Acta Geophysica
vol. 59, no. 4, Aug. 2011, pp. 674-700
DOI: 10.2478/s11600-011-0012-6
________________________________________________
© 2011 Institute of Geophysics, Polish Academy of Sciences
Statistical Tools for Maximum Possible
Earthquake Magnitude Estimation
Andrzej KIJKO
1
and Mayshree SINGH
2
1
Aon-Benfield Natural Hazard Centre, Pretoria University,
Pretoria, South Africa; e-mail: andrzej.kijko@up.a.za
2
School of Civil Engineering, Surveying and Construction,
University of KwaZulu Natal, Durban, South Africa
e-mail: singhm5@ukzn.ac.za
Abstract
Several procedures for the statistical estimation of the region-
characteristic maximum possible earthquake magnitude, m
max
, are cur-
rently available. This paper aims to introduce and compare the 12 exist-
ing procedures. For each of the procedures given, there are notes on its
origin, assumptions made in its derivation, condition for validity, weak
and strong points, etc. The applicability of each particular procedure is
determined by the assumptions of the model and/or the available infor-
mation on seismicity of the area.
Key words: m
max
, earthquake magnitude distribution, maximum magni-
tude.
1. INTRODUCTION
The region-characteristic, maximum possible earthquake magnitude, m
max
, is
required by the earthquake engineering community, disaster management
agencies and the insurance industry. Although the m
max
value is important, it
is astonishing how little has been done in developing appropriate techniques
for estimating this parameter. Presently, there is no universally accepted
technique for estimating the value of m
max
; however, the current procedures
for m
max
can be divided into two main categories: deterministic and probabil-
STATISTICAL TOOLS FOR m
max
ESTIMATION
675
istic. A presentation and discussion of deterministic techniques for the
assessment of m
max
can be found in, e.g., Wells and Coppersmith (1994),
Wheeler (2009), and Mueller (2010). This paper focuses only on the prob-
abilistic techniques.
The maximum regional earthquake magnitude, m
max
, is defined as the
upper limit of earthquake magnitude for a given region and is synonymous
with the magnitude of the largest possible earthquake. This definition is used
by earthquake engineers (EERI Committee on Seismic Risk 1984) and seis-
mologists (Hamilton 1967, Page 1968, Cosentino et al. 1977, Working
Group on California Earthquake Probabilities (WGCEP) 1995, Stein and
Hanks 1998, Field et al. 1999). It assumes a sharp cut-off magnitude at
a maximum magnitude, m
max
, so that, by definition, no earthquakes are to be
expected with magnitude exceeding m
max
.
This paper presents several statistical techniques for the evaluation of
m
max
, which can be used depending on the assumptions about the statistical
distribution model and/or the information available about past seismicity.
Certain procedures can be applied in extreme cases when no information
about the nature of the earthquake magnitude distribution is available. These
procedures are capable of generating an equation for m
max
, which is indepen-
dent of the assumed frequency-magnitude distribution. Some of the proce-
dures can also be used when the earthquake catalogue is incomplete, i.e.,
when only a limited number of the largest magnitudes are available.
The described procedures are available in a MATLAB toolbox called
MMAX. This toolbox can be obtained from the authors free of charge.
2. THEORETICAL BACKGROUND
The methodology assumes that in the area of concern, within a specified
time interval, T, all n of the main earthquakes that occurred with a magnitude
greater than or equal to m
min
, are recorded. The largest observed earthquake
magnitude in the area is denoted as
max
obs
m . Next assume that the value of the
magnitude
m
min
is known and is denoted as the threshold of completeness. It
is further assumed that the magnitudes are independent, identically distrib-
uted, random values with probability density function (PDF),
f
M
(m), and the
cumulative distribution function (CDF),
F
M
(m). The unknown parameter
m
max
is the upper limit of the range of magnitudes and is thus termed the
maximum regional earthquake magnitude,
max
ˆ
m
, that is to be estimated.
The estimation techniques are organized in three sections: 2.1 Parametric
estimators, 2.2 Non-parametric estimators, and 2.3 Fit of CDF of earthquake
magnitudes.
A. KIJKO and M. SINGH
676
2.1 Parametric estimators
Parametric estimators can be used when the parametric models of the fre-
quency-magnitude distributions are known. Five procedures are described.
The first procedure, denoted as T-P, is based on complex mathematical con-
siderations (Tate 1959), but is computationally very straightforward and
does not require extensive calculations. The next two, known as K-S proce-
dures, are based on the generic equation derived by Cooke (1979) and they
differ only in numerical details. In the derivation of the first procedure, the
exact distribution of the largest earthquake magnitude is replaced by its
Cramér’s approximation. The second K-S procedure is based on exact solu-
tion of Cooke’s (1979) generic equation; it is therefore able to provide a bet-
ter solution, particularly when the number of observations is limited. Finally,
the last two procedures are especially useful when only a rough knowledge
of the functional form of earthquake magnitude distribution is available. In
all five procedures, the largest observed earthquake magnitude,
max
obs
m , plays
a crucial role.
All the procedures presented in this section are based on the underlying
principle that the estimated m
max
value is equal to
max
obs
m
+
Δ , where Δ is a
positive correction factor. This principle is similar to the popular determinis-
tic procedure, where the increment Δ varies from 0.25 to 1.0 of a magnitude
unit (Wheeler 2009). Despite this similarity, there is a fundamental differ-
ence between the two approaches. In the deterministic approach, Wheeler’s
correction factor is a pure deterministic number, essentially just a guess;
however, in the probabilistic approach the correction factor is determined by
factors characterizing the seismicity of the area. The correction factor is a
function of
max
obs
m
the seismic activity rate, and the ratio between the number
of weak and strong events. The correction factor depends on seismic para-
meters supporting intuitive expectations that it is always positive and its val-
ue decreases as the time span of observation increases.
