# Using Ensemble-Based Systems with Near-Infrared Hyperspectral Data to Estimate Seasonal Snowpack Density

^{*}

## Abstract

**:**

^{2}= 0.90, RMSE = 44.45 kg m

^{−3}, BIAS = 3.87 kg m

^{−3}and NASH = 0.89).

## 1. Introduction

^{2}, Root mean square error (RMSE), BIAS, and Nash-criterion (NASH)).

## 2. Materials and Methods

#### 2.1. In Situ Measurements for the Calibration and Validation Database

^{2}, located in an open sector of the experimental site.

#### 2.2. Algorithm Development

#### 2.2.1. The Hybrid Model

- ❖
- Classification of the snow samples into one of the three snow classes using the classifier of the HM;
- ❖
- Estimation of each class’s density using the corresponding specific estimator [19].

#### 2.2.2. Development of the Ensemble-Based System

- ➢
- Parameterization of classifiers based on ensembles

_{bagg}was set to 25,000), which is one of the most commonly used algorithms to build EBSs [28]. It consists of randomly removing a part of the calibration database and to compute a new classifier using the remaining data using the CART algorithm. The result of each iteration is the calculation of a threshold. The end of this step is marked by the development of two random vectors (v1) and (v2) composed of “n = 25,000” classifiers. The appearance of the thresholds composing the random vectors made it possible to determine a threshold probability distribution for each of the discrimination variables of the HM classifier. This probability is characterized by a mean (μ) and a variance (σ). These two statistical moments are subsequently used to develop the ensemble-based classifier consisting of a nominal (N), lower (L) and upper (U) threshold. Based on these statistical moments, it was possible to quantify the classification uncertainty by using the following equations:

_{OT}). Each Optimal threshold (OT) is weighted according to its occurrence in the random vector (Table 1), fixed at three in our study. Table 1 summarizes the abscissa and weights related to each OT as proposed in the work of Tørvi and Hertzberg [30], where the mathematical details of the GQF demonstration and its validation can also be found. Thus, Equations (1) and (2) take the following forms:

_{OT})), n

_{OT}= 3.

- ➢
- Parameterization of estimators based on ensembles

_{OT}and j = 1: n

_{OT}; and where 1 refers to Lower (L), 2 refers to Nominal (N), and 3 refers to the Upper threshold (U); Figure 5); ωi and ωj are the weights associated with each particular optimal threshold for V1 and V2 (ω

_{1}= ω

_{3}= $\frac{1}{6}$ and ω

_{2}= $\frac{2}{3}$; Table 1); V1 and V2 are the discrimination variables computed by the CART method; OTV1i and OTV2j are the optimal thresholds calculated by the GQF for V1 and V2 (Equations (3) and (4)); ${\mathrm{Est}}_{\mathrm{OTV}{1}_{\mathrm{i}}^{\uparrow}}$ is the set of estimators (n

_{OT}) trained by the subgroups of the WMM snow class; ${\mathrm{Est}}_{\mathrm{OTV}{1}_{\mathrm{i}}^{\downarrow},\mathrm{OTV}{2}_{\mathrm{j}}^{\uparrow}}$ is the set of estimators (${\mathrm{n}}_{\mathrm{OT}}^{2}$) trained by the subgroups of the MHM snow class; ${\mathrm{Est}}_{\mathrm{OTV}{2}_{\mathrm{j}}^{\downarrow},\mathrm{OTV}{1}_{\mathrm{i}}^{\downarrow}}$ is the set of estimators (n

_{OT}) trained by the subgroups of the snow class HVM; and k and p are the indices of the OT related to V1 and V2, respectively (k ≤ n

_{OT}and p ≤ n

_{OT}). More details are available in Appendix A.

