Toward a source spectral model for ETS

1. Toward a source spectral model for ETS Heidi Houston University of Washington

2. Constructing spectral source models for ETS • Important tool for understanding physics and scaling - e.g., -2 models for earthquakes • Serve as reference to compare • - variability of ETS’s • - short- and long-period character of ETS • - different sizes of events - ETS, SSE, VLF, etc

3. Indications that high frequency spectrum falls off as f-1 Spectra of 20-min intervals on Sept. 10 2005 noise • Displacement spectra - • strong tremor day • Spectral slope near f-1 • Instrument removed 1016 tremor moment Moment-rate spectrum (N-m) slope f-1 1014 low -> high tremor moment 1012 10-1 1 10 Frequency (Hz)

4. Spectra of different duration tremor events on 9/8/2005 • Pieces of tremor 40 s to 1 hr • Spectral slopes near f-1

5. Ide et al.’s proposed scaling law for duration vs moment: duration ~ M01 • They used • LFE - low frequency eqs • VLF - very low frequency eqs • SSE - slow slip events • ETS - episodic tremor and slip • Silent earthquakes • Contrast with regular earthquakes (Houston, 2001) • duration ~ M01/3 Log duration Cascadian LFEs Log M0 Ide, Beroza, Shelly, Uchida, Nature, 2007

6. Assume that high-frequency spectral amplitudes ~ duration1/2 • Tremor signal incoherent at high frequency • High-frequency amplitude spectrum ~ duration1/2 ~ M01/2 (from Ide et al. 2007) • => faster increase w/M0 than for earthquakes which have h.f. amplitude spectrum ~ M01/3

7. 10^19 10^16 Moment rate spectrum (Nm/s/Hz) 10^13 0.01 1 10 Frequency (Hz) ETS spectral model regular w-2 model • Enforce high-frequency ampl. ~ M01/2 • then c ~ M0-1/2 • If • c = c vR (/M0)1/3 • must have vR1/3 ~ M0-1/6 • h.f. amplitude increases faster than for earthquakes

8. For -1 model with h.f. spect. amp. ~ M01/2 • need M() = M0c /(c+) for spectral slope • needc = c vR (/M0)1/3 to get units right • impliesc ~ M0-1/2 • Then vR1/3 ~ M0-1/6 • If  constant, vR ~ M0-1/6 • If vR constant,  ~ M0-1/2

9. Start to compare data to model • GPS moment • Get moment rate-spectrum at 1-10 Hz from tremor • Assumption: tremor consists mainly of direct S waves • Levels calibrated empirically by small earthquakes

10. Estimate band-limited moment-rate from tremor • displacement at station => tremor moment-rate • assume signal consists mostly of direct far-field S-waves • PA array in Olympic Mountains, almost directly over region of high slip • assume average radiation pattern • or assume shear slip in subduction direction • assume Q (~200), beta, distance to source

11. Empirical calibration - compare network moment to band-limited ‘tremor moment’ • 4 small quakes • M1.5 - M 2.9 • M2.9 on 20050913 • 60 km from array • Applying processing and assumptions (e.g., direct S-waves) obtain M4.0 • factor of ~40 overestimation of moment • probably due to reverberations at the site • Adjust band-limited moment estimate accordingly

12. Scaling of high-frequency spectral amplitudes • Tremor signal incoherent at high frequency • High-frequency amp spectrum ~ duration1/2 ~ M01/2 • High-frequency amplitude spectrum of N days ~ N1/2 amplitude spectrum of 1 day

13. 10^19 10^16 Moment rate spectrum (Nm/s/Hz) 10^13 0.01 1 10 Frequency (Hz) GPS moment • Estimates for 2005 Cascadia ETS • Spectral amplitudes corrected for 14-day duration • Tremor spectral amps corrected w/empirical calibration VLFs from Japan Tremor from Cascadia 0.0000001

14. 10^19 10^16 Moment rate spectrum (Nm/s/Hz) 10^13 0.01 1 10 Frequency (Hz) ETS spectral model GPS moment • 2005 ETS => • vR ~ 0.1 m/s •  ~ 0.1MPa • still underpredicts data • spectral bumps • shorter duration process (tides?) • calibration preliminary • gather data • compare data VLFs from Japan Tremor from Cascadia 0.0000001

