Forecasting, Prediction, and Testing David D. Jackson, UCLA

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Forecasting, Prediction, and TestingDavid D.
Jackson, UCLA

• Thanks to Yan Kagan, Yufang Rong, Zheng-kang
  Shen, Ned Field, Daniel Schorlemmer, Matt
  Gerstenberger, John Rundle, Don Turcotte, Volodya
  Kossobokov, Ilya Zaliapin

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Definitions

• Forecast specification of the probability per
  unit area, magnitude, time, focal mechanism,
  etc.
  
• Prediction special case of forecasting in which
  the probability in some region is much higher
  than normal, and high enough to justify
  exceptional .
• Notes a prediction must be temporary it also
  requires a definition of normal (i.e., a null
  hypothesis).
  

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Why forecast and test

• Test hypotheses of earthquake physics
• Quasiperiodic characteristic earthquakes
• Magnitudes limited by fault geometry
• Moment balance (tectonic in seismic out)
• Earthquake rate proportional to stress rate
• Inform decisions about earthquake risk
• Facility locations, building codes, insurance
  rates, retrofit, etc.
  
• Inspire envy

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Some testable statements

• At least one event will occur within given
  region, time interval, and magnitude interval
  with probability p.
  
• For many small intervals, this is RELM type
  forecast
  
• Replace region by segment and have WG88 type
  forecast. But earthquake on segment is
  subjective.
  
• N events will occur within given region, time
  interval, and magnitude interval with probability
  p(N).
  
• The largest event in given region and time
  interval will be m, with probability p(m). My
  recommendation for a usable and stable forecast.
  
• If an event occurs in region 1, time interval 1,
  and magnitude interval 1, it will be in the
  included region 2, time interval 2, and magnitude
  interval 2 with probability p. (KB type
  forecast)
• The next earthquake in a given region and
  magnitude interval will not be before time t with
  probability p(t). Replace region by segment and
  have WG88 type forecast.

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What we can and cant do now

• Can
• Forecast 90 of quakes in 10 of area.
• Predict aftershocks
• Forecast mag-freq relationship for N(m)10.
• Maybe can
• Forecast earthquake rate from deformation rate.
• Establish earthquake rate for normal
  conditions
  
• Forecast Mmax
• Cant
• Predict times of individual earthquakes

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Likelihood Testing

• Simulate catalogs using L(lat,lon)
• Compute log likelihood function for observed and
  simulated catalogs
  
• Sort L values in increasing order, plot order vs
  L.

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Practical problems in testing

• Many users many criteria
• Earthquakes not independent
• Testing should be rigorous, but also feel
  good.
  
• Data on rupture lengths, endpoints of rupture,
  slip distribution, etc., not formalized

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Ten-year Prospective Test of Seismic Gap and Null
(Poissonian Smoothed Seismicity) models
Rank zones by decreasing probability of
characteristic earthquake accumulate area,
predicted earthquake number, and actual
earthquake number.
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Properties of test methods

• Likelihood ratio test
• Includes absolute rates
• Useful for marked point process lat, lon, mag,
  etc.
  
• As used, assumes independent events
• Adaptable to time varying forecasts
• Could use declustered catalog
• Gives scalar score
• Likelihood ratio test on largest event in zone,
  time interval
  
• Reduces effect of dependent events
• May respond to long-term users investors,
  planners
  
• Molchan diagram
• Based on relative rates, as normally used
• Needs a scalar measure to reject a hypothesis
• Not convenient for marked point processes need
  to commit to magnitude threshold, e.g.
  
• Does not avoid problem of dependent events

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Testing scheme for elements of risk model
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Conclusions

• Many users Many criteria for optimality
• Absolute rate vs. relative rate
• Short time vs. long time
• Big events vs. smaller ones
• Biggest problem is interdependence of quakes
• Long term users want stability unconditional
  probabilities
  
• Scientists care about interactions conditional
  probabilities
  
• Solutions to interdependence
• Decluster catalog needs model
• Test using conditional probabilities updated
  automatically
  
• Test only largest earthquake in zone.