#### 2.3. Accuracy Assessment

^{2}), BIAS, Root mean square error (RMSE), and Nash–Sutcliffe efficiency (NASH). The NASH criterion assesses performance based on the estimated values and the mean of the in situ measurements. A negative NASH value indicates that the mean of the measurements is more accurate than the model estimates; a NASH value of 1.0 means that the model is perfect [34]. The mathematical equations for the statistical indices used are as follows:

## 3. Results and Discussion

#### 3.1. Analysis of In Situ Snow Data

#### 3.2. Estimator Calibration

^{2}= 0.77 for specific estimators 13 and 14) to high (R

^{2}= 0.97 for estimator 3). It is important to note that the specific estimators designed to estimate the three snow classes (WMM, MHM and HVM) were trained using linear functions.

#### 3.3. Evaluation and Validation of the Ensemble-Based System

^{2}, as it compares the estimates to the mean of the observed measurements and is therefore not influenced by extreme values. An RMSE of 44.45 kg m

^{−3}for such a range of densities is a very acceptable error. The slightly positive BIAS indicates that the EBS tends to overestimate the snow density. The robustness of the EBS was also confirmed by the scatterplot of the in situ measurements against their estimates (Figure 7), where all points are well distributed with respect to the 1:1 line. In summary, the EBS performs similarly to the HM.

#### 3.4. Reliability Test

_{NASH}= 0.18 and std

_{RMSE}= 15.04 kg m

^{−3}versus std

_{NASH}= 0.02 and std

_{RMSE}= 4.27 kg m

^{−3}for the HM and EBS, respectively). Furthermore, the boxplot results support the histogram results. The boxplots (NASH and MSE) for the MBME are narrower with less data falling outside the normal (boxplot) which means that the values, whether NASH or RMSE, belong to the same population, in other words, the EBS estimates for the 1K iterations do not vary much. This is in contrast to the HM results, where we notice more values that fall outside the boxplot and therefore reflect the non-robust nature of the HM (Figure 9).

## 4. Conclusions

^{2}= 0.90, despite some relatively low-density values (90–120 kg m

^{−3}). However, the validation process showcased that the EBS underestimated density values. Since this is a systematic error, it could be corrected during the modelling process.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Scheme of training subgroup selection in one of the moderate snow class cases (brown points).

## Appendix B

**Figure A2.**Flow chart of the developed algorithm for density estimation (Samp is the number of samples; OTV1 and OTV2 refer to the optimal thresholds of V1 and V2 and Est are the specific estimators as described in Table 3).

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**Figure 3.**NIR spectral reflectance acquisition device for the vertical profile of snow samples from the company RESONON (Resonon Inc., Bozeman, MT, USA).

**Figure 4.**Density class determination using the hybrid model. The light blue, the marine blue, and the dark blue dots indicate the WMM, MHM, and HVM classes, respectively, and represent the calibration database used to calibrate the corresponding estimators. The grey lines are the two discrimination thresholds used based on wavelengths V1 (1161 nm) and V2 (1024 nm).

**Figure 5.**The two-dimensional GQF scheme using variables V1 and V2 and its application to calibrate the EBS. The black, red, and blue lines represent the optimal thresholds (lower, nominal, and upper, respectively) for V1 (OTV1L = 0.632, OT1N = 0.648 and OTV1U = 0.664) and V2 (OTV2L = 0.468, OTV2N = 0.480 and OTV2U = 0.492). k (1–3) and p (1–3) are the indices of the optimal thresholds associated with V1 and V2. The light blue, marine blue, and dark blue points represent the data used for training (WMM, MHM, and HVM, respectively) to calibrate the specific estimators. At the top right is the flow chart of the simplified EBS operational mode. The light blue, marine blue, and dark blue boxes designate the areas with WMM, MHM, and HVM classes, respectively. The overlapping areas in light and dark grey indicate the transitions between low-moderate and moderate-high densities, respectively. The terms of the equations are detailed in Equation (6). A detailed flow chart is presented in Appendix B.

