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13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13

Seoul, South Korea, May 26-30, 2019

1

Ground motion sample size vs estimation uncertainty in seismic risk

Georgios Baltzopoulos

Assistant Professor, Dept. of Structures for Engineering and Architecture, Università degli studi di

Napoli Federico II, Naples, Italy

Iunio Iervolino

Full Professor, Dept. of Structures for Engineering and Architecture, Università degli studi di Napoli

Federico II, Naples, Italy

Roberto Baraschino

PhD Candidate, Dept. of Civil Engineering, National Technical University of Athens, Athens, Greece

ABSTRACT: In the context of seismic risk assessment as per the performance-based earthquake engineering

paradigm, a probabilistic description of structural vulnerability is often obtained via dynamic analysis of a non-

linear numerical model. It typically involves subjecting the structural model to a suite of ground-motions that are

representative, as a sample, of possible seismic shaking at the site of interest. The analyses’ results are used to

calibrate a stochastic model describing structural response as a function of seismic intensity. The sample size of

ground motion records used is, nowadays, usually governed by computation-time constraints; on the other hand,

it directly affects the estimation uncertainty which is inherent in risk analysis carried out in this way. Recent studies

have suggested methodologies for the quantification of estimation uncertainty, to be used as tools for determining

the appropriate number of records for each application on an objective basis. The present study uses one of these

simulation-based methodologies, based on standard statistical inference methods and the derivation of structural

fragility via incremental dynamic analysis, to investigate the accuracy of the risk estimate (e.g., the annual failure

rate) vs the size of ground motion samples. These investigations consider various scalar intensity measures and

confirm that that the number of records required to achieve a given level of accuracy for annual failure rate depends

not only on the dispersion of structural responses, but also on the shape of the hazard curve at the site. This indicates

that the efficiency of some frequently-used intensity measures is not only structure-specific but also site-specific.

1. INTRODUCTION

Performance-based earthquake engineering

(PBEE; Cornell and Krawinkler 2000), entails the

probabilistic quantification of structure-specific

seismic risk. This risk can be quantified by the

annual rate of earthquakes able to cause the

structure to violate a seismic performance

objective, which can be simply termed the failure

rate,

f

, given by Eq. (1):

f im

im

λ P f im dλ

(1)

where the conditional probability term

P f im

represents what is often known as a fragility

function, which provides the probability of failure

for various values of a seismic intensity measure

(IM), while

im

is the annual rate of earthquakes

exceeding the value of shaking intensity

im

and

therefore constitutes a measure of seismic hazard

at the site.

The state-of-the-art in PBEE is to analytically

estimate structure-specific fragility functions by

means of procedures that require multiple

dynamic analysis runs of a numerical model of the

structure. These analyses typically use a multitude

of acceleration records as input motion, in order

to map the record-to-record variability of inelastic

structural response (Shome et al. 1998). On the

other hand, the evaluation of

im

for various

intensity levels, which is known as the hazard

curve, is usually obtained by means of

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13

Seoul, South Korea, May 26-30, 2019

2

probabilistic seismic hazard analysis (PSHA; e.g.,

McGuire 1995), which typically employs

empirical ground motion prediction models

(GMPMs) to account for the attenuation of

shaking intensity.

In modern practice, the number of records

used for non-linear dynamic analysis of a

structure is typically limited due to the large

computation times required for running intricate

structural models at high non-linearity levels.

However, this number of records determines the

sample-size of seismic structural responses that is

used for fragility estimation and, eventually, the

failure rate. Since these descriptors of seismic

fragility and risk are inferred from finite-size

samples, they are only estimates of the

corresponding true values, and are therefore

affected by estimation uncertainty (Iervolino

2017). In fact, the estimator of

f

, obtained using

a specific sample of ground motions and denoted

using a hat symbol as

f

ˆ

, can be regarded as a

random variable (RV) whose distribution is a

function of the sample size. In other words,

computing

f

ˆ

over and over for a number of

times using different sets of accelerograms (equal

in number to the first one and equivalent in

characteristics) would lead to a different value for

the estimator each time around. Although

GMPMs are also based on samples of recorded

ground motion, these datasets are extensive

enough to allow the assumption that the

estimation uncertainty underlying

f

ˆ

is only due

to the fragility portion of Eq. (1).

