Empirical Ground Motion Prediction Equations
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Empirical Ground Motion Prediction Equations
Introduction to Strong Motion SeismologyDay 2, Lecture 3
Nay Pyi Taw, MyanmarTuesday, 29 January 2013
Empirical Specification of Ground MotionsEmpirical Specification of Ground Motions
• Ground-motion prediction equations • Collections of ground-motion values (and time series)
for magnitude and distance bins
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Overview
• Empirical ground-motion prediction equations (GMPEs)– What they are and how they are used– The database
• Processing of data
• Combining horizontal components
– Expected dependency on magnitude and distance– What functional form to use– What to use for the explanatory variables– Example: PEER NGA-West2 GMPEs
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The Engineering Approach
• Use physically based empirical relationships
• Take a large suite of recorded earthquake motions and perform regression analysis to obtain models that may be used for the estimation of the distribution of future ground-motions
• Simple models that only require knowledge of a few parameters
• Significant variability associated with the estimates of these equations– Partly reflects the simplicity of the models– Partly reflects the inherent variability of earthquake ground-
motions
• The treatment of this variability is crucial for hazard analysis
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Ground-Motion Prediction EquationsGround-Motion Prediction Equations
Gives mean and standard deviation of response-spectrum ordinate (at a particular frequency) as a function of magnitude distance, site conditions, and perhaps other variables.
10-1 1 101 102
0.01
0.1
1.0
Shortest Horiz. Dist. to Map View of Rupture Surface (km)
La
rge
rH
ori
zon
talP
ea
kA
cce
l(g
)
1992 Landers, M = 7.31994 NR, M=6.7 (reduced by RS-->SS factor)Boore et al., Strike Slip, M = 7.3, NEHRP Class D_+
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Call them “Ground-Motion Prediction Equations” (GMPEs)
• “Attenuation EquationsAttenuation Equations” is a poor term
– The equations describe the INCREASE of amplitude with magnitude at a given distance
– The equations describe the CHANGE of amplitude with distance for a given magnitude (usually, but not necessarily, a DECREASE of amplitude with increasing distance).
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Deriving the Equations
• Regression analysis of observed data if have adequate observations (rare for most of the world).
• Regression analysis of simulated data for regions with inadequate data (making use of motions from smaller events if available to constrain distance dependence of motions).
• Hybrid methods, capturing complex source effects from observed data and modifying for regional differences.
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1 10 100 1000
5
6
7
8
Mo
me
nt
Ma
gn
itud
e
Used by BJF93 for pga
Western North America
1 10 100 1000
5
6
7
8
Distance (km)
Mo
me
nt
Ma
gn
itud
e
AccelerographsSeismographic Stations
Eastern North America
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Observed data adequate for regression exceptclose to large ‘quakes
Observed data not adequate for regression, use simulated data
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What Measure of Seismic Intensity to Use?
• PSA (derived from SD)• SD? Need for displacement-based design
– Can be derived from PSA, so GMPEs in terms of PSA also give equations for SD
– If use SA, then need separate GMPEs for PSA and SD
• Usually horizontal component
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How to Use Two Horizontal Components
• Use both independently• Use larger component as recorded• Use larger component after rotation to find maximum• Use vector sum• Use geometric mean as recorded• Use orientation-independent geometric mean (gmroti50)• Use orientation-independent, non-geometric mean (rotd50)
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What to use for the Predictor Variables?
• Moment magnitude• Some distance measure that helps account for the
extended fault rupture surface (remember that the functional form is motivated by a point source, yet the equations are used for non-point sources)– Distance must be estimated for future event (leaves out distance to
energy center, hypocentral distance)
• A measure of local site geology
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Moment Magnitude
• Best single measure of overall size of an earthquake• Can be determined from ground deformation or
seismic waves• Can be estimated from paleoseismological studies• Can be related to slip rates on faults
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Many distance measures are used
• There is no standard, although the closest distance to the rupture surface is probably the distance most commonly used
• The distance measure must be something that can be estimated for a future earthquake
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Most Commonly Used:
Rrup
RJB
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How does the motion depend on magnitude?
