# Spectro-Temporal Heterogeneity Measures from Dense High Spatial Resolution Satellite Image Time Series: Application to Grassland Species Diversity Estimation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials

#### 2.1. Study Area

^{2}is characterized by a mosaic of crops, small woods and grasslands. It is dominated by mixed crop-livestock farming. Grasslands provide food for cattle by grazing and/or producing hay or silage. They range from monospecific grasslands sown with ryegrass (improved with mineral fertilizing and mown up to three times a year) to semi-natural grasslands composed of spontaneous plant species (not fertilized and mown once a year). Grasslands are mainly located on steep slopes, whereas annual crops are in the valleys on the most productive lands. The climate is sub-Atlantic with sub-Mediterranean and mountain influences (mean annual temperature, 12.5 ${}^{\circ}$C; mean annual precipitation, 750 mm) [27,28].

#### 2.2. Satellite Image Time Series

#### 2.3. Field Data

^{®}” French database of orthophotos, ©IGN). For this study, an inner buffer of 10 m was removed from all the grasslands’ polygons to avoid edge effects due to mixed pixels at the parcel edges. After rasterizing the polygons, only the grasslands composed of at least 10 pixels of 10-m resolution, i.e., having an area higher than 1000 m

^{2}, were kept to ensure a minimum number of pixels per grassland. After this treatment, the dataset is composed of 192 grasslands. Their location can be seen in Figure 1.

## 3. Method

#### 3.1. Measures of Spectral Heterogeneity in the Literature

#### 3.2. Spectral Clustering Algorithm for High Dimensional Data and Derived Measures of Spectral Heterogeneity

#### 3.2.1. Between- and Within-Class Variabilities

- ${\mathbf{B}}_{i}={\sum}_{c=1}^{{C}_{i}}{p}_{ic}({\mathit{\mu}}_{ic}-{\mathit{\mu}}_{i}){({\mathit{\mu}}_{ic}-{\mathit{\mu}}_{i})}^{\top}$ is the between-class covariance matrix,
- ${\mathit{\mu}}_{ic}$ is the spectro-temporal mean of pixels in ${g}_{i}$ assigned to cluster c,
- ${\mathit{\mu}}_{i}$ is the mean spectro-temporal value computed from all the pixels of ${g}_{i}$,
- ${\mathbf{W}}_{i}=\frac{1}{{n}_{i}}{\sum}_{c=1}^{{C}_{i}}{\sum}_{k\in c}({\mathbf{x}}_{ik}-{\mathit{\mu}}_{ic}){({\mathbf{x}}_{ik}-{\mathit{\mu}}_{ic})}^{\top}={\sum}_{c=1}^{{C}_{i}}{p}_{ic}{\mathbf{V}}_{ic}$ is the within-class covariance matrix,
- ${\mathbf{V}}_{ic}$ is the empirical covariance matrix of pixels of ${g}_{i}$ assigned to cluster c.

#### 3.2.2. Entropy

#### 3.3. Methodology

## 4. Results

#### 4.1. Univariate Correlation with Multitemporal Data

#### 4.2. Multivariate Correlation with Multitemporal Data

#### 4.3. Univariate and Multivariate Correlation with Monotemporal Data

## 5. Discussion

#### 5.1. Spectral Heterogeneity Measures

#### 5.2. Clustering

#### 5.3. Contribution of Multitemporal Imagery

#### 5.4. Limitations

#### 5.5. Outlooks

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

B | Log-transformed between-classes variability |

CNES | Centre National d’Etudes Spatiales (French spatial agency) |

E | Entropy computed from soft assignment |

GIS | Geographic Information System |

H | Shannon index |

HDDC | High Dimensional Discriminant Clustering |

ICL | Integrated Classification Likelihood |

MDC | Mean Distance to Centroid |

NDVI | Normalized Difference Vegetation Index |

NIR | Near-Infrared |

PCA | Principal Components Analysis |

SH | Spectral Heterogeneity |

SITS | Satellite Image Time Series |

STVH | Spectro-Temporal Variation Hypothesis |

SVH | Spectral Variation Hypothesis |

V | Log-transformed global variability |

W | Log-transformed within-class variability |

## Appendix A

**Table A1.**Braun-Blanquet abundance-dominance coefficients associated with each plant species recorded in three grasslands a, b and c having a Shannon index of 0.10, 1.57 and 2.89, respectively. “spp.” means that the species from the given genus was not identified.

