Chapter 7
Ground-Motion Characterization for the
Median
7.1
Introduction
A variety of terms have been used ground-motion equations: attenuation relations, ground-motion
prediction equations (GMPEs), and ground-motion models (GMMs). All of these terms refer to
the same thing. Early ground-motion equations focused on the distance scaling and were therefore
called attenuation relations. Currently, the preferred term is ”GMM” because it reflects that we
have a model for estimating the ground motion. The previous term ”GMPE” is out of favor because
the term “prediction” gets confused with earthquake prediction. In these class notes, I will use
the term GMM to describe the equations used to compute the median and standard deviation of
the ground motion.
GMMs describe the median and variability of the ground motion given the properties of the
earthquake source (magnitude, style-of-faulting, depth), the wave propagation (distance), and site
response (site class or VS30 ). Using our simple equation for hazard:
Haz(Y > z) =
nF
Xlt
Rate(Mi , Ri ) P (Y > z|Mi , Ri , S)
(7.1)
i=1
The conditional probability of exceeding the ground motion given the earthquake scenario is given
by:
Z ∞
P (Y > z|Mi , Ri , S) =
fY (Y, MI , Ri , S)dY
(7.2)
z
where fY (Y ) is the pdf of the ground motion. Typically, the pdf of ground motion is assumed to
follow a lognormal distribution. The GMM provides the median and standard deviation of the
lognormal distribution of the ground-motion intensity measure (IM) for a given scenario. Knowing
the median and standard deviation, we can compute the P (Y > z|Mi , Ri , S) term using the
standard normal distribution:
ln(z) − ln(Ymed (Mi , Ri , S))
(7.3)
P (Y > z|Mi , Ri , S) = 1 − Φ
σ(Mi , Ri , S)
7.1
where Φ(x) is the cumulative standard normal distribution.
In this chapter, a broad overview of GMM is given covering their use in PSHA. The seismological
basis for the development of the model for the median ground motion is described in detail in
Chapter 11. The standard deviation, which represents the aleatory variability of the ground
motion for a given scenario, is described in detail in Chapter 12.
7.2
Tectonic Regions for Ground-Motion Models
Three broad categories of tectonic regimes are typically used for GMMs for seismic hazard assessments: shallow crustal earthquakes in active tectonic regions (e.g. California, Japan, New Zealand,
Italy, Turkey), shallow crustal earthquakes in stable continental regions (e.g. Australia, eastern
North America, northern Europe), and subduction zone earthquakes (e.g. Japan, Chile, Alaska,
New Zealand). The magnitude and distance scaling of the ground motion is different between
these three broad tectonic categories.
Ideally, we would have sufficient data from earthquakes near the site to constrain the groundmotion scaling, but that is usually not the case. To have enough data to constrain a model,
the magnitude and distance scaling within each broad tectonic category is often assumed to be
the same. That is, we assume that the median and standard deviation developed for the broad
tectonic category is applicable to any specific location within the same broad tectonic category.
This is called the ”ergodic” assumption in which we trade time for space. That is, rather than
waiting a long time to get ground-motion recordings in small region near the site, we combine the
ground-motion data sets observed over a large region over a short time period.
Over the last decade, there has been a large increase in the number of ground-motion recordings
from earthquakes. In particular, we now have multiple recordings at specific sites from multiple
earthquakes in a specific region. With these large data sets, it has become clear that there are
significant systematic differences in the ground-motion scaling for different regions within each
broad tectonic category. That is, our ergodic assumption is not working well. While ergodic
ground-motion models are still commonly used in practice, recent ground-motion models have
begun to include region-specific aspects in the models. This move to nonergodic GMMs will be
discussed in detail in Chapter ??. In this chapter, we only address the global (ergodic) scaling of
ground motions. The ergodic ground-motion models will still be used to constrain basic scaling of
the regionalized models other than the region-dependent site and path effects.
Example of the differences in the median ground motions for the three main tectonic regions
are shown in Figure 7.1. For the SCRs, there is a large increase in the short-period ground motion
due to reduced material damping in the shallow crustal material. Intra-slab events are rich in
short period energy, partly related to the ray paths that are from deeper depths. In terms of the
frequency content, the crustal earthquake have the strongest long-period content.
Example of the differences in the aleatory variability of the ground motions for the three main
tectonic regions are shown in Figure 7.2. The aleatory variability for subduction earthquakes is
larger than for crustal earthquakes reflecting greater variability in the source and wave-propagation
effects for subduction earthquakes as compared to crustal earthquakes.
7.2
Spring 2023
7.3
Approaches for Developing Models for the Median Ground
Motion
GMMs can be developed using empirical recordings of ground motions from past earthquakes, numerical simulations of ground motions based on seismological theory, or a combination of empirical
and simulated ground motions. In regions with large data sets, empirical models are generally favored, whereas, in regions with sparse data sets, numerical simulations are generally favored.
