# Cross Sensor Simulation of Tomographic SAR Stacks
## 2 Wavenumbers in SAR Acquisitions
### Wavenumbers in SAR Tomography  
For the sake of simplicity, the discussion in this section hypothesizes the absence of a squint angle in the antenna pointing and in the focusing algorithm. Holding this condition, after azimuth compression this direction can be considered as decoupled from the other two. For this reason, the following analyses are carried out considering only the zero Doppler plane shown in Figure 1 rather than the whole three-dimensional space. This plane hosts many targets whose common feature is the position of the center of the synthetic antenna, this point also lies on the zero Doppler plane and it is often synthetically referred to _sensor position_. Whenever the area is imaged several times, each target can be associated with many sensor positions: a proper acquisition plan and processing steps ensure that every sensor position lies on the zero Doppler plane. One of them shall be chosen as reference sensor in order to define the nominal geometry (range axis, look angle and so on), this favored sensor is referred to as _master_ whereas the others as _slaves_. The definition of a nominal geometry enables to parametrize the zero Doppler plane by means of a pair of orthonormal unit vectors directed as the line joining master and target (slant range) and cross range as shown in Figure 1. As opposed to a \(\{y,z\}\) coordinate system, \(\{r,\zeta\}\) rotates with the range coordinate however it always highlights the directions along which a single SAR image has resolution capabilities (slant range) and where it lacks (cross range). Figure 2: Geometrical loci associated with the wavenumbers illuminated by a typical airborne SAR system. A small target, observed from a fixed azimuth position, is associated with the dashed segment here shown: the distance from the origin being proportional to the radar central frequency, its length to the system bandwidth and its orientation following the segment connecting sensor and target (the local line-of-sight). The set of segments associated with every position of the sensor within the synthetic aperture is shown as a gray surface. The set of patches associated with all the off-nadir angle \(\vartheta\) explored by the physical antenna gives rise to the volume whose edges are shown as thick lines. The vector joining any two sensors is known as baseline, it can be conveniently expressed as a sum of its components along the slant range and cross range; the former is referred to as _parallel baseline_ whereas the latter as _normal baseline_. The larger the baseline the further the two sensors, however only the normal component is responsible for a change in the look angle associated with the two positions. The existence of a difference in the look angles implies different directions of arrival of the wave when it impinges on the target. This, in turn, leads to different wavenumbers of the same target illuminated by the two sensors. The geometrical locus in the \(\{ky,kz\}\) plane associated with the tomographic acquisition is a set of segments [21], each of them with a different tilt; they are shown in Figure 3 in case of regularly spaced normal baselines and for a nominal look angle of 45\({}^{\circ}\). The spread of the spectrum provided by the various acquisitions results in the possibility of resolving targets along the cross range direction as well. The relationship between resolution and spectral width is expressed by Equation (5). \[\rho_{\zeta}=\frac{2\pi}{\Delta k_{\perp}} \tag{5}\]  
\(\Delta k_{\perp}\) can be expressed as a function of the central frequency and look angles as shown in Equation (6). \[\Delta k_{\perp}=\frac{4\pi}{c}f_{0}\Delta\theta \tag{6}\]  
where \(\Delta\theta\) is the largest difference of look angles.