If we build an ETAS model based primarily on information from smaller earthquakes, will it work for forecasting the larger (M≥6.5) potentially damaging earthquakes?

Test the Null Hypothesis that the same stationary ETAS model explains earthquakes of all magnitudes.

1. Fit ETAS parameters to the instrumental catalog (dominated by smaller earthquakes.)

2. Test whether the historical large earthquake catalog is fit with the same ETAS parameters.

a) Verify that the ETAS model removes all clustering, e.g. the transformed times are Poissonian.

b) Compare the triggering intensities of large and small earthquakes.

c) Decluster the historical large earthquake catalog using the ETAS model, and search for any residual spatial-temporal clustering.

UCERF2 California CatalogThe Uniform California Earthquake Rupture

Forecast, Version 2 (UCERF 2)

By the 2007 Working Group on California Earthquake Probabilities

USGS Open File Report 2007-1437

Appendix H: WGCEP Historical California Earthquake Catalog, by K.R. Felzer and T. Cao

Appendix I: Calculating California Seismicity Rates, by K.R. Felzer

- Updated through 5/20/2011 using the ANSS catalog.

- All catalogs are complete, based on Karen Felzer’s study of catalog completeness, Appendix I of the UCERF2 report.

FIT: Instrumental, small earthquake catalogs.

TEST: Historical, large earthquake catalogs.

Spatial-Temporal ETAS Model

ETAS equations (e.g. Ogata, 1988):

Best parameters (maximum likelihood):

Apparent ETAS branching ratio depends on Mmin:

- Relationship between true branching ratio, ntrue, and apparent branching ratio, napp, is given by (Sornette and Werner, 2005):

napp = ntrue(Mmax-Mmin)/(Mmax- M0)

- Ratio between napp and (Mmax-Mmin) is constant:

napp/(Mmax-Mmin)= ntrue/(Mmax-M0)

- For California catalogs, estimate n/(Mmax-Mmin)=0.14, assuming Mmax=7.9.

- 1932 catalog, n/(Mmax-Mmin) = 0.16

- 1957 catalog, n/(Mmax-Mmin) = 0.14

- 1997 catalog, n/(Mmax-Mmin) = 0.12

- Therefore, for California catalogs, use an apparent branching ratio of:

napp=0.14(7.9-Mmin)

Testing the fit of the ETAS model (temporal only):

Transformed time, tti = how many earthquakes are predicted by ETAS to have happened by the time of event i.

Test that events are Poissonian in transformed time:

- KS test of transformed interevent time distribution:

- Compare distribution of interevent times to an exponential distribution.

- Runs test of ordered transformed interevent times:

- Remove median interevent time from ordered interevent times.

- Compare number of runs of positive and negative values to theoretical distribution.

- Autocorrelation test of ordered transformed interevent times:

- Autocorrelate the ordered interevent times (mean removed) shifted by one.

- Compare with distribution of autocorrelations for multiple random resamplings of the interevent times.

Can’t reject at 95% the null hypothesis that all catalogs are Poissonian in transformed time (KS test; Runs test; and Autocorrelation test.)

Instrumental Catalogs:

Instrumental Catalogs, Large Earthquakes Only Subcatalogs:

Can’t reject at 95% the null hypothesis that all catalogs are Poissonian in transformed time (KS test; Runs test; and Autocorrelation test.)

Historical + Instrumental Catalogs:

ETAS models using median values of b,p,c,q,d from the fits to the instrumental catalogs, branching ratio n=0.14(7.9-Mmin).

Can’t reject at 95% the null hypothesis that all catalogs are Poissonian in transformed time (KS test; Runs test; and Autocorrelation test.)

Testing how well ETAS explains triggering of large vs. small earthquakes (spatial and temporal):

Triggering intensity = intensity at time and location of event due to other events:

Test that small and large earthquakes have similar triggering intensity:

- KS test of distributions of triggering intensity.

Can’t reject at 95% the null hypothesis that small and large earthquakes have the same distribution of triggering intensities (KS test of CDFs.)

Instrumental Catalogs:

Can’t reject at 95% the null hypothesis that small and large earthquakes have the same distribution of triggering intensities (KS test of CDFs.)

Instrumental Catalogs:

Small events resampled, same number as larger events.

Can’t reject at 95% the null hypothesis that small and large earthquakes have the same distribution of triggering intensities (KS test of CDFs.)

Full UCERF2 Historical + Instrumental Catalog:

Small events resampled, same number as larger events, same temporal distribution.

Declustered Historical + Instrumental Catalogs

Greatest excess of pairs at 50-150 km, 55-85 years. Rate in exactly that range significantly higher than synthetics (wrong way to do statistics), but about average for maximum in some range that size. Don’t reject null hypothesis.

Problem: Some large earthquakes that intuition tells us are triggered, the ETAS models assign a low probability of being triggered.

- Joshua Tree 1992 -> Landers 1992: ~12%

- San Simeon 2003 -> Parkfield 2004: ~6%

- San Fernando 1971 -> Northridge 1994: ~4%

- Landers 1992 -> Hector Mine 1999: ~3% (epicentral distance) - ~8% (finite fault distance)

Although it does get some obvious immediate large “aftershocks”:

- Landers 1992 -> Big Bear 1992: >99%

- Elmore Ranch 1987 -> Superstition Hills 1987: >99%

- The ETAS model is not modeling long-term, long-range triggering between large earthquakes. Rather, these large earthquakes appear Poissonian.

- Large earthquake catalogs in general tend to have small branching ratios (e.g. Sornette & Werner, 2005). There is not much clustering, hence large earthquake catalogs have limited usefulness for operational earthquake forecasting.

Problem: Some large earthquakes that intuition tells us are triggered, the ETAS models assign a low probability of being triggered.

Solution: Include smaller earthquakes, and therefore more secondary triggering of large earthquakes, in the ETAS model.

- If all earthquakes with M≥3 are included, the Landers and Hector Mine events appear triggered (these are the only two events I’ve tested so far.) Similar results found by Felzer et al. (2003).

- In general, catalogs with smaller earthquakes have larger branching ratios, and contain the small earthquakes that link the larger events together.

- Including smaller earthquakes should improve ETAS-based operational earthquake forecasting.

Apparent probability of the earthquake being triggered.

If we build an ETAS model based primarily on information from smaller earthquakes, will it work for forecasting the larger (M≥6.5) potentially damaging earthquakes?

Answer: A qualified “yes” for California.

We can’t reject the Null Hypothesis that the same stationary ETAS model explains earthquakes of all magnitudes. However, the California earthquake catalog contains few large earthquakes, which makes it harder to reject any null hypothesis related to the larger events.

Including smaller earthquakes in the ETAS model increases the level of clustering, and links together larger earthquakes, making the model more useful for operational earthquake forecasting.