# Snow Cover Reconstruction in the Brunswick Peninsula, Patagonia, Derived from a Combination of the Spectral Fusion, Mixture Analysis, and Temporal Interpolation of MODIS Data

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## Abstract

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## 1. Introduction

^{2}covering a large part of the Brunswick Peninsula (~53°S), Patagonia, was detected in the last 45 years (1972–2016 [3]), attributed to a significant increase in winter temperatures, estimated as 0.71 °C at the weather station of Punta Arenas (Figure 1).

## 2. Study Area

^{2}) (http://www.subdere.gov.cl/división-administrativa-de-chile/gobierno-regional-de-magallanes-y-antártica-chilena, accessed on 20 December 2020). The water balance in the Brunswick Peninsula is critically important regarding the water supply to the population. In addition, the presence of seasonal snow has allowed the development of the ski resort “Club Andino”, in 1938, located in Cerro Mirador, at an elevation of 350–625 m, 8 km from the Punta Arenas city center. Snow cover has shown a relevant reduction over the last several decades, mainly due to atmospheric warming [3], significantly reducing the operation of the ski center, which only managed to open for a period of less than 1 month in 2016–2019 (Table 1), whereas until the 1990s, the skiing season typically lasted for over 2 months (Rodrigo Adaros, personal communication). The lack of snow at Club Andino has caused the regional government to explore an alternative location at Tres Morros hill, 31 km southwest of Punta Arenas city, at an elevation of 550–820 m, as a possible alternative site for the development of a future skiing area.

## 3. Data and Methods

#### 3.1. Data

#### 3.1.1. Satellite Sensor Data

#### 3.1.2. Weather Stations

^{−2}), both incoming and outgoing.

#### 3.2. Methods

#### 3.2.1. Downscaling and Spectral Fusion

^{2}) and root mean square error (RMSE). In contrast to the work of Wang et al. (2015) [55], here we compared the different approaches for an image with snow and another without snow, since this study focused on improving the representation of snow cover and its spectral interactions with other land-cover types.

#### 3.2.2. Spectral Mixture Analysis

#### 3.2.3. Spatio-Temporal Snow Reconstruction

- (a)
- Cloud and snow masks

- (b)
- Temporal interpolation

#### 3.2.4. Ground Validation

#### 3.2.5. Snow Cover Variability

## 4. Results

#### 4.1. Spectral Fusion

#### 4.1.1. The First Term of Spectral Fusion: The Linear Relationship

_{500}), as indicated in Table 3. The distribution of the residuals shows a clear violation of heterogeneity, justifying the recommendation of Wang et al. (2015) [55] to use a GLS instead of an OLS model. Additionally, the existence of a spatial correlation was reviewed with a variogram analysis of its residuals (Figure 5), with a clear spatial dependence that was considered.

_{500}), with a selection of variance structures for each band, according to the protocol, as indicated by Zuur et al. (2009) [60]. For the example image (25 September 2015), the following variance structures were selected: Band_3

_{500}power of the covariate variance structure; Band_4

_{500}exponential variance structure; and Band_5

_{500}, Band_6

_{500}, and Band_7

_{500}constant plus-power values of the variance covariate function.

_{500}. The GWR model has the advantage of generating coefficients for each pixel. The coefficients of determination (R

^{2}) between the modeled coarse bands using GWR

_{500}and the original coarse bands varied between 0.97 and 0.99 were, Band_3

_{500}= 0.99, Band_4

_{500}= 0.99, Band_5

_{500}= 0.97, Band_6

_{500}= 0.98, and Band_7

_{500}= 0.97, with low RMSE values for all models (Table 4).

_{500}and GLS

_{500}, the key issue was the estimation of the regression coefficients for the coarse bands, assumed to be universal at different spatial resolutions. Consequently, the relationship built at coarse spatial resolutions could be applied at a higher spatial resolution [56]. This assumption was the same for the GWR

_{500}model; but, since there were coefficients for each pixel (β

_{0}and βi in Equation 4), these were assigned to the highest-resolution imagery on a pixel by pixel basis of the original data as an additional information layer.

