Imaging in shot-geophone space |
Let the geophones descend a distance into the earth.
The change of the travel time of the observed upcoming wave will be
Simultaneously downward project both the shots and
geophones by an identical vertical amount
.
The travel-time change is the sum
of (9.8) and (9.9), namely,
Three-dimensional Fourier transformation converts
upcoming wave data to
.
Expressing equation (9.12) in Fourier space gives
|
Equation (9.14) is known as the double-square-root (DSR) equation in shot-geophone space. It might be more descriptive to call it the survey-sinking equation since it pushes geophones and shots downward together. Recalling the section on splitting and full separation we realize that the two square-root operators are commutative ( commutes with ), so it is completely equivalent to downward continue shots and geophones separately or together. This equation will produce waves for the rays that are found on zero-offset sections but are absent from the exploding-reflector model.
Imaging in shot-geophone space |