A Multifactor Eigenvector Spatial Filtering-Based Method for Resolution-Enhanced Snow Water Equivalent Estimation in the Western United States

Abstract

Accurate snow water equivalent (SWE) products are vital for monitoring hydrological processes and managing water resources effectively. However, the coarse spatial resolution (typically at 25 km from passive microwave remote sensing images) of the existing SWE products cannot meet the needs of explicit hydrological modeling. Linear regression ignores the spatial autocorrelation (SA) in the variables, and the measure of SA in the data assimilation algorithm is not explicit. This study develops a Resolution-enhanced Multifactor Eigenvector Spatial Filtering (RM-ESF) method to estimate daily SWE in the western United States based on a 6.25 km enhanced-resolution passive microwave record. The RM-ESF method is based on a brightness temperature gradience algorithm, incorporating not only factors including geolocation, environmental, topographical, and snow features but also eigenvectors generated from a spatial weights matrix to take SA into account. The results indicate that the SWE estimation from the RM-ESF method obviously outperforms other SWE products given its overall highest correlation coefficient (0.72) and lowest RMSE (56.70 mm) and MAE (43.88 mm), compared with the AMSR2 (0.33, 131.38 mm, and 115.45 mm), GlobSnow3 (0.50, 100.03 mm, and 83.58 mm), NCA-LDAS (0.48, 98.80 mm, and 81.94 mm), and ERA5 (0.65, 67.33 mm, and 51.82 mm), respectively. The RM-ESF model considers SA effectively and estimates SWE at a resolution of 6.25 km, which provides a feasible and efficient approach for SWE estimation with higher precision and finer spatial resolution.

1. Introduction

Snow is an indispensable part of the hydrosphere and plays a critical role in the water cycle and water supply [1]. SWE refers to the equivalent water volume (after melt) of the snowpack, which is defined as the product of snow depth (SD) and its density. Owing to its importance in hydrological modeling [2,3,4], a reliable estimated SWE product with high precision is necessary for hydrology and global climate change analysis.

Passive microwave remote sensing (PMRS) can be less affected by cloud and supports all-weather detection [4], thereby providing an effective method to retrieve SWE observations. Microwave sensors receive microwave emission from the snow volume and its underlying surface, which is expressed by the brightness temperature values. Shorter wavelengths are used to detect the emission of the near-surface, while longer wavelengths offer more information on deeper snow. For example, Kelly et al. (2009) utilized TB of 10 GHz as an enhancement to penetrate and estimate deep snowpacks [5]. Most of the methods that estimate the SWE or SD using PMRS exploit the brightness temperature difference (TBD) between 19 GHz and 37 GHz [6]. The brightness temperature at 37 GHz is a measurement frequency considered to be sensitive to snow volume scattering, while that at 19 GHz is considered insensitive to snow. There are many methods that have been proposed to estimate SD or SWE using PMRS [6,7,8,9,10,11,12,13,14,15], which can be divided into four types according to Xiao et al.: (1) linear retrieval algorithm; (2) microwave emission-based models; (3) nonlinear regression algorithms; and (4) data assimilation [16]. A snow-parameter linear retrieval algorithm developed by Chang et al. (1987) is the most commonly used model because of its simplicity and good interpretability [10]. Specifically, the retrieval algorithm for estimating SD uses the TBD between 18 GHz and 37 GHz with horizontal polarization from the SMMR sensor. It is assumed that the grain size of snow is 0.3 mm and that the snow density is 0.3 g/cm³ in their research, so the SWE product is obtained using the product of the SD and the snow density [10]. However, the algorithm has limitations in mountainous areas where the sensitivity of microwaves will be reduced. For instance, snow data in mountainous terrain are also screened from the global AMSR-E algorithm [12]. The areas with heavy vegetation will also reduce the sensitivity of microwaves, especially the areas with dense tree cover, which tends to have lower albedo for snow-covered lands [13]. Foster et al. took the effects of forest cover and snow grain size on the microwave response into account, which significantly improved the estimation of snow mass in Eurasia and North America when compared to the algorithm of Chang et al. [10,13]. In 2005, Foster et al. further developed the microwave method by considering vegetation cover and snow morphology topography. They found that the topography and atmospheric conditions are the main sources of uncertainty.

Spatial autocorrelation (SA) is considered persistent in the SWE [17], which indicates that the SWE values at a given location tend to be similar (or dissimilar) to those at nearby locations due to the presence of spatial clustering. When a traditional linear regression model is performed on geographic datasets, the model residuals must have no SA, otherwise the results of the model are unreliable [18]. Thus, it is important to address the SA problem in linear regression models. SA is also considered in the SWE data assimilation algorithm. Takala et al. developed a data assimilation algorithm that assimilates weather station data on snow depth with satellite passive microwave radiometer data to produce a 30-year-long time series of seasonal SWE for the northern hemisphere [6]. In their study, SA is implicitly captured in the spatial variability autocorrelation function calculated by analyzing observational data from North America and Eurasia. Moreover, using geostatistical techniques that consider spatial autocorrelation has been shown to improve the snow depth estimates [19]. However, few studies have quantified the spatial autocorrelation of SWE in the traditional TB gradient linear algorithm. An Eigenvector Spatial Filtering (ESF) method was proposed to separate SA from variables effectively by adding Eigenvector Spatial Filters (ESFs) into linear regression models to improve their performance [20]. The ESFs are a set of eigenvectors extracted from the spatial weights matrix which tie geographic objects together in space. These vectors serve as control variables in the model to identify and isolate SA between observations so that the observations can be considered spatially independent when modeled [21].

Popular SWE products are divided into three categories in the study of Mortimer et al.: (1) products from reanalysis; (2) independent passive microwave estimates; and (3) passive microwave estimates combined with station observations [22]. Most existing SWE products are generated based on static retrieval algorithms and have a spatial resolution of 25 km because of the constraints from the spatial resolution of passive microwave radiometers, including AMSR-E, AMSR2, SMMR, SSM/I, and SSMIS [7]. These SWE products with coarse resolution cannot meet the needs of regional hydrological models, and hence, many studies have endeavored to produce finer-resolution snow products [23,24].

