Transcript On the Fluctuations of Seismicity and Uncertainties in

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)
Statistical Seismology
stochastic and clustered earthquakes
uncertain representations of
earthquakes in catalogs
scientific hypotheses, models, forecasts
Magnitude Fluctuations
b=1
Gutenberg-Richter Law
Relocated Hauksson Catalog, 1984-2002
Rate Fluctuations
7.1 Hector Mine 1999
7.3 Landers 1992
6.4 Northridge 1994
Rate
Triggered Events
6.6 Superstition Hills 1987
Days since mainshock
Magnitude
Relocated Hauksson Catalog, 1984-2002
Omori-Utsu Law
Productivity Law
Spatial Fluctuations
7.1 Hector Mine 1999
7.3 Landers 1992
6.4 Northridge 1994
5.4 Oceanside 1986
Relocated Hauksson Catalog, 1984-2002
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
complex
Current “gold standard” null hypothesis
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)
Earthquake forecasts
Experimental forecasts for California
based on the ETAS model
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 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.
Earthquakes, catalogs and models
?
Earthquakes
Measurement process
!
Earthquake catalog
New catalog data
Seismicity Model
Model parameters
!
Calibrated
seismicity model
!
Forecasts
neglected
!
exact
noisy
Evaluation of consistency
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
Moment Magnitude
Uncertainties CMT vs USGS
Distribution of magnitude estimate differences
“Hill” plot of scale parameter
Laplace distribution:
Short-Term Clustering Models
Omori-Utsu Law
Productivity 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), ...
A Simple Cluster Model
Earthquake
rate
mainshocks:
cluster centers
aftershocks:
clusters
Noisy magnitudes:
centers
aftershocks
What are the fluctuations
of the deviations?
Distributions of Perturbed Rates
PDF
PDF
PDF
PDF
Survivor function
Survivor function
Heavy Tails of Perturbed Rates
for
exponent
Productivity
Productivity
Noise scale
Noise scale
law of law
aftershocks
of aftershocks
parameter
parameter
Combination of
1. Power law tails
2. Catalog realization
3. Averaging according
to Levy or Gauss regime
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 test’s confidence is 90% (i.e. expect 1 in 10 rejected wrongfully)
Calibrate parameters of the experiment to mimic California
Numerical Experiment Results
Level of noise
Number of
rejected “models”
Violates assumed
90% confidence bounds
0/10
no
10/60
probably
9/10
yes
7/10
yes
10/10
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 processes
2. Use Bayesian framework for optimal use of information
3. 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
To obtain analysis : xka  (1  K k ) xkf  K k xko
t
t
as estimate of true state : xk  M k , k 1 xk 1  k 1
using model forecast : xkf  M k , k 1 xka 1
and observatio n : xko  H k xkt   k
Bayesian Data Assimilation
Unobserved states:
• This is a conceptual solution only.
Initial condition
Model forecast
Noisy observations:
Data likelihood
• Analytical solution only available under additional assumptions
• Kalman filter: Gaussian distributions, linear model
• Approximations:
Obtain posterior:
• local Gaussian: extended Kalman filter
Using •Bayes’
theorem:
ensembles
of local Gaussians: ensemble Kalman filter
• particle filters: non-linear model, arbitrary evolving distributions
Sequentially:
Prediction:
Update:
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
Numerical Experiment
Model:
Noisy observations:
Parameters:
Step 1
Step 2
Step 5
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)