129 5. Full waveform inversion results if dense frequencies cover the whole spectrum. The Parseval relationship indeed indicates that J m d m t d t d m d ( ) = ( )[ ]− [ ] = ( )[ ]− [ ] 1 2 1 2 2 2 calc obs calc obs ω ω , (5.7) where the dependency on w indicates the Fourier transform from time to angular frequency. We omit the dependency on the source and receiver positions. The result is different if one considers only a small number of discrete frequencies. There is currently some consensus that: • for many frequencies, the most efficient approach is time implementation. • for a limited number of frequencies in 3D, the time approach is also preferred, followed by a discrete Fourier transform. • for a limited number of frequencies in 2D, the frequency approach is more suited, but one has to solve a large (sparse) linear system, for example with a LU decomposition. Once this LU decomposition is performed, the imaging is very efficient. Beyond the time versus frequency approaches, the most important aspect is to avoid local minima. FWI = migration + tomography? This title is a reference to a publication by Mora in 1989. For an adequate application of FWI, it is essential to understand how FWI behaves. In particular, FWI has two main “modes”, i.e. different methods of updating the long and short wavelength components of the velocity model: • The tomographic mode means that the long wavelengths of the velocity model are updated, with an influence on the kinematics of wave propagation. In practice, this is the difficult part of FWI as this process is non-linear: if one multiplies the velocity model by 10%, then the data, for example reflected waves, are recorded at a different (shorter) time; • The migration mode indicates that if the data, up to a first-order approximation, linearly depend on the short wavelength components of the velocity model. After linearization, the data only contain reflected waves. If the velocity perturbations are increased by 10%, then the recorded pressure or displacement at the receiver is also multiplied by the same factor. There is no strict limit between the tomographic and migration modes in FWI, but if multiple frequencies are used for the inversion, the migration will dominate and it will be difficult to update the long wavelengths. The linearized version of FWI is called “iterative least-squares migration” (LSM). Standard migration corresponds to the first iteration only.
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