Smoothed Seismicity Rates
Karen Felzer
USGS
Smoothed seismicity
• Smoothed seismicity is used in many forecasts,
including the National Hazard Maps and UCERF2,
to help constrain the off-fault hazard.
• It has been statistically demonstrated that
smoothed seismicity is predictive of future
earthquakes (Kagan and Jackson, 2000; Kafka
2007). It works for small earthquakes and for
M>6 (Kafka, 2007). An upper magnitude limit for
applicability has not been demonstrated.
Potential smoothing methods for
UCERF 3
• National Hazard Map smoothing method
(Frankel, 1996)
• Helmstetter et al. (2007) smoothing method
(Currently winning RELM Southern California 5
year earthquake forecasting test)
• Modified Helmstetter et al. (this talk)
National Hazard Map smoothing
method
linear scale
• The catalog is declustered
using Gardner and Knopoff
(1975)
• The Weichert method is
used to calculate rates in
each bin from M≥4, M≥5,
and M≥6 earthquakes from
different periods.
• Rates are smoothed around
each bin using a Gaussian
kernel and a fixed 50 km
smoothing constant.
Map created from 1850-2010
catalog data
Helmstetter et al. (2007) smoothing
method
log10 scale
• The catalog is declustered
using Reasenberg (1985).
Remaining catalog still
has some clustering.
• M≥2 earthquakes are
used from >1981 only.
• A Gaussian or power law
kernel with an adaptive
smoothing constant is
expanded around each
hypocenter.
Map uses 1981-2005 catalog data
Modified Helmstetter et al. (2007)
smoothing method
log10 scale
1850-2010 catalog data
• No declustering.*
• Uses M≥4 seismicity back to
1850, all magnitudes treated
equally.*
• Uses power law kernels
centered at each hypocenter,
with the Helmstetter adaptive
smoothing constant.
• Calculates smoothed values at
bin centers rather than
integrating across bins.*
• Only relative rates have been
calculated for the current
implementation.
*Improves result *Makes life simpler
The different methods can be evaluated using
the MLE Gain given in Helmstetter et al. (2007)
G  exp(
L  L unif
)
N

G = Gain
L = log likelihood of forecasting map
Lunif = log likelihood of a uniform probability
map
N = Number of earthquakes
Evaluation is performed only within the UCERF polygon
Retrospective tests performed
• NHM vs. modified Helmstetter for forecast of
M≥6 earthquakes over 1957-2006 (50 yrs):
30% higher gain for Helmstetter.
• Modified Helmstetter with no declustering vs.
modified Helmstetter with Gardner and
Knopoff (1975) declustering : Non-declustered
tend to have a higher gain, but statistical
difference not established. Reasenberg
(1985) declustering may improve results.
Retrospective tests needed
• NHM vs. Helmstetter over multiple 1 and 5
year periods.
• Modified Helmstetter vs. full Helmstetter over
1 year, 5 year, and 50 year periods.
• More tests with declustering (discussion
coming up next!).
Arguments against declustering
• All declustering methods are to some degree
arbitrary and incomplete.
• Earthquakes continue in aftershock zones for
years. We would not want to miss the next
Hector Mine or Christchurch.
• Current declustering methods bias magnitudefrequency statistics by a-posteriori removing
the smaller earthquakes in a cluster. This is
not helpful for a-priori forecasting.
Darfield, M 7.1
Failing to predict aftershocks is not
helpful
Christchurch,
M 6.3
Sorry, but according
to our b value you
didn’t have an
earthquake!
Arguments for declustering
• Some declustered forecasts appear to perform better.
Why? Some thoughts:
• Declustering emphasizes larger earthquakes.
More aftershocks occur around larger
earthquakes => higher future risk in these areas.
• Declustering effectively decreases the hazard
from aftershock zones that may have been much
more active in the past than at present.
However, the risk from still-active aftershock
zones might be decreased too much by rigorous
declustering.
A proposed modified approach
1) Use ETAS, rather than straight smoothing, to
model very large/recent earthquakes that are
still producing aftershocks at a rapid rate. This
will give the larger earthquakes the extra risk, at
a presumably more correct rate.
2) Decrease the risk associated with earthquakes in
long-dormant aftershock zones, using empirical
measures or ETAS to estimate amount of
decrease.
3) Do not alter the magnitude-frequency
distribution
4) Test, Test, Test!!!
Decisions that need to be made
• Smoothing method: NHM, Helmstetter,
modified Helmstetter ?
• Declustering: Gardner and Knopoff,
Reasenberg , no declustering, or the modified
approach?
• Magnitude-frequency distribution:
Declustered distribution, or full catalog
magnitude-frequency distribution?
Decisions that need to be made
• What tests will be definitive for choosing one
method over another? What confidence level
do we want of improvement before selecting
a new method?
• Is there a measure of performance that we
want besides the Helmstetter MLE Gain?
Some differences between Helmstetter
et al. and NHM
Helmstetter
et al.
National
Hazard Map
Minimum
magnitude
2.0
(1981-2005)
4.0, 5.0, 6.0
(1850-2010)
Smoothing
constant
Distance to nth
neighbor
50 km
Binning
Smoothing kernel Smoothing kernel
drawn around
drawn around the
each hypocenter center of each bin