Analysis of euphotic depth in snow with SNICAR transfer scheme

Solar radiation in the visible spectrum can penetrate through snowpack to a considerable depth, which is named as the euphotic depth in snow. If the snow depth is no greater than the euphotic depth, the surface albedo is greatly affected by the underlying surface. This study defines the euphotic depth as the depth where the residual solar radiation in snow began to be less than 1.0 W m−2 and provides a convenient approach to estimate it. A two‐stream, multilayer radiation penetration model (SNow, ICe, and Aerosol Radiation) was applied to predict the vertical profiles of solar radiation based on regular measurements of snow pits and downward solar radiation at Col de Porte (CDP), France from 1993 to 2011. The euphotic depth in snow at CDP exhibits clear seasonal variations, with median values in winter months of 6.8, 8.8, and 10.5 cm, and those in spring remaining nearly on the same level (24.4 and 24.2 cm). The maximum euphotic depth at CDP reached as deep as 78 cm. Further analyses demonstrated that the euphotic depth in snow is related to not only the external initial solar irradiance but also the interior snow extinction coefficient. Stronger illumination and smaller snow extinction coefficients may correspond to greater euphotic depths in snowpack.


Introduction
Snow is translucent such that solar radiation in the visible spectrum can penetrate through the snowpack to a considerable depth (e.g. Giddings and Lachapelle, 1961;Baker et al., 1991;Flanner and Zender, 2005). The underlying surface has a great influence on surface albedo if the incident solar radiation can penetrate to the bottom of the snowpack (e.g. O'Neill and Gray, 1972). Wiscombe and Warren (1980) indicated that the surface albedo of a shallow snowpack is substantially different from that of deep snowpack, especially in the visible spectrum. Hence, special attention should be given to shallow snowpack cases in snow albedo and snow model studies. However, how to distinguish shallow snowpack from deep snowpack has been controversial.
Many general circulation models (GCMs) parameterize grid-average albedos ( ) as weighting functions (f s (h s )) of snow depth (h s ) (e.g. Hansen et al., 1983;Verseghy, 1991;Dickinson et al., 1993;Douville et al., 1995;Cox et al., 1999), where s is the albedo of deep, homogeneous snow, and 0 is the albedo of the snow-free surface. Equation (1) illustrates that albedo increases with the enhancement of snow depth in shallow snowpack, and that albedo tends to be independent of the underlying surface after the snowpack reaches a certain depth (h 0 , the threshold of deep snowpack). Various empirical expressions for f s (h s ) in different GCMs contribute to various thresholds (Armstrong and Brun, 2008, figure 4.13). Gray and Landine (1987) set 25 cm as a reference depth to discriminate between deep or shallow snow cover for the respective snow albedo decay rates. Baker et al. (1991) indicated that 5, 7.5, and 15 cm of snow over bare soil, sod, and alfalfa, respectively, is required to mask the underlying surface effectively. That is to say, there is no fixed standard to differentiate clearly between deep and shallow snowpack. In principle, the key is to compare the snow depth (h s ) with the euphotic depth (z eu ), which is the path depth to which incident solar radiation is transmitted in the snowpack. If h s > z eu , then the snowpack is classified as deep snowpack. h s is easily measured, and z eu can be calculated with knowledge of light transmission in the snowpack. Bohren and Barkstrom (1974) stated that incident solar radiation attenuates exponentially with increasing depth in the snowpack. The transmission can be expressed as follows: where k represents the 'asymptotic flux-extinction coefficient' at a particular wavelength , and F ↓ (z, ) denotes the spectrum solar radiation at level z (Brandt and Warren, 1993). Several studies have investigated the solar radiation transmission in snowpack. Järvinen and Leppäranta (2013) measured the solar radiation above and inside the snowpack, and found that the mean spectral diffuse extinction coefficient varied between 0.04 and 0.31 cm −1 (10-20 cm layer). They further inferred that the euphotic depth (z eu ) was approximately 50 cm at which the broadband irradiance was 1% of the downward irradiance at the surface. Baker et al. (1991) indicated that the extinction coefficient ranges from 0.07 to 1.5 cm −1 .
These investigations were based on direct measurements of solar radiation profiles. However, solar radiation transmission measurements through snow are notoriously difficult, since light levels beneath the snow are quite weak. Furthermore, spectral dependence complicates the measurements, not to mention the problem of inserting the sensor without disturbing the snowpack (Perovich, 2007). Model simulations represent an alternative approach and can provide a preliminary estimation of the light transmission profile.
In light of the above, this study aimed to investigate the euphotic depth in snowpack and to answer the question as to how thick a snowpack should be before it can be regarded as a deep or shallow snowpack. Specifically, a method of detecting the euphotic depth is proposed. The method applies a two-stream, multilayer radiation penetration model [SNow, ICe, and Aerosol Radiation (SNICAR)] to estimate the vertical distribution of solar radiation based only on regular measurements of snow pits and the initial downward solar radiation. Furthermore, seasonal variability in the euphotic depth and possible key factors are discussed.

