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Examples based on a 2D dome model

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. These figures display sampled versions of the models with $\Delta x = \Delta z =$ 5 m; the model fields are actually given analytically, and can be sampled at any spatial rate.

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. Figure 3 exhibits a shot gather, computed with a (2,4) (= 2nd order in time, 4th order in space) staggered grid scheme, $\Delta x = \Delta z =$ 5 m and an appropriate near-optimal time step, acquisition geometry as described in caption. The same gather computed at different spatial sample rates seem identical, at first glance, however in fact the sample rate has a considerable effect. Figures 4 and 5 compare traces computed from models sampled at four different spatial rates (20 m to 2.5 m), with proportional time steps. The scheme used is formally 2nd order convergent like the original 2nd order scheme suggested by Virieux (1984), but has better dispersion suppression due to the use of 4th order spatial derivative approximation. Nonetheless, the figures clearly show the first order error, in the form of a grid-dependent time shift, predicted by Brown (1984).

Generally, even higher order approximation of spatial derivatives yields less dispersive propagation error, which dominates the finite difference error for smoothly varying material models. For discontinuous models, the dispersive component of error is still improved by use of a higher order spatial derivative approximation, but the first order interface error eventually dominates as the grids are refined. Figure 6 shows the same shot gather as displayed earlier, with the same spatial and temporal sampling and acquisition geometry, but computed via the (2,8) (8th order in space) scheme. The two gather figures are difficult to disinguish. The trace details (Figures 7, 8) show clearly that while the coarse grid simulation is more accurate than the (2,4) result, but the convergence rate stalls out to 1st order as the grid as refined, and for fine grids the (2,4) and (2,8) schemes produce very similar results: dispersion error has been suppressed, and what remains is due to the presence of model discontinuities.

See (Symes and Vdovina, 2009) for more examples, analysis, and discussion, also (Fehler and Keliher, 2011) for an account of consequences for quality control in large-scale simulation.


next up previous [pdf]

Next: Creating the examples - Up: Using IWAVE Previous: Acoustodynamics

2012-10-17