Estimation uncertainty present in parametric

fragility models fitted from dynamic analysis

results has also been highlighted by other past

studies (Eads et al. 2015; Gehl et al. 2015; Jalayer

et al. 2015): in fact, a quantitative measure of the

effect of this uncertainty on the failure rate, can be

obtained according to Eq. (2):

f

f

ˆ

λ

f

ˆ

VAR λ

CoV ˆn

Eλ

(2)

where the notation

f

ˆ

λ

CoV

indicates the coefficient

of variation of

f

ˆ

,

f

ˆ

VAR λ

and

f

ˆ

Eλ

denote its

variance and expected value, respectively,

n

is

the sample size of accelerograms used to estimate

the fragility function and

is a parameter that

depends on the so-called efficiency of the IM

chosen to express structural fragility and also on

the shape of the site-specific hazard curve.

The objective of the present article is to

employ a simulation-based methodology for the

quantification of estimation uncertainty, which

was recently proposed as part of a broader-in-

scope study (Baltzopoulos et al. 2018a) and

investigate the efficiency of some commonly-

used scalar IMs, directly in terms of the ground

motion sample size required to contain the mean

relative estimation error, rather than in terms of its

frequently-used proxy; i.e., the dispersion of

response. This methodology is based on

incremental dynamic analysis (IDA; Vamvatsikos

and Cornell 2001) and involves using a relatively

large set of accelerograms to run dynamic

analyses for an assortment of simple inelastic

structures. The results of these analyses are then

used to fuel a procedure based on Monte-Carlo

simulation, where fragility estimates at various

limit states and using alternative IMs are

generated and statistics of the estimator of the

failure rate,

f

ˆ

, are extracted.

The structure of this article follows this

order: first there is a brief presentation of the

methodology for estimating structural fragility via

an IM-based procedure and of that for obtaining

statistics of the estimator of failure rate. Then

specific applications are given, considering

single-degree-of-freedom (SDOF) and simple

frame structures exposed to a variety of seismic

hazard conditions. Finally, the issue of record

sample size vs estimation uncertainty in the

estimate of the risk metric is discussed, in

conjunction with the choice of IM used as

interfacing variable, followed by some

concluding remarks.

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13

Seoul, South Korea, May 26-30, 2019

3

2. METHODOLOGY

In order to investigate the issue of ground

motion sample-size vs estimation uncertainty,

fragility is derived via dynamic analysis using the

so-called IM-based approach using IDA. IDA

consists of running a series of analyses for a non-

linear structure, using a suite of accelerograms

that are scaled in amplitude in order to represent a

broad range of IM levels. At each IM level, a

measure of structural response is registered

generically named an engineering demand

parameter (EDP). An exception to this are cases

where response approaches numerical instability,

which translates to lack of convergence in the

computer model (Shome and Cornell 2000). Thus,

at the conclusion of the dynamic analyses at an

adequate number of IM levels, a quasi-continuous

EDP-IM relationship is obtained, termed an IDA

curve (Figure 1).

Figure 1: IM-based derivation of seismic fragility via

incremental dynamic analysis. Set of generic IDA

curves and intersections of each curve with a vertical

line passing from the failure threshold (a);

parametric (lognormal) and non-parametric

representations of the fragility function derived from

the IDA results (b).

It can be assumed that violation of some limit

state of seismic performance (i.e., failure) occurs

whenever the EDP response exceeds a certain

threshold value, denoted as

f

edp

. In this context,

IM-based fragility entails the introduction of an

additional RV,

f

IM

, which is the lowest seismic

intensity that a record has to be scaled to, in order

to cause

f

EDP edp

. Thus,

f

IM

may be viewed

as the seismic intensity that causes structural

failure and, consequently, the fragility function

can be defined as the complementary cumulative

distribution function of

f

IM

i.e.,

f

P f im P IM im

.

