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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,
| |
(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

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(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