Tutorial: Hierarchical FWI: multiscale approach Jean Virieux Université de Grenoble I (Joseph Fourier) Before the emergence of wide-azimuth, long-offset data, the short offset range recorded by seismic reflection surveys and the limited frequency bandwidth of the active sources make seismic imaging poorly sensitive to intermediate wavelengths (Jannane et al., 1989), leading to a quite specific two-step workflow: first, the construction of the macromodel using kinematic information (essentially travel times of reflections), and then the amplitude projection (or simple inversion) through different types of migrations (Claerbout and Doherty, 1972; Gazdag, 1978; Stolt, 1978; Baysal et al., 1983; Nemeth et al., 1999; Yilmaz, 2001; Biondi and Symes, 2004). This procedure has turned out to be efficient for relatively simple geological targets in shallow-water environments, although more limited performances have been achieved for imaging structurally complex structures, such as salt domes, sub-basalt targets, thrust belts and foothills. In such complex geological environments, building an accurate and quantitative depth-dependent velocity model is challenging. With new available data, one may exploit the entire information contained in these time series for improving the seismic imaging procedure and this strategy has been labelled as the Full Waveform Inversion (FWI) (Lailly, 1983; Tarantola, 1984; Mora, 1987, 1988). This approach is a challenging data-fitting procedure for extracting quantitative information from seismograms at an ultimate resolution of half the propagated wavelength. At these earlier times, the acquisition design was not adequat and successful realistic results have only been obtained later under the impulsion of G. Pratt and co-workers (Pratt, 1990; Pratt and Worthington, 1990; Pratt et al., 1998; Pratt, 1999; Pratt and Shipp, 1999). Due to the nonlinearity of seismic waves with respect to model parameters, the method is sensitive to the initial smooth velocity macro-model. If the modeled data computed in this velocity model does not match the recorded data within a half cycle, the inversion may be easily trapped into a local minimum. Robust data-driven inversion strategies can be used to mitigate this issue by frequency, time (scattering-angle) and/or offset continuations (Bunks et al., 1995; Shipp and Singh, 2002; Sirgue and Pratt, 2004; Brossier et al., 2009). One can also consider more convex misfit functions such as those based on correlation (Luo and Schuster, 1991; van Leeuwen and Mulder, 2010; Luo et al., 2016), deconvolution (Luo and Sava, 2011; Warner and Guasch, 2014), dynamic warping (Hale, 2013; Ma and Hale, 2013) or optimal transport measurement (Engquist and Froese, 2014; Métivier et al., 2016). In spite of these intensive investigations, the long-wavelength content has been mainly brought by the transmission regime of early arrival data (diving waves, refracted waves, postcritical reflections). Waveform inversion of these early arrivals – EWI - (Shipp and Singh, 2002; Sheng et al., 2006; Sirgue, 2006; Shen, 2014) preferentially samples the small vertical wavenumbers (long-wavelength content) of the subsurface along sub-horizontal wavepaths (transmission path) (Sirgue and Pratt, 2004). On the other hand, updating the long-wavelength content of the velocity model by FWI is challenging in the deep part of the subsurface where the aperture illumination provided by reflection data becomes insufficient. FWI mostly behaves as a least-squares migration (Nemeth et al., 1999). Alternative imaging methods should be used to update the long-wavelength content of deep structures before considering the reconstruction of short-to-intermediate wavelength content by FWI. Conventionally, ray-based tomographic methods are widely implemented due to their computational efficiency for updating low-wavelength content at depths (e.g. Farra and Madariaga, 1988; Taillandier et al., 2009; Lambaré, 2008; Prieux et al., 2013). With limited-offset data, the tomographic methods mostly rely on reflection traveltimes (in opposition to first-arrival traveltimes) because the limited offset coverage prevents the diving waves from penetrating at sufficient depths. One pitfall of traveltime tomography is related to picking, which can be challenging for reflection phases performed either in the data domain or in the image domain for improving SNR. Alternatively, more automatic waveform-based approaches implemented in the image domain, such as differential semblance optimization (Symes and Carazzone, 1991), have met success and avoid picking issues. In these approaches, the reflection data are migrated to generate common image gathers and the velocity analysis process seeks to maximize the flatness of the reflectors along the offset or angle axis. Other approaches rely on an extended search space with space or time shifts where the velocity updating seeks to minimize the residual energy away from the zero shift (Sava and Fomel, 2006; Yang and Sava, 2011; Biondi and Almomin, 2012; Sun and Symes, 2012; Lameloise et al., 2015). Inspired by the pioneering work of Chavent et al. (1994); Clément et al. (2001), data-domain waveform inversion strategies, referred to as reflection waveform inversion (RWI), have been recently revisited to build the velocity macromodel (Xu et al., 2012; Zhou et al., 2012; Wang et al., 2013; Brossier et al., 2015; Staal, 2015; Guo and Alkhalifah, 2016). This method uses a prior short-scale reflectivity model and builds two-way reflection wavepath to update the smooth content of the velocity model. Therefore, RWI is suitable to sample small horizontal wavenumbers (long-wavelength content) along sub-vertical wavepaths (propagation path down before the reflection and up after the reflection). Combining EWI and RWI to sample a wider range of subsurface wavenumber spectrum seems to be quite attractive. Wang et al. (2015) presented a real data case study, in which both refracted waves and reflected waves are used to update the velocity macro- model. Zhou et al. (2015) have proposed a unified formulation, called JFWI, that naturally introduces diving waves into the RWI approach by reshaping the misfit function. In this lecture, the presentation of this JFWI formulation will illustrate the rather simple assumptions on which the FWI is based, assumptions related to the framework of diffraction tomography (Devaney, 1984). An introduction to full waveform inversion is kindly provided by SEG through a website of an open ebook project of Encyclopedia of Exploration Geophysics managed by V. Grechka and K. Wapenaar