# `MF3CF` (Model Free 3-Component decomposition for Full-pol data)

This functionality computes the model free 3 component scattering power decomposition for full polarimetric SAR data. The required input and the computed output are as follows:

    input : input_T3/C3_folder, window_size
    output: Ps_FP.bin, Pd_FP.bin, Pv_FP.bin, Theta_FP.bin

The formulation of the scattering powers (\(P_s\) : Surface, \(P_d\) : Double bounce, \(P_v\) : volume) is as follows:

\[\begin{split}P_{d}^{\text{FP}}=\frac{m_{\text{FP}}{\text{Span}}}{2}{\left(1-\sin2\theta_{\text{FP}}\right)}\\P_{v}^{\text{FP}}={\text{Span}}\left(1-m_{\text{FP}}\right)\\P_{s}^{\text{FP}}=\frac{m_{\text{FP}}{\text{Span}}}{2}\left(1+\sin2\theta_{\text{FP}}\right)\end{split}\]

where \(m_\text{FP}\) is degree of polarization, \(\theta_\text{FP}\) scattering type parameter, Span is the sum of the diagonal elements os coherence matrix (T3). The derivation of these parameters in-terms of coherancey matrix (T3) elements is as shown below. Further details can be obtained from [[4]](#4)

\[ \begin{align}\begin{aligned}m_{\text{FP}}=\sqrt{1-\frac{27|\mathbf{T3}|}{\big(\mathrm{Trace}(\mathbf{T3})\big)^3}};\qquad{}\tan\theta_{\text{FP}}=\frac{m_{\text{FP}}{\text{Span}}\left(T_{11}-T_{22}-T_{33}\right)}{T_{11}\left(T_{22}+T_{33}\right)+m_{\text{FP}}^{2}{\text{Span}}^{2}}\\\text{Span}=T_{11}+T_{22}+T_{33}\end{aligned}\end{align} \]