15. 10^19 10^16 Moment rate spectrum (Nm/s/Hz) 10^13 0.01 1 10 Frequency (Hz) Possibility that real spectrum is more complex GPS moment VLFs Tremor 0.0000001

16. Summary • Constructing spectral source models for ETS • understand physics • for reference in comparison • Underway: data comparison - spectral models provide reference

17. Useful things More studies should provide info about tremor amplitude Test assumptions of f-1 falloff M0 ~ duration1 Collect data on vR vs M0  vs M0

18. Tremor Earthquake seconds

19. Strong tremor (black) and LFEs (red) at BS01 2-8 Hz Counts LFEs much smaller than tremor M1.7 earthquake Time (s) Amplitudes not same

20. VLFs:Very low frequency events Not common Occur during some, not all, tremor events 30 minute record red traces filtered at 0.02 to 0.05 Hz black traces filtered at 2 to 8 Hz VLF Ito et al., Science, 2007

21. Tremor, VLF’s, and tilt migrate together in space and time Ito et al., Science 2006

22. VLFtime functionsfrom Ide et al. • onset and termination not abrupt Ide et al., GRL, 2008

23. M0 - durationX relation for VLFs • Slope of 1/3 to 1/2, not 1 • Is M0 ~ duration1 really true? Ide et al., GRL 2008

24. Menagerie of slow events • LFE - low frequency eqs ~.3 s, M1 • VLF - very low frequency eqs ~50 s, M4 • SSE - slow slip events ~4 days, M6 • ETS - episodic tremor and slip ~14 days, M6.6 • Silent earthquakes ~1 yr, M7 • All shear slip*, longer duration, less seismic energy • Ide et al.’s philosophy: lump all together, seek scaling relation tilt GPS GPS *a bit controversial

25. Moment-duration scaling for earthquakes:duration ~ M01/3 • M0 =  D L W • Quasi-constant aspect ratio => L ~ W • Quasi-constant stress drop => D ~ W • So M0 ~ L3 ~ (vRT)3 ~ T3 • T~ M01/3 Constant rupture velocity vR

26. Estimate spectral amp for 14 days of tremor Assume N days of tremor have N times energy as 1 day so spectral amp ~ N1/2 Spectra of 20-min intervals on 20050910 noise 1016 tremor moment Moment-rate spectrum (N-m) slope f-1 1014 low -> high tremor moment 1012 10-1 1 10 Frequency (Hz)

27. Spectra of different duration tremor events on 9/8/2005 Pieces of tremor 40 s to 1 hr Large variation in spectral ampl.

28. Suggested time functions of slow slip processes • Duration ~ moment1 • implies time function’s mean amplitude doesn’t grow with moment or time • Source time functions like boxcars

29. Suggested time functions of slow slip processes • Duration ~ moment1 • implies time function’s mean amplitude doesn’t grow with moment or time • Source time functions like boxcars • High frequencies generated only at beginning and end • would allow small and large events to have same high-frequency content

30. 33s 10s 3s 1s 100s 3s 0.1s Ide et al. proposed f-1 model • Solid: f-2 model for reference • Dashed: f-1 model Ide et al., Nature, 2007

31. 10^19 10^16 Moment rate spectrum (Nm/s/Hz) 10^13 0.01 1 10 Frequency (Hz) Similar smooth w-1 spectral model . • standard w-2 model • M() = • M0c2/(c2 + 2) • corner freq • c = c b (Dt/M0)1/3 • w-1 model • M() = • M0c1/(c1 + 1) • c = 2/T • Goal: compare model, high- and low frequency data . 0.0000001

32. 33s Can this “slow slip” source model be correct? 10s 3s • Only for boxcar time functions • Not for realistic t.f.s • 1 sec LFE does not have same spectral amplitude at high frequencies as slow-slip event lasting days • Real onsets and terminations not sufficiently abrupt 1s 100s 3s 0.1s Ide et al., Nature, 2007

33. duration ~ M01 & f-1 spectrum => boxcar t.f.s • implies “events” have abrupt onset and termination so that all/most high frequencies generated then • consider onsets and terminations of actual VLFs, tremor, and ETS • Not abrupt.

