Statistics of Seismicity and Uncertainties in Earthquake Catalogs Forecasting Based on Data Assimilation

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Statistics of Seismicity and Uncertainties in
Earthquake CatalogsForecasting Based on Data
Assimilation

• Maximilian J. Werner
• Swiss Seismological Service
• ETHZ

Didier Sornette (ETHZ), David Jackson, Kayo Ide
(UCLA) Stefan Wiemer (ETHZ)
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Statistical Seismology
stochastic and clustered earthquakes
uncertain representations of earthquakes in
catalogs
scientific hypotheses, models, forecasts
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Magnitude Fluctuations
b1
Gutenberg-Richter Law
Relocated Hauksson Catalog, 1984-2002
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Rate Fluctuations
7.1 Hector Mine 1999
7.3 Landers 1992
6.4 Northridge 1994
6.6 Superstition Hills 1987
Rate
Triggered Events
Days since mainshock
Magnitude
Relocated Hauksson Catalog, 1984-2002
Omori-Utsu Law
Productivity Law
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Spatial Fluctuations
7.1 Hector Mine 1999
7.3 Landers 1992
6.4 Northridge 1994
5.4 Oceanside 1986
Relocated Hauksson Catalog, 1984-2002
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Seismicity Models
simple

• Time-independent random (Poisson process)
• Time-dependent, no clustering (renewal process)
• Time-dependent, simple clustering (Poisson
  cluster models)
• Time-dependent, linear cascades of clusters
  (epidemic-type earthquake sequences)
• non-linear cascades of clusters

Current gold standard null hypothesis
complex
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A Strong Null Hypothesis
Epidemic-Type Aftershock Sequence (ETAS) model
Ogata (1988, 1998)
Gutenberg-Richter Law
Omori-Utsu Law
Productivity Law

Time-independent spontaneous events

Every earthquake independently triggers
events (of any size)
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Earthquake forecasts
Experimental forecasts for California based on
the ETAS model
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Effects of Undetected Quakes on Observable
Seismicity

• why small earthquakes matter
• why undetected quakes, absent from catalogs,
  matter
• using a model to simulate their effects
• implications of neglecting them

Sornette Werner (2005a, 2005b), J. Geophys. Res.
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Magnitude Uncertainties Impact Seismic Rate
Estimates, Forecasts and Predictability
Experiments

• Outline
• quantify magnitude uncertainties
• analyze their impact on forecasts in short-term
  models
• how are noisy forecasts evaluated in current
  tests?
• how to improve the tests and the forecasts

Werner Sornette (2007), in revision in J.
Geophys. Res.
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Earthquakes, catalogs and models
?
Seismicity Model
Earthquakes
Measurement process
Model parameters
!
Earthquake catalog
!
Calibrated seismicity model
New catalog data
!
neglected
Forecasts
!
exact
noisy
Evaluation of consistency
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Magnitude Noise and Daily Forecasts of Clustering
Models
Collaboratory for the Study of Earthquake
Predictability (CSEP) Regional Earthquake
Likelihood Models (RELM) Daily earthquake
forecast competition
I will focus on random magnitude errors and
short-term clustering models
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Moment Magnitude Uncertainties CMT vs USGS
Distribution of magnitude estimate differences
Hill plot of scale parameter
Laplace distribution
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Short-Term Clustering Models
Productivity Law
Omori-Utsu Law
Gutenberg-Richter Law
These 3 laws are used in models by Vere-Jones
(1970), Kagan and Knopoff (1987), Ogata (1988),
Reasenberg and Jones (1989), Gerstenberger et al.
(2005), Zhuang et al. (2005), Helmstetter et al.
(2006), Console et al. (2007), ...
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A Simple Cluster Model
mainshocks cluster centers
aftershocks clusters
Earthquake rate
Noisy magnitudes
centers
aftershocks
What are the fluctuations of the
deviations?
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Distributions of Perturbed Rates
PDF
PDF
PDF
PDF
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Heavy Tails of Perturbed Rates
for
Survivor function
exponent
Productivity law of aftershocks
Noise scale parameter
Productivity law of aftershocks
Noise scale parameter

• Combination of
• Power law tails
• Catalog realization
• Averaging according
• to Levy or Gauss regime

Survivor function
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Evaluating Noisy Forecasts
How important are the fluctuations in the
evaluation of forecasts?

