Physics of thg Earth and Planetary Interiors, 14 (1977) 97—108
© Elsevier Scientific Publishing Company, Amsterdam — Printed in The Netherlands
97
EARTHQUAKE RISK PREDICTION AS A STOCHASTIC PROCESS
1
Y. KAGAN and L. KNOPOFF
Institute of Geophysics and Planetary Physics, University of California, Los Angeles, Calif 90024 (USA.)
(Received August 28, 1976; revised and accepted November 22, 1976)
Kagan, Y. and Knopoff, L., 1977. Earthquake risk prediction as a stochastic process. Phys. Earth Planet. Inter., 14:
97— 108.
The extrapolation in time of an earthquake sequence considered as a multidimensional stochastic point process is
discussed. Estimates of seismic risk for both long- and short-term predictions are considered and an algorithm for the
calculations is proposed. Several examples of short-term extrapolations are carried out by means of Monte Carlo simulations of the process. An assessment of the predictability of the seismic process shows that the catalog of strong
earthquakes (M ~ 7.0) contains about 0.4 bits of information per earthquake for the particular model of the process
applied here.
1. Introduction
2. The stochastic model
In this paper we extend our recent investigations of
statistical features of earthquake catalogs (Kagan and
Knopoff, 1976, henceforth referred to as I). In I we
proposed a stochastic model for a process of space—
time interrelationships among earthquakes which permitted us to search for and identify these relationships
in the worldwide catalog of strong earthquakes (M ~ 7,
1905—1972). A model that describes the most significant features of the seismic process can be used to
extrapolate a catalog of earthquake data in time. In
other words we can use the model to estimate the seismic risk in a particular time—space—magnitude interval
iii the future. To be useful, such predictions should be
made using local catalogs having many more entries
than the worldwide catalog with M ~ 7. However, as
an introduction to the problem, we have attempted to
extrapolate the worldwide catalog of strong shallow
earthquakes to assess the feasibility of, and the difficulties involved in, such predictions.
Before proceeding to the main topic of this paper
it is convenient to discuss some modifications which
we have found necessary to introduce into the model
we used in 1. The notation here is the same as in I. The
first of these changes concerns the form of the distribution function .t[r, t) that governs the distribution of
secondary earthquakes in the space—time continuum
(r is the distance, t is the time interval between two
shocks). In its present form [see formulas (13) and (14)
in I], this function is dependent only on the average
seismicity i~around the main shock. Thus, in locations
with small seismicity adjoining a zone of high seismicity, the ratio of the number of stochastically dependent,
i.e. triggered, events to the number of independent
events will be higher than in the reverse case. It seems
appropriate to assume that this ratio is approximately
uniform for different parts of the worldwide seismic
belts. Hence, the functionf(r, t) should have the form:
j’(r, t) = c~ s(h)
Publication No. 1609 of the Institute of Geophysics and
Planetary Physics, University of California, Los Angeles,
Calif. 90024, U.S.A.
.
cl(r, t)
(1)
where v(h) is the level of seismicity at the site of a
dependent shock, ~(r, t) corresponds to the distribution of dependent shocks for homogeneous space and
98
c
is the normalization factor:
Therefore, we must calculate the normalization coefficient for each main earthquake separately; if we continue to use the smoothing function for the calculation
of the seismoactivity in its exponential form [see
r r
c=
J J ~‘(h)~(r, t) dtdh
(2)
~‘
TI!
‘FABLE I
Values of branching ratio
~i
(A) Foreshocks:
Distance
(km)
Time (years)
—-—----—————~-—-—
—6.0< T< --3.0
0—135
136—315
316—630
631—990
991—1,350
- ___________________________________________________________
—3.0< T< —0.9
—0.9< T< —0.15
—0.15< T< —0.03
—0.03< T< 0
0
0
0
50
117
119
0
0
0
78
113
111
347
313
309
0
0
0
31
50
39
0
39
39
147
143
0
47
0
18
0
0
0
61
0
0
8
3
0
0
0
144
0
0
945
257
0
0
0
265
140
23
0
0
37
46
0
0
0
16
8
24
57
55
0
0
0
0
0
0
40
0
0
9
43
(B) Aftershocks:
Distance
(km)
0—135
136—315
316-630
Time (years)
—
0.03 < T < 0.15
680
658
680
309
359
376
6
75
74
80
147
148
0
0
0
199
174
204
191
188
199
309
238
227
14
0
0
0
0
0
66
74
16
11
95
227
68
60
0
0
0
0
0
0
611
93
313
657
201
227
63
157
116
109
0
0
0
68
65
78
631—990
50
49
991—1,350
0
0
0
32
32
0.15 < T <
0.9
0< T<0.03
9
9
325
0.9 < T < 3.0
3.0 < T <
0
115
248
5. Distance and time intervals are shown for an earthquake with M =
All values of l~should be multiplied by i0
0
0
8.0
6.0
99
formula (11) in I], a large amount of calculation time
will be required.
