018), can be utilized as a mechanism to predict PAR transmittance values at the ice-ocean interface if the sea ice and snow optical properties are known.

PAR Transmittance Through Snow and Sea Ice
The TUV-snow model simulates the PAR intensity from the atmosphere, through a three-layer structure comprising a snowpack, a drained sea ice layer, and an internal sea ice layer for positive freeboard conditions.Twolayer negative freeboard conditions comprising a snowpack and internal sea ice layer are also considered for completeness and to explore a large range of sea ice and snow physical parameter space, as negative freeboards do occur in the Arctic, albeit less frequently than in the Southern Ocean (Li et al., 2021).The under-ice albedo in the model uses a wavelength-dependent albedo (<0.09) which is calculated with the COART model for a 100 m deep ocean with a typical Arctic Ocean chlorophyll level (1.44 mg m 3 ) (Jin et al., 2006;Lamare, 2017; PML (UK), 2022).Figure 5 shows that variation of PAR with depth is sensitive to the albedo of the underlying ocean for the layers that are not optically thick (i.e., the drained and interior sea ice layers).These layers would be considered optically thick at >3 e-folding depths, which typically exceeds ∼10 m for sea ice (e.g., France et al., 2011;Lamare et al., 2016;Redmond Roche and King, 2022).The evidence for this effect is manifested by a different PAR-depth profile for different under-ice albedos (0.1-0.7) for the drained and interior sea ice layer, but not the more scattering snow layer, which can be optically thick when it exceeds 10 cm (France et al., 2011;Marks and King, 2013).Many studies (e.g., Grenfell & Maykut, 1977;Lim et al., 2022;Stroeve et al., 2021) use a so-called Beer-Lambert exponential decay to calculate the propagation of PAR through the sea ice and snow layers.Whilst the attenuation of PAR with depth might first look like it is can be described by exponential functions, unfortunately, this would only be valid when all the layers are optically thick (e.g., King & Simpson, 2001).Moreover, an exponential is not necessarily the best fit for PAR as each wavelength  .Modeled values of PAR transmittance for a 1 m thick first-year sea ice with a 5 cm snowpack under the Spring scenario conditions.The wavelength-independent under-ice albedo changes from 0.1 to 0.7 and the effect that this has on PAR transmittance is shown in different colors and compared with a Beer-Lambert exponential decay (dashed and dotted) taken from Stroeve et al. (2021).As the sea ice layers are not optically thick (e.g., France et al., 2011;Lamare et al., 2016;Redmond Roche and King, 2022), the changing under-ice albedo affects the intensity of the PAR transmittance with depth.Therefore, the PAR profiles should be described using Equation 4.
Earth and Space Science 10.1029/2023EA002948 has a slightly different decay constant, and the sum of multiple exponentials is not an exponential.A comparison with a simple Beer-Lambert exponential decay used in Stroeve et al. (2021) is shown in Figure 5 using the values of the extinction coefficient for a two-level sea ice (1 m 1 ) and snow (10 m 1 ) model.The Beer-Lambert exponential decay does not correctly describe the value of PAR at the ice-ocean interface owing to the fact that the sea ice is (a) not optically thick, (b) the drained layer is not considered, and (c) different decay rates for the snow and sea ice are used.Investigating the value of PAR transmittance as a function of depth, through the snow and sea ice column indicated that the sea ice was sensitive to the under-ice albedo and did not obey a simple exponential decay.In an optically thick snowpack, the decay is similar through most of the snowpack and such behavior is limited to the snowpack base (King et al., 2005), whereas in sea ice, the albedo affects the intensity of PAR throughout the sea ice, as indicated by the variance in PAR transmittance in Figure 5. Therefore, it is recommended to use the algorithm provided here to predict the value of PAR transmittance at the ice-ocean interface, as the existing Beer-Lambert exponential decay method may be providing inaccurate measurements as it intrinsically considers the sea ice to be optically thick.

