# Earthquake population statistics and recurrence intervals

### How many earthquakes?

Here is an important observation about earthquake populations worldwide: earthquakes of a given magnitude happen about 10 times as frequently as those one magnitude unit larger.

Annual earthquake population statistics compiled by the USGS. Figure source: http://earthquake.usgs.gov/earthquakes/eqarchives/year/eqstats.php#table_2

Earthquake populations approximately follow this relationship:

log

N=a-bM.

This is a power-law equation in which *N* is the number of earthquakes whose magnitude exceeds *M* and *a* and *b* are constants. For the majority of earthquake catalogs, the constant *b* is approximately equal to 1. When b ≈ 1, this equation describes a line whose slope is about -1.

Seismologists can test the validity of the equation above using catalogs of earthquakes to make "frequency-magnitude diagrams." These diagrams show how many earthquakes of a given magnitude there are in a population of earthquakes.

Now observe the plot below. For two separate catalogs of earthquakes that occurred in the New Madrid region, magnitude is plotted vs. the mean annual occurrence of earthquakes greater than or equal to a given magnitude. This plot is only different from the example plot above in that the N values on the y-axis have been normalized to one year. This is so two catalogs that span different lengths of time can be compared directly.

The figure above explained out loud! (and a close-captioned version)

Both of the curves in the plot above deviate from a straight line relationship log *N* = *a* - *bM *at small magnitudes. For the Nuttli Catalog, the line has a slope of about -1 at magnitudes greater than 3.5 and for the NMSZ catalog, the line has a slope of about -1 for approximately magnitude 1.5 and greater. Doesn't it look like in the Nuttli Catalog, there are the same number of magnitude 2 earthquakes every year as there are magnitude 3 earthquakes? But didn't we say that there should be ten times more magnitude 2's? What's going on?

Furthermore, how come there aren't any big earthquakes in this plot? The New Madrid Seismic Zone (NMSZ) catalog peters out at about magnitude 5 and the Nuttli Catalog doesn't have anything much over magnitude 6. But we know there have been big earthquakes in this region in the past, (or else why argue about seismic risk here), so where are they?

The answer to both of these problems is simply that any catalog of earthquakes is limited in two ways. The first way is that not every piece of the Earth has a seismometer sitting on it, therefore there will be some small earthquakes that don't get recorded, even though they happened. For most catalogs, some standard is applied with regard to how many seismometers have to record an earthquake in order to include it in the catalog. This is for quality control reasons. It is hard to locate an earthquake and calculate its origin time within acceptable error limits if not enough stations recorded it. Therefore, the farther apart the seismometers are, the fewer small earthquakes will end up being included in the catalog. For the Nuttli Catalog, we can say that the catalog is incomplete below the threshold of M ≈ 3.5 because that is where the slope of the line (or the "b-value") begins to deviate from -1. The threshold for the NMSZ catalog is lower. Why do you think this is?

The second way a catalog is limited is that it is finite in time. Let's say for a given region, magnitude 8 earthquakes happen once every 1,000 years or so. If your catalog only spans 10 years, how likely are you to have a magnitude 8 in your catalog? For that matter, how likely are you to have a magnitude 7 in your catalog? How many magnitude 6's can you expect in 10 years? In the plot above, the time ranges for both catalogs are listed on the plot. Why does the NMSZ catalog have a lower maximum magnitude than the Nuttli Catalog?

### Calculating a recurrence interval from a seismic catalog

In order to assess seismic risk, we want to know how often a large earthquake happens in this region. How do we do that if our seismometers haven't ever recorded a big earthquake? We have to extrapolate using the data that we do have. Extrapolation is a tricky business, because small uncertainties turn into huge uncertainties the farther away you get from what you've actually measured. For a catalog of seismicity, we rely on the assumption that the relationship log *N* = *a* - *bM *holds true over all magnitudes and times. We then extend our catalog data into the realm of the unknown and predict how often large magnitude earthquakes are expected. In the plot above from Newman *et* *al*., 1999, they use dashed lines to show their extrapolations. How often do they predict a magnitude 7 will happen in the NMSZ? What about a magnitude 8? What uncertainties do they associate with these predictions?