In the context of IDA, the lowest IM value

for each record that causes the structure to reach

the performance threshold, can be calculated by

finding the height,

,fi

im

, where the i-th IDA curve

intersects the vertical line

f

EDP edp

,

1,2,...,in

,

n

being the total number of

records), as shown in Figure 1a. These values can

be considered as a sample of realizations of

f

IM

and, consequently, well-known statistical

methods (e.g., Baker 2015) can be used to fit a

parametric probability distribution model to that

sample. One frequently-used distribution is the

lognormal (Figure 1b), which is completely

defined by two parameters: the logarithmic mean

and standard deviation, whose point estimates

based on one sample,

f

IM

ˆ

η

and

f

IM

ˆ

β

respectively,

are given in Eq.(3):

1

2

1

Φ

1 log

11

ff

f

ff

IM IM

n

IM f ,i

i

n

IM f ,i IM

i

ˆ

ˆ

P f im logim ηβ

ˆ

η n im

ˆˆ

β n log im η

(3)

where

,fi

im

is the i-th record’s (lowest) scaled IM

value causing failure and

Φ

is the standard

(cumulative) Gaussian function. A non-

parametric alternative is to assume that the

observed sample values approximate the fragility

by defining a stepwise function, according to

Eq.(4):

1

1

f ,i

n

fim im

i

P f im P IM im I

n

(4)

where

f ,i

im im

I

is an indicator function that returns

1 if

f ,i

im im

and 0 otherwise. In either case, once

the fragility function has been estimated, the point

estimate of the failure rate

f

ˆ

can be obtained via

Eq. (1).

1.00.50

(b)

(a)

edpf

EDP

IM

Single-record IDA curve

imf,i , intersec tion with edpfLognormal fit

Non para metric fragility

P f im

Seoul, South Korea, May 26-30, 2019

4

As already mentioned, the estimation

uncertainty inherent in deriving the fragility from

a finite sample of structural responses is

propagated to the estimator of seismic risk

f

ˆ

,

which should be therefore regarded as a RV and a

function of the sample: assuming that one were to

perform a number of different IDAs, using each

time a set of accelerograms of the same size but

with different records than the previous ones, it is

to be expected that the estimated fragility curve

will differ from time to time, thus leading to

different estimates of the failure rate (i.e.,

different realizations of the RV

f

ˆ

).

One way of quantifying the estimation

uncertainty of

f

ˆ

is by means of the mean relative

estimation error,

f

ˆ

λ

CoV

, which can be regarded as

the coefficient of variation of the estimator. In the

case of IM-based fragility via IDA, the

relationship between

f

ˆ

λ

CoV

and the ground

motion sample size

n

can be approximated by

means of Monte-Carlo simulation (Baltzopoulos

et al. 2018a). This procedure begins with a

reference IDA that uses a relatively large amount

of records (

200n

is used herein) to derive a

reference fragility function, which can be either

lognormal or non-parametric. The simulation

entails randomly sampling

s

times from this

reference distribution of

f

IM

for different

sample sizes

2 3 200n , ,...,

(in the case of non-

parametric, empirical fragility, this translates to

resampling with substitution). At the next step in

the procedure, either new lognormal fragility

curves are fitted to each extracted sample

according to Eq.(3), or Eq.(4) is mustered to

directly express the fragility function. In either

case, integrating the fragility with a hazard curve,

according to Eq.(1), leads to a point estimate of

the failure rate at the j-th simulation, denoted

f , j

ˆ

λ

. As a last step, after

s

simulations have been

concluded at any given record sample size

n

,

ˆ

f

E

and

ˆ

f

VAR

can be approximated via

the first two moments of the Monte-Carlo-

generated sample of point estimates. By

substituting these values into Eq.(2), one obtains

Eq.(5):

2

11

1

11

1

1

f

ss

f ,j f ,k

jk

ˆs

λ

f ,j

j

ˆˆ

λλ

ss

CoV ˆ

λ

s

(5)

which provides the simulation-based

approximation for

f

ˆ

λ

CoV

.

3. APPLICATIONS

The methodology outlined in the previous

sections is applied to an assortment of simple

inelastic structures, which are assumed to be

located at three Italian sites that can be considered

representative of varying levels of seismic hazard

severity. The three sites considered are in the

vicinity of the cities of L’Aquila (representative

of a high seismic hazard site), Naples (medium

hazard levels) and Milan (low hazard) and are all

assumed to be characterized by firm soil

conditions. At each of the three sites a yielding

single-degree-of-freedom system is considered,

with natural vibration period

0.7 sT

and

viscous damping ratio

0.05

. Additionally, at

the L’Aquila site a four-story steel moment-

resisting frame is considered, with first mode

period

11.82 sT

.

