# Determining the AMSR-E SST Footprint from Co-Located MODIS SSTs
## 4 Results
### AMSR-E Footprint Estimation
In order to quantitatively compare the dependence of the footprint on cell position, latitude and year, we use two different statistics, one based on the mean absolute percentage deviation (MAPD), a measure of the deviation of a given footprint from a reference footprint as a percentage, and the other based on the aspect ratio of the best fit ellipse to the footprint, a measure of the deviation of a given footprint from a reference footprint based on shape. Both sets of statistics were determined for the footprints shown in Figure 9 (cell position), Figure 11 (latitude) and Figure 12 (year). We begin with the MAPD statistic, defined as follows:  
\[\text{MAPD}=\frac{\sum_{i=1}^{775}(|f(i)-\hat{f}(i))/\hat{f}(i)|)}{775}\times 1 00(\%), \tag{10}\]  
where \(f\) is the footprint to evaluate (e.g., retrieved at a particular cell position) and \(\hat{f}\) is the reference footprint, depicted in Figure 10. (MAPD was chosen as one of the two statistics because of the ease in its interpretation. Possible issues with very small values contributing an outsized share to MAPD were reduced by excluding the border of the footprint, where the weights are very low, before its estimation.) The results are shown in Figure 13a for the across-track dependence, Figure 13b for annual dependence and Figure 13c for the latitudinal dependence. In all cases, the amplitude of the deviations is below 17%. The deviations are found to be smaller for the year-by-year estimates. This was expected considering that the footprints correspond to an average over all the cell positions. The best yearly MAPD value is around 6% with an average of \(\approx 8\%\). The latitudinal footprints also result from across-track averaging but the associated deviations show more variability. This may be due to the variability in the SST fields; the smoother the field the larger the deviation of the retrieved footprint since the approach used here is not able to obtain an estimate of the footprint from a flat field. SST fields equatorward of \(30^{\circ}\text{N}/\text{S}\) tend to be more uniform that those in the subtropics. The across-track MAPD results tend to be relatively stable for negative cell positions and show an increasing trend with the increasing positive cell positions. The reason for this is not clear. Figure 8: Estimated AMSR-E footprints determined from matchups over the AMSR-E cell range indicated in the sub-caption of each frame. The axes of each of the footprints are in units of MODIS pixels (no smoothing applied). First bin, \(-115\) to \(-106\), with relatively few matchups (Figure 7), is not shown. Figure 9: Estimated AMSR-E footprints obtained by projecting the footprints in Figure 8 on regular 4 km \(\times\) 4 km grids aligned with the instantaneous AMSR-E field-of-view (a 4 \(\times\) 4 moving averaged filter was applied prior the calculation). A contour-line at 0.002 level (magenta) and its best fitting ellipse (black) are traced for each footprint. The second method for comparing footprints is based on the aspect ratio of the footprint determined as follows. For each estimated footprint, we extracted the 0.002 contour-line (magenta curves in Figure 9 for the along-scan footprints). An ellipse was then least squares fitted to the extracted contour (black curves in Figure 9) and the aspect ratio, \(\sigma_{1}/\sigma_{2}\), of this ellipse was calculated, where \(\sigma_{1}\) is the length of the semi-major axis and \(\sigma_{2}\) that of the semi-minor axis. The 0.002 contour line used in these fits corresponds to one half of the maximum value of the mean footprint (half of the fitted value of \(a\) in Equation (9)). The same procedure, applied to the mean footprint (Figure 10), resulted in an aspect ratio of \(1.58\pm 0.03\); somewhat smaller than the 1.74 value of the AMSR-E 6.93 GHz bands. Figure 14 shows the aspect ratio by (a) cell position, (b) year and (c) latitude. The footprint was stable for the entire mission at approximately the value of the reference footprint. There also appears to be little latitudinal dependence.