# Magnitude-Frequency Distributions (MFDs)

# 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:

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 (*M*_{max}), 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., *M*_{W}0.5) to *M*_{max} 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
*M*_{max} 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 (*M*_{max}) 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.