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 (q1, q2, q3) and ray parameter coordinate system (, , ).
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 q1, q2, q3 connected with the ray are defined in the following way: q3 corresponds to the traveltime along the ray . q1 and q2 form a 2D orthogonal Cartesian coordinate system in the plane perpendicular to , as shown in Figure 2.
Figure 2 The ray-centered coordinate system. q3 is tangent to the raypath. q1 and q2 forms a plane perpendicular to q3.
The physical meaning of the ray parameter coordinate system is very obvious. and are two ray parameters in the x1 and x2 direction, which are constant along the whole ray and variable from one ray to another. Their definitions are as follows
Dynamic ray tracing system can be expressed as
where Q, P, V are matrices.
The definition of each component is
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 q1, q2, P is a transformation matrix from the ray parameters , to the slowness vector component in the ray-centered coordinate system. Physically, matrix Q measures the derivations of paraxial rays from ray and is also referred to as the matrix of geometrical spreading. Matrix P has no obvious physical meaning. But we can define a matrix M of second derivatives of the traveltime field with respect to the ray-centered coordinate q1, q2 by the relationship
In order to fit the requirement of Runge-Kutta package, equation (8) is expressed as
The initial condition is chosen as
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
Sub-index denotes a value taken at the source. C is a factor dependent upon the source, and is constant along the ray.