With consideration of the filtering performance, which may vary in snow climates, eight seasonally snow-covered study sites with different snow climates were selected to implement numerical experiments in this study (Sturm et al., 1995; Trujillo and Molotch, 2014). These sites are distributed at different latitudes in the Northern Hemisphere, and the sites included the Arctic Sodankylä site (SDA, 179 m), located beside the Kitinen River in Finland, with the upper 2 m being frozen (Rautiainen et al., 2014); the Snoqualmie site (SNQ, 921 m) with a rain–snow transitional climate in the Washington Cascades of the USA, where the SD measured by snow stakes was employed (Wayand et al., 2015); the maritime Col de Porte (CDP, 1330 m) site in the Chartreuse Range in the Rhone-Alpes region of France; the Mediterranean-climate Refugio Poqueira site (ROPA, 2510 m) in the Sierra Nevada Mountains of Spain, which has a high evaporation rate (Herrero et al., 2009); the Weissfluhjoch site (WFJ, 2540 m) in Davos of Switzerland, with automatic SD observations being used at this site (Wever et al., 2015); the continental Swamp Angel Study Plot (SASP, 3370 m) site in the San Juan Mountains of Colorado, USA; and two sites from typical snow-covered regions in China, namely the Altay meteorological observation site (ATY, 735.3 m) in northern Xinjiang, China, where there is less wind in the winter season, and the Mohe meteorological observation site (MOHE, 438.5 m) in a county of northeast China, which has a cold temperate continental climate and is the northernmost part of China. Serially complete meteorological measurements are available and can certainly be used as forcing data in these sites; the downward longwave and shortwave radiation values of MOHE were extracted from the China Meteorological Forcing Dataset (CMFD) (Chen et al., 2011) since there are no radiation measurements at this site.

It is noteworthy that the spatial variance of the performance of the model is negligible since these sites themselves are flat and the surrounding vegetation types are uniform. We have used this dataset to examine the sensitivity of simulated SD to physics options, and the results show that the dataset has a reliable quality. In addition, the location, the detailed information of snow climates, and details about the dataset processing for the eight sites can be also referenced in You et al. (2020a).

The snow partial module within Noah-MP model can be divided up into to three layers depending on the depth of the snow (Niu et al., 2011). The SD hsnow is calculated by 1hsnowt=hsnowt-1+Ps,gρsfdt, where Ps,g is the snowfall rate at the ground surface, dt is the time step, and ρsf is the bulk density of the snowfall. When hsnow< 0.025 m, the snowpack is combined with the top soil layer, and no dependent snow layer exists. When 0.025≤hsnow≤ 0.05 m, a snow layer is created with a thickness equal to SD. When 0.05<hsnow≤0.1 m, the snowpack will be divided into two layers, each with a thickness of Δz-1=Δz0=hsnow/2. When 0.1<hsnow≤0.25 m, the thickness of the first layer is Δz-1=0.05 m, and the thickness of the second layer is Δz0=(hsnow-Δz-1) m. When 0.25<hsnow≤0.45 m, a third layer is created, and the three thickness are Δz-2=0.05 m and Δz-1=Δz0=(hsnow-Δz-2)/2 m. When hsnow>0.45 m, the layer thickness of the three snow layers are Δz-2=0.05 m, Δz-1=0.2 m, and Δz0=hsnow-Δz-2-Δz-1 m. Certainly, the snow cover is highly influenced by air and ground temperature, and the snow layer combines with the neighboring layer due to sublimation or melting and is redivided depending on the total SD. The snow module of the Noah-MP model provides an estimate of snow-related variables using energy and mass balance. This computing process requires a series of meteorological forcing data, such as near-surface air temperature, precipitation, and downward solar radiation. The snow accumulation or ablation parameterization of the Noah-MP model is based on the mass and energy balance of the snowpack, and the snow water equivalent can be calculated using the following equation: 2dWsdt=Ps,g-Ms-E, where Ws is the snow water equivalent (mm), Ps,g is the solid precipitation (mm s-1), Ms is the snowmelt rate (mm s-1), and E is the snow sublimation rate (mm s-1).