For shallow crustal earthquakes in active regions, the GMMs are primarily empirically-based
models because there is sufficient data to constrain the overall magnitude, distance, and site
scaling in the GMMs. Often there is some use of numerical simulation methods to help constrain
the scaling for large magnitudes (M > 7) and short distances (R < 15km).
For stable continental regions (SCRs), the GMMs are primarily simulation based due to the
sparse data: stable regions have few earthquakes so there are few recorded ground motions available. The point-source model (section 11.1) is the simplest and most commonly used numerical
simulation method for developing GMMs for SCRs. Recently, there has been a large increase in
seismicity from induced earthquakes in the central and eastern U.S. due to fracking and wastewater injection. As a result, the empirical data sets in this stable continental region have grown
significantly in the last decade, and they are being used to develop empirically-based GMMs for
the stable regions for moderate magnitudes (M < 6) at moderate and large distances (R > 15km).
The extrapolation of the GMMs to larger magnitudes (M > 6) still requires constraints from
numerical simulations or from seismological theory.
For subduction zones, the GMMs are primarily empirically-based models due to the high rate
of activity of subduction zones. Subduction zones generate most of the large earthquakes in the
world, but often they are recorded at large distances because the trench is located offshore and
the slab is at depth under the land. Numerical simulations are also used to constrain the scaling
for very large magnitudes (M8 to M9) and for short distances (R < 50km).
7.4
Empirical GMMs
Although the name ”empirical” GMM implies a fit to observed ground motions, empirically-based
GMMs are not simply statistical fits to the observed data. The development of empirically-based
GMMs is a model building process with an emphasis on how the model extrapolates outside
the empirical data range to the design scenarios. The key issue is that most of the data are
from moderate magnitude earthquakes recorded at moderate to large distances, but the scenarios
that control the design ground motions in active regions, such as California, are typically large
magnitude earthquakes at short distances. To constrain the extrapolation for the empirical models,
theoretical seismological and geotechnical methods are used.
A commonly used form of base magnitude, distance, and site scaling in empirical GMMs is
shown below:
ˆ
ln(SA(T
)) =c1 (T ) + c2 (T )M + c3 (T )M 2 + (c4 (T ) + c5 (T )M )ln (R + c6 (T )fF F (M, T ))) + c7 (T )R
+ c8 (T )FRV + c9 (T )FN M L + c10 (T )ZT OR + C11 (T )ln(VS30 )
(7.4)
7.3
Spring 2023
where M is the moment magnitude, R is a measure of the closest distance from the rupture to
the site, ZT OR is the depth the top of rupture, and VS30 is the site class in terms of the shearwave velocity over the top 30m. The FRV and FN M L are dummy variables for reverse and normal
faulting earthquakes (e.g. FRV = 1 for reverse and 0 otherwise). The c6 (T )fF F (M, T ) term is
a finite-fault term that accounts for the slip being distributed over the rupture dimension rather
than being concentrated in a point source by increasing the distance term. This adjusted distance,
R + c6 (T )fF F (M, T ), is sometimes called the ”effective distance”. In other words, we account for
the finite dimension of the rupture by entering an adjusted distance into the point-source form of
the model.
The coefficients in eq 7.4 may be period dependent. To simplify the notation, the period
dependence of the coefficients is not shown in the following discussion, but it is implied.
The basis for the functional form given in eq 7.4 is in the scaling of the simplest theoretical
seismological model that we have for ground motion, called the omega-squared point-source model.
The point-source model is described in Chapter 11.
7.5
Ground-Motion Model Parameters
The primary model parameters used in GMMs are the magnitude, distance, site condition, and
style-of-faulting. As the ground motion data sets increase, the dependence of the ground motion on
additional parameters can be estimated. The recent NGA-W2 models have included the following
additional model parameters: hanging-wall/footwall indicator (sign of Rx ), depth to top of rupture
(ZT OR ), fault dip, and depth of the soil. These parameters are explained in this section.
7.5.1
Magnitude
Section 6.3 discussed the alternative magnitude measures that are used in seismology. Since the
mid 1990s, moment magnitude has been used in nearly all GMMs. In these notes, we will only
consider moment magnitude. That requires that the source model developed in Chapter ?? should
be based on moment magnitude.
