## 2 Data and Methods
### Synthetic Aperture Radar (SAR) Derived Wind Field
The rectangular box in purple represents the transect “Ref” in an area without influence of atmospheric wakes, while the black box indicates the transect through the cluster N-4 (”N4”). The detailed procedures regarding the 10 m wind retrieval are given in [PERSON] et al. [17] and [PERSON] et al. [13]. The use of CMOD5.N for the SAR wind speed retrieval introduces some errors for the cases, where the boundary layer is not neutral. It was shown in [PERSON] et al. [15] that the standard deviation of the derived SAR wind speed in comparison to in-situ measurements is about 1.9 m/s. It is important to emphasize in this context that the present study is mainly concerned with relative changes of wind speeds, i.e., changes with respect to the wind speed over land. In a first approximation, the wind speed profile over the ocean can be modelled as  
\[u(z)=\Phi(z,u^{*},L)=\frac{u^{*}}{\kappa}\Big{(}\log(\frac{z}{z_{0}})-\Psi(z/L )\Big{)}, \tag{1}\]  
where \(u^{*}\) is the friction velocity, \(\kappa\) is the Karman constant, \(z_{0}\) is the ocean surface roughness length and \(L\) is the Obukhov length. The stability function \(\Psi\) has different forms for stable and unstable conditions and the version used here can be found in [PERSON] et al. [34]. An approximation of the roughness length \(z_{0}\) can be derived from the friction velocity using the Charnock relation [34]. For a given incidence angle and wind direction, the observed radar cross section is mainly determined by \(u^{*}\), i.e., the wind speed at 10 m height derived from CMOD5.N can be written as  
\[U_{10}=\Phi(z=10\;\text{m},u^{*},L=\infty) \tag{2}\]  
which is Equation (1) evaluated for neutral conditions and \(z=10\;\text{m}\). We now analyse the error in the estimation of the relative wind speed increase for the case that the boundary is not neutral, i.e., \(L=L^{\prime}<\infty\). Let us denote the saturated friction velocity far offshore as \(u_{OS}^{*}\). We know from our statistical analysis presented in Section 3.5, that the relative wind speed increase with respect to land conditions is typically about 40%; the friction velocity close to the land is therefore of the order of \(u_{land}^{*}=u_{OS}^{*}/1.4\). The correction factor that needs to be applied to derive the \"true\" \(U_{10}\) wind speed from the CMOD5.N estimate is \(\Phi(u^{*},L=L^{\prime})/\Phi(u^{*},L=\infty)\). The correction factor \(C\) for the ratio of offshore wind speed and land wind speed is  
\[C(L,u_{OS}^{*})=\frac{\Phi(u_{OS}^{*},L=L^{\prime})\;\Phi(u_{land}^{*}L= \infty)}{\Phi(u_{OS}^{*},L=\infty)\;\Phi(u_{land}^{*}L=L^{\prime})}. \tag{3}\]  
This function was evaluated for friction velocities \(u_{OS}^{*}\in[0.1\;\text{m}/\text{s},1\;\text{m}/\text{s}]\) and for \(L\in[8\;\text{m},1000\;\text{m}]\) (representing stable to neutral conditions) and \(L\in[-1000\;\text{m},\;-2\;\text{m}]\) (representing neutral to unstable conditions). The derived range for the ratio was \(C\in[0.979,1.000]\) for unstable conditions and \(C\in[1.000,1.030]\) for stable conditions. This means that the percentage error for the relative wind speed increase due to the use of CMOD5.N in non-neutral conditions is 3% at most, which is regarded as negligible for the presented analysis.