A snow interception model was implemented into the Noah-MP model to describe the process of snowfall intercepted by the vegetation canopy (Niu and Yang, 2004). Within this model, the snowfall rate at the ground surface Ps,g is then calculated by 3Ps,g=Ps,drip+Ps,throu, where Ps,drip (mm s-1) is the drip rate of snow, and Ps,throu (mm s-1) is the through-fall rate of snow. In the Noah-MP model, the ground surface albedo is parameterized as an area-weighted average of the albedos of snow and bare soil, and the snow cover fraction of the canopy is used to calculate the ground surface albedo, as shown in Eq. (4): 4αg=1-fsnow,gαsoil+fsnow,gα,snow where αsoil and αsnow are the albedo of bare soil and snow, respectively. fsnow,g is the snow cover fraction on the ground and is parameterized as a function of snow depth, ground roughness length, and snow density (Niu and Yang, 2006).

The Bayesian recursive estimation problem is solved by the Monte Carlo approach within the PF technique, making this scheme appropriate for a nonlinear system with a non-Gaussian probability distribution (Magnusson et al., 2017). The basic concept of the PF technique is to use a large number of randomly generated realizations (i.e., particles) of the system state to represent the posterior distribution. Meanwhile, the particles are propagated forward in time as the model evolves. The weights associated with the particles are updated based on the likelihood of each particle's simulated proximity to the real observation. The weight of the particles can be updated as follows: 5wti=wt-1ipztxti, where wt-1i is the weight of ith particle at time t-1, and the weight is updated by the likelihood function pztxti, which measures the likelihood of a given model state with respect to the observation zt. The observation errors are generally assumed to follow a Gaussian distribution, and the chosen likelihood function represents this assumption. In this study, we employed a normal probability distribution to serve as the likelihood function: 6pztxti=Nzt-xti,σ, where N represents the normal probability distribution of the residuals between observed zt and simulated xt. Finally, the weights of the updated model state would be normalized, and the assimilated value of the model state is the weighted average of all particles at time t. Although the particle filter has been widely applied in various nonlinear systems, the particle degeneracy and impoverishment in the particle filter are still the fatal limitations that need to be urgently addressed. To address the degeneration problem in the PF technique, traditional resampling methods like multinominal resampling and systematic resampling were employed to resample the particles if the effective sample size, 7Neff=1∑i=1Nwti2, fell below a specified number. In the above equation, N is the ensemble size, and wti is the normalized weight defined in Eq. (5). To be honest, traditional resampling methods can effectively mitigate the problem of particle degeneracy by resampling high-quality particles. However, after multiple iterations, these methods often lead to a serious lack of diversity among particles, which is known as the particle impoverishment problem. To mitigate both of these issues simultaneously, we employed the genetic algorithm (GA) to resample the particles, resulting in the genetic particle filter algorithm (GPF). The GA is inspired by Darwin's theory of evolution and emphasizes the principle of survival of the fittest. In fact, in the resampling phase, the fitness of particles should be reselected according to the theory of particle filtering. Selection, crossover, and mutation are major steps used to simulate population evolution. As shown in Fig. 1, these three operators are utilized to produce better offspring and improve the overall population fitness, with the aim of preventing particle degeneracy and impoverishment. These operators will be used to improve particle fitness when it falls below a threshold value. The three operators are described below.

In order to investigate the impact of meteorological perturbations on snow simulations, an ensemble containing 100 SD simulations derived from as many different meteorological conditions was analyzed. For the sake of concision and clarity, we considered only one winter season for implementing snow simulation experiments at each site, and the results are shown in Fig. 2. As shown in Fig. 2, the possible overestimation and underestimation of SD simulations produced by the perturbation forcing data were contained within the ensemble spread, which is a direct consequence of the perturbation of the forcing data. Since the meteorological perturbations are unbiased, the physical processes with nonlinear characteristics within the model are supposed to be the main reason for the uncertainty (Piazzi et al., 2018). During the winter season in the Northern Hemisphere, precipitation and air temperature are the primary factors that can determine the total amount of snow.

