## 1. Introduction

The spatiotemporal evolution of seasonal snowpacks is an important indicator of climate [

1]. The measurement, monitoring, and management of this water resource are of great interest to governments and the scientific community. Snow cover is the set of snow layers that accumulate on the ground throughout the winter [

2]. Each of these layers has a given density. Quantifying the variability of density in time and space is essential for estimating the water equivalent [

3,

4], hydroelectric power production [

5,

6], and assessing natural hazards (avalanches, floods, etc.) [

7,

8]. Indeed, this variable varies with changes in other physical properties such as grain size, grain shape, and liquid water content during the metamorphic transformation of the snowpack [

9]. The density of newly deposited snow is expected to have the lowest values and to increase and reach its highest values during the maturation phase. According to Pomeroy et al. [

10], the typical seasonal density of the snowpack ranges between 80 kg m

^{−3} and 600 kg m

^{−3}.

For a country like Canada, which covers a very large area with a vast expanse of snow cover, regular monitoring of snow density is important [

11,

12]. The density is measured using a variety of methods and technologies. These include manual measurements by taking core samples from the snowpack (such as ‘federal’ snow tubes, e.g., ESC-30) [

13], or the installation of devices that lie flat on the ground and weigh the snow as it accumulates on top (such as snow pillows) [

14,

15]. However, each of these methods has several drawbacks [

15,

16]. Snow core measurements are labor intensive, time-consuming, not feasible for 24-h data collection, and subject to human error. Snow pillows have measurement errors, logistical and transport problems for their installation, and can only measure an area of about 10 m

^{2} [

15]. There are also other methods for measuring snow density, including proximal remote sensing (such as the GMON (GammaMONitor) snow water equivalent probe (Campbell Scientific Canada, Edmonton, AB, Canada)) [

17], spatial remote sensing (microwave remote sensing) [

18]. However, these methods have some drawbacks; for example, they do not measure the density of each snow layer that makes up the vertical stratigraphy of the snowpack, but only the average density of the snowpack. Recently, optical sensor data have been used as an alternative to monitor snow cover over large areas and have led to improved monitoring and management of this water resource [

19]. To optimize the modeling process of the physical properties of snow and to develop new efficient models, a high spectral resolution is essential [

20].

Hyperspectral imaging technology is an innovative approach based on spectroscopic analyses. It is fast, non-invasive, and facilitates real-time measurements [

21,

22], which can be used in conjunction with traditional measurement methods. This technology has proven to be effective for field, laboratory, and industrial applications [

23,

24]. It provides detailed information about the physical and chemical components of a scanned sample due to its high spectral and temporal resolution [

23,

25,

26]. It has already been demonstrated that the near-infrared (NIR) spectrum is sensitive to the physical parameters of snow [

27,

28,

29,

30]. In fact, snow granulometry is clearly visible in the NIR and the short waves of infrared regions (SWIR) [

31,

32]. Eppanapelli et al. [

33] found that the spectral reflectance of snow in the NIR is inversely proportional to the liquid water content in the snow. In addition, the absorption of ice in the NIR is very high [

34], so the effect of impurities such as mineral dust and soot is negligible beyond 1000 nm wavelength. The above findings highlight the potential of hyperspectral NIR data to gather information on the physical properties of snow for modeling purposes [

35].

Indeed, several models and approaches designed to model snow density based on spectral data are now available, but none has yet achieved a satisfactory performance [

36,

37]. This is probably due to the fact that most models are based on the assumption that density measurements can be modeled using the same function. Even though the spectral reflectance of snow in the NIR depends on density, it is expressed by the size and shape of the grains (granulometry) and the liquid water content in the snow [

38]. Consequently, the optical properties of the physical parameters of snow influence one another and create a non-bijective (surjective (In mathematics, a surjective function (also known as surjection, or onto function) is a function f that maps an element x to every element y; that is, for every y, there is an x such that f(x) = y.)) relationship between snow density and reflectance in the NIR, resulting in poor correlations.