Hazard curves were calculated at these sites,

in terms of several different scalar IMs, using the

software REASSESS (Chioccarelli et al. 2018)

employing the seismic source model from Meletti

et al. (2008). Hazard was obtained at all three sites

for spectral pseudo-acceleration at the SDOFs’

period,

0.7 sSa T

, and also for peak ground

acceleration (PGA) and

1.8 sSa T

at

L’Aquila. Also considered, were two more

advanced IMs that implicitly account for spectral

shape (Bojórquez and Iervolino 2011; Eads et al.

2015), namely average spectral acceleration

avg

S

and

Np

I

, given by Eqs.(6) and (7), respectively:

Seoul, South Korea, May 26-30, 2019

5

1

T

T

n

n

avg i

i

S Sa T

(6)

0 40

11

.

Np avg

I Sa T S Sa T

(7)

where

T

n

is the number of periods,

i

T

, that are

used in the definition of

avg

S

. Hazard curves in

terms of

avg

S

and

Np

I

are obtained at all three

sites using

0.7s, 1.0s, 1.5s

i

T

for the seismic risk

assessment of the SDOF structures and at

L’Aquila, using

0.6s, 1.8s, 2.5s, 4.0s

i

T

for that of

the steel frame. These hazard curves are shown in

Figure 2.

Figure 2: Annual exceedance rates (hazard curves)

at the three Italian sites for all IMs considered;

hazard curves used for seismic risk assessment of the

SDOF structures (above) and for the four-story steel

frame presumed at L’Aquila (below).

In order to construct reference fragility

functions for all of the structures considered, via

IDA, a set of two-hundred records was selected

from the NESS flatfile (Pacor et al. 2018),

avoiding records that were likely affected by near-

source effects such as rupture directivity or by site

effects due to deformable soil deposits.

3.1. SDOF structures

The simplest structures used in this

application are yielding SDOF systems that

follow a peak-oriented hysteretic rule (Lignos and

Krawinkler 2011) that also considers in-cycle

strength degradation by including a softening,

negative-stiffness post-peak branch in their

monotonic pushover (backbone) curve, thus

permitting explicit consideration of the collapse

limit state in the numerical analyses. The yield

threshold and backbone characteristics of the

three SDOF oscillators have been tweaked to

render them ostensibly risk-equivalent; i.e., they

were determined so that each structure at its

presumed site exhibits the same estimated annual

collapse rate (

4

ˆ3.6 10

f

) when fragility at

collapse is calculated from the IDA flat-lines

(Vamvatsikos and Cornell 2004) with

200n

records, using

avg

S

as IM. The numerical model

of the oscillators and IDA analyses were set up in

the OPENSeeS analysis platform (McKenna

2011) using the DYANAS interface

(Baltzopoulos et al. 2018b). Both lognormal

models and non-parametric representations are

considered for collapse fragilities.

In Figure 3, the resulting values of the

relative mean estimation error

f

ˆ

λ

CoV

from the

Monte-Carlo simulation procedure – i.e., from

Eq.(5) – are plotted against record sample size

n

for all combinations of IM, structure-site pairing

and fragility model (eighteen cases in total), also

reporting the point estimates of

ˆ

f

at

200n

in

the legend. It is clear that these two-hundred-

record estimates shift when switching IM, but this

is mainly an effect of how sensitive structural

response is to seismological parameters when

records are scaled (Luco and Cornell 2007), and

not directly related to ground motion sample size

and estimation uncertainty. The figure also reports

the number of records required to limit

f

ˆ

λ

CoV

to

10 0

IM [g]

10-4

10

100

λim

-2

10-5

Sa (T=1.8 s)

PGA

Savg

ΙNp

Savg

Savg

Savg

ΙNp

ΙNp

ΙNp

Milan

Naples

L’Aquila

Sa (T=0.7 s)

Sa (T=0.7 s)

Sa (T=0.7 s)

10-4

10

100

λim

-2

Seoul, South Korea, May 26-30, 2019

6

20% and 10% for some cases. The most

immediate observation emanating from Figure 3,

is that, for risk-wise nominally equivalent

structures that express fragility in terms of the

same IM, the shape of the hazard curve makes a

difference on the number of records required to

limit estimation uncertainty to a desired level, as

verified also analytically in the past (Baltzopoulos

et al. 2018a). The parameter

, that summarizes

the combined effect of the shape of the hazard

curve and IM efficiency on the coefficient of

variation of

ˆ

f

according to Eq. (2), can be

evaluated by means of a least-squares fit of that

equation to the simulation data and is given in

Table 1.

