Earthquake forecasting: statistics and information


ANNALS OF GEOPHYSICS, 58, 6, 2015, S0658; doi:10.4401/ag-6816

S0658

Earthquake forecasting: statistics and information

Vladimir Gertsik1,*, Mark Kelbert1,2,3, Anatoly Krichevets4

1 Institute of  Earthquake Prediction Theory and Mathematical Geophysics RAS, Moscow, Russia
2 Swansea University, Department of  Mathematics, Swansea, United Kingdom
3 Higher School of  Economics, Moscow, Russia
4 Lomonosov MSU, Department of  Psychology, Moscow, Russia

ABSTRACT

The paper presents a decision rule forming a mathematical basis of  earth-
quake forecasting problem. We develop an axiomatic approach to earth-
quake forecasting in terms of  multicomponent random fields on a lattice.
This approach provides a method for constructing point estimates and
confidence intervals for conditional probabilities of  strong earthquakes
under conditions on the levels of  precursors. Also, it provides an approach
for setting a multilevel alarm system and hypothesis testing for binary
alarms. We use a method of  comparison for different algorithms of  earth-
quake forecasts in terms of  the increase of  Shannon information. ‘Fore-
casting’ (the calculation of  the probabilities) and ‘prediction’ (the alarm
declaring) of  earthquakes are equivalent in this approach.

1. Introduction
The methodology of  selecting and processing of

relevant information about the future occurrence of
potentially damaging earthquakes has reached a rea-
sonable level of  maturity over the recent years. How-
ever, the problem as a whole still lacks a comprehensive
and generally accepted solution. Further efforts for op-
timization of  the methodology of  forecasting would be
productive and well justified.

One of  our objectives here is to address some
drawbacks of  the approach to earthquake prediction
based on pattern recognition theory. It is developed by
“Keilis-Borok team” and presented in Keilis-Borok and
Soloviev [2003]. The pattern recognition is used in fore-
cast with the hope that ‘training sets’ in which strong
earthquakes have higher values of  precursors may help
to decide whether or not a strong earthquake will hap-
pen in the future. As it is demonstrated by below pub-
lished Krichevets’s theorem, this hope has no basis.

A comprehensive review of  the modern earth-
quake forecasting state of  knowledge and guidelines for
utilization can be found in Jordan et al. [2011]. Note

that all methods of  evaluating the probabilities of
earthquakes are based on a combination of  geophysi-
cal, geological and probabilistic models and consider-
ations. Even the best and very detailed models used in
practice are in fact only ‘caricatures’ of  immensely
complicated real processes. (For example, the equations
of  the theory of  elasticity are valid only for infinitesi-
mal strains, but used even at the edges of  the cracks
where strains tend to infinity. It is also clear that we do
not know about the inhomogeneity of  elastic proper-
ties in the real medium, and we have no way of  know-
ing about it with sufficient accuracy.)

A mathematical toolkit for earthquake forecasting
is well presented by Harte and Vere-Jones [2005]. This
work is based on the modeling of  earthquake sequences
in terms of  the marked point processes. However, the
mathematical technique used is quite sophisticated and
does not provide direct practical tools to investigate the
relations of  the structure of  temporal-spatial random
fields of  precursors to the appearance of  strong earth-
quakes.

The use of  the multicomponent lattice models (in-
stead of  marked point processes) gives a different novel
way of  investigating these relations by a more elemen-
tary way. Discretization of  space and time allows us to
separate the problem in question into two separate
tasks. The first task is the selection of  relevant precur-
sors, i.e., observable and theoretically explained physi-
cal and geological facts which are causally related to a
high probability of  strong earthquakes. Particularly, this
task involves the development of  models of  seismic
events and computing probabilities of  strong earth-
quakes in the framework of  these models. Such proba-
bilities are used as precursors in the second task.

The second task is the development of  methodol-

Article history
Received June 25, 2015; accepted December 21, 2015.
Subject classification:
Earthquake forecasting, Lattice, Random process, Information.



ogy of  working with these precursors in order to extract
the maximum information about the probabilities of
strong earthquakes. This is the main topic of  this paper.

Our approach allows us to obtain the following
results:

- Estimates of  probabilities of  strong earthquakes
for given values of  precursors are calculated in terms
of  the frequencies of  historic data.

- Confidence intervals are also constructed to pro-
vide reasonable bounds of  precision for point estimates.

- Methods of  predictions (i.e., binary alarm an-
nouncement [Keilis-Borok and Kossobokov 1990, Keilis-
Borok 1996, Holliday et al. 2007]) and forecasting (i.e.,
calculating probabilities of  earthquakes [Kagan and
Jackson 2000, Harte and Vere-Jones 2005, WGNCEP
2007, Jordan et al. 2011]) are equivalent in the following
sense: the setting of  some threshold for the probability
of  earthquakes allows to update the alarm level. On the
other hand, the knowledge of  the alarm domain based
on historical data allows us to evaluate the probabilities
of  earthquakes. In a sense, the prediction is equivalent
to hypothesis testing as well (see Section 11).

- In our scheme we propose a scalar statistic which
is the ratio of  the actual increment of  information to
the maximal possible increment of  information. This
statistic allows us to linearly order all possible forecast-
ing algorithms. Nowadays the final judgement about
the quality of  earthquake forecasting algorithms is left
to experts. This arrangement puts the problem outside
the scope of  natural sciences which are trying to avoid
subjective judgements.

The foundation of  our proposed scheme is the as-
sumption that the seismic process is random and can-
not be described by a purely deterministic model. Indeed,
if  the seismic process is deterministic then the inaccu-
racy of  the forecast could be explained by the non-com-
pleteness of  our knowledge about the seismic events
and non-precision of  the available information. This
may explain, at least in principle, attacks from the au-
thorities addressed to geophysicists who failed to pre-
dict a damaging earthquake. However, these attacks
have no grounds if  one accepts that the seismic process
is random. At the end of  the last century (February-
April 1999) a group of  leading seismologists organized
a debate via the web to form a collective opinion of  the
scientific community on the topic: ‘Is the reliable pre-
diction of  individual earthquakes a realistic scientific
goal?’ (see http://www.nature.com/nature/debates/
earthquake/).

Despite a considerable divergence in peripheral is-
sues all experts taking part in the debate agreed on the
following main principles:

- the deterministic prediction of  an individual

earthquake, within sufficiently narrow limits to allow a
planned evacuation programme, is an unrealistic goal;

- forecasting of  at least some forms of  time-depen-
dent seismic hazard can be justified on both physical and
observational grounds. The following facts form the basis
of  our agreement with this point of  view. The string-
block Burridge-Knopov model, generally accepted as a
mathematic tool to demonstrate the power-like Guten-
berg-Richter relationship between the magnitude and
the number of  earthquakes, involves the generators of
chaotic behaviour or dynamic stochasticity. In fact, the
nonlinearity makes the seismic processes stochastic: a
small change in the shift force may lead to completely
different consequences. If  the force is below the thresh-
old of  static friction the block is immovable, if  the force
exceeds this threshold it starts moving, producing an
avalanche of  unpredictable size.

This mechanism is widespread in the Earth. Suppose
that the front propagation of  the earthquake approaches
a region of  enhanced strength of  the rocks. The earth-
quake magnitude depends on whether this region will
be destroyed or remains intact. In the first case the front
moves further on, in the second case the earthquake re-
mains localized. So, if  the strength of  the rocks is below
the threshold the first scenario prevails, if  it is above the
threshold the second scenario is adapted. The whole
situation is usually labelled as a butterfly effect: infini-
tesimally small changes of  strength and stress lead to
macroscopic consequences which cannot be predicted
because this infinitesimal change is below any precision
of  the measurement. For these reasons determinism of
seismic processes looks more doubtful than stochasticity.

The only comment we would like to contribute to
this discussion is that the forecasting algorithms based
exclusively on the empirical data without consistent
physical models could hardly be effective in practice
(see Section 13 for more details).

Finally, note that our approach may be well-applic-
able for the space-time forecasting of  different extremal
events outside the scope of  earthquake prediction.

2. Critical analisis of  the prediction based on pattern
recognition

The main objective of  this paper is to present a
mathematical formalism of  the earthquake forecasting
alternative to the approach based on the pattern recog-
nition theory. So, it is appropriate to summarize the lat-
ter approach.

An advantage of  the formalism of  the Keilis-Borok
team [Keilis-Borok and Kossobokov 1990, Keilis-Borok
1996, Keilis-Borok and Solovoev 2003] is a separation
of  the forecast into two separate tasks: the selection of
relevant precursors and the technique of  their process-

GERTSIK ET AL.

2



3

ing. On the other hand, some drawbacks of  the scheme
should be pointed out.

