Computation of look cross-spectra (WV and IW products)

The main steps to compute the look cross-spectra are following the paper Engen and Johnsen [1995].

Extraction of azimuthal looks

The extraction of azimuthal looks is computed as follow:

  1. Taking the Inverse Fourier Transform of \(FT^{2D}\left[\widetilde{\underline{DN_c}}\right]\) in the range direction.

  2. Slicing the returned azimuthal Doppler bandwidth into \(n\) portions.

  3. Taking the Inverse Fourier Transform of each portion in the azimuthal direction.

  4. Normalizing each look energy. (if necessary)

  5. Detect the look

They are evaluated as follow:

\[FT^{1D}\left[\widetilde{\underline{DN_c}}\right](rg,f_{az}) = \dfrac{1}{2\pi}\int FT^{2D}\left[\widetilde{\underline{DN_c}}\right] e^{i2\pi f_{rg}rg} df_{rg}\]

The second and third step corresponding to the extraction of look \(i\) writes:

\[\widetilde{\underline{DN_c}}^i(rg,az) = \dfrac{1}{2\pi}\int FT^{1D}\left[\widetilde{\underline{DN_c}}\right](rg,f_{az})W_i(f_{az}) e^{i2\pi f_{az}az} df_{az}\]

where \(W_i\) is the weighting function corresponding to slice \(i\) in the azimuthal spectrum.

Figure ref{} shows \(\left|FT^{1D}\left[\widetilde{\underline{DN_c}}\right](rg,f_{az})\right|^2\) averaged over the range direction and the weighting function of a look.

Detecting look \(i\) and normalizing its energy finally writes:

\[look^i(rg,az)=\dfrac{\left|\widetilde{\underline{DN_c}}^i\right|^2}{\sum_{rg,az}{\left|\widetilde{\underline{DN_c}}^i\right|^2}}\]

In practice, the width of the slicing function \(W_i\) is defined relatively to the total frequency range of the azimuthal Doppler spectrum. The baseline processing relies on a division into 3 looks and each look contains 25% of the total Doppler frequency range. The remaining 25% are located at the two borders of the frequency axis (12.5% on each side).

Looks cross-spectra

Cross-spectra between look \(i\) and look \(i+n\) writes:

\[XS^{n\tau}(f_{rg},f_{az})=FT^{2D}[look^i]\cdot FT^{2D}[look^{i+n}]^\star\]

where the \(\star\) symbol stands for the complex conjugate and where the definition of the 2D Fourier Transform \(FT^{2D}\) is

\[F(f_{rg},f_{az}) \triangleq FT^{2D}[f(rg,az)] = \iint f(rg,az) e^{-i2\pi(f_{az}az+f_{rg}rg)} d_{az}\ d_{rg}\]

The time separation ‘’\(\tau\)’’ between two consecutive looks writes:

\[\tau = SaD\times look_{sep}\]

where \(SaD\) and \(look_{sep}\) are respectively the Synthetic aperture Duration [second] and the look separation.

They writes:

\[\begin{split}SaD = \dfrac{c\times s}{2f_{r}V_{sat} \Delta_{az}}\\ look_{sep} = look_{width}\times(1-look_{overlap})\end{split}\]

with \(c\), \(s\), \(f_r\), \(V_{sat}\), \(\Delta_{az}\) being respectively the speed of light, the slant range distance, the radar frequency, the satellite ground velocity and the azimuth spacing. In the baseline processing, \(look_{width}=0.2\) for IW, \(look_{width}=0.25\) for WV and \(look_{overlap}=0\).