## 2 Data and Methods
### Empirical Model for Horizontal Wind Speed Gradients  
In order to condense the information retrieved from SAR into a few parameters, an empirical model is used to describe the one-dimensional downstream horizontal wind speeds from the coastline towards the sea, assuming that the horizontal wind speed gradients have a exponential behavior. With the coordinate \(x\) defined as before, the model can be written as  
\[u^{f}(x)=u_{land}+\Big{(}u_{offshore}-u_{land}\Big{)}\Big{(}1-\exp(-\frac{x} {\sigma})\Big{)}. \tag{10}\]  
The parameter \(\sigma\) describes the distance at which about 63% of the final wind speed increase  
\[\Delta u=u_{offshore}-u_{land} \tag{11}\]  
is observed. In addition, the distance \(x_{95\%}\) will be considered, at which the wind speed has increased to 95% of its final value \(u_{offshore}\). This distance can be expressed as  
\[x_{95\%}=-\sigma\log\Big{(}\frac{0.05\,u_{offshore}}{\Delta u}\Big{)}. \tag{12}\]  
The model given by Equation (10) is fitted to observed wind speed gradients using a least-squares approach, i.e., the following cost function is minimised  
\[J(\sigma,\Delta u,u_{land})=\sum_{i=1}^{N}(u^{f}(x_{i})-u_{SAR}(x_{i}))^{2}, \tag{13}\]where \(u_{SAR}\) denotes the SAR wind measurements and the profile is sampled on \(N\) discrete points \(x_{1},\ldots,x_{N}\). A binsize of about 2 km was used for the subsequent analysis and the number of bins is around \(N=60\). The nonlinear minimisation problem defined by Equation (13) was solved using a Gauss-Newton iteration method to obtain estimates for the parameters \(\sigma,\Delta u,u_{land}\) and hence also \(u_{offshore}\) as well as \(x_{95\%}\). From Equation (10), the modelled wind speed increase \(\delta u^{f}\) and the normalised wind speed increase \(R^{f}\) can be derived as follows:  
\[\delta u^{f}(x)=\Big{(}u_{offshore}-u_{land}\Big{)}\Big{(}1-\exp(-\frac{x}{ \sigma})\Big{)} \tag{14}\]  
and  
\[R^{f}(x)=\Big{(}\frac{u_{offshore}}{u_{land}}-1\Big{)}\Big{(}1-\exp(-\frac{x}{ \sigma})\Big{)}. \tag{15}\]  
Given Equation (15) the expected maximum of the normalised wind speed increase \(R^{f}_{max}\) can be expressed as  
\[R^{f}_{max}=R^{f}(x=\infty)=\Big{(}u_{offshore}-u_{land}\Big{)}/u_{land}. \tag{16}\]  
Denoting the number of available SAR wind speed transects \(u_{10}(x)\) by \(N_{p}\) and the _ith_ transect by \(u_{10}^{(i)}(x)\), the mean normalised wind speed increase \(R_{u_{10}}(x)\) is estimated as follows:  
\[R_{u_{10}}(x)=100\%\times\frac{\frac{1}{N_{p}}\sum_{i=1}^{N_{p}}u_{10}^{(i)}( x)-\frac{1}{N_{p}}\sum_{i=1}^{N_{p}}u_{land}^{(i)}}{\frac{1}{N_{p}}\sum_{i=1}^{N_{p}} u_{land}^{(i)}}. \tag{17}\]  
Downstream the coast the power density (power per rotor disc area) associated to the wind speed gradient from SAR measurements can also be computed using \(u^{f}\) (Equation (10)) as:  
\[\psi(x)=\frac{1}{2}\,\rho\,\eta\,(u^{f}(x))^{3}, \tag{18}\]  
where \(\rho=\)1.225 kg/m\({}^{3}\) is the density of air and \(\eta\) the power coefficient (here \(\eta\) is taken as 0.4). In offshore, the average power density \(\psi\) is therefore expressed knowing the estimates of \(u_{offshore}\) as:  
\[\psi_{offshore}=\frac{1}{2}\rho\,\eta\,\frac{1}{N_{p}}\sum_{i=1}^{N_{p}}(u_{ offshore}^{(i)})^{3}. \tag{19}\]  
Scaling Equation (18) with the \(u_{land}\) in a similar way as in Section 2.3 and Equation (15), the average normalised power density transect \(R_{\psi}(x)\) is computed as follow:  
\[R_{\psi}(x)=100\%\times\frac{\frac{1}{N_{p}}\sum_{i=1}^{N_{p}}\Big{(}u_{(i)}^ {f}(x)\Big{)}^{3}-\frac{1}{N_{p}}\sum_{i=1}^{N_{p}}(u_{land}^{(i)})^{3}}{\frac {1}{N_{p}}\sum_{i=1}^{N_{p}}(u_{land}^{(i)})^{3}}. \tag{20}\]