2. Real-time earthquake source estimation 2.1 Review of Bayes’ Theorem3. Applications of the VS method to selected Southern California earthquake datasets3.2 16 October 1999 M7.1 Hector Mine, California earthquake: low station density4. How subscribers might use early warning information5. Station density and the evolution of estimate uncertainties6. Conclusions7. AcknowledgementsThe Virtual Seismologist (VS) method: a Bayesian approach to earthquake early warning by Georgia Cua1, Thomas Heaton2 1. Puerto Rico Seismic Network, Univeristy of Puerto Rico Mayagüez, Puerto Rico, USA. 2. Department of Civil Engineering, California Institute of Technology, Pasadena, USA.1. Introduction Subscribers attempting to use early warning information to reduce earthquake-related losses typically have a selected sequence of actions they would like to complete before damaging ground motions reach their site. These subscribers decide whether or not to initiate their damage-mitigating actions based on uncertain information. How to make optimal decisions with uncertain information is a fundamental question to these subscribers. Answering this question properly requires addressing a seismological and an economics problem in tandem. The seismological question is that of real-time earthquake source estimation, which can be phrased as follows: what are the best estimates of magnitude and location given the available data? The economics question is the user response problem, which can be phrased as follows: what is the optimal decision or course of action, given the current source estimates and its uncertainties? The Virtual Seismologist (VS) method is a Bayesian approach to earthquake early warning that addresses the source estimation and user response problems in tandem. In the source estimation problem, the VS method shares with other proposed methodologies (Nakamura, 1988; Allen and Kanamori, 2003; Wu and Kanamori, 2005a, 2005b) the use of relative frequency content or predominant period and attenuation relationships to estimate magnitude and/or location from available ground motion observations. The introduction of prior information into the earthquake source estimation problem distinguishes the VS method from other paradigms for early warning. Bayes’ theorem allows the use of a “background” state of knowledge to help in resolving trade-offs in magnitude and location that cannot be resolved due to the scarcity of observations at the initial stages of the earthquake rupture. Types of information that can be included in the Bayes prior include: state of health of the seismic network, previously observed seismicity, known fault locations, and the Gutenberg-Richter magnitude-frequency relationship. The benefits of prior information are largest in regions with low station density, where large inter-station distances result in initial early warning estimates based on a sparse set of observations. We illustrate the performance of the VS method with high station density using ground motions recorded from the M=4.75 Yorba Linda, California earthquake, and with low station density using data from the M= 7.1 Hector Mine, California earthquake. The examples shown in this paper approximate the earthquake as a single point source. While this seems adequate for earthquakes with M< 6, this methodology needs to be modified to be effective for long ruptures where near-source ground motions are expected even at large distance from the epicenter. Yamada and Heaton (2006) present a strategy for extending VS to handle long ruptures. Aside from a method to estimate magnitude and location from sparse information from the initial stages of the earthquake rupture, it is equally important for earthquake early warning studies to address how subscribers might make optimal decisions using early warning information. Different subscribers will require different types of early warning estimates, depending on subscriber tolerance to missed and/or false alarms. Ultimately, the subscriber requirements dictate how the source estimation problem must be phrased.This is not consistent with the traditional separation of the source estimation and user response problems in earthquake early warning research. The VS method facilitates an integrated approach that recognizes the role of the user decision-making process in formulating the source estimation problem. 2. Real-time earthquake source estimation 2.1 Review of Bayes’ Theorem Consider that we want to estimate the magnitude and location of an earthquake given an available set of observed ground motions. According to Bayes theorem, the state of belief regarding magnitude and location (M,loc) given a set of available observations Yobs is given by P(M,loc |Yobs) =P(Yobs| M,loc)×P(M,loc)P(Yobs) (1) where M is magnitude, loc is a location parameter (epicentral distance, or epicentral location), and Yobs is the available set of observed ground motions. We can use the proportional form of Bayes’ theorem since P(Yobs) is not a function of the parameters being estimated (M,loc): P(M,loc |Yobs) ∝ P(Yobs| M,loc)×P(M,loc) (2) P(M,loc|Yobs), is the posterior probability density function (pdf); it is the conditional probability that an earthquake of magnitude and location M,loc generated the set of observations Yobs. The VS estimates,(, are the most probable source estimates given the available observations; they maximize P(M,loc|YM,loc)VSobs). The spread of P(M,loc|Yobs) yields the uncertainties on the VS source estimates. On the right hand side, P(Yobs|M,loc) is the likelihood function; it is the conditional probability of observing a set of ground motions Yobs given an earthquake with magnitude and location M,loc. The likelihood function requires ground motion models relating source descriptions (M,loc) to observed ground motion amplitudes. The VS method uses 1) relationships between ratios of peak ground motions and magnitude, and 2) ground motion attenuation relationships describing observed amplitudes as functions of magnitude and distance, to define the likelihood function P(Yobs|M,loc). P(M,loc) is the Bayes prior; it represents a background state of knowledge, independent of the observations, on relative earthquake probabilities that we want to include in the estimation process. The degree of complexity that can be incorporated into the prior is flexible. The simplest prior we can use is the assumption that all magnitudes and all locations are equally