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One-way propagation (PDKO)

Let us consider the equation:
 (95)
at

kz=kz(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 ikz 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