SIGMA: What it is, why it ma-ers and what we can do with it

Next
Genera*on
A-enua*on
for
CEUS
(NGA‐East)
Project
 SIGMA:
 What
it
is,
why
it
ma-ers
and
what
 we
can
do
with
it 
 Julian
J
Bommer
 Imperial
Co...
Author: Elmer Cobb
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Next
Genera*on
A-enua*on
for
CEUS
(NGA‐East)
Project


SIGMA:


What
it
is,
why
it
ma-ers
and
what
 we
can
do
with
it



Julian
J
Bommer
 Imperial
College
London


PGA


PGV


M
=
4.3


M
=
6.0


M
=
7.6


(J. Steidl)

Empirical
Ground‐Mo*on
Predic*on
Equa*ons



(GMPEs)
 Relate
the
logarithm
of
the
chosen
ground‐mo*on
parameter,
 Y,
to
explanatory
variables
that
characterize:


EARTHQUAKE
SOURCE


Magnitude

 Style‐of‐Faul*ng


SOURCE‐to‐SITE
TRAVEL
PATH


Distance


RECORDING
SITE


Surface
geo‐materials


Empirical
GMPEs
 log(Y)
=
f(M,
F,
R,
S)
 e.g.,


Akkar
&
Bommer
(2010)


GMPE
coefficients
determined
by
regression
analysis
on
recorded
data


log(Yobs)


δ,
residual
 log(Y)


log(Ypred )


M
 log(R)


δ
=
log(Yobs)
–
log(Ypred)
=
log(Yobs)
–
f(M,
F,
R,
S)
 We
introduce
a
third
Greek
le-er,
ε,
to
represent
the
 residuals
normalized
by
the
standard
devia*on,
ε=
δ/σ


ε


Probability
of
 Exceedance





0















50%

 















(median)
 


1














16%
 

‐1














84%
 


2














2.3%
 


3














0.1%


The
logarithmic
 residuals
are
 generally
found
to
 conform
to
a
normal
 (Gaussian)
 distribu*on
with
 mean
0
and
 standard
devia*on
σ


The
distribu*on
of
the
ground‐mo*on
residuals
can
therefore
be
 completely
characterized
by
the
logarithmic
standard
devia*on,
σ


S




I

G




M



A
 log(PGA)


CATTER





N













ROUND














OTION









TTENUATION


N(0,σ)

M log(Distance)


log(PGA)


log(Y)
=
f(M,
F,
R,
S)
+
δ
=
f(M,
F,
R,
S)
+
ε.σ


 N(0,σ)

ε.σ


M
 log(Distance)


Soil‐site
recordings
of
September
2004
Parkfield
earthquake


ε
=
2
 ε
=
1
 ε
=
0


Strong‐Mo*on
Accelerogram


Value
of
σ
varies
with
treatment
of
horizontal
record
components


Beyer
&
Bommer
(2006)



σ[ln(Y)]
=
2.3σ[log10(y)]


~300
 records


Year


~3000
 records


0.15
to
0.35
log10


[0.35
to
0.80
ln]
 Strasser
et
al.
(2009)


Boore
et
al.
(1997)
equa*ons,
median
±σ


84‐percen*le
PGA
values
generally
about
80%
>
medians


Impact
of
σ
on
Seismic
Hazard
Analysis:
DSHA


(Kramer,
1996)


Impact
of
σ
on
Seismic
Hazard
Analysis:
DSHA
 log(Y)
 log(Y84)
 log(Y50)


84th
Percen*le


σlog(Y)
 Median


log(R)
 Figure
courtesy
of
F.O.
Strasser


Impact
of
σ
on
Seismic
Hazard
Analysis:
DSHA
 Rather
than
simply
using
the
median
or
the
84‐percen*le
PGA,
 it
would
be
more
ra*onal
to
select
ε
on
the
basis
of
the
 associated
probability
of
exceedance…..


(Strasser
et
al.,
2008)


But
that
choice
should
be
influenced
by
the
recurrence
rate
of
 the
scenario
earthquake,
in
which
case
we’re
doing
PSHA….