Species | a | b | c |
---|---|---|---|

Agrimonia eupatoria | + | ||

Agrostis capillaris | 1 | ||

Anthoxanthum odoratum | 1 | ||

Arrhenatherum elatius | 1 | ||

Bellis perennis | 1 | ||

Bromus erectus | 1 | ||

Carex divulsa | + | ||

Carex flacca | 1 | ||

Centaurea nigra | + | ||

Cirsium arvense | 1 | ||

Cirsium dissectum | 1 | ||

Cirsium vulgare | + | ||

Convolvulus arvensis | 1 | 1 | |

Crepis capillaris | 1 | ||

Crepis spp. | + | ||

Dactylis glomerata | 1 | 3 | |

Daucus carota | 1 | ||

Festuca arundinacea | 2 | 3 | |

Festuca rubra | 1 | ||

Galium mollugo | 1 | ||

Gaudinia fragilis | 1 | ||

Holcus lanatus | 1 | ||

Hypericum perforatum | + | ||

Hypochaeris radicata | 1 | 1 | |

Lathyrus pratensis | 2 | ||

Leucanthemum vulgare | 1 | ||

Linum usitatissimum | 1 | ||

Lolium perenne | 5 | ||

Lotus corniculatus | 1 | ||

Medicago spp. | + | ||

Muscari comosum | + | ||

Orchis purpurea | + | ||

Plantago lanceolata | 1 | ||

Poa pratensis | 2 | ||

Poa trivialis | + | 5 | |

Potentilla reptans | 1 | 1 | |

Prunus spinosa | 1 | ||

Rafanus spp. | + | ||

Ranunculus acris | 1 | ||

Ranunculus bulbosus | 1 | ||

Ranunculus repens | 2 | ||

Rasica oleacera | + | ||

Rhinanthus minor | + | ||

Rubus spp. | + | ||

Rumex acetosa | 1 | ||

Rumex crispus | 1 | + | |

Senecio jacobaea | 1 | ||

Sonchus asper | + | ||

Stachys officinalis | + | ||

Taraxacum officinalis | 1 1 | ||

Tragopogon pratensis | + + | ||

Trifolium dubium | 2 | ||

Trifolium pratense | 1 | 2 | |

Trifolium repens | 1 | ||

Verbena officinalis | 1 | ||

Veronica arvensis | + | ||

Veronica arvensis | + | ||

Vicia sativa | + | 1 |

## References

- Eriksson, A.; Eriksson, O.; Berglund, H. Species Abundance Patterns of Plants in Swedish Semi-Natural Pastures. Ecography
**1995**, 18, 310–317. [Google Scholar] [CrossRef] - Critchley, C.; Burke, M.; Stevens, D. Conservation of lowland semi-natural grasslands in the UK: A review of botanical monitoring results from agri-environment schemes. Biol. Conserv.
**2004**, 115, 263–278. [Google Scholar] [CrossRef] - O’Mara, F.P. The role of grasslands in food security and climate change. Anna. Bot.
**2012**, 110, 1263–1270. [Google Scholar] [CrossRef] [PubMed] - Werling, B.P.; Dickson, T.L.; Isaacs, R.; Gaines, H.; Gratton, C.; Gross, K.L.; Liere, H.; Malmstrom, C.M.; Meehan, T.D.; Ruan, L.; et al. Perennial grasslands enhance biodiversity and multiple ecosystem services in bioenergy landscapes. Proc. Natl. Acad. Sci. USA
**2014**, 111, 1652–1657. [Google Scholar] [CrossRef] [PubMed] - Magurran, A. Ecological Diversity and Its Measurement; Croom Helm: Kent, UK, 1988. [Google Scholar]
- Shannon, C.E. A Mathematical Theory of Communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Simpson, E.H. Measurement of diversity. Nature
