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Content uploaded by Paul Somerville

Author content

All content in this area was uploaded by Paul Somerville on Sep 29, 2015

Content may be subject to copyright.

EFFECTS OF RUPTURE DIRECTIVITY ON

PROBABILISTIC SEISMIC HAZARD ANALYSIS

Norman A. Abrahamson

1

ABSTRACT

For long period structures such as bridges that are near faults with high activity rates,

it can be important to explicitly include directivity effects in the attenuation relations for

either probabilistic or deterministic analyses. There are two rupture directivity effects. The

first effect is a change in strength of shaking of the average horizontal component of motion,

and the second effect is the systematic differences in the strength of shaking on the two

horizontal components oriented perpendicular and parallel to the strike of the fault. The new

San Francisco Oakland Bay Bridge is used as an example to show the effects of including

rupture directivity in probabilistic seismic hazard analysis. For this example, including both

effects results in about a 30% increase in the T=3 second spectral acceleration for a 1500

year return period as compared to traditional analyses.

Introduction

Rupture directivity effects can lead to large long period pulses in the ground motion.

Recently, models have been developed to quantify the directivity effect (e.g. Somerville et

al, 1997). With these models of the rupture directivity effect, directivity can be included in

either deterministic or probabilistic seismic hazard analyses. This paper demonstrates the

effect of rupture directivity on probabilistic seismic hazard analyses.

Attenuation Relations and Rupture Directivity

For design of long-period structures such as bridges, characterization of long-period

motion is essential. Attenuation relations commonly used in California do not explicitly

include rupture directivity effects but they can adjusted to account for near-fault directivity

effects using the Somerville et al. (1997) fault-rupture directivity model. The Somerville et

al. (1997) model comprises two period-dependent scaling factors that may be applied to

horizontal attenuation relationship. One of the factors accounts for the change in shaking

intensity in the average horizontal component of motion due to near-fault rupture directivity

effects (higher ground motions for rupture toward the site and lower ground motions for

rupture away from the site). The second factor reflects the directional nature of the shaking

intensity using two ratios: fault normal (FN) and fault parallel (FP) versus the average (FA)

1

Pacific Gas and Electric Company, 245 Market Street, Mail Code N4C, San Francisco, CA 94105

Proceedings of the Sixth International Conference on Seismic Zonation (6ICSZ)

EERI November 12-15, 2000 Palm Springs CA

component ratios. The fault normal component is taken as the major principal axis resulting

in an FN/FA ratio larger than 1 and the fault parallel component is taken as the minor

principal axis with an FP/FA ratio smaller than 1. The two scaling factors depend on

whether fault rupture is in the forward or backward direction, and also the length of fault

rupturing toward the site.

There are several aspects of the empirical model for the average horizontal

component scale factors developed by Somerville et al that needed to be modified to make

the model applicable to a probabilistic hazard analysis. As published, the directivity model

is independent of distance. The data set used in the analysis includes recordings at distances

of 0 to 50 km. A distance dependent taper function was applied to the model that reduces

the effect to zero for distances greater than 60km.

T

d

(r) = 1 for r < 30 km

1 - (r-30)/30 for 30 km < r < 60 km (1)

0 for r > 60 km

As published, the model is applicable to magnitudes greater than 6.5. A magnitude taper

was applied that reduces the effect to zero for magnitudes less than 6.0.

T

m

(m) = 1 for m = 6.5

1 - (m-6.5)/0.5 for 6 = m < 6.5 (2)

0 for m < 6

The empirical model uses two directivity parameters: x and θ where x is defined as

the fraction of the fault length that ruptures toward the site and θ is the angle between the

fault strike and epicentral azimuth (see Somerville et al for details). The worst case is x=1

and θ=0. The empirical model uses a form that increases a constant rate as x increases from

0 to 1. There is little empirical data with x cos(θ) values greater than 0.6, and the

extrapolation of the model to larger x cos(θ) values is not well constrained. Based on an

evaluation of empirical recordings and numerical simulations, the form of the directivity

function is modified to reach a maximum at x cos(θ) =0.4. The slope is greater than the

Somerville model, but it flattens out at a lower level. The T=3 second value is used to guide

the adjustment of the model at all periods. The resulting model is given by

y

Dir

(x,θ,Τ) = C

1

(T) + 1.88 C

2

(T) x cos(θ) for x cos(θ) = 0.4 (3)

C

1

(T) + 0.75 C

2

(T)

for

x cos(θ) > 0.4

where C

1

(T) + C

2

(T) are from Somerville et al. (1997), and are listed in Table 1.

