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 $F_j({\alpha }_{\varphi }(k_x,k_y))$ functions defined in the azimuth $k_y$ and range $k_y$ wave-number space.

This yields to the general formulation of CWAVE parameters $C_{ij}$:

$$C_{ij}=\sum _{k_x,k_y}\bar{P}(k_x,k_y)H_{ij}(k_x,k_y)d{k}_{x}d{k}_y,$$

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

The orthonormal functions are defined such as:

$$H_{ij}(k_x,k_y)=G_i({\alpha }_k)F_j({\alpha }_{\varphi })\eta (k_x,k_y),$$

where $\eta$ writes as:

$$\eta (k_x,k_y)=(\frac{2(a_2{k}_x^2+2a_1{k}_x^4+k_y^2)}{(k_x^2+k_y^2)(a_2{k}_x^2+a_1{k}_x^4+k_y^2)(\log k_{max}-\log k_{min})}{)}^2,$$

with

$$\begin{array}{r}\begin{array}{rl}\gamma & =2\\ a_1& =\frac{({\gamma }^2-{\gamma }^4)}{({\gamma }^2\ast k_{min}^2-k_{max}^2)}\\ a_2& =\frac{k_{max}^2-{\gamma }^4{k}_{min}^2}{k_{max}^2-{\gamma }^2{k}_{min}^2}\end{array}\end{array}$$

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 :

$$\begin{array}{rl}{G}_i^{(\lambda )}(x)& =\frac{1}{i}(2x(i+\lambda -1)G_{i-1}^{(\lambda )}(x)-(i+2\lambda -2)G_{i-2}^{(\lambda )}(x)),\text{ for }i\ge 2.\end{array}$$

Otherwise:

$$\begin{array}{r}\begin{array}{rl}{C}_0^{(\lambda )}(x)& =1\\ C_1^{(\lambda )}(x)& =2\lambda x\end{array}\end{array}$$

In this study $\lambda$ is set to $3/2$. $F_j({\alpha }_{\varphi }(k_x,k_y))$ writes :

$$\begin{array}{r}\begin{array}{rl}{F}_j(x)& =\sqrt{2/\pi }\sin (jx),\text{ for n>1, when i is even}\\ F_j(x)& =\sqrt{2/\pi }\sin ((j-1)x),\text{ for n>1, when i is odd}.\end{array}\end{array}$$

otherwise:

$$\begin{array}{rl}{F}_1(x)& =\sqrt{2/\pi }\end{array}$$

Finally ${\alpha }_k$ and ${\alpha }_{\varphi }$ write:

$$\begin{array}{r}\begin{array}{rl}{\alpha }_k& =2\frac{\log (\sqrt{a_1{k}_x^4+a_2{k}_x^2+k_y^2})-\log (k_{min})}{\log (k_{max})-\log (k_{min})}-1\\ {\alpha }_{\varphi }& =\arctan (k_x,k_y).\end{array}\end{array}$$