2.1. Pattern recognition
The method is relaying on the pattern recognition

technique that can provide only empirical but not the-
oretical foundation. The possibility to base a reliable
forcast on the idea of  supervised pattern recognition is
highly questionable in principle. Indeed, the features se-
lection based on the observations only and not related
to the physics of  earthquakes is not helpful and any
hopes for valid ‘training sets’ are not grounded. A suc-
cessful supervised recognition is possible only if  the fea-
tures has proved causal relation with pattern. This
principle is illustrated by a simple but important theo-
rem by A.N. Krichevets.

Theorem. Let A be a finite set, B1 ∪ B2 ⊂ A; B1 ∩ B2=
Ø. We say that B1 and B2 are finite educational samples. Let
X ∈ A; X ∉ B1∪B2 be a new object. Then among all classifi-
cations, i.e., subsets {A1, A2} such that B1 ⊂ A1, B2 ⊂ A2,
A1∪ A2= A, A1∩ A2 = Ø satisfying condition that either
B1 ∪ X ⊂ A1 or B2 ∪ X ⊂ A2 exactly a half  classifies X as an ob-
ject of  sample B1 and a half  classifies X as an object from B2.

Proof. It is easy to define a one-to-one correspon-
dence between classifications. Indeed, if  {A1, A2} A1∪ A2=
A; is a classification such that B1 ⊂ A1, X ∈ A1, B2 ⊂ A2,
one maps it into the unique classification {A'1, A'2} such
that B1⊂A'1, X∈A'2, B2⊂ A'2, where A'1= A1\X, A'2= A2∪X.

Corollary. A supervised pattern recognition is impos-
sible. After the training procedure the probability to classify
correctly a new object is the same as before training.

This result implies that a clairvoyance is involved
in a successful forecast: one should know in advance (or
guess correctly) a “relevant” set of  precursors.

2.2. Voting procedure
As a decision rule for a forecast a voting procedure

is adopted: an alarm is announced when the number
of  precursors taking abnormally large values exceeds
certain thershold. This procedure is well-justified only
if  the pattern in question is unique. Generically, this is
not true: if  two or more possible scenarios are present,
different precursors may be strongly affected in differ-
ent scenarious. In this case a combination of  abnormal
values for precursors related to different scenarios may
produce a false alarm.

2.3. Alarm domain
The values of  precursors are computed at the

point of  some lattice, and are assigned a domain sur-
rounding the point. The above scheme employs a circle
with a center at a lattice site and a radius proportional
to the minimal magnitude of  the forecasted earth-

quake. However, this choice leads to a contradiction.
Indeed, suppose we announce an alarm for earth-

quakes with minimal magnitude 6 in a domain A6. Note
that outside the alarm domain we expected that the
earthquake will not occur. But the announcement of
the alarm with minimal magnitude 7 is also correct for
this point and the alarm should be announced in the
domain A7 as well. By definition A6 ⊂ A7 and we expect
an earthquake with a magnitude of  at least 7 and do
not expect an earthquake with à magnitude of  at least
6 in the domain A7\ A6. This is absurd.

Futhermore, the alarm area is too large with this
approach.

2.4. Molchan’s error diagram
Below we demonstrate that the procedure of

alarm announcement is equivalent to the classical prob-
lem of  hypothesis testing. So, the standard error dia-
gram provides the dependence of  the second type error
(probability of  missing a target) from the first type
error (probability of  false alarm).

In line with the general case the well-known
Molchan’s error diagram [Molchan 1990] displays the
dependence of  the first kind error probability versus
the share of  alarms.

H0: the distribution of  the precursor is equal to its
unconditional distribution (earthquake “could either
happen or not happen”).

H1: the distribution of  the precursor is equal to its
conditional distribution under the condition that the
earthquake will happen.

Note that the rejection of  hypothesis H0 leads to
absurd results: “earthquake will not happen and will
happen”.

Of  course, Molchan’s error diagram can be inter-
preted not in terms of  hypothesis testing. Then there is
no contradiction, but the standard error diagram is dif-
ferent from the Molchan’s error diagram by a small
value proportional to the share of  cells occupied by the
strong earthquakes (in our approach the spacetime is
divided into disjoint cells).

2.5. The loss function, risk function and the economic
criterion for the quality of  the forecast

The Keilis-Borok team refers to the economic loss
function for the quantitative evaluation of  the quality
of  forecast. First, we calculate the expected loss cor-
rectly. A natural economic measure for a quality of  bi-
nary forecast is the economic risk function (it should
not be confused with seismic risk, there is the concept
of  the theory of  statistical decisions) or damage r re-
lated to the earthquakes and the necessary protective
measures. In mathematical statistics the risk function is

EARTHQUAKE FORECASTING: STATISTICS AND INFORMATION



defined as the expectation of  the loss function, which is
an expected losses for different outcomes. In our case
there are two types of  losses: damage and expenses re-
lated to protection. For each cell of  our grid (in our ap-
proach the space-time is divided into disjoint cells) the
risk may be specied by the formula

r = bPr {no earthquake; alarm} + aPr {earthquake;
no alarm} + cPr {earthquake; alarm}

here the symbol Pr {·} denotes a probability, a stands
for the average damage from a seismic event, b stands
for the average expenses for protection after a seismic
alarm is announced, c stands for the average damage
after the alarm, c = a + b − d, where d is the damage
prevented by the alarm. The coefficient in front of
Pr {no earthquake; no alarm}, obviously, equals 0, be-
cause in the absence both of  a seismic event and an
alarm there is no loss of  any kind. Clearly, only the case
when d > b is economically justified, i.e., the gain from
the prevention measures is positive. Obviously, d should
be less than a+b, i.e., an earthquake cannot be prof-
itable. Taking into account that a, b and c depend on
the geographical position of  the cell, we write the total
risk as the summation over all cells in the region of  a
given forecast. In the simplest case of  the absence of
the spacial component, when a single cell represents a
region of  forecast, the expression for the risk is simplied
as follows r = amo+bx+cm(1 −o), where x is a spatio-
temporal share of  alarm, m is a share of  cells occupied
by the earthquakes, o is the share of  missed targets.

In Molchan’s theory related to the one-dimensional
case, the third component is missing.

However, the risk function r, which is very useful
for economical considerations and as a basis for an ad-
ministrative decision, could hardly be used as a criteria
for the quality of  seismic prediction. First of  all, it can-
not be computed in a consistent way because the coef-
ficients a, b and c are not known in practice, and hence
no effective way of  its numerical evaluation is known.
The computation of  these coefficients is a difficult eco-
nomic problem and goes far beyond of  the competence
of  scientists. On the other hand, the readiness of  the au-
thority to commit resources to solving the problem de-
pends on the quality of  the geophysical forecast. This
situation leads to a vicious circle.

The second drawback of  the economic risk as a cri-
terion for the quality of  prediction is related to the fact
that it depends on many factors which have no relation
to geophysics or earthquake prediction. These factors
include the density of  population, the number and size
of  industrial enterprises, infrastructure, etc. It also de-
pends on subjective factors such as the willingness of

authorities to use resources for prevention of  the dam-
age from earthquakes. The natural sciences could
hardly accept the criteria for the forecast quality which
depend on the type of  state organization, priorities of
ruling parties, results of  the recent elections, etc.

In this paper, we introduce a scalar efficiency (in-
formation efficiency) forecast that only depends on in-
formation about future earthquakes contained in the
precursors.

2.6. The measure of  an area
In Molchan and Keilis-Borok [2008] the area of  the

alarm domain is defined in terms of  a non-homoge-
neous measure depending on spacial coordinates.
Specifically, the measure of  an area is proportional to
the number of  earthquake epicenters from a sample
catalog. This approach is used to eliminate the decrease
of  the share of  alarmed sites x with the extension of
the domain when a purely safe and aseismic territory is
included into consideration. It would be well-justied if
the quantity x could be accepted as an adequate crite-
rion of  the quality of  forecast in its own right. On the
other hand, it can be demonstrated that the our infor-
mation efficiency converges to a non-zero value 1 −o
when the number of  cells with an alarm is fixed but the
total number of  cells tends to infinity. An inhomoge-
neous area of  the territory under forecast which is pro-
portional to “activity” does not enable us to calculate the
informational efficiency. Moreover, it possesses a num-
ber of  unnatural features from the view of  evaluating
economical damage. A seismic event in the territory of
low seismicity is more costly because no precautions are
taken to prevent the damage of  infrastructure. How-
ever, in this inhomogeneous area an alarm announced
in an aseismic territory will have a smaller contribution
than an alarm in a seismically active territory where the
losses would be in fact smaller. We conclude that this
approach ‘hides’ the most costly events and does not
provide a reasonable estimate of  economic damage.

