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In two dimensions, considering one-way time, the wave equation is

| |
(1) |

where is the pressure wave field, and is the
velocity of the wave propagation in the medium.
We can make the following approximation of the wave field:

| |
(2) |

expressing the pressure wave field in term of amplitude and
phase , where is the
traveltime along the ray paths. Then replacing (2) in equation
(1) and making the high frequency approximation, we obtain the 2-D
Eikonal equation:
| |
(3) |

Among other uses, this equation is useful to compute traveltimes of rays
in a given earth model.
V.J. Khare introduced the following equation for the ray-theoretic
interpretation of *all-angle* time migration Khare (1991):

| |
(4) |

where and are respectively the
pressure wave field and the wave propagation velocity in the time-retarded
coordinate system Claerbout (1976),
| |
(5) |

with the time-like vertical coordinate Lowenthal et al. (1985),
| |
(6) |

If we make the following approximation of the wave field in the new coordinate
system, equivalent to the equation (2):

| |
(7) |

we can express the 2-D Eikonal equation for time migration giving the
traveltime in the frequency domain in the time-retarded coordinate
system:
| |
(8) |

Using equations (5) and (6), we obtain the 2-D Eikonal
equation for time migration in the physical coordinate system:

| |
(9) |

with
| |
(10) |

** Next:** USING 3-D EIKONAL EQUATION
** Up:** Berlioux: 3-D Eikonal equation
** Previous:** INTRODUCTION
Stanford Exploration Project

11/17/1997