High Precision/Fast Adaptive Step Size Ray-Tracing By Curvature Criteria

Here I present one version of Hamilton based ray-tracing equations solution (Cerveny, 1987) (Popov,<br>1996) aiming at increasing precision as well as reducing and controling the computation’s effort. The<br>errors in numerical evaluating the differential ray-tracing equation can be classified in two kinds. First<br>the error in interpolating the velocity field and its derivatives. Second the error in evaluating the next<br>ray’s coordinates and slowness vector by the Runge-Kutta method. The first one has well-known solutions<br>like fourth order precision Bi-cubic spline interpolation (William et al., 1994). The second, the<br>fourth order precision Runge-Kutta method, needs the time step. If we apply a constant time step, it<br>requires the use of a high-density coordinate’s sampling when used for a non-homogeneous medium.<br>The reason is that the precision of ray-tracing is directly related to the curvature radio of the ray and the<br>last with the gradient of the square slowness. Then to obtain a high precision ray-tracing we need to decrease<br>the size of the time step in regions where there are large variations of the square slowness field<br>or on the other hand increase where there are no such variations. The result is an adaptive algorithm,<br>which automatically adapts the size of the ray coordinates interval to the necessity of the medium by<br>geophysical control. Despite the existence of the “adaptive control for Runge-Kutta method” (William<br>et al., 1994) it’s based in numerical considerations. Here I developed an alternative based on solid<br>physical arguments most properly useful for wave propagation purposes which can easily control the<br>accuracy of ray’s coordinates and in consequence the ray’s amplitudes (Lambaré, G. and Lucio, P.S.<br>and Hanyga A., 1996) and time of processing by just changing a parameter.