Michell, Richter and Earthquakes

What are earthquakes?

An earthquake, also called a seism, is an abrupt and non-permanent shaking of the earth’s crust. It is produced when a substantial amount of energy is released and can be caused by the activity of geological faults, the friction of tectonic plates, volcanic processes, asteroid impacts or even as a consequence of nuclear detonations.

The point of origin of an earthquake is called the focus or hypocentre. The point on the earth’s surface directly above the hypocentre is called the epicentre.

Depending on its origin and intensity, an earthquake can cause crustal displacements, landslides, tsunamis or volcanic activity.

The three worst earthquakes recorded to date have been the following:

• Chile, 1960: It took place on 22 May 1960. Its magnitude was 9.5 Mw, the largest ever recorded. It caused between 6,000 and 10,000 deaths and more than 2 million people were affected.

• Indonesia, 2004: Occurred on 26 December 2004 off northern Sumatra. It killed 230,000 people and affected Thailand, Malaysia, India, Bangladesh, Sri Lanka, Maldives, Myanmar and Indonesia.

• United States, 1964: On 28 March 1964 in Anchorage, Alaska. Its magnitude was 9.2 Mw and its intensity caused the ground to rise up to 11.5~m over hundreds of thousands of square kilometres. It killed 128 people.

 

On average, there are about one million earthquakes per year on Earth, although the vast majority are so small that they go unnoticed. However, observations since 1990 indicate that at least 18 major earthquakes occur each year.

An earthquake can change the length of the day. Indeed, the energy released can alter the distribution of the planet’s mass and produce a small acceleration in its rotational speed. It is also known that standing water can give off an unpleasant smell and slightly increase in temperature immediately before an earthquake, due to underground gases released as the tectonic plates move. In fact, the presence of these gases can also contribute to local changes in the ecosystem.

Because of the spectacular effects they produce, earthquakes have been made into movies on numerous occasions. One example is the 1974 film “Earthquake”, produced and directed by Mark Robson and starring Charlton Heston and Ava Gardner. Much more recent is the 2018 film of the same name directed by John Andreas Andersen, set around Oslo.

The origin of earthquakes seems to have been discovered in the 18th Century by the British engineer John Michell. Michell designed a balance that made it possible to measure seismic effects for the first time (a primitive seismograph).

Michell is also known for introducing the concept of the dark star, the Newtonian version of what is now called the black hole. Michell’s idea comes from the concept of escape velocity, which on Earth is approximately 40,000 km/h. He imagined a star of enormous density, so much so that there would come a point at which the escape velocity would equal the speed of light. He deduced that, if the star were even heavier and denser, an object could not escape even when moving at the speed of light. This argument was later supported by Laplace, who presented it with details in the first two editions of his book Exposition du Système du Monde. However, in the presence of strong gravitational fields, Newtonian mechanics is not applicable. Thus, Michell’s ideas are now obsolete, superseded by the theory developed by Schwarzschild and Chandrasekhar. We will talk about all this some other time, perhaps in a future post…

Mathematics and earthquakes

Mathematics can help to describe and even predict earthquakes at various levels.

Thus, the probability of the occurrence of an earthquake is given by a Poisson distribution. More precisely, the probability to suffer \(k\) earthquakes of magnitude \(M\) in a period of amplitude \(T\) in a region is given by
$$
\mbox{Prob}\,(k,T,M) = \frac{1}{k!} \left( \frac{T}{T_r(M)} \right)^k
e^{-\frac{T}{T_r(M)}},
$$

where \(T_{r}(M)\) is the return time of an earthquake of intensity \(M\) in the region, i.e. the mean time between two earthquakes of that intensity.

If a detailed description of the evolution of an earthquake is required, models based on PDEs should be used. Accepting that a region of the earth’s crust occupies the set \(\Omega \subset \mathbf{R}^3\) and we are interested in the evolution of an earthquake during the time interval \((0,T)\), it is appropriate to assume that the corresponding displacement field \(u = (u_1,u_2,u_3)\) is a solution in \(\Omega \times (0,T)\) of the linear elastic system

$$\rho(x) u_{tt} – \nabla \cdot (\mu(x)(\nabla u + \nabla u^T) – \nabla (\lambda(x) \nabla \cdot u) = f(x,t)
$$

(together with appropriate initial and boundary conditions for \(u\)). Here, \(\nabla\) is the gradient operator, of components the spatial derivatives; \(\rho = \rho(x)\) is the mass density and \(\mu = \mu(x)\) and \(\lambda = \lambda(x)\) are the Lamé coefficients of the medium (the Earth’s crust); \(f = f(x,t)\) is the field of external forces acting on the region: gravity, thermal and/or electromagnetic effects, etc.