7.5.2
Distance
A single, consistent definition of the site-to-source distance has not been widely adopted by different
authors of ground-motion relations. In applications of the GMMs, the differences in distance
definitions are sometimes ignored. For example, it is common to see plots comparing the distance
scaling between different GMMs using a generic ”distance” on the x-axis. It is important to use
the appropriate distance measure with each GMM, particularly for short distances. Commonly
used distance measures in GMMs are listed in Table 7.1
The first two distances (Rhypo and Repic ) are point-source measures. The next two distances
measures (RJB and RRU P ) are extended versions of the two point-source measures to include the
finite dimension of the rupture. They are both measures of the distance from the closest point on
the rupture to the site. For large magnitude earthquakes (M > 6), the closest-distance metrics
are generally preferred over the point-source metrics. As the rupture dimension goes to zero,
RRB = Repi and RRup = Rhypo . Most modern GMMs use either RRB or RRU P .
7.4
Spring 2023
The distance metrics are attempting to capture the effect of the finite-rupture geometry. This
does not always work for sites located close to large earthquakes. For those cases, additional
distances metrics have been added that provide an improved description of the location of the site
relative to the source geometry for finite ruptures.
The Rx distance was introduced as part of the hanging-wall (HW) scaling. It indicates if the
site is on the HW or FW side of a dipping fault and, for sites on the HW side, it indicates where
the site is located over the rupture. Rx is positive for sites on the HW side of the rupture and is
negative for sites on the FW side of the rupture.
The Ry distance was introduced as part of directivity scaling. It indicates if the site is located
near the center of the rupture (Ry = 0) or near the end of the rupture. Often, the Ry value is listed
as positive because we are concerned with the site location relative to the center of the rupture,
regardless of the direction.
The Ry0 distance was introduced as part of HW scaling. It is the distance off the end of the
rupture. This is used to taper the HW scaling as the site moves further off the end of the rupture.
For sites located along the rupture (not off the end), then Ry0 = 0.
The reason for including so many distance metrics is to define the location of the site relative to
the location of the finite rupture in space. Using only one distance metric will not provide enough
information to define the source/site geometry for finite ruptures.
7.5.3
Style-of-faulting Parameter
There are five general styles-of-faulting: reverse, reverse/oblique, strike-slip, normal/oblique, and
normal. These are typically defined by the rake of the earthquake: rake angles between 60-120 are
classified as reverse; rake angles between -60 and -120 are classified as normal; rake angles between
-30-30 and between 150-210 are classified as strike-slip. Other rake are classified as normal/oblique
or reverse/oblique.
The style-of-faulting parameter is a proxy for the earthquake stress drop. That is, there is
a correlation between the style of faulting and the median stress drop. The style-of-faulting
parameter is used instead of stress drop because it is more readily available and is used in the SSC.
7.5.4
Hanging-wall / Footwall Parameter
Dipping faults have a hanging-wall side and a footwall side as shown in Figure 7.3. ”Hanging wall”
and ”footwall” are mining terms. If you were digging a mine down a fault, you would hang your
lantern on the hanging wall and walk on the foot wall. If yiou keep this in mind, you will not mix
up which is the HW side and which is the FW site.
Sometimes the terms upthrown block and downthrown block are used interchangeably for
hanging wall and footwall. This works for reverse faults, but it is not correct for normal faults.
For a normal fault, the hanging wall side is the downthrown side. Don’t use the upthrown and
downthrown blocks to decide on HW or FW classification.
For earthquakes that rupture to the surface, the boundary between the hanging wall and
footwall is clear; however, for ruptures that do not reach the surface, the separation is less obvious.
For most GMMs, the definition used for the separation between hanging wall and footwall for
buried ruptures is given by the vertical projection of the top of the fault and not by the projection
of the fault updip to the ground surface. This is shown in Figure 7.3 So, for the input to the
7.5
Spring 2023
GMM, the classification of HW and FW sites from buried ruptures can be inconsistent with the
geologic definition of HW and FW.
7.5.5
Depth-to-Top-of-Rupture Parameter
Recent ground-motion models have found that there is a dependence with the depth of the faulting.
Seismological studies of past earthquakes have also found that buried ruptures lead to strong
ground motions compared to ruptures that reach the surface. This effect is included through the
depth-to-top of rupture (called ZT OR ) as shown in Figure 7.3. Note: this parameter is the depth
to top of the rupture for a specific earthquake and not the depth to top of the fault. A fault may
reach the surface, but a the rupture may be buried.
7.5.6
Dip
The fault dip is the angle from the horizontal to the fault plane (see Figure 7.3), so a vertical
fault has a 90 degree dip. The dip is mainly used when computing the various distance metrics,
but recently, some GMMs have included dip as a predictive parameter for the ground motion from
small earthquakes. In such an application, the dip is serving as a proxy for the radiation pattern.
