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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.

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

Maximilian J. Werner

Swiss Seismological Service


Didier Sornette (ETHZ), David Jackson, Kayo Ide (UCLA)

Stefan Wiemer (ETHZ)

statistical seismology
Statistical Seismology

stochastic and clustered earthquakes

uncertain representations of

earthquakes in catalogs

scientific hypotheses, models, forecasts

magnitude fluctuations
Magnitude Fluctuations


Gutenberg-Richter Law

Relocated Hauksson Catalog, 1984-2002

rate fluctuations
Rate Fluctuations

7.1 Hector Mine 1999

7.3 Landers 1992

6.4 Northridge 1994

6.6 Superstition Hills 1987


Triggered Events

Days since mainshock


Relocated Hauksson Catalog, 1984-2002

Omori-Utsu Law

Productivity Law

spatial fluctuations
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
Seismicity Models


  • 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
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
Earthquake forecasts

Experimental forecasts for California

based on the ETAS model

effects of undetected quakes on observable seismicity

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

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, catalogs and models



Seismicity Model

Measurement process

Model parameters


Earthquake catalog



seismicity model

New catalog data







Evaluation of consistency

magnitude noise and daily forecasts of clustering models
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
Moment Magnitude Uncertainties CMT vs USGS

Distribution of magnitude estimate differences

“Hill” plot of scale parameter

Laplace distribution:

short term clustering models
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), ...

a simple cluster model
A Simple Cluster Model


cluster centers





Noisy magnitudes:



What are the fluctuations of the deviations?

heavy tails of perturbed rates
Heavy Tails of Perturbed Rates


Survivor function



law of aftershocks


law of aftershocks

Noise scale


Noise scale


Combination of

Power law tails

Catalog realization

Averaging according

to Levy or Gauss regime

Survivor function

evaluating noisy forecasts
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
Numerical Experiment Results

Level of noise

Number of

rejected “models”

Violates assumed

90% confidence bounds











  • 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:
      • 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.
earthquake forecasting based on data assimilation

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
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)

Generalize to multi-dimensional, marked point processes

Use Bayesian framework for optimal use of information

Provide sequential forecasts and updates

data assimilation
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 availableinformation”
  • 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
bayesian data assimilation
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:




sequential monte carlo methods
Sequential Monte Carlo Methods
  • flexible set of simulation-based techniques for estimating posterior distributions
  • no applications yet to point process models (or seismology)



temporal renewal processes
Temporal Renewal Processes


Renewal process:


Likelihood (observation):

Analysis / Posterior:

Werner, Ide and Sornette (2007), in prep

numerical experiment
Numerical Experiment


Noisy observations:


  • 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)