# Determining the AMSR-E SST Footprint from Co-Located MODIS SSTs
## 3 Methodology
### Footprint Estimation Approach  
To determine the AMSR-E SST footprint, we adopt a linear least-square method as described below, although other methods [18] may also be used. The method exploits the availability of cloud-free high-resolution MODIS SST data, considered as the true sub-resolution SSTs, that overlap each AMSR-E measurement. Consider an AMSR-E SST pixel as shown in Figure 2. The 25 \(\times\) 31, 4 \(\times\) 4 pixel averages of the original MODIS L2 SST field centered on the AMSR-E field-of-view, are displayed as red squares in the figure and the theoretical field-of-view of the AMSR-E 6.93 GHz channel is plotted in black. We express the MODIS pixels as a column vector \(M_{775\times 1}\). We express the AMSR-E pixels also as a column vector \(A_{N\times 1}\), where \(N\) is the number of match-ps used in the analysis. In perfect conditions (absence of measurement noises), the coarse-resolution AMSR-E SST measurement results from a weighted average of the sub-resolution SSTs (approximated here by the MODIS measurements). In this ideal case, the relation between the collection of all available AMSR-E SST pixel measurements (A) and MODIS mach-ups (M), can be written as:  
\[\left( \begin{array}{c}a_{1}\\ a_{2}\\ a_{3}\\ a_{4}\\ \vdots\\ a_{N-1}\\ a_{N}\end{array} \right)_{A}=\left( \begin{array}{cccccc}m_{1}^{1}&m_{2}^{1}&m_ {3}^{1}&\cdots&m_{775}^{1}\\ m_{1}^{2}&m_{2}^{2}&m_{3}^{2}&\cdots&m_{775}^{2}\\ m_{1}^{3}&m_{2}^{3}&m_{3}^{3}&\cdots&m_{775}^{3}\\ m_{1}^{4}&m_{2}^{4}&m_{3}^{4}&\cdots&m_{775}^{4}\\ &&\vdots\\ m_{1}^{N-1}&m_{2}^{N-1}&m_{3}^{N-1}&\cdots&m_{775}^{N-1}\\ m_{1}^{N}&m_{2}^{N}&m_{3}^{N}&\cdots&m_{775}^{N}\\ \end{array} \right)_{M}\left( \begin{array}{c}h_{1}\\ h_{2}\\ h_{3}\\ h_{7}\\ h_{7}\\ \end{array} \right)_{H}, \tag{1}\]  
or more compactly as:  
\[A_{N\times 1}=M_{775\times N}H_{775\times 1}, \tag{2}\]  
where \(H\) is the footprint vector containing the weighting elements. A more realistic representation, i.e., one including measurement noise \(\epsilon\), can be written as:  
\[A_{N\times 1}=M_{775\times N}H_{775\times 1}+\epsilon_{N\times 1}. \tag{3}\]  
Equation (3) simply identifies a regression relation between \(A\) and \(M\); i.e., the SST of every AMSR-E pixel is seen to be the H-weighted average of the sub-resolution MODIS SSTs. The problem of deriving the AMSR-E footprint is then equivalent to determining the correct form of \(H\), which minimizes the following difference:  
\[\arg\min_{H}\lVert M_{775\times N}H_{775\times 1}-A_{N\times 1}\rVert_{2}^{2} \tag{4}\]  
This is the standard linear least-square problem, and there are several methods [19] for solving this type of problem. However, the solution to this problem allows for both positive and negative values of the elements of H, but it is unlikely that any of the regions in the AMSR-E field-of-view have a negative weight, this would be equivalent to a negative number of photons arriving at the detector from the region. We therefore add the constraint that all elements of \(H\) must be positive:  
\[H_{i}\geq 0,\quad\forall i=1,\cdots 775. \tag{5}\]  
In addition, assuming that there is no bias between the AMSR-E field and the MODIS field, the elements of \(H\) should sum to one to preserve the mean value of the AMSR-E SST pixel--the temperature of an AMSR-E SST retrieval from a uniform field should be the temperature of the field. To facilitate the analysis, we fold any bias between the retrievals into the error term of Equation (3) and apply the following constraint:  
\[\sum_{i=1}^{i=775}H_{i}=1. \tag{6}\]  
Subject to these constraints our least squares problem becomes:  
\[\text{arg}\underset{H}{\text{min}}\|M_{775\times N}H_{775\times 1}-A_{N\times 1 }\|_{2}^{2}\quad\text{Subject to}\quad\sum_{i=1}^{i=775}H=1\quad\&\quad H_{i}\geq 0 \quad(\forall i=1,\cdots 775). \tag{7}\]  
We used the Matlab solver, lsqlin, to determine \(H\) subject to the given constraints and the known values \(A\) (AMSR-E SST pixels) and \(M\) (the corresponding MODIS pixels).