Table 1. Model Coefficients for the Modified Somerville et al. (1997) Directivity

Effects for the Average Horizontal Component

Period (sec) C1 C2

0.60 0.000 0.000

0.75 -0.084 0.185

1.00 -0.192 0.423

1.50 -0.344 0.759

2.00 -0.452 0.998

3.00 -0.605 1.333

4.00 -0.713 1.571

5.00 -0.797 1.757

Finally, including the directivity effect should results in a reduction of the standard

deviation of the attenuation relation. Based on an evaluation of the empirical data, at T=3

seconds, there is a reduction of the standard deviation of about 0.05 natural log units due to

adding the directivity term into the ground motion model. The period dependence of the

reduction was approximated by the period dependence of the slope of the directivity effect.

To account for the reduction in the standard deviation due to including the directivity effect

as part of the model, the standard deviations for the published attenuation relations were

modified for use in the hazard analysis using the following relation:

σ

dir

(M,T) = σ (M,T) - 0.05 C

2

(T)/1.333 (4)

where C

2

(T) is given in Table 1 and σ(M,,T) is the standard deviation from the published

attenuation relation (without directivity effects). The final modified Somerville model for

the average horizontal component for strike-slip faults is given by

ln Sa

dir

(M,r,x,θ,Τ) = ln Sa(M,r) + y

Dir

(x,θ,Τ) T

d

(r) T

m

(m) (5)

where Sa(M,r) is an empirical attenuation relation without directivity. This modified model

is shown in Figure 1a. The FN/Avg ratios are shown in Figure 1b for the modified

Somerville et al. (1997) model.

Incorporating Rupture Directivity Into Probabilistic Seismic Hazard Analysis

It is straightforward to include rupture directivity into a probabilistic hazard analysis.

The main change is that the location of the hypocenter on the rupture area needs to be

included as an additional source of (aleatory) variability (Figure 2).

In a standard hazard calculation, the hazard is given by:

ν

i

(A > z) = N

i

(M

min

)

Ey

1

∫

Ex= 0

1

∫

RA =0

∞

∫

f

m

i

(m) f

W

i

(m,W) f

RA

i

(m,RA)f

Ex

i

(x)f

Ey

i

(m,x)

m=M

min

M

max

i

∫

W= 0

∞

∫

P(A > z | m,r

i

(x,y,RA,W))dWdRAdxdydm

(6)

where N

i

(M

min

) is the rate of earthquakes with magnitude greater than M

min

from the i

th

source; m is magnitude; M

max

i

is the maximum magnitude (for the i

th

source); f

m

(m),

f

W

(m,W), f

RA

(m,RA), f

Ex

, and f

Ey

are probability density functions for the earthquake

magnitude, rupture width, rupture area, location of the center of rupture along strike and

location of the center of rupture down dip, respectively; and P(A>z|m,r,ε) is the probability

that the ground motion exceeds the test level z for a given magnitude and distance.

Including directivity effects for strike-slip faults, the hazard is given by

where r, X, and θ are computed from the rupture location and the hypocenter location. In

this calculation, we need to define and additional probability density function for the

location of the hypocenter on the rupture, f

hx

(h

x

). Here, I have assumed uniform

distributions for the hypocenter location (e.g. no preferred locations for the nucleation of the

rupture).

