# Seasonal Composite Landsat TM/ETM+ Images Using the Medoid (a Multi-Dimensional Median)

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. The Medoid and the Maximum NDVI Composite

_{1}-median, also known as the geometric median (this term was introduced by Haldane [14]). This is the point which minimizes the sum of the distances to all the points in the dataset. The medoid is defined in the same way, but with the extra constraint that the selected point be taken from the set of observations. The concept arises in the literature on clustering methods [15]. A formal definition is given in Equation (1).

^{n}is the set of points in n-dimensional space, the ‖ · ‖ operator is the Euclidean distance, and the arg min

_{xi∈X}operator selects the element of X which minimizes the given expression.

#### 2.2. Quantifying “Representativeness”

_{k}for that band in a single season, and seasonal value S which we have chosen as representative of that season for that band. We define the seasonal residual of

**S**

_{S}| over all seasons, per pixel, and then averaging over all pixels.

## 3. Data

## 4. Results

## 5. Discussion

## 6. Conclusion

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustrative example of medoid selection in a 2-dimensional space. The medoid, as selected by Equation (1), is circled in red. The data is randomly generated, purely to illustrate the medoid concept.

**Figure 2.**Australia, showing the state of Queensland, and the location of Landsat WRS-2 path/row 093/077 (in red).

**Figure 3.**Detail of Landsat WRS-2 path/row 093/077, showing the two points being used for illustrative time series. Backdrop image is for 1 July 2009, and is colored with Landsat TM bands 5, 4, 2 as red, green, blue respectively. Points are marked as red dots. Northern point is in open grassland, southern point is in forest.

**Figure 4.**Time series of reflectance for all Landsat TM/ETM+ reflective bands, for the forested example pixel. Seasonal values are marked as crosses. The green crosses are the seasonal reflectance values as selected by the maximum NDVI method, while the red crosses are the seasonal reflectance values selected using the medoid.

**Figure 5.**Time series of reflectance for all Landsat TM/ETM+ reflective bands, for the grassland example pixel. Seasonal values are marked as crosses. The green crosses are the seasonal reflectance values as selected by the max NDVI method, while the red crosses are the seasonal reflectance values selected using the medoid.

**Figure 6.**Time series of NDVI, for the two example pixels: (

**a**) forest; (

**b**) grassland. Seasonal values are marked as points. Seasonal NDVI is calculated directly from the seasonal reflectance.

**Figure 7.**Seasonal average residuals for all Landsat TM reflective bands, for the forest example pixel.

**Figure 8.**Seasonal average residuals for all Landsat TM reflective bands, for the grassland example pixel.

**Figure 9.**Mosaic of seasonal images for Queensland, for March–May 2008. (

**a**) Medoid composite; (

**b**) Maximum NDVI composite. Both images are stretched to the same maximum/minimum values. Colouring is with TM bands 5, 4, 1 as red, green and blue respectively. Null value is coloured as grey, showing areas where less than three values remained after masking for cloud and shadow.

Cover Type | Longitude | Latitude |
---|---|---|

Forest | 147.59884 | −24.74944 |

Grassland | 147.63865 | −24.58561 |

**Table 2.**Average values of residuals for medoid and MNC seasonal reflectance values, averaged over all pixels in WRS-2 path/row 093/077, for all Landsat TM/ETM+ reflective bands. First two rows compare mean residuals. Next two rows compare mean absolute residuals. The first four rows are all in reflectance units. Final row shows the percentage of seasons of the period March 2000 to November 2012 in which the medoid residual is larger (in absolute value) than the MNC residual.

TM1 | TM2 | TM3 | TM4 | TM5 | TM7 | |
---|---|---|---|---|---|---|

Mean medoid residual ( $\overline{{\u220a}_{\text{med}}}$) | 0.0009 | 0.0006 | 0.0001 | −0.0002 | −0.0011 | −0.0009 |

Mean MNC residual ( $\overline{{\u220a}_{\text{MNC}}}$) | 0.0051 | 0.0050 | 0.0096 | −0.0062 | 0.0106 | 0.0119 |

Mean absolute medoid residual $\left(\left|\overline{{\u220a}_{\text{med}}}\right|\right)$ | 0.0033 | 0.0032 | 0.0040 | 0.0065 | 0.0060 | 0.0060 |

Mean absolute MNC residual $\left(\left|\overline{{\u220a}_{\text{MNC}}}\right|\right)$ | 0.0063 | 0.0065 | 0.0103 | 0.0135 | 0.0149 | 0.0151 |

Percentage of seasons with |∊_{med}| > |∊_{MNC}| | 21 | 21 | 11 | 22 | 16 | 16 |

© 2013 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 license ( http://creativecommons.org/licenses/by/3.0/).

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Flood, N.
Seasonal Composite Landsat TM/ETM+ Images Using the Medoid (a Multi-Dimensional Median). *Remote Sens.* **2013**, *5*, 6481-6500.
https://doi.org/10.3390/rs5126481

**AMA Style**

Flood N.
Seasonal Composite Landsat TM/ETM+ Images Using the Medoid (a Multi-Dimensional Median). *Remote Sensing*. 2013; 5(12):6481-6500.
https://doi.org/10.3390/rs5126481

**Chicago/Turabian Style**

Flood, Neil.
2013. "Seasonal Composite Landsat TM/ETM+ Images Using the Medoid (a Multi-Dimensional Median)" *Remote Sensing* 5, no. 12: 6481-6500.
https://doi.org/10.3390/rs5126481