Marmousi acoustic velocity model. (a) True model. (b) Starting model. Insert shows mean power spectrum of the true synthetic data. The model includes a free surface; 91 sources and 187 receivers were used throughout. The velocity scale is identical for all subsequent Marmousi figures.
(a) Shot record generated using the true velocity model from Figure 1a. (b) Shot record generated using the starting model from Figure 1b; the first-arrival data are cycle skipped with respect to Figure 2a.
A portion of the shot record from Figure 2 showing the observed and predicted data interleaved. The observed data, from the true model, are indicated by yellow; the predicted data, from the start model, are indicated by green. The starting model generates first-arrival data that are late by about a cycle in the offset range 2000–3000 m, and that are early by about a cycle at longer offsets in the range 4000–5000 m. The starting model, therefore, generates data that are badly cycle skipped with respect to the true data.
Velocity model recovered by conventional FWI obtained when the starting data are badly cycle skipped. The deep structure is poorly focused; the intense low-velocity region within the shallow high-velocity fault block at center right is a symptom of cycle skipping — the FWI updates here have the wrong sign because they act to move the predicted data toward the wrong cycle.
Velocity model recovered by AWI when the starting-model data are badly cycle skipped. The AWI result is clearly superior to the conventional FWI result from Figure 4. The result shows no evidence of cycle skipping, and its limitations are caused principally by the limited bandwidth and limited coverage of the observed data.
Model evolution during AWI. The starting model is top left, the final model is bottom left, and the true model is bottom right. Intermediate models are spaced at approximately equal steps in the logarithm of the total number of iterations. The central shallow portion of the model is recovered early, and improvements in the model tend to spread outward and downward from that region. Note that the high-velocity layer at bottom left is formed initially at the wrong depth, but it evolves toward the correct depth as the shallow velocities improve.
Source wavelets used to test FWI and AWI. (a) Wavelet used to generate synthetic data, and to produce the inversions shown in Figure 12. (b) Incorrect wavelet used to produce the inversions shown in Figure 13.