Yan Y. Kagan

Dept. Earth and Space Sciences, UCLA, Los Angeles,

CA 90095-1567, ykagan@ucla.edu, http://eq.ess.ucla.edu/~kagan.html

NEGATIVE BINOMIAL DISTRIBUTION OF EARTHQUAKE

NUMBERS

http://moho.ess.ucla.edu/~kagan/NBD.ppt

Abstract. We discuss the application of the negative binomial distribution (NBD) to approximate

earthquake numbers in earthquake catalogs. Kagan (1973a,b) derived the NBD for the model of earthquake occurrence as the 'birth and immigration' population process. In particular, a point branching process along the magnitude (or seismic moment) axis with independent events (immigration) explains both the magnitude/moment frequency relation and the NBD of earthquake counts in large time/space windows as well as the dependence of the NBD parameters on the magnitude threshold. There are many different representations of the NBD, but most of them can be traced either to the Pascal distribution or to the mixture of the Poisson distribution with the gamma law. We discuss advantages and drawbacks of both representations in the statistical analysis of earthquake catalogs and the application of the NBD for earthquake forecasts. In contrast to the one-parameter Poisson distribution, which is widely used to describe the earthquake occurrence, the NBD has two parameters; the second parameter can be used to characterize the clustering or over-dispersion of a process (Kagan and Jackson, 2000). We determine the parameter values and their uncertainties for several local and global (CMT and PDE) catalogs, and their subdivisions in various time intervals, magnitude thresholds, spatial windows, and tectonic categories. These parameter values can be used in earthquake forecasting as it is now implemented in the CSEP. The theoretical model of the clustering parameter dependence on the corner (maximum) magnitude can be used to predict the future earthquake number distribution in regions where no very large earthquakes has yet occurred.

NBD Hilbe’s representation

NBD standard representation (Matlab)

NBD Evans’s representation

World seismicity: 1990 – 2000 (PDE)

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PARAMETER VALUES FOR VARIOUS SUBDIVISIONS OF CMT CATALOG, 1982--2007/03/31, Mw>=5.6

-------------------------------------------------------- All Subd. Orog. Inter. Fast Slow-------------------------------------------------------- 1. N 7720 5022 1004 279 584 831 2 Mmax 9.07 9.07 8.28 8.15 7.67 7.15 3. Inf/N 1.03 1.18 1.11 0.59 0.30 0.44 4. Ind/N 0.745 0.690 0.819 0.866 0.935 0.941 5. \mu 0.131 0.169 0.093 0.099 0.042 0.035 6. b 1.02 0.98 0.96 1.05 1.28 1.23 7. \delta 0.43 0.39 0.37 0.30 0.35 0.39 8. \theta 0.12 0.11 0.27 *0.1 0.27 *1.0 9. \sigma 0.30 0.29 0.15* 0.47 0.17 0.27 10. \eps_r 21.8 22.1 21.6 18.0 19.5 17.7 11. \eps_h 3.5 4.4 3.0* 5.1 3.0* 3.0*--------------------------------------------------------

\sigma – focal size for M4 EQ; \eps_r – horizontal error; \eps_h – vertical error

PARAMETER VALUES FOR VARIOUS SUBDIVISIONS OF

CMT CATALOG, 1982--2007/03/31, Mw>=5.6; close aftershocks removed (1.3%)

---------------------------- All All/c---------------------------- 1. N 7720 7471 2 Mmax 9.07 9.07 3. Inf/N 1.03 0.857 4. Ind/N 0.745 0.758 5. \mu 0.131 0.131 6. b 1.02 1.01 7. \delta 0.43 0.41 8. \theta 0.12 0.1* 9. \sigma 0.30 0.28 10. \eps_r 21.8 21.9 11. \eps_h 3.5 3.4----------------------------

\sigma – focal size for M4 EQ; \eps_r – horizontal error; \eps_h – vertical error

