124 Seismic Imaging shot gathers in order to mimic the physics of wave propagation. Finally, due to the limited data frequency band and limited data acquisition from the surface only, FWI does not necessarily lead to a unique solution. For example, if the user is interested in determining P-velocity and density models, there is an intrinsic tradeoff between the two quantities, especially for short offset data. This is not specific to FWI: other imaging techniques suffer from the same effect, but this is visible in the FWI context as FWI is expected to provide quantitative results. A large number of FWI results have been published on real data in seismology, as well as at the exploration scales, at least in the marine case. The use of the technique on land with onshore data, however, has only been proven for a limited number of applications due to the presence of strongly energetic surface waves. While at the geotechnical scale, it has only been tried on a few occasions (among others, Gao et al., 2007; Gélis, 2005; Pérez Solano et al., 2014). The chapter is organised as follows. We first give a brief overview of the history of Full Waveform Inversion (section 5.2). We then introduce the formalism, limiting the number of equations to the most important ones. We discuss the potential impact as well as the limitations of FWI. This is an important section for anyone who would like to evaluate the potential of FWI on a particular dataset (section 5.4). Finally, we present a few illustrations at the geotechnical scale. For an overview of FWI at the seismological and exploration scales, we refer to Virieux and Operto (2010), and to Fischtner (2010) as well as to three recent didactic papers (Louboutin et al., 2017; Louboutin et al., 2018; Witte et al., 2018). There are two main elements to consider before applying FWI: the first important aspect is that Full Waveform Inversion considers the full wave field (e.g. pressure field or vertical displacement at the receiver position), and does not decompose the data in terms of travel times and amplitude (more details in 5.3). The other important aspect is that FWI can provide high-resolution quantitative results if there is a correct understanding of the main phenomena influencing the wave propagation as well as a proper strategy to iteratively converge towards a meaningful solution. For these reasons, FWI is not an automatic process and is applied after more standard processing such as travel time tomography (Bishop et al., 1985). 5.2 History This section mainly refers to the exploration scale (imaging of the first few kilometres of the subsurface). It is perhaps surprising that the formalism was not derived until the 1980s, through the work of Tarantola and his group in particular (Tarantola, 1984; Mora, 1987). Imaging is an inverse problem: one seeks a model such that the differences between the modelled (computed) data and the observed data are minimum in the least-squares sense. This is a very standard technique in many physical fields. On the geophysical side, the new aspect discovered in the 1980s was a method to update the model, i.e. how to compute the gradient of the misfit
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