Forward modeling

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, $\Psi$, that is an (approximate) complex-valued solution to the Helmholtz equation,  

$${\bf L}\Psi({\bf s},{\bf x};\omega) = \left( \nabla^2 +\fra... ...ght) \Psi({\bf s},{\bf x};\omega) = -\delta({\bf s} - {\bf x}),$$ (1)

where ${\bf L}$ is the Helmholtz operator, $\nabla^2$ the Laplacian operator, $\omega$ angular frequency, $c({\bf x})$ the assumed velocity profile in spatial domain ${\bf x}$, ${\bf s}$ the source position, and $\delta$ the Dirac delta function operator. Note that the waveform inversion problem is non-linear in model parameters, $m ({\bf x}) = c^{-2}({\bf x})$, 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, $\Psi_{calc}({\bf s},{\bf r};\omega)$, to the observed data, $\Psi_{obs}({\bf s},{\bf r};\omega)$, where ${\bf r}$ is the receiver position. This procedure leads to a residual wavefield, $\Delta \Psi ({\bf s},{\bf r};\omega)$, defined as the difference between the two wavefields  

$$\Delta \Psi ({\bf s},{\bf r}; \omega) = \Psi_{calc}({\bf s},{\bf r};\omega) - \Psi_{obs}({\bf s},{\bf r};\omega).$$ (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.