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The reflectance factor of a surface is generally defined as the amount of radiation reflected from the surface divided by that reflected by an ideal Lambertian diffuser. In practice, the latter is estimated by measuring the reflected radiation from a reference material approximating that ideal. If both are measured exactly once concurrently in the same conditions, the estimateof the surface reflectance factor is trivial to compute. However, if we decide to do repeated measurements of the quantities, then - like the proverbial man with two watches - we already run into multiple choices for our estimate, and for our assumptions.
Do we assume the reflected light from either source to stay constant during our measurements? Is it enough to measure the reference just once, but keep measuring the surface? The assumptions we choose naturally lead to different ways to estimate the reflectance factor as either the ratio of the mean measurements, the slope from the linear regression to our measurements, or possibly the mean of the ratios for each pair of measurements (i.e. the mean reflectance factor). The latter estimate is especially prevalent in the field of hyperspectral imaging where the same choices arise from assumptions made about spatial instead of temporal uniformity, and many datasets commonly used for analysis consist only of the already computed ratios. This can present problems for accurate comparison of different instruments, since in general the different estimates can and do differ even on the same data due to their different behaviour with respect to the assumptions and instrument errors. Furthermore, different estimates also have different numerical characteristics, which should be considered especially when computing with discretized values from digital instruments.
In order to gain a rigorous understanding of the different estimates and their associated uncertainties, in this work we present a review of the different assumptions that can be made of such measurements from a statistical viewpoint. We will present a mathematical framework for evaluating the uncertainties of each estimate given the instrument characteristics and a statistical model for the measured quantities. The framework will then be used to map out the various combinations of assumptions and estimates in order to guide planning of measurements and analysis workflows. Furthermore, we discuss the suitability of each estimate in the context of comparison between instruments, which will be accompanied by concrete examples using real and simulated data.
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