Earthquake Magnitude Transition Probability and Causal Dependence for Mexico City Earthquakes

Abstract

The study of the dependence, if any, between the occurrence of earthquakes of different magnitude is an important facet of the problem of earthquake prediction. Recently, interest has concentrated on the subject of statistical dependence between earthquakes as a basis for the statistical prediction of earthquakes. For example, deviations from the Poisson process have been discussed by Aki (1956), Knopoff (1964), Ferraes (1967) and others. A concise summary of the statistical dependence between earthquakes may be found in Lomnitz (1966). The idea of earthquake independence is not yet well understood; however, probabilistic or statistical independence is not the same as causal independence and earthquake statistical independence is neither a necessary nor a sufficient condition for the existence of a causal relationship. In this light it seems worthwhile to report on the magnitude data on earthquakes in Mexico City as related to the problem of the causal dependence between earthquakes. Following Suppes (1970) causal earthquake magnitude independence is investigated by probability considerations. We attempt to test the hypothesis that one event is the cause of another if the first event of a given magnitude is followed with a high probability by a second of some other magnitude. To measure this probability we introduce the concept of earthquake magnitude transition probability and use standard probability theory. We define the conditional probability of the occurrence of an event of mean magnitude Mj given that an earthquake of mean magnitude Mi has occurred as that number of trials in which &Tj occurred following ATi. The conditional probability of Mj given Mi is denoted by the symbol P(Mj/Mi) which is given by the relation Provided that P(MJ > 0. If P(aJ = 0, then P(&Tj/Mi) is undefine. Here the set pij(Mi n Mj) is the observed number of earthquakes, with midpoint magnitude Mj, which occurred following an earthquake with midpoint magnitude Mi and ni(Mi) is the total number of earthquakes with midpoint magnitude KIP Analysis of Mexico City data The data used in this study were taken from the bulletins of the United States Coast and Geodetic Survey and those of the Servicio Seismologico Mexican0 for the period 1941-1968. The resulting catalogue includes 221 successive earthquakes felt