**Earthquake Forecast Models**

Earthquake sequences appear to be globally continuous over time. This suggests that seismic sources or faults may be part of a critically-stressed, self-organized network where earthquakes occur as a chain reaction with respect to one another. Earthquakes trigger other earthquakes as stresses move around in the fault network.

The assumption that the mainshock of an **earthquake** can be considered an aftershock of some other earthquake provides a simplification of the complex physical process of earthquakes in order to quantify the residual hazard following a mainshock.

A number of empirical and seismicity-based aftershock forecasting **models** have been proposed. The two best-known models are the Short-Term Earthquake Probability Model (STEP) and the Epidemic-Type Aftershock Sequence Model (ETAS).

The STEP model is a simple model used by the USGS for aftershock forecasting in California. The ETAS model was developed from observations of earthquakes in **Japan**. The ETAS model is currently being considered for implementation in the operational forecasting component of the updated Uniform California Earthquake Rupture Forecast (UCERF3) model, which is being developed by the USGS and others.

*Short-Term Earthquake Probability Model (STEP)*

The USGS’s STEP model is a relatively simple method of predicting the rate of aftershocks based on empirical statistical parameters of aftershock distributions. The method is a “one-generation” aftershock model, in which sequence is assumed to be independent of any seismicity prior to the mainshock. The STEP model does not predict time, location and magnitude dependency of aftershocks as they occur.

In the last two decades, the USGS has monitored aftershock activity in California and issued aftershock forecasts following significant earthquakes. Empirical coefficients of the Gutenberg-Richter magnitude-frequency relationship, **Omori’s time-decay law** and the distance-decay relationship have been calibrated to the California region.

Following the 1989 7.1 Mw Loma Prieta earthquake in California, the USGS issued updated aftershock forecasts using the STEP model twice daily for the first eight days, once daily for the following nine days and twice weekly for the following four weeks.

For example, the actual aftershock annual probabilities estimated from the STEP model for a greater than or equal to 7.0 Mw earthquake following the Loma Prieta earthquake dropped from 9 percent immediately following the earthquake to about 2.5 percent after the first five days, to less than 1 percent after 60 days following the earthquake.

**Epidemic-Type Aftershock Sequence Model (ETAS)**

The ETAS model is a multigenerational model in which aftershocks from one earthquake trigger their own aftershock sequences through multiple generations of earthquakes. The model predicts the time, space and magnitude dependence of observed seismicity triggering above some minimum magnitude threshold. The empirical premise of the model is that all earthquakes are dependent on one another.

Figure 3 shows the occurrence of one mother earthquake at a given time and the location of the daughter earthquakes in terms of their elapsed time, location and frequency.

**FIGURE 3: EPIDEMIC-TYPE AFTERSHOCK SEQUENCE MODEL (ETAS)**

*Source: Helmstetter, A., and D. Sornette (2002). “Subcritical and Supercritical Regimes in Epidemic Models of Earthquake Aftershocks”, Journal of Geophysical Research, V. 107, p. ESE 10-1 to 21.*

The frequency and magnitude distributions of the triggered events, while still fundamentally based on Omori’s law and the Gutenberg-Richter magnitude-frequency relationship, have more complex formulations to better represent the statistical distributions seen in instrumental earthquake catalogs.

The distance parameter quantifies the occurrence probability of a triggered event at a specified distance from the mother event, which accounts for spatial dependence of the induced static and dynamic stress of the mother event.

In the ETAS model a branching parameter establishes the number of generations of aftershock sub-sequences relative to the total number of aftershocks, allowing for the possibility that some aftershocks could be larger than the mother event.

Within the seismological community, it is thought that ETAS models hold the potential for making reliable intermediate-term earthquake forecasts on the order of about five or more years forward in active tectonic regions such as California.

Among the academic and scientific community, extensive testing and calibration of these models is currently taking place with application to various regions.

**Testing Earthquake Forecast Models**

A number of practical issues surround effective testing of earthquake forecast models. The Regional Earthquake Likelihood Models (RELM) working group was sponsored by the USGS to perform large-scale prospective tests of more than a dozen five-year earthquake forecasts for California from 2006 to 2011.

Preliminary results indicate that while most models are consistent with earthquake observations in California during the last 2.5 years, only the ETAS model passes all of the formal statistical acceptance criteria. It should be mentioned that the RELM-ETAS model was developed by some of the most experienced researchers worldwide.

On the other hand, researchers have not yet incorporated the spatial proportions of three-dimensional faults into ETAS modeling. Besides, earthquake forecast models cannot account for realistic geometrical constraints of finite fault rupture associated with all large earthquakes.

Within the seismological community it is thought that ETAS models have the potential for making reliable intermediate-term earthquake forecasts on the order of about five or more years forward in active tectonic regions such as California.

**Note:**

Paul C. Thenhaus, Kenneth W. Campbell and Dr. Mahmoud M. Khater, “Spatial and Temporal Earthquake Clustering: Part 2 – Earthquake Aftershocks,” EQECAT, February 27, 2012. http://www.eqecat.com/global-earthquake-clustering-whitepaper-part-2-2012-02.pdf

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