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, Gutenberg-Richter Law b=1

Relocated Hauksson Catalog, Northridge Hector Mine Landers Superstition Hills 1987 Rate Fluctuations Omori-Utsu LawProductivity Law Days since mainshock Rate Triggered Events Magnitude

Spatial Fluctuations Relocated Hauksson Catalog, Northridge Hector Mine Landers Oceanside 1986

Seismicity Models simple 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 LawOmori-Utsu LawProductivity Law Time-independent spontaneous events Every earthquake independently triggers events (of any size) + + Ogata (1988, 1998)

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 Seismicity Model Earthquakes Measurement process Earthquake catalog Model parameters Forecasts Evaluation of consistency New catalog data ? Calibrated seismicity model exact noisy ! ! ! ! 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 Earthquake rate Noisy magnitudes: What are the fluctuations of the deviations?

Distributions of Perturbed Rates PDF

Heavy Tails of Perturbed Rates Combination of 1.Power law tails 2.Catalog realization 3.Averaging according to Levy or Gauss regime for Productivity law of aftershocks Noise scale parameter exponent Productivity law of aftershocks Noise scale parameter Survivor function

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 noiseNumber of rejected “models” Violates assumed 90% confidence bounds 0/10 10/60 9/10 7/10 10/10 no probably 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

Bayesian Data Assimilation Initial conditionModel forecastData 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 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)