Synthetic Data

The fwimfs method is tested on synthetic data computed for the SEG/EAGE salt model with a marine geometry. The model is decimated by a factor of $3\times 3$ for less overall computational time, as shown in Figure 3.4(b). The source wavelet is a Ricker wavelet peaking at 8 Hz. There are 60 shot gathers evenly distributed across the top of the model with the shot spacing of 82.3 m, the receiver spacing is 27.4 m, and the line length is 2.3 km.

The FWI method uses a preconditioned conjugate gradient method, where the acoustic forward and backward solvers are a finite-difference solution to the 2D space-time wave equation of constant density. The fdtd algorithm is second-order accurate in time and fourth-order accurate in space, denoted as $O(2, 8)$ . The source wavelet for the proposed frequency selection method is a pure cosine wave, also employed in Nihei and Li (2007) and Sirgue et al. (2008), at a selected frequency.

Figure 3.4: (a) The initial velocity, (b) the true velocity, (c) the result of the proposed FWI at the 439th iteration, and (d) the result of standard FWI at the 69th iteration.

The starting model is shown in Figure 3.4(a) and the standard FWI tomogram after 69 iterations is shown in Figure 3.4(d). This result and the associated CPU time will serve as the standard metrics by which the fwimfs algorithm will be measured.

Figure 3.5: Normalized error of waveform (a) and velocity (b) as functions of iteration number, for the proposed method (denoted by `Sinusoidal') and the standard FWI (denoted by `Ricker'). Note that the iteration numbers for the red dashed curves are actually $6.362$ times those in display. See text for details.

The fwimfs strategy produces the tomogram shown in Figure 3.4(c). This result required 439 iterations to achieve the same accuracy as the Figure 3.4(d) result in 69 iterations. This amounts to a factor of $439/69 = 6.362$ . The convergence curves shown in Figure 3.5 are plotted in the way that the x-axes for the red dashed curves have been deliberately shrunk by $6.362$ . With this adjustment, the two convergence curves of velocity error in Figure 3.5(b) almost coincide with each other. Taking this factor into account, among others such as the overhead of increased runtime per finite-difference run, the speedup over the conventional FWI scheme is estimated to be about 4. This compares to the $8\times$ speedup reported by Huang and Schuster (2012) for RTM. One reason for this discrepancy is that the implementation in the time domain suffers from a $2\times$ overhead in order to reduce the effect of transient noise in the simulated sinusoidal seismograms.