**1949**, 163, 688. [Google Scholar] [CrossRef] - Rocchini, D.; Balkenhol, N.; Carter, G.A.; Foody, G.M.; Gillespie, T.W.; He, K.S.; Kark, S.; Levin, N.; Lucas, K.; Luoto, M.; et al. Remotely sensed spectral heterogeneity as a proxy of species diversity: Recent advances and open challenges. Ecol. Inform.
**2010**, 5, 318–329. [Google Scholar] [CrossRef] - Skidmore, A.K.; Pettorelli, N.; Coops, N.C.; Geller, G.N.; Hansen, M.; Lucas, R.; Mücher, C.A.; O’Connor, B.; Paganini, M.; Pereira, H.M.; et al. Environmental science: Agree on biodiversity metrics to track from space. Nature
**2015**, 523, 403–405. [Google Scholar] [CrossRef] [PubMed] - Rocchini, D.; Boyd, D.S.; Féret, J.B.; Foody, G.M.; He, K.S.; Lausch, A.; Nagendra, H.; Wegmann, M.; Pettorelli, N. Satellite remote sensing to monitor species diversity: Potential and pitfalls. Remote Sens. Ecol. Conserv.
**2016**, 2, 25–36. [Google Scholar] [CrossRef] - Pettorelli, N.; Laurance, W.F.; O’Brien, T.G.; Wegmann, M.; Nagendra, H.; Turner, W. Satellite remote sensing for applied ecologists: Opportunities and challenges. J. Appl. Ecol.
**2014**, 51, 839–848. [Google Scholar] [CrossRef] - Cord, A.F.; Brauman, K.A.; Chaplin-Kramer, R.; Huth, A.; Ziv, G.; Seppelt, R. Priorities to Advance Monitoring of Ecosystem Services Using Earth Observation. Trends Ecol. Evol.
**2017**, 32, 416–428. [Google Scholar] [CrossRef] [PubMed] - Kerr, J.T.; Ostrovsky, M. From space to species: Ecological applications for remote sensing. Trends Ecol. Evol.
**2003**, 18, 299–305. [Google Scholar] [CrossRef] - Gould, W. Remote Sensing of Vegetation, Plant Species Richness, and Regional Biodiversity Hotspots. Ecol. Appl.
**2000**, 10, 1861–1870. [Google Scholar] [CrossRef] - Palmer, M.W.; Earls, P.G.; Hoagland, B.W.; White, P.S.; Wohlgemuth, T. Quantitative tools for perfecting species lists. Environmetrics
**2002**, 13, 121–137. [Google Scholar] [CrossRef] - Wilson, J.; Fuller, S.J.; Mather, P.B. Formation and maintenance of discrete wild rabbit (Oryctolagus cuniculus) population systems in arid Australia: Habitat heterogeneity and management implications. Austral Ecol.
**2002**, 27, 183–191. [Google Scholar] [CrossRef] - Tews, J.; Brose, U.; Grimm, V.; Tielbörger, K.; Wichmann, M.C.; Schwager, M.; Jeltsch, F. Animal species diversity driven by habitat heterogeneity/diversity: The importance of keystone structures. J. Biogeogr.
**2004**, 31, 79–92. [Google Scholar] [CrossRef] - Oldeland, J.; Wesuls, D.; Rocchini, D.; Schmidt, M.; Jürgens, N. Does using species abundance data improve estimates of species diversity from remotely sensed spectral heterogeneity? Ecol. Indic.
**2010**, 10, 390–396. [Google Scholar] [CrossRef] - Möckel, T.; Dalmayne, J.; Schmid, B.C.; Prentice, H.C.; Hall, K. Airborne Hyperspectral Data Predict Fine-Scale Plant Species Diversity in Grazed Dry Grasslands. Remote Sens.
**2016**, 8, 133. [Google Scholar] - Oindo, B.O.; Skidmore, A.K. Interannual variability of NDVI and species richness in Kenya. Int. J. Remote Sens.
**2002**, 23, 285–298. [Google Scholar] [CrossRef] - Fairbanks, D.H.K.; McGwire, K.C. Patterns of Floristic Richness in Vegetation Communities of California: Regional Scale Analysis with Multi-Temporal NDVI. Glob. Ecol. Biogeogr.
**2004**, 13, 221–235. [Google Scholar] [CrossRef] - Rocchini, D. Effects of spatial and spectral resolution in estimating ecosystem α-diversity by satellite imagery. Remote Sens. Environ.
**2007**, 111, 423–434. [Google Scholar] [CrossRef] - Féret, J.B.; Asner, G.P. Mapping tropical forest canopy diversity using high-fidelity imaging spectroscopy. Ecol. Appl.
