|
|
|
| Imaging in shot-geophone space | |
|
Next: The DSR equation in
Up: SURVEY SINKING WITH THE
Previous: The survey-sinking concept
An equation was derived for paraxial waves.
The assumption of a
single
plane wave means that the arrival time
of the wave is given by a single-valued .
On a plane of constant , such as the earth's surface,
Snell's parameter is measurable.
It is
|
(2) |
In a borehole there is the constraint that measurements
must be made
at a constant , where the relevant measurement from an
upcoming
wave would be
|
(3) |
Recall the time-shifting partial-differential equation and its
solution as some arbitrary functional form :
The partial derivatives
in equation (9.4) are taken to be at constant ,
just as is equation (9.3).
After inserting (9.3) into (9.4) we have
|
(6) |
Fourier transforming the wavefield over , we
replace
by .
Likewise, for the traveling wave
of the Fourier kernel
,
constant phase means that
.
With this, (9.6) becomes
|
(7) |
The solutions to (9.7) agree with those to the scalar wave equation
unless is a function of , in which case
the scalar wave equation has both upcoming and downgoing solutions,
whereas (9.7) has only upcoming solutions.
We
go into the lateral space
domain by replacing by
.
The resulting equation is useful for superpositions of many local plane waves
and for lateral velocity variations .
|
|
|
| Imaging in shot-geophone space | |
|
Next: The DSR equation in
Up: SURVEY SINKING WITH THE
Previous: The survey-sinking concept
2009-03-16