Previous studies show that some factors have a great influence on SWE estimation. In many multifactor approaches, in order to obtain snow parameters (SWE or SD), geolocation (latitude and longitude) is considered and was proved to help improve the multiple linear regression model’s spatiotemporal estimates in the study of Wei et al. [23,25,26,27]. Solar radiation varies with latitude, leading to the latitudinal variation of SWE distribution [28]. Meanwhile, in the western United States, there are many north–south mountain ranges, which makes it possible for the SWE distribution to have a pattern of longitudinal divergence. The temperature has strong effects on snow by affecting snow density [29]. In addition, longwave and shortwave radiation fluxes also show a close relationship with the percentage of snow cover. The surface energy, measured with the surface heat flux, can influence the snow processes [30,31]. Moreover, elevation is a significant factor because relatively low elevation temperatures tend to reach the melting point more easily [32]. In addition to elevation, topographical features like slope are regarded as essential variables for snow parameters in some studies [26,27,33]. Zhong et al. (2021) reported that the aspect and slope have a similar influence on snowfall, and the maximum SWE values concentrate on the northern slopes of the Altai Mountains [34]. Forest reduces the microwave reflection from snow, resulting in underestimating of SD. Thus, some studies correct for these effects by introducing parameters such as forest cover or forest density [13,23,35,36]. In addition to the environmental factors and topographical features, snow factors, including snow cover fraction, snow precipitation rate, and snow melt, are also used to improve inversion accuracy [24,27].

Therefore, in this study, a Resolution-enhanced Multifactor Eigenvector Spatial Filtering (RM-ESF) model was developed based on a multiple linear regression model to estimate daily SWE distribution with a spatial resolution of 6.25 km in the western U.S. This method is based on the relationship between TBD and snow depth and considers factors including geological features, topographical features, snow and environmental factors to estimate SWE with finer resolution.

2. Study Area and Datasets

2.1. Study Area

The western U.S. roughly ranges between 99°43′W~125°58′W and 30°48′N~48°45′N, as shown in Figure 1. The north–south mountain ranges create wet areas on the western sides and dry rain shadows on the eastern sides, producing east–west precipitation gradients in the mountain ranges [37]. Across the western U.S., up to 70% of the water supplies come from the spring snowmelt [38].

2.2. Datasets and Preprocessing

This study is based on the winter months (December-January-February, DJF) in the United States. Due to the lack of data beyond 2016 in the NCA-LDAS V2.0 daily dataset, we used datasets in 2011–2014 for modeling and datasets in 2015 and 2016 for validation to test the robustness of our model. The datasets of the variables used for modeling are listed in

Table 1. Summary of datasets for modeling.

Category	Variable		Abbr.	Resolution	Source	Link
Station observation	Snow depth/Snow water equivalent		SD/ SWE	/	GHCNd	https://www.ncei.noaa.gov/ (accessed on 19 September 2022).
Geolocation factors	Latitude		LAT	/	GADM	https://gadm.org/ (accessed on 19 September 2022).
	Longitude		LON	/
Topographical features	Elevation		ELV	5 km	Global 1, 5, 10, 100-km Topography	https://www.earthenv.org/topography (accessed on 19 September 2022).
	Slope		SLP
Snow factors	Snowmelt		QSM	0.125°	NCALDAS_NOAH0125_D	https://ldas.gsfc.nasa.gov/NCA-LDAS (accessed on 19 September 2022).
	Snow precipitation rate		SPR
	Snow cover fraction		SCF
Environmental factors	Air temperature		AT	0.125°	NCALDAS_NOAH0125_D	https://ldas.gsfc.nasa.gov/NCA-LDAS (accessed on 15 September 2022).
	Soil temperature (underground 0–10 cm)		ST
	Heat flux		QG
	Wind speed		WS
	Tree cover		TC	30 m	Global Forest Change 2000–2021 Data	https://storage.googleapis.com/earthenginepartners-hansen/GFC-2021-v1.9/download.html (accessed on 15 September 2022).
Brightness temperature	Brightness temperature	19 (h, v) 22 (v) 37 (h, v) 91 (h, v)	TB	3.125 km	MEaSUREs	https://nsidc.org/data/nsidc-0630/versions/1 (accessed on 03 October 2022.
				6.25 km

.

2.2.1. Ground Observation Data

The ground SWE and SD observations were obtained from the daily Global Historical Climatology Network (GHCNd) [39]. We used the dataset “Global Landform classification” (https://esdac.jrc.ec.europa.eu/resource-type/soil-projects-data) (accessed on 27 May 2023) with a resolution of 1 km, which adopted Meybeck et al.’s definition of mountain and hill. The definition considers both the Relief Roughness (RR) value and elevation (RR = maximum minus minimum elevation per cell divided by half the cell length in meters/kilometer, or ‰) [40]. Records within the grid cells of mountainous areas (‘mountain’ and ‘hill’ in Meybeck et al.’s definition) were filtered to avoid the errors caused by deep snow [6,41]. The stations whose records are less than 15% of modeling days were also screened, so as to ensure that the modeling data come from sites with stable records. Finally, records from 564 stations served as the dependent variable in our models.

2.2.2. Geolocation Features

Longitude (LON) and latitude (LAT) data were derived from GADM. We generated a fishnet and extracted the geographic location information of each central point, then interpolated it to the whole study area to obtain the raster data of geolocation.

