# Scanning Inside Volcanoes with Synthetic Aperture Radar Echography Tomographic Doppler Imaging
## 2 Methodology
### Vibrational Model of the Earth
The vibrational force applied to the mass \(m1\) of Figure 3c is equal to [33]:  
\[F=-4\xi\mathbf{r}\bigg{(}1-\frac{L_{0}}{\sqrt{L^{2}+4\mathbf{r}^{2}}}\bigg{)}. \tag{1}\]  
if \(\mathbf{r}\ll L\), (1) is expanded in the following series:  
\[F=-4\xi\mathbf{r}(L-L_{0})\Big{(}\frac{\mathbf{r}}{L}\Big{)}-8\xi L_{0}\bigg{[} \Big{(}\frac{\mathbf{r}}{L}\Big{)}^{3}-\Big{(}\frac{\mathbf{r}}{L}\Big{)}^{5 }+\dots\bigg{]}, \tag{2}\]  
where a precise approximation of (2) is the following cubic restoring force:  
\[F=m\ddot{\mathbf{r}}\approx-4\xi\mathbf{r}(L-L_{0})\Big{(}\frac{\mathbf{r}}{L }\Big{)}\bigg{[}1+\frac{2L_{0}}{(L-L_{0})}\Big{(}\frac{\mathbf{r}}{L}\Big{)} ^{2}\bigg{]}. \tag{3}\]  
Considering (3), the non-linearity dominates when \(L\approx L_{0}\).If we define:  
\[\omega_{0}=\frac{4\xi}{m}\bigg{[}\frac{(L-L_{0})}{L}\bigg{]}, \tag{4}\]  
and  
\[\xi=\frac{2L_{0}}{L^{2}}(L-L_{0}). \tag{5}\]  
Considering (3), we have:  
\[\ddot{\mathbf{r}}+\omega^{2}\mathbf{r}\Big{(}1+\xi\mathbf{r}^{2}\Big{)}=0. \tag{6}\]if we consider damping and forcing (6) is modified as:  
\[\bar{\mathbf{r}}+\lambda\dot{\mathbf{r}}+\omega^{2}\Big{(}1+\xi\mathbf{r}^{2} \Big{)}\mathbf{r}=\mathbf{f}(\omega t), \tag{7}\]  
where \(\mathbf{f}(\omega t)\) is the forcing term and \(\lambda\) is the damping coefficient. If non-linearity of (7) is sufficiently low, it can be reduced to the following two-degrees-of-freedom linear harmonic oscillator:  
\[\mathbf{r}(t)=(a\cos\omega_{0}t,b\sin\omega_{0}t)\exp\bigg{(}\frac{-\lambda t }{2}\bigg{)}. \tag{8}\]  
In (8), \(\{a,b\}\) are the instantaneous shifts estimated by the coregistrator. The harmonic oscillator (8) is the displacement parameters \(\epsilon_{r_{1}},\epsilon_{r_{2}},\epsilon_{c_{1}}\) estimated by (A12). According to Figure 3d, the vector representation of \(k\) samples of the time-domain function (8) consisting in the following multi-frequency data input is considered:  
Figure 2:0  
The steering matrix \(\mathbf{A}(z)=[\mathbf{a}(z_{min}),\dots,\mathbf{a}(z_{MAX})],\in\mathbf{C}^{ k\times F}\) contains the phase information of the Doppler frequency variation of the sub-aperture strategy, associated with a source located at the elevation position \(\mathbf{z}\in\{z_{min},z_{MAX}\}\),  
\[\mathbf{A}(\mathbf{K}_{z},\mathbf{z})= \begin{bmatrix}1,\exp(j2\pi k_{z_{2}}t_ {20}),\dots,\exp(j2\pi k_{z_{k-1}}t_{20})\\ 1,\exp(j2\pi k_{z_{2}}t_{1}),\dots,\exp(j2\pi k_{z_{k-1}}t_{21})\\ \dots\\ 1,\exp(j2\pi k_{z_{2}}t_{F-1}),\dots,\exp(j2\pi k_{z_{k-1}}t_{2F-1})\end{bmatrix} , \tag{10}\]  
where \(\mathbf{K}_{z}=\frac{4\pi B_{z}}{4\pi B_{z}\sin\theta}\), \(i=1,\dots,k\), \(B_{\perp}\) is the \(i\)-th orthogonal baseline, which is visible in Figure 3d, and \(\mathbf{r}_{i}\) is the \(i-\)th slant-range distance. The standard sonic tomographic model is given by the following relation:  
\[\mathbf{Y} =\mathbf{A}(\mathbf{K}_{z},\mathbf{z})\mathbf{h}(\mathbf{z}). \tag{11}\]  
where in (11) \(\mathbf{h}(\mathbf{z})\in\mathbf{C}^{1\times F}\), inverting (11) I finally find the following tomographic solution:  
\[\mathbf{h}(\mathbf{z}) =\mathbf{A}(\mathbf{K}_{z},\mathbf{z})^{\dagger}\mathbf{Y}. \tag{12}\]  
In (12), the steering matrix \(\mathbf{A}(\mathbf{K}_{z},\mathbf{z})\) represents the best approximation of a matrix operator performing the digital Fourier transform (DFT) of \(\mathbf{Y}\). The tomographic image \(\mathbf{h}(\mathbf{z})\), which represents the spectrum of \(\mathbf{A}(\mathbf{K}_{z},\mathbf{z})\), is obtained by carrying out pulse compression. The tomographic resolution is equal to \(\delta_{T}=\frac{\lambda R}{2A}\), where \(\lambda\) is the sound wavelength over the earth, \(R\) is the slant range, and \(A\) is the orbit aperture considered in the tomographic synthesis; in other words, it consists in the Doppler bandwidth used to synthesize the sub-apertures. The maximum tomographic resolution obtainable using this SLC data, synthesized at 24 kHz, is as follows.