# About regularization cost function present in Hasselman's 91 (Retrieve wave spectra)

Good morning,

What I'm trying to understand is the part of minimizing the functional present in HH91 ("On the nonlinear mapping of an ocean wave spectrum into a synthetic aperture radar image spectrum and its inversion." By minimizing this cost function we are supposed to be able to invert the image spectrum and thus obtain the wave spectrum.

The cost function is $$J= \int [\Delta P^n-(P^{a}-P^n)]+\mu [\Delta F^n-(F^{a}-F^n)]d\mathbf{k}$$

where $$\mathbf{k}$$ is the wavenumber vector, $$\mu$$ is a constant, $$F^n$$ represents the SAR wave spectrum, $$P^n$$ represents SAR image spectrum and $$\Delta P^n= \frac{\exp{(k_x q)}}{2}[\vert T_{k}^S \vert^2 \psi(\mathbf{k})\exp{(j\omega \Delta t)}+\vert T_{-k}^S \vert^2 \psi(\mathbf{-k})\exp{(-j\omega \Delta t)}]$$ it is the quasi linear transformation and $$P^a$$ and $$F^a$$ are initial inputs on the iterative method and doesn't change.

The solution proposed on the article by variational method is:

$$\Delta F^n= \frac{A_{-k}(W_k\delta P+\mu\delta F_k )-B_k(W_{-k}\delta P+\mu \delta F_{-k})}{A_kA_{-k}-B_k^2}$$

$$\delta P= P^a(\mathbf{k})-P^n(\mathbf (k))=P^a(\mathbf{-k})-P^n(\mathbf (-k))$$ $$\delta F_k=F^a(\mathbf{k})-F^n(\mathbf (k))$$ $$A_k=W_k^2+2\mu$$ $$B_k=W_kW_{-k}$$ $$\vert T^S\mathbf (k) \vert ^2 \exp(-k_x^2 q)$$

Anyone who understands this part of the theory of SAR retrieve wave spectra could you explain to me how to obtain $$\Delta F^n$$ equation?

the part of the algorithm I was able to understand but I imagine that i'm doing something wrong. Right now i am trying to use variational calculus with Euler-Lagrange equation but i don't know really well this theory and i have doubts about if this is the right strategy.

I would like to thank you for any help