E. Varini and R. Rotondi - Copernicus.org · 2020. 5. 1. · E. Varini and R. Rotondi Istituto di...

Probabilistic damage scenarios from uncertainmacroseismic data

E. Varini and R. Rotondi

Istituto di Matematica Applicata e Tecnologie InformaticheConsiglio Nazionale delle Ricerche, Milano – Italy

EGU General Assembly: Sharing Geoscience Online4-8 May 2020

Motivations

We consider the beta-binomial model for macroseismic attenuation (Rotondi et al.,BSSA, 2009; Rotondi et al., Bull. Earthquake Eng, 2016), which:

• respects as far as possible the ordinal nature of the intensity scale,• allows for the assumption of spatial isotropy or anisotropy,• allows for the Bayesian treatment of uncertainties,• is a probabilistic tool to produce macroseismic scenarios.

Critical point:

The application of the beta-binomial model typically requires rounding-up or -down theobserved intensities to the nearest integer values (e.g. intensity VIII-IX must be setequal to VIII or IX).

Solution:

We propose an extension of the beta-binomial model in order to include in thestochastic modelling the uncertainty in the assignment of the intensities.

E.Varini, R.Rotondi 2 / 12

Original beta-binomial model of the intensity Is at site

J circular bins are drawn around the epicenter (isotropy)E

r_1

r_2

r_3

r_4

•

•

••

••

•

•

•

•

•

•

•

•

•

At a given bin j :

• Is = I0 −∆I follows the binomial distribution Binom(Is | I0, pj):

Pr {Is = i | I0 = i0, pj} = Binom {i | i0, pj} =

=

(i0i

)pij (1− pj)i0−i i ∈ {0, 1, . . . , I0}

• random variable pj ∼ Beta distribution

Beta(pj ; αj , βj) =Γ(αj + βj)

Γ(αj)Γ(βj)

∫ pj0

xαj−1(1− x)βj−1dx

to account for the variability in ground shaking

The prior hyperparameters αj , βj are assigned on the basis of the macroseismic fieldsbelonging to the same class, but of different I0.


New beta-binomial model of the intensity Is at site

J circular bins are drawn around the epicenter (isotropy)E

r_1

r_2

r_3

r_4

•

•

••

••

•

•

•

•

•

•

•

•

•

At a given bin j :

• Is = I0 −∆I follows the binomial distribution Binom(2Is | 2I0, pj):

Pr {Is = i | I0 = i0, pj} = Binom {2i | 2i0, pj} =

=

(2i02i

)p2ij (1− pj)2i0−2i i ∈ {0, 0.5, 1, 1.5 . . . , I0}

• random variable pj ∼ Beta distribution

Beta(pj ; αj , βj) =Γ(αj + βj)

Γ(αj)Γ(βj)

∫ pj0

xαj−1(1− x)βj−1dx

to account for the variability in ground shaking

The prior hyperparameters αj , βj are assigned on the basis of the macroseismic fieldsbelonging to the same class, but of different I0.


Updating parameters in the Bayesian framework(Fortran software)Given all the earthquakes with epicentral intensity I0 in a class,

let Nj be the number of felt intensities Dj in the j-th bin

Original beta-binomial model: αj = αj,0 +

Nj∑n=1

i (n)s βj = βj,0 +

Nj∑n=1

(I0 − i (n)s )

New beta-binomial model: αj = αj,0 +

Nj∑n=1

2i (n)s βj = βj,0 +

Nj∑n=1

(2I0 − 2i (n)s )

I0 VIII, class B

0 100 200

Distance [km]

0

0.2

0.4

0.6

0.8

1

Pa

ram

ete

r p

B We estimate the parameter pj through its posterior mean

p̂j = E (pj | Dj) =αj

αj + βjj = 1, . . . , J


Beta-binomial model with smoothed p = p(d)The estimates p̂j are smoothed by the inverse power function of the distance throughthe least squares method:

p(d) =

(c1

c1 + d

)c2Smoothed binomial distribution at any distance d

Pr(Is = i |I0 = i0, p(d)) =(

i0i

)p(d)i [1− p(d)](i0−i)

Pr(Is = i |I0 = i0, p(d)) =(

2i02i

)p(d)2i [1− p(d)](2i0−2i)

I0 VIII, class B

0 100 200

Distance [km]

0

0.2

0.4

0.6

0.8

1

Pa

ram

ete

r p

B

0 1 2 3 4 5 6 7 8

Is

pro

ba

bili

ty

0.0

0.1

0.2

0.3

0.4

Io VIII

p(d) = 0.9 at d = 10Km

0 1 2 3 4 5 6 7 8

Is

pro

ba

bili

ty

0.0

0.1

0.2

0.3

0.4

Io VIII

p(d) = 0.7 at d = 50Km

The mode ismooth of the smoothed binomial distribution is the predicted value of Is


Learning set

Analysis of 441 macroseismic fields (MFs) of “good quality” from the ItalianDBMI15 database:

− occurred since 1500,

− at least epicentral intensity V ,

− at least 40 felt reports.

Database DBMI15

Distribution of maximum macroseismic intensity


Hierarchical agglomerative clustering (R package cluster)

The learning set is first analyzed by the Ward’s hierarchical agglomerative clusteringmethod.