Impact
of
σ
on
Seismic
Hazard
Analysis:
PSHA


Calculate
the
mo*on
 at
the
site
due
to
 every
feasible
 scenario
(M‐R‐ε)
and
 calculate
the
 associated
frequency


Area
Source
2


PSHA




0.3g


R1A


Log(PGA)


Faul

rce u o S t


Area
Source
1


Log(N)


Annual
 Frequency


Log(R)


M

M

Area
Source
2


PSHA




0.3g


R1A


Log(PGA)


Faul

rce u o S t


+ε


Area
Source
1


Log(N)


Annual
 Frequency


Log(R)


M

M

Area
Source
2


PSHA




0.3g


R1B


Log(PGA)


Faul

rce u o S t


+2ε


Area
Source
1


Log(N)


Annual
 Frequency


Log(R)


M

M

Combina*ons
of
M‐R‐ε
to
Produce
0.3
g
at
the
Site
 Scenario


R
(km)


M


ε


f(M)


f(ε)


Frequency








1














15






7










0







0.002




0.50






0.00100














2














15






6










1


0.020




0.16






0.00320














3














25






6










2







0.020



0.023





0.00046








Σ
 0.00466




 Integrate
over
all
possible
magnitudes
at
all
possible
 loca*ons
over
all
sources,
and
consider
all
values
of
ε


Impact
of
σ
on
Seismic
Hazard
Analysis:
PSHA


PSHA
 Not
PSHA


(Abrahamson,
2000)


Value
of
σ
exerts
strong
influence
in
PSHA


Increasing
σ

Bommer
and
Abrahamson
(2006)


Can
the
influence
of
σ
be
reduced
by
trunca*ng
at
εmax?


Yes,
but
to
result
in
an
appreciable
reduc*on
of
hazard,
we
 need
to
truncate
at
3
standard
devia*ons
(εmax
=
3)



PSHA
for
Bay
Bridge


Figure
courtesy
of
Norm
Abrahamson


EPRI
study
in
 2006
concluded
 that
there
is
no
 sta*s*cal
basis,
 using
current
 strong‐mo*on
 datasets,
to
 truncate
at
less
 than
3
sigmas


Strasser
et
al.
(2008)


Uncertainty
in
Ground‐Mo*on
Predic*on
 Sigma
is
a
measure
of
ALEATORY
variability
 This
means
that
it
represents
inherent
RANDOMNESS
 (from
alea,
La*n
for
“dice”)


We
could
think
of
it,
however,
as
apparent
randomness
 since
it
the
aleatory
variability
w.r.t.
a
model
(GMPE)


Variability
in
Ground‐Mo*on
Predic*on
 GMPEs
are
very
simple
(crude)
models
for
very
complex
processes
 Therefore,
a
major
contribu*on
to
σ
is
the
absence
of
parameters
 that
influence
the
ground
mo*on
but
are
not
included
in
GMPEs
 e.g.,
SOURCE
SIZE


Magnitude
included
 Stress
drop,
direc*vity,
etc.,
etc.,
not


e.g.,
SITE
GEOLOGY


Vs30
used
to
characterize
site
effect
 Deeper
geological
structure
o~en
not


Sylmar
County
Hospital
 (Los
Angeles)


Nesher
Site
 (Haifa)


SCH


Vs30
=
280
m/s


NES


Vs30
=
284
m/s


Median
amplifica*on
func*ons
from
non‐linear
site
 response
analyses
with
120
records


Figure
courtesy
of
Myrto
Papaspiliou


Uncertainty
in
 Ground‐Mo*on
 Predic*on
 Unless
the
data
is
abundant
 and
well‐distributed
with
 respect
to
the
explanatory
 variables,
there
will
be
 uncertainty
regarding
the
 posi*on
of
the
median
 predic*on
of
ground
mo*on.



Bommer
&
Abrahamson
(2007)


Uncertainty
in
Ground‐Mo*on
Predic*on
 This
is
referred
to
as
EPISTEMIC
uncertainty
because
it
 reflects
our
lack
of
knowledge
regarding
earthquake
source
 processes
and
wave
propaga*on
in
the
region
under
study


(From
epistêmê
Greek


for
“knowledge”)


Celsus
Library,
 Ephesus


Epistemic
Uncertainty
in
GMPEs
 The
epistemic
uncertainty
in
the
median
ground‐mo*ons
is
 usually
incorporated
into
the
hazard
analysis
through
a
 logic‐tree,
with
branches

carrying
different
models
to
 which
weights
(reflec*ng
the
rela*ve
confidence
of
the
 analyst
in
each
model
being
the
most
appropriate
for
the
 region)
are
assigned

 Whereas
the
aleatory
variability
influences
the
shape
of
the
 hazard
curve,
the
epistemic
uncertainty
results
in
several
 hazard
curves


MEAN


Median
spectra
 for
strike‐slip
 earthquakes

 recorded
on
rock
 sites
at
10
km,
 from
NGA
models


Abrahamson
et
al.
(2008)


