# Characterization of the Tail of the Distribution of Earthquake Magnitudes by Combining the GEV and GPD Descriptions of Extreme Value Theory

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## Abstract

The present work is a continuation and improvement of the method suggested in Pisarenko et al. (Pure Appl Geophys 165:1–42, 2008) for the statistical estimation of the tail of the distribution of earthquake sizes. The chief innovation is to combine the two main limit theorems of Extreme Value Theory (EVT) that allow us to derive the distribution of T-maxima (maximum magnitude occurring in sequential time intervals of duration T) for arbitrary T. This distribution enables one to derive any desired statistical characteristic of the future T-maximum. We propose a method for the estimation of the unknown parameters involved in the two limit theorems corresponding to the Generalized Extreme Value distribution (GEV) and to the Generalized Pareto Distribution (GPD). We establish the direct relations between the parameters of these distributions, which permit to evaluate the distribution of the T-maxima for arbitrary T. The duality between the GEV and GPD provides a new way to check the consistency of the estimation of the tail characteristics of the distribution of earthquake magnitudes for earthquake occurring over an arbitrary time interval. We develop several procedures and check points to decrease the scatter of the estimates and to verify their consistency. We test our full procedure on the global Harvard catalog (1977–2006) and on the Fennoscandia catalog (1900–2005). For the global catalog, we obtain the following estimates: $$\hat{M}_{{\rm max} }$$ = 9.53 ± 0.52 and $$\hat{Q}_{10} (0.97)$$ = 9.21 ± 0.20. For Fennoscandia, we obtain $$\hat{M}_{{\rm max} }$$ = 5.76 ± 0.165 and $$\hat{Q}_{10} (0.97)$$ = 5.44 ± 0.073. The estimates of all related parameters for the GEV and GPD, including the most important form parameter, are also provided. We demonstrate again the absence of robustness of the generally accepted parameter characterizing the tail of the magnitude-frequency law, the maximum possible magnitude M max, and study the more stable parameter Q T (q), defined as the q-quantile of the distribution of T-maxima on a future interval of duration T.

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Correspondence to V. F. Pisarenko.

## Appendix: Proofs of the Three Corollaries

### Corollary 1

Let F H (x) be the GPD-distribution

$$\begin{array}{*{20}l} {F_{H} (x) = G_{H} (x\,|\xi ,\,s) = 1{-}(1 + \xi (x - H)/s)^{ - \, 1 \, /\xi } ,} & {x \ge H} \\ \end{array}.$$
(41)

In accordance with Lomnitz formula (7), up to terms of order exp(−λT), the distribution function of the T-maxima M T is given by

$$\begin{array}{*{20}l} {\varPsi_{T} (x) = { \exp }( - \lambda T \cdot (1 + \xi (x - H)/s)^{ - \, 1 \, /\xi } )} & {{\text{if}}\,\lambda T \gg1} \\ \end{array} .$$
(42)

If we set

$$\begin{array}{*{20}l} {\sigma = \sigma (T) = s \cdot (\lambda T);} & {\mu = \mu (T,H) = H{-}(s/\xi )[1 - (\lambda T)^{\xi } ]} \\ \end{array}$$
(43)

then we can rewrite (42) in the form of a GEV-distribution

$$\varPsi_{T} (x) = { \exp }\{ - [1 + \xi (x - \mu )/\sigma ]^{ - 1/\xi } \} .$$
(44)

It should be noted that, in Eq. (41), F H (x) is defined only for x ≥ H, while Ψ T (x) does not vanish at x = H, since:

$$\varPsi_{T} (H) = { \exp }( - \lambda T).$$

But according to the condition of Corollary 1, we can neglect terms of order exp(−λT). Therefore, one can complement the domain of definition of Ψ T (H) for x < H (as it is required for the GEV-distribution), since Ψ T (x) remains smaller than exp(−λT) for x < H.

Inversely, if we assume that M T have a GEV-distribution, then using the transformation law (43) for the parameters, we have

$$\begin{array}{*{20}l} {s = \sigma /(\lambda T);} & {H = \mu + (s/\xi )[1 - (\lambda T)^{\xi } ],} \\ \end{array}$$
(45)

and we get the distribution function of M T in the form (41), from which it follows that F H (x) is a GPD-distribution.

### Corollary 2

Let X be distributed according to the GPD-distribution:

$$\begin{array}{*{20}l} {F_{H} (x) = 1{-}(1 + \xi (x - H)/s)^{ - 1/\xi } ,} & {x \ge H.} \\ \end{array}$$
(46)

Then, for any K > H the conditional distribution of X under the condition X > K is

$$\begin{array}{*{20}l} {F_{K} (x) = \{ F_{H} (x){-}F_{H} (K)\} /\{ 1 - F_{h} (K)\} ,} & {x \ge K.} \\ \end{array}.$$
(47)

Putting (46) into (47), we get

$$\begin{array}{*{20}l} {F_{K} (x) = 1{-}(1 + \xi (x - K)/S)^{ - 1/\xi } ,} & {x \ge K,} \\ \end{array}$$
(48)

where

$$S = s + \xi (K - H).$$
(49)

### Corollary 3

This corollary follows from the proof of Corollary 1 above.

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Pisarenko, V.F., Sornette, A., Sornette, D. et al. Characterization of the Tail of the Distribution of Earthquake Magnitudes by Combining the GEV and GPD Descriptions of Extreme Value Theory. Pure Appl. Geophys. 171, 1599–1624 (2014). https://doi.org/10.1007/s00024-014-0882-z