Uncertainties
The optical properties of the sea ice and snow considered here with the TUV-snow model are taken from the literature (Fisher et al., 2005;France et al., 2011;Gerland et al., 1999;Grenfell & Maykut, 1977;King et al., 2005;Warren, 1982;Lamare et al., 2016;Light et al., 2008Light et al., , 2015;;Marks andKing, 2013, 2014;Perovich, 1990;Redmond Roche and King, 2022;Stroeve et al., 2021Stroeve et al., , 1996;;Verin et al., 2022.) and are known to have a natural variance in nature.For the three snowpacks, the scattering cross-section can vary from 15 to 25 m 2 kg 1 for the cold polar snow (0.1-0.5 mm) used in the Winter scenario, 5-10 m 2 kg 1 for the coastal windpack snow (0.5-2 mm) used in the Spring scenario, and 0.5-2 m 2 kg 1 for the melting snow (2-5 mm) used in the Summer scenario (Fisher et al., 2005;France et al., 2011;Gerland et al., 1999;Grenfell & Maykut, 1977;Verin et al., 2022;Warren, 1982).How this variance in snow scattering cross section affects PAR transmittance has additionally been modeled and is available in the online repository (Redmond Roche and King, 2023).The scattering crosssections of the interior (0.03 m 2 kg 1 ) and drained (0.3 m 2 kg 1 ) sea ice layers are fixed throughout this study to be consistent with the measured vertical profiles from the ICESCAPE cruises and Barrow, Alaska, presented in Figure 8 of Light et al. (2015).The scattering cross-sections of baer sea ice are known to vary significantly, particularly between first-year and multi-year sea ice (e.g., Grenfell & Maykut, 1977;Perovich, 1990Perovich, , 1996)), and are uncommon in the Arctic Ocean, so only snow-covered sea ice scenarios are considered here.
There is also a natural variation in sea ice density from 700 to 950 kg m 3 (Gerland et al., 1999;Grenfell & Maykut, 1977;Perovich, 1990Perovich, , 1996;;Timco and Frederking, 1996), however, this is fixed at 920 kg 1 m 3 (Zhang et al., 2020) so that the freeboard can be approximated using Equation 1 (Alexandrov et al., 2010).The predicted freeboard values correlate well with several of the measured values (Lund-Hansen et al., 2015;Matthes et al., 2019;Nicolaus et al., 2013), however, there is also disagreement with the measurements in two other studies where the freeboards are described as being anomalous (Light et al., 2008(Light et al., , 2015)).The simple freeboard allows for a three-layer model to be used which offers an improvement on previous two-layer modeling studies (e.g., Hancke et al., 2018).A broad range of conditions have been modeled (0.2-3.5 m thick ice and 0.01-1 m thick snow) which may offer extreme environmental conditions, such as a 0.2 m thick ice with a 1 m thick snow cover.Whilst these scenarios are highly unusual in nature, they have been modeled here for completeness to explore a large range of sea ice and snow physical parameter space, such as in the occurrence of extreme weather events.In general, the modeled Winter, Spring, and Summer scenarios show a strong correlation with field measurements derived from the literature.Negative freeboards, whilst uncommon in the Arctic, are considered in this study for completeness and to explore a broader range of conditions.The single comparison with a 1.1 m thick sea ice and exceptional 95 cm snow cover (Hancke et al., 2018) resulted in the modeled value of PAR (2.26E-4), giving a very close fit to the measured value of PAR (2.02E-4).However, for this study to be applied in Antarctic conditions where negative freeboards are more common (Li et al., 2021), including an additional layer that correctly captures the absorption and scattering of a flooded snowpack may offer more accurate results.
Diffuse light is mainly considered throughout the study to remove a solar zenith angle dependency and provide a more general and less specific algorithm.It is also representative of overcast Arctic weather.However, as the snowpack is efficient at extinguishing PAR, the effect of a varying solar zenith angle (45-84°) is not particularly significant on the PAR transmittance value at the ice-ocean interface.Nevertheless, an empirical reduction factor of ∼0.94 at 53°, ∼0.87 at 60°, ∼0.8 at 66°, ∼0.74 at 72°, ∼0.69 at 78°, and ∼0.68 at 84°has been established relative to diffuse conditions for the Winter, Spring, and Summer scenario.The under-ice albedo affects PAR transmittance through sea ice and the wavelength-dependent albedo used in this study is for typical Arctic Ocean conditions (Jin et al., 2006;Lamare, 2017).

