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![]() | Tomographic full waveform inversion and linear modeling of multiple scattering | ![]() |
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We can achieve accurate modeling of perturbed wavefield by solving
equations 5-6
instead of
equations 7-8.
Equations 5-6
can be solved numerically
with a simple explicit method; that is,
one that adds the scattered wavefield
up to time
to the right-hand side of
equation 6 to compute the scattered wavefield
at
.
Even in presence of large velocity variations,
the scattered wavefield has now the correct time shift.
Numerical solutions produce accurate results,
although the scattered wavefield is still propagated
with the background velocity,
because multiple scattering is taken into account of.
The challenge with using these equations
in a gradient-based inversion algorithm
is that the relation between the scattered
wavefield
and the velocity perturbations
is now nonlinear.
In the next section, I present a method for
linearizing this relation that is alternative
to the conventional one represented by
equations 7-8,
and is based on an extension of the velocity model in time.
FWI-Norm-new
Figure 1. FWI norm as a function of the true velocity, when the starting velocity is equal to 1.2 km/s. |
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FWI-Grad-new
Figure 2. FWI search direction as a function of the true velocity, when the starting velocity is equal to 1.2 km/s. |
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![]() | Tomographic full waveform inversion and linear modeling of multiple scattering | ![]() |
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