Computation of CWAVE parameters

CWAVE parameters is a concept described in the paper Schulz-Stellenfleth et al. [2007]. CWAVE parameters are based on the SAR image x-spectrum.

The spectral information of the normalized x-spectrum is decomposed according to orthonormal functions \(H_{ij}\) defined as tensor products of Gegenbauer polynomial \(G_i(\alpha_k(k_x,k_y))\) and harmonic \(F_j(\alpha_\phi(k_x,k_y))\) functions defined in the azimuth \(k_y\) and range \(k_y\) wave-number space.

This yields to the general formulation of CWAVE parameters \(C_{ij}\):

\[C_{ij} = \sum_{k_x, k_y}\overline{P}(k_x,k_y) H_{ij}(k_x,k_y)dk_x dk_y,\]

with \(i \in [1,n_k]\) and \(j \in [1,n_{\phi}]\). In this study \(n_k=4\) and \(n_{\phi}=5\).

The orthonormal functions are defined such as:

\[H_{ij}(k_x,k_y) = G_i(\alpha_k) F_j(\alpha_\phi) \eta(k_x , k_y),\]

where \(\eta\) writes as:

\[\eta(k_x,k_y) = \bigg( \frac{2(a_2k_x^2 + 2a_1k_x^4+k_y^2)}{(k_x^2 + k_y^2)(a_2k_x^2 + a_1k_x^4 + k_y^2)(\log k_{\max}-\log k_{\min})}\bigg)^2, %\log k_{\max} %(\log k_{\max}-\log k_{\min})\]

with

\[\begin{split}\begin{align} \gamma & = 2 \\ a_1 & = \frac{(\gamma^2 - \gamma^4) }{ (\gamma^2 * k_{\min}^2 - k_{\max}^2) }\\ a_2 & = \frac{ k_{\max}^2 - \gamma^4 k_{\min}^2 }{k_{\max}^2 - \gamma^2 k_{\min}^2} \end{align}\end{split}\]

In this study, \(k_{\min} = 2\pi/600\) and \(k_{\max} = 2\pi/25\) to take benefit of the improved resolution and size of Sentinel-1 SAR images.

\(G_i(\alpha_k(k_x,k_y))\) writes :

\[\begin{align} G_{i}^{(\lambda)}(x) & = \frac{1}{i} \bigg(2 x (i+\lambda-1) G_{i-1}^{(\lambda)}(x) - (i+2\lambda-2) G_{i-2}^{(\lambda)}(x) \bigg), \textrm{ for } i \ge 2. \end{align}\]

Otherwise:

\[\begin{split}\begin{align} C_{0}^{(\lambda)}(x) & = 1 \\ C_{1}^{(\lambda)}(x) & = 2 \lambda x \end{align}\end{split}\]

In this study \(\lambda\) is set to \(3/2\). \(F_j(\alpha_\phi(k_x,k_y))\) writes :

\[\begin{split}\begin{align} F_j(x) & = \sqrt{2/\pi}\sin\big(jx\big), \textrm{ for n>1, when i is even} \\ F_j(x) & = \sqrt{2/\pi}\sin\big((j-1)x\big), \textrm{ for n>1, when i is odd}. \end{align}\end{split}\]

otherwise:

\[\begin{align} F_1(x) & = \sqrt{2/\pi} \end{align}\]

Finally \(\alpha_k\) and \(\alpha_{\phi}\) write:

\[\begin{split}\begin{align} \alpha_k & = 2 \frac{ \log\bigg(\sqrt{a_1 k_x^4 + a_2 k_x^2 + k_y^2}\bigg) - \log(k_{\min}) }{ \log(k_{\max})- \log(k_{\min}) } - 1 \\ \alpha_{\phi} &= \arctan(k_x, k_y). \end{align}\end{split}\]