Modelling of snow interception on a Japanese cedar canopy based on weighing tree experiment in a warm winter region

Snow interception by tree canopy affects the water and energy budgets. Several snow interception models have been proposed to estimate the temporal change of intercepted snow on the tree canopy from meteorological data. However, most models are based on a few observational results at limited sites or some conceptual understandings and assumptions; they still need more observational evidence and more detailed descriptions in the parameterization model for snow accumulation and snow unloading response to various meteorological conditions. A weighing tree experiment that measures the intercepted snow on the cut Japanese cedar trees was conducted in Tokamachi, Japan, to investigate the relationship between meteorological conditions and the change of intercepted snow. The results showed that the interception efficiency, which shows the ratio of intercepted snow to the total precipitation, increased with increasing air temperature in the range from −4.2 to 0°C due to increased adhesion and cohesion of snow. The maximum interception capacity was not present in observations such as an air temperature range. The unloading of intercepted snow from the canopy induced by snowmelt was related to air temperature, solar radiation, and intercepted snow amount. The snow unloading caused by wind‐blowing intercepted snow off the tree, started to develop at wind speeds exceeding 0.8 m s−1. The wind‐induced snow unloading rate coefficient increased as the wind speed increased. This study proposed a new parameterization model of snow interception based on these observations. It is the only snow interception model to use solar radiation in the snow unloading parameterization. It found that simulated and observed temporal changes of intercepted snow correlated well. Nevertheless, there were significant errors between the simulated and observed intercepted snow on several snowfall events because the model could not accurately assess the occurrence of snow unloading due to snowmelt.


| INTRODUCTION
Snowfall in the forest is partially intercepted by the canopy. Snowfall interception can reduce the amount of subcanopy snow by increasing sublimation of intercepted snow (Lundberg & Halldin, 1994;Nakai et al., 1994;Schmidt, 1991) and meltwater drip from intercepted snow (Storck et al., 2002). The intercepted snow also increases albedo at the land surface (Betts & Ball, 1997). Snowfall interception affects the water and energy budgets in areas where snowfall occurs.
To demonstrate and model the behaviour of snow on the tree canopy, the weighing tree experiments that measure intercepted snow by measuring the change in weight of the entire cut tree have been conducted at several sites. Satterlund and Haupt (1967) showed that intercepted snow increases as the cumulative snowfall precipitation grow, and it asymptotically approaches the maximum values of intercepted snow, known as 'maximum interception capacity' that can be stored on the canopy. Based on the concept of maximum interception capacity, the relationship between intercepted snow and cumulative snowfall precipitation was modelled using a sigmoid curve (Moeser et al., 2015;Satterlund & Haupt, 1967;Schmidt & Gluns, 1991) or an exponential curve (Hedstrom & Pomeroy, 1998). On the other hand, Storck et al. (2002) found that the intercepted snow increased linearly with the increase in cumulative snowfall precipitation. The maximum interception capacity did not appear in a warmer region, even though the range of intercepted snow and cumulative snowfall precipitation was much greater than that observed by Satterlund and Haupt (1967) and Schmidt and Gluns (1991). The difference in the form of these functions induces a significant difference in the amount of intercepted snow, especially when the event total snowfall precipitation is large. Therefore, the function form and model parameters must be chosen carefully.
The maximum interception may vary depending on meteorological conditions, such as air temperature and wind speed. Schmidt and Gluns (1991) demonstrated that the maximum interception capacity is inversely proportional to the density of new snow. Hedstrom and Pomeroy (1998) developed an interception model based on a study by Schmidt and Gluns (1991). The maximum interception capacity was modelled as a function of leaf area index (LAI) and new snow density.
This model assumed a relationship in which the maximum interception capacity decreased with increasing air temperature. However, other research displayed snow cohesion and adhesion increase when the air temperature is >À3 C. As a result, snow accumulation on a narrow board increases with increasing air temperature (Kobayashi, 1987;Pfister & Schneebeli, 1999;Takahashi & Takahashi, 1952). Some models implement a parameterization model which expresses increasing the maximum interception capacity due to increasing air temperature (Andreadis et al., 2009;Lundquist et al., 2021;Roth & Nolin, 2019). Lundquist et al. (2021) suggested the need for its implementation. However, the response to increasing air temperature employed in these models is the opposite of the model parameterization developed by Hedstrom and Pomeroy (1998). Takahashi (1952) found that interception efficiency, which is the ratio of intercepted snow to snowfall precipitation, increases when air temperature ranges from À3 to 0 C. This suggests that the maximum interception capacity may also increase with increasing air temperatures. Although such observed data had been obtained during the initial study of snow interception, they were not formulated and were not adopted in subsequent models. Therefore, the behaviour of snow interception to changes in meteorological conditions remains partially unknown.