As Fig. 2 shows, the intervals of the SD ensemble are significantly different at different sites, although an identical meteorological perturbation method was used. At some sites, such as ATY, MOHE, WFJ, and CDP, larger SD ensemble spreads were obtained, and most of the SD observations were covered by the ensemble spread. In this case, high-quality particles can be directly selected from the ensemble. However, at some other sites, such as ROPA, SDA, and SASP, narrow SD ensemble spreads were obtained, and the uncertainty interval of simulated SD can hardly cover the observations. In this case, the so-called high-quality particles cannot even be found in the ensemble, and the model prior error becomes a prerequisite for successful assimilation at this time. Especially at the ROPA site, the snow cover was extremely unstable, resulting in difficulty in figuring out any variation rules of the SD. The narrow SD ensemble spread at this site also demonstrates that precipitation and air temperature were not the main factors causing snow change. According to the literature, sublimation losses at ROPA ranged from 24 % to 33 % of total annual ablation and occurred 60 % of the time during which snow was present. A high sublimation rate may be the main reason for snow instability (Herrero et al., 2016; You et al., 2020a). This directly leads to a perfect ensemble spread that can cover all observations and that cannot be produced by perturbing the air temperature and precipitation. Generally speaking, the ensemble produced by perturbing air temperature and precipitation does not contain high-quality particles at this site. It was found that the spread of SD ensembles increases when a snowfall event occurs because the perturbation in precipitation would provide different input snow rates for model realization at all sites. Despite this, we still found that the simulated SD deviated significantly from the observation. For example, at the SNQ site, the maximum value of simulated SD was almost half the maximum value of observed SD. In this case, it is impossible to obtain a simulated SD ensemble spread that can cover or nearly cover the observation through perturbing the meteorological-forcing data. On the one hand, precipitation and air temperature are not the dominant factors affecting snow cover change, which leads to a narrowed ensemble spread at these sites. On the other hand, although the variation trend of snow cover can be accurately expressed by the Noah-MP model, serious underestimation of the simulated SD shows that the snow simulation performance of Noah-MP is poor at these sites. Nonetheless, the simulated ensembles will be improved whenever the prior error of the model state is considered.

Generally, the ability of a model to simulate autonomously can be limited if observation data are assimilated too frequently, resulting in assimilation results that are essentially the same as the observations and do not reflect the differences among models. To address this, the site's SD measurements were assimilated into the Noah-MP model with an observation frequency of 5 d in this study, enabling the GPF to perform differently at distinct sites. Figure 3 shows the SD assimilation results across snow climates, indicating a substantial improvement in the SD simulations with satisfactory assimilation performance at all sites. The GPF algorithm can handle not only serious underestimations, such as at SNQ and SDA, but also overestimations during the snow ablation period, as seen at the CDP, SASP, ATY, and MOHE sites. These results demonstrate the effectiveness of the GPF algorithm as a snow data assimilation scheme and its ability to significantly improve SD simulations despite the numerous overestimations and underestimations that may occur in the Noah-MP model's snow simulation results across snow climates.

