## Travel times in the INRIA Marmousi models

Ludek Klimes

### INRIA bench-mark versions of the Marmousi model

Two smooth continuous bench-mark 2-D models without structural interfaces, based on the Marmousi model, have been prepared for the workshop "Computation of multi-valued traveltimes" held in INRIA Rocquencourt, France, on September 16-18, 1996. The "hard" model is globally quite rough and is smooth on the scales of the order of several metres. Let us point out once again that the "hard" model is a continuous model without interfaces. The "smooth" model is a considerably smoothed version of the "hard" model. See the figures of the models (24 kB).

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 m/s are stored with the inner loop from the top to the bottom and the outer loop from the left to the right.

### Source

The point source is situated at horizontal distance 6.000 km from the left-hand side of each 2-D model of dimensions 9.192 km times 2.904 km, at the depth of 2.800 km.

### Receivers

The receivers are located along the top of the model (depth 0 km), with spacing of 24 metres:
Receiver '001' is situated at horizontal distance 0.000 km from the left-hand side, '002' at 0.024 km, '003' at 0.048 km, ... '384' at 9.192 km.

### Methods

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 Klimes (1995) has been applied to the grid 9 times denser in both directions (spacing 24/9 m) than the given B-spline grid.

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 Klimes (1996) .

### Resulting travel times in the INRIA Marmousi models

The resulting travel times in the INRIA Marmousi models are stored in the ASCII files.
The header of each file consists of several strings in apostrophes terminated by a slash.
The body of each file consists of lines corresponding to the individual travel time, sorted (1) according to the receivers, (2) according to the increasing travel times at each receiver. Each line consits of:
'receiver index' , horizontal coordinate in km , depth in km = 0 km , remaining coordinate = 0 km , travel time in s , /
Each file is terminated by a slash.

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.

### Travel-time curves in the INRIA Marmousi models

top travel time=2.6 s

bottom travel time=0.9 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 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.

### Travel-time curves in the INRIA Marmousi models reduced to velocity 3.2 km/s

The above travel time curves are reduced here to velocity 3.2 km/s,
[reduced travel time] = [travel time] - [hypocentral distance] / [3.2 km/s],
and are displayed in 10 times enlarged scale.

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.

### Acknowledgements

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".

### Related:

Summary of the paper L. Klimes (1996): Grid travel-time tracing: second-order method for the first arrivals in smooth media. PAGEOPH, 148, 539-563.

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.

SW3D - main page of consortium Seismic Waves in Complex 3-D Structures .