# Pre-Flight SAOCOM-1A SAR Performance Assessment by Outdoor Campaign
## 2 Outdoor Experiment Concept, Design, and Simulations
The return from successive echoes measured at the target range can be modeled as a mono-dimensional time-variant signal, given by the superposition of the distributed target (or clutter), the signal from the trihedral target, and the noise \(w\), due to thermal and quantization:  
\[y(\tau)=\int_{\rho_{a}}a_{c}e^{i\phi_{APS}(\xi,\tau)}d\xi+a_{s}e^{i\phi_{APS}( \xi_{0},\tau)+\phi_{r}(\tau)}+w(\tau) \tag{4}\]  
where the first integral is extended to the unfocused azimuth aperture, the resolution \(\rho_{a}\). In the equation above, \(\tau\) represents the time and \(\xi\), the spatial coordinate. The phase noise term \(\phi_{APS}\) accounts for the propagation within the troposphere and it is discussed later on in the text. The two terms \(a_{c}\) and \(a_{s}\) are the returns, respectively, from the distributed target and the trihedral. The second term is the signal related to the target. The phase term \(\phi_{r}\) accounts for the motion of the target and represents an equivalent of the \"phase history\" of the target, to be compensated for (\"focused\"). In the case of linear motion along the line of sight with velocity \(v\), the phase history corresponds to a linear phase ramp:  
\[\phi_{r}(\tau)=-\frac{4\pi}{\lambda}R(\tau)=-\frac{4\pi}{\lambda}(R_{0}+v\cdot\tau) \tag{5}\]  
Figure 1: SAOCOM outdoor test (ODT) concept overall view. In the figure, P represents the moving target to be acquired and \(\delta\) the inclination of the Radar Line of Sight with respect to the local normal to the terrain. And hence the observed data sequence at the target range \(R_{0}\) is a sampled sinusoid with frequency:  
\[f=-\frac{2}{\lambda}v \tag{6}\]  
The term \(f\) is the Doppler Frequency created by the target motion. Unlike a spaceborne SAR, in which every target in the scene has its own Doppler, here all the fixed targets appear at Doppler zero except for the one of interest. Assuming a very good knowledge of the phase \(\phi_{r}\), the \"focused\" target response is retrieved by compensation of the phase on the signal \(y(\tau)\), performing what in SAR focusing is called \"backprojection\":  
\[y_{F}=\frac{1}{N}{\sum_{i}}y(\tau_{i})\cdot e^{-i\phi_{r}} \tag{7}\]  
In the following, we study the content of \(y(\tau)\), making different hypotheses on the disturbing phase noise \(\phi_{APS}\) and on the clutter statistics. The impact of propagation (the phase noise term \(\phi_{APS}\)), accounts for the additional delay introduced by the non-free space propagation within the troposphere. If the monochromatic approximation holds (a small bandwidth compared to the carrier frequency), the phase term \(\phi_{APS}\) is linearly related to the delay, which in turn depends on the integral of the refraction index along the path. \[\phi_{APS}=\frac{4\pi}{\lambda}d_{ATM}=\frac{4\pi}{\lambda}{\int_{Lip}N(r( \overrightarrow{l}))dl} \tag{8}\]  
where \(N\) is the space- and time-variant refractivity index of the atmosphere, a function of the space \(r\) and the time of acquisition. The statistics of the atmosphere have been widely studied in the past [6]. Limiting the temporal scope to one day (far longer than one SAR acquisition), we can well consider the Kolmogorov turbulence statistical model of the delay, represented by the power law:  
\[E\big{[}d(t+\Delta t)-d(t)\big{]}=\bigg{(}\frac{\Delta t}{\overline{\tau}} \bigg{)}^{\alpha}ad \tag{9}\]  
where \(\overline{\tau}\) is a temporal constant depending on the local atmospheric conditions (typical values are in the order of hours). The exponent \(\alpha\) is also defined as slope, as the function is a line in a log-log space, assuming values experimentally determined to be close to \(2/3\). The applicability of the model was verified with experimental ground-based radar measurements in Reference [6]. Numerical simulations were carried out to investigate the impact on the obtained signal-to-clutter ratio due to the atmospheric delay spatial and temporal variation.