Four attenuation classes are identified.

class A class B class C class D132 MFs 192 MFs 82 MFs 35 MFs

10.69°E 10.99°E 11.28°E 11.57°E 11.87°E 12.16°E 12.45°E

43.29°N

43.5°N

43.72°N

43.93°N

44.14°N

44.35°N

44.56°N

1919/06/29, Io X: observed

Is

IIIIIIIVVVIVIIVIIIIXX

15.1°E 15.39°E 15.69°E 15.98°E 16.27°E 16.57°E 16.86°E

37.65°N

37.87°N

38.1°N

38.33°N

38.56°N

38.79°N

39.02°N

1907/10/23, Io VIII−IX: observed

Is

IIIIIIIVVVIVIIVIIIIX

10.72°E 11.9°E 13.07°E 14.24°E 15.41°E 16.58°E 17.76°E

43.77°N

44.58°N

45.39°N

46.2°N

47.01°N

47.82°N

48.63°N

1895/04/14, Io VIII−IX: observed

Is

IIIIIIIVVVIVIIVIII

7.73°E 8.91°E 10.08°E 11.25°E 12.42°E 13.59°E 14.77°E

42.58°N

43.41°N

44.23°N

45.06°N

45.89°N

46.71°N

47.54°N

1914/10/27, Io VII: observed

Is

IIIIIIIVVVIVII

0 50 100 150 200 250

0

2

4

6

8

10

12

0 50 100 150 200 250

0

2

4

6

8

10

12

0 50 100 150 200 250

0

2

4

6

8

10

12

0 50 100 150 200 250

0

2

4

6

8

10

12


Smoothed posterior distribution of intensity Is for class A

Pr {Is = i | I0 = i0, pi0 (d)} = Binom {2i | 2i0, pi0 (d)} where pi0 (d) =(

c1c1 + d

)c2

I0 n.MFs c1 c2V 22 13846.42 219.98

V-VI 17 - -VI 21 113.96 2.16

VI-VII 3 - -VII 17 1989.44 29.01

VII-VIII 10 - -VIII 8 9674.80 126.72

VIII-IX 0 - -IX 14 3573.55 37.52

IX-X 4 - -X 10 27752.59 254.40

X-XI 2 - -XI 4 36102.69 332.45

The number of MFs having uncertain epi-central intensity I0 is sometimes very smallor even null; it follows that the correspond-ing estimated coefficients c1 and c2 mightbe quite unrealiable.

In order to obtain more reliable and stableestimates for uncertain epicentral intensityI0 = i0, its parameter pi0 (d) is chosen asfollows:

pi0 (d) =pdi0e(d) + pbi0c(d)

2


Probability of the intensity attenuation versus distance

Original model New model New model

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250I

II

III

IV

V

Italy: Io V , class A

Km

Is

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250

I

II

III

IV

V

Italy: Io V , class A

Km

Is

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250

I

II

III

IV

V Italy: Io V−VI , class A

Km

Is

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250I

II

III

IV

V

VI

Italy: Io VI , class A

Km

Is

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250

I

II

III

IV

V

VI

Italy: Io VI , class A

Km

Is

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250

I

II

III

IV

V

VI Italy: Io VI−VII , class A

Km

Is

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250I

II

III

IV

V

VI

VII

Italy: Io VII , class A

Km

Is

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250

I

II

III

IV

V

VI

VII

Italy: Io VII , class A

Km

Is

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250

I

II

III

IV

V

VI

VII Italy: Io VII−VIII , class A

Km

Is



Original model New model New model

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250I

II

III

IV

V

VI

VII

VIII

Italy: Io VIII , class A

Km

Is

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250

I

II

III

IV

V

VI

VII

VIII

Italy: Io VIII , class A

Km

Is

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250

I

II

III

IV

V

VI

VII

VIII Italy: Io VIII−IX , class A

Km

Is

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250I

II

III

IV

V

VI

VII

VIII

IX

Italy: Io IX , class A

Km

Is

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250

I

II

III

IV

V

VI

VII

VIII

IXItaly: Io IX , class A

Km

Is

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250

I

II

III

IV

V

VI

VII

VIII

IX Italy: Io IX−X , class A

Km

Is

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250I

II

III

IV

V

VI

VII

VIII

IX

X

Italy: Io X , class A

Km

Is

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250

I

II

III

IV

V

VI

VII

VIII

IX

XItaly: Io X , class A

Km

Is

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250

I

II

III

IV

V

VI

VII

VIII

IX

X Italy: Io X−XI , class A

Km

Is



Original model New model

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250I

II

III

IV

V

VI

VII

VIII

IX

X

XI

Italy: Io XI , class A

Km

Is

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250

I

II

III

IV

V

VI

VII

VIII

IX

X

XIItaly: Io XI , class A

Km

Is

References

Rotondi R., Zonno G. (2004), Bayesian analysis of a probability distribution for local intensityattenuation, Ann. Geophys., 47 (5), 1521-1540

Zonno G., Rotondi R., and Brambilla C. (2009), Mining macroseismic fields to estimate theprobability distribution of the intensity at site, BSSA, 99 (5), 2876-2892

Azzaro R., D’Amico S., Rotondi R., Tuvè T., ZonnoG. (2013), Forecasting seismic scenarios onEtna volcano (Italy) through probabilistic intensity attenuation models: A Bayesianapproach, J. Volcanol. Geotherm. Res., 125, 149-157

Special issue Bull. Earthq. Eng. (2016) 14, 7


MotivationsClustering

E. Varini and R. Rotondi - Copernicus.org · 2020. 5. 1. · E. Varini and R. Rotondi Istituto di...

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