Accurate estimates of the amount of intercepted snow necessitates a thorough understanding of both the response of snow accumulation and unloading phenomena to meteorological conditions. In colder continental climates, sublimation is a dominant process for intercepted snow unloading, and in warmer maritime climates, mechanical removal due to snowmelt and meltwater drip are the dominant processes (Lundquist et al., 2021). Mechanical removal has been expressed as a function of air temperature and/or wind speed, and the function has not considered solar radiation (Gregow et al., 2008;Liston & Elder, 2006;Roesch et al., 2001). Model parameters were determined based on certain conceptual understandings and assumptions (Miller, 1966;Nakai et al., 1994;Yamazaki et al., 1996). Takahashi (1952) demonstrated from measured data that intercepted snow is quickly unloaded through melt effects from solar radiation and sensible heat at air temperatures greater than 0 C. At wind speeds >1 m s À1 and air temperatures less than 0 C, snow unloading increases with increasing wind speed. However, these findings have not been developed and integrated into the model. There is a possibility that knowledge of the response of snow accumulation and snow unloading to meteorological conditions based on observational results may be insufficient to accurately estimate the behaviour of intercepted snow.
Based on studies of the difference in the event based snowfall depth differences between the forest and open sites, it has been demonstrated that interception efficiency and its variability are related to the forest canopy structure (Moeser et al., 2015;Roth & Nolin, 2019).
Those results allowed for the modelling of the spatial variability of snow interception caused by the structure of the forest canopy (Helbig et al., 2020;Moeser et al., 2016). However, these observations only provide event-based data and cannot measure changes in intercepted snow during a snowfall event or snow unloading following an event. Despite not being able to show the relationship between snow interception and forest canopy structure, traditional weighing tree experiments can be used to investigate the relationship between meteorological conditions and the behaviour of intercepted snow.
This study presented a novel parameterization model of intercepted snow, named the Japanese cedar snow interception model (JSIM), based on the findings of the analysed snow accumulation and unloading. This study aims to (1) investigate the relationship between meteorological conditions and the behaviour of intercepted snow on the tree canopy through direct measurements of intercepted snow on the tree canopy and (2) propose a new model for snow interception and unloading, that integrates meteorological data. A weighing tree experiment using cut Japanese cedar trees (Cryptomeria japonica) was conducted to directly observe the amount of intercepted snow over four snow seasons in a warm and heavy snow environment in Japan.
This study analysed the response of snow accumulation and unloading to temperature, wind speed, and solar radiation. The effects of the forest canopy structure and density, such as canopy gap, canopy height, and LAI, were not analysed in this study. to 2019-2020. The abovementioned study by Takahashi (1952) was carried out at this same site. The site is located in Tokamachi, Japan, a low-elevation mountainous area in central Japan facing the Japan Sea.
We observed the amount of intercepted snow on a cut tree in an open area for meteorological observation. Figure 1 shows a schematic of the measurement apparatus. We used a similar design from previous studies conducted by Takahashi (1952) and Kato (2000). Four flat concrete plates were placed on the ground and four load cells (Kyowa Co., LCN-A-5KN) were placed on top of these. A mount with a 16-cm inner sleeve was placed on these load cells to hold the tree. The load cells were positioned on the four corners of the mount. The cut tree was set using a crane into the inner sleeve of the mount. Anchor piles were inserted through the holes in the bottom of the mount to prevent it from tipping over due to wind. The apparatus was covered with wooden boards to prevent snow from accumulating on the mount. The snow around the apparatus was occasionally removed to prevent the branches from contacting the snowpack surface. The force applied to each load cell was recorded by a data logger (Kyowa Co., UCAM-60B) at 10-min intervals and a resolution of 1.25 N. After the cut-tree was set on the mount, the measured total force at this time was set as the correction value, and differences in total force from the correction value were recorded in subsequent measurements. The amount intercepted snow of the whole tree was obtained from the changes in measured force. The intercepted snow was expressed as the unit of snow water equivalent in millimetres for the vertical projection area of the tree canopy. When the measured intercepted snow had a negative value due to the drying of the cut-tree, the correction value was updated before the next snowfall event began. Table 1 shows the properties of the cut-trees used each snow season. The top of the Japanese cedar trees, which was about 6-7 m high, was used for the measurements. Different tree tops were taken each winter from the same forest near the observation site, where the trees are about 40 years old and 12 m tall. The tree tops were transported by a truck. Assuming that the vertical projection of tree crowns was circular, the vertical projection crown area was obtained from the average of the canopy radii measured in eight directions.

