Next: ENO and WENO for Up: Qian & Symes: Adaptive Previous: Paraxial eikonal equation

The advection equation for take-off angle

The amplitude satisfies the zeroth order transport equation Cervený et al. (1977):
 (3)

If the traveltime field has been found by solving the eikonal equation, then equation (3) is a first order advection equation. However, we see that the Laplacian of traveltime field is involved in the transport equation, which implies that we need a third order accurate traveltime field to get a first order accurate amplitude field El-Mageed et al. (1997); El-Mageed (1996); Symes (1995). To avoid this complexity, we use another approach to compute the amplitude.

In 2D isotropic media, the amplitude satisfies Cervený et al. (1977); Friedlander (1958) where J(x,z;xs,zs) is the Jacobian of the transformation from Cartesian coordinates (x,z) to ray coordinates , is the traveltime, is the take-off angle from source point (xs,zs) to a general point (x,z) in the subsurface: where and are the gradients of take-off angle and traveltime, respectively.

Since the take-off angle is constant along any ray,
 (4)
That is, the gradient is the wavefront normal which is tangential to the ray; the gradient is tangential to the wavefront.

However, the gradient of the take-off angle depends on the second order derivative of traveltime, so that we need third order accurate traveltimes to get a first order accurate gradient of take-off angle. Zhang 1993 used this equation in polar coordinates to compute the geometrical spreading factor, but his computation of the take-off angle was based on the first order traveltime field. Consequently, the gradient of take-off angle computed by his scheme was inaccurate. Vidale 1990 encountered a similar difficulty.

Next: ENO and WENO for Up: Qian & Symes: Adaptive Previous: Paraxial eikonal equation
Stanford Exploration Project
4/20/1999