# Scanning Inside Volcanoes with Synthetic Aperture Radar Echography Tomographic Doppler Imaging
## Appendix A Mathematical Formulation
### Doppler Sub-Aperture Strategy  
The decomposition of the SAR data into Doppler sub apertures is formalized in this subsection, which is performed starting from the spectral representation of the focused SAR data. To this end, notice that the generic \(i-\)th chirp sub-aperture 2-dimensional DFT of (11) is given by:  
\[\begin{split}& S_{SLC_{i}}(n,q)=DFT2\bigg{\{}2N\tau\exp\Big{[}- \frac{4\pi r}{\lambda}\Big{]}\text{sinc}\Big{[}\pi B_{c_{i_{x}}}\Big{(}k-L_{c_{ x}}\Big{)}\Big{]}\text{sinc}\big{[}\pi B_{c_{D}}\big{(}x-L_{D_{h}}\big{)} \big{]}\bigg{\}}\\ &=2N\tau\exp\Big{[}-\frac{4\pi r}{\lambda}\Big{]}\sum_{k=0}^{N-1} \sum_{\sum_{\bar{\
u}=0}^{M-1}\text{sinc}\Big{[}\pi B_{c_{\bar{\
u}_{\bar{\
u }}}}\Big{(}n-L_{c_{\bar{\
u}}}\Big{)}\Big{]}\text{sinc}\big{[}\pi B_{c_{D}} \big{(}q-L_{D_{h}}\big{)}\big{]}\\ &\exp\Big{(}-j\frac{2\pi n\pi}{N}\Big{)}\exp\Big{(}-j\frac{2\pi xq }{M}\Big{)}\\ &=2N\tau\exp\Big{[}-\frac{4\pi r}{\lambda}\Big{]}\frac{1}{\pi B_{ c_{i_{x}}}}\text{rect}\bigg{[}\frac{n}{\pi B_{c_{i_{x}}}}\bigg{]}\frac{1}{\pi B_{c_{D}}} \text{rect}\Big{[}\frac{q}{\pi B_{c_{D}}}\Big{]}\exp\Big{(}-j\pi nL_{c_{\bar{ \
u}}}\Big{)}\exp(-j\pi qL_{D_{h}}).\end{split} \tag{11}\]  
from the last equation, it turns out that a single point stationary target has a two-dimensional rectangular nature with total length proportional to the range-azimuth bandwidths respectively. The phase term \(\exp\Big{(}-j2\pi nL_{c_{\bar{\
u}}}\Big{)}\exp\big{(}-j2\pi qL_{D_{h}}\big{)}\) is due to the sinc function dislocation in range and azimuth when the SLC SAR data are considered. In the SAR, the movement of a point target with velocity in both range and azimuth direction is immediately warned by the focusing process, resulting in the following anomalies:  
* azimuth displacement in the presence of target constant range velocity;
* azimuth smearing in the presence of target azimuth velocity or target range accelerations;
* range-walking phenomenon, visible as range defocusing, in the presence of target range speed, backscattered energy can be detected over one or more range resolution cells.  
In practical cases, the backscattered energy from moving targets is distributed over several range-azimuth resolution cells. As a matter of fact, considering the point-like target \(T_{1}\) (of Figure 1) that is moving with velocity \(\vec{v}_{t}\) whose range-azimuth and acceleration components are \(\{v_{r},v_{a}\}\), and \(\{a_{r},a_{a}\}\), respectively, then it is possible to highlight  
\[\begin{split}& R^{2}(t)=(Vt-S_{a})^{2}+(R_{0}-S_{r})^{2}\text{ with }S_{r}=v_{r}t+\frac{1}{2}a_{r}t^{2}\text{ and }S_{a}=v_{a}t+\frac{1}{2}a_{a}t^{2}\\ &|R(t)|=|R_{0}-S_{r}|\bigg{\{}1+\frac{(Vt-S_{a})^{2}}{(R_{0}-S_{r })^{2}}\bigg{\}}^{\frac{1}{2}}.\end{split} \tag{12}\]considering the following Taylor expansion:  
\[(1+x)^{\beta}\approx 1+\beta x \tag{10}\]  
and that \(R_{0}-S_{r}\approx R_{0}\), and \((V_{t}-S_{a})^{2}\approx V^{2}t^{2}-2 VtS_{a}\), (10) can be written in the following form:  
\[|R(t)|=\left\{|R_{0}-S_{r}|+\frac{1}{2}\frac{\left(Vt-S_{a}\right)^{2}}{\left( R_{0}-S_{r}\right)}\right\}=|R_{0}-S_{r}|+\frac{V^{2}t^{2}}{2R_{0}}\bigg{(}1- \frac{2S_{a}}{Vt}\bigg{)} \tag{11}\]  
the term \(\frac{Va_{a}t^{3}}{2R_{0}}\) can be neglected and by approximating \(\big{(}V^{2}-2 Vv_{a}\big{)}\approx\big{(}V-v_{a}\big{)}^{2}\) Equation (12) can be written like:  
\[|R(Vt)|=R_{0}-v_{r}t+\frac{t^{2}}{2R_{0}}\Big{[}(V-v_{a})^{2}-R_{0}a_{r}\Big{]}. \tag{12}\]  
recasting (12) in terms of \(x=Vt\), We obtain [32]:  
\[|R(x)|=R_{0}-\epsilon_{r_{1}}x+\Big{[}(1-\epsilon_{c_{1}})^{2}-\epsilon_{r_{2 }}\Big{]}\frac{x^{2}}{2R_{0}},\ x=Vt. \tag{13}\]  
where:  
* \(\epsilon_{r_{1}}=\frac{v_{r}}{V}\) (due to range velocity);
* \(\epsilon_{r_{2}}=\frac{a_{r}R_{0}}{V^{2}}\) (due to range acceleration);
* \(\epsilon_{c_{1}}=\frac{V}{V}\) (due to azimuth velocity).  
Thus, the above terms modify the received signal, as shown in [32], and should be taken into account in Equation (10).