# Estimating Significant Wave Height from SAR with Long Integration Times
## 2 Estimating Significant Wave Height from SAR with Long Integration Times
### Ocean Wave Refocusing  
In order to solve the problems of invisible ocean waves and only a weak wave system in the case of multiple wave systems can be observed acquired by long integration time SAR, the ocean wave refocusing method based on optimum focus setting will be introduced in this section. Focus setting \(\Delta V\) is defined as the difference between platform speed \(V\) and azimuth matched filter speed \(U\), namely,  
\[\Delta V=V-U \tag{2}\]  
The focus setting that makes the target to be best-focused is called the optimum focus setting [20; 21]. The contrast of ocean waves can be improved by changing the focus setting, and the optimum focus setting is obtained according to the azimuth phase speed [22]. However, the ocean waves are usually invisible and obtained by the long integration time SAR, which will result in the optimum focus setting being unable to be calculated by the present methods. Accordingly, it is a failure to achieve the ocean wave refocusing. In addition, for the case of multiple ocean wave systems and only a weak wave can be observed, the present methods can only obtain the optimum focus setting of an ocean wave system, which will result in that only an ocean wave system can be refocused. In order to solve the above two problems, this section will improve the present methods. The optimum focus setting is calculated by setting a focus setting variation section and finding the local maximum value of the peak-to-background ratio (PBR), and further the refocused SAR images of ocean waves are obtained. Next, the improved method is described. In this subsection, in order to obtain the uncompressed data in Range-Doppler domain, the Range-Doppler domain data is multiplied by the complex conjugate \(h^{*}(\tau)\) of the azimuth matched filter used in imaging process. The complex conjugate \(h^{*}(\tau)\) of the azimuth matched filter is  
\[h^{*}(\tau)=\exp\left(-j\frac{2\pi V^{2}\tau^{2}}{\lambda R}\right) \tag{3}\]  
where \(V\) is the platform speed, \(\tau\) is the azimuth time, \(\lambda\) is the radar wavelength, and \(R\) is the slant range. The next step is to set a focus setting variation section \(\Delta V_{i}\in[V-50,V+50]\). And new azimuth matched filter functions \(h_{i}(\tau)\) are generated,  
\[h_{i}(\tau)=\exp\left(j\frac{2\pi U_{i}^{2}\tau^{2}}{\lambda R}\right) \tag{4}\]  
where the velocity parameters \(U_{i}=V-\Delta V_{i}\) of the azimuth matched filter. Refocusing results under different focus settings are obtained by means of the new azimuth matched filter functions \(h_{i}(\tau)\), and the refocused image is calibrated, which including slant-to-ground conversion, multi-look processing and range energy normalization [19]. Next, the calibrated images are compared to determine the optimum focus setting. Therefore, the SAR images corresponding to different focus settings \(\Delta V_{i}\) are transformed into the two-dimensional wavenumber domain, and the peak-to-background ratio (PBR) of the wavenumber spectral energy is calculated. The definition of PBR is as follows [23; 24]:  
\[PBR=\frac{(S_{I})_{\text{max}}}{\langle S_{noise}\rangle} \tag{5}\]  
among them, \(\left(S_{I}\right)_{\text{max}}\) and \(\langle S_{noise}\rangle\) represent the peak value and noise floor of the SAR image spectrum, respectively. The peak-to-background ratio \(PBR_{i}\) is normalized to obtain the curve of normalized \(PBR_{i}\) with focus setting \(\Delta V_{i}\). According to the focus setting corresponding to the \(n\)th local maximum value of \(PBR_{i}\), the optimum focus setting \(\Delta V_{opt}^{n}\) of the \(n\)th ocean wave system is obtained. The optimum focus setting of \(Num\) ocean wave systems is expressed as  
\[\mathbf{\Delta V_{opt}}=[\Delta V_{opt}^{n}],1\leq n\leq Num \tag{6}\]\(\mathbf{\Delta V_{opt}}\) will be used to solve the 180\({}^{\circ}\) ambiguity of wave propagation direction. In addition, the SAR images corresponding to the optimum focus setting and the number \(Num\) of local maximum value will be used in the subsequent estimating significant wave height.