Pauli Decomposition

The Pauli decomposition is a rather simple decomposition and yet it yield a lot of information about the data. Therefore I recommend to perform this little task as a standard quality check of your data. I assume your data set is given in the HV - basis (H = horizontal, V = vertical). In this case you will have 4 channels ($S_{HH}$,$S_{HV}$,$S_{VH}$,$S_{VV}$), corresponding to the 4 elements of the [S]-Matrix. Sometimes you will only get 3 channels because $S_{HV} \equiv S_{VH}$. If you have 4 channels you can combine the cross-polar channels into one channel

$$S_X = \frac{S_{HV}+S_{VH}}{2}$$ (B.1)

So now you end up with 3 channels ($S_{HH}$,$S_{X}$,$S_{VV}$. From these channels you compute the 3 Pauli components

$$Pauli1 = S_{HH}+S_{HH}$$ (B.2)

$$Pauli2 = S_{HH}-S_{VV}$$

$$Pauli3 = S_{HV}S_{VH}=2S_{X} \mbox{45$^o$ tilted even bounce component}$$

From each of the 3 components you calculate the absolute value and convert it into byte array (8 bit per pixel). Then you assign to each of the layers a color (usually Pauli1=blue (sea-surface), Pauli2=red (double bounce from walls), and Pauli3=green (forests will get green) and combine them in a 3 color composite with a program or some image manipulation software. In order to improve the contrast you might want so scale the images not linear. I usually use a square root like scaling or scaling with an exponent of 0.7. For details look in the source code of the program decomp.pro on http://epsilon.nought.de.