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 kg, the Fourier component along the geophone axis of the data volume in (s,g,t)-space. The other cosine, with ks, 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.