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The dome model - demo1

This simple 2D model embeds an anticline or dome in an otherwise undisturbed package of layers. The velocity and density models are depicted in Figures 1 and 2.

Symes and Vdovina (2009) use this model to illustrate the interface error phenomenon: the tendency, first reported by Brown (1984), of all finite difference schemes for wave propagation to exhibit first order error, regardless of formal order, for models with material parameter discontinuities. The shot record (Figure 3, acquisition geometry described in caption) looks perfectly normal. However the spatial sample rate of the model in Figures 1 and 2 has a considerable effect. The material parameter fields are constructed as functions of position in 2D space, hence can be sampled at any rate at all. Figures 4 and 5 compare traces computed from models sampled at four different rates. The scheme used is the 2nd order in time, 4th order in space staggered grid scheme, which is formally 2nd order convergent like the original 2nd order scheme suggested by Virieux (1984), but has better dispersion suppression. Nonetheless, the figures clearly show the first order error, in the form of a grid-dependent time shift, predicted by Brown (1984). See (Symes and Vdovina, 2009) for more examples, analysis, and discussion.

vp0
vp0
Figure 1.
Dome velocity model
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dn0
dn0
Figure 2.
Dome density model
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data0
data0
Figure 3.
2D shot record, 301 traces: shot x = 3300 m, shot z = 40 m, receiver x = 100 - 6100 m, receiver z = 20 m, number of time samples = 1501, time sample interval = 2 ms. Source pulse = zero phase trapezoidal [0.0, 2.4, 15.0, 20.0] Hz bandpass filter.
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trace
trace
Figure 4.
Trace 100 (receiver x = 2100 m) for $\Delta x = \Delta z = $ 20 m (red), 10 m (green), 5 m (blue), and 2.5 m (black). Note arrival time discrepancy after 1 s: this is the interface error discussed in (Symes and Vdovina, 2009). Except for the 20 m result, grid dispersion error is minimal.
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wtrace
wtrace
Figure 5.
Trace 100 detail, 1.8-2.5 s, showing more clearly the first-order interface error: the time shift between computed events and the truth (the 2.5 m result, more or less) is proportional to $\Delta t$, or equivalently to $\Delta z$.
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Inspection of the SConstruct file in demo1 will show that the modeling tool used is asg, the IWAVE acoustic modeling command. This command reads its parameters from a par file. Four par files are present in demo1, each one defining a modeling job, corresponding to a given level of grid refinement. The meaning of each parameter in the par file is described in the IWAVE web documentation: http://www.trip.caam.rice.edu/software/iwave/doc/html/index.html.


next up previous [pdf]

Next: Acknowledgements Up: IWAVE Demonstration Package - Previous: Acoustodynamics

2012-06-07