Ideally, the elastic wave equation should be used to model the propagation of seismic waves in the earth. But in exploration reflection seismology, except in very specific cases, only the vertical component of the propagated wavefield is recorded at the surface. This recorded wavefield is often assumed to be a scalar (converted modes projected in the vertical wavefield component are treated as noise). Thus a scalar wave equation is a good approximation of the physics involved in an exploration reflection seismology experiment.
Additionally, modeling or migrating with the scalar wave equation is computationally expensive. That is why it is rarely used for migration Claerbout (1985); Stolt and Benson (1986). Instead, two one-way wave equations are used to mimic the scalar wave equation solution Claerbout (1971); Stolt and Benson (1986). This approximation is reasonable in computational time and accuracy.
In the next three subsections we review the three steps involved in shot-profile modeling with the one-way wave equation: first, downward continuation of the source wavefield, second, upward continuation of the receiver wavefield to the surface, and third, truncation of the receiver wavefield at zero depth.