In this blogpost we will try and explain how to calculate earthquake epicenters with examples of measurement systems and methodologies.

## Tectonic plates

The Earth can be subdivided into different ‘floating’ parts called tectonic plates. This subdivision follows three different main mechanism as explained in the figure below outlining the tectonic plates.

These tectonic plates move around on the earth in unpredictable patterns that build up stress throughout the solid medium. When the built up stress releases the earth moves and we have an earthquake.

These event are extremely energetic releasing immense amounts of energy. The release of energy causes widespread destruction as seen in the Turkey earthquake from February 2023, the Kahramanmaras earthquake.

**Related read:** *top ten misconceptions about earthquakes.*, *Effective Investment strategies for climate adaptation*

## Definition of a p(ressure)-wave

Firstly, lets define the definitions of pressure and shear waves, the so-called p and s-waves.

The definition of the pressure, p-wave is a pressure type underlying wave moving through the solid medium. The direction of propagation is dominated by compression\expansion pressures. The particle path is first compressed and then elongated.

The p-wave is the fastest moving pressure wave throughout the solid medium. The initial disturbance moves quickly through the solid medium. It emanates from the area of interest; the epicenters of earthquakes.

## Definition of a s(hear)-wave

The shear type, s-waves, is different from the p-wave. Contrary to p-waves, shear waves moves in the direction of propagation through shearing motions. Shearing waves causes particle motions to move in a up-and-down type of fashion. In addition to this, the s-wave is slower than the pressure waves as shearing of solid medium poses more resistance.

The shearing causes changes in the underlying soil characteristic and in extreme cases causing earth ruptures. These waves emanate in all-directions arising from the epicenters of earthquakes.

## Definition of a love wave

Thirdly, we have the Love type surface waves. Contrary to p and s-type waves, this is a surface wave. In other words surface waves have particle motions varying with depth, where deeper particles move less, than upper surface particles. As an example of the L-type wave, see the figure below.

Important to realize is that Love type waves are slower than both p- and s-type waves and are present at the surface. In particular the particle motions which are horizontally varying with depth causes a diagonally shearing with depth. Thus L-waves require a solid material in order to be able to propagate. Conversely L-type waves are unable to propagate through liquid media. Therefore does the presence of L-type waves throughout the earth constitute an underlying measure for the existence of a liquid earth core. This is one of the reasons why we believe the core of the earth is liquid due to the absence of L-waves across the earths solid media.

## Definition of a Rayleigh wave

Fourthly, we have the Rayleigh wave. This type of wave is similar to the love wave in the sense that it is a surface wave traveling across the globe emanating from the epicenters of earthquakes.

The Rayleigh waves propagate in a fashion similar to ocean waves with horizontal displacements in direction of propagation varying with depth. An illustration explaining the wave propagation is shown in the figure below.

The Rayleigh waves are slower than love type waves but are also causing surface changes and therefore affects buildings, infrastructure and people alike.

The particle motion is orbital meaning that it changes with depth. The Rayleigh waves are able to happen across liquid and solid media alike, thus differing significantly from L-type waves.

Now that we have introduced the four types of earthquake waves we now move further into measuring these different types of waves. For this scientists and engineers utilize the physical instrument called the seismograph.

## Seismograph

The seismograph is an instrument utilized in measurement engineering when trying to estimate and predict the onset of earthquakes. It works by having giant ear-like sounding equipment directed towards the earth. The resulting vibrations ‘heard’ are measured with high accuracy and mapped onto a seismogram see Figure 2.

From the picture you can see the characteristic black line indicating differences in surface vibrations measured in time. The vibrations consist of a superposition of p-, s-, L- and R-waves all mixed together onto the seismogram. The amplitude and phases differs based on the type of waves allowing engineers and scientists to differentiate individual wave types.

## Seismogram

The seismogram allows us to analyze the differences in amplitude period and type of vibrations. In addition it allows interpretation by seismologists geologists or similar experts. Furthermore the interpretation include the characterization of individual waves. An example of wave characterizations is shown in Figure 3.

From Figure 3, we can see that the earliest sign of an earthquake is the p-wave, followed by the s- and surface waves (Love and Rayleigh). Amplitudes of the surface waves compared with the p- and s-waves are significantly larger making them easily felt by people on the ground. Furthermore the p and s-waves are barely measurable and have significantly faster arrival times compared with surface waves.

With the introduction of the seismograph and seismogram we now continue with the mathematical description of the distance calculations following the narrative with Euclidean distances.

## Euclidean distance

An Euclidean distance is the observed straight-line connection between points. It is calculated in two dimensions as the difference between points x1, x0 and y1, y0 denoting end and startpoints respectively.

(1)

Now as we live and breathe in three-dimensions, (four counting time) we need to include a third coordinate in our calculations of distance. For this purpose we include the coordinate z1 and z0 in a similar manner.

Now we can calculate the distance between two points in space. Finally we need to compare different locations in time to accurately calculate the earthquake epicenters.

## Triangulation methodology

With the knowledge of how to calculate a distance between points, now we can move on to calculating the epicenters location through use of time-dependent triangulation methodologies. The location of epicenters in three-dimensional space corresponds to accurately figuring out a location based on satellite measurements. As the medium we consider is solid state matter, we need estimations of sound speeds through the media and for this we need seismologists, geologists and similar experts.

## Seismologists, geologists and experts

An important parameter is the time-aspect of figuring out exactly how to do the determination of the individual epicenters. By comparing measurements of seismograms through time, one can calculate the arrival time of individual p-, s-, L-, and R-waves. As the speed of sound is assumed through specific soils this leaves the only unknown to be distance which is solved for numerically.

This type of analysis is carried out by seismologists, geologists and similar experts with knowledge about wave propagation speeds throughout solid state media and how to interpret the individual seismograms. Now lets consider the epicenters themselves

## Epicenters of earthquakes

With the advent of seismographs, seismograms and specialized scientific methodologies for accurately calculating the measured distances and time-variation of seismograms allows the calculation of epicenters of earthquakes.

The epicenters are important parameters to understand as they allow for the construction of the tectonic plates. They are characterized by experts whose knowledge is used to construct the tectonic plate theory. Finally understanding epicenters help when trying to understand the earths complex composition of solid state matter.