This paper presents an overview and a detailed description of the key logic steps and mathematical-physics framework behind the development of practical algorithms for seismic exploration derived from the inverse scattering series. There are both significant symmetries and critical subtle differences between the forward scattering series construction and the inverse scattering series processing of seismic events. These similarities and differences help explain the efficiency and effectiveness of different inversion objectives. The inverse series performs all of the tasks associated with inversion using the entire wavefield recorded on the measurement surface as input. However, certain terms in the series act as though only one specific task,and no other task, existed. When isolated, these terms constitute a task-specific subseries. We present both the rationale for seeking and methods of identifying uncoupled task-specific subseries that accomplish: (1) free-surface multiple removal; (2) internal multiple attenuation; (3) imaging primaries at depth; and (4) inverting for earth material properties. A combination of forward series analogues and physical intuition is employed to locate those subseries. We show that the sum of the four taskspecific subseries does not correspond to the original inverse series since terms with coupled tasks are never considered or computed. Isolated tasks are accomplished sequentially and, after each is achieved, the problem is restarted as though that isolated task had never existed. This strategy avoids choosing portions of the series, at any stage, that correspond to a combination of tasks,i.e.,

Seismic full-waveform inversion (FWI), as proposed by Lailly and Tarantola in the 1980s (Lailly, 1983, Tarantola, 1987), consists of minimizing the misfit between observed and computed data. This data fitting approach is attractive because, theoretically, it allows us to directly invert for the Earth parameters, without using concepts like angle reflectivity and scale separation between background velocity and reflectivity. However, the applicability of waveform inversion is limited by at least two factors: the numerical cost of the solutions of the elastodynamic equations needed to obtain the computed data and the ill-posedness of the inverse problem because of the presence of local minima caused by cycle skipping between observed and computed data. The large increase in computing power became an enabler for this technique, even if it is still expensive to numerically solve the visco-elastic wave equations and most of the current waveform inversion implementations make severe approximations, notably the acoustic assumption in exploration geophysics. Because of the computing cost of solving the wave equations, the minimization of the misfit function has to be carried out with a local (gradient) optimization technique. The current implementations to invert 3D seismic data are then sensitive to the initial values of the Earth parameters. This sensitivity depends on the type of data (Gauthier et al., 1986, and Virieux and Operto, 2009, for a review), which means that improvements in acquisition greatly impact the quality of the results obtained with FWI.

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