**2014**, 24, 1289–1296. [Google Scholar] [CrossRef] - Drusch, M.; Bello, U.D.; Carlier, S.; Colin, O.; Fernandez, V.; Gascon, F.; Hoersch, B.; Isola, C.; Laberinti, P.; Martimort, P.; et al. Sentinel-2: ESA’s Optical High-Resolution Mission for GMES Operational Services. Remote Sens. Environ.
**2012**, 120, 25–36. [Google Scholar] [CrossRef] - Hooper, D.U. The role of complementarity and competition in ecosystem responses to variation in plant diversity. Ecology
**1998**, 79, 704–719. [Google Scholar] [CrossRef] - Sakai, S. Phenological diversity in tropical forests. Popul. Ecol.
**2001**, 43, 77–86. [Google Scholar] [CrossRef] - Ryschawy, J.; Choisis, N.; Choisis, J.P.; Joannon, A.; Gibon, A. Mixed crop-livestock systems: An economic and environmental-friendly way of farming? Animal
**2012**, 6, 1722–1730. [Google Scholar] [CrossRef] [PubMed] - Carrié, R.; Andrieu, E.; Cunningham, S.A.; Lentini, P.E.; Loreau, M.; Ouin, A. Relationships among ecological traits of wild bee communities along gradients of habitat amount and fragmentation. Ecography
**2017**, 40, 85–97. [Google Scholar] [CrossRef] - Hagolle, O.; Huc, M.; Villa Pascual, D.; Dedieu, G. A multi-temporal method for cloud detection, applied to FORMOSAT-2, VENuS, LANDSAT and SENTINEL-2 images. Remote Sens. Environ.
**2010**, 114, 1747–1755. [Google Scholar] [CrossRef] [Green Version] - Eilers, P.H.C. A Perfect Smoother. Anal. Chem.
**2003**, 75, 3631–3636. [Google Scholar] [CrossRef] [PubMed] - Atzberger, C.; Eilers, P.H.C. Evaluating the effectiveness of smoothing algorithms in the absence of ground reference measurements. Int. J. Remote Sens.
**2011**, 32, 3689–3709. [Google Scholar] [CrossRef] - Braun-Blanquet, J.; Fuller, G.; Conard, H. Plant Sociology: The Study of Plant Communities: Authorized English Translation of Pflanzensoziologie; McGraw-Hill Publications in the Botanical Sciences; McGraw-Hill: New York, NY, USA, 1932. [Google Scholar]
- Parsons, L.; Haque, E.; Liu, H. Subspace Clustering for High Dimensional Data: A Review. Acm Sigkdd Explor. Newsl.
**2004**, 6, 90–105. [Google Scholar] [CrossRef] - Bouveyron, C.; Girard, S.; Schmid, C. High-dimensional data clustering. Comput. Stat. Data Anal.
**2007**, 52, 502–519. [Google Scholar] [CrossRef] - Lagrange, A.; Fauvel, M.; Grizonnet, M. Large-Scale Feature Selection with Gaussian Mixture Models for the Classification of High Dimensional Remote Sensing Images. IEEE Trans. Comput. Imaging
**2017**, 3, 230–242. [Google Scholar] [CrossRef] - Bouveyron, C.; Girard, S.; Schmid, C. High-Dimensional Discriminant Analysis. Commun. Stat. Theory Methods
**2007**, 36, 2607–2623. [Google Scholar] [CrossRef] - Girard, S.; Saracco, J. Supervised and Unsupervised Classification Using Mixture Models. EAS Publ. Ser.
**2016**, 77, 69–90. [Google Scholar] [CrossRef] - Neyrinck, M.C.; Szapudi, I.; Szalay, A.S. Rejuvenating the Matter Power Spectrum: Restoring Information with a Logarithmic Density Mapping. Astrophys. J. Lett.
**2009**, 698, L90. [Google Scholar] [CrossRef] - Hall, K.; Reitalu, T.; Sykes, M.T.; Prentice, H.C. Spectral heterogeneity of QuickBird satellite data is related to fine-scale plant species spatial turnover in semi-natural grasslands. Appl. Veg. Sci.
