(A.18) |

or compactly, in vector notation:

(A.20) |

where

It follows then that

where

Let us notice now that

so the perturbation in travel time becomes

According to Fermat's principle , which allows to infer the following system of equations

(A.21) |

The Hamiltonian can equally be rewritten in order to proceed with an integration along travel time,
and by substituting the components of sound slowness with the components of the wavenumber.
In fact, by taking into account that

and that

one can obtain the Hamiltonian[17]

(A.22) |

As shown by the two-dimensional case with cylindrical symmetry follows automatically from this case as

There are also alternativa approaches,
which consider a Hamiltonian written in terms of or .
For the first case and for two-dimensional case with cylindrical symmetry one can obtain the Hamiltonian

(A.23) |

as for the second case the Hamiltonian and associated system of equations correspond to[18]

(A.24) |

Orlando Camargo Rodríguez 2012-06-21