###### Abstract
Sea-surface movement produces projection components in four directions, where \(V_{S}\) and \(V_{G}\) represent horizontal radial velocity components along beams far from the ascending and descending satellites. \(V_{V}\) and \(V_{U}\) represent surface velocity components along the north-south and east-west directions, respectively. In addition, \(\alpha\) represents the intersection angle between the direction of the beam center line and the east-west direction on the horizontal plane. Figure 1: Schematic diagram of the geometric relationship between the sea-surface velocity and its components in the trajectory-crossing method. The simulation method proposed in this paper is mainly divided into three parts: radial velocity simulation, wave velocity simulation and current vector simulation. The flow chart is shown in Figure 2. The first step is to calculate the radial velocity using the SAR ocean imaging model, the second step is to calculate the wave motion component, and the third step is to calculate the two-dimensional current field combined with the trajectory-crossing method. The radar parameters include the signal frequency, look direction of the beam and incident angle. Platform parameters include the satellite orbit height and flight speed. Based on the SAR ocean imaging model [7; 8; 9] proposed by [PERSON], University of Hamburg, Germany, the sea-surface horizontal radial velocity components along the spaceborne SAR beam look in directions with the ascending and descending passes simulated. The model supports simulation in the frequency range of 0.4 to 35 GHz. The main processes include the calculation of the wave-current interaction model, sea-surface scattering model and sea-surface SAR Doppler model. Wave-current interaction is a physical process that must be considered in the real sea motion environment. We need to first give the sea-surface-current field and wind field parameters and then solve the action spectrum balance equation and calculate the modulated wave spectrum according to the weak hydrodynamic interaction theory. The action spectrum balance equation to be solved is [9; 13; 14]  
\[\frac{\partial N(\mathbf{x},\mathbf{k},t)}{\partial t}+\big{[}c_{g}(\mathbf{k})+U(\mathbf{x},t )\big{]}\frac{\partial N(\mathbf{x},\mathbf{k},t)}{\partial\mathbf{x}}-\mathbf{k}\frac{ \partial U(\mathbf{x},t)}{\partial\mathbf{x}}\frac{\partial N(\mathbf{x},\mathbf{k},t)}{ \partial\mathbf{k}}=Q(\mathbf{x},\mathbf{k},t) \tag{1}\]  
where \(t\) represents time, \(\mathbf{x}=(x,y)\) represents the sea-surface spatial position, \(\mathbf{k}=\big{(}k_{x},k_{y}\big{)}\) represents the wave number, and \(k_{x}\) and \(k_{y}\) represent the wave numbers in the X and Y directions, respectively. \(N\) represents the action spectral density of sea-surface micro scale wave, \(U\) represents the surface current field, and \(C_{g}\) represents the modulated wave group velocity. These variables about ocean waves are related to the conditions of sea-surface wind field. \(Q\) represents the source function:  
Figure 2: Flow chart of two-dimensional sea-surface-current-field simulation for trajectory crossing spaceborne SAR. \[Q(\mathbf{x},\mathbf{k},t)=\mu(\mathbf{k})N(\mathbf{x},\mathbf{k},t)\bigg{(}1-\frac{N(\mathbf{x},\mathbf{k},t)} {N_{0}(\mathbf{k})}\bigg{)} \tag{2}\]  
where \(N_{0}\) represents the action spectral density in the sea-surface equilibrium state, and \(\mu\) represents the relaxation rate. The relationship between the wave spectrum and action spectral density is [15]  
\[N(\mathbf{x},\mathbf{k},t)=\frac{\rho\omega_{0}(\mathbf{k})}{\mathbf{k}}\psi(\mathbf{x},\mathbf{k},t) \tag{3}\]  
where \(\psi\) represents the modulated wave spectrum. The natural angular frequency \(\omega_{0}(\mathbf{k})=\sqrt{g\mathbf{k}+(\tau_{s}/\rho)\mathbf{k}^{3}}\), \(g\) is the gravitational acceleration, \(\rho\) is the seawater density, and \(\tau_{s}\) is the seawater surface tension.