| Meteorological observations
Meteorological observations were conducted in the same open area as the weighing tree experiment. Air temperature (Yokogawa Electric Co., E-734) was automatically measured at 4 m above the ground, wind speed (Sonic Co., SAT-540H) at 10.5 m above the ground, and downward shortwave radiation (Eko Instrument Co., MS-402F) at 4.3 m above the ground. Hourly precipitation rate was observed using the SR2A precipitation gauge (Tamura, 1993), which has a resolution of 0.005 mm. The precipitation gauge was placed inside a wind fence 4 m high and 3 m above the ground. For backup observation of precipitation, we used the Japanese standard precipitation gauge RT-4 (Yokogawa Electric Co., B-071) at 3 m above the ground.
The precipitation gauge displayed an undercatch of snowfall induced by the wind; therefore, adjustment of precipitation measurements was necessary to obtain the true precipitation amount (Goodison et al., 1998).
The measured precipitation was calibrated by following Equation (1) of the catch ratio CR proposed by Yoshida and Saito (1956): where, m is a coefficient that depends on the gauge type and precipitation type and u G is wind speed (m s À1 ) at the height of the gauge opening. For calibration, we used m ¼ 0:6 and m ¼ 0:26 for dry and wet snow, respectively. These coefficient values were obtained from a comparison between the measured precipitation by SR2A and the Double Fence Intercomparison Reference (Masuda et al., 2018).
Moreover, m ¼ 0:128 was used to calibrate of precipitation rate observed by RT-4 (Yokoyama et al., 2003). We estimated the wind F I G U R E 1 (a) Photograph of intercepted snow on the cut tree taken on 13 February 2018, at Tokamachi, Japan representing a 15-mm snow load on the Japanese cedar canopy. (b) Schematic of measurement apparatus.
speed at the height of the precipitation gauge opening using Equation (2) of the log wind profile assumption: where, u H is the measured wind speed (m s À1 ), h G is the height of gauge opening (m), HS is the snow depth (m), H is the height of the anemometer (m), and z 0 is the aerodynamic roughness length (m). The depth of the snow was observed by using an ultrasonic snow depth sensor (Sonic Co., SL-350). We also used z 0 ¼ 0:005 m as a representative value of the roughness of the land surface covered with snow (Davenport et al., 2000).
The wind speed at one-third of the canopy length, which corresponds to the centre of gravity when the horizontal projection of the canopy is assumed to be a triangle, was used as the wind speed to describe the behaviour of intercepted snow against the wind in the analysis and the interception model. This wind speed was estimated by Equation (2), replacing the height of the gauge opening with this height.