#### 4.1.2. The Second Term of Spectral Fusion: The Kriging Interpolation

_{500}, GLS

_{500}, and GWR

_{500}), so as to redistribute the error at a resolution of 250 m. First, we needed to select the best model representing the residuals’ spatial correlations (variograms). Once this model was estimated for each band, we proceeded with implementing the ATAK interpolation to obtain the second term of Equation (2) (${Z}_{v2}^{l}$). This comparative process is detailed in Script N°1 of (Supplementary Materials and codes), and its results are shown in Table 4.

_{500}model were the best option; once implemented in the fine-resolution data (250 m), the three ATAK models achieved a similar behavior based on the QNR index. The best results correspond to the GLS_ATAK

_{250}model, with a QNR index of 0.949. However, regarding the results for each band’s UQIs, the GLS model was the best (Table 4). The QNR index represents the performance of each spectral fusion model, where, in our case, we considered the spatial (Ds) and spectral (Dγ) distortions with equal relevance. An experiment with the GWR

_{250}model without an ATAK interpolation was performed, obtaining a QNR index of 0.914, which increased to 0.946 after applying the ATAK interpolation (GWR_ATAK

_{250}). Obviously, the improvement was not so significant compared to the OLS and GLS models. This can be explained in part by the spatial correlations of its residuals (Equation (2)), which lacks a good model representation. On the other hand, the computation time of the GLS_ATAK

_{250}method was almost three-times faster than the GWS_ATAK

_{250}, which was preferable, given that one of our goals was to have an operational methodology for a large amount of data to be processed.

#### 4.2. SMA

#### 4.3. Spatio-Temporal Snow Reconstruction

#### 4.3.1. Cloud and Snow Masks

#### 4.3.2. Temporal Interpolation

#### 4.4. Ground Validation with AWS Data

^{2}increased to 0.48. AIC values were -63 (linear) and -249 (GAM), respectively, indicating that the GAM model provided the preferred approach. Nonetheless, for snow height values below 20 cm (presented in the relation GAM 20 cm), fSC values of 0 were found, suggesting that the threshold of the fSC in the snow mask (fSC > 0.2) for this pixel left some fSCs undetected. Table 6 shows the results of the AWS Tres Morros data vs. the reconstructed values. As for the confidence of snow detection at this pixel, we found a 98% coincidence between the fSC (with a threshold of 10%) and snow height measurement at AWS Tres Morros, using a threshold of 5 cm.

#### 4.5. Reconstructed Snow Cover Variability in the Brunswick Peninsula

^{2}per year, and after applying a Holt–Winters filter, it showed a significant (at 95%) decreasing trend of 4.7 km

^{2}per year. The reconstructed snow days (snow season days) for the whole period (2000–2020) showed a trend of +0.11 days/year (not significant); but, by applying an exponential filter, it presented a significant trend of +0.54 days/year in the Brunswick Peninsula (p-value < 0.005). However, for the last 10 years (2010–2020), a significant decreasing trend of −4.64 snow days/year was observed in the Brunswick Peninsula (p-value < 0.001).

## 5. Discussion

#### 5.1. Method Improvement

#### 5.2. Climatic Forcing

^{2}), reconstructed from the MODIS data, and the ERA5 Land climate variables for the cold season (April to October).

^{2}of 0.48 (p < 0.001). We then adjusted a linear and non-linear GAM model to explain the MODIS snow area based on the mean atmospheric temperature, liquid precipitation, and solid precipitation. In those two models, only the mean temperature proved to be significant, while the other two variables (liquid and solid precipitations) did not result in a significant relation to snow cover extent.

^{2}= 0.86) between ERA5 Land product and AWS Tres Morros data in 2018–2020. This correlation increases significantly to R

^{2}= 0.98 on a monthly scale (Figure 14b). With this validation, we used ERA5 land temperature data to check the performance of RegCM4 at AWS Tres Morros (2000–2005), obtaining R

^{2}= 0.53, which the GAM model adequately represented due to its non-linear behavior (Figure 14b).

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Study area within the Brunswick Peninsula, showing the 4 selected basins in latitude/longitude and UTM 19S WGS84 coordinates. White circles are runoff stations and yellow circles are referential sites for ski activities (Cerro Mirador and Tres Morros). Contour line interval (in orange) is 100 m. (

**b**) Reference location of the study area in southern South America. Figure designed using open-source software QGIS v_3.16 (QGIS Geographic Information System. QGIS Team (2017). QGIS Geographic Information System. Open Source Geospatial Foundation Project. Available Online at: https://qgis.org, accessed on 29 October 2020).