Next, we build a forecasting method free from this
kind of  flaws.

3. Events and precursors on the lattice
In order to define explicitly estimates of  probabil-

ities of  strong earthquakes we discretize the two-di-
mensional physical space and time, i.e., introduce a
partition of  three-dimensional space-time into rectan-
gular cells with the space partition in the shape of
squares and time partition in the shape of  intervals.
These cells should not intersect to avoid an ambiguity
in computing the frequencies for each cell. In fact, the
space cells should not be perfect squares because of  the
curvature of  the Earth’s surface, but this may be neg-

GERTSIK ET AL.

4



5

lected if  the region of  forecasting is not too large.
So, we obtain a discrete set XK with N = I × J × K

points which is defined as follows. Let us select a rec-
tangular domain A of  the two-dimensional lattice with
I × J points x = (xi, yj), xi = a × i, i = 1, ..., I and yj = a × j,
j = 1, ..., J, a is the step of  the lattice. A cell in XK takes
the shape of  parallelepiped of  height Dt with a square
base. Clearly, any point in XK has coordinates (xi, yj, tk);
tk = t + kDt, k = 0, ..., K.

We say that a seismic event happens if  an earth-
quake with magnitude greater than some pre-selected
threshold M0 is registered. For any cell in our space-
time grid we define an indicator of  an event, i.e., a bi-
nary function h (i, j, k). This function takes the value 1
if  at least one seismic event is registered in a given cell
and 0 otherwise. Suppose that for all points (xi, yj, tk)
the value of  a vector precursor f (i, j, k) = { fq(i, j, k),
q = 1, ..., Q} is given. The components of  the precursor
fq(i, j, k), q = 1, ..., Q are the scalar statistics constructed
on the base of  our understanding of  the phenomena
that precede a seismic event.

4. Mathematical assumptions
A number of  basic assumptions form the founda-

tion of  the mathematical technique of  earthquake fore-
casting. In the framework of  mathematical theory they
can be treated as axioms but are, in fact, an idealization
and simplication with respect to the description of  the
real phenomena. Below we summarize the basic as-
sumptions which are routinely used in existing studies
of  seismicity and algorithms of  earthquake forecasting
even the authors do not always formulate them explic-
itly. We accept the following assumptions or axioms of
the mathematical theory:

(i) The multicomponent random process {h (i, j, k),
f (i, j, k)}, describing the joint evolution of  the vector
precursors and the indicator of  seismic events, is sta-
tionary.

This assumption provides an opportunity to in-
vestigate the intrinsic relations between the precursors
and the seismic events based on the historical data. In
other words, the experience obtained by analysing the
series of  events in the past, is applicable to the future as
the properties of  the process do not depend on time.
In reality, this assumption holds only approximately and
for a restricted time period. Indeed, plate tectonics de-
stroys the stationarity for a number of  reasons including
some purely geometrical considerations. For instance,
the movements of  the plates leads to their collisions,
their partial destruction and also changes their shapes.
Nevertheless, the seismic process can be treated as a
quasi-stationary one for considerable periods of  time.
At the time when the system changes from one quasi-

stationary regime to another (say, nowadays, many re-
searchers speak about the abrupt climate change) the
reliability of  any prediction including the forecast of
seismic events is severely restricted.

(ii) The multicomponent random process {h (i , j, k),
f (i, j, k)} is ergodic.

Any quantitative characteristic of  seismicity more
representative than a registration of  an individual event
is, in fact, the result of  averaging over time. For instance,
the Gutenberg-Richter law, applied to a given region re-
lates the magnitude with the average number of  earth-
quakes where the averaging is taken over a specific time
interval. In order to associate with the time averaging
a proper probabilistic characteristic of  the process and
make a forecast about the future one naturally needs
the assumption of  ergodicity. This exactly means that
any averaging over a time interval [0, T] will converge
to the stochastic average when T → ∞. In view of  ergod-
icity one can also construct unbiased and consistent es-
timates of  conditional probabilities of  strong earthquakes
under the condition that the precursors take values in
some intervals. Naturally, these estimates are the fre-
quencies of  observed earthquakes, i.e., ratios of  the
number of  cells with seismic events and prescribed val-
ues of  precursors to the total number of  cells with the
prescribed values of  precursors. (Recall that an unbiased
point estimate î of  parameter i satisfies the condition
E [i] =  î and a consistent estimate converges to the true
value i when the sample size tends to infinity).

(iii) Any statement about the value of  the indicator
of  a seismic event h (i, j, k) in the cell (i, j, k) or its prob-
ability should be based on the values of  the precursor
f (i, j, k) only.

This assumption means that the precursor in the
given cell accumulates all the relevant information about
the past and the information about the local properties
of  the area that may be used for the forecast of  the seis-
mic event in this cell. In other words, the best possible
precursor is used (which is not always the case in prac-
tice). As in the other cases, this assumption is only an
approximation to reality, and the quality of  a forecast
depends on the quality of  the selection and accumula-
tion of  relevant information in the precursors.

Below we present some corollaries and further
specifications.

(iii-a) For any k the random variables h (i, j, k),
i = 1, ..., I, j = 1, ..., J are conditionally independent under
the condition that the values of  any measurable func-
tion u (f (i, j, k)) of  the precursors f (i, j, k), i = 1, ..., I,
j = 1, ..., J are fixed.

In practice this assumption means that the forecast
for the time tk = t0+ kDt cannot be affected by the values
related to the future time intervals (tk, tk+Dt]. In reality

EARTHQUAKE FORECASTING: STATISTICS AND INFORMATION



all of  these events may be dependent, but our forecast
does not use the information from the future after tk.

(iii-b) The conditional distribution of  the random
variable h at a given cell depends on the values of  the
precursors f at this cell and is independent of  all other
variables.

(iii-c) The conditional probabilities Prij {h|u ( f )} of
the indicator of  seismic events h in the cell (i, j, k), under
condition u ( f ) in this cell do not depend on the posi-
tion of  the cell in space (the time index k related to this
probability may be dropped due to the stationarity of
the process).

In other words, the rule for computing the condi-
tional probability Pr based on the values of  precursors
is the same for all cells, and the space indices of  proba-
bility Pr may be dropped. This condition is widely ac-
cepted in constructions of  the forecasting algoritms but
rarely formulated explicitly. However, the probability
of  a seismic event depends to a large extent on the local
properties of  the area. Hence, the quality of  the fore-
casting depends on how adequately these properties are
summarized in the precursors. This formalism properly
describes the space inhomogenuity of  the physical space
because the stationary joint distribution of  Prij {h, f ≤ x}
for an arbitrary precursor f depends, in general, on the
position of  the cell in the domain A. Below we will use
the distributions of  precursors and indicators of  seis-
mic events in domain A that do not depend on the spa-
tial coordinates and have the following form

(iii-d) Note that assumption (iii) implies that the
conditional probabilities Prij {h|u ( f )} are computed via
the probabilities PrA{h, f ≤ x} only. 

The properties listed above are sufficient to obtain
the point estimates for the conditional probabilities of  seis-
mic events under conditions formulated in terms of  the
values of  precursors. However, additional assumptions are
required for a testing of  the forecasting algorithm:

(iv) The random variables f (i, j, k) are condition-
ally independent under condition that h (i, j, k) = 1.

Again, these conditions are not exactly true, how-
ever they may be treated as a reasonable approxima-
tion to reality. Indeed, if  the threshold M0 is sufficiently
high then the strong earthquakes may be treated as
rare events, and the cells where they are observed are
far apart with a high probability. Any two events re-

lated to cells separated by the time intervals t are as-
ymptotically independent as t → ∞ because the seismic
process has decaying correlations (the mixing property
in the language of  random processes). The loss of  de-
pendence (or decaying memory) is related to the physical
phemonema such as healing of  the defects in the rocks,
relaxation of  strength due to viscosity, etc. As usual in
physical theories, we accept an idealized model of  the
real phenomena applying this asymptotic property for
large but finite intervals between localizations of  seismic
events. The independence of  strong earthquakes is not a
new assumption, in the case of  continuous space-time it
is equivalent to the assumption that the locations of  these
events form a Poisson random field. (Note that the dis-
tribution of  strong earthquake should be homogeneous
in space a priori, because by condition there is no infor-
mation about the heterogeneity outside precursors. The
information about the location of  epicenters should be
contained in the precursors) The Poisson hypothesis is
used in many papers, see, e.g. Harte and Vere-Jones
[2005]. It is very natural for the analysis of  the tails of  the
Gutenberg-Richter law for large magnitudes [Pisarenko
et al. 2008]. Summing up, the development of  the strict
mathematical theory of  earthquake forecasting does not
require any additional assumption except those routinely
accepted in the existing algorithms but usually not for-
mulated explicitly.

