Earthquake risk models used by the insurance industry for estimating the damage caused by an event typically use the mean of the ground motion intensity to predict the damage to a particular building. On the other hand, the ground motion intensity can be modelled as a ran- dom variable. Literature related to ground-motion predictive equation derivation includes, as a rule, the estimate of the standard deviation associated with the distribution of the intensity. The Study Group was asked to find a way to include the uncertainty associated with the predic- tion of the ground motion intensity contained in the standard deviation into the damage calculation, in a way in which the computational effort is not increased significantly

The Study Group proposed a way forward based on Bayes’ theorem for the marginal distribution of damage and found an analytical expression for the damage distribution function. However, the expression is an in- tegral that needs to be evaluated numerically and the Gaussian-Hermite quadrature was proposed to carry out the calculations. The approach seems plausible to be included in the existing models and the additional computational load is estimated as to be marginal relating to the current computational demands

Bandeira, Afonso

Cavaleiro, Marta

Davies, Russell

Hazelwood, Vera

Mondal, Anirban

Morales, Sergio

Piwarska, K.

Please, C.P.

Mathematics in Industry Information Service Eprints Archive

Earthquake risk ESGI73
Earthquake risk: Including Uncertainties in
the Ground Motion Calculations
Problem was funded by
Natural Environment Research Council
and presented by
John E. Alarcon and Shane Latchman
AIR Worldwide Ltd
Executive Summary
Earthquake risk models used by the insurance industry for estimating
the damage caused by an event typically use the mean of the ground
motion intensity to predict the damage to a particular building. On
the other hand, the ground motion intensity can be modelled as a ran-
dom variable. Literature related to ground-motion predictive equation
derivation includes, as a rule, the estimate of the standard deviation
associated with the distribution of the intensity. The Study Group was
asked to find a way to include the uncertainty associated with the predic-
tion of the ground motion intensity contained in the standard deviation
into the damage calculation, in a way in which the computational effort
is not increased significantly
The Study Group proposed a way forward based on Bayes’ theorem for
the marginal distribution of damage and found an analytical expression
for the damage distribution function. However, the expression is an in-
tegral that needs to be evaluated numerically and the Gaussian-Hermite
quadrature was proposed to carry out the calculations. The approach
seems plausible to be included in the existing models and the additional
computational load is estimated as to be marginal relating to the current
computational demands.
Version 1.0
28 May 2010
i
Earthquake risk ESGI73
Report coordinator
Vera Hazelwood (Knowledge Transfer Network for Industrial Mathematics)
vera.hazelwood@industrialmaths.net
Contributors
Afonso Bandeira (Universidade de Coimbra)
Marta Cavaleiro (Universidade de Coimbra)
Russell Davies (Cardiff University)
Anirban Mondal (Texas A&M University)
Sergio Morales (University of Warwick)
Karina Piwarska (University of Warsaw)
Colin Please (University of Southampton)
ESGI73 was jointly organised by
The University of Warwick
The Knowledge Transfer Network for Industrial Mathematics
and was supported by
The Engineering and Physical Sciences Research Council
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Earthquake risk ESGI73
Contents
1 Introduction 1
1.1 Catastrophe modelling . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 The work of the Study Group 2
2.1 Mathematical formulation of the problem . . . . . . . . . . . . . . . 2
2.2 Mathematical solution: marginal distribution . . . . . . . . . . . . . 3
2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Bibliography 6
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Earthquake risk ESGI73
1 Introduction
1.1 Catastrophe modelling
(1.1.1) AIR Worldwide is a catastrophe risk modelling company who provides
their clients in insurance and re-insurance markets with models that allow
earthquake damages, within other perils, to be estimated. Part of the
process carried out on earthquake risk assessment can be summarised, at
a high level, as presented below.
• A given geographical location, that could be a square with a side of
250 km is represented as a number of points on the grid. A typical
discretisation may use a resolution of 50 meters distance between
points in both latitude and longitude directions.
• For each earthquake event with magnitude M in the stochastic cat-
alogue, the induced ground motion intensities are calculated at each
point on the grid. A typical stochastic catalogue contains hundreds
of thousands of events.
• At each point and for each earthquake, Ground Motion Prediction
