# Bayesian Harmonic Modelling of Sparse and Irregular Satellite Remote Sensing Time Series of Vegetation Indexes: A Story of Clouds and Fires

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## Abstract

**:**

## 1. Introduction

## 2. Materials

#### 2.1. Study Areas

#### 2.2. Satellites Time Series

## 3. Methods

#### 3.1. Harmonic Models

#### Summary Descriptors of the Seasonal Dynamics

#### 3.2. Models Fitting

Algorithm 1: Pseudocode of fitting strategy for the Bayesian model. YSM and YAM are the harmonic model defined in the text, while TM is a simple trend model. When the model is fit with no explicit prior definition a flat prior was used. |

#### 3.3. Cost of Model Fitting

## 4. Results and Discussion

#### 4.1. Selection of Vegetation Index

#### 4.2. Simulation of Cloud Cover Experiment

#### 4.3. Testing over Forest Fires

#### 4.4. Effect of Land Cover on Vegetation Phenology

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Detailed Comparison of Fire Breaks

**Figure A1.**Comparison between estimation of maximum change near the first fire event. Histogram shows distribution of differences in day of the year between the actual fire event and the estimated one on the pixel that dNBR analysis identified as hit by this fire event. In the lower part, the two images show the spatial distribution in the two approaches. The projection used in the map is UTM zone 33N and extent and orientation is the same as Figure 2a in the main text.

**Figure A2.**Same as Figure A1 for the second fire event.

**Figure A3.**Same as Figure A1 for the third fire event.

**Figure A4.**Same as Figure A1 for the forth fire event.

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**Figure 1.**View of the three study sites over the backgrounds of Terrametrics TrueEarth 2019 images taken from google map services. In red the areas of interest, while in green the boundary of the protected areas. Peneda Gêres (

**a**) is show in UTM29N projection, while Murgia Alta and, just south, Bosco Difesa Grande are shown in panel (

**b**) using UTM33N projection. (

**c**) The location of the area of interest are shown within European continent.

**Figure 2.**Summary of the four fires that hit the “Bosco Difesa Grande” region as defined by dNBR analysis (difference of the Normalized Burning Index) and bounded by the fire cadastral delimitation. (

**a**) The four fires are overlapped to observe the number of different fire event that hit each pixel area; (

**b**) the view of each fire event. Very few experimented 3 fire events.

**Figure 3.**Time of execution evaluation. The x-axis is relative at the value of the 3 different parameters. When one parameter is changed the others are fixed to 36 and 500 for yearly sample and replicates, respectively.

**Figure 4.**Ratio between input and output noise for different values of surface reflectance NIR and RED. In panels (

**a**,

**c**) data are relative to NDVI, while the others to MSAVI2. In panels (

**a**,

**b**) data are obtained from numerical simulation with 500 replicates per pair of surface reflectance values, whereas (

**c**,

**d**) resulted from the derivative approach.

**Figure 5.**Distribution of difference between 423 EFA values across 6 years (2005–2010) in the Peneda Gerês data set, estimated with a full set of observations and applying on them 100 different cloud patterns. Estimates with modelled and raw data are respectively blue and orange. In subfigure (

**a**–

**c**) are shown yearly mean and stardard deviation, day of the year of maximum and number of observation per year, respectively.

**Figure 6.**Summary of the Breaks For Additive Season and Trend (BFAST) reconstruction over the pixel that experienced 2012 and 2017 fire based on dNBR analysis. In green the observation after Savitzky–Golay filter, in dark red the median of the BFAST expectation, in cyan the median of pre-2012 expectation. Vertical dark red line represent BFAST estimation of breaking point, whereas bright red is the actual fire event. Take note that median estimation and 2.5 and 97.5 quantile of breaking point estimation overlap in the figure.

Name Locality | Scenes | ${\mathit{n}}^{\xb0}$ X Cells | ${\mathit{n}}^{\xb0}$ Y Cells | Years Span |
---|---|---|---|---|

Peneda Gerês | 66 | 2521 | 2458 | 2005–2010 |

Murgia Alta | 538 | 1122 | 488 | 2000–2018 |

Bosco Difesa Grande | 192 | 87 | 96 | 2010–2017 |

**Table 2.**Bias and average deviation in days of the two methods (our proposed method, BM, and the reference, BFAST) compared to the true date of fire over the pixel characterized by dNBR smaller than −0.27, estimated using pairs of Landsat image before and after the known fire date. Negative value of bias indicates the estimated date prior true date. In each case the nearest break was used to estimate the statistics.

