# Spatial sampling

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

Spatial aliasing was discussed in detail in the 2-D Fourier transform and in relation to migration in further aspects of migration in practice. The spatial aliasing problem is caused by spatial undersampling of the wavefield to be migrated — for example, the stacked section. The spatial sampling of stacked data (without trace interpolation) is defined by the recording parameters. Therefore, receiver spacing, crossline spacing, and the crossline direction in relation to dominant dip direction used in the field must be chosen carefully.

Figure 1.2-21  A plane wave reflecting at normal incidence from a dipping reflector with a dip angle θ arrives at two consecutive receiver locations A and B at the surface with a separation Δx. Geometry of this plane wave is used to derive equation (6).

 ${\displaystyle \sin \theta ={\frac {v\Delta t}{2\Delta x}},}$ (6)

From Figure 1.2-21, note that a relationship exists between the trace spacing on a stacked section, dip, and the frequency at which spatial aliasing begins to occur. Imagine normal-incidence rays recorded at two receivers, A and B. In the constant-velocity case, the angle between the surface and the wavefront is the true dip of the reflector from which these rays emerged. There is a time delay equivalent to travelpath CB between the receivers at A and B. If this time delay is half the period of a given frequency component of the signal arriving at the receivers, then that frequency is at the threshold of being aliased.

From the relationship given by equation (1-7), note that the maximum frequency that is not aliased gets smaller at increasingly steeper dips, lower velocities, and coarser trace spacing. From this relationship, an optimum trace spacing can be derived for the inline and crossline directions based on the knowledge of a regional velocity field and subsurface dips. Typical trace spacings in the inline and crossline directions in 3-D surveys are 12.5 to 25 m, and 25 to 50 m, respectively. Even if the trace spacing in the crossline direction is as small as possible, for economic reasons, it usually is greater than that in the inline direction. Because of this, trace interpolation may be required along the crossline direction before migrating the data.