In the past few years we have demonstrated with blind-test synthetic examples and a 3D field-data example that a time-lag extension of the velocity model can lead to robust convergence of FWI (we called our method Tomographic FWI or TFWI) even when the long wavelengths of the starting model are far from the long-wavelengths of the true one. However, model-extension methods are still challenged in case of complex and strongly scattering media, such as salt bodies. Building on physical intuition, I will explain the current challenges and discuss possible remedies. A possible solution is to devise model-extensions that are more appropriate to the non-linearity of the problem at hand. Another approach is to introduce geologic and petro-physical constraints. For example, the geometrically complex and sharp nature of salt-sediment interface can be built into the problem by applying level-set methods. In other cases, like time-lapse FWI targeted to geo-mechanical effects, we can introduce constraints that force model differences, or their gradients, to be sparse by the addition of appropriate regularization terms.
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