2.2.3. Snow and Environmental Factors

Snow factors include snowmelt (QSM), snow precipitation rate (SPR), and snow cover fraction (SCF). Environmental factors include air temperature (AT), soil temperature (underground 0–10 cm) (ST), heat flux (QG), and tree cover (TC). QSM, SPR, SCF, AT, ST, and QG data were obtained from the dataset “NCALDAS_NOAH0125_D” in the National Climate Assessment—Land Data Assimilation System (NCA-LDAS) [42]. TC data were obtained from the “Global Forest Change 2000–2021 Data” dataset [43]. The “Tree canopy cover for year 2000” and “Year of gross forest cover loss event” datasets were used to generate tree cover data.

2.2.4. Topographical Features

Topographical features, including elevation (ELV) and slope (SLP), with spatial resolution at 5 km, were obtained from the “Global 1, 5, 10, 100-km Topography” dataset [44].

2.2.5. Brightness Temperature Product

SD can be described as a function of the brightness temperature difference (TBD). Brightness temperature (TB) data were selected from “NSIDC-0630”, which is part of the NASA Making Earth System Data Records for Use in Research Environments (MEaSUREs) dataset [45]. The “NSIDC-0630” is a calibrated enhanced-resolution passive microwave dataset. In this study, the Special Sensor Microwave Imager/Sounder (SSMIS) platform (F-18) was used, offering four channels, including 19, 37, and 91 GHz in horizontal and vertical polarization and 22 GHz in vertical polarization. In recent studies, the 22 GHz channel was proved to be sensitive to freshly fallen and fine-grained snow clusters [46], and it was used as a model input in many studies of SWE estimation [47,48,49]. Hence, we considered 22 GHz and used it as a model input because of the effect of liquid water in the afternoon. Only observations from descending passes (time from 00:00 to 12:00) were selected because the snowmelt in the ascending orbit data is more severe than in the descending orbit data, and the liquid water in the snowpacks may result in the underestimation of SWE [50]. The 21 variables of TBD (TBD37h37v, TBD37h91h, TBD37h91v, TBD37h19h, TBD37h19v, TBD37h22v, TBD37v91h, TBD37v91v, TBD37v19h, TBD37v19v, TBD37v22v, TBD91h91v, TBD91h19h, TBD91h19v, TBD91h22v, TBD91v19h, TBD91v19v, TBD91v22v, TBD19h19v, TBD19h22v, and TBD19v22v) were obtained through subtraction between every two frequencies in the TB data with a raster calculator tool.

2.2.6. Dataset Integration for Modeling

All of these independent variables were resampled to 6.25 km. Due to the need for more data with higher spatial resolution, some studies obtained the datasets by resampling [27,51]. Therefore, the datasets from ‘NCALDAS_NOAH0125_D’, which have a worse spatial resolution (0.125°) than 6.25 km, were resampled to 6.25 km with the nearest neighbor resampling method to match other datasets. Using the geographical coordinates of the 564 GHCNd stations (with map projection of ‘North America Lambert Conformal Conic Projection’), all the values of independent variables mentioned above were extracted to build models. The min-max normalization method was performed for individual months on all of the independent variables mentioned above.

2.2.7. SWE Dataset for Independent Validation

The SWE observations from the Snow Telemetry (SNOTEL) network were used for independent validation. The SNOTEL network consists of over 900 automated data collection sites in the western U.S. The records with snow density values of 0, infinity, and greater than 1 g/cm³ or with SD greater than 50 cm were deleted. It has been confirmed that the shallow snow (5 cm) was transparent to microwave radiation [5,52], so shallow snow was excluded from the validation. The threshold of SD of 5 cm was converted to SWE using 0.24 g/cm⁻³ [53], so SWE observations below 12 mm were removed in this study. Moreover, similar to the process in Section 2.2.1, SNOTEL sites in mountain areas were also filtered. Finally, 11,492 SNOTEL records from 143 stations in DJF months in 2015 and 2016 were selected. We made comparisons between the SWE estimates from our model and four existing SWE datasets (

Table 2. Summary of datasets for independent validation.

Category	Method	Abbr.	Resolution	Source	Link
Reference SWE data for independent validation	Station observations	SNOTEL	/	SNOTEL	https://www.nrcs.usda.gov/ (accessed on 19 November 2022).
Comparison SWE dataset	Products from reanalysis	NCA	0.125°	NCALDAS_NOAH0125_D	https://ldas.gsfc.nasa.gov (accessed on 15 September 2022).
		ERA5	0.1°	ERA5	https://cds.climate.copernicus.eu/ (accessed on 23 May 2023).
	Independent passive microwave estimates	AMSR2	25 km	AMSR-E SWE v2.0	https://nsidc.org/data/ae_dysno/versions/2 (accessed on 26 October 2022).
	Passive microwave estimates combined with station observations	GlobSnow3	25 km	GlobSnow v3.0	https://www.globsnow.info/ (accessed on 23 May 2023).

), which can be classified into three types according to the study of Mortimer et al. [22]. They are (1) products from reanalysis (NCA-LDAS and ERA5); (2) independent passive microwave estimates (AMSR-E SWE v2.0); and (3) passive microwave estimates combined with station observations (GlobSnow v3.0).

3. Methodology

The RM-ESF model includes six steps shown in Figure 2: (1) multifactor selection; (2) spatial weights matrix construction; (3) eigen decomposition and eigenvector selection; (4) parameter estimation for the RM-ESF model; (5) assessment for the RM-ESF model; and (6) estimation and assessment for SWE estimations. In the classic TB gradient retrieval algorithm, snow density is assumed to be a constant value for all calculations while it has spatial heterogeneity. Thus, we generated SWE by multiplying the estimated SD and interpolated snow density.

3.1. Multifactor Selection

In this study, it is assumed that the pattern of SA is similar within each month, so the RM-ESF model was constructed for each month individually. The Pearson correlation coefficient (Pearson’s r) can be applied for continuous data and tests for a linear relationship, which is the situation in this study [54]. Thus, the Pearson correlation coefficient analysis was conducted to select factors for each month.

TBDs whose correlation coefficients with ground SD observations are greater than 0.1 were selected. For the highly correlated TBDs with correlation coefficients greater than 0.8, only the TBD with a higher correlation with SD was retained. For topographical, snow, and environmental factors, we selected the factors whose correlation coefficients with SD were greater than 0.1.

