The ideas of 1-D complex seismic trace analysis extend readily to two dimensions. Two‐dimensional instantaneous amplitude and phase are scalars, and 2-D instantaneous frequency and bandwidth are vectors perpendicular to local wavefronts, each defined by a magnitude and a dip angle. The two independent measures of instantaneous dip correspond to instantaneous apparent phase velocity and group velocity. Instantaneous phase dips are aliased for steep reflection dips following the same rule that governs the aliasing of 2-D sinusoids in f-k space. Two‐dimensional frequency and bandwidth are appropriate for migrated data, whereas 1-D frequency and bandwidth are appropriate for unmigrated data. The 2-D Hilbert transform and 2-D complex trace attributes can be efficiently computed with little more effort than their 1-D counterparts. In three dimensions, amplitude and phase remain scalars, but frequency and bandwidth are 3-D vectors with magnitude, dip angle, and azimuth.
Norm variation of ergodic averages with respect to two commuting transformations
