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Imaging by wavefield extrapolation (WE) is based on recursive
continuation of the wavefields () from a given depth level
to the next by means of an extrapolation operator ():
| |
(45) |

Here and hereafter, I use the following notation conventions:
means operator applied to *x*, and
means function *f* of argument *x*.
The subscripts or indicate quantities corresponding
to the depth levels and , respectively.
This recursive down can also be explicitly written in
matrix form as
-_0-_1-_n-1 _0_1_2_n = _0 0 0 0 ,

or in a more compact notation as:

| |
(46) |

where the vector stands for data,
for the extrapolated wavefield at all depth levels,
for the extrapolation operator and
for the identity operator.
Here and hereafter, I make the distinction between
quantities measured at a particular depth level (e.g. ),
and the corresponding vectors denoting such quantities at all
depth levels (e.g. ).
After wavefield extrapolation, we can obtain an image
by applying, at every depth level, an imaging operator () to
the extrapolated wavefield :

| |
(47) |

where stands for the image at some depth level.
A commonly used imaging operator () involves summation
over the temporal frequencies.
We can write the same relation in compact matrix form as:
| |
(48) |

stands for the image, and
stands for the imaging operator which is
applied to the extrapolated wavefield at all
depth levels.
A perturbation of the wavefield at some depth level can
be derived from the background wavefield
by a simple application of the chain rule to down:

| |
(49) |

where represents the scattered wavefield
generated at by the interaction of the wavefield
with a perturbation of the velocity model at depth . is the accumulated wavefield perturbation
corresponding to slowness perturbations at all levels above.
It is computed by extrapolating the wavefield perturbation from the level above, plus the scattered wavefield at this level.
wavpert is also a recursive equation which can be
written in matrix form as
-_0-_1-_n-1 _0_1_2_n = _0 _1_n-1 _0_1_2_n ,

or in a more compact notation as:

| |
(50) |

The operator stands for a
perturbation of the extrapolation operator .The quantity represents a scattered
wavefield, and is a function
of the perturbation in the medium by the
scattering relations derived in Appendix C.
For the case of single scattering, we can write that
| |
(51) |

The expression for the total wavefield perturbation from
wavpert becomes

| |
(52) |

which is also a recursive relation that can be written in matrix
form as
-_0-_1-_n-1 _0_1_2_n = _0 _1 _n-1 _0 _1 _2 _n s_0s_1s_2s_n ,
or in a more compact notation as:

| |
(53) |

The vector stands for the slowness perturbation at all depths.
Finally, if we introduce the notation

| |
(54) |

we can write a simple relation between a slowness perturbation and the corresponding wavefield perturbation :
| |
(55) |

This expression describes wavefield scattering caused by the interaction
of the background wavefield with a perturbation of the medium.

** Next:** WEMVA objective function
** Up:** Wave-equation migration velocity analysis
** Previous:** Introduction
Stanford Exploration Project

11/4/2004