# Scanning Inside Volcanoes with Synthetic Aperture Radar Echography Tomographic Doppler Imaging
## 1 Introduction  
Vesuvius is an active volcano and dominates the central part of the Campanian coast, located in Italy. The morphology of the volcanic complex (including the Campi Flegrei) clearly reveals its active nature, characterized by eruptive cones and magmatic fluids flows [1]. From a morphological point of view, Vesuvius is a volcanic complex composed of Mount Somma, whose activity ended with the formation of a summit caldera. All this has led to Vesuvius being characterized as one of the most dangerous volcanoes in the world [2]. The reinterpretation of the volcanological and historical evidence shows that the eruption of Vesuvius in 79 AD consisted of two main phases. The initial Plinian phase, lasting 18-20 h, caused a large pumice fallout, resulting in the slow accumulation of a layer of pumice up to 2.8 m thick over Pompeii and other regions to the south [3]. Research [4] gave the general model for the behaviour of the Somma-Vesuvius, distinguishing three classes in the eruptive pattern. The authors of [5] gave the seismic evidence of an extended magmatic sill under Vesuvius. The authors found evidence of an extended (at least 400 km\({}^{2}\)) low-velocity layer at about 8 km depth. The inferred seismic velocities (about 2.0 km/s), as well as other evidence, indicate an extended sill with magma interspersed in a solid matrix. The velocity structure of the Somma-Vesuvius volcano, obtained by joint inversion of P-and S-wave arrival times from both local earthquakes and shot data collected during the TOMOVES 1994 and 1996 experiments, is presented in [6]. The entire seismic catalog available from 1987 to 2000 (1003 events) was relocated in the 3D velocity model obtained. The thermal model and the 2D numerical scheme of the Vesuvius magma chamber, locating it at a depth of approximately 6 km, has been studied in [7]. Given the high risk of the Vesuvius crater, an efficient numerical modeling and the methodology adopted to evaluate the resistance of buildings under the combined action of volcanic phenomena is developed in [8]. Numerous works have been carried out in the field of sonic-inspired imaging, especially in the medical field, through the use of ultrasound. Theoretical studies conducted in the early 1980s suggested that ultrasonic imaging using the correlation technique can overcome some of the drawbacks of classical pulse echography. An efficient high-resolution technique was developed in [9] by transmitting coded signals using high-frequency transducers, up to 35-50 MHz, for ophthalmic echography to image fine eye structures. The first high-frequency echographic images obtained with the prototype probe are presented in [10]. An attractive description of the first mechanical scanning probe for ophthalmic echography based on a small piezoelectric ultrasound motor is reported in [11]. A novel method for line restoration in speckle images addressing a sparse estimation problem using both convex and non-convex optimization techniques is reported in [12]. Lung ultrasound imaging is a fast evolving field of application for ultrasound technologies. The authors of [13] design an image formation process to work on lung tissue, and ultrasound images are generated with four orthogonal bands centered at 3, 4, 5, and 6 MHz which can be acquired and displayed in real time. In the field of seismic tomography, several works have been carried out. In [14], scientific methods are developed for the design of linear tomographic reconstruction algorithms with specified properties. An experimentation reconstruction method for seismic transmission travel-time tomography was developed in [15]. The method is implemented via the combinations of singular value decomposition, appropriate weighting matrices, and variable regularization parameters. A cross-correlation of environmental seismic noise from 1 month of recordings at USArray stations in California that produced hundreds of seismic wave group velocity measurements was implemented in [16].