Epistemic
Uncertainty
in
GMPEs
 In
addi*on
to
the
epistemic
uncertainty
in
the
 median
ground‐mo*ons
predic*ons,
there
is
also
 epistemic
uncertainty
associated
with
the
value
of
 sigma
for
each
equa*on
 For
example,
there
is
s*ll
uncertainty
about
 whether
sigma
is
dependent
on
earthquake
 magnitude
([email protected])
or
independent
of
 magnitude
([email protected])


NGA
models:
Magnitude
dependence
of
σ


Abrahamson
et
al.
(2008)


GMPEs
with
Heteroscedas*c
Sigma


Strasser
et
al.
(2009)


Akkar
&
Bommer
(2007)
European
GMPE


Pure
error
analysis,
 following
Douglas
&
 Smit
(2001),
revealed
 apparently
strong
 magnitude‐ dependence
of
 standard
devia*on


Akkar
&
Bommer
(2007)
European
GMPE


Akkar
&
Bommer
(2010)
Sigma
Values


Epistemic
Uncertainty
in
GMPEs
 Median,
μ


Sigma,
σ




2) . 0 = w ( 
 GMPE‐2 GMPE ‐3


(w=0 .4)


Sigma‐2
(w=0.4)


σµ

σσ

Can
σ
be
reduced?
 In
theory,
since
it
represents
inherent
randomness,
 it
is
irreducible
 But
σ
is
the
apparent
randomness
in
the
observa*ons
 with
respect
to
a
par*cular
model
that
a-empts
to
 explain
those
observa*ons
(i.e.,
it
is
the
part
that
 remains
unexplained)
 Therefore,
if
we
develop
models
that
be-er
explain
 the
data,
the
apparent
variability
should
decrease


Adding
Explanatory
 Variables
 In
addi*on
to
characterizing
 the
earthquake
source
only
 by
it
size
(magnitude),
we
can
 also
include
the
influence
of
 the
style‐of‐faul*ng


The
impact
on
σ
is
modest,
 but
nonetheless
worthwhile
 Bommer
et
al.
(2003)


Refine
 Explanatory
 Variables?


Courtesy
of
Dave
Boore


Courtesy
of
Dave
Boore


Values
of
σ
not
reducing
significantly
over
*me…..


……despite
increase
in
the
complexity
of
the
equa*ons


Data
courtesy
of
J.
Douglas


Empirical
GMPE
for
Italy


Bommer
&
Scherbaum
(2005)


Norm
Abrahamson
2009
EERI
Dis*nguished
Lecture


As
our
knowledge
of
the
genera*on
and
propaga*on
 of
earthquake
ground‐mo*on
improves….
 Epistemic
Uncertainty
will
be
REDUCED
 Aleatory
Variability
will
be
REFINED
 The
key
to
refining
σ
is
its
decomposi*on
into
 different
elements,
and
establishing
the
influences
 on
each
of
these,
and
indeed
whether
they
are
all
 purely
aleatory
or
if
some
of
the
components
of
σ
 are
actually
epistemic



σT is the total variability τ is the inter-event (earthquake-toearthquake) variability σ is the intra-event (record-to-record) variability Strasser
et
al.
(2009)


Akkar
&
Bommer
(2010)
European
GMPE


The
aleatory
variability
can
be
broken
down
into
Modeling
 and
Parametric
components,
σm
and
σp
respec*vely:


Bommer
&
Abrahamson
(2007)


Errors
in
metadata
(magnitudes,
depths,
distances,
etc.)
are
 propagated
into
the
total
variability.
 If
these
contribu*ons
to
the
variability
can
be
quan*fied,
they
 can
be
subtracted
from
the
appropriate
variability
component

 Abrahamson
&
Silva
(2008)
NGA
model


Strasser
et
al.
(2009)


The
Ergodic
Assump*on
 In
seismic
hazard
analysis,
we
are
interested
in
the
varia*ons
 in
ground‐mo*on
amplitudes
at
a
par*cular
site
over
*me
 (i.e.,
with
repeated
earthquakes).
 Since
in
general
we
do
not
have
observa*ons
over
long
 periods
at
any
site,
we
use
records
from
many
sites
(and
 regions)
to
represent
the
variability
of
the
ground
mo*on

 The
ergodic
assump*on
therefore
is
that
temporal
variability
 of
ground
mo*on
can
be
represented
by
spa*al
or
even
 regional
variability
(i.e.,
trade
space
for
*me)