It could be argued that the effect on directivity on the average horizontal component

is already included in the standard deviation of the ground motion of the published

attenuation relations. There are two reasons why the published standard deviations do not

adequately account for directivity effects. First, the standard deviation of most attenuation

relations is averaged over all distances. As a result, the standard deviation of the near fault

ground motion is underestimated by the average standard deviation over all distances.

Second, the size of the directivity effect can vary significantly for different locations that are

the same distance away from the fault. That is, a particular site-fault geometry may be more

likely to experience forward directivity effects than other sites.

Example Calculation

As an example, the probabilistic seismic hazard is computed with and without

directivity. As discussed above, we can consider the effects of directivity on the average

horizontal component or on the individual components oriented parallel and perpendicular

to the strike of the fault. The base case is using the attenuation relations as published.

These are for the geometric mean of the two horizontal components. This case is label as

“without directivity” in Figure 3. Including just the effect of directivity on the average

horizontal component increases the hazard at return periods greater than 500 years. The

hazard with directivity can be deaggregated in terms of the rupture directivity parameter,

Xcos(θ). The deaggregation at the 1500 year return period is shown in Figure 4. This figure

shows that most of the long period ground motion hazard is from forward directivity cases.

ν

i

(A > z) = N

i

(M

min

)

hx= 0

1

∫

Ey

1

∫

Ex= 0

1

∫

RA =0

∞

∫

f

m

i

(m) f

W

i

(m,W) f

RA

i

(m,RA)f

Ex

i

(x) f

Ey

i

(m,x)

m=M

min

M

max

i

∫

W= 0

∞

∫

f

h

x

(h

x

)

P(A > z | m,r,X,

θ

)dW dRAdx dydh

x

dm

(7)

Deaggregation on the directivity can guide the selection of time histories in terms of

selecting forward directicity time histories vs neutral or backward directivity time histories.

Conclusions

Rupture directivity has not been included in most probabilistic seismic hazard

calculations. To accurately estimate the hazard from long period ground motions, directivity

should be directly included in the hazard analysis as one of the important sources of

variability of the long period ground motion. I believe that including rupture directivity will

soon become the standard of practice for computing the hazard for long period ground

motions near active faults.

References

Somerville, P. G., N. F. Smith, R. W. Graves. and N. A. Abrahamson (1997). Modification

of empirical strong ground motion attenuation relations to include the amplitude and

duration effects of rupture directivity, Seism. Res. Let, Vol. 68, 199-222.

QGa

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Directivity Model

Xcos(Theta)

Modified Somerville et al.

(1997) directivity model

0

0.05

0.1

0.15

0.2

0.25

0.3

0 102030405060708090

Ln(FN/Avg)

Theta

Somerville et

al. (1997)

FN/Average

Horizontal

Model

1a.

1b.

Figure 1. Modified Somerville et al. (1997) rupture directivity model.

QGa

15

10

5

0

0 102030405060708090100

Depth (km)

Distance (km)

Figure 2. Example of the variability in hypocenter location over the rupture surface for a single rupture

location. The heavy line defines the rupture area of an earthquake scenario, the circle indicates the center of

the rupture, and the stars indicate the range of hypocenters.

QGa

0.0001

0.001

0.01

0.1

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Annual Probability of Exceedance

Spectral Acceleration (g)

Ave Horizontal without Directivity

Fault Parallel Component

Average Horizontal Component

Fault Normal Component

Figure 3. Effects of directivity on the hazard for T=3 seconds spectral acceleration for the new San

Francisco Bay Bridge.

Figure 4. Deaggregation of the hazard for a spectral period of 3.0 seconds at the 1500 year return period.

KCampbell@eqecat.com

To "Gilles Bureau" <gbureau@geiconsultants.com>

03/03/200510:12

AM

ee

john_barneich@geopentech.com,

kcampbell@absconsulting.com,

Nancy

_ Collins@urscorp.com,

bee

Subject Re: Correction for fault normal

I fault parallel effects

Gilles,

Note

that

there

is

an

error

in

the

strike-slip

equation

that

Norm

gives

in

his

paper

(personall

corom.,

2003).

