**Probabilistic Earthquake Hazard Analysis and Aftershocks**

The purpose of probabilistic seismic hazard analysis (PSHA) is to quantify the rate (or probability) of exceeding various ground-motion levels at a site or sites, given all possible **earthquakes**. PSHA involves the following three steps:

- Characterization of the seismic-hazard source model(s)
- Specification of the ground motion model(s) or attenuation relationship(s)
- Probabilistic calculation

In PSHA, the probabilistic calculation of the mean long-term frequencies of earthquakes is generally done by assuming a Poisson (time independent) recurrence distribution. Accordingly, it has been common practice to remove dependent earthquakes (foreshocks, aftershocks and swarms) from historical earthquake catalogs before estimating mean earthquake frequencies.

Some have argued that because aftershocks can cause damage, they should not be removed from the frequency calculations but rather included in a PSHA. However, there are two issues with this suggested approach:

- Since aftershocks are dependent events, they violate the Poisson distribution independence assumption that forms the basis of the PSHA methodology.
- Including aftershocks in a historical event catalog requires a long-term average frequency assumption, which violates the spatial-temporal clustering of aftershock sequences.

In summary, if dependent earthquake events are to be taken into account in probabilistic loss calculations, they need to be accommodated outside of the standard PSHA methodology by simulating short-term and medium-term earthquake clustering in the generation of the stochastic event set that is used in the loss calculations. **EQECAT** is currently developing such a model to test its impact on probabilistic estimates of portfolio loss.

**Aftershocks: Temporal and Spatial Properties**

Figure 1 shows the epicenter location of the mega 2004 Andaman-Nicobar (Sumatra) earthquake and the properties of the spatial and temporal distributions of the aftershocks that followed the event.

**FIGURE 1: TEMPORAL AND SPATIAL DISTRIBUTIONS OF EARTHQUAKES BEFORE AND AFTER THE DECEMBER 26, 2004, 9.1 Mw ANDAMAN-NICOBAR (SUMATRA) EARTHQUAKE**

*Source: Jordan, T.H., et al (2011). “Operational Earthquake Forecasting: State of Knowledge and Guidelines for Utilization”, International Commission on Earthquake Forecasting for Civil Protection, Annals of Geophysics, V. 54, p. 316-391.*

The mainshock was followed three months later by an 8.6 Mw earthquake on March 28, 2005, partially hidden from view but near the large green triangle.

The temporal distribution on the right shows the clustering of aftershocks of different magnitudes from 2004 through 2011 and reveals that aftershocks themselves are sometimes mainshocks to other aftershock clusters. The magnitude, timing and geographic distributions of aftershock clusters depend on the magnitude of the mainshock earthquake that produced them.

The properties of aftershock clusters mentioned above generally conform to the following four fundamental empirical laws displayed in Figure 2:

**Omori’s Time Decay Law** states that frequency of aftershocks decays in proportion to the reciprocal of the time since the mainshock occurred. That is, it estimates the frequency of earthquakes during a certain time (t) after the mainshock.

**Gutenberg-Richter Law** estimates the expected distribution of magnitudes during the aftershock sequence. It implies that the frequency of aftershocks diminishes by about a factor of 10 for each higher magnitude unit. Note that there is no constraint on when earthquakes of different magnitudes occur.

**Bath’s Empirical Law** defines the expected largest aftershock following the mainshock. Specifically, it states that the average difference of magnitude between the mainshock magnitude and its largest aftershock is about 1.2 units, and the largest aftershock is independent of the magnitude of the mainshock. It should be noted that an aftershock can be more severe than the original mainshock, and in these cases it would be redefined as the mainshock retrospectively.

**Spatial Decay (Aftershock Geographic Density)** estimates the probability of finding an aftershock with distance from a mainshock or mainshock fault rupture. This probability decreases in proportion to the reciprocal of distance from the mainshock.

**FIGURE 2: FUNDAMENTAL EMPIRICAL AFTERSHOCK MODELS**

*Sources: *

*Omori, F. (1894). “On the Aftershocks of Earthquakes”, Journal of the College of Science, Imperial University of Tokyo, Vol. 7, p. 111-200.**Gutenberg, R. and C.F. Richter (1944). “Frequency of Earthquakes in California”, Bulletin of the Seismological Society of America, Vol. 34, p. 185-188.**Bath, M. (1965). “Lateral Inhomogeneities in the Upper Mantle”, Tectonophysics, Vol. 2, p. 483-514.**Felzer, K.R., and E.E. Brodsky (2006). “Decay of Aftershock Density with Distance Indicates Triggering by Dynamic Stress”, Nature, Vol. 441, p. 735-737.*

To identify the primary seismicity region after an earthquake, the U.S. Geological Survey (USGS) regards earthquakes as aftershocks if they occur within a distance of about one fault-rupture length from the mainshock. In reality, however, some aftershocks themselves are mainshocks to sub-sequences of other aftershocks. Those so-called “triggered” earthquakes occur well beyond the “one or two rupture-length” distance criterion.

The nature of stress transfer responsible for observed patterns of distance decay is currently being debated in the scientific literature. A consensus model of spatial density decay of aftershocks with distance from the mainshock remains a very active research area.

**Notes: **

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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