| Analysis of snow interception
The temporal change in the intercepted snow is represented by the following components shown in Equation (3): where, I is the intercepted snow per unit vertical projection area of the canopy in units of snow water equivalent (mm), L is the snow loading rate (mm h À1 ), U is the snow unloading rate due to snow melt and wind (mm h À1 ), M is the unloading rate due to the drip of snow melting water (mm h À1 ), and S is the sublimation rate (mm h À1 ). This equation is a fundamental formula used by many snow interception models (Lundquist et al., 2021). The equation ignored the process of riming, which increases the intercepted snow independent of precipitation falling and affects wind-induced snow unloading (Lumbrazo et al., 2022). The difference between M and U is that M is dripping off liquid water producing due to snow melt, and U is slipping off intercepted snow due to snow melt or wind.
Snowmelt occurs when the air temperature is greater than 0 C, but under these conditions, the snow unloading due to drip off of snow melting water is also significant. Whether the intercepted snow is removed as snow or snow melting water affects the snow accumulation beneath the forest canopy (Storck et al., 2002). Distinguishing between these in snow interception models will lead to improved modelling of forest snowpack. However, it is not easy to separate the amount of the snow unloading rate due to snow melt and the drip of snow melting water from our observations. Then, the unloading rate due to the drip of snow melting water in Equation (3) was not explicitly described but was included in the snow unloading rate due to snow melt and wind.
Furthermore, sublimation increases as wind velocity increases (Lundberg & Halldin, 1994;Nakai et al., 1994), but under this condition, the snow unloading due to wind is also expected to increase. Even if we observe the behaviour of decreasing intercepted snow with increasing wind speed, for the same reason, it is not easy to separate the amount of the snow unloading rate due to wind and the sublimation from our observations. Then, sublimation in Equation (3) was also included in the snow unloading rate due to snow melt and wind.
The snow loading rate and the snow unloading rate can be simply shown by the following equations, respectively: where, dI dP is the interception efficiency, P is the total precipitation (mm), p is the precipitation rate (mm h À1 ), and f is the snow unloading rate coefficient for 1 h (h À1 ). As mentioned above, interception efficiency is a function of the maximum interception capacity (Satterlund & Haupt, 1967). The maximum interception capacity was obtained by gathering several data on intercepted snow and event total precipitation obtained immediately after storm events. Snow unloading can occur during storm events; however, it is not possible to determine from the data whether it occurred. Therefore, the maximum interception capacity may include snow unloading during storm events that cannot be explicitly stated. At the experimental site, air temperatures during snowfall are near 0 C, snow unloading due to snowmelt is likely to occur even during snowfall. Thus, we did not measure the maximum interception capacity; rather, we measured the interception efficiency at each time. In prior studies, snow unloading from snowmelt and snow unloading from wind have been parameterized as unique functions (Lundquist et al., 2021;Roesch et al., 2001). In accordance with these previous studies, snow unloading due to snowmelt f melt (h À1 ) and wind f wind (h À1 ) were obtained separately.
dI dP and f were obtained from the change in observed intercepted snow amount in 1 h and precipitation rate by the following Equations (6) and (7): T A B L E 1 Tree height, crown length, and vertical projection crown area used in the study. dI dP ¼ I n À I nÀ1 p n , ð6Þ where, I n and I nÀ1 are the intercepted snow (mm) for each time point and p n is the precipitation rate (mm h À1 ).
The interception efficiency was determined from the data when we observed an increase in intercepted snow and a precipitation rate of 1 mm h À1 or greater. Precipitation types such as snow or rain are related to air temperature, and a past observation at the Tohkamachi experimental station showed that the frequency ratio of snowfall was 50% at a temperature of 1.5 C (Takeuchi et al., 2016). Moreover, snow unloading due to snowmelt is more likely to occur during the daytime, and snow unloading due to wind occurs when wind speed is >1 m s À1 (Takahashi, 1952). Therefore, to obtain the interception efficiency, we used only the data gathered during the nighttime with a wind speed of ≤1 m s À1 and air temperature of ≤1.5 C. The snow unloading rate coefficient f was determined from the data when we observed a decrease in intercepted snow with an intercepted snow amount of ≥5 mm and no precipitation. f melt was obtained using data obtained during the daytime with wind speeds of ≤1 m s À1 . f wind was obtained from nighttime data with an air temperature of <0 C which snowmelt is negligible.

| Multiple linear regression analysis
We analysed the relation for the coefficient of dI dP , f melt , and f wind to mereological condition using the multiple linear regression analysis to make a parameterization model of each coefficient in JSIM. Air temperature, wind speed, and intercepted snow were used as potential explanatory variables in the analysis of dI dP . The p-value of each variable, coefficient of determination R 2 , and Akaike Information Criterion (AIC) of all seven possible regression equations with the combination of three variables were determined. The regression equation for the combinations of explanatory variables that consists only of statistically significant variables (p < 0:05) and minimizes the AIC was selected as the best regression equation from these seven equations. The root mean square error (RMSE) between the measured and modelled η was also determined to quantify the accuracy of the best regression equation. Previous studies demonstrated a relationship between the air temperature and the interception efficiency when the air temperature is <0 C and no relationship when it is >0 C (Takahashi, 1952).
Because the relationship of the interception efficiency to air temperature is likely to be different for temperatures above and below 0 C; therefore, the analysis of dI dP was conducted separately for air temperatures of 0 C or more and those less than 0 C. Air temperature, solar radiation, and the amount of intercepted snow were used as potential explanatory variables for f melt . The best regression equation for f melt was selected using the same procedure with the analysis of dI dP from the seven possible regression equations with the combination of these three variables. The intercept of the regression model for f melt was forced to zero because the unloading rate is assumed to be zero under conditions where the air temperature is 0 C, no solar radiation occurs and no intercepted snow is present. The regression model for f wind only considers wind speed as an explanatory variable, and the intercept of its model was also forced to zero. RMSE between the measured and modelled f melt , and f wind were determined to quantify their accuracy.

