This paper provides a generic equation for the evaluation of the maximum earthquake magnitude m max for a given seismogenic zone or entire region. The equation is capable of generating solutions in different forms, depending on the assumptions of the statistical distribution model and/or the available information regarding past seismicity. It includes the cases (i) when earthquake magnitudes are distributed according to the doubly-truncated Gutenberg-Richter relation, (ii) when the empirical magnitude distribution deviates moderately from the Gutenberg-Richter relation, and (iii) when no specific type of magnitude distribution is assumed. Both synthetic, Monte-Carlo simulated seismic event catalogues, and actual data from Southern California, are used to demonstrate the procedures given for the evaluation of m max .The three estimates of m max for Southern California, obtained by the three procedures mentioned above, are respectively: 8.32 ± 0.43, 8.31 ± 0.42 and 8.34 ± 0.45. All three estimates are nearly identical, although higher than the value 7.99 obtained by FIELD et al. (1999). In general, since the third procedure is non-parametric and does not require specification of the functional form of the magnitude distribution, its estimate of the maximum earthquake magnitude m max is considered more reliable than the other two which are based on the Gutenberg-Richter relation.

Most probabilistic seismic-hazard analysis procedures require that at least three seismic source parameters be known, namely the mean seismic activity rate λ, the Gutenberg-Richter b-value, and the area-characteristic (seismogenic source) maximum possible earthquake magnitude m max . In almost all currently used seismic-hazard assessment procedures that utilize these three parameters, it is explicitly assumed that all three remain constant over time and space. However, closer examination of most earthquake catalogs has indicated that significant spatial and temporal variations existed in the seismic activity rate λ, as well as in the Gutenberg-Richter b-value. In this study, the maximum likelihood estimation of these earthquake hazard parameters considers the incompleteness of the catalogs, the uncertainty in the earthquake magnitude determination, as well as the uncertainty associated with the applied earthquake-occurrence models. The uncertainty in the earthquake-occurrence models is introduced by assuming that both the mean seismic activity rate λ and the Gutenberg-Richter b-value are random variables, each described by the gamma distribution. This approach results in the extension of the classic frequency-magnitude Gutenberg-Richter relation and the

A new methodology for probabilistic seismic hazard analysis (PSHA) is described. The approach combines the best features of the ''deductive'' (CORNELL, 1968) and ''historical'' (VENEZIANO et al., 1984) procedures. It can be called a ''parametric-historic'' procedure. The maximum regional magnitude m max is of paramount importance in this approach and Part I of our work presents some of the statistical techniques which can be used for its evaluation. The work is an analysis of parametric procedures for the evaluation of m max , when the form of the magnitude distribution is specified. For each of the formulae given there are notes on its origin, assumptions made of its derivation, and some comparisons. The statistical concepts of bias and variance are considered for each formula, and appropriate expressions for these are also given. Also, following KNOPOFF and KAGAN (1977), we shall demonstrate why there must be a finite upper bound to the largest seismic event if the Gutenberg-Richter frequency-magnitude relation is accepted.

The Aki (1965) maximum likelihood estimate of the Gutenberg-Richter b-value is extended for use in the case of multiple catalogs with different levels of completeness. The most striking feature of this newly derived estimator is its simplicity-it is more manageable than the well-known and already easy to use Weichert (1980) solution to the analogs problem. In addition, confidence intervals for the newly derived estimator are provided.

A b s t r a c tSeveral procedures for the statistical estimation of the regioncharacteristic maximum possible earthquake magnitude, m max , are currently available. This paper aims to introduce and compare the 12 existing procedures. For each of the procedures given, there are notes on its origin, assumptions made in its derivation, condition for validity, weak and strong points, etc. The applicability of each particular procedure is determined by the assumptions of the model and/or the available information on seismicity of the area.