ISSN:
1420-9136
Keywords:
Apparent magnitude of earthquakes
Source:
Springer Online Journal Archives 1860-2000
Topics:
Geosciences
,
Physics
Notes:
Abstract The apparent magnitude of an earthquakey is defined as the observed magnitude value and differs from the true magnitudem because of the experimental noisen. Iff(m) is the density distribution of the magnitudem, and ifg(n) is the density distribution of the errorn, then the density distribution ofy is simply computed by convolvingf andg, i.e.h(y)=f*g. If the distinction betweeny andm is not realized, any statistical analysis based on the frequency-magnitude relation of the earthquake is bound to produce questionable results. In this paper we investigate the impact of the apparent magnitude idea on the statistical methods that study the earthquake distribution by taking into account only the largest (or extremal) earthquakes. We use two approaches: the Gumbel method based on Gumbel theory (Gumbel, 1958), and the Poisson method introduced byEpstein andLomnitz (1966). Both methods are concerned with the asymptotic properties of the magnitude distributions. Therefore, we study and compare the asymptotic behaviour of the distributionsh(y) andf(m) under suitable hypotheses on the nature of the experimental noise. We investigate in detail two dinstinct cases: first, the two-side limited symmetrical noise, i.e. the noise that is bound to assume values inside a limited region, and second, the normal noise, i.e. the noise that is distributed according to a normal symmetric distribution. We further show that disregarding the noise generally leads to biased results and that, in the framework of the apparent magnitude, the Poisson approach preserves its usefulness, while the Gumbel method gives rise to a curious paradox.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00877017
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