**2012**, 15, 145–157. [Google Scholar] [CrossRef] - Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V.; et al. Scikit-learn: Machine Learning in Python. J. Mach. Learn. Res.
**2011**, 12, 2825–2830. [Google Scholar] - Biernacki, C.; Celeux, G.; Govaert, G. Assessing a Mixture Model for Clustering with the Integrated Completed Likelihood. IEEE Trans. Pattern Anal. Mach. Intell.
**2000**, 22, 719–725. [Google Scholar] [CrossRef] - Price, J.C. How unique are spectral signatures? Remote Sens. Environ.
**1994**, 49, 181–186. [Google Scholar] [CrossRef] - Nagendra, H. Using remote sensing to assess biodiversity. Int. J. Remote Sens.
**2001**, 22, 2377–2400. [Google Scholar] [CrossRef] - Moog, D.; Poschlod, P.; Kahmen, S.; Schreiber, K.F. Comparison of species composition between different grassland management treatments after 25 years. Appl. Veg. Sci.
**2002**, 5, 99–106. [Google Scholar] [CrossRef] - Schäfer, E.; Heiskanen, J.; Heikinheimo, V.; Pellikka, P. Mapping tree species diversity of a tropical montane forest by unsupervised clustering of airborne imaging spectroscopy data. Ecol. Indic.
**2016**, 64, 49–58. [Google Scholar] [CrossRef] - Maeda, E.E.; Heiskanen, J.; Thijs, K.W.; Pellikka, P.K. Season-dependence of remote sensing indicators of tree species diversity. Remote Sens. Lett.
**2014**, 5, 404–412. [Google Scholar] [CrossRef] - Pärtel, M.; Zobel, M. Small-scale plant species richness in calcareous grasslands determined by the species pool, community age and shoot density. Ecography
**1999**, 22, 153–159. [Google Scholar] [CrossRef] - Bruun, H.H. Patterns of Species Richness in Dry Grassland Patches in an Agricultural Landscape. Ecography
**2000**, 23, 641–650. [Google Scholar] [CrossRef] - Franzén, D.; Eriksson, O. Small-scale patterns of species richness in Swedish semi-natural grasslands: The effects of community species pools. Ecography
**2001**, 24, 505–510. [Google Scholar] [CrossRef] - Bray, J.R.; Curtis, J.T. An Ordination of the Upland Forest Communities of Southern Wisconsin. Ecol. Monogr.
**1957**, 27, 325–349. [Google Scholar] [CrossRef] - Ustin, S.L.; Gamon, J.A. Remote sensing of plant functional types. New Phytol.
**2010**, 186, 795–816. [Google Scholar] [CrossRef] [PubMed] - Homolová, L.; Malenovský, Z.; Clevers, J.G.; García-Santos, G.; Schaepman, M.E. Review of optical-based remote sensing for plant trait mapping. Ecol. Complex.
**2013**, 15, 1–16. [Google Scholar] [CrossRef] - Jetz, W.; Cavender-Bares, J.; Pavlick, R.; Schimel, D.; Davis, F.W.; Asner, G.P.; Guralnick, R.; Kattge, J.; Latimer, A.M.; Moorcroft, P.; et al. Monitoring plant functional diversity from space. Nature Plants
**2016**, 2, 1–5. [Google Scholar] [CrossRef] [PubMed] - Abelleira Martínez, O.J.; Fremier, A.K.; Günter, S.; Ramos Bendaña, Z.; Vierling, L.; Galbraith, S.M.; Bosque-Pérez, N.A.; Ordoñez, J.C. Scaling up functional traits for ecosystem services with remote sensing: concepts and methods. Ecol. Evol.
**2016**, 6, 4359–4371. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Location of the study area in southwest France and of the grasslands within the study area. The background is an aerial photograph issued from the French orthophoto database “BD ORTHO

