Phase-rayfields

A monochromatic acoustic wavefield, ${\cal U}$, at frequency, $\omega$, and spatial location, ${\bf x}$, may be represented by,

$${\cal U}( {\bf x}, \omega) = A( {\bf x}, \omega) \, {\rm e}^{ {\rm i}\phi( {\bf x}, \omega) },$$ (1)

where $A({\bf x},\omega)$ and $\phi({\bf x},w)$ are the amplitude and phase functions, respectively. For monochromatic waves propagating through isotropic media, the gradient of phase function, $\nabla \phi({\bf x},w)$, represents the instantaneous direction of energy transport and is a characteristic to the solution of the governing Helmholtz equation Foreman (1989). Analogous to the ray precept in broadband theory, this vector quantity defines the instantaneous direction and magnitude of one ray in a continuous ray manifold. However, to differentiate between the broadband and monochromatic ray representations, we term the latter quantity phase-rays Shragge and Biondi (2003). The governing differential equations for a phase-ray, r_i, are presented in the Exact-ray formulation of Foreman (1989). In Cartesian coordinates, the subscript i on r refers to the projection of the ray along the x and z axes - r_x and r_z, respectively. The phase-ray equations, in summation notation, are,  

$$\frac{ {\rm d}r_i}{{\rm d}s} = \frac{ \partial\phi}{\partial... ...{ \partial \phi}{\partial x_k}\right) \right]^ {-\frac{1}{2}},$$ (2)

where $\phi$ is the above phase function, x_i is a coordinate of the underlying Cartesian grid, and the repeated index k here (and throughout the paper) represents a summation over all coordinate indices. Scalar step magnitude, ${\rm d}s$, is given by,  

$${\rm d}s({\bf x}) = v({\bf x}) \, {\rm d}\t,$$ (3)

where $v({\bf x})$ is the velocity in the neighborhood of ray, $r_i({\bf x})$,and ${\rm d}\t$ is an element of time along the ray.

Calculating phase-rays thus requires isolating the gradient of the monochromatic phase function. An efficient procedure is to calculate the ratio of the wavefield gradient to the wavefield itself,  

$$\frac{\nabla {\cal U}}{{\cal U}} = \frac{\nabla A}{A} + {\rm i}{\nabla \phi},$$ (4)

which eliminates the oscillatory nature of the wavefield. Taking the imaginary component of equation (4),  

$$\nabla \phi = \Im \left( \frac{\nabla {\cal U}}{{\cal U}} \right),$$ (5)

yields the required phase gradient. The right hand side of equation (5) is calculable only when a wavefield solution is known. The solution for a ray, r_i, is computed through integrating the right hand sides of equations (2) using a one-sided, non-stiff integration method (e.g. Simpson's 1/3 rule). Interestingly, ray solutions are uniquely determined given an initial starting position by reason that equations (2) form a decoupled system of differential equations of first-order. Accordingly, a phase-ray coordnate system is uniquely defined by specifying of a set of initial coordinate points and a frequency, $\omega$. Note that this specification makes the coordinate system frequency dependent. Additional information on the theory of phase-rays is discussed in both Shragge and Biondi (2003) and Foreman (1989).