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In 3D the eikonal equation is

u^{2}+v^{2}+w^{2}=s^{2}

(8) 
where
For a sphericalcoordinates system
 
(9) 
where
The cross derivative equation (4) is transformed into
the spherical coordinates system
 
(10) 
The finitedifference equivalent of equation (5) is
the system
 
(11) 
which is used to advance the stencil for a new radial increment.
Once the values of the functions and
are known on the spherical
front with constant radius (),
the third function can be calculated using the eikonal equation
 
(12) 
The value of the traveltime is found by integration:
In equation (11), the EngquistOsher scheme is applied twice, once for
calculating the values of across three points
of consecutive values of , and second for calculating
the values of across three consecutive values
of .The computational front advances in spherical shells, and on
each shell the computations advance a circle at a
time.
The angle is the horizontal angle while
the angle is the vertical angle.
The EngquistOsher scheme is applied along each
three consecutive points on the circle with constant vertical
angle to determine from the equation
.For each circle of constant vertical angle the
EngquistOsher scheme is applied for three points
(), and (), which are
perpendicular on the circle in the coordinates.
The scheme is completely vectorizable.
Next: 3D TRAVELTIME MAPS
Up: Popovici : FD Traveltime
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Stanford Exploration Project
12/18/1997