Acoustic Staggered Grid Modeling in IWAVE

All FD Schemes are First Order in Heterogeneous Media

The bulk modulus and buoyancy models depicted in Figures 5 and 6 embed an anticline or dome in an otherwise undisturbed package of layers. 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. The IWAVE utility $sfstandardmodel$ (in the Madagascar bin directory) builds this example and a number of others that can be sampled arbitrarily for grid refinement studies. See its self-doc for usage instructions.

Symes and Vdovina (2009) use the model depicted in Figures 5 and 6 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 7 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 (more than 20 gridpoints per wavelength at the wavelength corresponding to the highest frequency, 12 Hz, with significant energy, and the smallest $v_p=1.5$ km/s) 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 8 and 9 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 10 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 11, 12) 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 is 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.

Note that the finest (2.5 m) grid consists of roughly 10 million gridpoints. Consequently the modeling runs collectively take a considerable time, from a minutes to a substantial fraction of an hour depending on platform, on a single thread. This example is computationally large enough that parallelism via domain decomposition is worthwhile. IWAVE is designed from the ground up to support parallel computation; a companion report will demonstrate parallel use of IWAVE.

Acoustic Staggered Grid Modeling in IWAVE