Compared to photographs, reflection seismic datasets are poorly compressed with the singular value decomposition. Dipping events extend the range of singular values that contain significant energy, invalidating the assumption that only the small singular values are needed to reconstruct the image. Fine detail, including noise, tends to be associated with the small singular values and is lost using reconstructions with only the large singular values.

Dipping events do not fall along columns or rows of the original image and
are spread over a wide range of singular values, violating the
assumption that we can represent the input with only the first few
singular values. Examples shown in Figures 3 to 5 show the number of
significant singular values increases with dip.
Figure 3 shows a flat event and its decomposition.
This event has only one significant singular value and can be
reconstructed with a single row of *V* and a single column of *U*.
If a dipping event is decomposed as seen in Figure 4, the number of
singular values increases and the amount of data needed to
reconstruct the image is larger than that needed to reconstruct the
image of the flat event. If an event with even more dip
is decomposed as seen in Figure 5,
all the singular values must be used to reconstruct the image, and the
amount of data needed to store the *U* and *V* matrices
is greater than the size of the original image.

flat
A flat event and its
decomposition.
Notice that there is only one nonzero singular value.
Figure 3 |

dip
A dipping event and its decomposition. Notice that there are now many more nonzero singular values.
Figure 4 |

diag
An even steeper dipping event and its decomposition. Every singular value is now equal.
Figure 5 |

High frequency events in time and space containing the fine detail
of an image
tend to be distributed in the partial images corresponding to small singular
values. The decomposition of the photograph
shown in Figure 1 shows
high frequency events in the upper rows of
the *U* and *V*
matrices that correspond to the smaller singular values.
In photographs like Figure 1, much of the image is made up of large uniform
areas; in contrast, seismic images contain
mainly fine detail.
Figure 6 shows the decomposition of
a dipping event in a background of random numbers that simulates a seismic
file with one strong event and much small detail.
If the *U* and *V* matrices are examined carefully, low frequency
signals corresponding to the dipping event are seen in the bottom few
rows.
When the image is
reconstructed in Figure 7 with a few large singular values,
the background shows a loss of random
noise.
Even worse, the strong event is corrupted with
noise from outside the event, and the random noise is organized along
the columns and rows.

Figure 6

Figure 7

11/18/1997