5. The standard form for precursors
The correct solution of  the forecasting problem

given the values of  precursors f (i, j, k) is provided by the
estimate of  conditional probability Prij{h|f (i, j, k)} of  the
indicator of  a seismic event in the cell (i, j, k). In prac-
tice this solution may be difficult to obtain because the
number of  events in a catalog might be not sufficient.

Indeed, the range of  value of  a scalar precursor is
usually divided into a number M of  intervals, and only
a few events are registered for any such interval. For a
Q-dimensional precursor the number of  Q-dimensional
rectangles, covering the range, is already MQ, and ma-
jority of  them contains zero events. Only a small num-
ber of  such rectangles contains one or more events, that
is the precision of  such an estimate of  conditional prob-
ability is usually too low to have a practical value.

For this reason one constructs a new scalar precur-
sor in the form of  the scalar function of  component of
the vector precursor, and optimizes its predictive power.
This approach leads to additional complication as the
units of  measurement and the physical sense of  different
components of  a precursor are substantially different. In
order to overcome this problem we use some transfor-
mation to reduce all the components of  the precursor to
a standard form with the same sense and range of  values.

, ,PrA

PrA PrA Pri

PrA Pri

Prih f x

fE x I J fE x

I J h

I J h f x

x

p h

1

1
1

1

1

,

,

,

rA

rA rA rijij
i j A

rA rijij
i j A

A
rijij

i j

A

#

#

#

#

# #

#

/

/ =

=

=

= =

!

!

!

Q
Q

Q

Q
V

V

V

V

E

E

fE

E

E

fE

H

H

H

H

H

H|

|

|

GERTSIK ET AL.

6



7

Let us transform all the precursors fq(i, j, k), q = 1,
..., Q to variables with the values in [0, 1] providing es-
timates of  conditional probabilities. So, after some trans-
formation F we obtain an estimate of  Pr {h|u ( f (i, j, k) =
1}, where u is a characteristic function of  some interval B,
i.e., the probability of  event h (i, j, k) = 1 under condi-
tion that this precursor takes the value f (i, j, k)∈B.

The transformation F of  a scalar precursor f (i, j, k)
is defined as follows. Fix an arbitrary small number f.
Let L be a number of  cells (i, j, k) such that h (i, j, k) =
1, and Zl, l = 1, ..., L, be the order statistics, i.e., the val-
ues f (i, j, k) in these cells listed in non-decreasing order.
Define a new sequence zm, m = 1, ..., M, from the or-
dered statistics Zl by the following recursion: z0 = −∞,
zm is defined as the first point in the sequence Zl, such
that zm − zm−1 ≥ f. Next, construct the sequence zm* =
zm + (zm +1− zm )/2, m = 1, ..., M − 1, and add the auxil-
iary elements z0 = −∞, zM = 1. Define also a sequence
nm, m = 1, ..., M, where nm equals to the number of  val-
ues in the sequence Zl, such that zm* −1 ≤ Zl < zm*. Finally,
define the numbers Nm, m = 1, ..., M counting all cells
such that zm* −1 ≤ f (i, j, k) < zm*, m = 1, ..., M. Observe that
∑Mm−1 nm= L, ∑

M
m−1 Nm= N, and use the ratios

as estimate of  unconditional probability of  a seismic
event in a given cell

The transformation F is defined as follows

This definition implies that transformation F re-
places the value of  a precursor for the frequency, i.e.,
the ratio of  a number of  cells containing a seismic event
and the values of  a precursor from [zm* −1, zm* ) to the
number of  cells with the value of  precursor in this range.
These frequencies are the natural estimates of  condi-
tional probabilities Pr {h (i, j, k) = 1|zm* −1 ≤ f (i, j, k) < zm*},
m = 1, ..., M, computed with respect to stationary dis-
tribution PA(x):

(This conditional probability can be written as
Pr {h = 1|u ( f )}, where u is the characteristic function
of  interval [zm* −1, zm* ). The function g has a stepwise
shape, and the length of  the step is bounded from
below by f. It can be checked that there exists the limit
~g =  lim——

f→∞
lim——

K→∞
g = Pr {h = 1|u ( f )}.

The estimates of  conditional probabilities in terms
of  the function g are quite rough because typically the
numbers nm are of  the order 1. As a final result we will
present below more sharp but less detailed estimates
of  conditional probabilities and confidence intervals for
them.

6. Combinations of  precursors
There are many ways to construct a single scalar

precursor based on the vector precursor (F fq, q = 1, ...,
Q). Each such construction inevitably contains a number
of  parameters or degrees of  freedom. These parameters
(including the parameters used for construction of  the
precursors themselves) should be selected in a way to
optimize the predictive power of  the forecasting algo-
rithm. The optimization procedure will be presented
below, its goal is to adapt the parameters of  precursors
to a given catalog of  earthquakes, that is to obtain the
best possible retrospective forecast. However, this adap-
tation procedure creates a “ghost” information related
with the specic features of  the given catalog but not
present in physical propertities of  real seismicity. This
“ghost” information will not be reproduced if  the al-
gorithm is applied to another catalog of  earthquakes. It
is necessary to increase the volume of  the catalog and
to reduce the number of  free parameters to get rid of
this “ghost” information. Clearly, the first goal requires
the considerable increase of  the observation period and
may be achieved in the remote future only. So, one con-
centrates on the reduction of  numbers of  degrees of
freedom. The simplest variant including Q − 1 parame-
ters is the linear combination

As a strictly monotonic function of  a precursor is
a precursor itself  the log-linear combination is an
equally suitable choice

Here cq, q = 1, ..., Q − 1 are free parameters. The re-
sult of  the procedure has the form g = F f *.

7. Alarm levels, point and interval estimations
In view of  Equation (5.3) the precursor g is the set

of  estimates for probabilities

N
L

m=

; ;

; ;

PrA

Pr

p h i j k

h i j k x dPA x1

1A rA

PA

/

=

= =

=
3

3

-
#

Q
Q Q
V
V V

E
F

H
I

-
#

, ,

if ; ; , , ..., .

g F f i j k

N
n

z f i j k z m M1
m

m
m m1 1#

= =

= =) )- Q
Q
V
V

; ;

; ;

; ;Pr

Pr

h i j k

h i j k x dPA x

z f i j k z

dPA x

1

1

PA

m m

PA
z
# z

z
# z

1

m

m

m

m

1

1

1#

=

= =

=

) )
-

)

)

)

)

-

-

Q
Q

Q

Q
Q

V
V

V

V
V

F
F I

I

z
#

z
#

f) F f1
q

Q

f1
2

= +f)
=

| .c F fqq fq1
2
-|

.ln lnf) F f1 c F fqq
q

Q

fqf1 1
2

= +f) -
=

Q QV V|

EARTHQUAKE FORECASTING: STATISTICS AND INFORMATION

(5.1)

(5.2)

(6.1)

(6.2)

(5.3)

(5.4)



Its serious drawback is that typically { z*l−1 ≤ f (i; j;
k) < z*l } correspond to single events, and therefore the
precision of  these estimates is very low (the confidence
intervals discussed below may be taken as a convenient
measure of  precision). In order to increase the precision
it is recommended to use the larger groups of  cells con-
taining a larger number of  events, that is a more coarse
covering of  the space where the precursor takes its val-
ues. In a sense, the precision of  the estimation and the
resolving power are related by a kind of  “uncertainty
principle”: the more precise estimate one wants to get
the more coarse is the time-space range of  their values
and vice versa.

We adapt the following approach in order to achieve
a reasonable compromise.

(1) For fixed thresholds as, s = 1, ..., S +1, a1 = 1,
as< as+1; aS+1= 0, we define S possible alarm levels
as+1 ≤ g (i; j; k) < as and subsets Xs, s = 1, ..., S , of  the set
XK corresponding to alarm levels, i.e., Xs is a set of  cells
of  XK, such that as+1 ≤ g (i; j; k) < as.

There are different ways to choose the number S
of  alarm levels and the thresholds as, s = 2, ..., S . Say, fix
S = 5, and select as = 10

−a(s−1). This is a natural choice
of  the alarm level because at a = 1 it corresponds to
decimal places of  the estimate of  the conditional prob-
ability given by the precursor. The problem with S = 2,
i.e. two-level alarm, may be reduced to the hypothesis
testing and is discussed in more details below.

(2) Compute the point estimates of  probabili-
ties

obtained via the distribution PX(x) of  a precursor g in
the same way as in Equation (4.4). The property (iv)
implies that for any domain Xs the binary random vari-
ables h (i, j, k) are independent and identically distrib-
uted, i.e.

and the unbiased estimate of  ps takes the form

where ns stands for the number of  cells in domain Xs,
and by ms we denote the number of  cells in Xs contain-
ing seismic events.