PARAMETER VALUES FOR VARIOUS SUBDIVISIONS OF PDE CATALOG,

1968--2007/01/01, M>=5.0

-------------------------------------------------------- All Subd. Orog. Inter. Fast Slow-------------------------------------------------------- 1. N 45942 29980 7686 2191 3296 2789 2. Mmax 8.80 8.80 8.50 8.45 7.60 7.30 3. Inf/N 1.90 1.99 2.16 1.40 1.28 0.98 4. Ind/N 0.680 0.661 0.687 0.716 0.816 0.869 5. \mu 0.141 0.140 0.133 0.234 0.182 0.109 6. b 1.05 1.04 1.05 1.17 1.17 0.93 7. \delta 0.42 0.43 0.43 0.23 0.02 0.12 8. \theta 0.28 0.33 0.23 0.12 0.40 0.55 9. \sigma 0.38 0.37 0.31 0.24 0.15* 0.20 10. \eps_r 9.5 9.7 7.9 9.8 10.1 12.6 11. \eps_h 3.0* 4.5 4.3 3.0* 3.0* 3.0*--------------------------------------------------------

\sigma – focal size for M4 EQ; \eps_r – horizontal error; \eps_h – vertical error

PARAMETER VALUES FOR VARIOUS SUBDIVISIONS OF PDE CATALOG,1968--2007/01/01, M>=5.0;

CLOSE AFTERSHOCKS REMOVED (5.5%)-------------------------------------------------------- All Subd. Orog. Inter. Fast Slow-------------------------------------------------------- 1. N 42925 27648 7205 2127 3217 2728 2. Mmax 8.80 8.80 8.50 8.45 7.60 7.30 3. Inf/N 1.25 1.24 1.50 1.17 1.04 0.76 4. Ind/N 0.695 0.674 0.709 0.736 0.831 0.881 5. \mu 0.171 0.177 0.146 0.241 0.185 0.121 6. b 1.04 1.03 1.04 1.17 1.17 0.93 7. \delta 0.38 0.37 0.41 0.18 0.0* 0.0* 8. \theta 0.11 0.13 0.10 0.1* 0.32 0.39 9. \sigma 0.37 0.35 0.32 0.15* 0.15* 0.15* 10. \eps_r 9.7 9.8 7.8 9.7 10.0 12.5 11. \eps_h 3.0* 4.8 4.7 3.0* 3.0* 3.0*--------------------------------------------------------

\sigma – focal size for M4 EQ; \eps_r – horizontal error; \eps_h – vertical error

Stochastic models of earthquake occurrence and forecasting

• Long-term models for earthquake occurrence, optimization of smoothing procedure and its testing (Kagan and Jackson, 1994, 2000).

• Empirical branching models (Kagan, 1973a,b; Kagan and Knopoff, 1987; Ogata, 1988, 1998; Kagan, 2006).

• Physical branching models – propagation of earthquake fault is simulated (Kagan and Knopoff, 1981; Kagan, 1982).

Kagan, Y. Y., 1991. Likelihood analysis of earthquake catalogues,

Geophys. J. Int., 106, 135-148.

Earthquake intensity function

Time influence function (Omori’s law)

Space influence function

Moment likelihood map

Moment Magnitude Distribution

Challenges• We need to start by investigating how different

branching models of seismicity approximate global earthquake catalogs.

• Global catalogs have an advantage of being considerably less inhomogeneous in time and space than local catalogs, and there are no spatial boundary effects which greatly complicate the analysis of local catalogs.

• Local seismicity is controlled by a few aftershock sequences of strong events.

• It is important, however, to analyze local catalogs as well to see that model parameter values are similar for worldwide and local catalogs.

• See http://bemlar.ism.ac.jp/wiki/index.php/Bird%27s_Zones

(a) Earthquake catalog data Phenomenological

models:(b) Point process branching along magnitude axis, introduced by Kagan (1973a;b) (c) Point process branching along time axis (Hawkes, 1971; Kagan & Knopoff, 1987; Ogata, 1988)

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