Plane waves in three dimensions

WAVE INTERFERENCE AND TRACE SCALING

Although neighboring seismometers tend to show equal powers, the energy on one seismometer can differ greatly from that of a neighbor for both theoretical reasons and practical ones. Should a trace ever be rescaled to give it the same energy as its neighbors? Here we review the strong theoretical arguments against rescaling. In practice, however, especially on land where coupling is irregular, scaling seems a necessity. The question is, what can go wrong if we scale traces to have equal energy, and more basically, where the proper scale factor cannot be recorded, what should we do to get the best scale factor? A related question is how to make good measurements of amplitude versus offset. To understand these issues we review the fundamentals of wave interference.

Theoretically, a scale-factor problem arises because locally, wavefields, not energies, add. Nodes on standing waves are familiar from theory, but they could give you the wrong idea that the concept of node is one that applies only with sinusoids. Actually, destructive interference arises anytime a polarity-reversed waveform bounces back and crosses itself. Figure 3 shows two waves of opposite polarity crossing each other.

super Figure 3. Superposition of plane waves of opposite polarity.

Observe that one seismogram has a zero-valued signal, while its neighbors have anomalously higher amplitudes and higher energies than are found far away from the interference. The situation shown in Figure 3 does not occur easily in nature. Reflection naturally comes to mind, but usually the reflected wave crosses the incident wave at a later time and then they don't extinguish. Approximate extinguishing occurs rather easily when waves are quasi-monochromatic. We will soon see, however, that methodologies for finding scales all begin with deconvolution and that eliminates the monochromatic waves.

Plane waves in three dimensions