**Figure 3.**Snow spectral signatures of different snow grain sizes. MODIS bands B1 to B7 are shown as green vertical lines. Image modified from Painter et al. (1998) [47].

**Figure 4.**Reflectance values of MODIS (MOD09GA) bands in snow and snowless images. The reflectance values ([0, 1]) are stretched to match the MODIS reflectance scale ([−100, 16,000]).

**Figure 5.**Variograms of linear regression residuals for each coarse band in the Brunswick Peninsula watersheds. The colors in the upper-left image represent the reflectance values (yellow represents high and black represents low values) of band 1 in an image from 25 September 2015.

**Figure 6.**Spectral fusion results using the GLS-ATAK models. From left to right: residuals of ATAK at 250 m, GLS at 250 m, GLA-ATAK at 250 m, and coarse band at 500 m. The 16-bit reflectance units for each band are expressed at a scale of [−100, 16,000], and the coordinates of the map are in UTM. MODIS image of 25 September 2015.

**Figure 7.**Endmember signatures and their seasonal variations in the Brunswick Peninsula. Reflectances are in % and represent spectral albedos for each MODIS band.

**Figure 8.**Spectral mixture analysis (SMA): (

**a**) elevation model; (

**b**) fractional snow cover for each pixel; (

**c**) broadband albedo; and (

**d**) average grain size of snow for each pixel (images for B, C, and D dating from 25 September 2015).

**Figure 9.**Snow and cloud cover and new snow classifications for three types of cloud cover: (i) presence of both snow and clouds, including mist (6 August 2015), (ii) mostly cloudy (8 August 2015), and (iii) snow with clear skies (25 September 2015). The top row shows RGB true-color images; second row, cloud mask product of MODIS (MOD35_L2); third row, first snow classification based on NDSI (snow threshold ≥ 0.4); fourth row, second snow classification based on MADI (snow threshold ≥ 6); fifth row, third snow classification based on fractional snow cover fSC (snow threshold ≥ 0.2); bottom row, snow mask generated based on the condition that all three snow classification indexes must agree.

**Figure 10.**Snow fraction (fSC) spatial–temporal reconstruction, before (

**top**) and after the interpolation (

**bottom**).

**Figure 11.**Snow height measurements at AWS Tres Morros. Each black line represents the time when the snow height decreases to below 20 cm.

**Figure 12.**Snow cover variability in the Brunswick Peninsula from MODIS reconstruction for the entire study area. (

**a**) Daily snow cover area by year as boxplots, with empty circles representing outliers. (

**b**) Monthly snow cover area as both km

^{2}and percentage. The horizontal pink line represents 10% of the total study area as a definition of the snow season and the numbers in blue above represent the number of months during the snow season. (

**c**) The blue line represents the number of days per year that the snow cover exceeds 10% of the total study area. The orange dashed line represents the exponential filter of the number of days per year for the whole series, which shows a significant trend of +0.54 days/year. The red dashed line represents the exponential filter of the number of days per year for the 2010–2020 period, showing a significant decreasing trend of −4.64 day/year. The green line presents the average of the snow season days in 2000–2020.

**Figure 13.**Spatial and temporal variabilities of snow cover in the Brunswick Peninsula from the MODIS reconstruction. (

**a**) Spatial variability of snow days per year as an average for the period 2000–2020; (

**b**) yearly significant trend of reduction in snow days for each pixel (in days/year). (

**c**,

**d**) Show the elevation component of the generalized additive model (GAM). The red line is a spline fit, with confidence intervals (at 95%) shown as blue dashed lines. (

**c**) Model showing the altitudinal distribution of snow days as an average for the period 2000–2020 and (

**d**) model showing the altitudinal distribution of yearly trend of snow days.

**Figure 14.**Reanalysis and regional climate model validation. (

**a**) Scatter plot of daily mean temperatures between ERA5 Land and AWS Tres Morros (2018–2020); (

**b**) scatter plot of monthly mean temperatures between ERA5 Land and AWS Tres Morros (2018–2020); and (

**c**) scatter plot of monthly mean temperatures between ERA5 Land and RegCM4 (2000–2005).

Year | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 |
---|---|---|---|---|---|---|---|---|---|---|---|

