Smoothed seismicity model
K. R. Felzer
The smoothed seismicity model is based on the method of Helmstetter et al. (2007)
(updated in Werner et al. (2011)) which performed well in the RELM California
earthquake forecasting experiment (Schorlemmer et al., 2010). As in the smoothed
seismicity model used for UCERF2 and the 2007 National Hazard Maps (Frankel,
1995), observed seismicity is smoothed using a Gaussian kernel. Instead of using a
fixed smoothing constant of 50 km, however, we use a variable smoothing constant,
which is set equal to the distance to the nth closest earthquake. Werner et al. (2011)
found that n=6 produced optimal forecasts for California in retrospective testing in
which one part of the catalog was used to forecast subsequent years. So we use n=6
here. Other differences from the UCERF2 approach are that rather using the post1850 ≥M4 catalog we use the post-1984 M≥2.5 earthquakes based on the findings of
Werner et al. (2011) that using smaller and more recent earthquakes produces the
best forecasts. This was determined by Werner et al. (2011) again under
retrospective testing, in which the most recent part of the catalog was removed and
then forecast by earlier parts. The resulting map has a much finer structure than the
smoothed map produced for UCERF2. A finer scale map may be more accurate as
the UCERF2 map was inconsistent with the locations of lines of long standing
precarious rocks in Southern California (Lisa Grant, personal communication). In
particular Purvance et al., (2008) demonstrated inconsistency between the
precariously balanced rock locations and seismic shaking forecasts given by the
2002 USGS National Seismic Hazard Maps, which used the same background
seismicity smoothing methodology as UCERF2.
The seismic catalog was declustered before smoothing using the Gardner and
Knopoff (1974) routine, which has been traditionally used for the National Hazard
Maps. It has not been investigated, however, whether this declustering routine is
optimal for earthquake forecasting. If time permits we hope to perform some
optimization on the declustering.
Part of the Helmstetter et al. (2007) routine includes correcting for catalog
incompleteness at points that are not adequately covered by network stations. The
Helmstetter et al. (2007) routine for finding the completeness at each point,
however, produces results with so much variance over short distances that it is
strongly suggested that random variations in the data, rather than simply real
changes in completeness, are significantly influencing the result, and the authors
smooth their values before application. The entire procedure is somewhat complex,
and so we wrote our own completeness routine, described below, which produces
more stable results. Like the Helmstetter et al. (2007) routine our completeness
routine does assume a Gutenberg-Richter magnitude frequency distribution at each
point. We also assume that the b value for this distribution = 1.0 (see the Observed
Magnitude Frequency Distributions Appendix for calculation of the b value).
Completeness routine:
1) Select catalog earthquakes (M≥2.5) in a 25 km radius around each grid point.
2) If there are fewer than 10 earthquakes selected, increase the size of the
radius until there are at least 10 earthquakes or the radius is 50 km,
whichever comes first.
3) Measure the Spearman correlation coefficient between earthquake distance
from the grid point and magnitude. If there is a significant correlation this
indicates that the degree of catalog completeness is changing within the
radius selected. If this is seen we repeat the steps of removing the most
distant earthquake from the data set and re-measuring the correlation until
the correlation becomes insignificant or there are fewer than 10 earthquakes
left. The goal is for the earthquakes within the radius to represent a single
completeness threshold, which equals the completeness threshold at the grid
point as closely as possible.
4) We remove the largest earthquake from the remaining data set to account for
outliers, and then take the mean of the remaining magnitudes. If the mean is
consistent with the mean expected for a completeness magnitude of 2.5 (at
95% confidence) then a completeness of 2.5 is assigned to the point. If the
mean is too high then we find the completeness magnitude that the mean is
consistent with.
Maps of the completeness magnitudes solved for and the smoothed seismicity are
given below in Figures 1 and 2 respectively, and as attached ascii files. Catalog
incompleteness is corrected for by estimating the missing seismicity rate above M
2.5. For example if we solve for a completeness of M 2.8 we multiply the seismicity
rate at the grid point by 10(2.8-2.5).
For comparison the smoothed seismicity map used for UCERF2 is given in Figure 3.
It can be observed that the new map is much more detailed and has sharper
delineations of currently active fault zones. It is of particular note that the new map
shows a low risk area between the Elsinore and San Jacinto faults in Southern
California, where a band of precariously balanced rocks has been documented
(Brune et al., 2006). The UCERF 2 map forecasted high risk in this area.
References
Brune, J. N. and A. Anooshehpoor and M. D. Purvance and R. J. Brune (2006). Band of
precariously balanced rocks between the Elsinore and San Jacinto, California,
fault zones: Constraints on ground motion for large earthquakes, Geology, 34,
137-140, doi: 10.1130/G22127.1
Frankel, A. (1995). Mapping seismic hazard in the Central and Eastern United
States, Seis. Res. Lett., 66, 8-21.
Helmstetter, Agnes and Yan Y. Kagan and David D. Jackson (2007). High-resolution
time-indepent grid-based forecast for M≥5 earthquakes in California, Seis. Res.
Lett.., 78, 78-86.
Purvance, Matthew D., James N. Brune, Norman A. Abrahamson, and John G.
Anderson, Consistency of precariously balanced rocks with probabilistic seismic
hazard estimates in southern California (2008). Bull. Seis. Soc. Am., 6, 26292640, doi: 10.1785/0120080169.
Schorlemmer, Danjiel, J. Douglas Zechar, Maximilian J. Werner, Edward H. Field,
David D. Jackson, Thomas H. Jordan and The RELM Working Group (2010). First
results of the regional earthquake likelihood models experiment, Pure and Appl.
Geophys., 167, 859-876.
Werner, Maximillian J., Agnes Helmstetter, David D. Jackson and Yan Y. Kagan
(2011). High-resolution long-term and short-term earthquake forecasts for
California, Bull. Seis. Soc. Am., 101, 1630-1648.
Figure 1. Estimated catalog completeness for the 1984--9/30/2011 catalog. Dark
blue areas are outside of the UCERF3 zone or did not have enough data
todetermined completeness. Completeness magnitudes below M 2.5 were not
tested.
Figure 2. Smoothed seismicity map. The color scale gives log10 of normalized
earthquake nucleation intensities.
Figure 3. Smoothed seismicity map used for UCERF2, courtesy of Chuck Mueller.