Many finite difference methods have been introduced to compute the traveltime for isotropic media directly on a regular grid El-Mageed et al. (1997); El-Mageed (1996); Fomel (1997); Popovici and Sethian (1997); Qin et al. (1992); Reshef and Kosloff (1986); Schneider et al. (1992); Schneider (1995); Vidale (1988); van Trier and Symes (1991).
The traveltime field is mostly smooth, and the use of upwind differencing (in all of the cited methods) confines the errors due to singularities which develop away from the source point. The source point itself is, however, also a singularity. The truncation error of a pth order method is dominated by the product of (p+1)st derivatives of the time field and the (p+1)st power of the step(s). The (p+1)st derivatives of the time field, however, go like the (-p+1)th power of the distance to the source. Therefore, near the source -- when the distance is on the order of the step -- the truncation error is quadratic in the step, i.e., first order. This inaccuracy spreads throughout the computation, and renders all higher order methods first-order convergent unless the scheme is modified near the source. The issue is not academic: the first-order error is sizeable, as we shall show. Moreover, it prevents reliable computation of auxiliary quantities such as takeoff angle and amplitude.
In this paper, we show how to use adaptive gridding concepts commonplace in the numerical solution of ordinary differential equations to resolve this difficulty. This work refines and extends the method introduced in Belfi and Symes 1998. The efficiencies achieved by adaptive gridding are considerable -- usually more than an order of magnitude gain in computation time for problems of typical exploration size. We also obtain dramatic improvements in the accuracy of takeoff angle computations and, therefore, for other geometrical acoustics quantities as well.