| Modelling of snow interception
We formulated an equation for determining the time change of intercepted snow in JSIM as Equation (8).
We made a parameterization model to determine each coefficient of dI dP , f melt , and f wind in JSIM as a function of mereological variables based on multiple linear regression analysis results. The time series of intercepted snow was simulated from the observed meteorological data using Equation (8) and parameterized dI dP , f melt , and f wind . We analysed the relation between the measured and modelled snow interception. We used the correlation coefficient r between these data, the mean error (ME), and the RMSE to validate the developed model.

| Model intercomparison
We validated the performance of JSIM and discussed the adequate models to represent snow interception phenomena in warmer regions through an intercomparison of simulated results by the JSIM and other models. The intercomparison of the simulated results of the time series of intercepted snow between JSIM and previously developed models was conducted to validate the performance of JSIM. We compared the changes in the time series when the snow interception or the snow unloading parameterization model in JSIM were replaced with those employed in other models, respectively.

| Intercomparison of snow interception parameterization
Three models were selected as a snow interception parameterization for model intercomparison. The first model of snow interception is a model for the boreal forest developed by Hedstrom and Pomeroy (1998). It assumes an exponential curve of intercepted snow to maximum interception capacity with increasing cumulative snowfall precipitation, and the following equation expresses the interception efficiency: where, k is the proportionality factor, I max is the maximum interception capacity, and I 0 is the initial interception. k ¼ 1=I max was adopted based on a suggestion by Hedstrom and Pomeroy (1998), and I 0 was set to zero. I max was estimated from the following equations of LAI and new snow density: where, I is the maximum interception capacity on the tree branch.
I ¼ 6:6 (mm) for pine obtained by Schmidt and Gluns (1991) and LAI = 4.1 (m 2 m À2 ) for the black spruce stand used in the experiment by Hedstrom and Pomeroy (1998) were used. New snow density was estimated from the following relation of air temperature.
From Equations (10) and (11), Hedstrom and Pomeroy (1998) assume that maximum interception capacity decreases with increasing air temperature.

Second, the equation of maximum interception capacity in
Equation (9) was changed to the following Equation (12) by Lundquist et al. (2021), assuming that maximum interception capacity increases at an air temperature of >À3 C due to increased adhesion and cohesion of snow.
The parameterization adopts the opposite behaviour to the relationship of maximum interception capacity to air temperature assumed in Equations (10) and (11) by Hedstrom and Pomeroy (1998).
The third model was based on the assumption that intercepted snow increases at a constant interception efficiency until maximum interception capacity (Storck, 2000).
The following Equations (14) and (15) by Andreadis et al. (2009), which modified the equation of maximum interception capacity in Storck (2000), was used: where, L r is the leaf area ratio, and m is the empirical constant.
m Â LAI ¼ 10:0 was used according to the parameter setting of Storck (2000).

| Intercomparison of snow unloading parameterization
Two models were selected as snow unloading parameterization for model intercomparison. The first model is a parameterization model for boreal forests developed by Hedstrom and Pomeroy (1998), which assumes that intercepted snow decreases exponentially over time and does not separately describe sublimation, snow unloading due to snow melt and wind, or meltwater dripping off. The following snow unloading rate coefficient proposed by Mahat and Tarboton (2014), based on estimated results of the decrease in intercepted snow with time by Hedstrom and Pomeroy (1998), was used for intercomparison.
The second model is a parameterization model developed by Roesch et al. (2001), which has parameterized the snow unloading rate coefficient due to snow melt and wind: f T a ð Þ¼ f u ð Þ ¼ 3600 where, f T a ð Þ is the snow unloading rate coefficient due to snow melt as a function of air temperature, and f u ð Þ is the snow unloading rate coefficient due to wind.

| RESULTS
3.1 | Weighing tree experiment and meteorological observation Figure 2 shows the observed results of intercepted snow I and air temperature T a for each studied snow season. The period shown in grey in these figures indicates the period, during which there are missing data of precipitation rate observed using a precipitation gauge of SR2A due to equipment malfunction. The data observed during this period were not used to analyse the coefficients of dI dP , f melt , and f wind . From Figure 2, the intercepted snow repeatedly increased and decreased over a short time. The duration of the existing intercepted snow on the tree canopy was a few days for most snow interception events and 1 week at the longest. The intercepted snow decreased and approached zero after temperatures began to exceed 0 C for most snow interception events. The maximum intercepted snow value in the four snow seasons was 40 mm, which was observed on 14 January 2017. However, most of the maximum values of intercepted snow for each snow event ranged from 10 to 20 mm. temperatures ranging from À4.2 to 1.3 C, with most of them near 0 C. The observed interception efficiency ranged from 0.02 to 1.33, and values >1 were obtained for some of the data. The mean value interception efficiency was 0.54 (N ¼ 144) for air temperatures of ≥0 C and 0.63 (N ¼ 190) for air temperatures <0 C. Interception efficiency tends to decrease with increasing air temperature ≥0 C and tends to increase with increasing air temperature at <0 C (Figure 3a).