The preliminary nonspatial exploration was conducted to select variables from potential factors for each monthly model, as shown in Appendix A. Though the correlation coefficient of LAT did not reach the set threshold of 0.1, it was chosen as the dependent variable. This is because previous studies have illustrated that temperature and vegetation show a clear pattern of divergence in latitude and both are essential factors influencing snow mass distribution [28]. In general, there were positive correlations between LAT, LON, ELV, SLP, QSM, SPR, WS, TC, TBD19h22v, and SD. QG, ST, AT, TBD37v19v, and TBD91v19h were negatively correlated with SD.

3.2. Spatial Weights Matrix Construction

Spatial relationships can be expressed using a spatial weights matrix [55]. First, the K-Nearest Neighbors (KNN) algorithm was adopted to construct the spatial weights matrix for the N ground stations. In this study, k values of 5, 10, and 15 were tested, and 10 was chosen because the model performed best at this threshold. The parameter “k” was set to 10, which means the returned matrix W displays the adjacency of each point with the 10 nearest neighbors using Euclidean distance. Steps 1 and 2 are visualized in Figure 3.

3.3. Eigen Decomposition and Eigenvectors Selection

The decomposition of an initially hidden autocorrelation pattern in the binary spatial weights matrix W was needed to gather a more detailed interpretation of spatial autocorrelation [44]. First, $\mathrm{W}$ was turned into a centering matrix ${\mathrm{W}}_{\mathrm{c}}$ using Equation (1), where $\mathrm{I}$ refers to the identity matrix and ${11}^{\mathrm{T}}$ symbolizes a square matrix of size $\mathrm{n}$ with all “1’s” in it.

$${\mathrm{W}}_{\mathrm{c}}=\left(\mathrm{I}-\frac{{11}^{\mathrm{T}}}{\mathrm{n}}\right)\mathrm{W}\left(\mathrm{I}-\frac{{11}^{\mathrm{T}}}{\mathrm{n}}\right)$$

(1)

Next, the eigenvectors $({\mathrm{E}}_1,{\mathrm{E}}_2\dots {\mathrm{E}}_{\mathrm{n}})$ of ${\mathrm{W}}_{\mathrm{c}}$ were extracted. The corresponding eigenvalues ${(\mathsf{\lambda }}_1,{\mathsf{\lambda }}_2\dots {\mathsf{\lambda }}_{\mathrm{n}}$) can capture the nature and degree of the potential SA. To improve the sampling efficiency of potential SAs, only the eigenvectors for which $\mathsf{\lambda }/{\mathsf{\lambda }}_{\mathrm{m}\mathrm{a}\mathrm{x}}>0.25$ would be considered initially, given that the SA in SD is mainly positive [44].

3.4. Parameter Estimation for the RM-ESF Model

Eigenvectors and the selected factors were added by forward stepwise selection. To reduce the amount of computation, only the first 50 factors or eigenvectors which made the Akaike Information Criterion (AIC) of the present model lower were selected. The final RM-ESF model can be written as

$$\mathrm{Y}=\mathsf{\gamma }+\mathrm{X}\mathsf{\beta }+\mathrm{E}\mathsf{\alpha }+\mathsf{\epsilon }$$

(2)

where $\mathsf{\gamma }$ is a constant term; X represents an $\mathrm{n}\times \mathrm{i}$ matrix, in which the $\mathrm{i}$th column means the $\mathrm{i}$th independent variable and the $\mathrm{n}$th row refers to the $\mathrm{n}$th ground station; E represents an $\mathrm{n}\times \mathrm{j}$ matrix, in which the $\mathrm{j}$ columns are the $\mathrm{j}$ final filtered eigenvectors; $\mathsf{\beta }$ and $\mathsf{\alpha }$ represent the corresponding coefficient vectors; and $\mathsf{\epsilon }$ represents the residuals. The linear combination of selected eigenvectors, $\mathrm{E}\mathsf{\alpha }$, in Equation (2) is the ESFs that incorporate the spatial effects.

The RM model is a Resolution-enhanced Multifactor model, which is a multiple linear model without ESFs. It was built to test whether the model performance improves with the ESFs added in RM-ESF. The RM model can be written as

$$\mathrm{Y}=\mathsf{\gamma }+\mathrm{X}\mathsf{\beta }+\mathsf{\epsilon }$$

(3)

where the meaning of the letters is the same as those described in Equation (2).

3.5. Assessment for the RM-ESF Model

The assessment of the RM-ESF model contains two parts: (1) the VIF test and (2) the spatial autocorrelation analysis for the residuals. The Variance Inflation Factor (VIF) was introduced to measure the multicollinearity of the model. In general, VIF > 10 indicates a severe collinearity problem, and VIF > 5 is cause for concern [56]. The spatial autocorrelation analysis was conducted by calculating Moran’s I values of model residuals. The Moran’s I value in the model residuals will converge on 0 if spatial effects have been taken into consideration in the model. The VIF values of the RM-ESF model were all less than 5, which indicated that the model passed the VIF test and there was no collinearity problem (details in

Table A1. VIF of variables in RM-ESF models.

Variables	December	January	February
LAT	1.73	2.00	1.42
LON	2.15	2.20	2.13
ELV	3.96	4.53	/
SLP	1.51	1.55	1.30
QSM	/	/	1.13
SPR	1.09	1.09	1.06
AT	2.45	2.10	/
ST	/	/	2.90
QG	/	/	1.23
WS	1.17	1.23	1.07
TC	1.16	1.17	1.21
TBD19h22v	1.50	1.37	1.34
TBD19v22v	3.23	/	/
TBD37h91v	/	/	1.29
TBD37v19v	2.46	2.34	1.63
TBD37v91v	1.60	/	/
TBD91v19h	/	2.35	/

in Appendix D).

