A note on earthquake magnitudes

Matt Yglesias writes that he “hear[s]” the Chilean earthquake is 1000x more powerful than the Haitian one. I get the feeling that a lot of people know that the Richter scale is logarithmic, but it’s not clear that they always know how to convert that back into raw units. The estimable Mr. Yglesias, for instance, shouldn’t need to “hear” that it’s 1000x more powerful; he should be able to figure it out on his own. (I get similarly vexed when people can’t compute tips at restaurants on their own.)

The USGS pages on the Chilean earthquake and the Haitian one mention their magnitudes (8.8 and 7.0, respectively) and give a helpful explanation of what that means:

Seismologists indicate the size of an earthquake in units of magnitude. There are many different ways that magnitude is measured from seismograms because each method only works over a limited range of magnitudes and with different types of seismometers. Some methods are based on body waves (which travel deep within the structure of the earth), some based on surface waves (which primarily travel along the uppermost layers of the earth), and some based on completely different methodologies. However, all of the methods are designed to agree well over the range of magnitudes where they are reliable.

Preliminary magnitudes based on incomplete but available data are sometimes estimated and reported. For example, the Tsumani Centers will calculate a preliminary magnitude and location for an event as soon as sufficient data is available to make an estimate. In this case, time is of the essence in order to broadcast a warning if tsunami waves are likely to be generated by the event. Such preliminary magnitudes, which may be off by one-half magnitude unit or more, are sufficient for the purpose at hand, and are superseded by more exact estimates of magnitude as more data become available.

Earthquake magnitude is a logarithmic measure of earthquake size. In simple terms, this means that at the same distance from the earthquake, the shaking will be 10 times as large during a magnitude 5 earthquake as during a magnitude 4 earthquake. The total amount of energy released by the earthquake, however, goes up by a factor of 32.

So then the amount of shaking in a magnitude-7.0 earthquake is 107, which is 10 million. A magnitude-8.8 earthquake will feature 101.8 times as much shaking as the magnitude-7.0 one. 101.8 is less than 102, which is 100. So the amount of shaking is nowhere near the 1000x that Mr. Yglesias heard.

But then the USGS also notes that the amount of energy goes up by a factor of 32 for every 1-unit increase in the Richter scale. So then there’s 321.8, or 512x, as much energy in a magnitude-8.8 earthquake as in a magnitude-7 one.

P.S.: I found an Ezra Klein piece that I was looking for before, where he suggests that he also doesn’t know what “logarithmic scale” means:

The devastation in Haiti was not just because the earth shook, and hard. The quake there was 7.0. Harder than the 6.5 quake that hit Northern California a day before (remember, though, that the Richter scale is logarithmic, so 7 is many times harder than 6.5)

If we’re talking about the magnitude of the shaking, the Haiti quake was 10.5 times as strong as the California one. You may remember that “x to the 0.5 power” is the same as “the square root of x.” To get your back-of-the-envelope-math muscles working, recall that “the square root of x” means “the number which, when squared, equals x.” The square of 3 is less than 10, and the square of 4 is more than 10, so the Haiti quake shook things somewhere between 3 and 4 times as hard as the California quake. As measured by raw power, Haiti’s quake was 320.5 times as powerful as California’s, meaning somewhere between 5 and 6 times as powerful.

“Many” has no exact definition, of course, but I doubt most people would say that “many times harder” means “between 3x and 6x as hard.”

8 thoughts on “A note on earthquake magnitudes

  1. Ryan

    From the Wiki page: “The energy release of an earthquake, which closely correlates to its destructive power, scales with the 3⁄2 power of the shaking amplitude.”

    The energy released was ~500 times greater in the Chile earthquake than the Haitian, so my knowledge of earthquakes, which consists of the aforementioned quotation, suggests that 500 is the more operative scaling factor than 10^1.8, or 60. Granted, still not 1000x, but closer.

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  2. Adam Hyland

    Depending on your leanings, folks far more (less) esteemed than Matt Yglesias have fallen into the innumeracy trap. One prominent example in my mind is the majority opinion in Winter v. Natural Resources Defense Council. Chief Justice Roberts substantially misunderstands how energy propagates through water, claiming that the drop off is exponential and not polynomial. Some attention is lavished on the point in the opinion.

    http://scienceblogs.com/goodmath/2008/11/innumeracyandtheus_supreme.php

    I’m not justifying the slip with that example, but I find people aren’t too clear on what “logarithmic scale” means and folks who have some understanding retreat to the relative safety of comparing whole numbers. I remember running into heuristics for guessing at log-scale values while I was in the Navy. pH is logarithmic and sailors who needed to calculate pH of fluid systems would cluster guesses around whole numbers rather than compute the actual resulting ion concentration.

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  3. Adam Hyland

    Addendum to the comment above, the link SHOULD have the titles words separated by underscores, but just one of them.

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  4. Justin

    Not to get all picky on math here, but that you start out by being condescending to Matt Yglesias for suggesting the Chile earthquake was 1000x as powerful, and then concluding that it was actually 500x as powerful (since, really, if you’re going to be scientifically correct, power is linear with energy), you’re not exactly making a strong point. He never said “the Chile earthquake had 1000x as much shaking as the Haiti one,” he said it was “1000x more powerful.” And he’s not that far off.

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  5. slaniel Post author

    Justin: is it especially controversial that most people don’t know how to convert from a logarithmic scale to a linear one? Am I wide of the mark here if I suggest that, based on what he wrote, Yglesias himself doesn’t know how to do this? Is it not clear, based on what he wrote, that he himself couldn’t figure out how to do this himself, and had to rely on others to figure it out for him?

    I don’t think any of this is controversial, and I submit that there’s nothing really to debate here.

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  6. Alex

    While Justin is correct to say that Yglesias was talking about power, and that power is linear with respect to energy, that does not mean that the Chilean quake was “500x as powerful”. Power is the rate of change of energy with time i.e. only if the energy released during the two quakes was done over the same time period could you say that it was “500x as powerful”.

    Saying that, Yglesias was using “powerful” in the colloquial sense, and no the physics sense. I guessed that he was, but after heading over there, he’s updated with a link to here, quoting your shaking calculation (although he hasn’t copied the superscript so it looks daft). So power=amount of shaking.

    Either way, Yglesias got it wrong.

    And, if Wikipedia is to be believed, the Richter scale isn’t used so much now:

    http://en.wikipedia.org/wiki/Momentmagnitudescale

    But it is in the media for some reason.

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  7. Jake Russ

    Yglesias and Klein hold everyone else to a higher standard, so they themselves should be no different. But it is understandable how they got it wrong. And CNN. And myself.

    If you blindly trust a source that says the difference between a 7.0 and 8.0 earthquake is 32 times. Then getting to 1000x is easy. 8.8 is getting automatically rounded to 9.0. And then in a matter of seconds inside one’s head you think the following equation 32 (32x) = 1024*x.

    The math has been done “right” even though as you show, the math doesn’t make sense for the application to quakes and the scale.

    But really you don’t even need to know anything about logs to get this done. Wikipedia lists the energy yields at the page you link to above: http://en.wikipedia.org/wiki/Richtermagnitudescale

    In the second table it lists Haiti’s earthquake at 32 megatons. And the Chile earthquake at 15.8 gigatons. A simple conversion gets this done for you. 15.8 gigatons –> 15800 megatons. So now it’s just simple arithmetic, (15,800 megatons/32 megatons) = 493.75.

    Or about ~500x more energy yield. Whatever that means for shaking power is up to the rest of you.

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