Single
Sta*on
Sigma
 When
we
do
have
many
recordings
from
a
single
sta*on,
it
is
 seen
that
the
variability
is
smaller
than
the
total
sigma
values
 calculated
for
standard
GMPEs
 Using
records
from
the
LA
basin,
Atkinson
(2006)
found
that
 single‐sta*on
sigma
values
were,
on
average,
10%
smaller
 than
the
σ
calculated
using
all
the
sta*ons



Atkinson
(2006)


Single
Source‐Sta*on
Sigma
 Using
mul*ple
recordings
from
a
single
site
of
earthquakes
in
 only
one
par*cular
source
region
(i.e.,
sampling
a
single
travel
 path),
much
larger
reduc*ons
in
σ
have
been
found

 For
LA
basin
case,
Atkinson
(2006)
found
that
reduc*ons
of
up
 to
40%
compared
to
σ
values
calculated
using
all
the
sta*ons
 and
records
from
mul*ple
sources



Midorikawa
et
al.
(2008)
 7,753
K‐Net
and
Kik‐Net
records
from
50
 earthquakes
(Mw
>
5.0)
in
6
source
zones



Lin
et
al.
(2009)
 Using
recordings
 from
single‐sta*ons
 and
single‐paths
in
 Taiwan


Figure
courtesy
of
Norm
Abrahamson


Region

Total

Single Site

Chen & Tsai (2002)

Taiwan

0.73

0.63

Atkinson (2006)

Southern CA

0.71

0.62

Morikawa et al (2008)

Japan

0.78

Lin et al (2009)

Taiwan

0.73

Single Path and site

0.41 0.36

0.62

0.37

Table
courtesy
of
Norm
Abrahamson


But,
this
reduc*on
in
Aleatory
Variability
can
only
be
invoked
if
 the
median
mo*ons
for
the
site/path
known
with
confidence;
 otherwise,
there
is
penalty
to
be
paid
in
terms
of
increased
 Epistemic
Uncertainty


σ‐σμ
trade‐off


Strasser
et
al.
(2009)


Es*ma*ng
σ
from
numerical
simula*ons
 The
aleatory
variability
cannot
be
obtained
simply
by
calcula*ng
 the
residuals
of
the
data
with
respect
to
the
model
 The
variability
needs
to
be
determined
from
the
variability
of
the
 parameters
in
the
simula*on
models
(taking
account
of
their
 correla*ons
to
avoid
over‐es*ma*on)
and
the
misfit
of
model
to
 observa*ons,
reflec*ng
the
influence
of
parameters
not
included
 Useful
to
dis*nguish
between
MODELING
uncertainty
(due
to
the
 difference
between
the
actual
physical
process
genera*ng
 ground
mo*ons
and
the
simplified
model
represen*ng
this
 process)
and
PARAMETRIC
uncertainty
(in
the
values
of
the
 parameters
in
the
model
for
future
earthquakes)


The
modeling
and
parametric
components
of
uncertainty
can
 each
be
broken
down
into
ALEATORY
and
EPISTEMIC
 components,
leading
to
four
components
of
the
total
uncertainty
 in
ground‐mo*on
predic*ons:



Toro
et
al.
(1997)


e.g.,
variability
in
Δσ
 for
CEUS
earthquakes
 e.g.,
uncertainty
in
 median
Δσ
and
its
 variability
for
CEUS
 earthquakes


ALEATORY:

 Parametric
uncertainty
in
stress
drop,
focal
depth,
κ
and
Q,
 and
from
aleatory
modeling
uncertainty.

 EPISTEMIC:
 Epistemic
parametric
uncertainty
in
stress
drop,
and
from
 epistemic
modeling
uncertainty.



Toro
et
al.
(1997)


Concluding
Remarks
 Sigma
is
the
aleatory
variability
in
ground‐mo*on
predic*ons
 It
is
part‐and‐parcel
of
the
GMPE;
the
median
predic*on
 alone
is
not
a
full
representa*on
 Sigma
cannot
be
wished
away,
must
always
be
taken
into
 account
and
it
has
a
major
impact
on
the
results
PSHA
 Trunca*ng
sigma
is
not
a
feasible
op*on
to
reduce
its
 influence
in
PSHA
at
the
current
*me
 The
most
promising
prospects
for
reducing
sigma,
and
its
 impact,
is
to
break
it
down
into
components,
iden*fy
those
 that
are
actually
epistemic,
take
them
out
and
deal
with
 them
separately
through
data
collec*on/modeling
or
logic‐ tree
branches


Decomposing
σ
into
Aleatory
&
Epistemic
Components


PEGASOS
Refinement
Project
mee*ng,
London,
January
2010



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