Amplitudes

Collecting the leading terms of amplitude in the ray series expansion yields the zeroth order transport equation (Cervený et al., 1977):  

$$\nabla\tau\cdot\nabla A+{1 \over 2}\left[\nabla \ln \rho \cdot \nabla \tau+ \nabla^2 \tau \right]A=0.$$ (10)

If we assume that the traveltime function has been found by solving the eikonal equation, then equation (10) is a first order, linear partial differential equation (PDE). We see that equation (10) contains the term $\nabla^2 \tau$that is related to the derivatives of slowness function with respect to the spatial coordinates. Thus, the validity of this equation requires the slowness function to be differentiable. If this requirement is met, we can presumably solve this equation using a finite difference method. An easier approach (Cervený et al., 1977) is to solve the transport equation in ray coordinates and transform the analytical solution of the resulting ODE to Cartesian coordinates, which yields  

$$A(x,z) = R(\gamma(x,z))\sqrt{\rho(x_0,z_0)m(x,z) \over \vert J(x,z)\vert \rho(x,z) m(x_0,z_0)},$$ (11)

where R is the radiation pattern of the source, $\gamma(x,z)$ is the take-off angle of the ray that reaches point (x,z), J(x,z) is the Jacobian of the transformation from Cartesian coordinates to ray coordinates.  

$$J(x,z) = {\pmatrix{\displaystyle{\partial x \over \partial s... ...amma} & \displaystyle{\partial z \over \partial \gamma} \cr}}.$$ (12)

In Appendix A, I show that for a 2-D medium and a line source (the line source is aligned along the third dimension on which the slowness does not depend),  

$$\vert J(x,z)\vert = {1 \over \sqrt{\left({\partial \gamma \o... ...right)^2+ \left({\partial \gamma \over \partial z}\right)^2}}.$$ (13)

Also in Appendix A, I show that function $\gamma(x,z)$ can be computed by solving the first order linear PDE  

$$\tau_x\gamma_x+\tau_z\gamma_z=0.$$ (14)

with the initial condition that $\gamma(x,z)$ is equal to the take-off angle at the source position.

In seismic exploration, point sources may be more appropriate. Cerveny (1981) shows that for point source, function J for line source should be multiplied by a factor $\eta$ to take account the geometric spreading in the third dimension. He showed that  

$$\eta(\hat{x}(s,\gamma),\hat{z}(s,\gamma))= m(x_0,z_0) \int^s_{s_0}{1 \over m(\hat{x}(\xi,\gamma),\hat{z}(\xi,\gamma))}d\xi$$ (15)

where (x₀,z₀) is the coordinate of the source position. I show, in Appendix B, that function $\eta(x,z)$ can be calculated by solving a first order linear PDE:  

$$\tau_x \eta_x+\tau_z \eta_z= m(x_0,z_0),$$ (16)

The initial condition for solving equation (16) is $\eta(x_0,z_0)\rightarrow 0$.The algorithm used for solving the eikonal equation, after some modifications, can be used to solve PDEs (14) and (16). It should be noted that all these formulas are valid only in a regular region in which no caustics exist.