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![]() | 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
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![]() | Tomographic full waveform inversion and linear modeling of multiple scattering | ![]() |
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