# Potential of a SAR Small-Satellite Constellation for Rapid Monitoring of Flood Extent
## 2 Methods
### Design of SAR Small-Satellite Constellation
#### 2.1.3 SAR Satellite Observation  
SAR satellites observe the area of interest by adjusting the direction of their sensors. The incidence angle \(\phi\) and the off-nadir angle \(\psi\) are defined as shown in Figure 1. \(\phi\) is typically larger than \(\psi\) as a result of the curvature of the Earth. Supposing that the ground area that can be observed by that adjustment is an observable range \(L\) [km] and assuming that the satellite orbit is circular and that the Earth's surface is spherical, \(L\) is calculated as follows [16]:  
\[L=R_{e}(\phi_{ub}-\psi_{ub})-R_{e}(\phi_{lb}-\psi_{lb}) \tag{1}\]  
where \(R_{e}\) [km] denotes the radius of the Earth, \(\phi_{ub}\) [rad] is the upper bound of incidence angles, \(\psi_{ub}\) [rad] is the upper bound of off-nadir angles, \(\phi_{lb}\) [rad] is the lower bound of incidence angles and \(\psi_{lb}\) [rad] is the lower bound of off-nadir angles. A relationship between an incident angle and an off-nadir angle may be observed as follows [16]:  
\[\frac{sin\phi}{sin\psi}=\frac{R_{e}}{R_{e}+a} \tag{2}\]  
where \(a\) [km] denotes the satellite's altitude. It is derived from Equations (1) and (2) that an observable range is getting wider as a satellite's altitude becomes higher.  
To define the incidence angle \(\phi\) range and the satellite altitude \(a\), the orbit and SAR property information of a SAR-equipped small satellite (ICEYE-X2) was used (detailed in Table 1) [17]. As shown in Table 1, an incidence angle from \(15^{\circ}\) to \(30^{\circ}\) may be used in its Strimpap imaging mode, and it operates at an altitude of 570 km, so we determined \(\phi_{lb}=15^{\circ}\), \(\phi_{ub}=30^{\circ}\), and \(a=570\) km for the calculation of \(L\). Regarding the inclination, we supposed a general use of satellites including other applications in this paper, so a polar orbit was adopted, not a lower orbital inclination. Current SAR systems are capable of operating in different imaging modes by controlling the antenna radiation pattern, which practically results in different combinations of the swath width and resolution [18]. The most fundamental mode is the Stripmap mode, where the ground swath is illuminated with a continuous sequence of pulses while the antenna beam is fixed in its orientation, thus imaging a long strip parallel to the flight direction [18]. This imaging mode should be ideal in this analysis because it has 30 km swath and 3 m resolution properties that are suitable for the requirements for flood detection. For a better azimuth resolution, the Spotlight mode is utilized, but this operation is usually at the expense of spatial coverage (1-m resolution and \(5\times 5\) km\({}^{2}\) scene in the reference SAR-equipped small satellite). For a wider swath, the system can be operated in the ScanSAR mode, but the azimuth  
Figure 1: Schematic illustration of SAR satellite observation.  
when compared to the Stripmap mode (this mode is currently under development in the reference SAR-equipped small satellite, but with 100 m resolution and 350 km swath in a traditional SAR satellite ALOS-2) [17].