Recently, it has been demonstrated that three optical classes of snow with different degrees of metamorphosis (weakly to moderately metamorphosed (WMM), moderately to highly metamorphosed (MHM), and highly to very highly metamorphosed (HVM)) can be identified and discriminated against without prior recognition, based only on NIR hyperspectral data [

39]. This study showed that the spectra of snow density are similar within the same optical class and significantly different from one optical class to another. In other words, densities expressed in terms of grain size, shape, and spectral response were discriminated and grouped into three different homogeneous subclasses [

39]. With this finding, it is possible to train estimators specific to the identified homogeneous classes, which are governed by a bijective (In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements.) relation between the density and the hyperspectral NIR reflectance.

The objective of this study is to develop a hybrid model (HM) to estimate the snow density using proximal NIR hyperspectral data. The HM is a combination of a classifier and specific estimators associated with three density classes (WMM, MHM, and HVM). The HM was calibrated and validated using a data set collected from a sampling site located in Quebec City, Canada. The performance of the HM was assessed using the leave-one-out cross-validation technique and independent validation data using a systematic data splitting technique. Four statistical evaluation indices (the coefficient of determination (R^{2}), root mean square error (RMSE), the bias (BIAS), and Nash-criterion (NASH)) were used to assess the model’s performance.

## 4. Conclusions

The objective of this study was to test the performance of a hybrid model (HM) designed to estimate the density of the seasonal snowpack using hyperspectral NIR imaging (900–1700 nm) at a spectral resolution of 5.5 nm. The hybrid model is a combination of a classifier and three specific estimators (weakly to moderately metamorphosed snow (WMM), moderately to highly metamorphosed snow (MHM), and highly to very highly metamorphosed snow (HVM)). The hybrid model was evaluated at two levels: using the leave-one-out cross-validation (LOOCV) algorithm and using the systematic division validation technique (SSV). The LOOCV technique was used to assess the three specific estimators, and the SSV data were used to assess the performance of the HM.

The calibration step, based on a stepwise multivariate regression, showed that the three classes of snow are sensitive to different regions of the NIR spectrum, limited to the short and long wavelengths. The WMM was sensitive to the wavelengths at 1265 nm and 941 nm, the MHM was sensitive to wavelengths at 1617 nm and 941 nm, and the HVM was sensitive to wavelengths at 1424 nm and 1188 nm. The LOOCV technique highlighted that the specific estimators of all classes tend to slightly overestimate the snow density (BIAS < 0.1 kg·m^{−3}). When the HM was challenged with SSV data, the modeling results were satisfactory with an R^{2} = Nash = 0.93, and the snow density was slightly underestimated (BIAS = 1.03 kg·m^{−3}).

The objective of this study was to develop a method based on the optical properties of snow to be used conjointly with conventional density measurement methods with the aim of alleviating field operations. The critical step in estimating snow density using the HM is the selection of the final specific estimator. Indeed, classification algorithms (such as CART) are known to be local and unstable. This instability can significantly affect the accuracy of the density using the specific estimators of the HM. In other words, for an ideal modeling process using the HM, the sample to be modeled must be well classified so that the specific estimator corresponding to that class is used for optimal density estimation. Otherwise, a wrong specific estimator will be selected, and consequently, the estimation will not be optimal, which could affect the accuracy. For example, for a measured density of 581 kg m^{−3} (classified as HVM), the relative errors vary by 5%, 39%, and 75% when estimated using specific estimators of HVM, MHM, and WMM, respectively. On the other hand, another obstacle associated with this method is the correct selection of homogeneous snow layers in the field and on the recovered hyperspectral image. For this reason, additional field campaigns need to be conducted to collect more data to overcome this weakness and allow proper field implementation. The HM provides an improved tool to monitor the evolution of seasonal snowpacks, with a satisfactory level of performance even for low to moderate snow densities. We conclude that our results are an important first step toward the development of an effective method for continuous monitoring of snow density profiles in the field.