Equations (GMPEs) are used to calculate the motion intensity (e.g.
Peak Ground Acceleration) y. A typical GMPE has a form
log10 y = c1 + c2M + c3 log10(R + c4) + c5S + σGMPE, (1)
where y is the intensity of ground motion, M is the magnitude, R
is the source-to-site distance, S is the site factor, which depends on
the type of soil, and σGMPE is the standard deviation. In order to
capture the epistemic uncertainty in ground motion prediction, a
number of equations are used in a logic tree framework. For a given
case, seven GMPEs are used to calculate the intensity at each point.
1 A weighted averaged value of y is used for subsequent calculations.
• Finally, the value of y is used to estimate the damage. In Figure 1
the blue curve is the mean of the β-distribution (red curve ) used to
model the damage distribution for a given y. Each β curve corre-
sponds to a single value of the ground motion intensity y.
It is worth pointing out that the calculations are very computationally
demanding. For instance, the earthquake catalogue for a given region
contains about 900,000 events, and the ground motions induced by each
event are calculated for the entire geographical grid previously mentioned
(i.e., 25,000,000 points in the grid). Then 7 equations are used at each
point, which overall gives over 150 trillion calculations before averaging
or calculating the damage resulting from all simulated earthquakes for all
points on the grid.
1Please see exact examples of GMPEs and their derivation in references [1] and [2].
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Earthquake risk ESGI73
Figure 1: Mean value of damage is plotted (blue curve) for each value of mean
ground motion (intensity) y. Damage is modelled to have a β-distribution (red
curve) around its mean.
1.2 Problem
(1.2.1) Equation 1 includes σGMPE, a standard deviation that describes uncertain-
ties associated with a particular type of GMPE. The most recent equations
present σGMPE as a function of earthquake magnitude M and site proper-
ties. As an approximation to the distribution of ground motion intensity,
the mean value of intensity from each GMPE is used.
(1.2.2) AIR would like to investigate whether the information about a distribution
of ground motion intensity described by σGMPE can be incorporated in the
damage calculations without significant increment in the computational
effort.
2 The work of the Study Group
2.1 Mathematical formulation of the problem
(2.1.1) First, let us formulate the problem in mathematical terms. Let Y be
the random variable describing the Ground Motion Intensity. Let Y to
be distributed as log-normal2: Y ∼ N(log y;m,σ), where m is the mean
value of logy distribution, σ is its standard deviation3 and y is a particular
realisation of Y .
(2.1.2) For each value y of Y the damage caused by this intensity is a conditional
probability D|{Y = y}. It is β-distributed with mean µ, which depends
on y, µ = µ(y), and standard deviation, which also depends on µ.
2Please note that here and everywhere else in the text where the base of logarithm is not
specified, we mean the natural logarithm (note that logarithm base 10 is used in GMPE).
3Please note that σ is different from σGMPE as the former is defined for the log-normal distribu-
tion while the latter is specified for the log10 distribution. The relationship is σ = log(10)σGMPE .
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Earthquake risk ESGI73
(2.1.3) The problem is to find the marginal distribution of D. To clarify, let us
look again at the vulnerability curve at the damage vs intensity graph.
(2.1.4) Figure 2 shows µ(y), which is the vulnerability curve, i.e. it gives the
mean value of damage for any given intensity. A vulnerability curve is a
continuous, monotonically increasing function bounded between 0 and 1
that gives the damage to a particular structure given the ground motion at
that structure. There are different vulnerability curves for different types
of buildings and for each building type and each intensity y, µ(y) is the
mean of the damage extent.
Figure 2: Different β-distributions for each mean value of intensity (horizontal axis)
(2.1.5) In reality, intensity is not a single point on the horizontal axis but each
value of intensity has its own (known) distribution, where for log y distri-
bution the mean is m and standard deviation is σ. So the question is how
to combine a β-distribution of the damage around a particular mean inten-
sity with the distribution of intensity itself? The answer to this question
would give us a marginal distribution of damage D.
2.2 Mathematical solution: marginal distribution
(2.2.1) When the σGMPE of the ground motion is included into the damage assess-
ment, the distribution of damage becomes a marginal distribution given
by Bayes’ theorem (previously the distribution of damage was given by
pdfD|{Y=y}(t) in the integral below).
pdfD(t) =
∫
pdfY (y)pdfD|{Y=y}(t)dy. (2)
(2.2.2) After filling in the distributions of the GMPE (given as lognormal distri-
