Traveltime computation with the linearized eikonal equation |

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

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 , which can be computed
from equation (1) as

or, taking into account equality (2),

Neglecting the squared terms, we arrive at the equation

which is the linearized version of the eikonal equation (1). The accuracy of the linearization depends on the relative ratio of the slowness perturbation and the true slowness model . 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.

Traveltime computation with the linearized eikonal equation |

2013-03-03