# Effects of Wave-Induced Processes in a Coupled Wave-Ocean Model on Particle Transport Simulations
## 3 Evaluation of Model Simulations  
### Methodology  
The evaluation of the model results consisted of two parts. The first part (Section 3.2) aimed at studying the model runs statistically with direct comparisons of the simulations with the drifter data. In the follow-up part (Section 5), the drifter trajectories were modelled with OpenDrift to investigate the importance of the wave effects, e.g., for search-and-rescue applications. The drifter velocities along the trajectories are calculated by dividing the distance between each drifter position by its time difference. The velocity is located in the middle between the two positions in time and space. Afterwards, the model velocities are interpolated trilinearly (lat, lon, time) to the drifter velocity positions and times. The root-mean-square error (_rnse_), standard deviation (_std_), bias (_bia_) and linear correlation coefficient (_cor_) are calculated to assess the model and drifter currents. Trilinear interpolation to the drifter position is performed for the wind and water depth and for each model experiment. For the analysis of meteorological conditions, periods with weak winds (25-27 June 2015) and strong southerly winds (01-03 June 2015) were chosen. In order to assign the errors of model velocities to different sources, a multi-linear regression was performed, solving the following equations:  
\[ \begin{array}{l}u_{drifter}=u_{model}a_{1i}+u_{wind}b_{1i}+c_{1i},\\ v_{drifter}=v_{model}a_{2i}+v_{wind}b_{2i}+c_{2i},\end{array}  \tag{3}\]  
Here, u\({}_{\text{model}}\) is the model velocity, u\({}_{\text{wind}}\) is the wind velocity, \(U=\sqrt{u^{2}+v^{2}}\) denotes the velocity magnitude, \(i\) corresponds to the drifter number and \(a\), \(b\), \(c\) are the coefficients. Coefficient \(b\) represents direct windage acting on the GPS drifters. The coefficient \(a\) can indicate if the model under- or over-estimates surface velocity. Coefficient \(c\) includes deviations that cannot be explained by the model currents or the wind, like inaccurate directions; it can be considered an \"offset\". The drifter starting positions are used as initial positions in OpenDrift. In the trajectory simulations, the particles are initialized at the start location and start time of the Albatros drifters and move under the influence of the forcing until the real drifters expire. Drifter simulations of 25 h drift paths that were initialized every day from 0 to 53 at 13:00 UTC were previously performed [27]. It is important to stress that inall of our experiments, after starting the drifter simulations, we did not re-initialize the modelled drifters. The time steps in the OpenDrift experiments and the output frequency are both 10 min. In the first set of experiments, only the currents of REF and CPL were used in OpenDrift. To study \begin{table}
\begin{tabular}{l l l l l} \hline
**Experiment** & **REF** & **CPL** & **WD-Ref** & **WD-CPL** \\ \hline NEMO-only & Yes & No & Yes & \\ \hline NEMO-WAM & No & Yes & No & \\ \hline Windage & No & No & 0.1\%/0.5\%/1\% & 0.1\%/0.5\%/1\% \\ \hline \end{tabular}
\end{table} 
Table 1: List of experiments. the impact of the wind, REF and CPL current data were used together with the windage coefficients WD-REF and WD-CPL, respectively. As a measure of the modelled trajectories, the skill score ss [4] is chosen. First, an index \(s\) is estimated as  
\[s=\sum_{i=1}^{N}d_{i}/\sum_{i=1}^{N}l_{oi} \tag{4}\]  
where \(N\) is the total number of time steps, \(d_{i}\) is the distance of real and modelled drifter at time step \(i\) and \(l_{oi}\) is the total length of the trajectory of the real drifter at time step \(i\). This index is used to calculate \(ss\) with  
\[ss=\left\{ \begin{array}{c}1-\frac{s}{n},\ (s\leq n)\\ 0,\ (s>n)\end{array} \right. \tag{5}\]  
where \(n\) is the tolerance threshold, equal to 1 [4], which means that the cumulative separation distance is not larger than the cumulative trajectory length. If \(s\) and \(ss\) are close to 1, the modelled trajectory is close to the observed trajectory.