## 3 Results
### In-Water Irradiance
#### 3.2.2 K-Functions in the UV  
In-water attenuation of \(E_{d}\) and \(E_{o}\) with depth is characterized by the diffuse attenuation coefficient (K-function):  
\[K(z)=-d\ln(E)/dz, \tag{5}\]  
where \(E\) is either \(E_{d}\) or \(E_{o}\), \(z\) is the depth and the wavelength dependence is omitted for briefness. Then, the propagation of \(E\) to depth is described by the exponential function:  
\[E(z)=E(0)\ \exp(-\int_{0}^{z}K(z)dz), \tag{6}\]  
where \(E(0)\) is the irradiance value at the surface. A penetration depth, \(z_{p}\), can be defined as a depth at which the quantity \(E\) decreases to a pre-defined fraction of its surface value \(\varepsilon=E(z_{p})/E(0)\). For instance, in the case that \(K\) equals a constant and the commonly used value \(\varepsilon=0.1\) for defining the euphotic zone depth, \(z_{10\%}\)[50], the penetration depth is \(z_{p}=-\ln(0.1)/K\). Figure 4: DNA-weighted planar (solid lines) and scalar (dashed lines) irradiances at two depths: 2 m (red lines) and 20 m (blue lines). Figure 3: Spectral planar (solid lines) and scalar (dashed lines) irradiances. **Left**: Absolute values of the irradiances at two depths: 2 m (red lines) and 20 m (blue lines). **Right**: Planar and scalar irradiances normalized over their values at 400 nm. The black line shows the DNA damage action spectrum. The diffuse attenuation coefficient at a given depth depends on the angular distribution of radiance and IOPs, e.g., [51]. Here, we compare the diffuse attenuation coefficients for downwelling planar irradiance, \(K_{d}\), and scalar irradiance, \(K_{o}\), as a function of different input variables. Figure 5 shows the dependence of \(K_{d}\) and \(K_{o}\) on depth, wavelength, chlorophyll concentration and SZA. The data in Figure 5 show that the dependences of the diffuse attenuation coefficient of scalar and planar irradiance on depth were quite close to each other. Given that the average cosine of the downwelling radiance is defined as \(\mu(z)=E_{d}(z)/E_{0}(z)\), this means a weak dependence of \(\mu\) in UV on depth in the upper layer of the ocean due to the following relationship \(d\mu/dz=\mu(K_{d}-K_{o})\). The weak dependence of \(\mu\) on depth results in a weak dependence of the \(K\) functions on depth (see Figure 5b) because we assume a vertically uniform distribution of IOPs. This assumption is well justified for the well-mixed upper layer of the ocean. The weak dependence of the \(K\) functions on depth in the upper layer allows for the use of a single value of the \(K\) functions averaged over the upper layer for predicting the penetration depths in the UV. This is particularly important for satellite estimates of the diffuse attenuation coefficients [16] because the satellite estimates are based on the measurement of water-leaving radiance, which is primarily formed in the upper layer of the ocean. The wavelength dependence of the \(K\) function shown in Figure 5a was determined by the IOP wavelength dependence. The dominant role in this dependence is played by the spectral dependence of the total absorption coefficient, which significantly increases with decreasing wavelength. Similarly, the dependence of the \(K\) functions on chlorophyll (Figure 5c) is explained by the increase in the total absorption coefficient with the increasing chlorophyll concentration (see Appendix A). Figure 5d shows that the \(K\) functions increase slightly with increasing SZA.