Conclusions
Sea ice and particularly snow, strongly attenuates visible light and reduces the amount of PAR transmitted to the ice-ocean interface.The value of PAR transmittance at the ice-ocean interface is significantly affected by the seasonal changes in the optical properties of sea ice and snow.When characteristic Arctic Ocean conditions (2 m thick sea ice and ∼0.2 m snowpack) are considered there is roughly a two-fold difference in PAR transmittance value at the ice-ocean between the Winter (0.003) and Spring (0.007) scenarios and an order of magnitude difference with the Summer scenario (0.04).Snow plays a significant role in decreasing PAR transmittance, with an increase in snow thickness from 5 to 10 cm resulting in a decrease in the value of PAR transmittance to ∼47% for the Winter scenario, ∼51% for the Spring scenario, and ∼63% for the Summer scenario, relative to the original values.Sea ice is less significant, with a 10 cm increase in thickness resulting in a relative decrease in the value of PAR transmittance to ∼96% of the original values for the Winter, Spring, and Summer scenarios.The effect of the solar zenith angle (45-84°) in clear sky conditions is also considered, and empirical reduction factors relative to diffuse conditions are established for a snowpack of at least 5 cm thickness.The drained and interior sea ice layers are not optically thick or quasi-infinite, and the PAR depth profile is affected by the albedo of the underlying ocean (<0.1), demonstrating that a simple Beer-Lambert exponential decay may give inaccurate values of PAR transmittance at the ice-ocean interface.The modeled values correlate within one standard deviation of measured values from across the Arctic Ocean throughout the year, taken from the literature; therefore, the different sea ice and snow types comprising each scenario are appropriate for the sea ice and snow conditions explored in the study presented here.Fundamentally, this study offers a simple empirical technique to predict the PAR transmittance value at the ice-ocean interface for the Winter, Spring, and Summer scenarios, so long as the sea ice and snow thickness and downwelling PAR at the surface are known.

Appendix A
See Table A1 and Table A2.

Figure 1 .
Figure 1.PAR transmittance values at the ice-ocean interface for the Winter, Spring, and Summer scenarios at varying thicknesses of sea ice (0.2-3.5 m) and snow (0.01-1 m).The blue circles represent positive sea ice freeboard conditions, and the red squares represent negative sea ice freeboard conditions.There is roughly a two-fold difference in PAR transmittance values between the Winter and Spring scenarios and an order of magnitude difference with the Summer scenario.

Figure 2 .
Figure 2. Panel (a) shows a comparison of the measured values of PAR transmittance from the literature as red squares with modeled values as blue squares; the Winter, Spring, and Summer scenarios comprise columns a, b, and c, respectively.Panel (b) plots the difference between the modeled and measured PAR transmittance values (i.e., the log measured and log modeled average (x-axis) versus log measured log modeled (y-axis) log measured log modeled); all values are within one standard deviation of the difference (marked as ± STDV).The modeled values average 144% of the measured values, and the standard deviations equate to up to 17% underestimations and a factor of two overestimations.The measurement data for points 1, 5, and 11 were taken from Figure 4 in Nicolaus et al. (2013); point 3 was taken from the "raw PAR measurements" on the 24th April in Figure 4, panels b and d in Campbell et al. (2016); point 4 was taken from data fitted from Figure 2 in Perovich et al. (1993); point 6 was taken from data fitted from Figure 6 in Perovich et al. (1998b); point 10 was taken from data fitted from Figure 3, panel d (21st July), in Light et al. (2008); point 12 was taken from data fitted from Figure 9, panel b (baer ice, 19th July) inLight et al. (2015).Points 2(Leu et al., 2010), 7(Matthes et al., 2019), 8(Hancke et al., 2018), 9(Matthes et al., 2019), and 13 (Lund-Hansen et al., 2015)  are all clearly described in the literature as integrated PAR (400-700 nm) values.All information on the measured and modeled data can be found in TableA1.

