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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
| |
(1) |

where , *k*_{z} 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,

| |
(2) |

where is an identity operator, and fields without subscripts
(e.g. and ) refer to complete
wavefields. Equation 2 is written more compactly
as
| |
(3) |

where is a Green's function between levels and generated by wavefield
extrapolation. The Green's function satisfies the following adjoint
definitions,
| |
(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,

| |
(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 ,
| |
(7) |

where is receiver location.