130 Seismic Imaging Avoiding local minima A standard FWI application, without careful pre-processing or meticulous parameter selection, would not typically lead to the expected solution. The algorithm may easily converge into a local minimum, which is possibly far from the global minimum (Bunks et al., 1995). To analyse this, it is useful to refer to the tomographic and migration modes explained in the previous section. Seismic data are oscillatory signals by nature, as the zero-frequency part is not recorded. The FWI misfit function (Eq. (5.1)) evaluates the least-squares distance between calculated and observed data, both being oscillatory signals. If the initial velocity model provides the correct kinematics, then FWI mainly operates in the migration mode and the events are correctly positioned. If the initial kinematics are not correct, then the algorithm will match incorrect phases between the synthetic and observed data. This is known as the cycle-skipping effect: it means that the algorithm converges in a local minimum (Bunks et al., 1995). We illustrated this effect using a very simple example, representative of what happens in practice. We considered a single trace. The blue trace corresponds to the observed data (Figure 5.1). An incorrect velocity model has an effect on the kinematic (position) and on the dynamic (amplitude) of the events. We studied here the kinematic effect because of the non-linearity. The red trace is the one in the inverted model (Figure 5.1, top). It does not perfectly match the observed trace because of the presence of noise. We then shifted the red trace to mimic the kinematic effect, for half a period and for a period (Figures 5.1, middle and bottom). Figure 5.1 Observed trace (blue) and synthetic trace (red), for different shift values applied to the red trace (from top to bottom: 0, -21 and -42 ms, respectively).
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