3.6. SWE Estimation and Assessment

Increases in the snow grain size or SWE will lead to increases in the TBD. Therefore, if fixed snow grain size and density were adopted to estimate SWE, there may be underestimation in shallow SWE and overestimation in deep SWE [1]. However, the snow grain size is difficult to measure due to its heterogeneity at different depths. Thus, the daily snow density for every station was calculated using SWE divided by SD from GHCNd records and interpolated. We tried the Kriging interpolation method, but it is difficult to determine the search radius and the semivariogram model. Moreover, due to the uneven distribution of the stations, the snow density interpolation maps generated by the Kriging method are unsmooth. Therefore, a simple and efficient inverse distance weighting (IDW) method was used, and the spatial resolution was set to 6.25 km. The SD was estimated by the RM-ESF model and multiplied by spatially interpolated snow density to obtain SWE estimates. Moreover, the SWE estimates were multiplied by the snow cover fraction to correct the snow mass in every grid cell.

The model was calibrated with the SWE observations from GHCNd in 2011–2014 and was used to estimate SWE in 2015 and 2016 using multifactor data. By comparing the SWE estimates and the SWE observations from SNOTEL in 2015 and 2016, we tested the performance of the RM-ESF model. Moreover, the accuracy of SWE estimates from the RM-ESF model and five SWE products in DJF months in 2015 and 2016 was also calculated. The SWE estimates and product estimates with different resolutions were extracted at the SNOTEL points through the “Extract Multi Values to Points” tool and compared with the SWE observations at the SNOTEL sites.

Accuracy was assessed using several evaluation metrics, including the correlation coefficient (Pearson’s r), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Positive Mean Error (PME), and Negative Mean Error (NME).

4. Results

4.1. Descriptive Statistics of Snow Parameters

Figure 4 shows the snow parameters (SD, SWE, and snow density) of the ground observations used for modeling in this study. The box plots have five sections according to the three-month winter (DJF) each year from left to right, with blanks indicating months not included in the modeling. A black line in each of the purple, green, and yellow bars represents the mean values of the snow parameters in December, January, and February, respectively. From Figure 4a,b, we can infer that the trend of SD and SWE is very similar. In every three-month winter period from 2011 to 2015, the SD and SWE keep increasing from December of that year to February of next year, except for January and February in 2011. The possible reason may be that in the three-month winter from December 2010 to February 2011, above-normal average temperatures were reported across the southwest of the U.S., which might lead to an acceleration of snowmelt [57]. As the snow depth increases, the depth’s average grain size increases. Thus, there is an increasing trend in the snow density from December to February in every winter, as can be inferred from Figure 4c. Specifically, snow density values are concentrated in the range of 0.20–0.28 g/cm³. Overall, Figure 4 shows that the snowpack condition is similar in the corresponding months of every winter, so we constructed three models (December, January, and February models) for the respective months to estimate snow mass.

The adoption of spatial methods relies on the premise that the SD displays spatial autocorrelation, so the Moran’s I values for SD observations were calculated for each month. The Moran’s I values are 0.601, 0.651, 0.649 in December, January, and February, with all of the p-values below 0.001, indicating that the SD is spatially autocorrelated.

4.2. Assessment of The Spatial Autocorrelation

When a variable is missing from a regression model and that variable accounts for a significant portion of the variation in the dependent variable, the unexplained variation may appear in the model residuals as SA [21]. If a missing variable has significant SA, the residuals are likely to exhibit a significant level of SA. Whether SA comes from the TB or other variables, only the SA among the residuals of the dependent variable is of interest, which implies that the variable is missing in the regression model. In this case, ESFs are proposed as a set of synthetic proxy variables for the unexplained variation. The ESFs are eigenvectors extracted from the spatial weights matrix, which ties geographic objects together in space. Then, the ESFs serve as control variables in the RM-ESF, identifying and separating the spatial dependencies from snow observations.

As shown in

Table 3. Moran’s I of the residuals in the RM and the RM-ESF model.

Time	RM		RM-ESF
	Moran’s I	p-Value	Moran’s I	p-Value
December	0.205	<0.001	−0.035	0.890
January	0.270	<0.001	−0.024	0.790
February	0.274	<0.001	0.068	0.008

, the RM-ESF model shows better performance than the RM model after ESF was introduced. Moran’s I values are high and significant for RM, while the Moran’s I values for the RM-ESF model are lower and insignificant. This indicates that spatial autocorrelation is filtered out, and the residuals can be regarded as random errors in the RM-ESF model.

4.3. Independent Validation of SWE Estimation

4.3.1. Overall Accuracy

Table 4. The criteria of the SWE estimates from the AMSR2, GlobSnow3, NCA, ERA5, the RM, and the RM-ESF model.

Criteria	AMSR2	GlobSnow3	NCA	ERA5	RM	RM-ESF
Pearson’s r	0.33 **	0.50 **	0.48 **	0.65 **	0.67 **	0.72 **
RMSE (mm)	131.38	100.03	98.80	67.33	62.57	56.70
MAE (mm)	115.45	83.58	81.94	51.82	48.72	43.88
PME (mm)	19.70	30.65	30.29	30.53	27.14	26.64
NME (mm)	−116.09	−87.69	−85.64	−56.99	−54.93	−49.74

Note: “**” means significant at 0.01 level.

shows the overall accuracy of the SWE estimates from all 11,492 records. The statistics illustrate that AMSR2 has the worst similarity with the SNOTEL SWE observations (r = 0.33), followed by NCA (0.48), GlobSnow3 (0.50), ERA5 (0.65), the RM model (0.67), and the RM-ESF model (0.72). ERA5 has the best performance among the SWE products, while AMSR2 is the worst. Compared with other products, the RMSE and MAE of the RM-ESF model are the lowest, with 56.70 mm and 43.88 mm, respectively. There is an obvious underestimation in all products and models, especially the AMSR2 product. The underestimation of the RM-ESF model is probably because GHCNd sites are typically located at low elevations. Because the GHCNd sites used for modeling are all located at relatively low elevations, similar to the study of Liu et al. [58]. Overall, the SWE estimates from the RM-ESF model achieve more accurate results than the other products.