Definition of the euphotic depth in snow
This study defines the euphotic depth in snow as the depth at which the residual solar radiation begins to be <1.0 W m −2 . If the residual solar radiation at the bottom of the snowpack is ≥1.0 W m −2 , which means the light has penetrated through the snowpack completely, then it is a shallow snowpack. In this case, the euphotic depth is equal to the snow depth. Reason for this definition is given in Section 3.3 with detailed examples.

SNow, ICe, and Aerosol Radiation
The vertical profiles of solar radiation in snowpack are predicted by the SNICAR model Zender, 2005, 2006), which employs the theory proposed by Wiscombe and Warren (1980) and a two-stream radiative transfer solution from Toon et al. (1989). Because the attenuation of solar radiation in snow varies strongly across the solar spectrum, the solar fluxes are computed in five spectral bands: one in the visible (0.3-0.7 m) and four in the near-infrared spectra (i.e. 0.7-1.0, 1.0-1.2, 1.2-1.5, and 1.5-5.0 m, respectively). In SNICAR, snow is considered to be made up of a collection of ice spheres, with a lognormal distribution of the optical effective radius over 50-1500 m. Ice Mie scattering parameters, which depend on grain radius and spectral bands, are derived offline for computational efficiency (Oleson et al., 2010). A number of validations and evaluations of SNICAR have been conducted and have demonstrated that SNICAR is solid and robust in calculating snow albedo and radiation transmission (e.g. Hadley and Kirchstetter, 2012;Meinander et al., 2013;Zhong et al., 2017). In addition, SNICAR has been incorporated into the Community Land Model (CLM) version 4 (Oleson et al., 2010).
The required input variables mainly include: incident radiation, solar zenith angle, snowpack depth and density, and snow grain optical effective radius. Most of these variables can be obtained from measurements directly, as described in Section 2.3. The snow grain optical effective radius can be calculated from the snow specific surface area (SSA), as described in Section 2.4.

Data from Col de Porte, France
The data used in this study are derived from the snow and meteorological research station Col de Porte (CDP), France (43.3 ∘ N, 5.77 ∘ E, ∼1325 m altitude). This site has been used for over 50 years for development and evaluation of snow models (e.g. Brun et al., 1989;Chen et al., 2014). The measurements in this study were collected in 1993-2011, including the following: (1) manual snow pit observations containing vertical profiles of snow density and snow grain type. Each snow pit (a total of 302 pits) observation was carried out approximately weekly throughout the duration time of snow cover, covering snow accumulation, and melting season; (2) corresponding surface incident direct and diffuse shortwave radiation.
More details can be found in Morin et al. (2012) and are available on the following website: http://epic .awi.de/30060/. Figures 1(a) and (b) show an example of the profiles of snow density and snow grain type, respectively, at CDP in winter 2010.

Optical effective radius
The other required inputs for SNICAR are the snow grain optical effective radius (r opt ). This variable is the surface area-weighted mean radius of an ensemble of ice spheres and is directly related to the SSA (Flanner and Zender, 2006), as follows: where ice denotes the density of ice (917 kg m −3 ). Domine et al. (2008) proposed a parameterization scheme of the snow SSA from an examination of snow crystal types and a measurement of snow density. Morin et al. (2013) verified this parameterization scheme and found that it performed rather satisfactorily, both qualitatively and quantitatively. Hence, following the scheme proposed by Domine et al. (2008), this study estimated the value of the snow SSA from snow type and snow density, which are included in the snow stratigraphy  Table 1. Box in white indicates missing data. measurements in Section 2.2. Details of the snow SSA parameterization scheme are listed in Table 1. Then, the effective optical radius can be calculated by Equation (3).
In this way, Figure 1(c) shows the corresponding snow grain effective radius. Vertically, snow at upper layers tends to have a lower density and smaller effective grain radius than that at lower layers. Snow compaction accounts for these phenomena. Horizontally, younger snow shows the similar characteristics, by contrast with the older snow. Snow metamorphism contributes to the differences between young and old snow.