Bias | Std | |||
---|---|---|---|---|

Method Fire Event | BM | BFAST | BM | BFAST |

27 June 2011 | 33.3 | 118.2 | 51.2 | 196.1 |

30 June 2012 | 21.4 | −29.9 | 47.6 | 110.6 |

15 August 2013 | −17.7 | 15.7 | 62.5 | 245.6 |

12 August 2017 | −3.1 | −422.1 | 51.4 | 271.1 |

**Table 3.**Results from 4 linear models for yearly mean and yearly standard deviation estimated with our approach (BM) and BFAST that use year of observations and fire events (estimated from dNBR) as predictors. We report the coefficient and the variance explained, adjusted by number of parameters (Rsq adj).

mean_mean | stdintra_mean | maxpos_mean | ||||
---|---|---|---|---|---|---|

BM | BFAST | BM | BFAST | BM | BFAST | |

Intercept (2011:nofire) | 0.330 | 0.322 | 0.090 | 0.080 | 144.206 | 145.140 |

2012:nofire | −0.038 | −0.053 | −0.007 | −0.006 | −4.431 | −13.457 |

2013:nofire | −0.053 | −0.041 | −0.005 | 0.004 | −8.722 | 3.898 |

2017:nofire | −0.011 | 0.006 | 0.006 | 0.007 | 5.722 | −8.013 |

fire | −0.076 | −0.069 | −0.003 | −0.000 | −26.258 | −29.640 |

2012:fire | 0.023 | 0.029 | 0.004 | 0.011 | 13.924 | 34.929 |

2013:fire | 0.071 | 0.077 | 0.012 | −0.008 | 4.703 | 35.068 |

2017:fire | 0.027 | 0.013 | 0.018 | −0.006 | 19.348 | 26.071 |

Rsq_adj | 0.162 | 0.182 | 0.048 | 0.013 | 0.065 | 0.041 |

**Table 4.**Spatial and time local variation of estimates across methods (our proposed ones, BM, and BFAST), using standard deviation as metric. A kernel of 9 pixels and a moving window of 3 were used respectively for space and time domains. For each summary statistic, the mean kernel, the overall and the ratio of the two values are reported.

Space | Time | ||||
---|---|---|---|---|---|

BFAST | BM | BFAST | BM | ||

maxpos | kernSD | 18.289 | 5.945 | 1.247 | 4.490 |

TotSD | 27.787 | 28.748 | 6.819 | 6.986 | |

Rate | 0.658 | 0.207 | 0.183 | 0.643 | |

mean | kernSD | 0.021 | 0.021 | 0.027 | 0.022 |

TotSD | 0.067 | 0.066 | 0.036 | 0.029 | |

Rate | 0.319 | 0.321 | 0.747 | 0.766 | |

std intra | kernSD | 0.014 | 0.014 | 0.007 | 0.007 |

TotSD | 0.036 | 0.038 | 0.009 | 0.013 | |

Rate | 0.386 | 0.378 | 0.708 | 0.560 | |

std inter | kernSD | 0.006 | 0.006 | - | - |

TotSD | 0.036 | 0.038 | - | - | |

Rate | 0.153 | 0.146 | - | - |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Vicario, S.; Adamo, M.; Alcaraz-Segura, D.; Tarantino, C. Bayesian Harmonic Modelling of Sparse and Irregular Satellite Remote Sensing Time Series of Vegetation Indexes: A Story of Clouds and Fires. *Remote Sens.* **2020**, *12*, 83.
https://doi.org/10.3390/rs12010083

**AMA Style**

Vicario S, Adamo M, Alcaraz-Segura D, Tarantino C. Bayesian Harmonic Modelling of Sparse and Irregular Satellite Remote Sensing Time Series of Vegetation Indexes: A Story of Clouds and Fires. *Remote Sensing*. 2020; 12(1):83.
https://doi.org/10.3390/rs12010083

**Chicago/Turabian Style**

Vicario, Saverio, Maria Adamo, Domingo Alcaraz-Segura, and Cristina Tarantino. 2020. "Bayesian Harmonic Modelling of Sparse and Irregular Satellite Remote Sensing Time Series of Vegetation Indexes: A Story of Clouds and Fires" *Remote Sensing* 12, no. 1: 83.
https://doi.org/10.3390/rs12010083