Tomographic full waveform inversion and linear modeling of multiple scattering |

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The problem is even deeper. When causes large time shifts by multiple scattering, there is no perturbation that can model those time shifts by single scattering; that is,

The non linearity of the modeling operator makes the objective function equation 2 to be non convex when the velocity perturbations are sufficiently large. Figure 1 shows an example of non-convexity of the objective function. The result correspond to several 1D transmission problems sharing the same starting velocity (1.2 km/s) and with different true velocities. For all these experiments the source-receiver offset is 4 km and the source function is a zero-phase wavelet bandlimited between 5 and 20 Hz. The FWI norm is plotted as a function of the true velocity. If the true velocity is lower than km/s or larger than km/s a gradient based method will not converge to the right solution, even in this simple and low-dimensionality example.

The challenges of solving the optimization problem in equation 1 by gradient based optimization can be alternatively represented by graphing, as a function of the initial velocity error, the search direction (opposite sign of the gradient direction) of the objective function with respect to velocity square. Figure 2 display this function computed by applying the adjoint of the linear operator to the data residuals; that is

For a gradient based method to converge, the search direction should be always negative when the true velocity is lower than 1.2 km/s, and positive when the true velocity is higher.

Tomographic full waveform inversion and linear modeling of multiple scattering |

2012-10-29