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In meteorology, engineering and computer sciences, data assimilation isroutinely employed as the optimal way to combine noisy observations with priormodel information for obtaining better estimates of a state, and thus betterforecasts, than can be achieved by ignoring data uncertainties. Earthquakeforecasting, too, suffers from measurement errors and partial model informationand may thus gain significantly from data assimilation. We present perhaps thefirst fully implementable data assimilation method for earthquake forecastsgenerated by a point-process model of seismicity. We test the method on asynthetic and pedagogical example of a renewal process observed in noise, whichis relevant to the seismic gap hypothesis, models of characteristic earthquakesand to recurrence statistics of large quakes inferred from paleoseismic datarecords. To address the non-Gaussian statistics of earthquakes, we usesequential Monte Carlo methods, a set of flexible simulation-based methods forrecursively estimating arbitrary posterior distributions. We perform extensivenumerical simulations to demonstrate the feasibility and benefits offorecasting earthquakes based on data assimilation. In particular, we show thatthe forecasts based on the Optimal Sampling Importance Resampling (OSIR)particle filter are significantly better than those of a benchmark forecastthat ignores uncertainties in the observed event times. We use the marginaldata likelihood, a measure of the explanatory power of a model in the presenceof data errors, to estimate parameters and compare models.

Earthquake Forecasting Based on Data Assimilation: Sequential Monte Carlo Methods for Renewal Processes