^{®}” (©IGN).

**Figure 3.**SPOT5 NDVI temporal profiles of all the pixels belonging to three grasslands along the H gradient: (

**a**) grassland with a low level of biodiversity, (

**b**) grassland with a medium level of biodiversity and (

**c**) grassland with a high level of biodiversity. The floristic record of these three grasslands can be found in Appendix A, Table A1. The x-axis corresponds to the day of the year of 2015, and the y-axis corresponds to the NDVI. Grasslands have been voluntarily chosen by their high number of pixels for better visualization.

**Figure 4.**Simulated pixels’ distributions for two different plots (

**a**) and (

**b**). Pixels are displayed in blue, and the centroids of the plots are displayed in red. The estimated MDC are very close while the spectral distributions of the plots are clearly different.

**Figure 5.**(

**a**) Simulated distributions of three spectral species (blue, yellow, red) in a 3-dimensional space. (

**b**) Clustering of the three distributions with PCA and k-means. (

**c**) Clustering of the three distributions with Gaussian mixture models.

**Figure 6.**Grassland clustered with an initial clustering of the landscape into 8 clusters. (

**a**) Hard assignment of the pixels. One color corresponds to one cluster (orange, yellow, blue). (

**b**–

**d**) Soft assignment of the pixels. The grey-scaled color corresponds to the assignment probability ${\pi}_{ick}$ to cluster (

**b**) orange, (

**c**) yellow and (

**d**) blue.

**Figure 7.**Overview of the method to compare the Spectral Heterogeneity (SH) measures (explanatory variables) to the Shannon index (response variable). Square rectangles correspond to data, and rounded rectangles correspond to a process.

**Figure 8.**(

**a**) False color image of a grassland acquired on 30 April 2015. The same grassland clustered using HDDC on multitemporal data with an initial clustering into (

**b**) 8 clusters and (

**c**) 150 clusters. Each cluster is represented by one color.

**Figure 9.**Adjusted coefficient of determination in the multivariate linear regression between different combinations of SH measures (V: log-transformed global variability or MDC, W: log-transformed within-class variability, B: log-transformed between-class variability, E: entropy) computed from multitemporal data and the Shannon index (response variable) depending on the number of clusters.

**Figure 10.**Shannon index (H) best univariate linear correlations with different SH measures (E: entropy, V: log-transformed global variability or MDC, W: log-transformed within-class variability, B: log-transformed between-class variability) computed from multitemporal data. C is the corresponding number of clusters, ${\overline{R}}^{2}$ is the adjusted coefficient of determination and ** signifies p-value <0.001. The black line is the linear regression line.

**Figure 11.**Adjusted coefficient of determination in the multivariate linear regression using one image acquired on (

**a**) 30 April and (

**b**) 29 June between different combinations of SH measures (V: log-transformed global variability or MDC, W: log-transformed within-class variability, B: log-transformed between-class variability, E: entropy) and the Shannon index (response variable) depending on the number of clusters.

**Figure 12.**Maps of spectral heterogeneity inside three grasslands (

**a**–

**c**). The first row shows the grasslands’ polygon limits in yellow on the SPOT5 false color image acquired on 10 May 2015. The second row shows the clusters after an HDDC clustering into eight clusters using multitemporal data. The color scale corresponds to the log-transformed variability of each cluster c in the grassland ${g}_{i}$. (

**a**) H = 0.10, E = 0, V = 10.13, W = 10.13, B = 0; (

**b**) H = 1.57, E = 0.68, V = 10.06, W = 9.41, B = 9.33; (

**c**) H = 2.89, E = 1.06, V = 9.58, W = 9.22, B = 8.42. The floristic record of these three grasslands can be found in the Appendix A, Table A1.