• Source scaling theory predicts a general increase with magnitude for a fixed distance, with more sensitivity to magnitude for long periods and possible nonlinear dependence on magnitude
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How does the motion depend on distance?
• Generally, it will decrease (attenuate) with distance• But wave propagation in a layered earth predicts
more complicated behavior (e.g., increase at some distances due to critical angle reflections (“Moho-bounce”)
• Equations assume average over various crustal structures
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M and R dependence shown by data
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• The scatter remaining after removing magnitude and distance dependence is large
• Can it be reduced by introducing other factors?• The most obvious additional factor tries to capture
the effect of site response
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Uncertainty after Mag & Dist Correction
0.1
1
10
4503001500
Observation Number (no particular order)
95-percent within factor of 3.5 of average
(sigma = 0.63 (natural log))
(E. Field)
Simplest: Rock vs Soil
0.1
1
10
4503001500
Sorted: Soil to left - Rock to right
motion is ~1.5 times greater than Soil Rock
(E. Field)
0 ( )
SZ z
S
zV
dV
0
1 1( )
z
SZ SSZ
S S dV z
Time-averaged shear-wave velocity to depth z:
Average shear-wave slowness to depth z:
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Site Classifications for Use WithGround-Motion Prediction Equations
• Rock = less than 5m soil over “granite”, “limestone”, etc.• Soil= everything else
2. NEHRP Site Classes (based on VS30)
3. Continuous Variable (VS30)
1. Rock/Soil
620 m/s = typical rock
310 m/s = typical soil
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200 1000 20000.40.5
1
2
345
V30, m/sec
Am
plifi
catio
n2003-06-08 23:12:15
T=0.10 s
slope = bv, where Y (V30)bV
200 1000 20000.40.5
1
2
345
V30, m/sec
T=2.00 s
VS30 as continuous variable
Note period dependence of site response24USGS - DAVID BOORE
Why VS30?
• Most data available when the idea of using average Vs was developed were from 30 m holes, the average depth that could be drilled in one day
• Better would be Vsz, where z corresponds to a quarter-wavelength for the period of interest, but
• Few observations of Vs are available for greater depths
• Vsz correlates quite well with Vs30 for a wide range of z greater than 30 m (see next slide, from Boore et al., BSSA, 2011, pp. 3046—3059)
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Effect of different site characterizationsfor a small subset of data for which V30values are available
• No site characterization• Rock/soil• NEHRP class• V30 (continuous variable)
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-0.5
0
0.5
1
1.5
Obs
-B
JF(M
,R)
RockSoil
T = 2.0 secResiduals computed as obs-bjf (with no site correction,which is the same as assuming BJF class A or V30 = VA).
200 300 400 1000-1
-0.5
0
0.5
1
V30 (m/sec)
Obs
-B
JF(M
,R)
-S
ite(r
ock,
soil)
RockSoil
T = 2.0 secrock & soil site classification
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σ=0.25
σ=0.25
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-1
-0.5
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Obs
-B
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-S
ite(B
,C,D
)
RockSoil
T = 2.0 secNEHRP B,C,D site classification
200 300 400 1000-1
-0.5
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V30 (m/sec)
Obs
-B
JF(M
,R)
-S
ite(V
30) Rock
Soil
T = 2.0 sec
V30 site classification (continuous)
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σ=0.21
σ=0.20
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Nonlinear Soil Response
• Clearly seen in studies of pairs of records• Not as large as found in lab experiments• Account for in regression equations by looking for
systematic variation of soil amplitudes relative to predicted rock amplitudes at given distance and magnitude
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2-stage Regression
• Use 2 stage regression• Use weighted least-squares (random effects model)
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Stage 1 Regression: Determine the distance function and event offsets
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Stage 2 Regression: Determine the magnitude scaling
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Quantifying the Uncertainty
• The GMPEs predict the distribution of motions for a given set of predictor variables
• The dispersion about the median motions can be crucial for low annual-frequency-of-exceedance hazard estimates (rare occurrences for highly critical sites, such as nuclear power plants, nuclear waste repositories)
• Must be clear on type of uncertainty• The scatter is very large; can it be reduced?