bution) and the β-distribution of damage for a given intensity, we get the
expression:
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Earthquake risk ESGI73
pdfD(t) =
∫
R+
Γ(α + β)
Γ(α)Γ(β)
tα−1(1− t)β−1 1
yσ
√
2pi
exp
(
−(log y −m)
2
2σ2
)
dy,
(3)
where α and β are functions of µ(y).
(2.2.3) The above integral can be evaluated numerically; however, this method
may be computationally- and time-intensive.
(2.2.4) Alternatively, an approximation of the intergral can be done using the
Gaussian-Hermite Quadrature4, such that we can re-arrange equation 3 to
be the integral of an exponential distribution of the form below multiplied
by a function, f(x). This is approximated through a sum of polynomial
functions.∫ ∞
−∞
e−x
2
f(x)dx ≈ w0f(x0) +
n∑
i=1
wi(f(xi+) + f(xi−)), (4)
where wi =
2n−1n!
√
pi
n2Hn−1(xi)2
and xi are roots of Hermite Polynomial. wi and xi
can be easily evaluated to any order n by software packages such as Maple
or Mathematica. The values are also available online.
(2.2.5) It can be shown that for odd (2n+ 1), x0 = 0 and xi− = −xi+ .
(2.2.6) Let us approximate the integral 3 by 5 terms, so using equation 4, we
obtain
pdfD(t) ' W2pdfD|{Y=y−2 }(t) +W1pdfD|{Y=y−1 }(t) + (5)
W0pdfD|{Y=y0}(t) +W1pdfD|{Y=y+1 }(t) +W2pdfD|{Y=y+2 }(t),
where
y±1 = e
√
2σx1±+m,y±2 = e
√
2σx2±+m,y0 = exp(m)
and we can calculateW0 = 0.533333375,W1 = 0.22207585W2 = 0.01125728, x1± =
±0.958572, x2± = ±2.0201828.
(2.2.7) The approximation can be done using 3 to 5 terms (or more depending on
how many more steps can be accommodated without greatly increasing
runtimes). A comparison of the 5-term approximation, the 3-term approx-
imation, the full approximation and the original distribution of damage
can be found in Figure 3.
4Please see reference [4] for proof and more information and also
http://en.wikipedia.org/wiki/Gauss%E2%80%93Hermite quadrature.
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Earthquake risk ESGI73
Figure 3: Comparison between a 1-term approximation - purple curve (current
AIR’s approach), full approximation calculated by Mathematica - blue curve, and
a 5-term approximation - green curve.
(2.2.8) Since in practice a weighted average of several GMPEs is used, a weighted
average of the probability distributions for each of the approximated GM-
PEs can be taken. This would ultimately give the overall distribution
of damage resulting from the combinations of the distribution of each
GMPE pdfY (y) and the associated damage distributions given each GMPE
(pdfD|{Y=y}(t)).
2.3 Conclusions
(2.3.1) The Study Group proposed a way to incorporate the uncertainty associ-
ated with the intensity of ground motion to the calculations of damage.
An analytical expression has been obtained for the marginal distribution
of damage, expressed as an integral.
(2.3.2) The integral 3 has to be evaluated numerically. There may be many
different approaches and the Study Group proposed to use the Gaussian-
Hermite quadrature.
(2.3.3) The additional computational load will be 3 to 5 extra calculations which
need to be saved (depending on the number of terms using to approximate
the above integral). This is due to the requirement of saving the standard
deviation of intensity and also the extra distributions in order to compute
the marginal distribution of damage.
(2.3.4) The results obtained by evaluating the integral made physical sense and
corresponded to the expectations for the marginal distribution of damage.
For instance, the resulting distribution was wider reflecting more uncer-
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Earthquake risk ESGI73
tainty contained in marginal distribution in comparison to the conditional
distribution. Initial observations show that the mean of the marginal
distribution of damage and the mean of the conditional distribution of
damage are quite similar despite the inclusion of the distribution around
intensity.
Bibliography
[1] Akkar, S. & Bommer, J.J. (2007). Prediction of elastic displacement
response spectra in Europe and the Middle East. Earthquake
Engineering and Structural Dynamics 36, 1275-1301.
[2] Atkinson, G.M. & D.M. Boore (2003). Empirical Ground-Motion
Relations for Subduction-Zone Earthquakes and Their Application
to Cascadia and Other Regions. Bulletin of the Seismological Society
of America 93(4), 1703- 1729.
[3] See for example, ”Bayes’ Theorem” article by James Joyce
in the Stanford Encyclopedia of Philosophy, which pro-
vides a comprehensive introduction to Bayes’ theorem.
http://plato.stanford.edu/entries/bayes-theorem/
[4] Abramowitz, M & Stegun, I A, Handbook of Mathematical
Functions, 10th printing with corrections (1972), Dover, ISBN
978-0-486-61272-0. Equation 25.4.46.
[5] Khater, M et al (2003) , ”Loss Estimation” in W.F. Chen and C.
Scawthorn (eds), Earthquake Engineering Handbook pp 1-23 (Boca
Raton: CRC Press)
6


Earthquake risk: Including Uncertainties in the Ground Motion Calculations

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