The

equation

(in

different

format)

should

be:

(Embedded

image

moved

to

file:

pic03902.jpg)

Ken

Kenneth

W.

Campbell,

PhD

Vice

President

EQECAT

Inc

(503)

533-4359

(503)

533-4360

(fax)

kcampbell@eqecat.com

http://www.eqecat.com

"Gilles

Bureau"

<gbureau@geiconsu

Itants.com>

To

<kcampbell@absconsulting.com>,

03/03/2005

08:55

<KCampbell@eqecat.com>,

AM

<john_barneich@geopentech.com>,

<Phalkun_Tan@geopentech.com>,

<yoshi_moriwaki@geopentech.com>,

<Nancy_Collins@URSCorp.com>

<Paul_Somerville@URSCorp.com>

Subject

Re:

Correction

for

fault

normal

/

fault

parallel

effects

To

all:

Thank

you

for

the

paper

and

for

clarifying

this

situation.

I

have

deeply

appreciated

the

fast

response

from

Paul

and

Ken,

after

learning

one

was

on

Australia

and

the

other

in

Italy.

cc

Best

regards,

Gilles

Gilles

Bureau,

P.E.,

G.E.

Senior

Consultant

GEl

Consultants,

Inc.

2201

Broadway,

Suite

321

Oakland,

CA

94612-3017

510-835-9838

x

105

510-835-9842

(fax)

gbureau@geiconsultants.com

»>

<Nancy_Collins@URSCorp.com>

3/1/2005

5:07:18

PM

»>

Gentlemen:

Attached

is

the

Abrahamson

paper

of

which

Paul

speaks

below

(See

attached

file:

Abrahamson_6lCSZ.pdf)

Nancy

F.

Collins,

Project

Scientist

URS

Corporation

566

EI

Dorado

Street,

Suite

200

Pasadena,

CA

91101

(626)

449-7650;

FAX

(626)

449-3536

Forwarded

by

Nancy

Collins/Pasadena/URSCorp

on

03/01/2005

04:52

PM

Paul

Somerville

To:

KCampbell@eqecat.com,

robert_graves@urscorp.com,

03/01/2005

yoshi_moriwaki@geopentech.com,

PM

john_barneich@geopentech.com

<gbureau@geiconsultants.com>,

01:54

nancy_collins@urscorp.com,

cc:

"Gilles

Bureau"

kcampbell@absconsulting.com,

naa2@pge.com,

paul_somerville@urscorp.com

fault

normal/fault

parallel

Subject:

effects

Re:

Correction

for

Gilles

et

al.

Ken

has

correctly

described

the

situation.

My

assistant

Nancy

Collins

is

going

to

fax

you

the

paper

by

Norm

Abrahamson

that

describes

the

tapering

of

the

model

for

small

magnitude

and

large

distance.

Paul

I,

Paul

Somerville

To: KCampbell@eqecat.com, robert-.9raves@urscorp.com,

:td::~t-

nancy-collins@urscorp.com, yoshi_moriwaki@geopentech.com.

-,

-',

03/01/200501 :54 PM

john_barneich@geopentech.com

cc: "Gilles Bureau" <gbureau@geiconsultants.com>,

kcampbell@absconsulting.com, naa2@pge.com,

pauLsomerville@urscorp.com

Subject: Re: Correction for fault normal

I fault parallel effects

I+/s.o

t/-v

-j>

-PM

I~

UlVI

J

CUA.I

Gilles et al.

1/i-j-

79b-

9/9/

Ken has correctly described the situation.

My

assistant Nancy Collins is going to fax you the paper by

Norm Abrahamson that describes the tapering

of

the model for

small magnitude and large distance.

