Ray tracing in random media

Plane-wave ray tracing has been performed through 2-D random media with Gaussian and exponential autocorrelation functions of the slowness perturbations. The standard deviation is ε, correlation length a and propagation distance L; computations have been performed for ε=1 and 3 per cent and for L/a=2.5 to 60. Up to L/a=5 or 10, regular focusing and defocusing of the rays is observed. Then, irregular behaviour develops, apparently without foci, but with increasing deviations q from the straight rays of a homogeneous medium. Two neighbouring rays at the top of a rectangular cross-section usually arrive far from each other at the bottom, and rays can even turn. The standard deviation σ of q, normalized by L, is approximately σ/L=ε(L/a)
^1/2
.From the traveltimes of the rays at the bottom of the cross-section, the first arrivals were determined and compared with the first arrivals according to the Huygens method of Podvin & Lecomte (1991). Both ray tracing and the Huygens method are high-frequency methods, but their traveltimes do not always agree. The reason is that the Huygens method gives first arrivals including diffractions around the slow parts of the structure, whereas ray tracing, in the sense of initial-value ray tracing, gives only transmitted rays. As a consequence, traveltimes calculated by ray tracing are systematically late with respect to the Huygens-method traveltimes. The difference becomes significant for L/a greater than 10 to 20, it increases with ε, and it is more pronounced for exponential than for Gaussian media. For instance, the velocity shift (with respect to the inverse of the volume average of the slowness) for an exponential medium, L/a=60 and ε=3 per cent, is −0.5 per cent according to ray tracing, whereas it is + 1.3 per cent according to the Huygens method. For ε=1 per cent the corresponding numbers are 0 and +0.3 per cent, i.e. the difference between ray tracing and the Huygens method is still significant.A conclusion from our calculations is that, even in the mildly laterally heterogeneous earth models of seismic tomography, ray tracing may give traveltimes that are significantly biased, i.e. overestimated. Another result is that, in random media, ray-tracing first-arrival traveltimes often have much stronger short-scale spatial variations than Huygens-method traveltimes which include wavefront healing due to diffractions. Both effects, traveltime bias and short-scale variations, limit the applicability of ray tracing already for weak random media.