127 5. Full waveform inversion minimization under constraints, with the introduction of Lagrangian multipliers λ s x, ,t ( ). In practice, the derivation of the gradient requires three elements: • computation of the forward wave field d (Eq. (5.2)) for each source; • computation of the backward residual wave field λ, also for each source. It is obtained by solving the adjoint wave equation (for the constant density acoustic wave equation, the same wave equation applies) for a source term being the residual wave field at the receiver position d d calc obs − . • cross-correlation between d and λ, with a summation over all times, but for fixed spatial x positions. The final gradient is obtained by adding the contribution of all sources. In practice, the derivation of d and λ are very similar: the gradient thus requires two modelling steps. The CPU cost associated to the cross-correlation is much less than that for solving the forward or backward problem. Once the gradient is computed, the new model is updated with the typical strategy m m J m n n + = − ∂ ∂ 1 α , (5.5) where the gradient is ∂ ∂ J m / and α>0 a scalar step length. More advanced methods such as quasi-Newton or Newton approaches take into account the curvature of the objective function (Hessian) for a faster convergence (Nocedal, 1980). As mentioned before, the user should provide an estimation of the source wavelet. The most popular strategy is to consider the direct arrival between a source and a receiver (Pratt, 1999). In practice, such determination depends on the unknown velocity model in the shallow part: the source wavelet as well as the model itself are together iteratively determined. By definition, 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. Travel time is a notion associated to high frequency approximation, also known as geometrical optics. FWI is thus considered as a “wave equation approach”, in the sense that it takes into account effects related to finite frequencies, such as the diffraction on a scatter. In practice, FWI can partly select data around certain windows, for example to only include a zone around the direct arrivals or to remove ground roll. This means that equation (5.1) is modified according to J m M d m d ( ) = ( ) − ( ) 1 2 2 calc obs , (5.6) where M is a mask in the data domain. With the adjoint state approach to derive the gradient, only a minor change is needed in the so-called adjoint source. In the next section, we illustrate the behaviour of FWI on a 2D synthetic dataset and discuss the impact of several factors, such as the initial velocity model and the role of the frequency content of the data.
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