128 Seismic Imaging 5.4 Applicability and practical aspects Below we provide some practical solutions for a number of key issues that users must address when applying FWI to specific datasets (Virieux and Operto, 2009; Operto et al., 2013; Basker et al., 2016; Raknes et al., 2017). Model parameterization Defining the model parameterization, i.e. the quantity FWI should determine, is the first issue to be considered. This may seem curious but is in fact essential in the case of multi-parameter estimation. We considered a practical example here: suppose that the model unknowns are the pressure velocity vp model and the density model ρ. The P-impedance is the product of the density and velocity (I v p p =ρ ). We compared two strategies: • In the first case, vp and density ρ were determined by FWI. Then the impedance was deduced from the product of the final vp and ρ models; • In the second case, vp and I p were determined by FWI. The density model was determined a posteriori by dividing I p by vp. The two approaches do not lead to the same result, as illustrated by Operto et al., 2013. This should not be surprising because for zero-offset data there is a trade-off between a velocity and density contrast associated to diffraction. With larger offsets, the diffractions have different responses. This means that the algorithm converges to different solutions, all in the null space, depending on the initial model. Choice of wave equation Selection of the appropriate wave equation is of course critical. It must be sensitive to the model to be determined. For example, if the attenuation factor is of interest, then the wave equation should contain visco-acoustic or visco-elastic terms. If the analysis of surface waves is important, then acoustic modelling is not sufficient and elasticity should be considered. The determination of a vp model from marine acoustic data is a subtler example. An acoustic framework would usually be sufficient in this situation, unless the data contain converted waves, even if only the pressure is recorded at the receiver position. For pure acoustic data, is there a need to consider density? The amplitudes of the transmitted (diving) arrivals are not sensitive to density, whereas this is the not the case for reflected waves. Such questions are still not fully solved. Time versus frequency Two main approaches are possible to solve the direct problem (Eq. (5.4)) and the inverse problem: either in the time or the frequency domain (Virieux and Operto, 2009; Raknes et al., 2017). Many scientific papers were published on this topic in the 2010s. From a theoretical point of view, the two approaches lead to the same
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