125 5. Full waveform inversion objective function. Similar expressions existed before, for example proposed by Claerbout (1968) but not necessarily leading to quantitative results. The definitions of the objective function and the associated gradient are detailed in section 5.3. In practice, the applications were limited to small size 2D datasets due to limited computer capabilities (more details on this aspect in section 5.4, “memory and CPU requirements”). New perspectives appeared around 2000, not only because computers were more powerful, but also because Sirgue and Pratt (2004), among others, proposed a practical strategy for the applicability of FWI. This strategy is further discussed in section 5.4 (“how to avoid local minima”). The main idea is to start from low frequency data. Following a few FWI iterations, the inverted model will contain the large-scale structure. Higher frequencies are then progressively introduced and the model is refined accordingly. The strategy of directly considering the full bandwidth may lead to an incorrect solution: it means that the objective function is multimodal (i.e. contains local minima) due to the non-linearity between the data and the model (more details in section 5.4). Spectacular results were obtained by Pratt and his group, especially on synthetic data for which the observed data contain low enough frequencies to enable the use of the increasing frequency strategy. The 2004 EAGE workshop demonstrated the fundamental role played by the low frequencies during the first iterations, and consequently triggered renewed interest in FWI (Billette and Brandsberg-Dahl, 2005). Since the early 2000s, FWI has been developed from 2D to 3D, from acoustic to elastic and visco-elastic, as well as in anisotropic contexts, from offshore to onshore datasets. It is clear that the development of computer facilities has supported this trend. FWI remains under development, particularly for multi-parameter estimation (i.e. not only pressure velocity models vp from body waves, but also shear velocity models vs from surface waves, as well as anisotropy or attenuation parameters). The difficulty is to extract more than one parameter (Operto et al., 2013). Since the 1980s, the challenge has been to incorporate higher frequencies from the data, initially in 2D and now in 3D, as well as more physics (more details in section 5.4). 5.3 Formalism For reasons of clarity, we limit the number of equations presented in this section, and aim to give a physical interpretation of the different quantities introduced. The objective of FWI is to minimize the least-squares misfit function J m d m d ( ) = ( ) − 1 2 2 calc obs , (5.1) where m x( ) is the model to be determined, which is a function of the spatial coordinates x =( ) x y z , , , d t obs s r, , ( ) the observed data at source position s, receiver
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