4.3.2. Accuracy Evaluation under Different Land Cover Types

Land cover type is critical in snow redistribution and microwave emissions of the snowpack [59], so the estimates were tested under different land cover types. The annual IGBP classification in the “MCD12C1 v006” land cover dataset with a resolution of 0.05° × 0.05° was used in this study (https://lpdaac.usgs.gov, accessed on 16 October 2022).

Table 5. The criteria of SWE estimates from the AMSR2, GlobSnow3, NCA, the RM, and the RM-ESF model using different land cover types.

Criteria	SWE	All	Forest	Savanna	Grassland
		(11,492)	(330)	(5053)	(6109)
Pearson’s r	AMSR2	0.33 **	−0.05 **	0.30 **	0.42 **
	GlobSnow3	0.50 **	0.30 **	0.56 **	0.48 **
	NCA	0.48 **	0.25 **	0.49 **	0.48 **
	ERA5	0.65 **	0.87	0.70 **	0.59 **
	RM	0.67 **	0.44 **	0.72 **	0.65 **
	RM-ESF	0.72 **	0.54 **	0.77 **	0.69 **
RMSE (mm)	AMSR2	131.38	156.66	142.64	119.66
	GlobSnow3	100.03	135.64	103.50	94.70
	NCA	98.80	120.15	94.62	100.88
	ERA5	67.33	40.33	61.64	72.79
	RM	62.57	101.77	61.55	60.60
	RM-ESF	56.70	93.26	52.57	57.37
MAE (mm)	AMSR2	115.45	137.76	126.58	105.03
	GlobSnow3	83.58	113.82	87.22	78.94
	NCA	81.94	95.17	76.62	85.63
	ERA5	51.82	25.76	47.11	57.13
	RM	48.72	78.21	48.10	47.64
	RM-ESF	43.88	70.92	40.86	44.93
PME (mm)	AMSR2	19.70	/	11.52	20.65
	GlobSnow3	30.65	88.69	28.92	29.30
	NCA	30.29	4.50	33.65	25.47
	ERA5	30.53	19.09	29.20	33.59
	RM	27.14	10.06	24.12	29.45
	RM-ESF	26.64	10.62	25.76	27.64
NME (mm)	AMSR2	−116.09	−137.76	−126.76	−106.00
	GlobSnow3	−87.69	−115.81	−89.71	−84.34
	NCA	−85.64	−97.14	−81.02	−88.63
	ERA5	−56.99	−30.44	−52.19	−61.68
	RM	−54.93	−82.14	−54.40	−53.55
	RM-ESF	−49.74	−76.29	−46.12	−50.96

Note: “**” means significant at 0.01 level. The number of records is in the bracket.

shows that in the forest, ERA5 has the best similarity with the SNOTEL SWE observations (r = 0.87). However, it is insignificant. AMSR2 even exhibits negative correlations (r = −0.05) with SNOTEL SWE observations. The RM-ESF has relatively low correlations in forests when compared to savanna and grassland. ERA5 has the best performance in the forest for the lowest RMSE and MAE. The RM-ESF model surpasses the other products in all criteria in savanna and grassland, especially in savanna, with the strongest correlations (r = 0.77), lowest RMSE (52.57 mm), and lowest MAE (40.86 mm). From the PME and NME results, there is an obvious underestimation in the forest for all SWE datasets except for the ERA5 product.

4.3.3. Accuracy Evaluation across Different Months

Table 6. The criteria of SWE estimates from the AMSR2, GlobSnow3, NCA, the RM, and the RM-ESF model in December, January, and February months.

Criteria	SWE	All	December	January	February
		(11,492)	(4424)	(4236)	(2832)
Pearson’s r	AMSR2	0.33 **	0.24 **	0.08 **	0.08 **
	GlobSnow3	0.50 **	0.34 **	0.41 **	0.32 **
	NCA	0.48 **	0.55 **	0.55 **	0.36 **
	ERA5	0.65 **	0.64 **	0.59 **	0.51 **
	RM	0.67 **	0.61 **	0.68 **	0.47 **
	RM-ESF	0.72 **	0.66 **	0.75 **	0.52 **
RMSE (mm)	AMSR2	131.38	99.05	136.67	163.65
	GlobSnow3	100.03	86.89	94.62	124.15
	NCA	98.80	63.35	99.60	136.02
	ERA5	67.33	48.11	67.90	88.80
	RM	62.57	45.10	61.14	84.37
	RM-ESF	56.70	42.64	53.19	77.28
MAE (mm)	AMSR2	115.45	86.47	122.74	149.79
	GlobSnow3	83.58	72.84	78.47	108.02
	NCA	81.94	51.90	87.24	120.95
	ERA5	51.82	36.50	54.50	71.75
	RM	48.72	34.76	50.20	68.31
	RM-ESF	43.88	33.13	43.68	60.97
PME (mm)	AMSR2	19.70	7.11	22.50	37.87
	GlobSnow3	30.65	30.44	28.83	35.63
	NCA	30.29	31.31	27.45	30.42
	ERA5	30.53	20.56	34.82	46.07
	RM	27.14	23.21	24.54	38.96
	RM-ESF	26.64	22.54	23.35	37.87
NME (mm)	AMSR2	−116.09	−87.05	−123.38	−150.51
	GlobSnow3	−87.69	−75.71	−83.51	−112.26
	NCA	−85.64	−54.35	−89.81	−125.03
	ERA5	−56.99	−41.37	−58.75	−76.68
	RM	−54.93	−39.79	−54.76	−75.97
	RM-ESF	−49.74	−38.02	−48.27	−69.13

Note: “**” means significant at 0.01 level. The number of records is in the bracket.

shows the independent validation results of SWE estimates from DJF in 2015 and 2016. In December, AMSR2 has the highest Pearson’s r of 0.24 compared with other months, although it is still very low. The ERA5 and the RM-ESF models exhibit similar high correlations in DJF, and the RM-ESF model has higher correlations, especially in January (r = 0.75). The RMSE and MAE increase from December to February, probably because the SWE also increases. From the PME and NME results, there is an obvious underestimation for all SWE datasets in all three months, especially in February.

