# Cross Sensor Simulation of Tomographic SAR Stacks
## 3 Cross Sensor Simulation  
This section illustrates a procedure for the simulation of a tomographic SAR stack starting from another one with possibly different carrier frequency, bandwidth and geometry of acquisition. Figure 4: Wavenumbers enlightened by an airborne (green) and spaceborne SAR systems (purple) in the \(\{ky,kz\}\) domain. The horizontal axis represents the transformed variable of the ground range coordinate whereas the vertical axis of the elevation; \(y\) and \(z\) in Figure 1 respectively. The airborne system here represented spans a wider range of look angles, has a greater bandwidth and a lower central frequency than the spaceborne system. Figure 5: Impulse response functions (IRFs) in the zero-Doppler plane varying the distance from the radar. The IRFs of the airborne system changes significantly (mainly its rotation) from near to far range whereas the spaceborne system is almost constant. The horizontal and vertical axes refers to the ground range (\(y\)) and elevation (\(z\)) axes shown in Figure 1 respectively. The height is here referred to a constant elevation taken as a reference. Quantities associated with the first system will be labeled with a \(s\), the second with a \(A\); this follows the practical need of simulating a spaceborne system starting from an airborne one although the algorithm here illustrated can be applied to any systems pair. The algorithm here presented relies on the exploitation of Equation (12) [22; 23; 24] for each azimuth position. \[I(r)=\int\int s\left(y,z\right)g\left(r-R\left(y,z\right)\right)e^{\frac{4\pi}{ \lambda}R\left(y,z\right)}dydz \tag{12}\]  
where \(s\left(y,z\right)\) represents the complex reflectivity distribution in the \(\left\{y,z\right\}\) plane, \(g\left(\cdot\right)\) is the waveform envelope, \(r\) is the range coordinate and \(R\) is the distance between each point scatterer and sensor. Equation (12) models the generation of any range line of a SLC SAR image: the integral expresses the coherent sum of the scattering coefficients associated with targets placed at the same distance from the sensor, the complex exponential describes the demodulation of the target spectrum in accordance with the radar central frequency and the convolution represents the filtering due to the limited signal bandwidth. The knowledge of system parameters and acquisition geometry leads to two unknowns only in Equation (12), they are the scatterers' distribution \(s\left(y,z\right)\) and the SAR image \(I(r)\). The availability of a stack of SAR images allows SAR tomography to be implemented, through which an estimation of the reflectivity map can be obtained. On the other hand, by feeding Equation (12) with any scatterers' distribution, a synthetic SAR image can be straightforwardly produced. The generation of a synthetic stack starting from real tomographic data relies on these two steps. Firstly, the reflectivity profile is estimated by performing SAR tomography on real data and then it is used, along with the parameters of the system to be simulated, to produce the synthetic images. However, due to the difference between the two acquisition systems, an extra processing step is needed to make the reconstruction provided by the first compatible with the second. The complex exponential in Equation (12) is responsible for a demodulation of the spectrum of the target before being filtered by \(h(r)\); as a consequence, only the passband components of \(s\left(y,z\right)\) contribute to \(I(r)\). However the spatial direction along which this demodulation takes place is ruled by the range axis, whose direction changes from near to far range. Or rather, the wavevectors of \(s\left(y,z\right)\) that contribute to \(I(r)\) have constant magnitude for each value of \(r\) but they change direction in accordance with the look angle. It follows that \(s\left(y,z\right)\) should lie on the spectral support defined by the acquisition system to be simulated for generating correct range lines.