For this reason we decided, in our first attempt, to
use the simplest form of this function: the seismicity
corresponding to each independent earthquake is distributed uniformly over a circle with radius 160 km.
If we use the function ~(r, t), in the form of several
concentric rings (see I), we may calculate the values of
c with relative ease. For the likelihood function, we
will get the following expression [compare formula
(6) in 1]:
I = logL
—A
~
— ~
y— 1
~
s=1 t=1
--
j=1
the fact that the normalization in I emphasized spatially-distant dependent seisniicity (as described above)
by neglecting the level of seismicity in the neighborhood of possible dependent shocks.
To check the stability of our solution, we have
added earthquake data for 1973 and 1974 to the catalog and repeated the calculations. The results are shown
in the third row of Table I. The changes in the entries
are small. We also note that the iteration for the new
mode of calculation converges much more quickly
than in the case considered in I. In general, only two
iterations are necessary to ensure convergence.
k~)-F
3. Estimation of seismic risk
rAy’~i—’
+Li log r’(h~)d/i [fl(H)T dr
~1
dr
+ k-<k
1 Pts
~Ttn-(~H5)j
/
+~
(3)
In this formula, t and s are the indices of the time and
distance intervals between the main and dependent
shocks. z~T~
is the tth time interval and l1(L~H~)
is the
number of independent shocks in a particular ring surrounding the main shock, 11(A) = IA r(h) dh, dr is a
small time interval and Cis coiistant. The quantity k~
is the numerical magnitude interval for the ith earthquake, k1 = 1
K; i = 1, ..., n; k = 1 corresponds to
M= 8.9;K = 20 is the total number of magnitude
intervals and n is the total number of earthquakes
(934 for the catalog 1905—1974). A andy correspond
roughly to the parameters “a” and “b” in the
magnitude—frequency law but, in this case, for independent earthquakes; ji is the branching coefficient defining the number of dependent shocks.
We can now show what influence this model will
have on the values of the branching coefficient p.
Table I shows values of p,.5 for three rounds of calculations: the values from table XIII (I), are given in the
first row; in the second row the new values, calculated
by maximizing (3) are given. The results in the second
row include the effects of some minor corrections to
the earthquake catalog but these caused only slight
changes in the value of Pts• As may be seen from the
table, the effect of the new model is almost negligible
for both close fore- and aftershocks. For more distant
earthquakes, the discrepancies are larger, but the general picture remains the same. This difference is due to
We consider how the available earthquake data contamed in the catalogs may be used to estimate seismic
risk. By “seismic risk” we mean the rate of occurrence
of earthquakes of a given magnitude at a particular
point of space. We may calculate this rate by taking
into account the known history of the process and
possible temporary interactions between the earthquakes, i.e., by calculating the conditional rate of
occurrence. Risk, as thus defined, is a function of
time. As another option, we may be interested in the
unconditional rate of occurrence, or the seismicity
level. It may be noted that the unconditional rate is
equal to the limiting value of the conditional rate in
the distant future of the process.
To calculate the seismicity, we need to know the
probability that each earthquake is independent, i.e.,
that it is a main shock [see formula (12) in 1], and to
smooth those probabilities over the space with an
appropriate smoothing function. Although different
forms of the smoothing function greatly influence the
values of v(dh), they have only marginal effects on the
values of the branching ratio which define the interactions between shocks; this may be seen from formula
(3) for the likelihood function.
As we have discussed in I (section 3.3), the
smoothing function that we have used has one serious
drawback: it is isotropic while the seismicity shows
clear signs of anisotropy. Therefore, this function
will either oversmooth the details of the seismicity or
it will not use the valuable information contained in
more distant earthquakes.
We may try to avoid this difficulty simply by de-
100
laying the smoothing to a later stage of the evaluation
of seismic risk. Seismic activity maps have been used to
calculate economic effects of earthquakes; in the
process of this latter calculation it is also necessary to
smooth maps by all possible epicentral intensity configurations (see, e.g., Molchan et al., 1970) and this
smoothing will be much stronger than anything we
have proposed here. In this paper we define the unconditional seismic risk by calculating the probability that
each earthquake in the catalog is independent. These
probabilities will be shown below in the examples of
Tables II and IV.
If we know the position of all n earthquakes in some
space—time—magnitude volume B and want to estimate
the probability of occurrence of an earthquake in another presumably small, space—time—magnitude
volume A we may apply the following Bayesian procedure: we let N(A) be the number of shocks in the intervalA. Then:
a
{N(A)
=
=
I}
=
—
N(dx2)
1,
1
—
~
A~
-
~x. x dx
—
~“~‘
J
.‘ —
~
a)
~‘
a
A
In eqs. 7 the first term in P(a1) takes into account the
possibility that the shock in A is an aftershock of some
earthquake in B (in our model summation is taken over
those shocks in B which are stronger than those in A),
and the second term sums up all possible contributions
for the unknown part of the space X; we use B = X B.
