# Determining the AMSR-E SST Footprint from Co-Located MODIS SSTs
## 4 Results
### AMSR-E Footprint Estimation
The pixel intensity of the retrieved footprint at a given location is determined by the contribution of the sub-resolution MODIS SST element to the AMSR-E SST at the same location. Consequently, the larger the pixel size  
Figure 7: Histogram of the cell positions for the matchups used in this paper. the more intense the contribution. Reassuringly, the footprints do not show a significant change in size in the along-track direction. In the previous paragraph, we found that the retrieved footprint depended on the AMSR-E cell position and on the size of MODIS pixels. In order to remove these dependencies, so that the across-track footprint properties can be better studied and a mean footprint derived, the following correction procedure was undertaken. First, we applied a \(4\times 4\) pixel moving average to the across-track footprints (Figure 8) in order to reduce the pixel-to-pixel noise. We then used the Matlab griddata function to transform the MODIS grid (with the x-axis parallel to scan lines, the y-axis parallel to the nadir track and grid spacing in pixels), in which the footprints are produced, to a Euclidean grid in kilometers, aligned with the instantaneous AMSR-E field of view. The results of this procedure are shown in Figure 9. The transformations applied have clearly removed the cell position dependence of the retrieved footprints, with all of the footprints now generally similar. This is not surprising given that the angle between nadir and the direction to the AMSR-E samples is very nearly independent of cell position. The similarity of the retrieved footprints suggests that a more accurate canonical footprint can be obtained by averaging the footprints of Figure 9. Figure 10 presents this average. A significant improvement was observed--the elliptical shape was considerably clearer and much smoother in comparison with the individual footprints. The standard error of the mean of the weights does not exceed \(9\times 10^{-5}\); the mean of these values over the entire footprint is \(3\times 10^{-5}\), well below estimated weights on the outer edge of the footprint. In what follows, the across-track averaged footprint, \(f\), is considered as the reference AMSR-E SST footprint to be used for the latitudinal and temporal analyses. This is also the footprint that we recommend using in other studies requiring knowledge of the characteristics of the AMSR-E SST footprint. (The change in notation from \(H\) used to represent the footprint in Equations (2)-(7) to \(f\) used in Equations (9)-(10), reflects the change from the footprint in satellite coordinates to the footprint in cartesian coordinates discussed above.) To facilitate quantitative comparisons, we render the reference footprint in functional form. Specifically, a curve of the form:  
\[\left\{ \begin{aligned} f(x,y)&=a\exp\left(-\frac{1}{2}( A(x-x_{0})^{2}+B(y-y_{0})^{2}+C(x-x_{0})(y-y_{0})\right)\\ A&=\left(\frac{\cos^{2}(\theta)}{\sigma_{x}^{2}}+ \frac{\sin^{2}(\theta)}{\sigma_{y}^{2}}\right)\\ B&=\left(\frac{\cos^{2}(\theta)}{\sigma_{y}^{2}}+ \frac{\sin^{2}(\theta)}{\sigma_{x}^{2}}\right)\\ C&=2\sin\theta\cos\theta\left(\frac{1}{\sigma_{x}^{2 }}-\frac{1}{\sigma_{y}^{2}}\right)\end{aligned} \right. \tag{9}\]  
has been fit, in a least squares sense, to the retrieved mean AMSR-E footprint (Figure 10). The best fit parameters are: \(a=0.004\), \(x_{0}=y_{0}\approx 0\), \(\theta=1.5736\), \(\sigma_{x}=19\) km and \(\sigma_{y}=29\) km. The fitted model is represented as white contour lines superimposed on the mean footprint in Figure 10. The parameter \(\theta\approx\pi/2\) indicates that the semi-major axis of the retrieved footprint is normal to the local tangent to the AMSR-E scan line. The angles obtained when averaging over bins on one side or the other of the nadir track were not statistically different from \(\pi/2\) either. footprint, by approximately a factor of four in the latitudinal case and a factor of ten in the annual case, but the resulting footprints are still visually comparable to the reference footprint. Statistical comparisons.