126 Seismic Imaging position r and for time t. The calculated data d t calc s r, , ( ) are a function of the m model and are the solution of the wave equation L m d t t ( ) ( ) = − ( ) ( ) s x s x , , , δ Ω (5.2) d t d t calc s r s x r , , , , , ( ) = = ( ) (5.3) where L is the wave equation operator, d the wave field, Ω t( ) the seismic source wavelet, and δ the dirac distribution. It means that the wave field d is the solution of the wave equation for a point source located at x s = and for a seismic source wavelet. The calculated data are obtained by sampling the wave field d at the receiver position (Eq. (5.3)). The simplest case corresponds to the constant density acoustic wave equation, with m being the pressure velocity model vp, yielding L m v t p ( ) = ( ) ∂ ∂ − 1 2 2 2 x ∆. (5.4) The Laplacian operator D is the sum of the second-order derivatives in space. There are thus three elements to evaluate the quality of a given model m x( ): • Determination of the source wavelet Ω t( ) (Pratt, 1999); • Choice of the wave equation operator L; • Resolution of the forward modelling (equations (5.3) and (5.4)) (Louboutin et al., 2017; Louboutin et al., 2018). In section 5.4, we provide more information on this aspect. The evaluation of J m( ) is a first step, but one usually needs to determine a more suitable model. There are a few recent examples of global search methods (e.g. Sajava et al., 2016). For this strategy, the model space is explored with a Monte Carlo approach or with genetic algorithms. Such an approach would be feasible if the number of model unknowns was small, but this is not typically the case: the model space normally contains millions of unknowns as it is finely discretised along the x, y and z-axis. As discussed in section 5.4, the CPU cost for solving a single equation (5.2) is the main limiting factor for a global approach (Raknes et al., 2017). The only practical approach is to use a gradient-based inversion, where the model is iteratively determined. The requirements for this are: • an initial model, usually determined by a standard tomographic approach (Bishop et al., 1985); • computation of the gradient of the objective function (Plessix, 2006). Formally, the gradient of the objective function is written as the derivative of the objective function with respect to the model parameters. It has the same (large) size as the model space. This is not a trivial task: J depends on m through the dependency on dcalc, and the relation between dcalc and m is given by the wave equation and is clearly non-linear (Eqs. (5.2–5.4)). As mentioned in the introduction, Lailly (1983) and Tarantola (1984) established an efficient way to compute the gradient, with the “adjoint state method” (Plessix, 2006). This approach is related to the
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