The other terms in eq. 6 require additional multiple integrations over the interval B.
The conditional probability P(bla) may also be decomposed into the following series:
—
(4)
N(B0) = 0}
“
—
‘~‘
where
.
where dx~is a small interval of space, time and magnitude surrounding a given earthquake (X = H X TX M
= U X Al) and:
B0 B
1)cx
(7)
P(bla) = [F(a°) P(bja°)+ P(a’) P(bla’) + P(a”)
b= {N(dx1)= 1,
N(dxn)
r
cD(x1,A) + J ~,L’(x,
A) dx
P(a
1=1
P(b!a”)+...]
1
(8)
P(a)
Here, a means that the shock in A does not produce
any dependent shock in B, a’ produces one shock, and
so on. P(b a°)is the conditional probability that n
earthquakes will occur in B if a single shock has occurred in A. This probability may be calculated in a
manner analogous to formula (3), with the difference
that, in the second term, we replace:
We write the conditional probability F(alb) as:
(K
P(aflb)
~a~b)
-
P(b)
~a)
P(a)P(bla)
P(bla) +~) ~bIä)
(5)
a0 +a1
+a2 +
...
k)
with
This is Bayes’ formula where a is the non-occurrence of
the event a. For the conditional rate of occurrence, we
may obtain a similar formula simply by replacing P()
with the moment of the process m(~).We may
describe the event a as a sum of events with different
histories of parental dependence:
a
--
i=1
(6)
n
(K
-—
ka)
f(~a, B~)+ ~
(K
--
k1)
(9)
where
fi~Ua,B0)
=
fJ(r,
t)
dhdt
and
a1
where a0 indicates that the shock occurs independently,
indicates that it is a first generation aftershock of
some main shock, and so on. Thus, for example:
B B0 X Bm
and include the influence of an additional earthquake
gao) ~ IA I
in
consider
the third
theterm
formula
of the
forformula.
events a’,But
a”,when
we we
find
begin
that,to
—
—
...
101
while the formal structure of the formula remains the
same, we must integrate them over the entire interval
LI to take into account all possible positions of secondary shocks. These integrations cannot be performed
analytically. We can overcome this difficulty by simulating the earthquake sequences by a Monte Carlo
procedure. From a number of different realizations
of the process of an earthquake sequence, including
clustering phenomena and estimating the corresponding
values P(b a), we actually perform the necessary integrations. The same procedure can be used to estimate
the integrals in expression (6).
There exist some differences in the simulation procedure for these two cases. For the P(bla) integration,
we know in advance the location of the main earthquake and may proceed to generate earthquake
clusters when the main shock is in the interval A alone.
In the second calculation, we do not know which
earthquake will produce the dependent shock so we
must generate clusters for each possible location of the
main shock, and perhaps intermediate shocks as well.
This may require lengthy computations so, as a first
trial, in the examples below, we attempted to calculate only the term P(bla) i.e., we consider the shock
mA as a main one. It may be noted that, for very
strong earthquakes, expression (6) reduces simply to
a0 since, according to the model, a strong (k = 1) shock
cannot be an offspring of other earthquakes.
We consider the simulation of the earthquake process. First, we assume that future seismic activity will
have the same behavior as the smoothed version of the
past spatial distribution of earthquakes, and that the
interaction of earthquakes is described by the third
row in Table I. Thus, if we want to estimate the conditional rate of occurrence, we choose some location
of an epicenter and take the time and magnitude of the
future shock at will. Then, from the calculation of:
5
5=1
10
—
ptsf(Ua,
B)
(10)
we define the branching rate for this earthquake. A
Poisson distribution of a random variable with this
rate will give us the number of first generation “offspring”. With this number for a particular realization,
we may then simulate the magnitudes and times of
occurrence of the offspring. From the distribution of
seismicity in the rings surrounding the main earthquake,
we may define the spatial neighborhood of the earthquake in the catalog to which this offspring corresponds
as a grid point, and then simulate the exact position
of the shock. This procedure is repeated for each offspring until all subsequent subclusters are calculated;
an example of this calculation is provided below.
We next proceed to calculate the logarithm of the
conditional probability P(bla); after subtracting from
it the value of the likelihood function, which is analogous to the logarithm of the probability P(b), we obtam the ratio of conditional to unconditional rates of
occurrence of a future shock; we call this ratio the
predictive ratio. When this ratio is approximately one,
the conditional rate of occurrence is about the same
as that expected from a random process; when it is
large, the risk is high.
