# How to compare earthquake magnitudes

On the USGS website, it says

Because of the logarithmic basis of the scale, each whole number increase in magnitude represents a tenfold increase in measured amplitude;

However my book says this:

Like many other scales in science (decibel is another example), Richter scale is not linear. This means that an earthquake of magnitude 6 does not have one and half times the destructive energy of an earthquake of magnitude 4. In fact, an increase of 2 in magnitude means 1000 times more destructive energy. Therefore, an earthquake of magnitude 6 has thousand times more destructive energy than an earthquake of magnitude 4.

But based upon the USGS information, an increase in 2 magnitude would be 10 x 10 = 100 stronger.

Now which one is correct?

Taken straight-out from Prof. L. Braile's collection of earthquake hazard information, that explains it quite nicely:

The magnitude scale is really comparing amplitudes of waves on a seismogram, not the STRENGTH (energy) of the earthquakes.

While one unit of magnitude is 10 times the amplitude on a seismogram, one unit of magnitude represents $10^{1.5}$ times (approximately 32 times) the energy, based on the long-standing empirical formula

$$\log{E} \propto 1.5M$$

thus

$$E \propto 10^{1.5M}$$

where $E$ is energy and $M$ is magnitude.

The example set in the webpage is to compare how much bigger would a 9.7 magnitude earthquake be in comparison to a 6.8 magnitude earthquake:

1. The magnitude scale is logarithmic, so a magnitude 9.7 earthquake is $\frac{10^{9.7}}{10^{6.8}} = 794.328$ times bigger on the seismogram than a magnitude 6.8 earthquake.

2. Measuring the change in energy, however, shows that the 9.7 earthquake is $\frac{{10^{1.5}}^{9.7}}{{10^{1.5}}^{6.8}} = 10^{1.5 \times (9.7-6.8)} = 22~387$ times the energy of a 6.8 earthquake.

Comparison of earthquake "size" on seismograms: this image from Nevada Seismological Lab's Living with Earthquakes in Nevada

A graphical comparison of earthquake energy release, a video by the US NWS Pacific Tsunami Warning Center (PTWC).

The USGS has a calculator for magnitude difference.

And there's a calculator for seismic energy and several equivalences. Unfortunately, most of the equivalences are related to the USA.

See Brian Hill's answer that explains that with the original Richter scale, one unit of magnitude is apprimately $31^{1}$ times the energy, and two units of magnitude are $31^{2}$ times the energy, and so on.

In short: you are half-right (one unit of magnitude it is 10 times bigger on the seismogram, not in energy), and the book is right (two units of magnitude are 1000 times stronger).

• So what you're saying is that a magnitude difference is indeed 10 times the shaking, but 32 times the energy/damage? Sounds reasonable to me. Similar to a small increase in wind leading to a much greater increase in damage. – JeopardyTempest Feb 13 '17 at 0:42

The base of the Richter scale is empirical. In terms of energy released, the best fit for the base turns out to be about 31.

Since 31**2 is close to 1000, two points on the Richter scale e.g., from 4 to 6 or from 5 to 7, corresponds to about 1000 times the energy release. Each point piles on another factor of (empirically about) 31.

This is so widely misquoted on the web that it is becoming sort of urban legend that the base of the logarithm is 10. To see people trying the empirical best fit and arriving at a base of about 31, you need to dig into the scholarly literature.

UPDATE: Although the main wikipedia titled "Richter magnitude scale" isn't very helpful, it turns out there is another article titled "Moment magnitude scale" that has the quantitive explanation. It might be helpful when reading that article to keep in mind that although e and 10 are the most common logarithmic bases, you can define a logarithm with any base. For example, software people might use that log base 2 of 64 is 6 (this is 100% equivalent to saying 2^6 = 64). The upshot is that seismologists have found that using 10^1.5 as the base for the modern scale fits the original Richter scale well, and 10^1.5 is about 31.62.

## protected by Community♦Dec 30 '17 at 4:15

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