Polarimetry scattering matrix

Hello everybody,
I have some questions about polarimetry SAR.

What is the exact definition of ‘scattering matrix’ such as Shh, Shv, Svv ?
Also, how can I get it using SNAP?

Actually I think it is difficult, is it possible to extract scattering matrix from the images only have intensity(in dB)? of course intensities on hh, hv, vh, vv polarization respectively
Truly, I’d like to make Pauli decomposition RGB images.

Let me know if you have any solutions.
Thanks to your attention.

Have you seen this tutorial? It was recently created and should answer many of your questions:
SAR Polarimetry and Analyses

Genreally, for most polarimetric operations, including Pauli decomposition, you need complex quad-pol data. Complex means that the product contains i and q as the real and imaginary part of the complex signal information.
Quad-pol means that you need for polarizations HH+HV+VH+VV to generate the combinations from the i and q parts. Sentinel-1, for instance, only has VV+VH.

If you have data in db, it means that you have already radiometrically calibrated it (to Sigma0, for example) and then converted it to the log scale. The i+q information is no longer present then. Accordingly, you cannot perform any polarimetric decompositions, matrices ect from it any longer.

On the difference between S (scattering), T (coherence) and C (covariance) matrices, please have a look at this document: Polarimetric Decompositions

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How could we get the scattering matrix element like the S11, S12, S21, and S22 elements in SNAP??