4. Examples
We first sought to determine if it is possible to
“predict” a past strong earthquake if the catalog is
imagined to have been truncated at some time before
this earthquake had occurred, using an application of
the method described above. For example, we consider
the distribution of the predictive ratio for times shortly before the Chilean earthquake of May 22, 1960
(nominal magnitude 8.3). This large earthquake was
preceded by two foreshocks, 33 and 9 h before the
main shock, both with magnitude 7.2.
The geographical region under consideration is
shown in Fig. 1. The numbered locations are sites
where other earthquakes have occurred in the catalog
and are introduced as convenient grid points for the
calculations; the numbers are the sequence numbers in
our catalog. The predictive ratio has been calculated at
each of these points at various times, using the entire
seismic history of South America (M ~ 7), including
events outside the specific geographical region (30—
45°Slatitude) from the time of start of the catalog
until the time of the main shock.
Shortly before the two foreshocks occurred, the
value of the predictive ratio was roughly 0.85—0.90,
almost uniformly over the entire southern part of
South America. This relatively low level of the ratio is
due to the fact that this earthquake had no remote
in time and space foreshocks.
The predictive ratio was recalculated immediately
—
—
102
0°
I
295
•846
.177
22 ~[~763
~~732
768~à92305
707
.859
__________
_________
888~~
618
839
.93!
73•4’t\~_ 222
2!T~.762
~30_
20°
30°__-
--
!35 ~~8~3385Q
758~~4!5
737•634
84!—.
267~1~84 586
223—. ~224
23~4 p536
I
j~-
346 -•~.-896690
311315
138 ~_453
79293
78i>7~ ~—l95,!96 ç—
7T88\47v~7~8l
~0
299~
40°_779~~~==~~~~~z:-~~
3~
~ 209
\
\
~
~‘
\ ,)
~(
50
90°
80°
70° Fig. 1. Epicenter map of South America.
6O~
after the first foreshock occurred and the rate was
also projected for 0.1 year after the first foreshock,
of course without the presumption that the second
foreshock or the main shock was about to occur. The
results of these two calculations are shown in columns
4 and 5 of Table II. Immediately after the first foreshock takes place at grid point 776, the predictive ratio
becomes larger in the neighborhood of 776, with a
maximum of almost 11 at grid point 453. The predictive ratio for 0.1 year after the first foreshock diminishes
from the large values immediately after the first foreshock, to values no more than 1.7.
We recompute the predictive ratio for times 0+ and
0.1 year after the second foreshock takes place at grid
point 777 (see columns 6 and 7). Immediately after
the second foreshock, the predictive ratio has risen
over the entire geographic region with maximum of
81.2 at grid point 453. One-tenth year after the second
foreshock, the predictive ratio has diminished over the
entire region, but virtually the entire region has values
significantly different from unity.
The main shock occurred at location 778 when the
predictive ratio, based on all the earlier history, was
near 13.9. Of course, we will have asimilar increase in
the ratio after every earthquake, unless this earthquake
is a demonstrably close aftershock or foreshock of some
other main event. In the latter case, the value of the
argument of the logarithm in formula (3) would be
sufficiently large that an additional term would not
change the value of the logarithm significantly. Column
8 gives the values of the probability for a shock being
independent; these values may be used, as we discussed
earlier, to construct an unconditional, i.e., long-term,
map of seismicity.
We now consider a second example which will elucidate the method in greater detail. A chronological list
of foreshocks of the Japanese earthquake of December
20, 1946, identified according to the model presented
in I, is given in Table III. The listing includes events that
are either identified as direct foreshocks (underlined)
of this earthquake or as more remote foreshock offspring. Note that one of these foreshocks is an
event withM= 8.3. We have calculated the predictive
ratio at grid point 599, the site of the large (M 8.4)
earthquake, as a function of time for several times of
cutoff of the full catalog. In Fig. 2, we show the predictive ratio for this point (599), for the catalog cut
off 4 years before the main shock; using the data only
up to time 4 years before, the predictive ratio is calculated for a subsequent period of 5 years by 0.5-year
intervals. The process is repeated for catalogs cut off
—3, —2, years before the main shock; each line begins from the time of cutoff (see Fig. 2).
We see that if the catalog is cut off 4 years before
the main shock, we get a ratio of less than one at time
t = —4 years. This is due to the absence of foreshocks
immediately before the cut-off date. The additional
negative term (see eq. 9) is relatively large, taking into
account both nearby and remote foreshocks, and, if
foreshocks do not take place, the resulting value of
the ratio is less than one. If they do take place, as we
have seen in the Chilean example and as we shall see
...
103
TABLE II
Seismic risk estimates during the Chilean earthquake sequence, May 22, 1960
No. of
earthquake
in catalog
Lat.