Extinction coefficient
In solar radiation observations, the transmission of the incident radiation can be simplified as follows (Brandt and Warren, 1993): where F ↓ (h) denotes the net solar radiation at depth of h, F ↓ (0) represents the net solar radiation at the sur- , it denotes the 'local bulk-extinction coefficient' for the total solar radiation at level z in the snowpack.
To concisely reflect the snow optical properties of a whole snow layer, a mean extinction coefficient (k) of the whole snow layer (from the surface to depth h) is better than a local extinction coefficient (k (z)). The mean extinction coefficient (k) is defined as follows: Combining Equation (4) with Equation (5), it can be expressed as, ) .

Euphotic depth at CDP and its seasonal variability
Euphotic depths in the snowpack at CDP in 1993-2011 were estimated as described in Section 2.1. In addition, the required vertical distributions of solar radiation are predicted by SNICAR, using the incident radiation, solar zenith angle, snow depth and density, and optical effective radius as inputs. Figure 2(a) presents the euphotic depth of 302 snow pits at CDP in 1993-2011. The red points (25 in total) mean that the light has already penetrated through the snowpack, while the blue points (277 in total) are the normal cases. The euphotic depths at CDP exhibit a concentrated distribution in winter days, but show a more varying distribution in spring days. Figure 2(b) presents a boxplot for the euphotic depth grouped by month. The ranges of distributions of monthly z eu become wider gradually, and the extremes become greater. The monthly maximum values of z eu in December, January, February, March, and April are 32.4, 46.0, 57.0, 67.0, and 78.0 cm, respectively. Baker et al. (1991) indicated that shortwave radiation can penetrate to a depth of 100 cm. The monthly median values of z eu in winter months (December, January, and February) are very close to each other (6.8, 8.8, and 10.5 cm, Table 1. Snow SSA parameterizations based on snow type and snow density, following Domine et al. (2007) and Morin et al. (2013).

Surface net solar radiation
The pronounced differences in the euphotic depths between winter and spring may be partly attributed to the seasonal evolution of the surface net solar radiation. Figure 2(c) presents the scatterplot of the relevant surface net solar radiation (Q net ), and Figure 2(d) shows the corresponding boxplot. Apparently, z eu exhibits a similar seasonal trend with Q net . Furthermore, z eu is proportional to Q net , as confirmed by a scatter plot of z eu versus Q net in Figure 2(e). The correlation coefficient between z eu and Q net reaches 0.874, which is significant at the 95% confidence level according to the two-tailed Student's t test. This finding implies that surface net solar radiation explains approximately 76.4% of the variation in euphotic depth. In general, with stronger illumination, more energy penetrates through the snowpack, and a deeper euphotic depth is required to attenuate the energy. However, there are exceptional cases since incident solar radiation is not the only contributor. Some interesting cases [marked as A, B, C, D, and E in Figure 2 To identify other factors that may influence the euphotic depth, these cases were collected in a randomized and controlled manner. B, C, and D exhibit almost the same degree of initial illumination, but their z eu values are quite different. With regard to D and E, the initial Q net of E is nearly three times as much as that of D. However, the z eu of E is nearly equal to that of D. The snow optical properties account for these exceptional cases. Figure 3(b) presents the mean extinction coefficient (k) over increasing depth (Equation (6)).k decreases with increasing snow depth due to the change in the spectral composition. On the threshold of euphotic depth, F ↓ (z eu ) = 1.0 W m −2 , hence, Equation (6) can be expressed as:

