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Let us consider the equation:

| |
(95) |

at
*k*_{z}=*k*_{z}(*z*).

For homogeneous medium this equation can be obtained by splitting the wave
equation operator . Equation (95)
has the solution:

which describes upgoing waves.
The operator
| |
(96) |

is kinematically equivalent (in the sense of the classical eikonal equation), but
there is no partial derivative equation in space-time domain that has equation (96)
as a solution. This operator is an example of PDKO (Pseudo-Differential
K-Operator).
But we can find a sequence of equations
for which:
- each equation permits only one-way propagation along the z-axis.
- the characteristic equation for approximates the classical
eikonal equation.
- The accuracy of the approximation increases with the number
*j*.

The way to perform it is the expansion
If we substitute the nth approximation of *ik*_{z} into equation (95)
and return to the (**r**,t) domain, remembering the correspondence

we get the equation which we looked for.
The first approximation:
| |
(97) |

yields to Jon Claerbout's famous 15-degree migration algorithm. With accordance
to the formula (24) (in Chapter 3), the characteristic equation which
corresponds to equation (97) is as follows:
| |
(98) |

It differs from classical eiconal's equation for isotropic medium.

** Next:** 9: INTEGRAL OPERATORS OF
** Up:** 8: CLASSIFICATION OF K-OPERATORS
** Previous:** Differential K-operators
Stanford Exploration Project

1/13/1998