Dynamic ray tracing will provide us dynamic
information like amplitude and phase-shift, not only kinematic information
like raypaths and traveltimes. More specifically, we can estimate the
traveltime, amplitude, and phase-shift in the vicinity of the central ray.
In order to do this, we need to introduce another two coordinate systems, i.e.,
ray-centered coordinate system (*q _{1}*,

The ray-centered coordinate system is a curvilinear orthogonal
coordinate system. Let us consider an arbitrarily selected ray ,
which is specified by ray parameter , . The ray-centered
coordinates *q _{1}*,

ray-center
The ray-centered coordinate system. Figure 2 q is tangent to the raypath. _{3}q and _{1}q forms a plane perpendicular to _{2}q.
_{3} |

The physical meaning of the ray parameter coordinate system is very obvious.
and are two ray parameters in the *x _{1}* and

(7) |

Dynamic ray tracing system can be expressed as

(8) |

where **Q**, **P**, **V** are
matrices.

(9) |

The definition of each component is

(10) |

where is the ray-centered coordinate. is the ray parameter associated with each ray. is the medium velocity.

Mathematically, **Q ** and **P** can be interpreted as
transformation matrices:
**Q** is a transformation matrix from the ray parameters ,
to the ray-centered coordinate *q _{1}*,

(11) |

(12) |

In order to fit the requirement of Runge-Kutta package, equation (8) is expressed as

(13) |

The initial condition is chosen as

(14) |

Here we set matrix and to be identity matrix. The reason for choosing an identity matrix is to make sure that is a symmetric matrix.

The paraxial ray tracing solution of the pressure field can be written as

(15) |

Sub-index denotes a value taken at the source. *C* is a factor
dependent upon the source, and is constant along the ray.

11/11/1997