Sections 2.1.1 to 2.1.5 provide a detailed description of the five proce-
dures introduced above.
2.1.1 Tate–Pisarenko procedure
This procedure is very straightforward and does not require extensive calcu-
lations. It can be shown that the procedure attempts to correct the bias of the
classical maximum likelihood estimator (Pisarenko et al. 1996), but it fails to
provide an estimator having a smaller mean-squared error.
After the transformation y = F
M
(m), the CDF of the largest among
(Y
1
, …, Y
n
), that is Y
n
, is equal to y
n
, and its expected value E(Y
n
) = n/(n + 1).
One of the possibilities for obtaining the estimator of
max
ˆ
m
is to introduce
STATISTICAL TOOLS FOR m
max
ESTIMATION
677
the condition E(Y
n
) = y
n
, from which we obtain the following equation
(Gibowicz and Kijko 1994):
max
()
1
obs
M
n
Fm
n
=
+
. (1)
Thus, the estimator of m
max
becomes a function of
max
obs
m and n, and is ob-
tained as a root of eq. (1). The above result is valid for any CDF of earth-
quake magnitude, F
M
(m), and that does not require the fulfilment of the
truncation condition. From eq. (1), Kijko and Graham (1998) derived an al-
ternative equation, which is approximate, but which demonstrates the re-
quired value of m
max
in a more explicit way (Pisarenko et al. 1996)
max max
max
1
()
obs
obs
M
mm
nf m
=+ . (2)
It should be noted that in eq. (2), the desired m
max
appears on both sides:
left, simply as m
max
, and on the right side as the unknown parameter of the
probability density function (PDF) of earthquake magnitude, f
M
(m). Howev-
er, from this equation an estimated value of m
max
,
max
ˆ
m ,
can be obtained
through iteration.
The estimator (2) was probably first derived by Tate (1959). If applied to
the Gutenberg–Richter magnitude distribution with PDF (Page 1968),
min
min
min max
max min
max
0for
exp[ ( )]
() for ,
1exp[ ( )]
0for
M
mm
mm
fm m mm
mm
mm
ββ
β
⎧
<
⎪
−−
⎪
=≤≤
⎨
−− −
⎪
⎪
>
⎩
(3)
it takes the form
max min
max max
max min
1exp[ ( )]
exp[ ( )]
obs
obs
mm
mm
nmm
β
β
−− −
=+
−−
, (4)
where β
=
b
ln(10), and b is the parameter of the frequency-magnitude
Gutenberg–Richter relation. With small modifications, eq. (4) is equivalent
to Tate’s (1959) estimator. It was used for the first time by Pisarenko et al.
(1996). The solution of eq.
(4) provides the estimated value of m
max
,
max
ˆ
m
,
and
in this paper is referred to as the Tate–Pisarenko estimator of m
max
or in short
as T-P. The approximate variance of the T-P estimator, which contains both,
aleatory and epistemic components, is of the form (Kijko and Graham 1998)
A. KIJKO and M. SINGH
678
2
2
max min
max
3
max min
1 exp[ ( )]
1
ˆ
Var( ) ,
exp[ ( )]
obs
M
obs
mm
n
m
nmm
β
σ
ββ
⎡
⎤
−− −
+
=+
⎢
⎥
−−
⎣
⎦
(5)
where σ
M
denotes the standard error in the determination of the largest ob-
served magnitude,
max
obs
m .
2.1.2 Kijko–Sellevoll procedure (Cramér’s approximation)
The largest observed magnitude, m
n
, which is also denoted as
max
obs
m , has the
cumulative distribution function (CDF)
[]
min
min max
max
0for
() () for .
1for
n
n
MM
mm
Fm Fm m mm
mm
<
⎧
⎪
=≤≤
⎨
⎪
>
⎩
(6)
After integrating by parts, the expected value of M
n
, E(M
n
), is
max max
min min
max
() d() ()d
nn
mm
nM M
mm
EM m F m m F m m==−
∫∫
. (7)
Hence
[]
max
min
max
() ()d
m
n
nM
m
mEM Fmm=+
∫
. (8)
This expression, after replacement of the expected value of the largest ob-
served magnitude, E(M
n
), by the largest magnitude already observed, pro-
vides the equation
[]
max
min
max max
()d
m
obs n
M
m
mm Fmm=+
∫
, (9)
in which the desired m
max
appears on both sides. An estimated value of m
max
,
max
ˆ
m , can be obtained through iteration.
Cooke (1979) was probably the first to obtain the estimator of the upper
bound of a random variable similar to eq. (9). The difference between eq. (9)
and estimator by Cooke (1979) is that the former provides an equation in
which the upper limit of integration is
max
obs
m , not m
max
. Therefore, for large n,
the two solutions are asymptotically equivalent. If Cooke’s estimator is ap-
plied to the assessment of m
max
, eq. (9) states that the maximum regional
earthquake magnitude, m
max
, is equal to the largest observed magnitude,
max
obs
m , increased by an amount
[]
max
min
()d.
m
n
M
m
F
mmΔ=
∫
Similar to eq. (2),