# Why the sizes of earthquakes multiply rather than add up

## Magnitude is derived from a formula involving a logarithm, which makes the scale exponential rather than linear.

A Mexican woman in her home devastated by the September 19 earthquake. (Source: AP)

A series of earthquakes in Mexico this month has led to inevitable comparisons of size. At magnitude 8.1, the September 8 earthquake was bigger than the September 19 earthquake (7.1), which was bigger than two earthquakes that followed on September 24 (6.1 and 4.5). But how much bigger than the other was each? And what does this magnitude express?

The earthquake magnitude scale gives a measure of the energy released, although the reading in itself is not of energy. Magnitude is a dimensionless number — no physical units — that is derived from a formula involving a logarithm, which makes the scale exponential rather than linear. The difference between 6.1 and 7.1, therefore, is not the same as the difference between 7.1 and 8.1.

In terms of the size of seismic waves, an increase of 1 on the scale corresponds to a tenfold increase in a quantity called wave amplitude. In other words, an earthquake of magnitude 8.1 is 10 times larger than one of magnitude 7.1, and 100 times larger than one of magnitude 6.1. In terms of energy released, on the other hand, magnitude 8.1 is 31.623 times stronger than magnitude 7.1, and 1,000 times (31.623 times 31.623) stronger than magnitude 6.1.

What it isn’t

It’s important not to confuse an earthquake’s magnitude with its intensity. They are different measures. And while the Richter scale is widely quoted, the modern magnitude measurement uses a different scale. The difference between intensity and magnitude is explained in clear, simple terms in a document released by IIT Kanpur and Building Materials and Technology Promotion Council, Delhi. Magnitude is a quantitative measure of the size of an earthquake, it says, while intensity is a qualitative measure of the shaking at a given location.

Two scales are commonly used for intensity, the Modified Mercalli Intensity scale and the MSK scale, both of which classify earthquakes from I (least perceptible) to XII (most severe). These readings are based on factors such as how people perceive the shaking. The same earthquake will have different intensity readings at different places; the farther one moves away from the epicentre, the less intense the shaking.

To understand how the modern magnitude scale works, it helps to look at the Richter scale, even though it has gone out of fashion. The reading on the Richter scale too is derived from a formula that involves a logarithm (base 10). It takes into account wave amplitude, and variations in distance between various seismographs and the earthquake epicentre. It is because of the base-10 logarithm that the “largeness” of an earthquake — its wave amplitude — multiplies in 10s when the magnitude reading rises in 1s.

What it is

The accuracy of the Richter scale, however, is limited to medium-sized earthquakes. Besides, the measurement depends on distance from the epicentre. The modern scale seeks to overcome these shortcomings while maintaining as much parity with the Richter scale as possible. It measures a quantity called “moment magnitude”, which is based on variables such as the area of the fault’s rupture, slippage along the fault and the size of the seismic waves.

For medium-sized quakes, say magnitude 5, the reading will be similar on the Richter and the moment magnitude scales. Also, as with the Richter scale, an increase of one step on the moment magnitude scale corresponds to an energy-release increase of 31.623 times (10 raised to the power 1.5). And when the magnitude increases two steps, it corresponds to an increase of 10 raised to the power 3, or 1,000 times in released energy.

A magnitude-6.3 quake releases as much energy as the atomic bomb dropped on Hiroshima. Only a fraction of it, however, will actually be felt. As the IITK-BMTPC document notes, most of the energy released during an earthquake goes into heat and fracturing rocks. More than the actual energy, the magnitude scale’s first utility is to indicate how one earthquake compares with another.

While one can compare the largeness and strength of two earthquakes with a calculator that can work with exponents, online tools make it easier. The US Geological Survey web site has a facility where one can input the magnitudes of two earthquakes, after which it returns figures for how many times one is larger, and stronger, than the other.