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Two types of Aleatory Uncertainty
• Within event (intraevent) (φ) (event-to-event variation has been removed)
• Event-to-event (interevent) (τ): generally smaller than φ
• Total (σ): 2 2 σ φ τ
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What functional form to use?
• Motivated by waves propagating from a point source• Add more terms to capture effects not included in
simple functional form
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Modeling Considerations
• Not too simple• Not too complex (over paramerized)• Reasonable extrapolations to data-poor but
engineering-important regions of predictor variable space (e.g., close to M~8 earthquakes)
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Represent M-scaling by joined quadratic and linear or simple quadratic?
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Pacific Earthquake Engineering Research Center (PEER) NGA Projects:
NGA (2008)NGA-West2 (2013)
• Abrahamson & Silva• Boore, Stewart, Seyhan, and Atkinson• Campbell & Bozorgnia• Chiou & Youngs• Idriss
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NGA Project Details
• All developers used subsets of data chosen from a common database
– Metadata (e.g., magnitude, distance, etc.)
– Uniformly processed strong-motion recordings
– U.S. and foreign earthquakes
– Active tectonic regions
• The database development was a major time-consuming effort
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• 1264 (173) worldwide shallow crustal events from active tectonic regions
• 19409 (3551) recordings (mostly 3-components each) uniformly processed strong motion stations
• M 3.0 (4.2) to 7.9 (7.9)
PEER NGA-West2 Strong-Motion Database
Blue = Previous NGA
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BSSA13 GMPEs
Dave Boore, Jon Stewart, Emel Seyhan, Gail Atkinson
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Our equation for predicting ground motions is:
1 1 2 2ln ln lnY w Y w Y (1)
where iw are weights that sum to unity. We use an average of two
predicted motions as a way of dealing with possible uncertainties
in the functional forms and the derived coefficients.
Boore, Stewart, Seyhan, and Atkinson (BSSA13) Ground-Motion Prediction Equations (GMPEs)
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49
In equation (1) ln iY is given by
30ln ( ) ( ) ( , ) ( )i i i ii D JB S S JBY F F R F V R M M , M , M M , (2)
In this equation, iFM , iDF , and i
SF represent the magnitude scaling,
distance function, and site amplification, respectively, for the ith set of coefficients (we drop the superscript in subsequent
equations). M is moment magnitude, JBR is the Joyner-Boore
distance (defined as the closest distance to the surface projection of
the fault), and 30SV is the time-averaged shear-wave velocity over
the top 30 m of the site. The predictive variables are M, JBR and
30SV ; the fault type is an optional predictive variable that enters
into the magnitude scaling term as shown in equation (6) below.
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50
is the fractional number of standard deviations of a single
predicted value of lnY away from the mean value of lnY (e.g.,
1.5 would be 1.5 standard deviations smaller than the mean
value). All terms, including the coefficient , are period
dependent. is computed using the equation:
2 2(M) (M) , (3)
where is the within-event (intraevent) aleatory uncertainty and is the event-to-event (interevent) aleatory uncertainty.
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The Distance Function
51
The distance function is given by:
1 2 3( , ) [ ( )]ln( / ) ( )D JB ref ref refF R c c R R c R R M M M , (4)
where
2 2JBR R h (5)
and 1c , 2c , 3c , refM , refR , and h are the coefficients to be
determined in the analysis.
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52
The magnitude scaling is given by:
a) hM M
20 1 2 3 4 5( ) ( ) ( )M h hF e U e SS e NS e RS e e M M M M M ,
b) > hM M
0 1 2 3 6( ) ( )M hF e U e SS e NS e RS e M M M ,
where U, SS, NS, and RS are dummy variables used to specify
unspecified, strikeslip, normal slip, and reverse slip fault type.