(3) The random variable ms takes integer values be-
tween 0 and ns. The probabilities of  these values are
computed via the well-known Bernoulli formula

Let us specify the confidence interval covering the
unknown parameter ps with the confidence level c. In
view of  the integral Mouvre-Laplace theorem for ns
large enough the statistics

is approximately Gaussian N (0, 1) with zero mean and
unit variance. Note that the values ns increase with
time. Omitting straightforward calculations and re-
placing the parameter ps by its estimate is we obtain
that is

− < is < +is
+; where

and tc is the solution of  equation U(tc) =
c
—
2

. Here U
stands for the standard Gaussian distribution function.

(4) As a result of  these considerations we intro-
duce ‘the precursor of  alarms’ which indicates the
alarm level: R ( f (i, j, k) ) = s (i, j, k) . It will be used for cal-
culations of  point estimates and the confidence interval
in the form

This result will be used for a prospective forecast-
ing procedure.

8. The information gain and the precursor quality
The construction of  a ‘combined’ precursor R in-

volves parameters from formula (6.1) or (6.2) as well as
parameters which appear in the definition of  each in-
dividual precursors fq. It is natural to optimize the fore-
casting algorithm in such a way that the information
gain related to the seismic events is maximal. In a one-
dimensional case the information gain as a measure of
the forecast efficiency was first intoduced by Vere-Jones
[1998]. Here we exploit a modification of  his ideas in
the case of  multi-dimensional space-time process. Re-
mind the notions of  the entropy and information. Put-
ting aside the mathematical subtlety (see Kelbert and
Suhov [2013] for details) we follow below an intuitive
approach by Prohorov and Rozanov [1969]. The infor-
mation containing in a given text is, basically, the length
of  the shortest compression of  this text without the loss
of  its content. The smallest length S of  the sequence
of  digits 0 and 1 (in a binary code) for counting N dif-
ferent objects satises the relations 0 ≤ S − log2 N ≤ 1. So,

; ; , , ..., .Pr h z f i j k z l L f1 1l l1 1#= =
) )
- Q QV VF I

, , , , ..., .Pr h a g i j k a s S1 1s s1 1#= =+ Q VF I

, , ,Pr h a g i j k a p1 s s s1 1# /= + Q VF I

n
m

s

s
i=

.Pr m k
n
k

p p1s
s

s
k

s
nV ks= = - -Q S QV X nV

p

p p

n

1
s s

s s

si -

-
Q
Q
V
V

,
n

t

n

t

1

1

s s
s

s s

s s
s

s s

i i
i i

i i
i i

= -
-

=
-

+

c

c

-

+

Q

Q

V

V

, , , ,s i j k s i j1i i
-
Q QVF , , ,k s i j k1 i+QV VI

GERTSIK ET AL.

8

(7.1)



9

the quantity S ~ log2 N characterizes the shortest length
of  coding the numbers of  N objects.

Consider an experiment that can produce one of  N
non-intersecting events A1, ..., AN with probabilities
q1, ..., qN, respectively, q1+ ...+ qN = 1. A message in-
forming about the outcomes of  n such independent
identical experiments may look as a sequence (Ai1, ...,
Ain), where Aik is the outcome of  the experiment k. But
for long enough series of  observations the frequency
ni/n of  event Ai is very close to its probability qi. It
means that in our message (Ai1, ..., Ain) the event Ai ap-
pears ni times. The number of  such messages is

By the Stirling formula the length of  the shortest
coding of  these messages

The quantity Sn measures the uncertainty of  the
given experiment before its start, in our case we are look-
ing for one of  the possible outcomes of  n independent
trials. The specific measure of  uncertainty for one trial is

known as Shannon’s entropy of  distribution q1, ..., qN
(in physical literature it is also known as a measure of
chaos or disorder). After one trial the uncertainty about
the future outcomes decreases by the value S = Sn− Sn−1,
this decrement equals to the information gain I = S, ob-
tained as a result of  single trial. The quantity

is the (unconditional) entropy of  distribution for the in-
dicator of  the seismic event h in a space-time cell in the
absence of  any precursors. The conditional entropy
S (h|as+1 ≤ g < as) under condition that in the cell (i, j,
k) the alarm level s is set up equals

The expected conditional entropy SR(h) of  indica-
tor of  seismic events where the averaging in performed
by the distribution of  precursors R takes the form

We conclude that the knowledge of  the precursor
values helps to reduce the uncertainty about the future

experiment by S (h) − SR(h) which is precisely informa-
tion I (R,h) obtained from the precursor. Taking into ac-
count Equations (8.1), (8.2) and the fact that

we specify the information gain as

By analogy with the one-dimensional case [Kol-
mogorov 1965] the quantity I (R,h) may be called the
mutual information about the random field h that may
be obtained from observations of  random field R. It is
known that the information I (R,h) is non-negative and
equals to 0 if  and only if  the random fields h and R are
independent. This mutual information I (R,h) takes its
maximal value S (h) in an idealized case of  absolutely
exact forecast. The mutual information is the informa-
tion that the distributions of  precursors contribute to
that of  the indicator of  a seismic event. For this reason
it may be considered as an adequate scalar estimate for
the quality of  the forecast.

The quantity I (R,h) depends on the cell size, i.e.,
on the space discretization length a and time interval
Dt. We need a formal test to compare precursors de-
fined for different size of  the discretization cells. For
this aim let us introduce the so-called ‘efficiency’ of  pre-
cursors as the ratio of  information gains

This efficiency varies between 0 and 1 and serves
as a natural estimate of  information quality of  precur-
sors. It allows comparing different forecasting algo-
rithms and select the best one.

A natural estimate of  S (h) based on Equations (5.1)
and (5.2) is defined as follows

Taking into account Equation (8.1) and using an
estimate of  PA(as+1 ≤ g < as) in the form of  ratio xs=

ns—
N

,
we construct an estimate of  I (R,h) as follows

Remark. It seems reasonable to introduce a penalty

!... !
!N n n

n
m

N1
= ! .

.log logS Nn n q qn Nn i
i

N

i2
1

2. .-
=

|

, ..., logn S n S q q q q
1 1

n n N i
i

N

i1 2
1

= =-
=

Q V |

log logS h pp- p p 11 AA A A2 2 -= p- - -Q Q QV V V

.

log

log

S h

p

p! p
p PA a g a11

R

s

s
s

S

s

s P$ A s s
1

2

2 1 1#-

=- +

+ -
=

+Q
Q

Q Q
V
V V V

p!
P$

|

p p PA a g aA s
s

S

PA s s
1

1 1#=
=

+Q V|

,

.

R log

log

I h

p
p

p# p
p

p PA a g a1
1

1
A

s

s
s

S

A

s

s PA s s

1
2

2 1 1#-
-

= +

+ -

=

+

Q

Q Q

V

V V

p#

&

|

log logS h a g a pp- p p 11s s ss s s1 2 21# -= p- - -+R Q QW V V

,R
R

r h
S h

I
=Q QQV

,
.

h
h
V
V

.log logS h 1 12 2m m m m=- - - -tQ Q QV V V

, ,R log logI h 1 1
1

s
s

s
s

s

S

s2 2
1

i
m

i
i

m

i
x= + - -

-

=

tQ QV V# &|

;
,R

r R h
S h

I
=t t
tQ QQV

,
.

h
h
V
V

EARTHQUAKE FORECASTING: STATISTICS AND INFORMATION

(8.1)

(8.2)

(8.3)

(8.4)

(8.5)



related to the number of  superficious parameters in eval-
uating the quality of  forecast pointing to the natural
analogy with the Akaike test [Akaike 1974] and similar
methods in information theory. In our context the main
parameter of  importance is r̂ (R; h) and its limit as t →∞.
This quantity does not involve the number of  parame-
ters directly. Probably, the rate of  convergence depends
on the number of  parameters but this dependence is
not studied yet.

9. The forecasting procedure
The number of  time intervals, i.e., the number of

observations N used in the construction of  estimates
increases with the growth of  observation time. So, the
computation procedure requires constant innovations.
On the other hand some computation time is required
to ‘adapt’ the model parameters to the updated infor-
mation about seismic events via an iterative procedure.
For these reasons we propose the following forecasting
algorithm.

(1) Given initial parameter values at the moment
tK−1= t0 + (K − 2)Dt we optimize them to obtain the
maximum of  efficiencys r̂ (R; h) of  a precursor in do-
main XK−1. For this aim the Monte-Carlo methods is
helpful: one perturbs the current values of  parameters
randomly and adapts the new values if  the efficiency
increases. The process continues before the value of  ef-
ficiency stabilized, this may give a local maximum, so
the precedure is repeated for a sufficient number of
times. The choice of  the initial value on the first step of
the optimization procedure is somewhat arbitrary but
a reasonable iteration procedure usually leads to con-
sistent results. The optimization procedure takes the
period of  time tK−1< t ≤ tK.

