## Introduction

An earthquake is a naturally occurring shudder or movement of the earth’s crust. It mostly originates below the surface of the earth due to a swift release of energy along faults (Lutgens, Tarbuk, & Tasa, 2010). The two common types of earthquakes are related to their mode of formation. Tectonic earthquakes are prompted as a consequent of movement of the earth’s crust because of the strain. Tectonic earthquakes are the most common types of earthquakes. Volcanic earthquakes, conversely, are caused by explosive eruption of volcanoes and are less common than tectonic earthquakes (*What causes earthquakes?* n.d.).

Earthquake statistics are the documented facts in the aftereffects of an earthquake. Such information is of utmost importance to the work of many professionals including “seismologists, geologists, engineers, government officials, and statisticians” (Brillinger, n.d., p. 1). Seismologists employ numerical models to illustrate seismicity with high precision.

According to Wiemer, there are two laws that aid in the characterization of independent and dependent elements of seismicity (n.d.). These are the Gutenberg-Richter law and the Omori law. The Gutenberg-Richter law illustrates the magnitude and distribution of earthquakes, whereas the Omori law depicts the decay of aftershock activities. However, despite the existence of these laws, it is practically impossible to envisage the happening of an earthquake. Earthquake statistics, therefore, give numerical information concerning earthquakes with the aim of improving the “detection, location, and quantification of seismic events; risk assessment; prediction of earthquakes; the distinguishing of earthquakes from nuclear explosions; and learning about the Earth’s interior” (Brillinger, n.d., p. 2).

## Frequency of Occurrence of Earthquakes

Earthquakes occur in varying magnitudes all over the world. The USGS National Earthquake Information Center detects and records numerous earthquakes globally. It approximates that millions of earthquakes take place worldwide annually, but countless go unobserved since they strike isolated areas or have very small magnitudes (*Earthquake facts and statistics,* n.d.). This implies that earthquakes with small magnitudes are more common than those with large magnitudes. The following table summarizes the average number of earthquakes experienced annually.

*Table 1: Estimated Frequency of Occurrence of Earthquakes (Earthquake facts and statistics, n.d.).*

The USGS National Earthquake Information Center reports an increase in the number of detection and location of earthquakes with the increased installation of seismographs (*Earthquake facts and statistics*, n.d.).

## Types of Data Used In Earthquake Statistics

Seismograms, documented time series of dislodgement, pace, or increase in velocity of particles on the earth’s surface provide the numerical figures used in computing the statistics of earthquakes. Observation plays a key role in the collection of information. Values obtained directly from a seismogram include the “first arrival time, direction of first motion, first motion amplitude, signal duration, maximum overall amplitude, and oscillation periods” (Brillinger, n.d.). The instruments can be arranged in a way that makes it possible to visualize the arrival of an earthquake as well as the way it changes its conformation.

Seismograms, however, do not provide all the seismological data. In some instances, the subjective evaluation of destruction after the occurrence of an earthquake provides useful information. These descriptions make use of Mercalli intensities (MM). Mercalli intensities are ordinal values with a specific set of descriptions for each level of earthquake intensity. For example, the description of MM intensity VI begins with the phrases “Felt by all; many frightened and run outdoors. Some heavy furniture moved…” while that of intensity VII begins with “Everybody runs outdoors. Damage negligible in buildings of good design…” (Brillinger, n.d., p. 3).

## Models Used In Earthquake Statistics

Models employed in earthquake statistics vary from simple ones such as the Gutenberg-Richter relation (an exponential distribution of magnitudes) to sophisticated models. The following models are commonly used in earthquake statistics:

### Aftershock Statistics

This is concerned with the distribution of aftershocks in time, space and magnitude. It comprises of several models that describe the standard activity of earthquake series. The Omori formula, for example, explains the frequency of aftershocks per unit time interval. It is in the form n (t) = K/ (t+c)^{p} where t stands for the elapsed time since the occurrence of the aftershock (Ogata, 1988). The value of K relies on the lower limit of the intensity of aftershocks estimated in n (t). A plot of the observed n (t) against t (time) on a log-log scale produces a straight line graph whose slope is an approximate value of p (Ogata, 1988). According to Ogata, such a plot done by Utsu in 1969 reveals that the aftershock of the Nobi earthquake progressed for 80 more years with decreasing intensity (1988).

The Gutenberg-Richter law of magnitude of frequency exponentially distributes the magnitudes of the earthquake and takes the form Log_{10}F (M) =a-bM where F is the frequency of earthquakes whose magnitudes are greater than M (Ogata, 1988). Some authors propose a non stationary Poisson process for the construction of statistical models concerning the occurrence of earthquakes. This model assumes no interrelationship between the occurrence times of aftershocks.

### Trigger Models

The trigger models suggest that main shocks are haphazardly disseminated in time and can produce a secondary sequence of events, the aftershocks. The model uses a conditional probability to predict the occurrence of an aftershock as triggered by the main earthquake. This is expressed in the equation σt_{o }(t) = ε.f (t-t_{o}) and t ≥t_{o }where (t-t_{o}) is the time interval between the earthquakes (Ogata, 1988).

### Epidemic-Type Model

This model resembles a population genetics model by Kendal that explains the age-dependent birth and death process. It likens the birth process to the generation of aftershocks.

## Statistical Methods

Statistical methods used in the analysis of data from seismograms include “maximum likelihood errors in variables, robust regression, nonlinear regression, probability analysis, Fourier inference, discrimination, array analysis, point processes, moment functions, inverse problems, bootstrap, and sensitivity analysis” (Brillinger, n.d., p.3).

## Conclusion

Earthquake statistics is an important field in the understanding earthquakes. Despite the impossibility of predicting an earthquake, earthquake statistics can establish the probability of an earthquake occurring. This is useful in reducing the magnitude of losses that result from earthquakes.

## References

Brillinger, D. R. (n.d.). *Statistics of earthquakes*. Web.

*Earthquake facts and statistics.* (n.d.). Web.

Lutgens, F. K., Tarbuk, E.J., & Tasa, D. (2010). *Foundations of earth science* (6^{th} ed.). New Jersey: Pearson College Division.

Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. *Journal of the American Statistical Association*. *83*(401), 9-27.

*What causes earthquakes?* (n.d.). Web.

Wiemer, S. (n.d.). *Earthquake statistics and earthquake prediction research.* Web.