**Figure 7.**Snow density estimated by the two models (

**a**) EBS and (

**b**) HM compared to the in situ measurements for the independent database.

**Figure 8.**Histograms showing the robustness test between EBS and HM: (

**a**) standard deviation (NASH

_{EBS}) = 0.02, (

**b**) standard deviation (NAS

_{HHM}) = 0.18, (

**c**) standard deviation (RMSE

_{EBS}) = 4.27 kg m

^{−3}, (

**d**) standard deviation (RMSE

_{HM}) = 16.04 kg m

^{−3}.

**Figure 9.**Boxplots representing the distribution of estimated snow density values expressed in (

**a**) NASH and (

**b**) RMSE, according to the two models EBS and HM.

Optimal Threshold (OT) | Abscissa (Z_{i}) | $\mathbf{Weight}\left({{\omega}}_{{i}}\right)$ |
---|---|---|

1 | 0 | 1 |

2 | −1, +1 | $\frac{1}{2}$$,\frac{1}{2}$ |

3 | $-\sqrt{3}$$,0,+\sqrt{3}$ | $\frac{1}{6}$$,\frac{2}{3}$$,\frac{1}{6}$ |

**Table 2.**Distribution of snow density as a function of snow grain type and size (field data: 2018, 2019 and 2020 [15].

Snow Class | Type of Grain | Grain Size (mm) | Number of Samples (N) | Density (kg m^{−3}) |
---|---|---|---|---|

WMM | $\left(\begin{array}{c}+\\ \mathsf{\lambda}\end{array}\right)$ | <1 mm | 19 | 100–250 |

MHM | $\left(\begin{array}{c}\square \\ \u2022\end{array}\right)$ | 1–2 mm | 59 | 150–400 |

HVM | $\left(\begin{array}{c}\u02c4\\ \u1d3c\end{array}\right)$ | >2 mm | 36 | 350–650 |

**Table 3.**Calibration equations for each specific estimator. R

^{2}is the coefficient of determination of the multivariate regressions; Samp is the size of the training dataset of each estimator; ${\mathrm{Exp}}_{\mathrm{var}}$ is the explanatory variable; U, N, and L are the Upper, Nominal and Lower thresholds; the arrow (↑) indicates an estimator trained with data superior to the threshold; and the downward arrow (↓) indicates an estimator trained with data inferior the threshold; the colors light blue, marine blue, and dark blue refer to the experts used to calculate the snow density of the WMM, MHM and HVM classes, respectively.

Snow Class | Estimator ID | Specific Estimator | Calibration Equation | R^{2} | Samp | ${\mathbf{Exp}}_{\mathbf{var}}\left(\mathbf{nm}\right)$ |
---|---|---|---|---|---|---|

WMM | 1 | ${\mathrm{Est}}_{\mathrm{TV}{1}_{\mathrm{L}}^{\uparrow}}$ | −1119.75 × SI_{SUB} (1,282,941) − 167.59 | 0.91 | 17 | 1282–941 |

2 | ${\mathrm{Est}}_{\mathrm{TV}{1}_{\mathrm{N}}^{\uparrow}}$ | −877.36 × SI_{SUB} (1,452,968) − 433.25 | 0.95 | 15 | 1452–968 | |

3 | ${\mathrm{Est}}_{\mathrm{TV}{1}_{\mathrm{U}}^{\uparrow}}$ | −967.69 × SI_{SUB} (1,666,935) − 425.24 | 0.97 | 13 | 1666–935 | |

MHM | 4 | ${\mathrm{Est}}_{\mathrm{TV}{1}_{\mathrm{U}}^{\downarrow},\mathrm{TV}{2}_{\mathrm{U}}^{\uparrow}}$ | −1491.40 × SI_{NOR} (1,600,946) − 951.09 | 0.83 | 45 | 1600–946 |

5 | ${\mathrm{Est}}_{\mathrm{TV}{1}_{\mathrm{U}}^{\downarrow},\mathrm{TV}{2}_{\mathrm{N}}^{\uparrow}}$ | −1491.40 × SI_{NOR} (1,600,946) − 951.09 | 0.83 | 45 | 1600–946 | |