Season length | 78 | 85 | 65 | 76 | 57 | 95 | 10 | 0 | 30 | 30 | Closed due to pandemic |

**Table 2.**The results of the OLS regression proposed by Wang et al. (2015) in the Brunswick Peninsula.

Snow Image | Snowless Image | |
---|---|---|

Band_{i} | Band_{i}~Band_{1} | Band_{i}~Band_{1} |

Band_{3} | R^{2} = 0.96RMSE = 0.229 | R^{2} = 0.91RMSE = 0.066 |

Band_{4} | R^{2} = 0.99RMSE = 0.094 | R^{2} = 0.84RMSE = 0.074 |

Band_{5} | R^{2} = 0.39RMSE = 0.264 | R^{2} = 0.22RMSE = 0.165 |

Band_{6} | R^{2} = 0.54RMSE = 0.229 | R^{2} = 0.73RMSE = 0.133 |

Band_{7} | R^{2} = 0.41RMSE = 0.409 | R^{2} = 0.93RMSE = 0.075 |

**Table 3.**Downscaling using spectral fusion, showing the first term (Equation (2)) as a new regression relationship based on coarse MODIS bands 3 to 7 at 500 m, using independent variables as the coarse-band 1 and the resampled DEM at a 500 m resolution (${\mathrm{D}\mathrm{E}\mathrm{M}}_{500}$).

Coarse Band | New Relationship Proposed | OLS Regression Results |
---|---|---|

Band 3 | ${\mathrm{B}\mathrm{a}\mathrm{n}\mathrm{d}}_{3}~{\mathrm{B}\mathrm{a}\mathrm{n}\mathrm{d}}_{1}$ | R^{2} = 0.96RMSE = 0.230 |

Band 4 | ${\mathrm{B}\mathrm{a}\mathrm{n}\mathrm{d}}_{4}{~\mathrm{B}\mathrm{a}\mathrm{n}\mathrm{d}}_{1}$ | R^{2} = 0.99RMSE = 0.094 |

Band 5 | ${\mathrm{B}\mathrm{a}\mathrm{n}\mathrm{d}}_{5}{~\mathrm{B}\mathrm{a}\mathrm{n}\mathrm{d}}_{1}+{\mathrm{D}\mathrm{E}\mathrm{M}}_{500}$ | R^{2} = 0.52RMSE = 0.234 |

Band 6 | ${\mathrm{B}\mathrm{a}\mathrm{n}\mathrm{d}}_{6}{~\mathrm{B}\mathrm{a}\mathrm{n}\mathrm{d}}_{1}+{\mathrm{D}\mathrm{E}\mathrm{M}}_{500}$ | R^{2} = 0.68RMSE = 0.319 |

Band 7 | ${\mathrm{B}\mathrm{a}\mathrm{n}\mathrm{d}}_{7}{~\mathrm{B}\mathrm{a}\mathrm{n}\mathrm{d}}_{1}+{\mathrm{D}\mathrm{E}\mathrm{M}}_{500}$ | R^{2} = 0.61RMSE = 0.333 |

**Table 4.**Models and downscaling results for subset selections in the snow-covered-image areas in the Brunswick Peninsula. All indexes are dimensionless: spatial distortion index (Ds), spectral distortion index (Dγ), quality index without reference (QNR index), and universal quality index (UQI).

OLS_{500} | GLS_{500} | GWR_{500} | OLS_ATAK_{250} | GLS_ATAK_{250} | GWR_ATAK_{250} | |
---|---|---|---|---|---|---|

Band_3_{500} | R^{2} = 0.97RMSE = 0.230 AIC = −175 | R^{2} = 0.97RMSE = 0.372 AIC = −2877 | R^{2} = 0.99RMSE = 0.110 AIC = −2500 | UQI = 0.92 | UQI = 0.94 | UQI = 0.93 |

Band_4_{500} | R^{2} = 0.993RMSE = 0.085 AIC = −3713 | R^{2} = 0.993RMSE = 0.101 AIC = −6053 | R^{2} = 0.99RMSE = 0.038 AIC = −6296 | UQI = 0.93 | UQI = 0.93 | UQI = 0.93 |

