## 5 Discussion  
This work addresses three questions:  
1. How much of the dimensionality of hyperspectral data is captured in multispectral data? 2. Is the characteristic pyramidal structure of the multispectral feature space also present in the low order dimensions of the hyperspectral feature space at comparable spatial scales? 3. How much variability in Substrate EMs present hyperspectral data is captured by multispectral sensors? The first question of this study is addressed quantitatively by the hyperspectral and multispectral partitions of variance given in Figure 4. It is further addressed qualitatively through the topologies of the low-order feature spaces shown in Figures 5 and 6. The spectral feature spaces show a striking similarity in the dimensionality, topology, and EM spectra of the coincident hyperspectral and multispectral datasets used in this study. This is observed despite a factor of 4 difference in the sensor spatial resolution, a factor of 30 difference in the spectral resolution, and further differences in atmospheric correction procedures between the products. At least 97% of the variance in each dataset is present in the first three dimensions and >99% is present in the first four dimensions for both sensors. This result is in accord with the multispectral findings of [14]. It suggests a substantially lower dimensionality than the previous hyperspectral studies of [19; 27; 28], but this is because our metrics for dimensionality differ. A primary purpose of dimensionality estimation by these previous AVIRIS studies was to quantify the number of unique materials that the hyperspectral sensor can image above noise level. This is not our purpose. Rather, we seek to describe the number of independent dimensions in which the majority of the spectra reside when decomposed by variance, the number of EMs which bound the space, and the way in which these EMs trade off. This is a fundamentally different question and is why we examine the topology of the point cloud so closely in addition to the partition of variance. The information content of a dataset is not a trivial quantity to estimate. We are in full accord with [18] in the opinion that \"measuring dimensionality in multivariate data sets is a slippery slope  
Figure 8: Spectral diversity in a 9 \(\times\) 15 km region of intensive agriculture in the Great Central Valley of California. Considerably more spectral diversity is present in this hyperspectral dataset than can be adequately represented by any false color composite image Locations of spectra a-h are shown on the AVIRIS feature space and on the image. **(A)**. The AVIRIS feature space for this spatial subset (**B**, **left column**) clearly shows more prominent clustering than the Landsat feature space (**B**, **right column**). Green fields are characterized by a wide range of red edge and absorption properties (**C1,C2**). Bare or senescent fields are characterized by an even greater range of spectral shapes (**C3–C8**). This high concentration of spectral diversity is consistent with the similarity in dimensionality and feature space topology between the limited area of this study and the much more extensive global studies of Small & Milesi (2013) and Sousa & Small (2017). and an approximation at best\". We also agree with the findings of [19], that the overdetermined nature of hyperspectral imaging \"is highly desirable since it offers numerical leverage while also providing the capability to measure unexpected phenomena and falsify modeling assumptions\". It is additionally clear that much of the value of hyperspectral data resides in the precise measurement in the location, depth, and breadth of absorption features. This analysis shows that these features are not well-represented in dimensionality estimates given by the PC transform, nor by deviations in the topology of the point cloud, at least in the first four dimensions. Variance is an imperfect metric for information content, especially in hyperspectral imagery.