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Following Sava and Biondi (2004), I develop equations for imaging by
wavefield extrapolation based on recursive continuation of
the wavefields
from a given depth level to the next by means of
an extrapolator operator
| ![\begin{displaymath}
{\cal U}_{z+\Delta z}= {\mathbf E}_z[{\cal U}_z],\end{displaymath}](img3.gif) |
(1) |
where
, kz is
extrapolation wavenumber, and
is the depth step.
Throughout this paper, I use a notation where
denotes
that operator
is applied to a field x. Subscripts z and
correspond to quantities associated with depth levels z and
, respectively.
Using this operator notation, a data wavefield
can be recursively
extrapolated through a medium described by model parameters
(i.e. slowness). This operation can be written explicitly in matrix
form,
| ![\begin{displaymath}
\left[ \begin{array}
{cccccc}
\mathbf{1} & 0 & 0 & ... & 0 &...
...thcal D}_0 \\ 0 \\ 0 \\
\vdots \\
0 \\ \end{array}\right],\end{displaymath}](img10.gif) |
(2) |
where
is an identity operator, and fields without subscripts
(e.g.
and
) refer to complete
wavefields. Equation 2 is written more compactly
as
| ![\begin{displaymath}
\left( \mathbf{1} -{\mathbf E}\right) {\mathcal U}= {\mathcal D},\end{displaymath}](img12.gif) |
(3) |
where
is a Green's function
between levels
and
generated by wavefield
extrapolation. The Green's function satisfies the following adjoint
definitions,
| ![\begin{eqnarray}
(\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}](img17.gif) |
(4) |
| (5) |
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,
| ![\begin{displaymath}
{\mathcal U}({\bf x},{\mathbf s}) = \left( \mathbf{1}-{\math...
...bf x}^\prime-{\mathbf s}) = {\mathbf G}_0({\bf x},{\mathbf s}),\end{displaymath}](img19.gif) |
(6) |
where
describes the propagation from source point
throughout the domain denoted by
. Note that the choice of
is arbitrary and an equivalent development applies for a
receiver Green's function
,
| ![\begin{displaymath}
{\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}](img23.gif) |
(7) |
where
is receiver location.