Question

The scattering matrix for a fully polarimetric synthetic aperture radar pixel is given below. The \( C_{11} \) element of the covariance matrix computed with a \( 1 \times 1 \) window will be __________? (rounded off to 2 decimal places).

Here, \( i = \sqrt{-1} \).

\[ \begin{bmatrix} 0.1 + 0.5i & 0.1 - 0.1i \\ 0.1 + 0.1i & 0.3 - 0.5i \end{bmatrix} \]

Hint:

To compute the \( C_{11} \) element of a covariance matrix in polarimetric SAR, square the magnitude of the corresponding scattering matrix element: \( C_{11} = |S_{HH}|^2 \).

Answer 1

The covariance matrix \( \mathbf{C} \) is computed as: \[ \mathbf{C} = \mathbf{S} \cdot \mathbf{S}^H \] But here, for a full polarimetric SAR system, the **covariance matrix** is often computed as: \[ \mathbf{C} = \langle \mathbf{k} \cdot \mathbf{k}^H \rangle \] where \( \mathbf{k} = [S_{HH}, S_{HV}, S_{VV}]^T \). For a simplified 2×2 matrix with just HH and HV, we compute \( C_{11} = |S_{HH}|^2 \). Given: \[ S_{HH} = 0.1 + 0.5i \] Then, \[ C_{11} = |S_{HH}|^2 = (0.1)^2 + (0.5)^2 = 0.01 + 0.25 = 0.26 \] \[ \boxed{0.26} \]