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Of course, the linear theory is not limited to discrete grids. It is
interesting to consider the continuous case, because of its connection
to the linear integral operators commonly used in seismic imaging.
Indeed, in the continuous case, linear decomposition (7)
takes the form of the nonstationary convolution integral
| |
(22) |

where *x* is a continuous analog of the discrete coefficient *k* in
(7), the continuous function *m* (*x*) is analogous to the
coefficient *c*_{k}, and *G* (*y*; *x*) is analogous to one of the basis
functions . The linear integral operator in
(22) has a mathematical form similar to the form of
well-known integral imaging operators, such as Kirchhoff migration or
``Kirchhoff'' DMO. Function *G* (*y*; *x*) in this case represents the
Green function (impulse response) of the imaging operator. Linear
decomposition of the data into basis functions means decomposing it
into the combination of impulse responses (``hyperbolas''.)
In the continuous case, formula (13) transforms to

| |
(23) |

where refers to the inverse of the ``matrix''
operator
| |
(24) |

When the linear operator, defined by formula (22), is
*unitary* ,
| |
(25) |

and formula (23) simplifies to the single integral
| |
(26) |

With respect to seismic imaging operators, one can recognize in the
interpolation operator (26) the generic form of azimuth
moveout Biondi et al. (1996), which is derived either as a cascade
of adjoint () and forward (*G* (*x*;*y*)) DMO or as a
cascade of migration () and modeling (*G* (*x*;*y*))
Fomel and Biondi (1995). In the first case, the intermediate variable
*y* corresponds to the space of zero-offset data cube. In the second
case, it corresponds to a point in the subsurface.

** Next:** Asymptotic pseudo-unitary operators as
** Up:** Fomel: Interpolation
** Previous:** Discrete Fourier basis
Stanford Exploration Project

11/11/1997