THEORETICAL APPROACH

Wave-equation migration techniques consist of two distinct steps:
1. Extrapolation of the wavefield recorded at the surface.
2. Imaging.

The extrapolation process is the solution of the homogeneous scalar wave equation. It calculates the wavefield at any position and time from the knowledge of the wavefield at other positions and times. The imaging process determines the position and time at which the extrapolated wavefield is required.

For a two-dimensional constant velocity medium, the extrapolation formula imaged to time t=0 yields the following equation of migration,

$$P_{out}(x_0,z_0,0)= \left[ \left( \frac{\partial}{\parti... ...}}P_{in} \left( x-x_0,0,t-\frac{r}{v} \right) dx \right],$$ (1)

where x₀ and z₀ are the spatial location and depth of the image point. P_in(x-x₀,0,t-r/v) is the unmigrated wavefield measured at the earth's surface, P_out(x₀,z₀,0) is the migrated wavefield, $cos \theta$ is the obliquity factor,$cos \theta = z_0/r$, and r² = (x-x₀)² + z₀².

Thus the Kirchhoff method implied by the above equation reflects a summation over a surface. It contains the half-derivative $(\partial /\partial t)^{\frac{1}{2}}$.This operator is equivalent to a filter with a constant $\pi /4$ phase delay and an amplitude spectrum proportional to the square root of frequency. The migration equation shows that the effect of migration in the (x,t) domain is an amplitude-weighted summation combined with a frequency filtering that incorporates a constant phase shift. The amplitude weighting decays as $1/{r^{\frac{1}{2}}}$, as shown by the integrand of equation (1).