Download Earthquake Magnitude Transition Probability and Causal

Geophys. J. R. astr. SOC.(1974) 38, 315-316.
Earthquake Magnitude Transition Probability
and Causal Dependence for Mexico City Earthquakes
Sergio G. Ferraes
(Received 1974 April 24)*
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 M j 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 M j given Mi is denoted by the symbol P(Mj/Mi) which
is given by the relation
Provided that P ( M J > 0. If P ( a J = 0, then P(&Tj/Mi)is undefine. Here the set
pij(Mi n M j ) is the observed number of earthquakes, with midpoint magnitude M j ,
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
* Received in original form 1973 July 25
315
316
S.G. Ferraes
in Mexico City, all greater than magnitude 4.6 and less than 70km deep. The
epicentres are concentrated in the narrow seismic region which extends from
longitude 96" W to 105" W and from latitude 16"N to 19" N. This catalogue and
equation (1) were used to compute the conditional probabilities P(&Ij/ATi)
shown in
Table 1. The earthquake magnitudes were grouped into intervals AM = 0.4 (Richter
scale) width; as Furumoto (1966) has pointed out the scatter is reduced by using
magnitude intervals of AM = 0.5 units and for the purpose of prediction there is not
much loss of information by using intervals of such size.
Table 1
Observed distribution of earthquake transition probabilities estimated by direct
enumeration of successive earthquakes near Mexico City.
To
MJ
From
Mi
4.8
5.3
5-8
6.3
6.8
7-3
4-8
5.3
5.8
6.3
0-59
0.23
0.27
0.38
0.36
0.20
0.00
0.13
0-20
0.18
0.07
0.20
0.17
0.03
0.01
0.07
0.02
0.12
0.06
0.07 ..O*OO
0.20
0.00
0.17
0-17
0.42
0.24
0.43
0.40
0.17
6.8
7.3
Totals
0.01
0.02
0.02
0.07
0.00
0.33
1.00
1.00
1-00
1-00
1.00
1.01
The following inferences concerning earthquake occurrence in Mexico City area
may be drawn from Table 1.
1. There is a high probability that a small earthquake (AT, G 5) will be followed
by an earthquake of about the same or higher magnitude.
2. There is a low probability that a large earthquake (Mi > 5) will be followed
by an earthquake of the same or higher magnitude. There is a high probability that
this earthquake will be followed by a smaller earthquake.
3. A single exception shows that a very large earthquake
= 7-3) has a high
probability to be followed by an earthquake of similar size.
This study indicates that there is a causal dependence (not statistical, Suppes 1970)
between the magnitudes of successive earthquakes around Mexico City. This
suggests that there may be some regular law governing the sequence of earthquakes in
this region.
(a,
Institute of Geophysics,
National University of Mexico,
Ciudad Universitaria,
Mexico City, Mexico
References
Aki, K., 1956. Some problems in statistical seismology, Zisin, 8, 205-228.
Ferraes, S . G., 1967. Test of Poisson process for earthquakes in Mexico City,
J. geophys. Res., 12 (14), 3741-3742.
Furumoto, A. S., 1966. Seismicity of Hawaii, part I: Frequency energy distribution
of earthquakes, Bull. seism. SOC.Am., 56, 1-12.
Knopoff, L., 1964. The statistics of earthquakes in Southern California, Bull.
seism. SOC.Am., 54, 1871-1873.
Lomnitz, C., 1966. Statistical prediction of earthquakes, Rev. Geophys., 4, 377-393.
Suppes, P., 1970. A probabilistic theory of causality, North Holland Publishing Co.,
Amsterdam.