## 3 Target Response Model and Analysis
### Acquisition Geometry and Response Model  
External calibration entails the comparison of the measured target response with the expectation. The expected response depends on sensor and target properties, but also on the acquisition geometry. For the purposes of calibration, it is convenient to describe the geometry in a rest frame of the instrument. This frame is a Cartesian coordinate system that translates and rotates with the sensor platform. A fixed point \(\vec{x}_{global}\) in a global coordinate system transforms into a time-variant coordinate \(\vec{x}_{instr}(t)\) in the instrument coordinate system as  
\[\vec{x}_{instr}(t)=\mathbf{R}_{G\to I}(t)\left(\vec{x}_{global}-\vec{N}(t) \right), \tag{1}\]  
where \(\vec{N}(t)\) is the position of the instrument reference point in global coordinates, as measured by the navigation subsystem, and \(\mathbf{R}_{G\to I}(t)\) denotes the rotation into the rest frame and depends on the time variant sensor attitude angles. In instrument coordinates, the phase centre of an antenna element \(e\in\{p,q\}\) is constant and is given by the sum of \(\vec{L}_{e}\) and \(\Delta\vec{L}_{e}\). These former represents the assumed phase center position. The latter denotes the correction to this position introduced in the process of sensor calibration. It is initially assumed to be zero. The position of the target \(\vec{\mathbf{T}}^{s}(t)\) is obtained from the known global target coordinate via the transform (1). The position of the target relative to a phase centre \(e\) is then defined as  
\[\vec{r}_{e}^{s}(az)=\vec{\mathbf{T}}^{s}\big{(}t_{pq}(az)\big{)}-\left(\vec{L}_{e }+\Delta\vec{L}_{e}\right), \tag{2}\]  
where \(t_{pq}(az)\) is the channel dependent mapping from azimuth sample index \(az\) onto the continuous time axis \(t\). Note that this formulation assumes that the _start-stop_ effect can be ignored, as is the case for airborne SAR but not for spaceborne platforms and high spatial resolutions [20]. Normalizing the relative position of the target gives the azimuth variant line of sight (LOS) vectors towards the target:  
\[\vec{los}_{e}^{s}(az)=\frac{\vec{r}_{e}^{s}(az)}{|\vec{r}_{e}^{s}(az)|} \tag{3}\]  
In a given acquisition geometry, the expected target response \(M_{pq}^{s}(az,f_{r})\) varies as a function of the azimuth sample index \(az\) and the physical range frequency \(f_{r}\). This response has an amplitude that is closely related to the radar equation for real aperture radar [21] as well as a phase related to the spatial seperation of sensor and target. The combination of these terms is the response model underlying the calibration process:  
\[M_{pq}^{s}(az,f_{r})=\frac{cA_{pq}^{s}(az,f_{r})}{\sqrt{4\pi f_{r}}}\frac{R_{ pq}^{s}(az,f_{r})}{4\pi\prod_{e\in\{p,q\}}|\vec{r}_{e}^{s}(az)|}e^{ip_{pq}^{s}( az,f_{r})}, \tag{4}\]  
where \(c\) is the speed of light and \(j\) denotes \(\sqrt{-1}\). The phase of the reference target response is related to the target range and can be written as  
\[\phi_{pq}^{s}(az,f_{r})=-2\pi\frac{f_{r}}{(1+\Delta c)}\sum_{c\in\{p,q\}}| \vec{r}_{e}^{s}(az)|\,, \tag{5}\]  
where \(\Delta c\) is a tropospheric propagation delay correction that is introduced in the calibration process and is initially assumed zero. The power of the response, meanwhile, is given by the received signal power as defined in the context of the radar equation for real aperture radar [21]. It includes the free space propagation loss in the denominator as well as the azimuth variant antenna amplification \(A^{s}_{pq}(az,f_{r})\) and the target backscatter amplitude \(R^{s}_{pq}(az,f_{r})\). The antenna amplification is given by  
\[A^{s}_{pq}(az,f_{r})=\prod_{e\in\{p,q\}}A_{e}\Big{(}\alpha\Big{(}\mathbf{R}_{I\to e }\mathbf{\widetilde{los}^{s}_{e}(az)}\Big{)}\,,\beta\Big{(}\mathbf{R}_{I\to e}\mathbf{ \widetilde{los}^{s}_{e}(az)}\Big{)}\,,f_{r}\Big{)}\, \tag{6}\]  
where \(A_{e}(\alpha,\beta,f_{r})\) is the complex valued antenna diagram of element \(e\), which is a function of propagation direction (angles \(\alpha\) and \(\beta\)) and frequency \(f_{r}\).