## 2 Background
### Measuring Spectral Variability  
A reflectance spectrum is a characteristic of a material. Reflectance is the continuous, wavelength dependent function which describes the fraction of incident light which is reflected in a given direction. One profitable way of conceptualizing a Visible to Shortwave Infrared (VSWIR) reflectance spectrum is as a composite of two signals: the continuum and the absorptions [16]. Hyperspectral imagers generally oversample the reflectance spectrum, allowing for the analysis of both continuum and absorptions. Broadband instruments undersample the spectrum unevenly, often blurring together the continuum with the absorptions.  
One way of quantifying the variability of reflectance (or radiance) vector sets is through their covariance matrix. For an image with \(n\) spectral bands, the spatial variance and covariance of each pair of bands forms an \(n\) square positive semi-definite matrix with \(n\) eigenvectors, representing uncorrelated modes of spectral variability, and \(n\) eigenvalues, representing the fraction of total variance described by each eigenvector and its corresponding spatial Principal Component, or PC.  
Because of spatial and spectral redundancy in large numbers of reflectance measurements, the actual dimensionality of the spectral feature space may be less than the number of spectral dimensions provided by the sensor. The uncorrelated modes of spectral variability given by the eigenstructure of the covariance matrix generally contribute unequally to the total variance of the data. As a result, the most important modes (i.e., those contributing the most variance) can be used to represent the underlying structure of the spectral feature space. This provides an optimal representation of the diversity and interrelationships (e.g., mixing) of the most disparate spectral patterns in the data. In this context, spectral dimensionality refers to the distinction between the most informative dimensions, describing the physically meaningful structure of the spectral feature space, and the remaining dimensions that describe stochastic variance like sensor noise, atmospheric effects (or correction artifacts), and natural spectral variability with a less straightforward physical meaning than that represented by the lower dimensions.  
This technique has been used to estimate the information content of hyperspectral [17; 18; 19; 20] imagery, as well as to describe the information loss when reflectance spectra are undersampled by multispectral imagers [21; 22]. Eigenvalues have also been used to directly quantify the dimensionality of multispectral data without reference to hyperspectral imagery [15]. Finally, the approach has also been used in the spatiotemporal analysis of image time series [23; 24; 25].  
The key assumption behind the use of this approach to estimate data dimensionality is that variance directly corresponds to information. As noted by [18], this is a key limitation for its use with hyperspectral data because the location and depth of specific absorptions provide substantial information in a small amount of overall variance. Despite this limitation, we use this method in our analysis because it is a reliable, well-understood tool with both precedent and broad applicability. The purpose of this work is to present a comparative analysis of the characteristics of coincident hyperspectral and multispectral observations of a wide range of diverse land covers.