131 5. Full waveform inversion For each possible shift, we computed the standard least-squares misfit (Eq. (5.1)) between the observed (blue) and synthetic trace (red) (Figure 5.2, blue solid line). The objective function oscillates, with a minimum for zero-shift as expected (blue dot). The shift of half a period corresponds to a local maximum (red dot) as the two signals are out of phase (Figure 5.1, middle), whereas the shift of a period is associated to a local minimum (cyan dot) for which the blue and red traces are mainly in phase. The maximum possible shift to ensure a convergence towards the global minimum (blue dot) is indicated by the red dot. This phenomenon is known as the cycle-skipping effect. For the same signals, if the central frequency had been halved, then the shape of the objective function would be within the blue dashed line (Figure 5.2). It means that the basin of attraction is twice as large (Bunks et al., 1995). Figure 5.2 Shape of the objective function for different shift values. The blue, red and cyan dots are associated to the traces in Figure 5.1 (from top to bottom). When the central frequency is divided by two, the objective function has a basin of attraction (dashed line) that is twice as large. In conclusion, there are two main strategies to ensure a proper convergence: • Start with an initial velocity model that is not too far away from the correct solution. • Use low frequencies at first, then progressively increase the frequency content. Low frequencies have larger basins of attraction than higher frequencies, as illustrated in Figures 5.1 and 5.2. Different authors have proposed practical rules (Sirgue and Pratt, 2004). The current practice is to consider a frequency band f f 0 − [ ], where f0 is the minimum reliable frequency and where f increases progressively. The alternative is to let f0 also increase. Note that for a given frequency band, at least a few non-linear iterations are performed before modifying the frequency range. Then the same process is repeated. Beyond these two points, the redundancy of the data is an important aspect to constrain the inversion. The null space refers to equivalent models associated to the
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