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The eikonal equation, describing the traveltime propagation in an
isotropic medium, has the form

| |
(1) |

where is the traveltime (eikonal) from the source to
the point with the coordinates (*x*, *y*, *z*), and *n* is the slowness at
that point (the velocity *v* equals 1/*n*.) In Appendix A, I review a
basic derivation of the eikonal and transport equations. To formulate
a well-posed initial-value problem on equation (1), it is
sufficient to specify at some closed surface and to choose one
of the two branches of the solution (the wave going from or to the
source.)
Equation (1) is nonlinear. The nonlinearity is essential for
producing multiple branches of the solution. Multi-valued eikonal
solutions can include different types of waves (direct, reflected,
diffracted, head, etc.) as well as different branches of caustics. To
linearize equation (1), we need to assume that an initial
estimate of the eikonal is available. The traveltime
corresponds to some slowness *n*_{0}, which can be computed
from equation (1) as

| |
(2) |

Let us denote the residual traveltime by and
the residual slowness *n* - *n*_{0} by *n*_{1}. With these definitions, we
can rewrite equation (1) in the form
| |
(3) |

or, taking into account equality (2),
| |
(4) |

Neglecting the squared terms, we arrive at the equation
| |
(5) |

which is the linearized version of the eikonal equation (1).
The accuracy of the linearization depends on the relative ratio of the
slowness perturbation *n*_{1} and the true slowness model *n*. Though it
is difficult to give a quantitative estimate, the ratio of 10% is
generally assumed to be a safe upper bound.
The intimate connection of the linearized eikonal equation
and traveltime tomography is discussed in Appendix B.

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Stanford Exploration Project

11/11/1997