## 2 Data and Methodology
### Statistical Data Analyses  
Descriptive statistics of spatially-disaggregated and temporally-aggregated time series MODIS VI data were estimated using ArcGIS 9.2 [32] as variability measures to distinguish between an unusual event and an event within the normal range of variability. Smoothing, and moving averages of the MODIS VI data were not used since VI data with sharp peaks or broad plateaus herald cases of such human-induced and/or natural disturbances as alteration of land cover and land use, defoliation, diseases, and herbivory, in their use for real-time or forecast applications. Correlations of the MODIS NDVI and EVI data were explored using Minitab 15.1 [Minitab Inc., State College, PA] according to the four land covers, six biogeoclimate zones, four seasons, and seven years. As for the removal of periodic noise patterns in the seasonal VI time series by filtering, a discrete Fourier transformation (DFT) was adopted to decompose complex waveform domain into frequency domain. The seasonal VIs were thus separated into the signal and noise spectrums, based on the application developed by [PERSON] and [PERSON] [36] with the selection of the optimal weights according to Gaussian distribution [33-36]. Fourier filtering enabled the provision of continuous time series data to estimate missing values as well as the impact of noise on the seasonal NDVI time series to be smoothed without adversely affecting the periodicity of seasonal vegetation change and the clearness of phenological characteristics [36]. The complex NDVI time series data (\(V_{(t)}\)) are written in a form of discrete Fourier series as follows:\[V_{(t)}=\frac{1}{N}\sum_{x=0}^{N-1}f_{(x)}.\exp(\ -\ i\frac{2\pi}{N}\ xt)\,, \tag{5}\]  
where \(N\) is the number of samples in the time series; \(x\) is an index representing the sample number; \(f_{(x)}\) is the \(x\)th sample value; \(t\) is the time variance in the discrete unit of season; and \(i\) is imaginary unit. Comparisons between the raw and Fourier-filtered (FF) NDVI data for each NTE were made using simple linear regression (SLR) models in Minitab 15.1 [Minitab Inc., State College, PA]. The SLR models of the FF VI can be used to estimate inflection and maximum points in the FF VI time series and to delimit growing seasons. VI time series-based methods, such as sum of positive VI values over a given period, maximum value of VI over a year, (Maximum VI value - Minimum VI value) / integrated VI, slope between two VI values at two defined dates, slopes of logistic curves fitted to VI time series, threshold models, moving average procedures, number of days where NDVI \(>\) 0, number of days between the estimated date of green-up and end of the growing season, and date when the maximum VI value occurs within a year, have been commonly used to quantify annual production rate and amount of vegetation biomass, rate of spring or fall phases, start of green-up, and timing of the maximum availability of vegetation [44]. Spatial signal-to-noise ratio (SNR) variations in the MODIS VI time series were quantified to reveal efficiency of VI data in discrimination of subtle spectral responses, as follows [35; 36]:  
\[\text{SNR}=\frac{\text{mean VI}_{\text{signal}}}{\text{SD}_{\text{noise}}}\,, \tag{6}\]  
where SDnoise is the standard deviation of noise spectrum separated by Fourier filtering. Decomposition of the MODIS VI time series with a seasonal length of four for each NTE was performed to separate the time series into the components of their linear trends and seasonally additive or multiplicative models as well as to examine the nature of the component parts in Minitab 15.1 [Minitab Inc., State College, PA]. Multi-year trends calculated for the seasonal VI time series of the 16 spatially-disaggregated ecosystem classes serve as an indicator of the direction and rate of mean seasonal change in the VI values, after the removal of seasonal effects. Similarly, seasonal indices reflect the difference of average responses for particular seasons from the overall average, after the removal of trend effects.