Band_5_{500} | R^{2} = 0.64RMSE = 0.152 AIC = −1654 | R^{2} = 0.63RMSE = 0.153 AIC =−1727 | R^{2} = 0.97RMSE = 0.047 AIC = −5404 | UQI = 0.89 | UQI = 0.98 | UQI = 0.89 |

Band_6_{500} | R^{2} = 0.69RMSE = 0.304 AIC = 822 | R^{2} = 0.65RMSE = 0.392 AIC = 222 | R^{2} = 0.98RMSE = 0.074 AIC = −3838 | UQI = 0.96 | UQI = 0.99 | UQI = 0.91 |

Band_7_{500} | R^{2} = 0.58RMSE = 0.302 AIC = 794 | R^{2} = 0.53RMSE = 0.359 AIC = 498 | R^{2} = 0.97RMSE = 0.080 AIC = −3545 | UQI = 0.97 | UQI = 0.99 | UQI = 0.89 |

Ds | 0.042 | 0.043 | 0.039 | |||

Dγ | 0.015 | 0.009 | 0.016 | |||

QNR Index | 0.943 | 0.949 | 0.946 |

Southern Hemisphere Season | Image Date |
---|---|

Summer | 17 January 2015 16 January 2016 |

Autumn | 29 April 2016 5 May 2016 |

Winter | 3 and 11 September 2016 |

Spring | 16 and 18 October 2016 |

**Table 6.**AWS Tres Morros snow height measurements (Snow_h) vs. fSC reconstruction from MODIS imagery (SF_Rec) in 2018–2020.

Relations Evaluated | |
---|---|

LinearSF_Rec ~ Snow_h | R^{2} = 0.20RMSE = 0.106 AIC = −63 |

GAMSF_Rec ~ s (Snow_h) | R^{2} = 0.45RMSE = 0.083 AIC = −249 |

GAM 20 cmSF_Rec ~ s (Snow_h) | R^{2} = 0.05RMSE = 0.067 AIC = −251 |

**Table 7.**Pearson’s cross-correlation between monthly snow cover area from MODIS data at Brunswick Peninsula and ERA5 land monthly variables from ERA5 data at AWS Tres Morros station for the period of 2000 to 2020, between April to October. ERA 5 variables are liquid and solid precipitation levels; mean, minimum, and maximum temperatures; and monthly degree hours.

ERA5 Climate Variables | Liquid Precipitation | Solid Precipitation | Mean Temperature | Maximum Temperature | Minimum Temperature | Degree Hours |
---|---|---|---|---|---|---|

Cross Pearson correlation with MODIS snow cover area | −0.49 | 0.48 | −0.70 | −0.65 | −0.60 | −0.70 |

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## Share and Cite

**MDPI and ACS Style**

Aguirre, F.; Bozkurt, D.; Sauter, T.; Carrasco, J.; Schneider, C.; Jaña, R.; Casassa, G.
Snow Cover Reconstruction in the Brunswick Peninsula, Patagonia, Derived from a Combination of the Spectral Fusion, Mixture Analysis, and Temporal Interpolation of MODIS Data. *Remote Sens.* **2023**, *15*, 5430.
https://doi.org/10.3390/rs15225430

**AMA Style**

Aguirre F, Bozkurt D, Sauter T, Carrasco J, Schneider C, Jaña R, Casassa G.
Snow Cover Reconstruction in the Brunswick Peninsula, Patagonia, Derived from a Combination of the Spectral Fusion, Mixture Analysis, and Temporal Interpolation of MODIS Data. *Remote Sensing*. 2023; 15(22):5430.
https://doi.org/10.3390/rs15225430

**Chicago/Turabian Style**

Aguirre, Francisco, Deniz Bozkurt, Tobias Sauter, Jorge Carrasco, Christoph Schneider, Ricardo Jaña, and Gino Casassa.
2023. "Snow Cover Reconstruction in the Brunswick Peninsula, Patagonia, Derived from a Combination of the Spectral Fusion, Mixture Analysis, and Temporal Interpolation of MODIS Data" *Remote Sensing* 15, no. 22: 5430.
https://doi.org/10.3390/rs15225430