The Magnitude Function
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Mechanism U SS NS RSunspecified 1 0 0 0strikeslip 0 1 0 0normal 0 0 1 0reverse 0 0 0 1
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The Magnitude Function
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ln lnS LIN NLF F F
The Site Amplification Function
30ln ln( / )LIN lin S refF b V V
Linear Amplification:
The coefficient blin depends on period. We use the Stewart and Seyhan (2013) model.
( 760 m/s)refV
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ln lnS LIN NLF F F
The Site Amplification Function
760m/s 0.1ln ln
0.1NL nl
PGA gF b
g
Nonlinear Amplification:
The coefficient bnl depends on period and VS30. We use Stewart and Seyhan’s (2013) model.
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Amplification using the Stewart & Seyhan (2013) Site Amplification Model
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Not Included• Directivity• Basin depth• Hanging wall (using RJB accounts for this to
some extent)• Other possible predictor variables (e.g., dip,
Ztor, etc.)
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Determination of Coefficients (2-stage regression)
• Select data: – no basement or large structure records, etc;– RJB < 80 km– event class 1, as determined by CRJB=10 km
• Adjust observations to Vs30=760 m/s using SS13 site amps • Constrain c3 (anelastic term) and Mh to values from
special study• Regress for other coefficients, including pseudodepth h
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Constrained Coefficients
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I don’t want you to think that BSSA13 is the only GMPE available, or that it is superior to others. There are 100s of GMPEs worldwide.
The large epistemic variations in predicted motions are not decreasing with time
From Douglas (2010)
Factor ~ 20
Mw6 strike-slip earthquake at rjb = 20 km on a NEHRP C site
Figure 2. Predicted PGA and SA(1 s) (unfilled black circles) for a Mw6 strike-slip earthquake at 20 km on a NEHRP C site against publication date for over 250 models published in the literature. Filled red circles indicate models published in peer-reviewed journals and for which basic information on the used dataset is available. Also shown are the median PGA and SA(1 s) within five-year intervals (black line) and the median ±1 standard deviation (dashed black lines) based on averaging predictions. From Douglas (2010).
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Within the PEER NGA project there are significant differences between the functions and data used. The BSSA13 GMPEs are probably the simplest, but there may be situations where they should be used with caution.
Here are some comparisons for the first set of GMPEs.
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Strike-Slip Fault Mechanism @ 10 kmMagnitude = 6.5, 7.5
NEHRP BC Site Conditions (VS30=760 m/s)
Slide from Yousef Bozorgnia74USGS - DAVID BOORE
45 degree dip to right, to 15 km depth 75USGS - DAVID BOORE
Differences after 10+ years
• SRL 1997
• Magnitude (~ 5-7.5)• Distance (~ 0-100km)• Period (~ 0-4/5s)• Site Class (1 Vs30)• Hanging Wall (simple)• Nonlinear site class (1 model)• Style-of-faulting• Geometric Mean• Magnitude Dependent Sigma
• NGA 2007, 2013
• Magnitude (~ 3-8+)• Distance (~ 0-200km)• Period (~ 0-10s)• All Vs30• Hanging Wall & Footwall
(complex)• All Nonlinear Site models• Depth to Top of Rupture• Sediment Depth• Style-of-faulting• Orientation independent GM• Magnitude Independent Sigma
P. Stafford76USGS - DAVID BOORE
Do the NGA models have it all?
• NGA models have advanced a lot in terms of consideration of new parameters and the capturing of effects that were previously not modelled by empirical equations
• So, have they captured everything? No.
• What remains to be done?
– Damping (in NGA-West2)
– Directivity (some accounting for this in NGA-West2)
– Fault-normal/Fault-parallel; SaMax
– Vertical component
– Correlations between periods
P. Stafford77USGS - DAVID BOORE
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