Statistics of Seismicity and Uncertainties in Earthquake Catalogs Forecasting Based on Data Assimilation Maximilian J. Werner Swiss Seismological Service ETHZ Didier Sornette (ETHZ), David Jackson, Kayo Ide (UCLA) Stefan Wiemer (ETHZ)
Dec 24, 2015
Statistics of Seismicity and Uncertainties in Earthquake Catalogs
Forecasting Based on Data Assimilation
Maximilian J. Werner
Swiss Seismological Service
ETHZ
Didier Sornette (ETHZ), David Jackson, Kayo Ide (UCLA)Stefan Wiemer (ETHZ)
stochastic and clustered earthquakes
uncertain representations of earthquakes in catalogs
scientific hypotheses, models, forecasts
Statistical Seismology
Magnitude Fluctuations
Relocated Hauksson Catalog, 1984-2002
Gutenberg-Richter LawGutenberg-Richter Law
b=1
Relocated Hauksson Catalog, 1984-2002
6.4 Northridge 1994
7.1 Hector Mine 19997.3 Landers 1992
6.6 Superstition Hills 1987
Rate Fluctuations
Omori-Utsu Law Productivity Law
Days since mainshock
Rat
e
Trig
gere
d E
vent
s
Magnitude
Spatial Fluctuations
Relocated Hauksson Catalog, 1984-2002
6.4 Northridge 1994
7.1 Hector Mine 19997.3 Landers 1992
5.4 Oceanside 1986
Seismicity Modelssimple
complex
• 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
A Strong Null Hypothesis
Epidemic-Type Aftershock Sequence (ETAS) model:
Gutenberg-Richter Law Omori-Utsu Law Productivity Law
Time-independent spontaneous events
Every earthquake independently triggers events(of any size)
+
+
Ogata (1988, 1998)
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.
Magnitude Uncertainties Impact Seismic Rate Estimates, Forecasts and
Predictability ExperimentsOutline
• 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.
Earthquakes, catalogs and models
Seismicity ModelEarthquakes
Measurement process
Earthquake catalog
Model parameters
Forecasts
Evaluation of consistency
New catalog data
?
Calibrated seismicity model
exactnoisy
!
!
!
!
neglected
Magnitude Noise and Daily Forecasts of Clustering Models
I will focus on random magnitude errors and short-term clustering models
Collaboratory for the Study of Earthquake Predictability (CSEP)Regional Earthquake Likelihood Models (RELM)
Daily earthquake forecast competition
Moment Magnitude Uncertainties CMT vs USGS
Distribution of magnitude estimate differences “Hill” plot of scale parameter
Laplace distribution:
Short-Term Clustering Models
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), ...
Omori-Utsu Law Productivity Law Gutenberg-Richter Law
A Simple Cluster Model
mainshocks:cluster centers
aftershocks:clusters
centers
aftershocks
Earthquakerate
Noisy magnitudes:
What are the fluctuations of the deviations?
Heavy Tails of Perturbed Rates
Combination of1. Power law tails2. Catalog realization3. Averaging according to Levy or Gauss regime
for
Productivitylaw of aftershocks
Noise scaleparameter
exponent
Productivitylaw of aftershocks
Noise scaleparameter
Sur
vivo
r fu
nctio
nS
urvi
vor
func
tion
Evaluating Noisy 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 test’s confidence is 90% (i.e. expect 1 in 10 rejected wrongfully)• Calibrate parameters of the experiment to mimic California
How important are the fluctuations in the evaluation of forecasts?
Numerical Experiment Results
Level of noise Number of rejected “models”
Violates assumed90% confidence bounds
0/10
10/60
9/10
7/10
10/10
no
probably
yes
yes
yes
Implications• Forecasts are noisy and not an exact expression of the model’s 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: 1. Simple bootstrap: Sample from past data distributions to generate many
forecasts.2. Data assimilation: correct observations by prior knowledge in the form of a
model forecast.
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.
Existing Methods in Earthquake Forecasting
1) The Benchmark:
• Ignore uncertainties
• Current “strategy” of operational forecasts (e.g. cluster models)
2) The Bootstrap: • Sample from plausible observations to generate average forecast• Renewal processes with noisy occurrence times• Paleoseismological studies (Rhoades et al., 1994; Ogata, 2002)
3) 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 processes2. Use Bayesian framework for optimal use of information3. Provide sequential forecasts and updates
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
Data Assimilation
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Bayesian Data Assimilation
Initial condition Model forecast Data likelihood
Unobserved states: Noisy observations:
Obtain posterior:
Using Bayes’ theorem:
Sequentially:
Prediction:
Update:
• 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
• 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
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
...
Temporal Renewal Processes
Noise:
Renewal process:
Forecast:
Likelihood (observation):
Analysis / Posterior:
Werner, Ide and Sornette (2007), in prep
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)