The Coordinate System

Abstract - In the following section we introduce the coordinate system, which is generally used in the SAR case. We will outline the difference between the wave and the antenna coordinate system and introduce the IEEE coordinate system convention for Back-Scatter Alignment (BSA) and Forward Scatter Alignment (FSA)

keywords: coordinate system, Back-Scatter Alignment (BSA), Forward Scatter Alignment (FSA)

The scattering process can be seen as a transformation of the incident wave into a scattered wave. In the general case the polarization state of the incident wave changes under this process. For remote sensing applications the Earth's surface is usually represented by a Cartesian-coordinate system (see Fig. 2.3) with the origin at point O on the surface of the scatterer, while the polarization vectors of incident and scattered waves are represented in terms of spherical coordinates $r_i$, $\theta_i$, $\phi_i$ and $r_s$, $\theta_s$, $\phi_s$, as shown in Fig. 2.3. For SAR applications we may assume that $0 < \theta_i$ and $\theta_s < \pi / 2$. The propagation directions of the incident ( $\vec{k}_{i}$) and scattered wave ( $\vec{k}_{s}$) are then given by

$$\vec{k}_{i} = - \cos \phi_i \sin \theta_i \vec{e}_{x} - \sin \phi_i \sin \theta_i \vec{e}_{y} - \cos \theta_i \vec{e}_{z}$$ (2.25)

$$\vec{k}_{s} = \cos \phi_s \sin \theta_s \vec{e}_{x} + \sin \phi_s \sin \theta_s \vec{e}_{y} + \cos \theta_s \vec{e}_{z}$$ (2.26)

where $\vec{k}_{i}$ is directed from the transmitting antenna to the origin O and $\vec{k}_{s}$ from the origin O towards the receiving antenna. In order to describe the polarization states of the incident and scattered wave we apply two local coordinate systems with the origins on the transmitting and receiving antenna locations. The IEEE standard definitions of terms for antennas [IEEE79] yields two different definition conventions for these local coordinate systems. The first one, the Forward Scattering Alignment (FSA), is a wave oriented system which defines the local right-handed coordinate system with respect to the propagation direction of the wave. The second one, the Backscatter Scattering Alignment (BSA) is an antenna oriented system, defining the local coordinate system with respect to the antenna polarization. The antenna polarization is defined as the polarization of a wave transmitted by the antenna with propagation direction $\vec{p}$ pointing away from the antenna, even when the antenna is used as a receiver. Using this convention the two local coordinate systems coincide when both antennas are located at the same position, which is favorable for backscatter applications like SAR. Therefore we will use the Backscatter Scattering Alignment in this treatment. The local coordinate system at the position of the transmitting antenna is then, with respect to the base vectors of the reference Cartesian coordinate system ( $\vec{e}_{x},\vec{e}_{y},\vec{e}_{z}$), given by

$$\vec{e}_{p}^{ i} = \vec{k}_i \sin(\theta_i)\cos(\phi_i) \vec{e}_x + \sin(\theta_i)\sin(\phi_i) \vec{e}_y + \cos(\theta_i)\vec{e}_z$$ (2.27)

$$\vec{e}_{h}^{ i} = \frac{\vec{e}_z \times \vec{e}_{p}^{ i}}{\... ...times \vec{e}_{p}^{ i}\vert}= -\sin(\phi_i) \vec{e}_x + \cos(\phi_i)\vec{e}_y$$ (2.28)

$$\vec{e}_{v}^{ i} = \vec{e}_{h}^{ i} \times \vec{e}_{p}^{ i}... ...i_i) \vec{e}_x + \cos(\theta_i)\sin(\phi_i) \vec{e}_y - \sin(\theta_i)\vec{e}_z$$ (2.29)

Analogously the local coordinate system at the position of the receiving antenna is given by

$$\vec{e}_{p}^{ s} = \vec{k}_{s} \sin(\theta_{s})\cos(\phi_{s}) \vec{e}_x + \sin(\theta_{s})\sin(\phi_{s}) \vec{e}_y + \cos(\theta_{s})\vec{e}_z$$ (2.30)

$$\vec{e}_{h}^{ s} = \frac{\vec{e}_z \times \vec{e}_{p}^{ s}}{\... ...s \vec{e}_{p}^{ s}\vert}= -\sin(\phi_{s}) \vec{e}_x + \cos(\phi_{s})\vec{e}_y$$ (2.31)

$$\vec{e}_{v}^{ s} = \vec{e}_{h}^{ s} \times \vec{e}_{p}^{ s}... ...vec{e}_x + \cos(\theta_{s})\sin(\phi_{s}) \vec{e}_y - \sin(\theta_{s})\vec{e}_z$$ (2.32)

Figure 2.3: Coordinate system for remote sensing applications

As mentioned above, for the applications discussed in this treatment only the backscattering situation is important, where the transmitting and receiving antennas are co-located, or at least so close to each other, that the bistatic angle is very small. In that case the coordinate systems associated to the antenna which transmits the incident wave is the same as that for the receiving antenna $\{\vec{e}_{p}^{ i},\vec{e}_{v}^{ i},\vec{e}_{h}^{ i}\}= \{\vec{e}_{p}^{... ...ec{e}_{v}^{ s},\vec{e}_{h}^{ s}\} = \{\vec{e}_{p},\vec{e}_{v},\vec{e}_{h}\}$.