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## Snell wave information in field data

The superposition principle allows us to create an impulse function by a superposition of sinusoids of all frequencies. A three-dimensional generalization is the creation of a point source by the superposition of plane waves going in all directions. Likewise, a plane wave can be a superposition of many Huygens secondary point sources. A Snell wave can be simulated by an appropriate superposition, called a slant stack, of the point-source data recorded in exploration.

Imagine that all the shots in a seismic survey were shot off at the same time. The downgoing wave would be approximately a plane wave. (Let us ignore the reality that the world is 3-D and not 2-D). The data recorded from such an experiment could readily be simulated from conventional data simply by summing the data field P(s,g,t) over all s. In each common-geophone profile the traces would be summed with no moveout correction.

To simulate a nonvertical Snell wave, successive shots must be delayed (to correspond to a supersonic airplane), according to some prescribed ps = dt/ds.

What happens if data is summed over the geophone axis instead of the shot axis? The result is point-source experiments recorded by receiver antennas that have been highly tuned to receive vertically propagating waves. Time shifting the geophones before summation simulates a receiver antenna that records a Snell wave, say, pg = dt/dg upcoming at an angle .

Integration over an axis is an extreme case of low-pass filtering over an axis. Between the two extremes of the point-source case and the plane-wave case is the case of directional senders and receivers.

The simple process of propagation spreads out a point disturbance to a place where, from a distance, the waves appear to be nearly plane waves or Snell waves. Little patches of data where arrivals appear to be planar can be analyzed as though they were Snell waves.

In summary, a downgoing Snell wave is achieved by dip filtering in shot space, whereas an upcoming Snell wave is achieved by dip filtering in geophone space.

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Stanford Exploration Project
10/31/1997