Question about half duration, smallest period and numerical noise

[szz326]

Question Short version

When the source frequency (half duration = 0.21 s) is higher than the grid-resolvable smallest period (2.65 s), how can one safely eliminate numerical noise?

Description

I am conducting an earthquake simulation for a point source Mw = 3.8 using a CMTSOLUTION file. Since there are no inversion results available for this event, I estimated the half duration from empirical magnitude–half-duration relationships and set the half duration of this Mw 3.8 earthquake to 0.21 s.

The simulation is 220 km × 180 km. I include a shallow low-velocity zone (Vs_min about 200 m/s). However, due to computational limitations, I cannot refine the shallow mesh sufficiently, so the smallest period of my mesh is about 2.65 s. This is far longer than the source’s dominant period. Someone pointed out that because my source frequency is higher than the smallest period, even if I low-pass filter the final seismograms to 2.65 s, the numerical noise generated by the unresolved high-frequency source cannot be safely removed.

I checked the SPECFEM3D manual, which states:

“Often, it is preferable to run the solver with half duration set to zero and convolve the resulting synthetic seismograms in post-processing after the run, because this way it is easy to use a variety of source-time functions (see Section 15.1). Komatitsch and Tromp [2002a] determined that the noise generated in the simulation by using a step source time function may be safely filtered out afterward based upon a convolution with the desired source-time function and/or low-pass filtering.”

Therefore, I would like to make sure:
If I follow this recommendation—run the simulation with a step source, then convolve the synthetics with the real source time function (half duration = 0.21 s), and finally apply a low-pass filter at 2.65 s—can I safely remove numerical noise?

If this approach is not sufficient, I would also like to ask: in (Komatitsch et al., 2004, BSSA) and (Lee et al., 2008, BSSA), they also simulated small-magnitude earthquakes with relatively coarse meshes. How did they address this issue?

Additionally, does anyone know how to properly filter snapshots? Lee et al. (2008, BSSA) mention filtering snapshots to 1 Hz. The only method I can think of is to output the time series at every spatial grid point, apply a low-pass filter to the smallest period, and then manually reconstruct the snapshots. But this would generate an extremely large volume of data.

My question is quite long — I would greatly appreciate it if someone could spare some time to help. Thank you very much.

SPECFEM3D version

Specfem3d Cartesian

Reference

Komatitsch, D. (2004). Simulations of Ground Motion in the Los Angeles Basin Based upon the Spectral-Element Method. Bulletin of the Seismological Society of America, 94(1), 187–206.
Lee, S.-J., Chen, H.-W., Liu, Q., Komatitsch, D., Huang, B.-S., & Tromp, J. (2008). Three-Dimensional Simulations of Seismic-Wave Propagation in the Taipei Basin with Realistic Topography Based upon the Spectral-Element Method. Bulletin of the Seismological Society of America, 98(1), 253–264.

[jpampuero]

Your interpretation of the manual seems correct. Your source duration being 10 times shorter than the filter cutoff period, it's basically as if you had "half duration set to zero" (a flat source spectrum): your source spectrum is flat within the low-pass filter band.

I did not find specifications of source duration in Komatitsch et al (2004). I guess they did as in the manual, which makes sense.

To get filtered snapshots, I would set a source time function (STF) with similar spectrum as your desired filter. The spectrum of a triangular STF decays as 1/f^2 beyond its corner frequency, similar to the spectrum of a second-order filter, maybe not as steep as you would like. A Ricker STF has an exponentially decaying spectrum, maybe too steep for your purposes. You might want to use as STF the time reponse of your (causal) filter.