The volume integral representation of the Porter-Bojarski integral allows
the backpropagation of the homogeneous portion in the measured data. We can
*not* backpropagate the inhomogeneous portion. Nevertheless, if the
inhomogeneous waves are weak in amplitude compared to the homogeneous waves,
we can get a good subsurface image.
Although we didn't use any weak scatterer approximation
for the derivation of the Porter-Bojarski integral, the image obtained
from the integral will not be complete if there are strong scatterers near
our geophones. The seismic exploration industry solves this problem by
cutting off evanescent energy and being content with the homogeneous
wave image. It is more clever to continue downward the homogeneous waves to
a certain reflector, let the reflector explode, and use the difference
of
synthetic and registered data to guess the evanescent energy and to update
the reflection coefficients. This can be done iteratively.

Programming the Porter-Bojarski integral for the backpropagation from an
arbitrary surface *S* leads to an algorithm called
*generalized holography*. It performs the backpropagation of
homogeneous waves with a constant frequency.

Let us for a moment consider the Kirchhoff integral for
the special case
of a planar
registration surface *S _{1}*.
We assume Cartesian coordinates and some scatterers only in the lower halfspace.
We construct a halfsphere

(11) |

(12) |

If we add the integrals for above and below the observation plane, we get the
*Raleigh I* integral

(13) |

Equivalent Raleigh formulations for the Porter-Bojarski integral
are obtained by
inserting the conjugate complex Green's functions *G ^{*}*. If we consider the
registration surface of reflection seismic surveys to be a plane, it will be
sufficient to backpropagate only monopoles with a Raleigh II type
representation of the Porter-Bojarski integral. We don't have to know the
gradient of the wavefield. This shouldn't be news for you, but the
explanations given for this fact by Schneider (1978) and others are
rather heuristical.

Programming the Raleigh I or Raleigh II integrals leads to the so-called
*Raleigh Sommerfeld Holography* (Langenberg, K.L., 1986).
It performs the backpropagation of
homogeneous waves emitted by dipoles or monopoles
for constant frequencies.

Until now we always considered holography in the space. Let us
study our problem in the frequency wavenumber domain. A homogeneous plane
wave is in the *k*-space located on the *Ewald sphere* (Ashcroft, N.W.,
Mermin, N.D., 1981).The
Ewald sphere is a sphere in the Cartesian *k*-space with the radius

(14) |

To image we have to superpose as many intersections of Ewald spheres and object functions as possible to get a high resolution. There are basically two possibilities to get different intersections

- 1.
- frequency diversity
- 2.
- angular diversity .

Superposition of all possible backshifted intersections, i.e., a complete
coverage of the *k*-space, yields to an ``optimal'' image with homogeneous waves.
But we still
miss the information of inhomogeneous waves due to strong scatterers
or sources near the geophones.

Doing generalized holography or Raleigh Sommerfeld Holography for different frequencies leads to the *
frequency diversity filtered backpropagation* algorithm:

- measure the intersection of the Ewald sphere with the shifted object function for different frequencies but constant incidence angle
- make a temporal deconvolution
- shift the Ewald sphere intersections back to the origin (backpropagation)
- superpose the images for different frequencies
- superpose the backpropagated and incident wavefields

- measure the cut of the Ewald sphere with for different angles of incidence but for the same frequency. (We apply
a lowpass because the intersections are within the sphere 2|
*k*_{i}|.) - shift the Ewald sphere intersections back to the origin (backpropagation)
- superpose the images for different angles
- superpose the backpropagated and incident wavefields

Either frequency diversity or angular diversity can image the object if we
have a complete coverage of the *k*-space. In practice, our data is
bandpassed and registration on a plane limits the angular resolution.
The seismic processing will be more sucessful if we can combine all the
information we have. Prestack migration yields to better results than
Zero-offset migration because the bandpassed frequency diversity of Zero-offset
migration is
completed by a limited angular diversity.

If our data is recorded on a plane, we should speak of the
Raleigh Sommerfeld frequency diversity or angular diversity
algorithms. For a constant velocity medium, Raleigh Sommerfeld angular
diversity holography is nearly the same as *diffraction tomography* that
will be studied in the next section.

1/13/1998