Exploding-reflector imaging will be replaced
by a broader imaging concept,
*
survey sinking.
*
A new equation called the
double-square-root (DSR) equation will be developed
to implement survey-sinking imaging.
The function of the DSR equation
is to downward continue an entire seismic survey,
not just the geophones but also the shots.
After deriving the DSR equation, the remainder of this chapter
will be devoted to explaining
migration, stacking, migration before stack, velocity analysis,
and corrections for lateral velocity variations in terms of the DSR equation.

Peek ahead at equation (35)
and you will see an equation with two square roots.
One represents the cosine of the wave
*arrival*
angle.
The other represents the
*takeoff*
angle at the shot.
One cosine is expressed in terms of *k*_{g}, the Fourier component
along the geophone axis of the data volume in (*s*,*g*,*t*)-space.
The other cosine, with *k*_{s}, is the Fourier component
along the shot axis.

Our field seismograms lie in the (*s*,*g*)-plane.
To move onto
the (*y*,*h*)-plane inhabited by seismic interpreters
requires only a simple rotation.
The data could be Fourier transformed with
respect to *y* and *h*, for example.
Then downward continuation would proceed
with equation (48) instead of equation (35).

The DSR equation depends upon the reciprocity principle
which we will review first.

- Seismic reciprocity in principle and in practice
- The survey-sinking concept
- Review of the paraxial wave equation
- The DSR equation in shot-geophone space
- The DSR equation in midpoint-offset space

10/31/1997