# Improving Spatial-Temporal Data Fusion by Choosing Optimal Input Image Pairs
## 3 Methods
### Quality Assessment  
For each study area, the fused Landsat-MODIS results were compared to the observed Landsat images. Only clear and valid pixels were included for the evaluation. Statistics were computed over the entire study area based on the land cover type provided by the NASS CDL.  
Three assessment metrics are used to evaluate data fusion accuracy and the consistency of images. These include mean absolute difference (MAD), Pearson correlation coefficient (R), and root mean square error (RMSE) as computed using Equations (2)-(4):  
\[\text{MAD}(t)=\frac{1}{N}{\sum\limits_{i=1}^{N}}\Big{|}r_{i}^{p}(t)-r_{i}^{o}( t)\Big{|} \tag{2}\]  
\[\text{R}(t)=\frac{\sum\limits_{i=1}^{N}\Big{(}r_{i}^{p}(t)-\overline{r^{p}(t) }\Big{)}\Big{(}r_{i}^{o}(t)-\overline{r^{o}(t)}\Big{)}}{\sqrt{\sum\limits_{i=1} ^{N}\Big{(}r_{i}^{p}(t)-\overline{r^{p}(t)}\Big{)}^{2}\times\sum\limits_{i=1}^ {N}\Big{(}r_{i}^{o}(t)-\overline{r^{o}(t)}\Big{)}^{2}}} \tag{3}\]  
\[\text{RMSE}(t)=\sqrt{\frac{1}{N}{\sum\limits_{i=1}^{N}}\Big{(}r_{i}^{p}(t)-r_ {i}^{o}(t)\Big{)}^{2}} \tag{4}\]  
where superscripts \(p\) and \(o\) represent the predicted and observed data, respectively. \(r\) is the reflectance or normalized difference vegetation index (NDVI) from the predicted or observed image on date \(t\). The subscript of \(i\) denotes pixels in each predicted and observed image with \(N\) pixels.  
NDVI is calculated based on reflectance in the red (Red) and near-infrared (NIR) spectral wavebands:  
\[\text{NDVI}=(NIR-Red)/(NIR+Red) \tag{5}\]  
MAD represents a spatially-averaged absolute difference between observed and predicted data on the same date (\(t\)). A smaller MAD means a better prediction. RMSE and R are used to evaluate the spatial consistency between MODIS and Landsat images, and also between the predicted and observed Landsat images at the date \(t\).  
Figure 3: Schematic diagram for the nearest date (ND) and higher correlation (HC) pair selection strategies. The black line and red line represent Pearson correlation coefficient (\(R\)) values between MODIS images at pair base dates \(t_{1}\) and \(t_{2}\), respectively, and the prediction dates. \(R\) generally decrease with increasing MODIS image time separation. \(t_{m}\) and \(t_{n}\) are the split dates determined by the HC and ND strategies, individually.  
For each Landsat-MODIS pair image, the MAD, R, and RMSE can be computed for the dates that have Landsat observations. To assess the year-long predictability of this pair image and the consistency of the different Landsat-MODIS combinations, the temporally averaged MAD (<MAD>), R (<R>) and RMSE (<RMSE>) were computed using all MAD, R, and RMSE values on Landsat observation dates from the entire year. Similarly, the smaller <MAD> and <RMSE>, and the larger <R> means a better predicted result and thus a better year-long predictive skill for the specific input image pair.  
In order to evaluate the spatial variability of reflectance or vegetation index at MODIS spatial resolution, the mean squared displacement (MSD) is computed using Landsat pixels within the MODIS pixel:  
\[\text{MSD}=\frac{1}{N_{r}}\sum_{i=1}^{N_{r}}\sqrt{\sum_{k=1}^{m}(r_{k}-\bar{ \tau})^{2}} \tag{6}\]  
The study area is divided into \(Nr\) small regions/windows with the window size of \(m\). In Equation (5), \(r_{k}\) is the value of the pixel \(k\) in a small window, and \(\bar{\tau}\) is the mean value within the window. In this study, the window size is fixed at 16 pixels (480 m), which is similar to the resolution of MODIS combined NBAR product. Therefore, \(m\) equals 256 here. MSD represents the spatial variability of surface reflectance or NDVI at MODIS spatial resolution. Higher MSD means a more spatially heterogeneous land cover.  
Both the fused surface reflectance and the derived NDVI were used for the evaluation. In this paper, fused NDVI refers to the NDVI derived from the fused reflectance in red and NIR wavebands rather than a fusion of MODIS and Landsat NDVI images.