34. Further implication of model: LFEs have same amplitude (moment-rate) as VLFs or ETS tremor • Not realistic.

35. 24 hours of tremor from Sept 2005 20050911 • Stack of 12 envelopes of • horizontal displacement • 6 array stations • 1-8 Hz • Gradual onsets and terminations

36. 3 Cascadia ETS show gradual onsets • July 2004 Sequim array • Sept 2005 Port Angeles array • January 2007 Price Lake array - near initiation of ETS 15 days Creager and colleagues

37. Tremor “events” Sept 8, 2005 24 hours

38. 33s Other implications of f-1 spectral model 10s 3s • energy/moment ratio varies greatly with moment • does not apply above some frequency • need falloff faster than f-1.5 to avert energy catastrophe • implies a smallest LFE because above some f spectral falloff must exceed 1.5 1s 100s 3s 0.1s Ide et al., Nature, 2007

39. 24 hours of tremor from Sept 2005 ETS 20050911 • Stack of 12 envelopes of • horizontal displacement • 6 array stations • 1-8 Hz

40. VLFtime functionsfrom Ide et al. • 2-3 SSEs / yr • Estimate total moment found in active day • VLF moment in each SSE • ~1-5 e15 Nm Ide et al., GRL, 2008

41. Possible pitfalls • Lumping together and comparing LFEs, VLFs, SSEs, ETSs, silent earthquakes, afterslip • amounts to the conflation of an event with a series ofevents • like comparing a mainshock to the subevents in it • Moment ~ duration1 may not hold

42. What spectral model could work for “slow-slip processes”? 1 to 10 Hz tremor has spectral fall-off near -1 Is there a simple f-1 source model that fits tremor, VLFs and ETS? In what sense can these slow-slip processes be self-similar? e.g. LFE, VLF, SSE, ETS?

43. Thank you!

44. Are stations in near- or far-field of tremor radiation? • Pujol (eq 9.5.20) implies if /r >> 1 near-field terms dominate •  = 2  c/ =c / f • c could be P or S wave velocity 3 - 6 km/s • f could be 1 to 8 Hz • r ranges from 25 to 80 km • Therefore, /r at most about 6/25 ~ .25 • => near-field terms not important • However, at periods > 3 s, near-field could become important!

45. Comparison of original (green) and alternate (cyan) spectral models for slow slip

46. 10^19 10^16 Moment rate spectrum (Nm/s/Hz) 10^13 0.01 1 10 Frequency (Hz) Alternate end-member spectral model • Enforce high-frequency ampl. ~ M01/2 • then c ~ M0-2 • OK fit leads to stress drop ~3000Pa c ~ c (/M0)1/2 • compared to 40,000Pa for entire ETS • need more data

47. 10^19 10^16 Moment rate spectrum (Nm/s/Hz) 10^13 0.01 1 10 Frequency (Hz) 2005 Cascadia ETS • Need larger corner freq, shorter time • Tides modulate tremor in Cascadia and Japan (Rubenstein et al, 2007; Nakada etal, 2008) • Dashed green line with c appropriate for 12 hours duration 0.0000001

48. high-frequency spectral behavior • Regular earthquakes w/ -2 model • M0 = DLW & quasi-const vR=> dur ~ M01/3 • h.f. spectral amp ~ dur ~ M01/3 • Slow-slip phenomena • M0 ~ dur (really??) • h.f. spectral amp ~ dur1/2 ~ M01/2

49. LFE (red), VLF (orange), and SSE (green) occur in the Nankai trough while ETS (light blue) occur in the Cascadia subduction zone. These follow a scaling relation of M0 ~t, for slow earthquakes. Purple circles are silent earthquakes. Black symbols are slow events listed in the bottom half of Table 1. a, Slow slip in Italy23, 24, representing a typical event (circle) and proposed scaling (line). b, VLF earthquakes in the accretionary prism of the Nankai trough26. c, Slow slip and creep in the San Andreas Fault21, 22. d, Slow slip beneath Kilauea volcano25. e, Afterslip of the 1992 Sanriku earthquake27. Typical scaling relation for shallow interplate earthquakes is also shown by a thick blue line.

50. Slow events on/near plate interface below locked zone Ito et al., Science, 2007