Figure 3 .
Figure 3.A comparison between modeled and measured values of PAR transmittance, with Winter scenarios presented as circles, Spring scenarios as squares, and Summer scenarios as triangles.The measurements from Barrow, Alaska (gray points) were taken in March, May, and June 2010 (Nicolaus et al., 2013) and were correspondingly modeled as the Winter, Spring, and Summer scenarios.The measurements from the Arctic Ocean (green points) were taken in August and September 2012 (LOMROG III; Lund-Hansen et al., 2015) and were modeled as the Summer scenario.The measurements from near Qikiqtarjuaq Island (red points for 2015 and blue points for 2016) were taken in April and June 2015 and 2016 (GreenEDGE; Oziel et al., 2019) and were modeled in red and blue, respectively, as the Winter, Spring or Summer scenarios, corresponding with the air temperature being <<0°C, <0°C, or ≥0°C, respectively.The error bars are based on the variation in light scattering cross sections for snow described in the literature, and the upper and lower modeled values of PAR are available in the online repository (Redmond Roche and King, 2023).

Figure 4 .
Figure 4. Modeled PAR transmittance through a 1 m thick sea ice with 5 (red), 10 (green) and 20 cm (blue) snowpacks under the Spring scenario conditions on a log-log plot.Values of PAR transmittance under different solar zenith angles (45-84°) and diffuse conditions are displayed by different symbols.The line of equality represents perfect agreement between the diffuse values and the different solar zenith angles.The poorest agreement is at 78-84°for the three snowpacks, as these are the furthest from the line of equality and represent an empirical reduction factor of ∼0.68 from the diffuse value.Relative PAR transmittance information for all three scenarios is detailed in TableA2.

Figure 5
Figure5.Modeled values of PAR transmittance for a 1 m thick first-year sea ice with a 5 cm snowpack under the Spring scenario conditions.The wavelength-independent under-ice albedo changes from 0.1 to 0.7 and the effect that this has on PAR transmittance is shown in different colors and compared with a Beer-Lambert exponential decay (dashed and dotted) taken fromStroeve et al. (2021).As the sea ice layers are not optically thick (e.g.,France et al., 2011;Lamare et al., 2016;Redmond Roche and King, 2022), the changing under-ice albedo affects the intensity of the PAR transmittance with depth.Therefore, the PAR profiles should be described using Equation4.

Table 1
Scattering Cross-Sections, σ scatt , of Each Sea Ice and Snow Layer for the Winter, Spring, and Summer Scenarios

Table 3
(Winter scenario), Table4(Spring scenario), and Table 5 (Summer scenario); the user inputs their given sea ice thickness as the t value in units of centimeters.The PAR transmittance value calculated is the amount of PAR at the ice-ocean interface relative to the amount of PAR at the snow surface (i.e., PAR base /PAR surf ).The Bland-Altman comparison with the measured values shown in Figure2b

Table 2
Coefficient Values for Equation 3 141% of the measured values, indicating a slight overestimation from the measured values.All of the values lie within one standard deviation of the differences, equating to up to 20% underestimations and up to 58% overestimations.The modeled PAR transmittance values compared with measured values in Figure are

Table 3
Coefficient Values for Equation 4 to Calculate the PAR Transmittance Value at the Ice-Ocean Interface for the Winter Scenario

Table 5
Coefficient Values for Equation 4 to Calculate the PAR Transmittance Value at the Ice-Ocean Interface for the Summer Scenario

Table A2
PAR Transmittance Values at Different Sea Ice (1 m) and Snow Thicknesses (5, 10, and 20 cm)for Diffusein the Winter, Spring and Summer Scenarios Note.The empirical reduction factors relative to diffuse conditions for all solar zenith angles and scenarios are presented.PhD study undertaken as part of the Centre for Doctoral Training (CDT) in Geoscience and the Low Carbon Energy Transition.It is sponsored by Royal Holloway, University of London, via their GeoNetZero CDT Studentship, whose support is gratefully acknowledged.The authors would like to thank the technical staff in the Department of Earth Sciences at Royal Holloway, University of London, who supported this computational project.