Hand migration

Given a seismic event at ( x₀ , t₀ ) with a slope $p \, = \, dt / dx$,let us determine its position ( x_m , t_m ) after migration. Consider a planar wavefront at angle $\theta$ to the earth's surface traveling a distance dx in a time dt. Assuming a velocity v we have the wave angle in terms of measurable quantities.  

$$\sin \, \theta \eq { v \ dt \over dx } \eq p \, v$$ (3)

The vertical travel path is less than the angled path by  

$$t_m \eq { t_0 } \ { \cos \, \theta } \eq { t_0 } \ { \sqrt { 1 \ -\ p^2 \, v^2 } }$$ (4)

A travel time t₀ and a horizontal component of velocity $v\,\sin\,\theta$ gives the lateral location after migration:  

$$x_m \eq x_0 \ -\ t_0 \,v\,\sin\,\theta \eq x_0 \ -\ t_0 \,p\,v^2$$ (5)

Consideration of a hyperbola migrating towards its apex shows why (5) contains a minus sign. Equations (4) and (5) are the basic equations for manual migration of reflection seismic data. They tell you where the point migrates, but they do not tell you how the slope p will change.