Figure 3: Mean relative estimation error,

f

ˆ

λ

CoV

,

calculated via Monte Carlo simulation for the three

SDOF structures considered, plotted against ground

motion sample size

n

.

3.2. MDOF steel frame structure

For the MDOF structure, that is the steel

moment-resisting frame presumed built at the

higher-hazard-level location of L’Aquila, the

same procedure was followed as for the three

simpler SDOF systems. In this case, a center-line,

non-linear finite-element model of the structure

created in the OPENSeeS environment was used

to run IDA using the same two-hundred record

set. Differently from the previous applications,

the collapse limit state was not considered;

instead, fragility was derived for two generic limit

states whose violation can be conventionally

defined by exceedance of some threshold in terms

of maximum inter-story drift ratio,

IDR

:

1.5%IDR

was considered for the first limit

state and

2.5%IDR

for the second. As in the

previous case, the simulation-based values of

f

ˆ

λ

CoV

were calculated for

2,3,...,200n

using

all four IMs for which hazard curves had been

derived and the results are plotted in Figure 4. The

corresponding

values, i.e., the site-and-

structure-specific parameter that allows the mean

relative estimation error to be expressed as a

function of record sample size as

ˆ

f

CoV n

,

is also reported in Table 1.

3.3. Discussion of the results

A cursory examination of the results from the

two examples, already reveals that adoption of a

traditional IM such as PGA, can be inadequate for

risk analysis of a flexible structure, since the

number of records required to limit

f

ˆ

λ

CoV

to an

(arbitrary) value as low as 10% verges on the

impracticable. In certain cases, such as the case of

estimating annual collapse rate and especially at

low-seismicity areas, the same can be said even

for first mode spectral acceleration

1

Sa T

; in

fact, even for these simple inelastic structures, the

number of records required to limit the mean

relative estimation error below 10% exceeds fifty.

Finally, it is interesting to compare the

relative efficiency of the geometric mean spectral

acceleration

avg

S

vs

Np

I

, the weighted geometric

mean; i.e., compare their ability to reduce

estimation uncertainty for a fixed record sample

43

10-2

10-1

100

101

Milan

λ

4

f3.7 10

Naples

λ

4

f3.7 10

L’Aquila

λ

4

f3.6 10

Milan

λ

4

f6.4 10

Naples

λ

4

f5.9 10

L’Aquila

λ

4

f4.7 10

Milan

λ

3

f1.1 10

Naples

λ

3

f1.0 10

L’Aquila

λ

4

f6.2 10

174

8 35

13 50

IM: Sa (T )

1

IM: INp

fragility

non-parametric

non-parametric

fragility

IM: Savg

non-parametric

fragility

10%

20%

20%

10%

20%

10%

10 41

25 100

18 73

Milan

λ

3

f1.2 10

Naples

λ

3

f1.0 10

L’Aquila

λ

4

f6.3 10

Milan

λ

4

f6.5 10

Naples

λ

4

f6.1 10

L’Aquila

λ

4

f4.8 10

Milan

λ

4

f3.6 10

Naples

λ

4

f3.7 10

L’Aquila

λ

4

f3.6 10

102

n

IM: Sa (T )

1

lognormal fragility

IM: INp

IM: I

lognormal fragility

IM: Savg

lognormal fragility

20%

10%

10%

20%

10%

20%

24 80 167

12 49 116

30

521 57

14 102

n

10-2

10-1

100

101

10-2

10-1

100

101

Seoul, South Korea, May 26-30, 2019

7

size. From the calculated

values, it can be

observed that, for these specific applications,

Np

I

is somewhat more efficient than

avg

S

at limit

states corresponding to lower level of inelasticity,

while

avg

S

overcomes

Np

I

in efficiency near

collapse. More elaborate discussion of the issue

can be found in the article from which this study

was inspired (Baltzopoulos et al. 2018a).

