*
Spatial aliasing
*
means insufficient sampling of the data along the space axis.
This difficulty is so universal, that all migration methods
must consider it.

Data should be sampled at more than two points per wavelength.
Otherwise the wave arrival direction becomes ambiguous.
Figure 11 shows synthetic data that is
sampled with insufficient density along the *x*-axis.

alias
Insufficient spatial sampling of synthetic data.
To better perceive the ambiguity of arrival angle,
view the figures at a grazing angle from the side.
Figure 11 |

You can see that the problem becomes more acute at high frequencies and steep dips.

There is no generally-accepted, automatic method for migrating spatially aliased data. In such cases, human beings may do better than machines, because of their skill in recognizing true slopes. When the data is adequately sampled, however, computer migration based on the wave equation gives better results than manual methods. Contemporary surveys are usually adequately sampled along the line of the survey, but there is often difficulty in the perpendicular direction.

The hyperbola-sum-type methods run the risk
of the migration operator itself becoming spatially aliased.
This should be avoided by careful implementation.
The first thing to realize is that you should be
*integrating*
along a hyperbolic trajectory.
A summation incorporating only one point per trace is a poor approximation.
It is better to incorporate more points,
as depicted in Figure 12.

lineint
For a low-velocity hyperbola,
integration will require more than one point per channel.
Figure 12 |

The likelihood of getting an aliased operator increases where the hyperbola is steeply sloped. In production examples an aliased operator often stands out above the sea-floor reflection, where--although the sea floor may be flat--it acquires a noisy precursor due to the steeply flanked hyperbola crossing the sea floor.

10/31/1997