Paul

Paul G. Somerville

URS Corporation

566

EI

Dorado Street

Pasadena,

CA

91101-2560

Voice: (626) 449-7650; FAX: (626) 449-3536

email: PauLSomerville@urscorp.com

-KCampbell@eqecat.com

wrote: -----

To: "Gilles Bureau" <gbureau@geiconsultants.com>

From: KCampbell@eqecat.com

Date:

03/0112005

11

:42AM

cc: kcampbell@absconsulting.com, naa2@pge.com, pauLsomerville@urscorp.com

Subject: Re: Correction for fault

normal/fault

parallel effects

Gilles,

I was in Italy and unable to respond to your question before now. I have

carefully reviewed the 1997 SRL paper and I believe that I have made a

mistake in representing the directivity model in

my

CRC Handbook (and two

other chapters that I have written). Even though they refer to the

FN

to

FP factor in many places throughout the text, including the equations

themselves, the authors state that the coefficients listed in the tables

are for the FN to

AV

(average) components. I believe that I was confused

by the ambiguity in their statements and I apologize for the error.

Therefore, I believe that

my

Equation (5.106) should not have the 1/2 in

the first two terms, Furthermore, the last term (0) should also apply when

M > 6, according to the authors. I also neglected to put that constraint

in as well. Furthermore, since the FN and FP relationships are not

numerically constrained at M > 6 and Rrup > 50, I would suggest that a

magnitude and distance taper similar

to

the one Norm applied to the spatial

directivity term

(my

f1) should also be applied to my f2 term, except that

the taper should be applied between M

= 5.5 and 6.5, centered on 6.0.

Paul and/or Norm, can you please confirm this and respond to my taper

suggestion?

Ken

cc

Kenneth

W.

Campbell, PhD

Vice President

EQECAT

Inc

(503)

5334359

(503)

5334360

(fax)

kcampbell@eqecat.com

http://www.egecat.com

"Gilles Bureau"

<gbureau@geiconsu

Itants.com>

To

<kcampbell@absconsulting.com>,

02/24/200512:52 <naa2@pge.com>,

PM <pauLsomerville@urscorp.com>

Subject

Correction

for

fault normal 1

fault parallel effects

Gentlemen:

Greetings.

I am facing an issue with the use

of

the correction factor for fault normal

(FN)

I fault parallel effects (FP)

for

ground motion estimates based on

your attenuation equations.

Paul and Norm, in their 1997, paper talk on page 214 (Seismological

Research Letters Vol. 68, No.1) about fitting the strike-normal to

strike-parallel ratio y

(y

being defined as the natural logarithm

of

the

strike-parallel to strike normal ratio

at

a given period), with an equation

of

the form:

y

=cos 2 ?

[ci

+ c2 In(Rrup + 1) + c3(Mw -

6)]

However, elsewhere in the text and in Table 6 and 7

of

the 1977 reference,

the c coefficients are defined for the strike-normal to average ratio.

In

Chapter 5

of

the

CRe

Handbook (2003), Ken uses a similar equation

for

FN

(and specifically refers to Norm's 2000 model) as follows:

f2(Rrup, Mw,

?) =% (cos 2 ?) [c3 +

c4ln(Rrup

+ 1) + c5

(Mw

- 6)] [5.106]

The

c3, c4 and

c5

are defined in Table 5.25 (Campbell, 2003) and are the

very same values as c1, c2 and c3 in the aforementioned Table

7.

Hence, a factor % (half the fault-normal effects) has been introduced in

Equation 5.106. Equation [5.106] also provides an equal in amplitude but

opposite sign correction

(-1/2 etc.) for FP effects when compared with the

average estimates.

My

question is:

How

where the c coefficients developed? Do they relate FN to FP,

or

FN to

average? Is a 0.5 factor correctly applied by Ken, or does it result from

the language ambiguity regarding y in the 1977 paper?

Thank you all very much for clarifying how to correctly apply the FN

correction. Your help will be greatly appreciated.

Best regards

to

all three

of

you,

Gilles.

Gilles Bureau, P.E., G.E.

Senior Consultant

GEl Consultants, Inc.

2201

Broadway, Suite

321

Oakland,

CA

94612-3017

510-835-9838 x 105

510-835-9842 (fax)

gbureau@geiconsultants.com