Figure 4 : Mean relative estimation error, calculated

via Monte Carlo simulation for the steel frame

assumed at L’Aquila, plotted against ground motion

sample size

n

.

Table 1: Dispersion of intensity causing failure

and site-/structure- specific parameter

of the mean

relative estimation error (

ˆ

f

CoV n

),provided for

each site, structure, IM, limit state and assumption

about the fragility function (LogN – lognormal, Emp

- empirical).

Site

Stru-

cture

Limit

State

IM

f

IM

β

Fragi-

lity

L’Aquila

Four-story steel moment resisting frame

T1=1.82s

IDR>1.5%

PGA

0.608

LogN

Emp

1.148

1.078

Sa(T1)

0.253

LogN

Emp

0.438

0.561

INp

0.220

LogN

Emp

0.408

0.490

Savg

0.265

LogN

Emp

0.530

0.563

IDR>2.5%

PGA

0.639

LogN

Emp

1.536

1.357

Sa(T1)

0.333

LogN

Emp

0.715

0.694

INp

0.267

LogN

Emp

0.628

0.605

Savg

0.241

LogN

Emp

0.622

0.628

Milan

Inelastic SDOF (T=0.70s)

Collapse

Sa(T)

0.475

LogN

Emp

1.839

1.476

Naples

0.444

LogN

Emp

1.318

0.998

L’Aquila

0.441

LogN

Emp

0.922

0.712

Milan

INp

0.378

LogN

Emp

1.468

1.383

Naples

0.341

LogN

Emp

1.099

0.855

L’Aquila

0.337

LogN

Emp

0.701

0.580

Milan

Savg

0.273

LogN

Emp

1.039

1.119

Naples

0.223

LogN

Emp

0.760

0.641

L’Aquila

0.214

LogN

Emp

0.457

0.409

4. CONCLUSIONS

The introduction of ever more realistic, and

thus complex, numerical structural models in

probabilistic seismic risk analysis, renders the

topic of constraining the appropriate size of the

input ground motion set to use, ever topical. This

study builds on recent proposals to base the

determination of the number of records on the

notion of limiting estimation uncertainty of the

risk metric to desired levels. A general rule-of-

thumb emerging from these results, is that using

record sample sizes in the thirty-to-fifty range, in

combination with adoption of more advanced,

10 -1

100

PGA

λ

3

f2.0 10

λ

3

f7.7 10

λ

4

f8.3 10

λ

4

f3.7 10

λ

4

f5.6 10

λ

3

f2.5 10

>

fragility >

16439

10 58

20%

n102n10 2

λ

4

f8.3 10

λ

4

f3.7 10

λ

4

f5.7 10

λ

3

f2.5 10

IDR %2.5>

fragility

non-parametric

λ

3

f2.6 10

λ

3

f1.6 10

λ

3

f2.0 10

λ

3

f7.7 10

non-parametric

fragility

IDR %1.5>>

IDR %2.5

lognormal

fragility

λ

3

f1.6 10

λ

3

f2.6 10

IDR %1.5

lognormal

10%

10% 10%

10%

20%

20% 20%

24 121 6 30 35 189

946

130

33

ΙNp

Savg

Sa (T )

1

PGA

ΙNp

Savg

Sa (T )

1

PGA

ΙNp

Savg

Sa (T )

1

PGA

ΙNp

Savg

Sa (T )

1

10 -1

100

Seoul, South Korea, May 26-30, 2019

8

efficient IMs, tends to keep the mean relative

estimation error at 10% or below. The higher end

of that range is needed for cases that combined

limit states corresponding to larger inelastic

excursions with site subjected to lower hazard

levels. It became apparent that the so-called

efficiency of seismic intensity measures, i.e., their

ability to keep estimation uncertainty to the

desired levels using smaller-size samples of

ground motions, is in fact site- and structure-

dependent, as recent research has shown.

5. ACKNOWLEDGEMENTS

The study presented in this paper was developed within

the activities of ReLUIS (Rete dei Laboratori

Universitari di Ingegneria Sismica) for the project

ReLUIS-DPC 2014–2018, as well as within the H2020-

MSCA-RISE-2015 research project EXCHANGE-Risk

(Grant Agreement Number 691213).

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