# Scanning Inside Volcanoes with Synthetic Aperture Radar Echography Tomographic Doppler Imaging
## 2 Methodology
### Vibrational Model of the Earth  
The proposed vibrational model of the Earth's surface is schematically shown in Figure 3a,b. The geometrical reference system for both sub-pictures is the range, azimuth and altitude three-dimensional space. For the present case, the vertical dimension represents the depth below the topographic level (for this specific case the medium boundary is represented by the green plane). The tomographic line of interest is constituted of the series of contiguous pixels laying on the green plane. As can be seen from Figure 3a, on each pixel belonging to the tomographic line, a mass is hanging using a spring. This system is now induced to oscillate harmonically, helped by the Earth magma instability. These oscillations are schematized as the vibration energy function visible in Figure 3b. In this context, the radar instantaneously perceives this coherent harmonic oscillation. From a mathematical point of view, the Earth's displacement is perceived as a complex shift belonging on each pixel of interest. Each instantaneous displacement is estimated between the master image with respect to the slave, where oversimplification shifts are estimated through the pixel  
Figure 3: Tomographic acquisition geometry. (**a**) Orbital geometry with the schematic representation of moving masses on the considered pixels. (**b**) Orbital geometry with the measured vibrations present on the considered pixels. (**c**) Details of the spring-mass scheme. (**d**) orthogonal baseline scheme for tomography. Figure 2: Doppler sub-aperture strategy. (**1**) Spectrum explored from the beginning. (**2**) Spectrum explored from the middle. (**3**) Finally explored spectrum. tracking technique [28; 29]. The number of tomographic independent looks (depending on the total number of Doppler sub-apertures) is defined by the parameter \(k\). We suppose now the spring being perturbed by an impulsed force. According to this perturbation, the rope begins to vibrate, describing a harmonic motion (in this context we are not considering any form of friction). The resulting perturbation moves the rope through the space-time in the form of a sinusoidal function. The seismic wave will then reach a constraint end that will cause it to reflect in the opposite direction. The reflected wave will then reach the opposite constraint that will make it reflect in the original direction and return to the initial location, maintaining the same frequency and amplitude. According to classical physics principles, the rebounding wave is superimposed on the arriving wave, and the interference of two sine waves with the same amplitude and frequency propagating in opposite directions leads to the generation of an ideal and perpetual standing wave on the spring. Each vibrational channel is now considered when the spring is able to oscillate into the three-dimensional space, according to specific perturbation nature. When the Earth vibrates, it happens that the length of the spring must also fluctuate. This phenomenon causes oscillations in the tension domain of the spring. It is clear that these oscillations (i.e. the longitudinal ones) propagate through a frequency approximately twice as high as the frequency value of the transverse vibrations. The coupling between the transverse and longitudinal oscillations of the spring can essentially be modeled through non-linear phenomena. Figure 3c,d illustrates the oscillating model in the Euclidean space-time coordinates (x,y,z,t), where the satellite motion has been purified from any orbital distortions, so that the geometric parameters used to perform the tomographic focusing can be rigorously understood. From Figure 3c, \(L\) is the length of the spring when it is at its maximum tension while \(L_{0}\) is its length when no mass is present. Finally, the spring has been considered to have an elastic constant equal to \(\xi\).