| Interception efficiency
The mean value of interception efficiency increased from 0.31 to 0.57 about two times when air temperature increased from À4 to À3 C.
For air temperature ≥0 C, the interception efficiency when intercepted snow ranged between 5 and 10 mm was greater than when intercepted snow was above or below this range (Figure 3b). For air temperature <0 C, the interception efficiency did not change with increasing intercepted snow (Figure 3d). The interception efficiency tended to decrease with increasing wind speed for both temperature conditions (Figure 3c,e).
From the linear regression analysis results with a single explanatory variable and intercepted term shown in Table 2, the estimated regression coefficient on air temperature was À0.54 for air temperature ≥0 C and 0.062 for air temperature <0 C. That trends were statistically significant (p < 0.001) for both air temperature conditions.
The linear regression analysis also showed no statistically significant relation between the interception efficiency and the amount of intercepted snow under both temperature conditions ( p ≥ 0.05). The estimated regression coefficient on wind speed was À0.23 for air temperature ≥0 C and À 0.21 for air temperature <0 C. There were both found to be statistically significant ( p < 0.01).
The results of the multiple regression analysis for air temperature ≥0 C showed that the best regression model was the one in that combined air temperature ( p < 0.001) and intercepted snow ( p < 0.05). For air temperature <0 C, the combination of air temperature ( p < 0.001) and wind speed (p < 0.001) was shown to be the best regression model. The best regression model predicted interception efficiency with an R 2 of 0.38, AIC of À48.5 for air temperature ≥0 C and with R 2 of 0.16 and AIC of À49.9 for air temperatures <0 C. Figure 4 shows the scatterplot between the measured and modelled interception efficiency. The RMSE between these data were RMSE = 0.20 for air temperature ≥0 C and RMSE = 0.21 for air temperature <0 C. When these best regression models for air temperature <0 C are used to predict the interception efficiency outside the range of the air temperature used in this analysis, the predicted interception efficiency can be less than zero. The interception efficiency is expected to be constant value at temperatures below À3 C in previous studies (Andreadis et al., 2009;Kobayashi, 1987;Lundquist et al., 2021). Based on such assumption and results of the multiple regression analysis, the following Equation (20) was employed to express the interception efficiency in Equation (8) 3.3 | Snow unloading due to snowmelt Figure 5 shows the relationship between the snow unloading rate coefficient due to snowmelt f melt (h À1 ) calculated using Equation (7) from the changes in the amount of intercepted snow for 1 h against air temperature T a ( C) (Figure 5a), solar radiation S # (MJ m À2 h À1 ) (Figure 5b), and the amount of intercepted snow I (mm) (Figure 5c). Table 3 shows the results of the multiple regression analysis with these data for all combinations of these three variables as explanatory variables (N ¼ 103). Snow unloading due to snowmelt was present in air temperatures that ranged between À3.7 and 3.5 C, and was close to 0 at temperatures ≤À3 C. The results of the linear regression analysis with a single explanatory variable shown in Table 3 indicate a statistically significant relationship between air temperature ( p < 0.001), solar radiation (p < 0.001), and the amount of intercepted snow (p < 0.001) on f melt . For all three variables, the estimated regression coefficient on these variables had a positive value. For solar radiation, the highest coefficient of determination and the lowest AIC was F I G U R E 3 Scatterplot and binned scatterplots of interception efficiency dI dP against (a) air temperature T a , (b) intercepted snow I at air temperatures ≥0 C, (c) wind speed u at air temperatures ≥0 C, (d) intercepted snow at air temperatures T a < 0 C, and (e) wind speed u at air temperatures <0 C. The open circles are the means of each bin, and the error bars represent the standard deviations. The data were obtained at night with a precipitation rate of ≥1 mm h À1 , a wind speed of ≤1 m s À1 , and an air temperature of ≤1.5 C.
3.4 | Snow unloading due to wind Figure 6a shows the relationship between the snow unloading rates for 1 h due to wind U wind (mm h À1 ) and wind speed u (m s À1 ) (N ¼ 108). As mentioned above, U wind and f wind includes the decrease in intercepted snow resulting from sublimation. The maximum sublimation rate from the intercepted snow in previous observational studies was 0.61 mm h À1 (Storck et al., 2002), which is shown as a dashed line in Figure 6a. From Figure 6a, most of the observed snow unloading rates were smaller than the maximum sublimation rates observed in previous studies. Only four data points exceeded these maximum sublimation rates, and they were clearly identifiable as the snow unloading due to wind. The smallest wind speed data among these four data was 0.8 m s À1 , and it showed that the snow unloading due to wind is possible to occur at least above this wind speed. Figure 6b shows the relationship between the snow unloading rate coefficient due to wind f wind (h À1 ) and wind speed. From Figure 6b, most of the observed snow unloading rate coefficients due to wind were less than F I G U R E 4 Comparison between modelled and measured interception efficiency dI dP for all four snow seasons. The solid line shows 1:1 relationship with a R 2 of 0.38, and RMSE of 0.20 at air temperatures ≥0 C and R 2 of 0.16, and RMSE of 0.21 for air temperatures <0 C.
F I G U R E 5 Scatterplot and binned scatterplot of snow unloading rate coefficients due to snowmelt f melt against (a) air temperature T a , (b) solar radiation S # , and (c) intercepted snow I. The open circles indicate the means of each bin, and the error bars represent the standard deviations. The data were obtained during the day with an intercepted snow amount of ≥5 mm, a wind speed of ≤1 m s À1 , and no precipitation.
T A B L E 3 Results of multiple regression analysis for variables predicting the unloading rate coefficient due to snowmelt f melt . Note: The estimated regression coefficient for air temperature T a , coefficient for solar radiation S # and coefficient for intercepted snow I, coefficient of determination (R 2 ), and AIC for each model. The numbers in bold indicate the best regression model. Significance codes: ***p < 0.001; **p < 0.01; *p < 0.05; n.s. Not significant.
0.1. These coefficients were smaller relative to the snow unloading rate coefficient due to melt. The obtained snow unloading rate coefficients due to wind included many effects of sublimation. However, the snow unloading rate coefficient due to wind appeared to be increasing with an increase in wind speed, and the following Equation (22) was obtained from these data. The coefficient of determination and the AIC for this regression equation was R 2 of 0.43 and AIC of À486.6, respectively.
3.5 | Model validation Figure 7a shows the scatterplot between the measured and modelled snow unloading rate coefficient due to snowmelt, and Figure 7b shows the scatterplot due to wind. The RMSE values were 0.097 (h À1 ) due to snowmelt and 0.025 (h À1 ) due to wind. began to appear. However, after that, there were times when the model did not correctly simulate the occurrence and amount of snow unloading, as a result, a large difference in the amount of intercepted snow appeared. They occurred during the daytime when the air temperature is around À2 C, or during the nighttime when the air temperature is around 0 C. Figure 9 shows the correlation between the measured and modelled intercepted snow for the entire period of analysis. The statistical measures of the correlation coefficient r, ME, and RMSE for each studied snow season are shown in Table 4. The r for the entire period of analysis was 0.91, which showed a good correlation between the observed and modelled intercepted snow. The ME and RMSE for the entire period of analysis were À0. F I G U R E 6 Scatterplot of (a) snow unloading rate due to wind U wind against wind speed u and (b) snow unloading rate coefficient due to wind f wind against wind speed u. The dashed line in (a) shows 0.61 mm h À1 or the maximum sublimation rate reported by Storck et al. (2002) and the solid line in (b) represents Equation (11). The data were obtained at night with an intercepted snow amount of ≥5 mm, an air temperature of <0 C, and no precipitation.
F I G U R E 7 Scatterplot between modelled and observed (a) snow unloading rate coefficient due to snowmelt f melt (R 2 of 0.78, RMSE of 0.097 h À1 ) and (b) snow unloading rate coefficient due to wind f wind (R 2 of 0.43, RMSE of 0.025 h À1 ) for all four snow seasons. The solid line shows a 1:1 relationship. The parameterization model by Lundquist et al. (2021), which implements the dependency of maximum interception capacity increases with air temperature increasing, improved the underestimation of intercepted snow by the parameterization model of Hedstrom and Pomeroy (1998) during the snow interception event of 10-18 December 2017. It showed a smaller RMSE than the parameterization model by Hedstrom and Pomeroy (1998); however, it estimated the largest ME of intercepted snow over the entire period than any other models (RMSE = 3.8, ME = 0.8). The parameterization model by Andreadis et al. (2009), which assumes the constant interception efficiency and the maximum interception capacity depend on air temperature, showed favourable simulated results (RMSE = 3.7, ME = 0.2) and close behaviour to the simulated results by the JSIM. Comparing the RMSE of these models to that of JSIM (RMSE = 3.2, ME = À0.7, shown in Table 4), the RMSE of JSIM was the smallest, followed by F I G U R E 9 Scatterplot between measured and modelled intercepted snow I for all four snow seasons. The solid line shows a 1:1 relationship. r = 0.91, ME = À0.3 mm, and RMSE = 2.3 mm for the entire period.

