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WEMVA Forward Modeling

Following Sava and Biondi (2004), I develop equations for imaging by wavefield extrapolation based on recursive continuation of the wavefields ${\mathcal U}$ from a given depth level to the next by means of an extrapolator operator ${\mathbf E}$  
{\cal U}_{z+\Delta z}= {\mathbf E}_z[{\cal U}_z],\end{displaymath} (1)
where ${\mathbf E}_z[]={\rm e}^{{\rm i}k_z \Delta z}$, kz is extrapolation wavenumber, and $\Delta z$ is the depth step. Throughout this paper, I use a notation where ${ \bf A}[x]$ denotes that operator $\bf{A}$ is applied to a field x. Subscripts z and $z+ \Delta z$ correspond to quantities associated with depth levels z and $z+ \Delta z$, respectively.

Using this operator notation, a data wavefield ${\mathcal D}$ can be recursively extrapolated through a medium described by model parameters (i.e. slowness). This operation can be written explicitly in matrix form,  
\left[ \begin{array}
\mathbf{1} & 0 & 0 & ... & 0 &...
 ...thcal D}_0 \\ 0 \\ 0 \\  
\vdots \\  
0 \\  \end{array}\right],\end{displaymath} (2)
where $\mathbf{1}$ is an identity operator, and fields without subscripts (e.g. ${\mathcal U}$ and ${\mathcal D}$) refer to complete wavefields. Equation 2 is written more compactly as  
\left( \mathbf{1} -{\mathbf E}\right) {\mathcal U}= {\mathcal D},\end{displaymath} (3)
where $\left( \mathbf{1} -{\mathbf E}\right)$ is a Green's function ${\mathbf G}_0({\bf x}^\prime,{\bf x})$between levels ${\bf x}$ and ${\bf x}^\prime$ generated by wavefield extrapolation. The Green's function satisfies the following adjoint definitions,
(\mathbf{1}-{\mathbf E}) = {\mathbf G}_0({\bf x}^\prime,{\bf x}...
 ...{\bf x^\prime},{\bf
 x}) = {\mathbf G}_0( {\bf x},{\bf x^\prime}),\end{eqnarray} (4)
where superscripts -1 and $^
$ indicate the inverse and adjoint operation (i.e. complex transpose), respectively.

Source wavefields well-modeled by a delta function exhibit the following relationships,  
{\mathcal U}({\bf x},{\mathbf s}) = \left( \mathbf{1}-{\math... x}^\prime-{\mathbf s}) = {\mathbf G}_0({\bf x},{\mathbf s}),\end{displaymath} (6)
where ${\mathbf G}_0({\bf x},{\mathbf s})$ describes the propagation from source point ${\mathbf s}$throughout the domain denoted by ${\bf x}$. Note that the choice of ${\mathbf s}$ is arbitrary and an equivalent development applies for a receiver Green's function ${\mathbf G}_0({\bf x},{\mathbf r})$,  
{\mathbf G}_0({\bf x},{\mathbf r}) = {\mathbf G}_0({\bf x},{...
 ...1}-{\mathbf E}\right)^{-1} \delta({\bf x}^\prime-{\mathbf r}), \end{displaymath} (7)
where ${\mathbf r}$ is receiver location.