The effectiveness of GPF in updating SD simulations is demonstrated by the KGE values of the DA simulations with perturbed meteorological-forcing data, as shown in Fig. 4. Although the mean ensemble simulations of SD exhibit substantial improvement at all sites, not all ensemble members were improved, as per the distribution of GPF-DA KGE values. Some ensemble members achieved significant improvement at sites like SDA, SASP, MOHE, and SNQ, while others showed only slight improvement at sites like ATY and WFJ. Figure 4 also reveals that updating SD model simulations at the ROPA and WFJ sites is more challenging. Snow simulation performance at the ROPA site is known to be poor due to the high sublimation rate. Certainly, the median value of SD ensemble prediction KGE values is expected to be below zero at this site, indicating that there are few qualified simulations in the prediction ensemble. While the GPF succeeds in enhancing the SD simulations at ROPA, the distribution of GPF-DA KGE values is not concentrated enough, with the 25th percentile being at approximately 0.2 and the 75th percentile at about 0.7, indicating that the GPF assimilation algorithm cannot enhance all members but can raise the mean level and obtain an approximation of the optimal posterior estimation. Conversely, the assimilation of snow measurements at the CDP site resulted in a poor quality of the SD simulations compared to the open-loop ensemble simulations. The median value of the GPF-DA KGE was lower than the median value of the open-loop (OL) KGE, indicating that a considerable number of ensemble simulations failed to capture the observed values after assimilating snow measurements. However, Fig. 3 shows that the mean ensemble simulations after assimilating snow measurements are much closer to SD observations. Thus, it underscores the importance of the ensemble mean in characterizing the filter effectiveness and the approximate value of the optimal posterior estimation of model state. Additionally, the scale of the model ensemble spread was found to be the determinant factor that significantly affects assimilation results. A large ensemble spread can adjust the simulations toward the observed system state even if the model predictions are heavily biased.

Figure 5 displays the CRPSS value of GPF-DA at different sites. The smaller the CRPSS value, the worse the probabilistic simulation (with an optimal score of 1). The highest CRPSS score of 0.91 was achieved at SASP, while the lowest score of 0.44 was observed at CDP. These results indicate that the GPF enhances the overall accuracy of ensemble simulations most at SASP and least at CDP with respect to the open-loop ensemble simulation. Certainly, this cannot be illustrated by the mean ensemble simulations (Fig. 3) but is consistent with the KGE statistical results (Fig. 4). Although the open-loop simulations at SNQ exhibited serious underestimation, a satisfactory assimilation result was obtained at this site with a CRPSS score of 0.87. At the SNQ site, the snow simulation performance of the Noah-MP model is poor, and the model shows serious underestimation during the snow stable phase. Implementing a data assimilation experiment in this case is a tricky business since it is difficult to obtain a suitable simulated ensemble by perturbing the meteorological forcings. However, since the model prior error was considered in the GPF algorithm, the overall accuracy of the ensemble simulations will be substantially enhanced, and this is the reason why a satisfactory assimilation result at the SNQ site can be obtained. ROPA was found to be a difficult site to enhance the overall accuracy of the ensemble simulations, with a CRPSS score of only 0.58. The snow cover was highly unstable, and the variation of SD exhibited extreme irregularity; these may be the main obstacles to snow data assimilation at this site.

Based on these findings, we conclude that the effectiveness of GPF varied among snow climates: it can be employed as a snow data assimilation scheme across snow climates; however, its performance varied across different sites. It is necessary to explore the sensitivity of the measurement frequency and ensemble size for the GPF assimilation scheme at various sites.

For complex land–snow process models, model errors can gradually lead to the system deviating from the true value. Therefore, it is necessary to continuously incorporate observations into the model framework to adjust the operating trajectory of the state. Obviously, the frequency of incorporating observations, that is, the assimilation interval, has an important impact on the assimilation system. To investigate the effect of the SD measurement frequency on the performance of GPF, we conducted a sensitivity experiment at eight sites. We aimed to determine how reducing the frequency of SD measurements affects the DA simulations. As expected, a decrease in SD measurement frequency led to a reduction in the impact of the GPF updating on the model simulations, resulting in a gradual increase in the mean RMSE value. Figure 6 illustrates the RMSE ensembles of SD simulations resulting from assimilating different-frequency SD measurements over the snow period at each site. Higher-frequency SD assimilation improves the accuracy of the simulated SD, as shown by the lower RMSE value achieved when the frequency of the SD measurements was set to 5 d. This means that more frequent SD measurements improve the accuracy of the model, which is particularly useful in regions where snow conditions can change rapidly. The range of RMSE values at different sites varied significantly as it was related to the maximum value of SD. For instance, thick snow at the SNQ and WFJ sites during the snow period led to larger RMSEs of SD simulations. Notably, an increase in the length of the assimilation window generally resulted in a significant increase in the RMSE value. However, an abnormal occurrence was observed at the SDA site, where the assimilation effect of 20 d of SD measurements was significantly better than that of 15 d. Although the RMSE distribution of SD assimilation results with 20 d of observations appeared to be superior to that of 15 d, the RMSE mean values of the two were very close: 0.08 and 0.07 m, respectively. Therefore, this anomaly can be ignored. These results indicate that the frequency of SD observations has a significant impact on the effectiveness of the GPF algorithm and that a dense amount of observational data can effectively improve the assimilation results.