In simplified models, it is assumed that \(\rho\), \(\mu\) and \(\lambda\) are positive constants and \(f \equiv 0\). Then \(u\) takes the form

$$
u = \nabla \phi + \nabla \times \psi \ \text{ with } \ \nabla \cdot \psi = 0,
$$
i.e., a sum of a gradient and a rotational, with \(\phi\) and \(\psi = (\psi_1,\psi_2,\psi_3)\) solutions of wave PDEs in \(\Omega \times (0,T)\):
$$
\phi_{tt} – \alpha^2 \Delta \phi = 0, \quad \psi_{tt} – \beta^2 \Delta \psi = 0,
$$
where \(\alpha^2 = (\lambda+2\mu)/\rho\), \(\beta^2 = \mu/\rho\).

It is often said that \(u\) is the sum of a P-type wave (the \(\nabla\phi\) component, also called a longitudinal or primary wave) and an S-type wave (the \(\nabla\times\psi\) component, a transverse or secondary wave).

Examples of \(\phi\) and \(\psi\) functions are

$$
\phi(x) = A \cos(x_1 – \alpha t), \quad \psi_1(x) = 0, \quad \psi_2(x) = \psi_3(x) = A \cos(x_1 – \beta t).
$$
This gives rise to the following P- and S-waves:
$$
\nabla\phi = (-A \sin(x_1 – \alpha t), 0, 0), \quad \nabla \times \psi = (0,A \sin(x_1 – \beta t),-A \sin(x_1 – \beta t)).
$$
In the first case, we are considering a wave motion propagated in the direction of the \(x_1\) axis, the same direction in which the particles vibrate, at velocity \(\alpha\) (which in practice ranges from 8 to 13 km/s), of amplitude \(A\). These waves travel through the interior of the Earth, through liquids and solids, and are the first to be recorded by measuring devices or seismographs.

In the second case, we are referring to a slower wave, of speed \(\beta\) (4 to 8 km/s) and equal amplitude, propagating in directions contained in the \(x_2\,x_3\) plane, perpendicular to the direction of vibration of the particles, capable of propagating only through solids.

The interaction of a P-type wave and an S-type wave often produces waves of a third type on the Earth’s surface. The latter propagate with a lower speed (about 3.5 km/s) but, because of their location, produce the greatest damage. The determination of \(u\) is in many cases possible with suitable numerical methods.

What a seismograph can tell us

Since they propagate at different speeds, P-type waves arrive earlier than S-type waves at the observation points.

Actually, by measuring the time interval from the arrival of a P-type wave to an S-type wave, we can easily find the distance from the seismograph to the epicentre. Naturally, by means of a classical triangulation procedure, the measurements of three unaligned seismographs allow us to locate this point.

On the other hand, the magnitude of an earthquake is a measure of the energy it releases. It is quantified by a value on the logarithmic Richter scale, usually between \(2\) and \(9\). The procedure followed to calculate this value is as follows:

• The value \(3.0\) is set for a seismic wave that can be detected with amplitude \(1\) mm at a distance of \(100\) km from the epicentre.

• Any other measurement of magnitude is carried out with the above data as a reference.

• For example, an earthquake detected at the same distance from the epicentre with amplitude \(10\) mm is assigned the value \(4.0\).

Earthquakes of magnitude \(8.0\) or greater are rare and very destructive.

Knowing the location of the epicentre and the magnitude of the earthquake, we can get an idea of its intensity or severity at each point.

Figures – 1 and 2: The effects produced by an earthquake; 3: John Michell (1734-1793); 4: Charles Ritcher (1900-1985); 5: Triangulation process and determination of the epicentre; 6: Determination of the magnitude of an earthquake on the Ritcher scale.

Learn more

• R. Ashler, “The seismic wave equation”, https://fliphtml5.com/kjno/fdic/basic

• C. H. Chapman, “Fundamentals of seismic wave propagation”, Cambridge University Press, Cambridge, 2004.

• Handbook of geomathematics. Volume 1, 2, Edited by Willi Freeden, M. Zuhair Nashed and Thomas Sonar. Springer-Verlag, Berlin, 2010.

• Kennett, B. L. N. The seismic wavefield. Vol. I, Introduction and theoretical development. Cambridge University Press, Cambridge, 2001.