# Types of MFDs

In probabilistic seismic hazard analysis (PSHA), source models require a defined occurrence rate for earthquakes of each considered magnitude, e.g., a magnitude-frequency distribution (MFD). These rates are determined either by statistically analysing the observed seismicity over instrumental and historic time scales, or-for well characterized sources—by using the fault dimensions and slip rates to model recurrence.

Regional models built by GEM use the following common approaches to characterize seismicity rates.

## Gutenberg-Richter

The Gutenberg-Richter MFD allows earthquake sources to generate earthquakes of different magnitudes. Gutenberg and Richter (1944) were the first to develop a functional form for the relationship between earthquake magnitude and occurrence rate, resolving a negative exponential distribution:

\begin{align} log N = a-bm \\ \end{align}

where N is the annual rate of earthquakes with M > m, a is the rate of all earthquakes, and b is the relative distribution of earthquakes among magnitudes. A higher b-value indicates a larger proportion of seismic moment released by small earthquakes. a and b are resolved from the available observations. Usually, b is close to 1.0.

### Truncated Gutenberg-Richter

A traditional Gutenberg-Richter MFD allows for earthquakes of any magnitude, but in reality, the source in question may not be capable of producing earthquakes beyond a certain threshold. For example, fault dimensions physically limit earthquake magnitude, or the observed earthquake magnitudes saturate. To account for these constraints, a truncated MFD is used to specify a maximum magnitude (Mmax), simply by cutting the MFD at this magnitude. The MFD is additionally cut at a minimum magnitude ("double-truncated"), below which earthquakes are not contributing to the hazard in ways significant to engineering.

Truncated Gutenberg-Richter MFDs are commonly used in hazard models build by the GEM Secretariat. Where MFDs are produced for a source zone, such as for distributed or inslab seismicity, the upper magnitude is usually determined by adding a delta value (e.g., MW0.5) to Mmax in the earthquake catalogue or subcatalogue used to produce the MFD. This is based on the premise that the observation period is too short to have experienced a true Mmax earthquake.

GEM models typically use the methodology of Weichert (1985) to compute double-truncated Gutenberg-Richter MFDs for seismic source zones, which allows for the use of different observation periods for different earthquake magnitudes (e.g., a completeness threshold).

If a seismicity distribution is not explicitly available, an MFD of this form can also be computed from a seismic moment budget using strain rates, fault dimensions, and assumed magnitude ranges and b-values. For models built internally by GEM, we apply this to faults with available slip rates. This methodology is described here.

## Characteristic

Some sources do not produce earthquakes that follow the Gutenberg-Richter distribution, but instead tend to host earthquakes of nearly the same magnitude, e.g., a characteristic earthquake. In this case, an earthquake with a moderate to high magnitude occurs more frequently than would be suggested by a Gutenberg-Richter MFD. For sources of this type, the MFD reveals more frequent occurrences concentrated around the most-likely/characteristic magnitude earthquake, for example using a boxcar or Gaussian distribution (e.g., Youngs and Coppersmith, 1985, or Lomnitz-Adler and Lomnitz, 1979).

Though the Youngs and Coppersmith (1985) MFD is technically a hybrid MFD, incorporating both a characteristic component and a Gutenberg-Richter component at lower magnitudes, it is typically often categorized as a characteristic MFD. GEM uses this MFD in a few models built in-house, such as the Philippines model, where sensitivity testing indicated that it produced a better fit to the regional seismicity than a double-truncated GR for crustal faults.

## Hybrid types

Some subduction interface source models built by the GEM secretariat use a hybrid approach that combines the Gutenberg-Richter MFD with a characteristic MFD. The latter approach derives a double truncated Gaussian distribution to model occurrence of the maximum magnitude (Mmax) earthquake that an interface segment can theoretically support (herein called the "characteristic earthquake").

The magnitude and occurrence rate of the characteristic earthquake for an interface segment are based on the fault area (e.g., from the complex fault output by the Subduction Toolkit), the convergence rate, and a seismic coupling coefficient. We choose between three recent scaling relationships for subduction interfaces that compute magnitude from fault area: Strasser et al. (2010), Allen and Hayes (2017), and Thingbaijam and Mai (2017). We use published convergence rates and seismic coupling coefficients to determine the time needed to accumulate enough strain for the characteristic earthquake. The coupling parameter is often challenging, in large part due to the scarcity of land and thus GPS measurements in close proximity to subduction zones. Where no other model is available, we take values from Heuret et al. (2011) or Scholz and Campos (2012), but cautiously, as many sometimes these values are suspiciously low (e.g., <0.1 where instrumentally recorded earthquakes M>8.0 have occurred.)

The characteristic MFD is combined with the Gutenberg-Richter MFD into a hybrid MFD by finding the intersection point of the two MFDs, and taking the Gutenberg-Richter occurrence rate below the intersection magnitude, and the characteristic rate above that magnitude.

## References

Allen, T. I., & Hayes, G. P. (2017). Alternative rupture‐scaling relationships for subduction interface and other offshore environments. Bulletin of the Seismological Society of America, 107(3), 1240-1253.

Gutenberg, B., & Richter, C. F. (1944). Frequency of earthquakes in California. Bulletin of the Seismological Society of America, 34(4), 185-188.

Heuret, A., Lallemand, S., Funiciello, F., Piromallo, C., & Faccenna, C. (2011). Physical characteristics of subduction interface type seismogenic zones revisited. Geochemistry, Geophysics, Geosystems, 12(1).

Lomnitz-Adler, J., & Lomnitz, C. (1979). A modified form of the Gutenberg-Richter magnitude-frequency relation. Bulletin of the Seismological Society of America, 69(4), 1209-1214.

Scholz, C. H., & Campos, J. (2012). The seismic coupling of subduction zones revisited. Journal of Geophysical Research: Solid Earth, 117(B5).

Thingbaijam, K. K. S., Martin Mai, P., & Goda, K. (2017). New empirical earthquake source‐scaling laws. Bulletin of the Seismological Society of America, 107(5), 2225-2246.

Strasser, F. O., Arango, M. C., & Bommer, J. J. (2010). Scaling of the source dimensions of interface and intraslab subduction-zone earthquakes with moment magnitude. Seismological Research Letters, 81(6), 941-950.