6 | ${\mathrm{Est}}_{\mathrm{TV}{1}_{\mathrm{U}}^{\downarrow},\mathrm{TV}{2}_{\mathrm{L}}^{\uparrow}}$ | −1432.65 × SI_{NOR} (1,617,946) − 880.87 | 0.84 | 48 | 1617–946 | |

7 | ${\mathrm{Est}}_{\mathrm{TV}{1}_{\mathrm{N}}^{\downarrow},\mathrm{TV}{2}_{\mathrm{U}}^{\uparrow}}$ | −1397.68 × SI_{NOR} (1,617,941) − 854.96 | 0.80 | 43 | 1617–941 | |

8 | ${\mathrm{Est}}_{\mathrm{TV}{1}_{\mathrm{N}}^{\downarrow},\mathrm{TV}{2}_{\mathrm{N}}^{\uparrow}}$ | −1397.68 × SI_{NOR} (1,617,941) − 854.96 | 0.80 | 43 | 1617–941 | |

9 | ${\mathrm{Est}}_{\mathrm{TV}{1}_{\mathrm{N}}^{\downarrow},\mathrm{TV}{2}_{\mathrm{L}}^{\uparrow}}$ | −1427.73 × SI_{NOR} (1,617,941) − 877.38 | 0.82 | 46 | 1617–941 | |

10 | ${\mathrm{Est}}_{\mathrm{TV}{1}_{\mathrm{L}}^{\downarrow},\mathrm{TV}{2}_{\mathrm{U}}^{\uparrow}}$ | −1480.06 × SI_{NOR} (1,600,946) − 940.11 | 0.80 | 41 | 1600–946 | |

11 | ${\mathrm{Est}}_{\mathrm{TV}{1}_{\mathrm{L}}^{\downarrow},\mathrm{TV}{2}_{\mathrm{N}}^{\uparrow}}$ | −1480.06 × SI_{NOR} (1,600,946) − 940.11 | 0.80 | 41 | 1600–946 | |

12 | ${\mathrm{Est}}_{\mathrm{TV}{1}_{\mathrm{L}}^{\downarrow},\mathrm{TV}{2}_{\mathrm{L}}^{\uparrow}}$ | −1419.73 × SI_{NOR} (1,617,946) − 868.75 | 0.82 | 44 | 1617–946 | |

HVM | 13 | ${\mathrm{Est}}_{\mathrm{TV}{2}_{\mathrm{U}}^{\downarrow}}$ | −26,859.26 × SI_{NOR} (979,974) + 82.90 | 0.77 | 28 | 979–974 |

14 | ${\mathrm{Est}}_{\mathrm{TV}{2}_{\mathrm{N}}^{\downarrow}}$ | −26,859.26 × SI_{NOR} (979,974) + 82.90 | 0.77 | 28 | 979–974 | |

15 | ${\mathrm{Est}}_{\mathrm{TV}{2}_{\mathrm{L}}^{\downarrow}}$ | −1378.90 × SI_{SUB} (14,411,122) + 1207.81 | 0.86 | 22 | 1441–1122 |

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**MDPI and ACS Style**

El Oufir, M.K.; Chokmani, K.; El Alem, A.; Bernier, M. Using Ensemble-Based Systems with Near-Infrared Hyperspectral Data to Estimate Seasonal Snowpack Density. *Remote Sens.* **2022**, *14*, 1089.
https://doi.org/10.3390/rs14051089

**AMA Style**

El Oufir MK, Chokmani K, El Alem A, Bernier M. Using Ensemble-Based Systems with Near-Infrared Hyperspectral Data to Estimate Seasonal Snowpack Density. *Remote Sensing*. 2022; 14(5):1089.
https://doi.org/10.3390/rs14051089

**Chicago/Turabian Style**

El Oufir, Mohamed Karim, Karem Chokmani, Anas El Alem, and Monique Bernier. 2022. "Using Ensemble-Based Systems with Near-Infrared Hyperspectral Data to Estimate Seasonal Snowpack Density" *Remote Sensing* 14, no. 5: 1089.
https://doi.org/10.3390/rs14051089