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The initial step in frequency-domain waveform inversion is to prescribe the forward model. I assume that wave propagation is adequately governed by the acoustic wave equation; thus, any forward-modeling procedure will generate a monochromatic scalar wavefield,
, that is an (approximate) complex-valued solution to the Helmholtz equation,
| ![\begin{displaymath}
{\bf L}\Psi({\bf s},{\bf x};\omega) =
\left( \nabla^2 +\fra...
...ght) \Psi({\bf s},{\bf x};\omega) = -\delta({\bf s} - {\bf x}),\end{displaymath}](img3.gif) |
(1) |
where
is the Helmholtz operator,
the Laplacian operator,
angular frequency,
the assumed velocity profile in spatial domain
,
the source position, and
the Dirac delta function operator. Note that the waveform inversion problem is non-linear in model parameters,
, which I will solve using an iterative inversion approach. Discussion of the specific approach to solving equation 1 being presented is deferred to the following section.
The next step is to compare the modeled wavefield solutions,
, to the observed data,
, where
is the receiver position. This procedure leads to a residual wavefield,
, defined as the difference between the two wavefields
| ![\begin{displaymath}
\Delta \Psi ({\bf s},{\bf r}; \omega) = \Psi_{calc}({\bf s},{\bf r};\omega) - \Psi_{obs}({\bf s},{\bf r};\omega).\end{displaymath}](img16.gif) |
(2) |
The residuals are a measure of waveform fit and will be back-projected to generate a velocity model update. Note that no assumption is explicitly made about a linear relation (i.e. the Born approximation is explicitly avoided in the forward modeling problem) Sirgue and Pratt (2004); however, if model parameters are too far removed from the true velocity model, then the monochromatic wavefields in equation 2 will cycle-skip giving erroneous residuals. However, because cycle-skipping is more likely at higher frequencies, the approach is generally more stable at lower frequencies.
Next: The Inverse Problem
Up: Review of Frequency-domain waveform
Previous: Review of Frequency-domain waveform
Stanford Exploration Project
1/16/2007