# Rangeland Productivity Partitioned to Sub-Pixel Plant Functional Types
## 2 Materials and Methods  
### NDVI Disaggregation  
The MOD17 model incorporates two remote sensing-based variables of plant canopy dynamics [15]; the fraction of photosynthetically active radiation (FPAR), and leaf area index (LAI). The Landsat derived 30 m GPP/NPP product [16] calculated both FPAR and LAI from the normalized difference vegetation index (NDVI) using established empirical relationships. While the NDVI has certain limitations (e.g., saturation in densely vegetated areas and variable responses to soil characteristics) it is the most widely utilized and validated remote sensing-based vegetation index. The NDVI values utilized were derived from a Landsat NDVI compositing methodology [20], whereby spatially contiguous NDVI composites were produced every 16 days for each 30 m \(\times\) 30 m pixel. These single pixel level values generalized the variability of the land surface occurring at sub-30 m scales. The NDVI constitutes both the relative area and the unique phenological characteristics of each representative PFT within a single pixel, which varies substantially among PFTs and through time (Figure 2). In order  
Figure 1: Example demonstrating sub-pixel variability of 30 m normalized difference vegetation index (NDVI) derived from Landsat (**a**) and 1 m National Agriculture Imagery Program (NAIP) NDVI (**b**); the distribution of NAIP NDVI values within the Landsat pixel of interest (**c**). to disaggregate pixel NDVI to sub-pixel components, we used linear mixing theory, where the NDVI of a mixed pixel was calculated as the mean NDVI of the individual PFTs within the pixel weighted by their fractional cover [21]. This is expressed for each pixel (_i_) by (Equation (1)):  
\[NDVI(i,\ t)=\ \sum_{c=1}^{k}\biggl{(}fc(i,\ c)\cdot\widehat{NDVI_{i}(c,\ t)} \biggr{)}+\ \varepsilon(i,\ t) \tag{1}\]  
where NDVI(_i_, _t_) is the pixel NDVI at time \(t\), fc(_i_, _c_) is the fractional cover of each PFT \(c\), \(k\) is the number of PFTs in the pixel, NDVI_i(_c_,_t_) is the mean NDVI of PFT \(c\) at time \(t\), and _\(\varepsilon\) is the error. While this approach is predominantly used with actual reflectance values, the assumptions hold using NDVI [21,22]. Using the ordinary least squares technique, it was possible to estimate the mean NDVI for each PFT through space [23, 24, 25]. By knowing the fractional cover of each PFT within a pixel, along with the NDVI value, a set of overdetermined linear equations was developed (Equation (2)):  
\[\left[ \begin{array}{c}NDVI(1,\ t)\\ \vdots\\ NDVI(i,\ t)\\ \vdots\\ NDVI(N,\ t)\\ \end{array} \right]=\left[ \begin{array}{ccccc}fc(1,\ 1)&\cdots&fc(1,\ c)&\cdots&fc(1,\ K)\\ \vdots&\ddots&\vdots&\vdots&\vdots\\ fc(i,\ 1)&\cdots&fc(i,\ c)&\cdots&fc(i,\ K)\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ fc(N,\ 1)&\cdots&fc(N,\ c)&\cdots&fc(N,\ K)\\ \end{array} \right]\left[ \begin{array}{c}\widehat{NDVI(1,\ t)}\\ \vdots\\ \varepsilon(i,\ t)\\ \vdots\\ \varepsilon(N,\ t)\\ \end{array} \right] \tag{2}\]  
Solving this system of equations resulted in a single estimate of mean NDVI for each PFT across the spatial extent of sampled pixels. Estimating the NDVI value of each PFT within a specific pixel, however, was problematic given the spatial heterogeneity of rangelands. To address these issues,  
Figure 2: Conceptual differences between observed Landsat NDVI for a single pixel (**a**) and the subsequent phenological profile throughout a year (**b**), and a disaggregated pixel (**c**) with unique phenological profiles for each PFT (**d**). The pixel level NDVI for any point in time is a function of both the fractional area of the representative PFTs and their phenological state. techniques have been developed to iteratively solve these equations for each pixel while weighting the input pixels by distance and spectral similarity [21; 26; 27]. Here we build on these methods, integrating the rangeland PFT dataset [19] and the 16-day Landsat NDVI composites to disaggregate pixel level NDVI to key PFTs across western rangelands of the conterminous United States.