## 4 Scatterometry Simulations of Ocean Reflected GPS Signals, Sea Surface Roughness, and Wave Heights  
The wave propagation is performed using a solution to the parabolic equation approximation to the electromagnetic wave equation [14; 16; 25; 29; 30]. The parabolic equation in the simulator is solved using the split-step sine transformation, where the Earth's surface is modeled using an impedance model [20; 27; 30; 31]. This impedance concept gives an accurate lower boundary condition in the determination of the electromagnetic field. The ocean waves are characterized by a semi-isotropic Phillips spectrum [23; 26]. In the simulator, this field is represented as the amplitude and the phase of the waves as a function of time, corresponding to the in-phase \(I\) and the quadrature \(Q\) components [20; 32]. The multiphase screen wave propagation uses a detailed refractivity model for the neutral atmosphere, the boundary layer, and the ionosphere. The ionosphere is modeled by  
Figure 7: Calculated surface roughness parameters as a function of the observed wind fields for all datasets. The blue curve is the mean estimate of the relation, based on all the observed GPS surface reflection signals and meteorological wind fields in the reflection zone. The gray curves represent the envelopes of all retrieved roughness parameters for all datasets. The orange curve is the relation derived from the above equations. the NeQuick model [33]. These models make it possible to simulate multipath phenomena in the neutral atmosphere and scintillations in the ionosphere. The mixed Fourier transformation and the impedance for rough surfaces (ocean waves) are used to describe the interacting between the electromagnetic wave and the ocean [16]. Equation (5) relates the rough surface impedance \(\delta\) to the smooth surface impedance \(\delta_{0}\)[20]. \[\delta=\sin(\theta)\frac{(1+\rho)\delta_{0}+(1-\rho)\sin(\theta)}{(1-\rho) \delta_{0}+(1+\rho)\sin(\theta)} \tag{5}\]  
Values of the smooth surface impedance can be found in the literature for different radio wave frequencies and polarizations [34]. \(\theta\) in Equation (5) is the angle between the wave vector and ground, known as the grazing angle. The symbol for the roughness reduction factor is \(\rho\). The roughness reduction factor can be expressed as a function of the Rayleigh roughness parameter using the following equation:  
\[\rho=I_{0}(\frac{\gamma^{2}}{2})e^{\frac{-\gamma^{2}}{2}} \tag{6}\]  
where \(I_{0}\) is the modified Bessel function of the first kind of order 0, while the Rayleigh roughness parameter \(\gamma\) is given by the following expression:  
\[\gamma=2 kh\sin(\theta) \tag{7}\]  
where \(k\) is the wave number of the electromagnetic wave, while \(h\) is the root-mean-square (rms) height of the ocean waves. It is possible to relate the rms of the ocean wave-heights to the wind speed. A detailed analysis of this can be found in [34]. The electromagnetic field at the antenna of the GPS receiver is composed of both a direct wave from the GPS satellite and an ocean reflected wave. The simulations in this study were performed using the L5 GPS signal, having a frequency of 1176.45 MHz. The output from the wave propagator constitutes the amplitudes and phases of the received GPS signal as function of time. The observations presented in the previous section used the L1 and the L2 GPS signals for the data retrievals (L1: 1575.42 MHz; L2: 1227.60 MHz). Earlier studies have shown no significant changes when using L5 rather than L1 or L2 in the simulations for the estimated ocean wave heights and surface roughness. Short-time Fourier transforms (spectrograms) were applied to the simulated \(I\) and \(Q\) signals using the meteorological conditions in the 95 cases [24; 25; 32; 35]. The total field decrease as expected for an increasing elevation angle and the boundaries of the reflected field become relatively sharp [24; 34; 35; 36].