4.4. Mapping of the SWE Estimation

The daily SWE maps obtained from the RM-ESF model and other products are shown in Figure 5. The maps whose SWE are estimated by descending orbital brightness temperature data have large strip gaps across the study area, resulting in missing SWE information. All the SWE maps show similar overall temporal characteristics, with a broad snow cover area in December and a gradual decline in January and February. The snow tends to be shallow in December and gets deep from January to February. The heavy snow mass is usually distributed over the high elevation area in February.

It can be seen from the maps that compared with AMSR2 and GlobSnow3, the NCA, ERA5, and the RM-ESF models provide more details and show clearer spatial distribution of snow mass. The SWE high values in these three sources are concentrated in high-altitude mountains. The difference is that the spatial distribution of the NCA and the RM-ESF model is more fragmented and natural than that of ERA5. In addition, the overall value of NCA is low, followed by the RM-ESF model and ERA5, with a broader area of deep snow (SWE of 300–400 mm).

5. Discussion

5.1. Model Accuracy

ERA5 has the best performance among the SWE products, according to Table 6. For this reason, Figure 6 compared scatterplots of the SWE values of SNOTEL and ERA5 and the RM-ESF model (for the scatterplot images for other products, see Appendix B). The different colors of the points represent the different densities of points. The red dashed and solid lines are straight lines through the origin, with a slope of “1” and the corresponding correlation coefficient, respectively. The similarity with the reference SWE is relatively high in December and January, and low in February for both sources. The ERA5 and the RM-ESF model exhibited heavy underestimation in February. The point distribution for the RM-ESF model is more concentrated and closer to a straight line than that of ERA5, especially in January months.

The independent validation and the scatterplots above confirmed that the SWE estimates from the RM-ESF model significantly outperform AMSR2, GlobSnow3, NCA, and ERA5. However, the ERA5 performs slightly better than the RM-ESF model in the forest area. There are two possible reasons: one is that the forest sample is too small to build a stable model (only 8.91% of the modeling data is in the forest), and the other is that the forest canopy can reduce the reflection from the snow surface, thus weakening the brightness temperature difference, which causes the estimation bias [4]. The AMSR2 data perform the worst in all accuracy tests, which is similar to the conclusions of the work of Mortimer et al. [22].

From the maps of SWE, it can be found that SWE products from reanalysis (NCA and ERA5) show more information on SWE distribution, while the others (AMSR2 and GlobSnow3) fail to present more spatial details. For SWE estimates from only passive microwave or passive microwave with surface observations it can be difficult to generate accurate SWE data, though they cover a wide area.

As pointed out in the independent validation of this study, all SWE products suffer from the problem of having an underestimation of SWE. Similarly, Mortimer et al. found that there is an underestimation of SWE across a larger range in ERA5, GlobSnow2, and AMSR2 [22]. There are several possible explanations for this bias:

First, the negative bias may come from the meteorological forcing data. It was found that severe underestimation of precipitation by various meteorological forcing data is the leading cause of SWE underestimation in a recent study [60]. More reliable meteorological forcing data are required to improve the SWE estimation. Second, since thick snow is radiologically very similar to perennial ice, microwave remote sensing methods at snow depths can lead to an underestimation of snow depth [61]. Last, the mountainous areas have low SWE confidence when using passive microwave methods due to the change in the relative satellite look angle and angle of incidence for polarization in mountainous areas with complex terrain [62]. This requires strengthening surface observation networks to develop new methods for SWE estimation.

5.2. Brightness Temperature Difference in the RM-ESF Model

The selection of TBDs in the model varies with different time periods, according to

Table 7. Coefficients of the RM-ESF model.

Variables	December	January	February
Intercept	1076.06	663.04	1191.22
TBD19h22v	−507.86	−588.95	−221.73
TBD19v22v	−58.49	/	/
TBD37h91v	/	/	−27.72
TBD37v19v	−948.37	−668.26	−707.8
TBD37v91v	−163.54	/	/
TBD91v19h	/	190.95	/
LAT	89.59	226.38	−87.91
LON	−310.44	−397.88	−22.42
ELV	741.36	1045.9	/
SLP	81.17	54.33	258.64
QSM	/	/	391.52
SPR	696.05	655.36	497.26
AT	−404.84	−234.58	/
ST	/	/	−708.35
QG	/	/	22.1
WS	106.34	81.94	195.52
TC	33.66	75.92	118.71

, while the TBD of 37 GHz and 19 GHz and the TBD of 22 GHz and 19 GHz are selected every month. The best set of TBDs is 37 GHz and 19 GHz, which is the same as the set (or 37 GHz and 18 GHz for SMMR data) commonly used in several studies made at the regional scale [6,10]. Moreover, the TBD of 22 GHz and 19 GHz is unusually used in previous studies because, at frequencies above 19 GHz, radiation is expected to emanate only from thin surface layers. The 22 GHz channel is considered sensitive to water vapor [63], while the recent study by Golunov concluded that the 22 GHz is sensitive to freshly fallen and fine-grained snow clusters [46]. Moreover, because the water vapor is contained under low temperatures in the air and may eventually condense and fall as snow, the TB of the 22 GHz channel can capture the atmosphere conditions for snowfall. More physical principles between SWE and brightness temperature in different channels need to be studied in the future.

5.3. The Influence of Variables on Snow Mass Distribution

The coefficients in the RM-ESF model reflect the influence of individual factors on snow mass distribution, as shown in Table 7. The influence of factors varies in different months. We discussed the impact of the variables in this section.