**Figure 13.**Mean NDVI temporal profiles of each cluster from the clustering into $C=8$ clusters using multitemporal data. The x-axis is the month of year 2015, and the y-axis is the NDVI.

**Figure 14.**Clustering of the same grassland (false color image of (

**a**) 30 April and (

**d**) 29 June) with an initial clustering into 150 clusters, using (

**b**) the image of 30 April and (

**c**) the full SITS, and into 20 clusters, using (

**e**) the image of 29 June and (

**f**) the full SITS.

Pixel Size | 10 m |
---|---|

Spectral bands | B1 “Green” (500–590 nm) |

B2 “Red” (610–680 nm) | |

B3 “Near-Infrared” (780–890 nm) | |

B4 “Short Wave Infrared” (1580–1750 nm) | |

Acquisition dates | 20-04-2015, 25-04-2015, 30-04-2015, 10-05-2015, 20-05-2015, 04-06-2015, |

24-06-2015, 29-06-2015, 04-07-2015, 09-07-2015, 14-07-2015, 19-07-2015, | |

24-07-2015, 13-08-2015, 18-08-2015, 28-08-2015, 02-09-2015, 07-09-2015 |

**Table 2.**Multivariate linear models for $C=8$ to explain the Shannon index (H) from the SH measures (V: log-transformed global variability or MDC, W: log-transformed within-class variability, B: log-transformed between-class variability, E: entropy) computed from multitemporal data. Reg. Coeff. is the regression coefficient; Std Err. is the standard error; F stands for the F-value with degrees of freedom in brackets; ${R}^{2}$ is the coefficient of determination; and ${\overline{R}}^{2}$ is the adjusted coefficient of determination.

Response Variable | Explanatory Variables | Reg. Coeff. | Std Err. | p-Value |
---|---|---|---|---|

H | W | 0.29 | 0.14 | 0.04 |

B | 0.01 | 0.02 | 0.61 | |

V | −0.15 | 0.14 | 0.30 | |

E | 0.40 | 0.13 | 0.003 | |

intercept | 0.73 | 0.51 | 0.16 | |

Model summary: ${F}_{(4,\phantom{\rule{3.33333pt}{0ex}}187)}$ = 8.0, p-value <0.001, ${R}^{2}$ = 0.145, ${\overline{R}}^{2}=0.127$ | ||||

H | W | 0.16 | 0.06 | 0.005 |

E | 0.37 | 0.09 | <0.001 | |

intercept | 0.65 | 0.51 | 0.20 | |

Model summary: ${F}_{(2,\phantom{\rule{3.33333pt}{0ex}}189)}$ = 15.4, p-value <0.001, ${R}^{2}$ = 0.140, ${\overline{R}}^{2}=0.131$ |

© 2017 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/).

## Share and Cite

**MDPI and ACS Style**

Lopes, M.; Fauvel, M.; Ouin, A.; Girard, S.
Spectro-Temporal Heterogeneity Measures from Dense High Spatial Resolution Satellite Image Time Series: Application to Grassland Species Diversity Estimation. *Remote Sens.* **2017**, *9*, 993.
https://doi.org/10.3390/rs9100993

**AMA Style**

Lopes M, Fauvel M, Ouin A, Girard S.
Spectro-Temporal Heterogeneity Measures from Dense High Spatial Resolution Satellite Image Time Series: Application to Grassland Species Diversity Estimation. *Remote Sensing*. 2017; 9(10):993.
https://doi.org/10.3390/rs9100993

**Chicago/Turabian Style**

Lopes, Mailys, Mathieu Fauvel, Annie Ouin, and Stéphane Girard.
2017. "Spectro-Temporal Heterogeneity Measures from Dense High Spatial Resolution Satellite Image Time Series: Application to Grassland Species Diversity Estimation" *Remote Sensing* 9, no. 10: 993.
https://doi.org/10.3390/rs9100993