| Model intercomparison
T A B L E 4 Statistical measures of the correlation coefficient r, ME, and RMSE between measured and modelled the interception efficiencies for each study snow season.

| Interception efficiency
Statistical analysis showed that the interception efficiency at air temperature <0 C tended to increase with increasing air temperature and decrease with increasing wind speed. Similar patterns have been observed in earlier measurements of snow interception on tree canopy (Takahashi, 1952). In prior studies, interception efficiency, determined for each snowfall event in the snow accumulation on a narrow board, increased with an increase in the air temperature >À3 C due to increasing snow cohesion (Kobayashi, 1987;Pfister & Schneebeli, 1999;Takahashi & Takahashi, 1952) and holds constant below À3 C (Kobayashi, 1987). In our experiment, the interception efficiencies were obtained in the air temperature range of À4.2 to 1.3 C, and it is consistent with the transitional air temperature that changes the behaviour of the interception efficiency mentioned in previous studies. However, since only a few data points were obtained at air temperatures <À3 C, it was not possible to determine if the interception efficiency would hold a constant value at air temperatures <À3 C. Takahashi (1952) and Storck et al. (2002) demonstrated that the interception efficiency decreases as wind speed increases due to the snow being removable by the wind. Snow unloading due to wind and sublimation is expected to increase with increasing wind speeds, but these cannot be separated from the observations. It is impossible to determine which is more dominant, but either effect would decrease interception efficiency with increased wind speeds. Alternatively, snow accumulation occurs when snow particles bounce on the snow surface, stop moving, and stay (Kobayashi, 1987). If snow particles are moving at a high velocity before they impact on the snow surface, they are expected to bounce significantly, making snow accumulation less likely to occur. Then, it can be assumed that snow accumulation is less likely to occur due to increased wind velocity, decreasing the interception efficiency.
The observed interception efficiency for air temperature <0 C was constant with increasing of the amount of intercepted snow. It had no statistically significant relationship to the amount of intercepted snow. This is similar to the results of warmer maritime climates shown by Storck et al. (2002) and Roth and Nolin (2019). If it can be assumed that the amount of intercepted snow increases with increasing cumulative snowfall precipitation and that there is an asymptotic change approaching the maximum interception capacity as represented by a sigmoidal or exponential curve, then the interception efficiency would be expected to be not constant with increasing intercepted snow. The result indicates that within the measured range of the intercepted snow amount, the maximum interception capacity does not exist, or it is sufficiently large to have no impact on the interception efficiency. This is most likely due to the air temperature at the time of snowfall. At temperatures >À3 C, the angle of repose for snow reaches 90 (Eidevåg et al., 2022;Kuroiwa et al., 1967), and the snow accumulation on objects is expected to grow vertically, maintaining its initial shape of the object's projection. In such cases, accumulated snow will continue to grow with continued snowfall, and the maximum capacity of accumulated snow will not appear unless broken by external forces. Suppose this same behaviour occurs in snow accumulation on the tree canopy. In that case, the maximum interception capacity is not expected to appear, and the interception efficiency does not change with increasing cumulative snowfall precipitation. On the other hand, it is not certain whether the maximum interception capacity exists at air temperatures <À4 C, which is lower than the temperature we observed. The angle of repose for snow particles tends to rapidly decrease with a decrease in air temperature under À3.5 C (Eidevåg et al., 2022;Kuroiwa et al., 1967). From data obtained during nine storm events at air temperatures ranging from À12.1 to À1.9 C, Moeser et al. (2015) suggested that the interception efficiency increases with increasing event precipitation and decreases after the interception efficiency reached maximum value. This suggests the existence of a maximum interception capacity under colder conditions.
The interception efficiency for air temperature ≥0 C tended to decrease with increasing air temperature and amount of intercepted snow. At temperatures ≥0 C, snow particles partially melt during their fall and contain water. The liquid water fraction of a snow particle relates to air temperature, relative humidity, and precipitation rate (Misumi et al., 2014). As air temperature increases, the fraction of liquid water contained in the snowfall particles increases, while the amount of solid precipitation decreases. As a result, it is assumed that even if the same amount of precipitation occurs, higher temperatures will reduce the amount of snow supplied on the tree canopy, corresponding to a decrease in the interception efficiency. Furthermore, melting of intercepted snow is likely to occur at air temperatures ≥0 C. Equation (10) shows that snow unloading due to snowmelt can occur even at nighttime if the air temperature is >0 C. However, in our measurements, the obtained interception efficiency unavoidably includes the effect of snow unloading. Based on these considerations, if the amount of intercepted snow is large, it is assumed that the snow unloading rate due to snowmelt will also be large, decreasing the interception efficiency as intercepted snow increases. In contrast, as shown in Figure 3b, the interception efficiency appears low when the amount of intercepted snow is less than 5 mm. At this experimental site, air temperatures are often higher in the early stages of a snowfall event, and it decreases with time. Figure 11 shows the relationship between air temperature and the amount of intercepted snow when the interception efficiency is observed. In order to discuss the effect of the air temperature at the early stages of a snowfall event on the interception efficiency when snowfall occurs at temperatures ≥0 C, we will take a typical example of intercepted snow of 0 and 5 mm. As Figure 11 shows, when the amount of intercepted snow is 0 and 5 mm, the mean air temperature at the time the interception efficiency data were observed is approximately 0.5 and 0.2 C, respectively. According to Equation (20), the change in interception efficiency due to the difference in the air temperature is estimated to be 0.18 lower when the air temperature is 0.3 C higher. In Figure 3b, the mean interception efficiency is 0.39 for intercepted snow of 0 mm and 0.62 for 5 mm, respectively. The difference is 0.23, which is approximately the same as the difference in interception efficiency due to air temperature differences determined from Equation (20). It seemed that the difference in the interception efficiency under these two conditions is mainly caused by the difference in air temperature.
Therefore, it is assumed that the smaller interception efficiency for the amount of intercepted snow <5 mm was influenced by the higher air temperature during the early stages of the snowfall event, and it is not due to the sigmoidal curve behaviour of intercepted snow growth.
The coefficient of determination R 2 and RMSE between the measured and modelled interception efficiency were 0.38, and 0.20, respectively at air temperature ≥0 C and 0.16, and 0.21, respectively at air temperature <0 C. The results show that the modelled interception efficiency is uncertain, and the explanatory variables used in this analysis do not fully explain the interception efficiency. Schmidt and Gluns (1991) demonstrated that the bouncing of snow particles increased as the new snow density increased, reducing the amount of interception. New snow density relates to the dominant shapes of crystals (Ishizaka et al., 2016), and the fall speed of snow particles relates to the dominant shapes of crystals (Locatelli & Hobbs, 1974).
These findings indicate that air temperature and wind speed alone is insufficient for modelling the interception efficiency and that information about snowfall particles may improve the prediction accuracy of interception efficiency.

