Abstract
The maximum likelihood estimation of theb parameter in the Gutenberg-Richter relation is extended to the case of uncertain magnitude. An interval which contains the real unknown magnitude is used rather than the uncertain magnitude itself. The proposed approach is very flexible, it allows for the combination of the parts of a catalog of different quality into a single minimally biased set of recurrence parameters.
Similar content being viewed by others
References
Aki, K. (1965),Maximum likelihood estimate of bin the formula logN=a−bmand its confidence limits. Bull. Earthquake Res. Inst., Tokyo Univ.43, 237–238.
Campbell, K. W. (1982),Bayesian analysis of extreme earthquake occurrences. Part I. Probabilistic hazard model, Bull. Seism. Soc. Am.72, 1689–1705.
Cosentino, P., V. Ficara andD. Luzio (1977).Truncated exponential frequency-magnitude relationship in earthquake statistics, Bull. Seism. Soc. A.67, 1615–1623.
Dong, W. M., Bao, A. B. andShah, H. C. (1984),Use of maximum entropy principle in earthquake recurrence relationships. Bull. Seism. Soc. Am.74, 725–737.
Kijko, A. (1983),A modified form of the first Gumbel distribution: Model for the occurrence of large earthquakes. Part II. Estimation of parameters, Acta Geophys. Pol.31, 147–159.
Kijko, A. andT. Ahjos (1985),Seismic hazard in Finland: Evaluation of M max Geophysica21, 39–50.
Kijko, A. andM. M. Dessokey (1987),Application of extreme magnitude distribution to incomplete earthquake files, Bull. Seism. Soc. Am.77, 1429–1436.
Kijko, A. andM. A. Sellevoll (1986),Bayesian estimation of seismic parameters for extreme historical and complete instrumental data. Seismo Series No. 7 (Seismological Obs. Bergen University, Norway) 26 pp.
Kijko, A. andM. A. Sellevoll (1987),Estimation of earthquake hazard parameters from incomplete data files, Seismo Series No. 11. (Seismological Obs. University of Bergen, Norway) 23 pp.
Kulldorff, G. (1961),Contribution to the theory of estimation from grouped and partially grouped samples (Almqvist and Wiksell, Stockholm, Göteborg, Uppsala) 144 pp.
Lee, W. H. K. andD. R. Brillinger (1979),On Chinese earthquake history-and attempt to model an incomplete data set by point process analysis, PAGEOPH117, 1229–1245.
Page, R. (1968),Aftershocks and microaftershocks, Bull. Seism. Soc. Am.58, 323–335.
Tinti, S. andF. Mulargia (1985a),Effects of magnitude uncertainties on estimating the parameters in the Gutenberg-Richter frequency-magnitude law, Bull. Seism. Soc. Am.75, 1681–1697.
Tinti, S. andF. Mulargia (1985b),Application of extreme value approaches to the apparent magnitude distribution of the earthquakes, PAGEOPH123, 199–220.
Utsu, T. (1965),A method for determining the value of bin a formula log n=a−bMshowing the magnitude-frequency relation for earthquakes. Geophys. Bull. Hokkaido Univ.13, 99–103.
Weichert, D. H. (1980),Estimation of the earthquake recurrence parameters for unequal observation periods for different magnitudes. Bull. Seism. Soc. Am.70, 1337–1346.
Author information
Authors and Affiliations
Additional information
On leave from the Institute of Geophysics, Polish Academy of Sciences, 00-973 Warsaw, Pasteura 3, Poland
Rights and permissions
About this article
Cite this article
Kijko, A. Maximum likelihood estimation of Gutenberg-Richterb parameter for uncertain magnitude values. PAGEOPH 127, 573–579 (1988). https://doi.org/10.1007/BF00881745
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00881745