The slowness in both smooth models without interfaces is interpolated by natural bicubic splines in the regular grid of 122 points vertically times 384 points horizontally. The grid spacing is 0.024km in both directions. The size of the spline grid is thus 2.904km vertically and 9.192km horizontally. The grid velocities in both models, in m/s, are available at WWW address http://www-rocq.inria.fr/~benamou/testproblem.html . The grid velocities are stored with the inner loop from the top to the bottom and the outer loop from the left to the right.
The first-arrival travel times are, as a rule, calculated
on regular rectangular grids of points,
using the discretized versions of seismic models.
For the calculations presented here, the accurate second-order
method by
The ray-theory travel times at given receivers in the "smooth" model were calculated by the shooting method. Even the "smooth" model is not sufficiently smooth for the shooting method: the geometrical spreading, especially at the leftmost part of the receiver profile, is so great that there is no take-off angle available within the single precision real numbers to shoot the ray to that region, see the travel time curves below. Thus, the shooting method cannot catch all the two-point rays in this model. It may miss about 20% two-point rays. On the other hand, if the two-point ray is found, the calculated ray-theory travel time is probably very accurate.
For more details refer to the manuscript by
Note that we have supplemented the receiver names (integer indices in apostrophes) with the receiver coordinates in order to reliably distinguish the applied receiver indexing from indexing '000' to '383'.
You may download the resulting first-arrival travel times marsf.dat in the "smooth" INRIA Marmousi model.
You may download the ray-theory travel times marsr.dat , found by the shooting method in the "smooth" INRIA Marmousi model.
You may download the resulting first-arrival travel times marhf.dat in the "hard" INRIA Marmousi model.
The ray-theory travel time curve in the "hard" model is so complex that the two-point ray tracing makes no sense, see the figures below.
top travel time=2.6 s
bottom travel time=0.9 s
The two-point ray-theory travel times (red) plotted over
the first-arrival travel times (blue)
in the "smooth" INRIA bench-mark version of the 2-D Marmousi model.
The first-arrival travel times at receivers where the shooting method
failed remain blue.
top travel time=5.2 s
bottom travel time=0.9 s
136818 initial-value ray-theory travel times (red) plotted over
the first-arrival travel times (blue)
in the "hard" INRIA bench-mark version of the 2-D Marmousi model.
136818 of 244302 initial-value rays,
shot with the angular increment of 0.000010 radians,
hit the receiver profile.
top reduced travel time=0.50 s
bottom reduced travel time=0.07 s
The first-arrival travel times in the "smooth" (upper curve)
and "hard" (lower curve)
INRIA bench-mark versions of the 2-D Marmousi model.
top reduced travel time=0.50 s
bottom reduced travel time=0.07 s
The two-point ray-theory travel times (red) plotted over
the first-arrival travel times (blue)
in the "smooth" INRIA bench-mark version of the 2-D Marmousi model.
The first-arrival travel times at receivers where the shooting method
failed remain blue.
top reduced travel time=0.50 s
bottom reduced travel time=0.07 s
136818 initial-value ray-theory travel times (red) plotted over
the first-arrival travel times (blue)
in the "hard" INRIA bench-mark version of the 2-D Marmousi model.
Reduced travel times over 0.5 s are clipped off.
The research has been supported by the Grant Agency of the Czech Republic under Contract 205/95/1465, and by the members of the consortium "Seismic Waves in Complex 3-D Structures".
Summary of the manuscript L. Klimes: Travel times in the INRIA Marmousi models, Seismic Waves in Complex 3-D Structures, Report 4, pp. 53-60, Dep. Geophys., Charles Univ., Prague 1996.
Abstract of the contribution L. Klimes: Traveltimes and their computation presented at the workshop "Computation of multi-valued traveltimes" held in INRIA Rocquencourt, France, on September 16-18, 1996.