# Estimating Underwater Light Regime under Spatially Heterogeneous Sea Ice in the Arctic
## 3 Results
### 3D Monte Carlo Numerical Simulations  
Figure 6 shows cross-sections of the simulated downward irradiance and upward radiance. A key difference for the upcoming discussion is that the simulated upward radiance was more homogeneous compared to the simulated downward irradiance. Figure 7 shows the reference irradiance, \(E_{d}(z)\), and reference radiance, \(L_{u}(z)\), profiles. The highest irradiance and radiance occurred when the melt pond occupied 25% of the sampling area, allowing for more light to propagate in the water column. None of the \(E_{d}(z)\) and \(L_{u}(z)\) reference profiles showed subsurface light maxima. Figure 8 shows the 50 simulated local downward irradiance and upward radiance profiles evenly spaced by 1 m in the horizontal distance from the melt pond centre. Local downward irradiance profiles under the melt pond (0-5 m) showed a rapid decrease with increasing depth described by a monotonically exponential or quasi-exponential decrease. Local simulated downward irradiance profiles just outside the melt pond (5-10 m from the melt pond centre) were characterized with subsurface light maxima occurring at a depth of between approximately 5 and 10 m. Further away from the melt pond centre, downward  
Figure 5: Scatter plots showing the relationships between the measured \(K_{d}\) and \(K_{Lu}\) in the spectral range between 400 and 580 nm at different depths (numbers in gray boxes). Red lines represent the regression lines of the fitted linear models. Regression equations and determination coefficients (\(R^{2}\)) are also provided in each plot. Dashed lines are the 1:1 lines. irradiance profiles followed a monotonically exponential or quasi-exponential decrease. None of the simulated upward radiance profiles presented subsurface light maxima (Figure 8). From local simulated irradiance and radiance profiles (Figure 8), \(K_{d}\) and \(K_{Lu}\) were calculated by fitting Equation (1) between the depths of 0 m and 25 m. Results are presented in Figure 9. \(K_{d}\) varied between 0.065 and 0.157 m\({}^{-1}\) and \(K_{Lu}\) between 0.079 and 0.116 m\({}^{-1}\). These \(K_{d}\) and \(K_{Lu}\) were used to propagate light downward from surface reference values \(E_{d}(0^{-})\). Figure 10 shows the profiles resulting from this calculation. A greater dispersion around the reference profiles (thick black lines in Figure 10) occurred when using \(K_{d}\) compared to the profiles generated with similarly derived \(K_{Lu}\) values. The relative differences between the depth-integrated values of each local profile (coloured lines in Figure 10) and the depth-integrated values of the reference profiles (thick black lines in Figure 10) were used to quantify the error of using either \(K_{d}\) or \(K_{Lu}\) as a proxy to predict downward irradiance in the water column (Figure 11). Below the melt pond, \(K_{d}\) overestimated the total downward irradiance by up to 40% when the melt pond occupied 1% of the surface area. In this region, the local \(K\) coefficients are inflated. In the transition region, at a horizontal distance of 5 and 10 m from the centre of the melt pond, where subsurface maxima are observed, \(K_{d}\) underestimated the downward irradiance by up to 35% when the melt pond occupied 25% of the surface area. Further away from the edge of the melt pond, the errors saturated to a maximum of \(-\)25%. The same behaviour is observed for \(K_{Lu}\) but with about two times less amplitude. The mean relative errors were lower by approximately a factor of two when using \(K_{Lu}\) (\(-\)7%) compared to \(K_{d}\) (\(-\)12%). Also, the prediction errors stabilized at a shorter horizontal distance from the centre of the melt pond when using \(K_{Lu}\) (\(\approx\)10 m) compared with using \(K_{d}\) (\(\approx\)20 m). Figure 6: Cross-sections of simulated downward irradiance and upward radiance fields under a melt pond with a 5 m radius. The logarithm of the normalized number of photons has been used to create the scale for visualization.