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Let the geophones descend a distance dz_{g} into the earth.
The change of the travel time of the observed upcoming wave will be
 
(8) 
Suppose the shots had been let off at depth dz_{s} instead of at z = 0.
Likewise then,
 
(9) 
Both (8) and (9)
require minus signs because the travel time
decreases as either geophones or shots move down.
Simultaneously downward project both the shots and
geophones by an identical vertical amount dz=dz_{g} = dz_{s}.
The traveltime change is the sum
of (8) and (9), namely,
 
(10) 
or
 
(11) 
This expression for may be substituted
into equation (4):
 
(12) 
Threedimensional Fourier transformation converts
upcoming wave data u(t,s,g) to .Expressing equation (12) in Fourier space gives
 
(13) 
Recall the origin of the two square roots in equation (13).
One is the cosine of the arrival angle at the geophones
divided by the velocity at the geophones.
The other is the cosine of the takeoff angle at the shots
divided by the velocity at the shots.
With the wisdom of previous chapters
we know how to go into the lateral space domain by
replacing i k_{g} by and
i k_{s} by .To incorporate lateral velocity variation v(x),
the velocity at the shot location must be distinguished
from the velocity at the geophone location.
Thus,
 
(14) 

Equation (14) is known as
the doublesquareroot (DSR) equation in shotgeophone space.
It might be more descriptive to call it the surveysinking equation
since it pushes geophones and shots downward together.
Recalling the section on splitting and full separation
we realize that the two squareroot operators are commutative
(v(s) 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 zerooffset sections
but are absent
from the explodingreflector model.
Next: The DSR equation in
Up: SURVEY SINKING WITH THE
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Stanford Exploration Project
12/26/2000