(2) Next, we construct the forecast in the follow-
ing way. At the moment tK the values of  precursor ĝ in
each cell (i, j, K + 1) is computed with optimized pa-
rameters. Based on these parameter values the alarm
levels, the point estimates and confidence intervals are
computed in each cell as well as the values of  efficiency
of  precursors.

(3) The estimates of  stationary probabilities of
seismic events in the cell î(i, j) are defined as follows:

they can be used for creation of  maps of  seismic danger
(but not “hazard”, this term is reserved) in the region.

10. Retrospective and prospective informativities
The efficiency of  precursor which is achieved as a

result of  parameters optimization could be considered
as retrospective as it is constructed by the precursors

adaptation to the historical catalogs of  seismic events.
The prospective efficiency for the space-time domain
containing the cell in the ‘future’ is based on the forecast.
It is computed via formulas (8.3), (8.4), (8.5) with the only
difference to the retrospective efficiency, however, ap-
proaches this value with time. In principle, the prospec-
tive efficiency is an ultimate criterion of  the precursors
quality and the retrospective efficiency could serve only
for the preliminary selection of  precursors and their
adaptation to the past history of  seismic events.

11. Testing of  the forecasting algorithm
The efficiency of  precursor could be computed ex-

actly only in an idealized case of  infinite observation
time. However, its estimate may be obtained based on
the observation over a finite time interval. So, if  an es-
timate produces a non-zero value not necessarily the
real effect is present. It may be simply a random fluc-
tuation even if  the precursor provides no information
about the future earthquake. For this reason we would
like to check the hypothesis H0 about the independence
of  a precursor and an event indicator with a reasonable
level of  confidence. In case the hypothesis is rejected
one has additional assurance that the forecasting is real,
not just a “ghost”. So, consider the distributions

and

The function PA(PA
−1(y)) = y of  variable y = PA(x)

provides an uniform distribution F*(y) = Pr {p ≤ y} of
some random variable p on [0,1]. Next, consider a dis-
tribution function G(y) = P'A(PA

−1(y)) on [0,1], and use a
parametric representation for abscissa PA(x) and ordi-
nate P'A(x). If  random fields g and h are independent the
distribution functions PA(x) = P'A(x) and G(y) are uniform.
So, the hypothesis about the absence of  forecasting, i.e.,
about the independence of  g and h is equivalent to the
hypothesis H0 that the distribution G(y) is uniform.

The empirical distribution GL(y) related to G(y) is
defined as follows. Denote by ul, l = 1, ..., L the values
of  the function g (i, j, k) sorted in the nondecreasing
order and belonging to the cells where h (i, j, k) = 1. Let
nl be the numbers of  cells such that h (i, j, k) = 1, g (i, j,
k) = ul. Denote by m (ul) the numbers of  cells such that
g (i, j, k) < ul, and define the empirical distribution GL(y)
as a stepwise function with GL(0) = 0 and positive
jumps of  the size

nl—
L

at points yl = 
m (ul)——

N    
. 

The well known methods of  hypothesis testing re-
quires that the function GL(y) has the same shape as for

;i j K
1

k

K

1

i =
=

tQ V | ,, ,s i j k
1

i Q V|

, , ,PriPA x g i j kI J x
1

;

PA
A

rijij
i j

#=
!

Q Q
Q

V V
V

E H|

, , , , .PrPA x I J g i j k x h i j k
1

1
;

PA ijij
i j A

#= =
!

l Q Q Q
Q

V V V
V

F I|

GERTSIK ET AL.

10

·

·



11

independent trials, i.e., random variables ul, l = 1, ..., L
are independent in view of  axiom (iv). Naturally, we ac-
cept the precursors such that the hypothesis H0 is re-
jected with the reasonable level of  confidence. (Remind,
that the hypothesis is accepted if  and only if  its logical
negation could be rejected based on the available ob-
servations. The fact that the hypothesis cannot be re-
jected does not mean at all that it should be accepted, it
only means that the available observations do not con-
tradict this hypothesis. Say, the well-known fact that “The
Sun rises in the East” does not contradict to our hypoth-
esis, however it may not be considered as a ground for its
acceptance.) For large values of  L the Kolmogorov sta-
tistics [Kolmogorov 1933a] is helpful for this aim

with an asymptotic distribution

or Smirnov’s statistics [Smirnov 1939]

with asymptotic distribution

The asymptotic expressions for these statistics can
be used for L > 20 [Bolshev and Smirnov 1965].

12. The binary alarm and the hypothesis testing
The prediction is the form of  forecast when an

alarm is announced in a given cell without a prelimi-
nary evaluation of  the probability of  a seismic event.
In this case we can estimate the probabilities of  events,
too. (If  the alarm is announced in an arbitrary domain
X we set up an alarm if  at least half  of  the cell of  our
model is occupied by alarm.)

Let M be the number of  cells in X which are in the
state of  alarm, M0 be the number of  cells where the
seismic event is present but no alarm was announced
(the number of  ‘missed targets’). Denote by x = M—

N
the

share of  the cells with alarm announced, m = L—
N

the share
of  the cells with seismic events, and o = M0—

M
the share of

missed targets. Let a random variable h (i, j, k) equal 1
if  an alarm is announced in the cell (i, j, k) and 0 other-
wise. Obviously, the estimate of  conditional probability
Pr {h (i, j, k) = 1|h (i; j; k) = 1} of  the seismic event
under the condition of  alarm is  m——(1 −o)———

x
; and the esti-

mate of  conditional probability Pr {h (i, j, k) = 1|h (i; j;
k) = 0} of  the seismic event under the condition of  no
alarm is   mo———

1−x 
.

If  the alarm is announced according to the pro-
cedure described in Section 5 the threshold a1 specify-
ing the acceptable domain of  values for g (i, j, k) should
be treated as a free parameter and selected by maxi-
mizing the information efficiency r̂ (h; h). The esti-
mate of  information increase for given values of  x and
o equals

The value of  h (i; j; k) characterizes the results of
checking two mutually exclusive simple hypothesis:

H0: the distribution of  f
*(i, j, k) has the form

implying ‘no seismic events’, or
H1: the distribution of  f

*(i, j, k)) has the form

implying the presence of  seismic event.
Statistics for checking of  these hypothesis is the

precursor g (i, j, k), and the critical domain for H0 has
the form {g (i, j, k) ≥ a1}. (If  usual method of  alarm an-
nouncement is used the relevant precursor plays the
role of  statistics and the critical domain is defined by
the rule of  the alarm announcement). The probability
of  a first type error

it is estimated as   x−m———
1−

(1−o)
———
m

. The probability of  second
type error

is estimated as o. (Note that due to condition (iii) any
test used for checking these hypothesis should not de-
pend on the coordinates of  the cell).

The Neyman-Pearson theory allows to define the
domain of  images of  all possible criteria: in coordinates
(a, b) it is a convex domain with a boundary C which
corresponds to the set of  uniformly most powerful tests.
This family may be defined in terms of  the likelihood
ratio K(x) =

p1A(x)———
p0A(x)

under condition that the distribu-

supD GL L= Q yy -V

( ) , ,lim Pr L D z e z1 0
L L

k

k

k z2 2 2
2# = -

"3
3

3

=-

-F I |

sup

inf

D G

D G
L L

L L

=

=-

+

-

Q!
! ,$

,$y
y y

y

L -

-Q
Q
V
V

,$
,$

, .

lim Pr lim PrL D z L D z

e z1 0

L L L L

z2 2 2

# #= =

= -

" "3 3

+ -

-

F FI I

,

.

log log

log

log

I h 1

1 1
1

1 1

1
1

1

1

2 2

2

2

h m o
x
o

mo
x
o

x m o
m x

x m o

x mo
x m

x mo

-

- -
- -

= - + - +

+ - -
-

- -
+

+ - -

tQ

Q

Q
Q

Q
Q
Q

Q
V V

V
V

V
V
V

V! $

, , , , ,PrAPA x f
) i j k i j kx h 0PA rA

0
#= =fF )Q Q QV V VfF I

, , , , ,PrAPA x f
) i j k x h i j k 1PA rA

1
#= =fF )Q Q QV V VfF I

, , , , ,Pr g i j k h i j k1 0a = = =Q QV VF I

, , , , ,Pr g i j k h i j k0 1b= = =Q QV VF I

EARTHQUAKE FORECASTING: STATISTICS AND INFORMATION



tions P1A(x) and P
0
A(x) have the densities p

1
A(x) and

p0A(x).

where ~ denotes the threshold. In Gercsik and Kelbert
[2004] we demonstrated that among all the tests with
the images on the boundary C there exists three different
best tests. Here the term “best” may be understood in
three different senses, i.e., maximizing the variational,
correlational and informational efficiency. The most rel-
evant criteria is the informational efficiency  r̂ (h; h).

