## 2 Materials and Methods  
The dynamic coherent radar detection diagram is demonstrated in Figure 1, in which the satellites are modeled on a sphere. In order to demonstrate the satellite constellation configuration clearly, the Vehicle Velocity Local Horizontal (VVLH) coordinate system is employed, where the red plane is parallel to the orbital plane, the purple arrow represents the direction of platform motion, the white arrow points to the center of Earth, and the green arrow is perpendicular to the orbital plane. We assume that the distributed satellite constellation is composed of \(N\) satellites and the subarray pattern for each radar is given by \(F_{\rm e}(\theta,\phi)\). The mathematical expression of the distributed array pattern is defined as  
\[F(\theta,\phi)=\sum_{n=0}^{N-1}F_{\rm e}(\theta_{n},\phi_{n})\mathrm{e}^{ \frac{\left|2\pi n\right|}{\lambda}\Delta R_{n}} \tag{1}\]  
where \(\theta_{n}\), \(\phi_{n}(n=1,2,\cdots,N)\) denotes the azimuth angle and the elevation angle of the nth satellite and \(\lambda\) is the signal wavelength. \(\Delta R_{n}\) represents the wave path difference among different satellites. The elevation and azimuth angles of these spaceborne radars can be approximately considered to be equivalent, that is, \(\theta_{n}\approx\theta\), \(\phi_{n}\approx\phi\). Therefore, Equation (1) can be changed to  
\[F(\theta,\phi)=F_{\rm e}(\theta,\phi)\sum_{n=0}^{N-1}\mathrm{e}^{\frac{2\pi n }{\lambda}\Delta R_{n}} \tag{2}\]  
For coherent radars [7], the whole field of the array manifold is given by  
\[\mathbf{w}=\left[1,\mathrm{e}^{\frac{\left|2\pi n\right|(\sqrt{R^{2}+d^{2}-2 \delta_{n}\cos\theta}-R)}{\lambda}},\cdots,\mathrm{e}^{\frac{\left|2\pi n \right|(\sqrt{R^{2}+d^{2}_{n}-2\delta_{n}\cos\theta}-R)}{\lambda}}\right]^{ \mathrm{T}} \tag{3}\]  
where \(d_{n}\) stands for the array element spacing between the reference radar and the nth radar. \(R\) represents the distance from the antenna phase center to the target and \(\psi\) is the scanning angle. In particular, if the detection region belongs to the far-field, Equation (3) can be rewritten as  
\[\mathbf{w}=\left[1\,\mathrm{e}^{\frac{\left|2\pi\mathrm{d}\cos\theta}{ \lambda}},\cdots,\mathrm{e}^{\frac{\left|2\pi\mathrm{d}\cos\theta}{\lambda}} \right]^{\mathrm{T}} \tag{4}\]  
From Equations (2) and (3), the detection space is generally divided into two parts, i.e., the near field and the far-field, according to the range difference between the target and the satellite constellation. The distributed coherent radar is confronted with resource scheduling problems when it operates under the unit scanning mode in the near field. As a result, the combination strategy, including range scanning and angle scanning, should be conducted to solve the variant issue of spatial steering vectors for different range cells in a single beam, which would not meet the requirement of real-time detection. The spatial  
Figure 1: The dynamic coherent radar detection diagram. steering vectors of different range cells, with regard to a specific beam, are considered to be approximately constant in the far-field, and hence only the beam scanning strategy is applied here. Therefore, the latter operation mode is adopted in the wide-range scanning process based on a distributed satellite constellation. The target searching diagram of the coherent radar in the whole field is shown in Figure 2.  
In this paper, a transmit beampattern optimization method for dynamic coherent radar based on a distributed satellite constellation is presented to deal with the moving target detection problem. Here, three main steps are given below. Step 1 Detection range design for distributed coherent radar. For the distributed satellite constellation, its detection performance can be equivalently regarded as a single large satellite by employing the cooperative operation mode for multiple small satellites. However, the coherent detection area is restricted due to its distributed configuration.