(°S)
Long.
(°W)
513
346
896
31
138
315
453
701
30.8
32.5
32.5
33.0
35.0
35.0
36.3
36.5
776
37.5
777
792
779
781
930
37.5
37.8
38.0
38.5
38.5
72.0
72.0
71.2
72.0
73.0
72.0
72.3
72.5
73.5
73.0
72.5
73.5
75.1
73.4
209
39.0
73.0
778
53
847
195
470
299
39.5
39.5
40.1
41.0
41.5
44.5
74.5
73.0
74.5
73.5
74.5
73.0
Predictive ratio
Unconditional seismic
risk (probability of
being independent)
time after first foreshock ~t (years)
time after second foreshock i~t(years)
0+
0.1
0+
0.1
1.19
1.17
1.16
1.54
2.83
2.91
10.96
8.08
6.97
7.19
7.01
0.94
0.96
0.93
0.96
1.06
1.10
1.68
1.64
1.54
1.51
1.54
1.45
1.62
1.40
1.41
1.11
1.10
1.11
1.09
1.07
0.94
1.67
1.55
1.62
1.60
9.23
12.65
81.21
73.77
57.86
53.40
61.68
51.73
31.54
58.89
49.45
13.94
12.55
15.00
11.79
11.66
2.29
0.95
0.94
0.93
1.25
1.30
2.66
0.586
1.0
1.0
0.274
1.0
1.0
3.01
2.95
2.69
1.0
0.986
0.247
0.247
6.30
8.07
5.97
6.09
3.46
3.25
3.53
3.16
3.10
1.39
2.43
3.34
2.52
1.87
2.22
2.20
1.40
1.32
1.36
1.30
1.28
0.95
1.0
0.148
0.941
1.0
1.0
1.0
0.930
1.0
0.002
0.992
0.992
TABLE III
Foreshocks of the earthquake of December 20, 1946
No. of earthquake in catalog
Date
377
401
425
427
483
486
489
490
501
503
505
519
522
14
18
21
20
18
16
21
8
26
15
553
556
559
560
561
562
GMT
(h
m
s)
1934
1935
1937
1937
1941
1941
1942
1942
1942
1942
3
11
7
11
16
19
7
15
21
17
59
5
2
59
46
19
7
40
9
12
34
23
35
16
22
39
43
24
13
0
19 Dec. 1942
23
10
40
13
10
7
12
10
18
26
11
5
8
4
18
4
10
22
21
11
36
35
38
57
8
14
37
49
53
42
26
56
7
27
50
Feb.
Oct.
Feb.
Aug.
Nov.
Dec.
Feb.
Apr.
Oct.
Nov.
Jun. 1942
Sep. 1943
Dec. 1944
Jan. 1945
Feb. 1945
Feb. 1945
Feb. 1945
Mar.1945
Lat.
(°N)
Long.
(°E)
Magnitude
17.5
12.5
44.5
14.5
32.0
21.5
38.0
13.5
45.5
37.0
31.5
42.8
35.3
33.8
34.8
41.3
42.0
26.0
37.0
119.0
141.5
149.5
121.5
132.0
120.5
142.0
121.0
151.5
141.5
142.5
143.2
134.0
136.0
136.8
142.5
143.0
143.5
142.0
7.9
7.1
7.4
7.5
7.9
7.1
7.1
7.9
7.2
7.0
7.0
7.4
7.4
8.3
7.1
7.3
7.0
7.1
7.2
104
Iogr
5,0
I
475
~
0,5
..—.-
,.
-~
-“~/
‘~.-~
-.
-~
----
-
0,25
-.
—~
I
TIME (years)
Fig. 2. Predictive ratios (r)
for different cutoffs of the catalog,
later, they boost the conditional rate significantly.
Thus, there is a trade-offbetween the relatively rare
appearance of close foreshocks and their strong influence on the conditional rate. For later times the
values of the conditional rate become about 15—20%
higher than the unconditional rate. This reflects the
influence of the distant foreshocks (nos. 505 and
earlier events). As the time interval between the
cutoff of the catalog and the prediction increases, we
take into account only distant foreshocks in eq. 9; if
they take place, the value of the ratio is raised. The
curve with cut-off time 3 years before the main shock
displays similar behavior,
The curve with a cutoff date 2 years before the
main shock starts off with a very high level of the
conditional rate, namely about 92 times more than
the unconditional rate. This huge value has been caused
by foreshock number 553; this earthquake has M
= 8.3 which obviously has an extremely low probability of occurrence. Since the logarithmic term of formula (3) compares the rates of independent and dependent
occurrence, an event that has a small chance of independent occurrence will boost the conditional rate
much more than, for example, an earthquake with magnitude 7.1. The rest of this curve (t~~10~~
= —2) as well
as the other curves display the combined influence of
all foreshocks, which now become “distant” shocks.
