Time-dispersion correction¶
Deepwave’s propagators step through time with an explicit, second-order-accurate leapfrog scheme. Being only second order in time, it introduces temporal dispersion: each temporal frequency \(\omega\) propagates as if it were the slightly lower \((2/\Delta t)\sin(\omega\Delta t/2)\). Raising the spatial accuracy order does nothing to this error — it depends only on the time step. The Koene–Robertsson remappings in deepwave.time_dispersion can reduce temporal-dispersion error when using a larger dt. Finite-record correction is approximate and the result depends on the retained band and windowing. See Time-dispersion correction for the derivation and transform directions.
The recipe is: pre-transform the source wavelet with deepwave.time_dispersion.correct_source() before modelling, and post-transform the recorded data with deepwave.time_dispersion.correct_receivers() afterwards. Both take the propagator’s internal time step internal_dt = dt / step_ratio (the step that governs the dispersion). Choosing a CFL-stable dt makes the propagator run with step_ratio == 1, so internal_dt == dt; deepwave.time_dispersion.cfl_internal_dt() returns it from the same inputs the propagator uses:
internal_dt = cfl_internal_dt(grid, dt_coarse, v0)
wavelet_corrected = correct_source(wavelet_coarse, internal_dt)
rec_corrected = correct_receivers(
run(dt_coarse, wavelet_corrected), internal_dt
)
We model the same homogeneous 2-D scalar problem at a coarse time step (near the CFL limit) and at a six-times-finer reference time step. Using the same continuous source wavelet for both (band-limited-resampled to the fine grid), the only difference between the runs is the time step, so the discrepancy is temporal dispersion. Without correction the coarse-dt trace differs from the fine-dt reference by a large relative L2 error; applying both corrections brings it into closer agreement (a more than twenty-fold error reduction in this example; this case-specific result is not a general accuracy bound):
relative L2 error, uncorrected: 0.5453
relative L2 error, corrected: 0.0226
improvement factor: 24.1x
