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
(heteroscedas@c)
or
independent
of
magnitude
(homoscedas@c)
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