13. The choice of  precursors
We use the term ‘empiric precursor of  an earth-

quake’ for any observable characterisric derived from
the catalog only, which provides for this catalog a rea-
sonable retrospective forecast of  seismic events, and
which is not derived from basic physical concepts of
seismicity (say, the periods of  relatice calm, deviation
of  some basic characteristic from a long-time average ,
etc). In contrast, the physical precursors are derived
from some physical processes and characterize physi-
cal quantities (stress fields, strength, concentration of
cracks, etc.) or well-defined physical processes (i.e.,
phase transitions, cracks propagations, etc.) In the me-
teorological forecast the danger of  using empirical pre-
cursors was highlighted by A. Kolmogorov in 1933
[Kolmogorov 1933b]. From that time the meteorolog-
ical forecast relies on the physical precursors which are
theoretically justified by the models of  atmospheric dy-
namics. Below we will present Kolmogorov’s argument
adapted to the case of  seismic forecast. This demon-
strates that the purely empirical precursors work well
only for the given catalog from which they are derived.
However, their efficiency deteriorates drastically when
they are applied to any other independent catalog.

Consider a group of  k empirical precursors used
for a forecast and selected from a set of  n such groups.
According to Kolmogorov’s remark the number k is
typically rather small. This is related to the fact that a
number of  strong earthquakes in catalog is unlikely to
exceed a few dozen. As the values of  precursors are ran-
dom there exists a small probability p that the efficiency
of  the forecast exceeds the given threshold C. Then the
probability of  event r̂ (R, h) ≤ C equals 1 − p, and the
probability of  event r̂ (R, h) > C for at least one collec-
tion of  precursors equals P = 1 − (1 − p)n and tends to
1 as n → ∞. (According to Kolmogorov some arbitrari-
ness of  the assumption of  independence is compen-
sated by the large number of  collections.)

Summing up, if  the number of  groups is large
enough with probability close to 1 it is possible to find

a group giving an effective retrospective forecast for a
given catalog. In practice this is always the case as the
number of  empirical precursors could be increased in-
definitely by variation of  real parameters used in their
construction. It is important to note that for such a
group, which is highly efficient for the initial catalog,
the probability that the efficiency is greater than C is
still equal to p for any other catalog. In other words: the
larger the number of  the groups of  empirical precur-
sors the less reliable the forecast is. So, the collection of
a large list of  the empirical precursors is counter-pro-
ductive. Much more reliable are the physical precursors
intrinsicly connected with the physical processes which
preserve their values with the change of  sample. The
probability to find such a set of  precursors by pure em-
pirical choice is negligible because they are very rare in
the immense collection of  all possible precursors.

14. Demonstration of  algorithm
A preliminary version of  the forecast algorithm de-

scribed above was used in the paper [Ghertzik 2008] for
Califorrernia and the Sumatra-Andaman earthquake re-
gion. These computations serve as a demonstration of
the efficiency of  the method but their actual results
should be taken with a pinch of  salt because the selec-
tion of  precursors does not appear well-justied from
the modern point of  view: the number of  free param-
eters (34) to be adapted in the precursor “stress indica-
tor” is too large. For this reason the method was not
fully exploited. Nowadays, the progress is resumed
with a particular emphasis on the selection of  a new set
of  precursors strongly related to the critical phenom-
ena in geophysics.

The following selection of  precursors was used:
(1) The stress indicator is a scalar predictor char-

acterizing the local level of  the stress field. It is con-
ceivable as a stress tensor component. The predictor
increases due to tectonic movements, decreases with
viscosity and with increasing fracturing, and is affected
by earthquake sources considered as cracks.

(2) The damage indicator is a predictor character-
izing the stress-induced microfracturing level. The pre-
dictor increases with the value of  applied stresses and
decreases as cracks are healed.

(3) The third predictor is the concentration
Zhurkov-Sobolev criterion of  seismically active faults
based on the calculation of  the parameter concentra-
tion (density) seismogenic ruptures and modified in
such a way that it can be calculated for a local cell.

(4) The far-order predictor is the integral of  the ra-
dial correlation function for seismic energy released in
a spatiotemporal cell. Its construction is based on the
idea of  the strong earthquake as a phase transition be-

, ,
, ,

i j k
i j k i

i1
0

h

h =
=Q

Q
V
VG iff x

iff x 2
1

~

~

K

K

Q
Q
V
V

GERTSIK ET AL.

12



13

tween consolidation and fracture states of  a medium.
The phase transition is preceded by the appearance of
a far order, an abrupt increase in the correlation radius
for the energy of  shocks released in a cell.

For detailed description of  precursors we refer to
original, for it is only an illustration but not the topic of
this work. It is useful, however, to summarize the main
results of  this paper.

California region. The catalog Global Hypocen-
ter Data Base CD-ROM NEIC/USGS, Denver, CO, 1989,
together with data from the site NEIC/USGS PDE
(ftp://hazard.cr.usgs.gov) for earthquakes with magni-
tudes M ≥ 4.0 with epicenters between 113°- 129° of
western longitude and 31°- 43° of  northern latitude
was used for parameter adaptation. The initial time t0
was selected by subtracting from the time of  actual
computation, 08.03.2006, an integer number of  half-
year intervals such that t0 fits the first half  of  the year
1936. (The final time 08.03.2006 could be considered as
an initial moment for constructing half  year forecast
forward up to the date of  the latest earthquake avail-
able in the catalog). During the computation the time
interval from the first half  of  1936 to the first half  of
1976 was used for relaxation of  the zero initial data
used for precursors. After this date the catalog for the
earthquakes with magnitude M ≥ 4.0 was used to esti-
mate the probabilities of  strong earthquakes with mag-
nitude M ≥ 6.0 up to the moment of  actual forecast.
Note that the adapted restriction to include into con-
sideration only earthquakes with magnitudes M ≥ 4.0 is
a severe restriction. It decreases the precision of  pre-
cursor computation and therefore, if  a prediction is suc-
cessful, increases the degree of  condence in the
predictor choice. We choose a = 150 km as a size of  the
spacial lattice, and Dt = 6 months as a time-step. Retro-
spective forecast was performed with 5 alarm levels de-
fined by the thresholds as = 10

−a(5− s), s = 1, ..., 4 and
a = 0.75. (Due to too short time step no alarms was
registered on the lowest level when parameter a = 1
was selected). In order to reduce the influence of  the
boundary conditions the large square covering all the

seismic events in the catalog used in the computations
was reduced by two layers of  elementary cells from
each boundary. As a result of  optimization the forecast
information efficiency of  0.526 was achieved, i.e., the
forecasting algorithm applied to the given catalog ex-
tracted from it about 53% of  all available information
about seismic events. It seems that this result could be
only partially explained by a lucky selection of  precur-
sors: another contributor to the high efficiency of  the
algorithm is the adaptation of  the parameters to the
features of  the specific catalog. The influence of  this ar-
tificial information may be reduced only with the in-
crease of  the observation interval. Accepting the rule
of  binary alarm announcement in the cells from group
1 and 2 from 5 levels possible one obtains that the
space-time share of  alarmed cells is 3.4% and the share
of  missed targets is 18.2%. This result is comparable
with the best forecasts available in the literature and ob-
tained by other methods (in the cases when the quan-
titative parameters of  algorithms are presented in the
publications). When the forecast was constructed in the
future we obtained that the estimate of  probability of
a strong earthquake anywhere in the area under study
is 0.174, and the maximal point estimate of  an event in
any individual cell is 0.071. As a whole the seismic situ-
ation in California did not look too alarming. Indeed,
there were no strong earthquakes in California in the
next half-year.

EARTHQUAKE FORECASTING: STATISTICS AND INFORMATION

Alarm
level

Number
of  cells

Number
of  events

Left boundary
of  interval

Point
estimate

Right boundary
of  interval

1 43 25 0.4186 0.5814 0.6998

2 155 11 0.0376 0.0710 0.1172

3 457 6 0.0063 0.0131 0.0291

4 1152 2 0.0005 0.001 0.0071

5 4049 0 0 0 0.001

Figure 1. Mapped areas of  five levels of  half-year forward forecast-
ing for California on March 8, 2006.

Table 1. Number of  events and alarm cells for each level of  the retrospective analysis for California.