| Snow unloading
For snow unloading due to snowmelt, the coefficients of determination for the linear regression analysis with a single explanatory variable were the largest for solar radiation, followed by intercepted snow quantity, and the smallest for the air temperature. Multiple regression analysis yielded the best regression model using all three of these variables. The snow unloading rate coefficient due to snowmelt f melt is related to the amount of snowmelt. Therefore, we consider the implications of the best regression model obtained from the energy balance of a snow-covered canopy. The albedo of snow-covered canopies is low (<0.2), and the net radiation above the canopy during the daytime remains positive (Pomeroy & Dion, 1996). The net radiation is greater than the sensible heat flux, and it is dominant for energy balance above the canopy (Nakai et al., 1999). When the air temperatures is slightly above 0 C where snow unloading was mainly observed, the sensible heat is small, and the effect of air temperature on the snow unloading due to snowmelt is considered smaller than the effect of solar radiation. Furthermore, snowmelt-induced mass releases cause a significant amount of intercepted snow to be removed from the canopy all at once. During such a snow large unloading event, the unloading snow induces further unloading in the lower canopy (Takahashi, 1952). The magnitude of the impact force seemed to be related to the mass of the intercepted snow released at one time. The existence of a correlation between the unloading rate coefficient and the amount of intercepted snow may be related to such a falling snow phenomenon. Therefore, in modelling the snow unloading rate coefficient due to snowmelt, it is not sufficient to use only air temperature as an explanatory variable. The solar radiation and the amount of intercepted snow should also be considered. It is the only snow interception model to take account the effect of solar radiation in the snow unloading parameterization. In this experiment, the snow unloading rate coefficient due to snowmelt was modelled using a cut tree set up in an open area that was exposed to greater solar radiation than the forest tree canopy. The canopy structure and solar angle affect shortwave transmissivity through the canopy and the net radiation above the canopy (Pomeroy & Dion, 1996). As a result, when Equation (22) is adapted to the simulation of the intercepted snow on the forest canopy, the parameters may need to be adjusted to account for the tree species and the canopy structure.
Snow unloading due to wind, which exceeded the maximum sublimation rates observed in previous studies, began to appear at wind speeds above 0.8 m s À1 . The snow unloading rate coefficient due to wind increased with increasing wind speed. These results are similar to those observed by Takahashi (1952). However, only a few data were identifiable as the snow unloading due to wind, and most data could not distinguish whether it is caused by the snow unloading due to wind or sublimation. Most of these may represent a decrease in intercepted snow due to sublimation. These results suggest that the F I G U R E 1 1 Scatter plot and binned scatter plot air temperature T a of against intercepted snow I when the interception efficiency is observed at air temperature T a ≥ 0 C. The open circles are the means within each bin and the error bars represent the standard deviations.
snow unloading due to wind is smaller than the sublimation rate or does not occur in such a range of wind speeds. For these reasons, Equation (22) does not appear to accurately describe wind-driven unloading. An observation method that exactly separates the snow unloading due to wind or sublimation is necessary to improve our understanding and modelling of these phenomena.

| Model validation and intercomparison
Although the modelled interception efficiency contains high uncertainty, the simulated behaviour of intercepted snow had a good corre- curves. These facts may indicate that a snow interception parameterization in which the snow interception efficiency increases with increasing air temperature is appropriate for simulating snow interception phenomena in warmer regions like those in this study. We do not recommend, but suppose, a snow interception parameterization that assumes the sigmoidal or exponential curve of intercepted growth will be used to simulate intercepted snow in warmer regions. In that case, the parameterization model should include the effect of increasing maximum interception capacity and interception efficiency due to increasing air temperature for air temperature >À3 C, and its model parameters should be chosen carefully.
From the intercomparison results for snow unloading, the model by Mahat and Tarboton (2014), which represents snow unloading as a function of time, significantly overestimated the intercepted snow for the entire period. The model was developed for cold boreal forests where snow unloading due to snow melt is insignificant. The results indicate that it is not appropriate to adapt this model to warmer regions where snowmelt frequently occurs during winter. The JSIM or model by Roesch et al. (2001), which incorporates snow unloading due to snow melt, produced simulated results close to the observed intercepted snow time series. The model represents snow unloading due to snow melt would be particularly effective in modelling snow interception in warm regions where snow melt is significant during winter. The JSIM produced better-simulated results of intercepted snow close to observations than the model by Roesch et al. (2001), which uses only air temperature to represent snow unloading due to snow melt. It indicates that the model can be improved by incorporating the effect of solar radiation into a parameterization model of snow unloading due to snow melt.
However, 15-20 mm difference appeared between the simulated by the JSIM and observed intercepted snow in some snowfall events.
These occurred during the daytime when the air temperature was around À2 C or during the nighttime when the air temperature was around 0 C. These conditions correspond to the boundaries that are likely to start snow unloading due to snowmelt. Therefore, it is assumed that snowmelt initiated the snow unloading errors. It seems that the miss estimation of snow unloading occurring by the model is sensitive to the simulation results of the amount of intercepted snow.
To obtain favourable simulated results, both the snow accumulation and snow unloading schemes must work properly. This is especially important in simulating the temporal change of intercepted snow in warmer regions where snowmelt is more likely to occur. Because snowmelt causes a mass release of intercepted snow, misestimation of snow unloading due to snowmelt can lead to critical errors in the simulation of intercepted snow. Thus, careful considerations will be needed to develop a model that represents snow unloading due to snowmelt and to determine its model parameters.
In this study, the snow interception phenomena have been discussed without regard to the effects of canopy structure. Recent observations suggested that the forest canopy structure influences interception efficiency (Moeser et al., 2015;Roth & Nolin, 2019).
Snow interception is probably impacted because canopy structure affects microclimate, such as wind speed and solar radiation within the canopy. Canopy structure may help distribute point-based models to the surrounding forest landscape. Future testing of the developed model using data observed in different meteorological conditions and forest canopy structures is needed to improve the model and better understand the snow interception phenomenon.

| CONCLUSIONS
A new parameterization model, which represents snow interception and unloading, was developed based on observed results. A weighing tree experiment using cut cedar trees investigated the relationship between meteorological conditions and intercepted snow on the tree canopy at Tokamachi, which located in a relatively warm and snow rich area in Japan. The interception efficiency at air temperatures <0 C increased with increasing air temperature and decreasing wind speed. It was not related to the amount of intercepted snow, and the maximum interception capacity was not present at air temperatures from À4.2 to 0 C. Due to a lack of data, it was not possible to show whether the maximum interception capacity existed at temperatures <À3 C. At air temperatures ≥0 C, the interception efficiency decreased with increasing air temperature and the amount of intercepted snow. precipitation gauge. We also thank the members of the Forestry Insurance Center for their passionate discussions on the forest insurance and risk assessment.