Mean extinction coefficient
That is to say, there is an inverse proportional relationship between the euphotic depth (z eu ) and the mean extinction coefficient of the whole euphotic zone (k | | |z eu ). When the light is transmitted to the same depth level, thek of B (orange line) is greater than those of C (green line) and D (blue line). Hence, under the same level of initial illumination (F ↓ (0)), the z eu of B is less than those of C and D. Similarly, even though the F ↓ (0) of E is nearly three times as much as that of D, thek of E is greater than that of D at the same depth level, resulting in nearly the same value of z eu . The euphotic depth  is not only related to the initial solar irradiance but is also attributed to its own snow extinction coefficient.
The detailed values are listed in Table 2. The mean extinction coefficient here is derived from the profile of solar radiation in snow, which follows the exponential law of attenuation. It should be noted that the extinction coefficient is actually a proxy for the effective radius of single particle and the layer mass (snow density) of multiple scattering according to the cause-and-effect reasoning (Wiscombe and Warren, 1980). Both the snow grain effective radius and snow density have a seasonal evolution, as shown in Figure 1, which intensifies the seasonal variability of the euphotic depth from an interior view.

Explanation for the definition of euphotic depth
Figure 3(c) illustrates the residual percentage over snowpack depth, which is an auxiliary explanation for the definition of z eu . The residual percentage is the proportion of the remaining downward solar irradiance to the initial solar irradiance beneath the surface, as follows: In aquatic ecology, the euphotic depth is defined as the depth at which photosynthetically available radiation is 1% of value beneath the surface (Lee et al., 2007). For the euphotic depth in snow, it would be nice if we could learn from this definition. However, among the measurements of the beneath surface solar radiation at the CDP station in 1993-2011, the maximum value was 233.0 W m −2 , and the 25th quantile value was 9.2 W m −2 , and if following the definition from acoustic ecology, the 1% levels were 2.33 and 0.092 W m −2 , respectively, then these values would be inadequate. As they are too lenient for strong radiation cases and too stringent for weak radiation cases. Therefore, we suggest using the definition of euphotic depth in snow as stated in Section 3.1.
On the threshold of euphotic depth, F ↓ (z eu ) = 1.0 W m −2 , and in this way, the remaining percentages for snow pits A, B, C, D, and E are 4.0, 1.8, 1.8, 1.9, and 0.66%, respectively. By cross-referencing these values, the euphotic depth of these pits can also be obtained. Strong initial illumination corresponds to a small residual percentage and vice versa. If the euphotic depth is defined as the depth at which the residual solar radiation is a fixed percentage of the initial irradiation, such as 1.0%, then the results would be strongly biased.

Conclusions
Solar radiation in the visible bands penetrates through snowpack to a considerable depth, which is called as euphotic depth in this study. The depth is defined here as the depth at which the residual solar radiation began to be <1.0 W m −2 . The vertical profiles of solar radiation transmitted in the snowpack can be predicted by SNICAR. Most of the variables required by SNICAR are available from regular measurements except for the snow grain effective radius. However, this variable can be estimated from the snow SSA, which can be parameterized using snow type and snow density.
Pronounced differences in the medians, means, and maximums between winter and spring indicated that there was a clear seasonal variation in euphotic depth at CDP. This result may partly owe to the seasonal evolution of net solar radiation. The euphotic depth in snow is proportional to the surface net solar radiation, with the correlation coefficient being as high as 0.874. Exceptional cases were further analyzed and demonstrated that the snow extinction coefficient was the interior contributor influencing the euphotic depth. Stronger illumination and smaller extinction coefficients lead to greater euphotic depths. Chen et al. (2014) inspected the euphotic depths at the same site (CDP, France), using a statistical method (LOcally WEighted Scatterplot Smoothing, LOWESS). The results were 21 and 33 cm in the dry and wet seasons, respectively, which also indicated a seasonal variation in the euphotic depth. Our results are in accordance with theirs but are more detailed and precise. More measurements and investigations are still needed to validate our method.
This study defined the euphotic depth in snowpack and provided a convenient approach to estimating it. This approach may improve our understanding of the differences in the albedo between deep and shallow snowpack in GCMs. In particular, it is important for investigations in regions with relatively shallow snowpack and strong solar radiation, where the euphotic depth is more likely to be great and the albedo is easily affected by the underlying surface. This study left aerosol effects and vegetation canopy out of consideration. As next step, more efforts are needed to understand the relationship between snow depth and snow fraction and the effects of light absorbing aerosols on solar radiation transmission in the snowpack.