The irregularities in the graphs are due to the use of
the unsmoothed values of the branching ratio from
Tablel.
We discuss the calculation of the point at t = +0.5
year of the last curve of Fig. 2, i.e. with cut-off time
equal to zero. In 14 cases out of 20 realizations of the
process, the main earthquake did not produce any dependent shocks; in four cases one dependent shock
was produced and five and two dependent shocks were
generated once each. This is in good agreement with
the value of the rate of occurrence of dependent shocks;
the additional term in eq. 8 is 0.494, i.e. we may expect dependent shocks in less than half (3 9%) of the
cases.
Diagrams of the interaction of earthquakes for two
different realizations are shown in Fig. 3. Grid point
numbers in parentheses indicate the occurrence of a
simulated shock. The list of predicted earthquakes is
given in Table IV, with the coordinates of the nearest
grid points. The predictive ratio for the first realization is 1 .399 and is smaller than that for a single earthquake in B without dependent shocks, namely 1.484:
the additional terms connected with the influence of
new foreshocks at 519, 559, 560 and 561 are small due
to the relatively low magnitudes of these four earthquakes; they cannot compensate for the additional
negative term, connected with the aftershock at point
483. In the second case, the two shocks nos. 483, 490
are calculated to be foreshocks of the aftershock at
161 realized by the Monte Carlo procedure. In this
case, the predictive ratio is greater than that for a single
independent earthquake since the two foreshocks have
relatively high magnitudes; their contribution is greater
than the negative contribution from the five offspring
which were generated. The values of the predictive
ratio shown in Fig. 2 are the average of all 20 relizations.
As a further example, we calculate the risk today for
a large earthquake in South America. Table V shows
the predictive ratio for a very strong earthquake
(M = 8.9), if it is presumed to occur at any one of the
locations of previous earthquakes in South America.
The grid points are shown in Fig. 1. The catalog has
been cut off at the end of 1974, and the prediction is
made for the first day of 1975, 1976 and 1977. The
highest values of the ratio are connected with the influence of the most recent earthquakes, and especially
nos. 930 (Aug. 18, 1974) and 931 (Oct. 3, 1974). As
in the first example, the predictive ratio is somewhat
erratic geographically due to our use of the unsmoothed
105
M
(c)
~
(483)
486/”~
483
~~9”
559
56!
556
560~
~2~06I)
486/
490
(9~0)~
b(
556/
7,0
7.0
I
-5
5
0
TIME
!20°E
/
-5
150°
“~(377)
(377~)
‘0
I
0
TIME (years)
(years)
135°
73)
!20°E
5
j350
45°N—
-~,
5,9
560
30°
N56,
~-
(bi
483
30°
~id
~~(l6l)
486
~
IS”
15° —
—
~‘(377)
J~.
~49~
Fig. 3. Two realizations (a, b and c, d) of the stochastic interaction between shallow earthquakes. Plots a, c show time—magnitude
and b, d —space connections. Dashed lines show
0) [see
interactions
formula (8)1.
generated by Monte Carlo procedure, solid lines show the relations
that are included in the calculation ofP(bIa
TABLE IV
F.arthquakes generated after cut-off of the catalog
No. and coordinates of grid points
_____________________________
No.
161
173
377
377a
483
599
910
lat.
long.
Coordinates of offspring shocks
________________________________
lat.
long.
(°N)
(°E)
(°N)
(°E)
24.0
25.0
17.5
124.0
123.0
119.0
32.0
32.5
23.6
132.0
134.5
12-1.6
24.572
24.742
17.694
17.898
32.708
32.500
23.341
124.825
123.756
120.187
119.548
133.153
134.500
121.270
Time after
cut-off
(years)
Magnitude
0.880
2.014
4.251
4.649
2.901
0.500
0.850
8.0
7.2
7.3
7.1
8.1
8.4
7.4
106
TABLE V
Seismic risk estimates for South America
No. of
earthquake
in catalog
Lat.
295
177
846
763
22
732
5.0°N
4.0°N
2.9°N
1.5°N
l.0°N
0.5°S
768
4.0°S
311
597
466
931
45
368
850
758
634
737
223
724
690
896
315
777
5.5°S
8.5°S
10.5°S
12.3°S
0S
15.0
16.5°S
18.0°S
20.0°S
21.8°S
23.5°S
24.5~S
25.5°S
28.0°S
30.0°S
31.5°S
32.5°S
35.0°S
37~5o5
930
38.5°S
53
470
41.5°S
499
217
299
395°s
44.5°s
Long.
