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)

Earthquake forecasts

Experimental forecasts for Californiabased 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 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?

Distributions of Perturbed Rates

PDF

PDF

PDF

PDF

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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xKxKx

ε+==

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:nobservatio and :forecast model using

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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

Numerical ExperimentModel:

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)