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Numerical test

We test our inversion scheme on the shallow part of the Hess synthetic anisotropic model, as denoted by the gray square in Figure 1. The initial model is a 1D gradient isotropic model from the seabed. Figure 2 compares the inversion results with the true models. The ratio of intial velocity and inverted velocity over the true stretched NMO velocity are shown in Figure 2(a) and 2(b), respectively. The error in the initial NMO velocity is up to 25%, which is far beyond the tolerance of FWI. The anisotropic WEMVA successfully reduces the error in velocity down to less than 5%. Notice that the error in velocity generally follows the dip in the image. This suggests that we should use better smoothing operators such as dip (steering) filters (Hale, 2007; Clapp, 2000) to regularize the inversion.

On the other hand, the $\eta$ update [Figure 2(d)] is in general larger than the true $\eta$ model [Figure 2(c)]. A trade-off is observed below 1,600 m, where the inverted velocity is smaller but $\eta$ is much larger than the true values. This result illustrates the null space of our inversion problem, since the reflector around 2,200 m is well-focused (although not perfectly focused) in the final image obtained with the inverted model [Figure 3(b)]. This problem can presumably be resolved by increasing the angle coverage at depth and allowing more iterations in the inversion.

Figure 3 compares the subsurface-offset images using the initial model (a), the updated model (b), and the true model (c). After the inversion, the reflectors are focused at zero subsurface-offset, and the depths of the reflectors are closer to the true depths. The focused image shows that we are dealing with a non-linear problem with a large null space. To reduce the size of the null space, and hence the uncertainty in the inverted model, other information such as checkshots or rock-physics prior knowledge is needed (Li et al., 2011b,a).


init-over-true,inv-over-true,true-eta,inv-eta Figure 2. (a) Ratio of initial velocity over true velocity; (b) ratio of inverted velocity over true velocity; (c) true $\eta$ model; (d) inverted $\eta$ model.