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Recently a renewed interest has arisen on the old problem of magnitude scale conversions (Castellaro et al., 2006; Deichmann, 2006; Utsu, 2002). The use of earthquake data which has been compiled at different time stages, subjected to different magnitude scales or from different station networks requires dealing with the concept of magnitude homogeneity. It is clear that any statistical analysis which is carried out on flawed data can yield biased or useless results. For reliable estimates of seismicity is also necessary to make use of as much data as possible in light of the short time span of modern observations based on digital records. Nevertheless, many people still employ magnitude conversions based on linear regressions without regard for the type of original observations which they stem from. This situation can be exacerbated when dealing with magnitude scales which are drawn upon incompatible measurements at a particular magnitude range. Such is the case, for example, with body wave magnitude and surface wave magnitude regressions at the M~4 level. In this study we compare relations obtained through direct linear regressions with those based on the preservation of Gutenberg-Richter Law (i.e. a and b values). The idea is to provide a relation which preserves the same Gutenberg Richter (Ishimoto-Ida) relation as that of the most robust magnitude set.

In what follows we show results of the tests for the three mentioned regions. For every case we first we show epicentral maps of the data (mb and Ms from ISC) and then we show comparisons of G-R distributions for the original magnitude data (A), and for a conversion of mb into Ms following a linear regression (B) and the G-R approach (C).



New Zealand








We base the study on the method proposed by Zúñiga and Wyss (1995) which was originally intended for finding suitable corrections to temporal magnitude variations introduced artificially. This is based on finding a correction which follows a linear relationship between two magnitudes:

M’ = c ∙ M + d(1)

while preserving the Gutenberg-Richter (G-R) relation of the magnitude we regard as most robust (M’). M is the magnitude we wish to convert.

It follows that constants c and d in (1) can be extracted from the a and b values of the G-R distributions for both magnitudes under consideration:

Log N = a – b∙M

Log N’ = a’ – b’∙M’, with:

The method rests on the reliability of the a and b value estimates, so we tested several approaches including Maximum Likelihood, Least Squares and a combination based on the different determinations of the minimum magnitude of completeness (EMR, Shi and Bolt, maximum curvature of histogram, etc). Notice that we do not require that the catalogue include both magnitude estimates for every event.

Converting Magnitudes based on the preservation of the

Gutenberg-Richter relation as compared to linear regression

Ramón Zúñiga

Centro de Geociencias, National Autonomous University of Mexico (UNAM), Juiriquilla, Mexico.




As a final test we employ event data which includes magnitude estimates for both scales, and observe the fit of the two relations (linear and preserved G-R). In frame D, the results for Italy show how there is a tendency for the fit to be best at the largest Ms values. The case of Mexico (E) does not show a large deviation between the direct linear regression and the relation obtained y G-R preservation, probably due to the large amount of data in the reliable range for both magnitude scales. For the New Zealand case (F), the two relations depart widely, but the one based on the G-R approach follows closely the larger Ms data.





We found that the method provides an excellent way to deal with the problem of magnitude conversions, specially for those situations where we do not count with two magnitude estimates for the same event data (which occurs most of the time). By employing different ways to calculate a and b values we can also test the completeness of the magnitude data since in some instances the results do not provide a good match to the magnitude data. Care has to be exercised when employing least squares so the appropriate magnitude range is selected.


We tested the procedure with data sets drawn from the International Seismological Centre (ISC) catalogue for three distinct regions: Italy, Mexico and New Zealand, which have different properties, helping to illustrate the advantages and drawbacks of the method.