Table 1 presents the number of  events and the
total number of  alarm cells for each level of  the retro-
spective analysis, as well as their ratios, which are well-
grounded point estimates of  the probability that at least
one earthquake with a magnitude M ≥ 6.0 will occur in
a cell at a given level of  alarm. The 5% condence intervals
are also given in the table for each of  these probabilities.
The true values of  event probabilities lie within these
intervals with a 95% probability (condence coefficient).
In all, 5856 cells participated in the prediction, and 44 of
these cells (0.75% of  all cells) contained events.

SAE region. We have conducted a retrospective
forecast of  strong earthquakes with magnitudes M ≥ 7.0
for the whole region where the Sumatra-Andaman earth-
quake (SAE) happened on 26.12.2004 with magnitude
M = 9.3. The catalog Global Hypocenter Data Base CD-
ROM NEIC/USGS, Denver, CO, 1989, together with
the data from the site NEIC/USGS PDE (ftp://hazard.
cr. usgs.gov) for earthquakes with magnitudes M ≥ 5.5
with epicenters between 84.3°- 128° of  eastern longitude
and 20°- 26° of  northern latitude was used for the pa-
rameters adaptation. The initial moment of  time t0 was
selected by subtracting from the time of  actual compu-
tation, 10.11.2004, an integer number of  half-year in-
tervals such that t0 fits the first half  of  the year 1936.
(The final time was selected in such a way that the next
half-year period covers SAE and its powerful after-
shock). During the computation the time interval from
the first half  of  1936 to the first half  of  1976 was used
for relaxation of  the zero initial data used for precur-
sors. After this date the catalog was used to estimate
the probabilities of  strong earthquakes with magnitude
M ≥ 7.0 up to the moment of  actual forecast. (For mag-
nitude M ≥ 7.5 the number of  seismic events was not
sufficient for reliable forecast because the 5%-confi-

dence intervals strongly overlapped). In this case the re-
striction to include into consideration only earthquakes
with magnitudes M ≥ 5.5 was adapted. As before, it de-
creases the precision of  precursor computation and
therefore, if  the prediction is successful, increases the
degree of  confidence to the predictor choice. We se-
lected the size of  the spacial grid as a = 400 km and the
size of  time-step Dt = 6 months. Retrospective analysis
was conducted following the same scheme as in the
previous case. In order to reduce the influence of  the
boundary conditions the large square covering all the
seismic events in the catalog used in the computations
was reduced by two layers of  elementary cells from
each boundary. As a result of  optimization the forecast
information efficiency was 0.549, i.e., the forecasting al-
gorithm extracted around 55% of  all available infor-
mation about seismic events when applied to the given
catalog. In the case of  binary alarm announcement the
space-time share of  alarmed cells was 3.1% and the
share of  missed targets was 8.3%. This result is compa-
rable with the best forecasts available in the literature
and obtained by the other methods (in the cases when
the quantitative parameters of  algorithms are presented
in the publications). In the case of  forward forecast the
two most powerful earthquakes, i.e., SAE and its major
aftershock, happened in the alarm zone of  the second
level, and two other events with smaller magnitudes in
the zone of  alarm level 4.

The results of  the half-year forward prediction be-
ginning from November 10, 2004, and M ≥ 7.0 earth-
quakes that occurred in this period are plotted in Figure
2. As seen from the figure, the two strongest events (the
SAE and its strongest aftershock) occurred in the level-
2 alarm zone, while two weaker events occurred in the
level-4 alarm zone. It is noteworthy that, if  monitoring

GERTSIK ET AL.

14

Figure 2. Mapped areas of  five levels of  half-year forward forecasting for the SAE region on November 10, 2004.



15

in this region were conducted according to the pro-
posed scheme, a nine-cell square cluster containing one
cell of  the level-1 alarm would be regarded as haz-
ardous, giving every reason for anxiety in relation to
the impending SAE.

Table 2 presents the number of  events and the
total number of  alarm cells for each level of  the retro-
spective analysis, as well as their ratios, which are well-
grounded point estimates of  the probability that at
least one earthquake with a magnitude M 7.0 will
occur in a cell at a given level of  alarm. The 5% con-
dence intervals are also given in the table for each of
these probabilities. In all, 8352 cells participated in the
forecasting and 36 of  these cells (0.43% of  all cells) con-
tained events.

Figure 3 plots the dependence of  the fraction of
target misses on the spatiotemporal fraction of  alarms
for points at which the boundaries between alarm lev-
els divide the set of  all cells into two classes: union of
more anxiety cells and union of  less anxiety cells. It is
an estimate for Molchan’s error diagram.

15. Conclusions
This article demonstrates that the prediction

method based on the pattern recognition does not have
sufficient foundation and contains a number of  errors.
We propoused here the alternative approach that is free
of  this errors and enhances the computing capability
of  the forecasting. Discretization of  spacetime, using
elements of  the theory of  multicomponent random
processes and axiomatization of  the problem allows to
drastically simplify the procedure of  earthquake fore-
casting and to obtain the following results:

(1) The method of  announcement to declare a
multi-level alarm was found.

(2) The method of  calculating the point estimate
of  probability of  an event for each alarm level was
found.

(3) The method of  calculating interval estimates
of  event probability for each level of  anxiety was found.

(4) The equivalence of  the announcement of  a
two-phase alarm and solving the problem of  testing hy-
potheses was shown.

(5) The method of  testing an algorthim of  forecast
was found.

(6) The scalar criterion for comparison of  different
algorithms of  forecast was found.

(7) The equivalence of  the method of  the announce-
ment of  alarms and method assessing the probability of
event was shown.

It appears to the authors, that a satisfactory math-
ematical formalism of  earthquake forecasting based on
arbitrary set of  precursors is fully fledged. The most im-
portant result we believe is the discovery of  the scalar
quality criterion of  forecast algorithms. The criterion
allows to find out which one is the best from existing al-
gorithms and to focus on finding the most effective pre-
cursors.

Acknowledgements. We would like to thank V. Pisarenko
and G. Sobolev for stimulating discussions that gave us the impulse
for writing this paper. M. Kelbert thanks the HSE for the support in
the framework of  the implementation of  the Global Competitive-
ness Program.

EARTHQUAKE FORECASTING: STATISTICS AND INFORMATION

Alarm
level

Number
of  cells

Number
of  events

Left boundary
of  interval

Point
estimate

Right boundary
of  interval

1 36 18 0.3300 0.500 0.6700

2 222 15 0.0400 0.0676 0.1052

3 220 2 0.0015 0.0091 0.0281

4 602 1 0 0.0017 0.0070

5 7272 0 0 0 0.0006

Figure 3. Empirical Molchan's diagram: the share of  missed targets
vs the share of  alarms.

Table 2. Number of  events and alarm cells for each level of  the retrospective analysis for SAE region.



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GERTSIK ET AL.

16

















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  /EncodeColorImages true
  /ColorImageFilter /DCTEncode
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  /ColorACSImageDict <<
    /QFactor 0.15
    /HSamples [1 1 1 1] /VSamples [1 1 1 1]
  >>
  /ColorImageDict <<
    /QFactor 0.15
    /HSamples [1 1 1 1] /VSamples [1 1 1 1]
  >>
  /JPEG2000ColorACSImageDict <<
    /TileWidth 256
    /TileHeight 256
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  >>
  /JPEG2000ColorImageDict <<
    /TileWidth 256
    /TileHeight 256
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  >>
  /AntiAliasGrayImages false
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  /EncodeGrayImages true
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  /GrayACSImageDict <<
    /QFactor 0.15
    /HSamples [1 1 1 1] /VSamples [1 1 1 1]
  >>
  /GrayImageDict <<
    /QFactor 0.15
    /HSamples [1 1 1 1] /VSamples [1 1 1 1]
  >>
  /JPEG2000GrayACSImageDict <<
    /TileWidth 256
    /TileHeight 256
    /Quality 30
  >>
  /JPEG2000GrayImageDict <<
    /TileWidth 256
    /TileHeight 256
    /Quality 30
  >>
  /AntiAliasMonoImages false
  /CropMonoImages true
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  /MonoImageMinResolutionPolicy /OK
  /DownsampleMonoImages true
  /MonoImageDownsampleType /Bicubic
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  /MonoImageDepth -1
  /MonoImageDownsampleThreshold 1.08250
  /EncodeMonoImages true
  /MonoImageFilter /CCITTFaxEncode
  /MonoImageDict <<
    /K -1
  >>
  /AllowPSXObjects false
  /CheckCompliance [
    /None
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  /PDFX1aCheck false
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  /CreateJDFFile false
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  /Description <<
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  >>
>> setdistillerparams
<<
  /HWResolution [2400 2400]
  /PageSize [595.000 842.000]
>> setpagedevice