As a premise for discussion, we divided DJF into two stages: the accumulation period (December and January) and the ablation period (February), according to the study of Li et al. [64], because the statistics show that the snow cover fraction value generally increases from December to January and starts to decrease in February, with mean values of 0.51, 0.63, and 0.59 for the respective months. Figure 7 shows the box plot of daily mean SCF values for each month.

The contribution of latitude to snowpacks is weak, because the impact of latitude turns from positive to negative from the snow accumulation to the ablation period, showing a complex correlation with snowpacks. The longitude affects the snow mass negatively in this study, while the results of other regional studies have not reached a consistent conclusion between longitude and snowpacks, which shows that the influence of longitude varies with different regions [23,26,27].

In all winter months, the elevation coefficient values remain high, indicating that elevation is an important factor in snow mass distribution. The moist air masses are forced to ascend the mountain slopes when the mountains block them, and as the elevation increases, the air temperature falls. Once the air temperature is low enough that the saturation level is attained, then the snowfall increases [37]. The slope has the greatest influence in February. Because, in the ablation period, the seasonal shallow snow tends to melt, and most of the perpetual snow concentrates on high-elevation areas with high slopes, showing a high interaction, which can also be inferred in Appendix A, where the two factors show a high correlation coefficient (about 0.5). Previous studies have also reported that elevation and slope had high interactions with other factors, such as wind speed [64].

The amount of snow depends on the difference between snowfall and snowmelt; both the snowmelt and snow precipitation rate positively contribute to snow mass. The amount of snow depends on the difference between snowfall and snowmelt, and both the snowmelt and snow precipitation rate have a positive contribution to the snow mass. It is common sense that the more snow melts, the less snow there is. However, snowmelt shows a positive effect, probably because during the ablation period, there is too little snow in the shallow snow areas to reach high snowmelt, and the areas with more snowmelt are the areas with deep snow.

Thermal factors, including air temperature and soil temperature, are the most direct factors influencing snow accumulation and ablation. The influence of surface heat flux is actually positive in the ablation period due to the spatial heterogeneity of snow depth observations. The wind is known to cause snow redistribution, which has a positive contribution to snow mass. Tree cover is also a significant factor, especially in the ablation period. This is because the tree canopy can reduce the wind speed and, thus, the magnitude of sensible and latent heat fluxes, which reduces snowmelt [65]. Many empirical studies have identified tree cover as a driver of snowmelt because the snow melts slower in forests than in open areas, lowering the snowmelt rate by up to 70% [66].

In conclusion, snow mass distribution is greatly influenced by topographical features, snow factors, and thermal factors. The effects of these attributes on snow mass distribution exhibited temporal heterogeneity.

5.4. ESFs in the RM-ESF Model

It was proved that SWE has a strong positive SA in this study, which means the snowpack distribution shows a spatial pattern by which high values tend to be located near other high values, and low values tend to be located near other low values. The SA in the SWE has two possible origins: one is from the geospatial distribution of snowpacks, and another may come from the optimal interpolation method of Poe in the MEaSUREs dataset, in which the antenna gain function is interpolated to produce higher sampling frequency data using the Backus–Gilbert theory [67]. The ESFs used to consider the SA in the RM-ESF model (Equation (2)) were interpolated, as shown in Figure 8. The spatial patterns of the interpolated ESFs in the accumulation period are very similar, showing positive and high values in the northeast and negative and low values in the southwest. However, in February, the ablation period, the negative and low values are mainly distributed in the northwest and east of the study area, which indicates that the spatial autocorrelation quantified by the ESFs values varies in different periods.

5.5. Limitations and Future Enhancements

The RM-ESF model assumed that the coefficients of each variable are fixed, which cannot capture the spatial heterogeneity caused by complex terrains. In future works, the ESF-based spatially varying coefficients model, which considers both spatial autocorrelation and spatial heterogeneity, can be used to improve estimation accuracy over the present model with constant coefficients.

In this study, it is assumed that there is a linear relationship between all independent variables and SWE, and the traditional retrieval algorithm is processed based on the assumption that the relationship between TBD and SWE is linear as well. The assumption of normal distribution is the key to the linear model. However, the relationship can be more complex and nonlinear. Applying the box-cox or log transformation to normalize the variable distribution may contribute a lot to our future research on simulating the complex nonlinear relationship. Moreover, nonlinear approaches can be developed to explore the nonlinear SWE estimation in the future.

Both the passive microwave data and nonpassive microwave variables have contributed to the RM-ESF model. In future works, models can be built by only using nonpassive microwave variables in these mountainous terrains as a control experiment to assess the additional contribution of passive microwave data. Moreover, the resolution of the SWE estimates from the RM-ESF model in this study depends on the resolution of the dataset used for modeling. Surface assimilation data with higher quality and finer spatial resolution are required to improve the proposed RM-ESF model and further enhance SWE estimates in the future.

6. Conclusions

In this study, a Resolution-enhanced Multifactor Eigenvector Spatial Filtering (RM-ESF) model was proposed to obtain resolution-enhanced SWE estimates by combining PMRS brightness temperature data, geolocation, topographical features, snow factors, and environmental factors. The estimated SWE data in the western U.S. outperformed the SWE products (AMSR2, GlobSnow3, NCA-LDAS, and ERA5) in both resolution and accuracy. Meanwhile, the RM-ESF model was proven to consider spatial autocorrelation and achieved promising results compared to the RM model. ESFs served as spatial effects when added to the RM-ESF model so that the spatial autocorrelation of snow distribution is considered. The coefficients of the RM-ESF model reveal the different influences of the independent variables on snow mass distribution. Although this work focuses on the western U.S., the model can be applied to other regions where finer spatial resolution SWE data are needed. This study offered a new approach for generating finer-resolution SWE data products with spatial autocorrelation considered. Finer-resolution SWE estimates from the proposed model will help to explain the response mechanism of SWE to environment and climate change.