# Earthquake frequency and magnitude relationship

The VG properties of the seismic sequences have been put in relationship with the typical seismological parameter, the b-value of the. The relationship between frequency and magnitude in homogeneous series of earthquakes is studied. The linear relation of Gutenberg and Richter between the . Statistics of earthquakes in California show linear frequency‐magnitude relationships in the range of M to M for various data sets.

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 curve labeled "Nuttli Catalog" shows historical earthquake data magnitudes mostly estimated from shaking reports and the curve labeled "NMSZ catalog" is digital data from the CERI catalog you are working with in this exercise. The figure above explained out loud! For the Nuttli Catalog, the line has a slope of about -1 at magnitudes greater than 3. 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?

### Gutenberg–Richter law - Wikipedia

Furthermore, how come there aren't any big earthquakes in this plot? But we know there have been big earthquakes in this region in the past, or else why argue about seismic risk hereso 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.

• Earthquake population statistics and recurrence intervals
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• Gutenberg–Richter law

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.

Plotting earthquake frequency vs magnitude in Excel 2013 (Windows)

Let's say for a given region, magnitude 8 earthquakes happen once every 1, years or so. If your catalog only spans 10 years, how likely are you to have a magnitude 8 in your catalog? The relation a has been obtained carrying out a simple model based on a number of assumptions, among which the more char- acterizing is the existence of a maximum regional finite magnitude value Mp.

This assumption, derived by an evidence recognized by most seismologists, allows a simple explanation of the known behavior of the experimental cumulative frequency-magnitude graphs. Finally, some results of application of the model to six seismic regions are presented. This relation generally agrees quite well with the experimental data, although in a narrow magnitude range Cosentino, The above relation also has been ex- tensively applied to cumulated experimental data.

Furthermore, many works carried out in order to explain the physical meaning of b parameter Mogi, ; Scholz, ; Wyss, have shown the importance of its reliable evaluation. Lately, it has been observed that the relation 1 could not be tested in all instances.

In particular, the general behavior of the experimental data in simple frequency form shows dis- crepancies in low magnitude range, while that in cumulative frequency representation allows observation of deviations in the high magnitude range, which also can be noticed by microfracturing in rock experiments Scholz, In order to account for the above mentioned misfit in low magnitude range, the B.

This relation does not account for the lately observed falling off of the earthquake frequencies as the highest observed magnitudes are ap- proached Richter, The use of a threshold magnitude value, equal to or higher than 4. Therefore, a simple exponential model is proposed in order to modify this relation, starting from a suitable threshold magnitude value M0 and taking into account the behavior of the experimental data in the high-magnitude range.

The model is based on the following assumptions: Assumption a is obviously stringent in the light of present knowledge. Assumption b is acceptable enough if the aftershocks are eliminated from the statistics Powell and Duda,in agreement also with the evidence that the rate of occurrence of earthquakes is well approximated by a Poisson distribution Lomnitz, ; Epstein and Lomnitz, ; Radu, In fact, the long-term stability of mean local magnitude of earthquakes has been shown by many authors Lomnitz, Assumption d has been well stated by most seismologists.

As a matter of fact, Richterin considering the behavior of the frequency-magnitude experi- mental data, asserts that "a physical upper limit to the largest possible magnitude must be set by the strength of the crustal rocks, in terms of the maximum strain which they are competent to support without yielding. Such an upper limit is likely to be a function of maximum source size in the Earth's crust and the upper mantle.

Assumption ddiscussed also by Riznichenko, is actually the more characterizing assumption in our model: It must be pointed out that, if a maximum possible magnitude in each seismic region exists, it should be introduced in the probability density distribution function. Derivation of the distribution function.

Let us have a set of N independent seismic events referring to the seismic region R during AT years, the lower magnitude value being M11,i, and the highest M. The following conditions must be satisfied. The first truncation point M0, representing the threshold magnitude value chosen for the statistics, should satisfy the above mentioned characteristics, while the second truncation point Mp is an unknown parameter representing the maximum possible magnitude in the region R as stated in the physical assumption d.

The characteristic function of the distribution, say C i vis given by: LUzIO from which it is possible to derive the first and the second moments: The visualization of relations 5 and 10 is given on semi-logarithmic representa- tion in Figure 1.

Furthermore, in order to visualize the behavior of relation 10Figures 2 and 3 show two sets of this function obtained for several values of fl and Mp parameters.

From these figures it is possible to observe the influence of each parameter on the shape of the curve.

## Gutenberg–Richter law

Estimation of the parameters. B e h a v i o r of the cumulative function F M for several values. H i g h e r values cor- respond to higher slopes of the curve in the low m a g n i t u d e range, and produce a t r e n d to the linearity in the same range.

These equations are soluble, by means of numerical B. Behavior of the cumulative function F M for several values. Higher Mv values produce a trend to the linearity, with lower slopes of the curve, in the low magnitude range.