2.2.3. Band Selection

Once the spectral library was finalized, the uSZU band selection technique [47] was performed to identify bands that optimize endmember class discrimination. uSZU is an automated band selection technique which attempts to select bands which maximize variation between endmember classes in a spectral library while minimizing the correlation between selected bands. uSZU assigns each band a Stability Index (SI) value based on the interclass variability divided by the intraclass variability for each class [47]:

$$SI\_i = \frac{\Delta inter\_i i}{\Delta intra\_i i} = \frac{\left| \sum\_{j=1}^k R\_{mean,j,i} \right|}{1.96 \times \sum\_{j=1}^k \sigma\_{j,i}} \tag{1}$$

where *k* is the total number endmembers, *Rmean,j,i* is the mean reflectance for endmember class *j* at wavelength *i*, and <sup>σ</sup>*j,i* is the standard deviation of class *j* for wavelength *i*. The band with the highest SI value is selected. Then a spectral correlation value (*Corr*) is calculated:

$$Corr(X, Y) = \frac{cov(X, Y)}{\sigma\_X \sigma\_Y} \tag{2}$$

where *cov*(*X*,*Y*) is the covariance between the selected waveband, *X* and each remaining waveband, *Y*; and *σ* is the standard deviation. All bands with *corr* values above a predetermined threshold *c* are then discarded. The process is repeated, with the threshold for correlation needed to discard a band decreasing by the value *i* with each iteration. For this study, values of *c* = 0.99 and *I* = 0.001 were used, both of these values were tested in Somers and Asner [47] and found to give acceptable accuracy. MESMA that was based on uSZU band reduced spectral libraries will be noted by including the term "uSZU" in the name.

## 2.2.4. Endmember Selection

Several approaches have been developed for determining the relative value of individual endmembers for representing their endmember class. One method for doing this is a count-based (CoB) approach, where each endmember is selected iteratively, and, using simple SMA, tested to see how many endmembers it can successfully model within the spectral library, using a predetermined Root Mean Square Error (*RMSE*) threshold to define success [52]. When this approach is applied to an endmember within its own endmember class, it is called (In-CoB); a desirable endmember will have a high In-CoB number (indicating a large number of other endmembers within the endmember class are derivative). This technique can also be applied against all the endmember classes which the endmember does not belong (Out-CoB), in this case, a desirable endmember will have a low Out-CoB number, indicating that this endmember would minimize confusion with other endmember classes.

Another approach for selecting endmembers is Endmember Average Root Mean Square Error (*EAR*), which evaluates each endmember's ability to model all other within class endmembers based on a summed RMSE [33]. EAR is calculated using the following formula:

$$EAR\_{A\_i} = \frac{\sum\_{j=1}^{n} RMSE\_{A\_i A\_j}}{n-1} \tag{3}$$

where *A* is an endmember class, *Ai* is the selected single endmember, and *Aj* are each of the other endmembers within the endmember class, and *n* is the total number of spectra in class *A*. A smaller EAR value is more desirable. Another approach, Minimum Average Spectral Angle (MASA, [34]), is similar to EAR but evaluates the summed spectral angle [53] instead of the RMSE fit. Both EAR and MASA evaluate only within an endmember class and do not evaluate interclass confusion.

Two techniques based on EAR, MASA, and CoB were used in this study. The first technique used a combination of EAR, MASA, and CoB to select three endmembers, it will therefore be abbreviated to "EMC". The combined EMC technique selected three endmembers for each endmember class: an endmember which minimizes the EAR value, an endmember which minimizes the MASA value, and an endmember which maximizes the In-CoB value. If multiple endmembers had the same In-CoB value, the endmember with the smallest Out-CoB value was selected. If the same endmember is selected through multiple EMC criteria (for example, the spectra that minimizes MASA also minimizes EAR), then fewer than three endmembers were used for that endmember class. The In-CoB [35] technique was also used in this study. For this technique, the In-CoB value of each endmember in an endmember class was evaluated, and any endmember with a unique In-CoB value was selected. If multiple endmembers have the same In-CoB value, the endmember with the minimum EAR value was selected. Another approach to endmember selection is Iterative Endmember Selection (IES) [31], which first picks two endmembers that maximize the performance of two-endmember SMA classifying the entire spectral library, as determined by using Cohen's kappa [54]. The algorithm then iteratively adds and removes endmembers in order to maximize kappa until the smallest spectral library that maximizes kappa is developed. While the other endmember selection approaches focus on within-class variability, IES explicitly considers confusion of endmembers between classes. In this study, the parameter of 0.025 was used as an RMSE constraint for the two-endmember SMA needed to classify the spectral library [32]. Spectral libraries processed in this way will be referred to as "IES" for the remainder of this study.

IES generally results in relatively large spectral libraries that can make the resultant MESMA analysis computationally expensive. Roberts et al. [55] proposed a method for reducing the size of IES generated spectral libraries through an iterative process, wherein the brightest endmember in each endmember class is initially chosen and all endmembers are modeled as a mixture of that endmember and shade are subsequently removed from the spectral library. Following the first pass, a new set of bright endmembers are selected from the reduced library, targeting endmembers that are most spectrally distinct from the first set. The process is repeated iteratively until either there are no more endmembers within a class in the reduced library, or new endmember selections fail to further reduce the library. The goal of this process is to identify the smallest set of endmembers for each class that also fully characterizes the spectral diversity of that class. The final result is a spectral library that retains the high accuracy of IES, while significantly reducing the number of endmember combinations. Spectral libraries processed in this way will be referred to as "Reduced IES".
