## 2 Background
### Spectral Variability and the Spatial Dimension  
Image dimensionality is an attempt to measure the spectral diversity captured by a dataset. However, the spectral diversity of the Earth surface is spatially heterogeneous. The local variations that can exist in Earth surface reflectance, as well as the dimensionality of a large, diverse set of hyperspectral data, are well illustrated by the recent study of [19]. One might expect imagery with a higher spatial resolution to resolve more spectral diversity because of fewer mixed pixels. While this is generally true, the increase in spectral diversity (and dimensionality) with spatial resolution depends on the characteristic spatial scale of the reflectance of the landscape relative to the resolution of the sensor. This dependence of dimensionality on spatial resolution and SNR has long been recognized [22; 26; 27; 28]. While it is not the focus of this study, it is noteworthy that the AVIRIS data we use have roughly half the Full Width at Half Maximum (and thus four times the spatial resolution) of the Landsat data. All else equal, more spatially extensive image domains might also be expected to include a wider range of reflectance spectra and so have higher dimensionality--up to a point. As with spatial resolution, the dependence of spectral diversity on domain size depends on the characteristic spatial scale of the landscape reflectance. However, if the spectral diversity within an area approaches the global spectral diversity of the Earth, imaging a larger area should not add additional dimensionality. The total area of land surface required to approach the level of global spectral diversity is clearly location dependent. However, the relationship between spatial resolution, extent of spatial domain, and the characteristic spatial scale of Earth's spectral diversity is important because it determines which combination of extent and resolution may provide the most informative depiction of the spectral feature space as imaged by the sensor. This is important because the spectral feature space provides a physically-based depiction of the properties of a landscape that a given sensor is able to distinguish. Part of the answer to this question lies in the overall global spectral diversity of the Earth. Previous work has attempted to address this. A global analysis of total Earth radiance as measured by the hyperspectral Scanning Imaging Absorption Spectrometer for Atmospheric Cartography (SCIAMACHY) was conducted by [29], finding 99.5% of variance explained in six dimensions. However, the spatial resolution of ~30 km attenuates much of the spectral variability of interest by spectral mixing. Due to the current absence of systematic global hyperspectral imagery of the land surface, no analog exists to these studies. The largest area study of Earth surface hyperspectral imagery to date was performed by [19]. This study used 15 m resolution AVIRIS imagery of a wide range of land cover types in California and estimated the overall spectral dimensionality at 50 dimensions. The broadband spectral dimensionality of a wide range of environments has been characterized for six-band multispectral Landsat imagery [13; 14]. Because Landsat resolves only the spectral continuum, it can represent 99% of its spectral variability in only three dimensions. The structural similarities of the low order feature spaces in [PERSON] and [PERSON]'s 2000 analysis of 510 AVIRIS scenes [18] and [PERSON]'s 2004 analysis of 100,000,000 Landsat spectra [14] suggest that multispectral and hyperspectral feature spaces may share a common structure, despite the vastly greater information content of hyperspectral data. The analysis of 2.5 \(\times\) 10\({}^{9}\) AVIRIS spectra by [PERSON] et al. [19] suggests that the spectral diversity observable within California can represent all of the canonical land cover types included in the global MODIS land cover classification.