# ORTHOGONALITY OF CROSSING PLANE WAVES

As an alternative to the customary approach of defining and analyzing aliasing by Fourier analysis, I suggest that spatial aliasing may be defined and analyzed with reference to plane waves. Consider two sinusoidal signals of different frequencies. Since they have different frequencies they should be orthogonal, but they need not be orthogonal if there is aliasing, because aliasing can make a high frequency look like a low one. Now let us see if we can substitute the word dip'' for the word frequency.'' Can we say that plane waves with different dips should be orthogonal?

Normally, waves do not contain zero frequency. Thus the time integral of a waveform normally vanishes. Likewise, for a dipping plane wave, the time integral vanishes. Likewise, a line integral across the (t,x)-plane along a straight line that crosses a plane wave or a dipping plane wave vanishes. Likewise, two plane waves with different slopes should be orthogonal if one of them has zero mean. The other wave, however, need not have zero mean. This other wave'' need not be a wave at all, but could be an impulse function, say ,spread over a mesh. Once it is spread out, it will look like an impulsive plane wave. The purpose of this impulsive plane wave is to multiply onto data, thereby enabling us to do a line integral of the data. What is important is how we represent this impulse function on the mesh. Theoretically, we have line integrals that should vanish when they cross a zero-mean plane wave. When in practice they do not, it means that we have not done a good job of representing the line integral.

Fourier analysis and sinc functions play an important role in integrals along straight lines. Production reflection seismic data processing involves much effort with line integrals along hyperbolic curves and other shapes--rarely straight lines. Hopefully, the theoretical concepts above will be suggestive of further theoretical and practical developments.