We have two problems to solve:

- I.
- The choice of the operator
**L**. - II.
- The choice of the boundary conditions.

**I.**- Let us take the classical eikonal equation

(80) |

1. Absence of wave dispersion. Let the coefficient of the equation (80) be constant. After substitution

where and . The equation is reduced to where .Looking for the solution in the form

we obtain the dispersion equation Since in nondispersive media ,only in the case when or

2. Amplitude equivalence. This term means that while
discontinuities propagate, amplitudes *A* change along rays *s* following the transport
equation (see Chapter 4)

*c*=0

3. Nonscattering vertical propagation. If in the amplitude-equivalent equation

(81) |

**II.** Boundary conditions. There are few types of conditions that can be used
for the wave-field continuation with the help of equation (80).

1. Mixed problem with boundary and initial conditions

where
This problem has a unique and stable solution (in a restricted domain). It is
restricted at (for direct continuation) and (for inverse continuation) if *h*>0 and (or ).

Representations of the solution:

Finite-difference technique which can be applied for any inhomogeneous media.

Kirchhoff's type integral:

(82) |

where , *I _{+}* =(0,

(83) |

For inhomogeneous media Green's function usually is unknown (although one can calculate ray zero-approximation of ). In the case of wave equation Green's function for homogeneous medium is

(84) |

(85) |

At big values of *R* the first term in square brackets can
be neglected:

(86) |

These expressions are valid only in 3D case. In 2D case Green's function is

(87) |

Inserting this into the equation (82) we obtain for the reverse continuation

(88) |

(89) |

Let the field *u _{0}* (

Taking into account all these considerations and neglecting the term with fast attenuating at , we derive from equation (88):

(90) |

Spectral representation in ()-domain (the wave equation, homogeneous media):

(91) |

2. The Cauchy problem with respect to z in the domain for the equation

(92) |

The natural way to stabilize Cauchy's problem is to cut off all frequencies at
*k ^{2}*

3. Reconstruction-type wave-field extrapolation. In this case we pose the conditions

and(93) |

There is only one reverse type regular continuation into homogeneous medium which satisfies the condition:

u|_{z=0} =u
_{0} |
(94) |

In inhomogeneous media different conditions produce different solutions. Figure shows an example of different solutions: (a) is the reversed continuation of a single event at for mixed conditions, and (b) for the Cauchy problem.

In Figure a we use reverse time -*t* instead of *t* (we can do it with
accordance with relation (51) in Chapter 5) because it promotes better understanding of
reflection-refraction pattern at the interface *z*=*d*. In reverse time we have the
usual process of wave propagation. Dashed lines show some ghost waves.

1/13/1998