Predictive ratio
(°W)
_~
1/1/75
1/1/76
1/1/77
Unconditional seismic
risk (probability of
being independent)
82.5
74.0
74.9
79.5
81.5
80.5
81.5
79.0
77.5
77.0
77.8
76.0
0.83
1.05
0.92
0.85
0.86
0.88
1.11
1.02
1.38
1.48
1.42
1.37
0.92
0.99
0.97
0.93
1.00
1.31
1.06
1.05
1.00
1.02
0.99
0.98
0.94
1.01
0.98
0.98
0.95
0.97
1.01
1.28
0.99
1.03
1.02
0.96
1.0
1.0
1.0
1.0
1.0
0.807
0.914
0.948
1.0
0.917
1.0
1.0
73.0
71.0
71.0
70.0
71.5
65.0
71.0
72.0
72.0
67.5
71.2
72.0
73.0
73.4
73.0
74.5
73.0
0.93
0.92
0.88
0.90
0.86
0.81
0.88
0.97
0.87
0.85
0.86
0.99
0.95
1.00
0.96
0.97
0.94
0.94
0.96
0.98
0.98
0.97
0.96
0.98
0.95
0.96
0.99
1.07
1.01
0.99
0.96
0.98
0.91
0.97
0.99
0.94
0.94
0.95
0.96
0.98
1.01
0.98
0.99
1.01
1.10
0.99
0.94
0.99
0.96
1.01
0.97
0.815
0.678
0.924
0.913
1.0
0.946
1.0
0.135
1.0
0.996
1.0
1.0
0.247
1.0
0.930
0.992
0.992
version of table Pts (Table I), as well as the fact that
these values are results of the stochastic process, i.e. a
Monte Carlo simulation, which is necessity includes
some random components.
continuous. It is possible to define the entropy of a
point process (see, e.g., McFadden, 1965), but in our
case we are concerned with the contributions from
events such as those treated in the expression for b in
formula (4); the entropy of such events goes to infinity
as the size of the interval dx 0 (Yaglom and Yaglom,
1973, pp. 112—114). However, we may use an
approach suggested by the Yaglorns (see also Gel’fand
and Yaglom, 1959) to define the information content
of a point process for a particular non-Poissonian
model, in comparison with that for a purely Poissonian
process. In this case we treat the difference between
the logarithms of the likelihood functions
10 for
the non-Poissonian model (I) and the Poissonian model
(Ia) as a statistical estimate of the information content
—~
5.
Information content of the seismic process
The foregoing discussion of the predictive ratio
raises a question concerning just what information we
are able to get from an earthquake catalog. We can
try to approach this problem from the point of view of
information theory (cf. Vere-Jones, 1975). There are
some difficulties connected with this method which is
concerned with point processes, because our process is
~—
107
000
3
750
200
--
N
______ N
500
-
250
- -
200
i
,,ç—”~________
1
00
-
901
910
920
930
1940
950
960
970
which is up to 16 years forM= 8.9. Other reasons for
fluctuations in the slope may be due to better determination of epicenters and magnitudes in the later parts
of the catalog. In addition, we have used the values of
parameters defined by the whole catalog, which may
not be exact for smaller time spans of the catalog. It is
probably best to estimate the information content of
the catalog from the slope of the curve in the righthand part of Fig4 This gives us an information rate
A possibly more useful estiii~ateof the information
TIME (years)
Fig. 4. The number of shallow strong earthquakes (N) and the
likelihood function (1) [information content ~I)1for the subcatalogs of increasing time spans.
of the catalog (cf. Kuliback, 1959, p. 94). In calculating the likelihood function we weight the shocks that
occur with the same probability p by the logarithm of
the probability of their occurrence in the real catalog
[see the third term of formula (3)]. Thus we have
formed the sum of quantitiesp log p; for a process
with continuous state variables, this sum is defined up
to an additive constant depending on the values of dt
and dh we use [see formula (3)].
The difference 1 1o is essentially independent of
the intervals dt and dh. Fig. 4 shows this difference for
subcatalogs of increasing time spans. We see that the
likelihood function increases roughly linearly with
time or number of shocks. Sudden changes of the likelihood function are easily identified with clusters of
nearby earthquakes. For example, the increase in bit
content of the catalog which took place in 1919 is
caused mostly by a pair of earthquakes, a foreshock and
a main shock, that occurred in March, 1919 in the
southern part of South America. The jump occurring
in 1931 is connected with several clusters of earthquakes: New Zealand (March), in the Altai Range of
Central Asia (August), and near the Solomon Islands
(October). The jump occurring in 1957 was associated
mainly with a sequence of aftershocks of the Aleutian
earthquake of March 9, 1957 (M= 8.25).
The slope of the curve (Fig. 4) increases with increasing length of catalogs. In our calculation we did
not take into account the interaction between the
earthquakes outside the catalog with those inside it.