• Conduct a numerical experiment
• Simulate earthquake reality according to our
  simple cluster model
• Make reality noisy
• Generate forecasts from noisy data
• Submit forecasts to mock CSEP/RELM test center
• Test noisy forecasts on reality using currently
  proposed consistency tests
• Reject models if tests confidence is 90 (i.e.
  expect 1 in 10 rejected wrongfully)
• Calibrate parameters of the experiment to mimic
  California

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Numerical Experiment Results
Level of noise
Number of rejected models
Violates assumed 90 confidence bounds
no
0/10
probably
10/60
yes
9/10
yes
7/10
10/10
yes
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Implications

• Forecasts are noisy and not an exact expression
  of the models underlying scientific hypothesis.
• Variability of observations consistent with model
  are non-Poissonian when accounting for
  uncertainties.
• The particular idiosyncrasies of each model also
  cannot be captured by a Poisson distribution.
• But the consistency tests assume Poissonian
  variability!
• Models themselves should generate the full
  distribution.
• Complex noise propagation can be simulated.
• Two approaches
• Simple bootstrap Sample from past data
  distributions to generate many forecasts.
• Data assimilation correct observations by prior
  knowledge in the form of a model forecast.

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Earthquake Forecasting Based on Data Assimilation

• Outline
• current methods for accounting for uncertainties
• introduction to data assimilation
• how data assimilation can help
• Bayesian data assimilation (DA)
• sequential Monte Carlo methods for Bayesian DA
• demonstration of use for noisy renewal process

Werner, Ide Sornette (2008), in preparation.
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Existing Methods in Earthquake Forecasting

• The Benchmark
• Ignore uncertainties
• Current strategy of operational forecasts (e.g.
  cluster models)

• The Bootstrap
• Sample from plausible observations to generate
  average forecast
• Renewal processes with noisy occurrence times
• Paleoseismological studies (Rhoades et al., 1994
  Ogata, 2002)

• The Static Bayesian
• consider entire data set and correct observations
  by model forecast
• Renewal processes with noisy occurrence times
• Paleoseismological studies (Ogata, 1999)

1. Generalize to multi-dimensional, marked point
   processes
2. Use Bayesian framework for optimal use of
   information
3. Provide sequential forecasts and updates

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Data Assimilation

• Talagrand (1997) The purpose of data
  assimilation is to determine as accurately as
  possible the state of the atmospheric (or
  oceanic) flow, using all available information
• Statistical combination of observations and
  short-range forecasts produce initial conditions
  used in model to forecast. (Bayes theorem)
• Advantages
• General conceptual framework for uncertainties
• Constrain unknown initial conditions
• Account for observational noise, system noise,
  parameter uncertainties
• Deal with missing observations
• Best possible recursive forecast given all
  information
• Include different types of data

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Data Assimilation
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Bayesian Data Assimilation
Unobserved states
Noisy observations

• This is a conceptual solution only.
• Analytical solution only available under
  additional assumptions
• Kalman filter Gaussian distributions, linear
  model
• Approximations
• local Gaussian extended Kalman filter
• ensembles of local Gaussians ensemble Kalman
  filter
• particle filters non-linear model, arbitrary
  evolving distributions

Initial condition
Model forecast
Data likelihood
Obtain posterior
Using Bayes theorem
Sequentially
Prediction
Update
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Sequential Monte Carlo Methods

• flexible set of simulation-based techniques for
  estimating posterior distributions
• no applications yet to point process models (or
  seismology)

particles
weights
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Temporal Renewal Processes
Noise
Renewal process
Forecast
Likelihood (observation)
Analysis / Posterior
Werner, Ide and Sornette (2007), in prep
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Numerical Experiment
Model
Noisy observations
Parameters
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Step 1
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Step 2
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Step 5
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Outlook

• Data assimilation of more complex point processes
  and operational implementation (non-linear,
  non-Gaussian DA)
• Including parameter estimation
• Estimating and testing (forecasting) corner
  magnitude,
• based on geophysics, EVT
• including uncertainties (Bayesian?)
• Spatio-temporal dependencies of seismicity?
• Estimating extreme ground motions shaking
• Interest in better spatio-temporal
  characterization of seismicity (spatial, fractal
  clustering)
• Improved likelihood estimation of parameters in
  clustering models
• (scaling laws in seismicity, critical phenomena
  and earthquakes)