Do epicentres migrate on the San Andreas Fault? 

WOOD AND ALLEN(1) have suggested that seismicity on a portion of the San
Andreas Fault can be used in support of a model of recurring migration of
epicentres. They considered 27 earthquakes of magnitude, M => 5 within 30 km of
the fault between 35.5'N and 38.5'N which occurred between 1930 and 1972. Five
parallel line segments are drawn through the time-latitude plot of 27 points;
the common slope of the lines is presumed to give the velocity of migration of
earthquakes. Several of these events correlate with one another; any line in
the neighbourhood of events that are close together in latitude and time will
provide a satisfactory fit. Obvious clustered events are the quadruplet
(1,2,3,4) and the pairs (15,16; 5,6; 19,20; and 25,26) (Table 1 of Wood and
Allen). Probably, other correlated events could be found were a different model
used. Further, event 13 seems not to have been used in the fit. 

Thus, there are 19 or fewer independent events in the data set. If a model is
used in which the slopes of the line segments are fixed, then there are 19 or
fewer degrees of freedom in the data, which are used to fit the intercepts of
the segments. 

The number of degrees of freedom in the model used by Wood and Allen may be
about 15 in addition to the common slope: that is, the intercepts and end
points of each of the five lines. The number of degrees of freedom in the end
points is a function of the length of the segments and is difficult to
evaluate. Thus, the number of degrees of freedom in the model is roughly equal
to the number of degrees of freedom in the data, and we conclude that there are
inadequate data to support the parametric complexity of the migration model. We
believe that the apparent migration of epicentres along part of the San Andreas
Fault is an artefact of Wood and Allen's model. 

Wood and Allen forecast an earthquake with M => 5 in a restricted time-space
interval along the San Andreas Fault. Though the complexity of the model makes
it easy to give qualitative predictions, quantitative predictions in terms of
probabilities remain difficult. Wood and Allen have avoided the question of
estimating the risk that is: the probable number of shocks that will occur in
the time-space region identified as dangerous. We can calculate the risk on the
basis of a simpler model involving a considerably smaller number of degrees of
freedom. Suppose that earthquakes are Poissonian between 36.75' and 37.05'N.
Seven earthquakes are then found in the 25 yr x 33 km time-space region; the
area identified as dangerous by Wood and Allen is about 6 yr x 17 km. Thus,
0.86 +/- 0.31 shocks can be expected in the defined time-space interval.
Dropping the constraint that the earthquake catalogue has a space-time
envelope, and assuming instead that shocks occurred randomly between 1930 and
1972, then 0.50 +/- 0.19 shocks with magnitudes of more than five will occur in
the smaller time-space interval. 

Y. Y. KAGAN 

L. KNOPOFF 

Institute of Geophysics and Planetary Physics, University of California, Los
Angeles, California 90024 

(1) Wood, M. D., and Allen, S. S., Nature, 244, 213 (1973).