These end effects decrease the value of the likelihood
function when the time span of the catalog is comparable with the maximum time span of the interaction,
—
rate is to present it in units of bits per earthquake. This
measure gives us the possibility of comparing different
catalogs with various magnitude and time—distance
limits. For this calculation, this measure is about 0.4
bits per earthquake. What can such a quantity mean
practically? Let us assume, for example, that we wish
to develop a strategy for the application of earthquake
mitigation measures. In the absence of any additional
information, i.e. on the Poissonian model, we apply
the mitigation measures uniformly in time. If we use
the non-Poissonian model described above, we would
distribute the same amount in proportion to the computed conditional rate of occurrence, assuming there
is no overhead incurred in relocating mitigation resources as a function of space and time. If we assume
that these nmeasures will reduce losses from earthquakes
in proportion to the rate of application, then the total
loss can be calculated by a formula which is analogous
to formula (3). On the average, the losses will be reduced by a factor of 20.4 in comparison with the
random application of the mitigation measures. However, the value 0.4 bits per earthquake is appropriate
for the case in which all the earthquakes before and
after a “predicted” event are known. For the onesided situation, in which we only have data for the
past and/or in which we are interested in possible extrapolations to times significantly beyond the end of the
catalog, the amount of available information will be
less than the value above. On the average, the calculated conditional rate will differ from the unconditional
one by only about 5—15%.
We can compare the amount of information supplied
by different models of dependent shocks. ln Table VI
we have summarized the results from I (section 4.2)
by reproducing the values of the maxima of the likelihood functions. These were obtained by approximating the interaction of the earthquakes by sums of
108
TABLE
VI
Values of the likelihood function for the different sets of
approximating functions
Type of
Number of
dependent
functions
shocks
in the sum
___________—~__________
Value of the maximum
of the likelihood
function
________________
Aftershocks
112.5
123.5
129.0
1
2
3
Foreshocks
1
62.6
2
70.3
different sets of functions: the first function takes
into account nearby dependent shocks and the second
and third functions consider more distant ones. The
information provided by aftershock sequences is about
twice that of foreshocks, and nearby fore- and aftershocks account for about 90% of the entire value of
the likelihood function; most of the information content in our calculation is contained in interactions between nearby events and only a small part in long-range
interactions.
We have not discussed the way this information is
distributed in regard to the magnitudes of earthquakes.
For practical purposes, the risk prediction of the làrgest earthquake is the most important; probably this
will account for the largest part of the foreshock information (see discussion above) and almost none of
the aftershock information,
This discussion may seem to present a discouraging
picture, but we remark that for none of the other methods for the possible prediction of earthquakes is a similar numerical estimate of the predictability of the seismic process available at present. An increase in the magnitude span of the catalog will raise the power of this
method though the greatest part of the improvement
must perforce go to the weakest shocks. As we have
seen in the previous investigation (I), the branching
ratio of foreshocks decreases with increasing difference
of magnitudes; thus, for more complete catalogs,
the number of foreshocks of very large shocks will not
be increased significantly.
We hope these results can be used to develop new
models of the stress fields which allow one earthquake
to influence the time of occurrence of another. The
usefulness of these results in actual prediction programs
will depend strongly on the strategy of the program
itself.
ln conclusion we may remark that we have not tried
to estimate the conditional second-order moments of
the process in this work; second-order moments will
enable us to construct error brackets for the risk values.
These estimates may be obtained by more or less standard methods, but of course, they will require extensive computation.
Acknowledgements
This research was supported by Grant ENV 76-0 1706
of the RANN program of the National Science Foundation. The assistance of Dr. John K. Gardner in helping
with programming problems is gratefully acknowledged.
References
Gel’fand, l.M. and Yaglom, AM., 1959. Calculation of the
amount of information about random function contained
in another such function. Transi. Am. Math. Soc., Ser. 2,
12: 199--246.
Kagan, Y. and Knopoff, L., 1976. Statistical search for nonrandom features of the seismicity of strong earthquakes.
Phys. Earth Planet. Inter., 12: 291—318.
Kuilback, S., 1959. Information Theory and Statistics, Wiley,
New York, N.Y., 395 pp.
McFadden, J.A., 1965. The entropy of a point process. J. Soc.
Ind. Appi. Math., 13: 988—994.
Molchan, G.M., Keiis-Borok, V.1. and Vil’kovich, G.V., 1970. Seismicity and principal seismic effects. Geophys. JR. Astron.
Soc., 21: 323—335.
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Yaglom,
AM.
and Yaglom,
mation.
Nauka,
Moscow,l.M.,
512
1973.
Probability
and Inforpp. (3rd
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[Translation 1st Russian ed.: Wahrscheinlichkeit und Information, VEB Deutscher Verlag der Wissenschaften,
Berlin, 1960 (in German).]