The results of the experiment aimed at evaluating the impact of particle number on the assimilation performance of GPF are presented in Fig. 7. As expected, increasing the particle number up to the threshold leads to a significant improvement in the percent effective sample size. However, the filter performance does not improve significantly when the particle number exceeds the threshold. Figure 7 shows that the GPF algorithm yields the minimum error at all sites when the particle number is set to 100, indicating that 100 particles can optimize the performance of the GPF algorithm. Although a large particle number can enhance particle diversity and prevent filter divergence, it increases the computation burden without reducing the system error. As illustrated in Fig. 7, the RMSEs are generally at the same level when the particle number equals 120 and 160, and they are significantly larger than the RMSE when the particle number is equal to 100. The slight impact of the change in the particle number on the performance of GPF when the particle number is below the threshold indicates low system sensitivity to the ensemble size, and this is observed at all sites. Essentially, blindly increasing the particle number does not guarantee a better DA performance of the GPF algorithm. As demonstrated in Fig. 7, the RMSEs of simulated snow depth are virtually unchanged at all sites despite an increase in the particle number from 120 to 160. This suggests that blindly increasing the ensemble size only increases the computational burden without improving the performance of the GPF.

To demonstrate the effectiveness of using genetic algorithms for particle resampling, we compared the results of our genetic algorithm (PF-G) to those of traditional resampling methods: systematic resampling (PF-S) and multinomial resampling (PF-M), which are both commonly used in particle resampling. The calculation process for these methods is detailed in the references of the particle filter introduction. Figure 8 shows the RMSE values for SD simulations obtained using these three methods. We found that the PF-G outperforms PF-M and PF-S at all sites, as evidenced by the significantly smaller mean and median RMSE values. This indicates that the PF-G is suitable for snow data assimilation in various snow climates and is somewhat superior to traditional particle filters. At most sites (MOHE, ATY, SDA, and ROPA), PF-M and PF-S showed similar performance, meaning that these methods did not produce a significant difference in the assimilation results. This is because these traditional resampling methods can only mitigate particle degeneration by resampling particles but are unable to prevent particle impoverishment. Therefore, they are unable to select high-quality particles and keep the particles that have variety. Significantly, the mean and median RMSE values for PF-G were lower than those of PF-M and PF-S at several sites (SASP, SNQ, and WFJ) where the snow cover was relatively thick, with maximum SD during the snow period reaching 2.45, 2.95, and 2.40 m, respectively. This suggests that PF-G performs better in assimilating data from thick snow covers.

The multinomial and systematic resampling methods select particles from the original particle set at different levels or based on the accumulation of particle weights. Both of the resampling methods extract particles from the entire particle set, and the corresponding particle values do not undergo any essential changes. However, when compared to the two traditional particle resampling methods, the genetic algorithm first uses the fitness function to calculate the survival rate of each particle one by one and then performs crossover, mutation, and other operations on the selected particles. This approach ensures that the resampled particles are high-quality particles, which is the main reason why genetic particle filtering has an advantage in the snow data assimilation experiments. As Fig. 8 shows, the assimilation error of the genetic particle filter is the smallest at all sites. From the results of the real assimilation experiment, it can be seen that genetic particle filtering has more advantages over the other two methods.