7.5.7
Site Classifications
There are many site classification schemes used in different GMMs for different regions of the
world. The site classification ranges from qualitative descriptions of the near-surface material
to quantitative definitions based on shear-wave velocity or site period. Without consistent site
classifications for the GMMs, it is often difficult to know how to apply the GMMs to a specific
site. Site classifications can vary from country to country because the site classifications used in
global GMMs may not fit into the site classification system used a particular region.
Site classification based on the shear-wave velocity removes a significant source of uncertainty
in the application of the ground motion models that existed when broad categories were used. For
example, many GMMs used in California in the 1990s were based on site categories of rock, soil
and soft-rock. In the 1990’s classification, the “soil” class was typically used for deep soil sites and
thin soil sites were often grouped into the rock category because their response at long periods
was closer to rock than to deep soil. As a result, the “rock” class was a “soft-rock” category that
included thin soil sites; however, the engineers using the results often interpreted “rock” to mean
engineering rock (800-1000 m/s). This categorization often led to misuse in forward application
such that the boundary between ‘soil’ and ‘rock’ became very subjective and inconsistent between
users.
The use of VS30 as a quantitative measure for site classification was promoted by the National
Earthquake Hazard Reduction Program (NEHRP) and the published site classes for the seismic
design provisions for new buildings (Martin and Dobry, 1994). The choice of VS30 to replace the
descriptive geological categorization was based on extensive empirical work done by Borcherdt et
al. (1994; 1979) and Fumal (1985), who showed good correlation of VS30 with local amplification.
The 30 m depth used in VS30 is not based on an optimized depth range for site amplification.
Rather, it was a pragmatic choice based on the depths ranges of available profiles in the 1980s:
most of the available profiles went to at least 30 m. Because of this initial selection, the 30 m depth
7.6
Spring 2023
range has become a standard. An unintended result is that new VS measurements are often only
collected down to 30 m because the GMM use VS30 as the input parameter. The top 30 m is not
the best depth range to use for characterizing the site VS profile from a site response perspective,
but the VS30 has taken on a life of its own.
As the VS profile was measured at more sites, we found that the generic “rock” category used
in the GMMs in the 1980s and 1990s corresponded to a VS30 of about 550 m/s, which was much
softer than typically assumed. Idriss and Silva (1999) evaluated suites of the recorded generic
“rock” ground motions with measured shear-wave velocity profiles to determine the effect of the
weathered rock and thin soil layers. They found that at high frequencies, the typical “rock” ground
motions are 20 to 30 percent larger than true outcrop ground motions. It is common practice to
ignore this difference between “rock” ground motions from ground-motion models and outcrop
ground motions. As a result, the effects of the weathered rock/thin soil layers are often double
counted in site response calculations.
The VS30 is not a simple average of the VS in the top 30 m. Instead, it is defined as the
time-averaged shear-wave velocity measured over the top 30 m and is given by:
30m
VS30 = PN
i=1 ti
(7.5)
where ti is the travel time through the ith layer, given by
ti =
Hi
V si
(7.6)
and Hi is the thickness of the ith layer.
While VS30 has been widely used for classifying sites for GMMs, the VS30 itself is not a fundamental parameter of site amplification. The site amplification depends on much more than the
VS30 value, but empirically, we have found that the VS30 is correlated with the deeper part of the
VS profile that does control the site amplification. The VS30 value should be thought of as simply
an index for the full VS (z) profile.
To see how well VS30 works as a proxy for the full VS profile for site amplification, Figures
7.6 and 7.7 show the residuals of a ground-motion model with discrete site terms for each VS30
range. The residuals are normalized to the amplification for VS30 = 450 m/s. These figures show
that there is a strong correlation between the site amplification and the VS30 . In particular, the
correlation is stronger for the longer periods (the slope of the residuals is stronger at T=3 sec than
at T=0.1 sec), even though the VS in the top 30 m only controls the short-period amplification.
Because the long-period amplification will depend on the deeper part of the profile, the stronger
correlation at long periods implies that the VS30 is correlated with the deep VS profile.
In most ground-motions models, a linear relation is used between ln(VS30 ) and ln(amp) for the
low strain (linear range) of the input motion. The key conclusion from Figures 7.6 and 7.7 is that
V30) can capture the site amplification for some data sets.
The correlation between VS30 and the VS profile is expected to vary from region to region,
depending on the geologic history of the site. Figures 7.8 and 7.9 show the VS profiles for sites in
California and Japan with similar VS30 values. The Japan profiles have steeper velocity gradients
than the California profiles. Therefore, we should expect that the scaling relation between VS30
and site amplification will be different for Japan and California.
7.7
Spring 2023
A key benefit of using VS30 scaling in ground-motion models is that it leads to a clear definition
of the site class and is a less ambiguous hand off between the hazard analysis and the site response
for defining the input motions as compared to previous generic “rock” and generic “soil” categories;
however, in addition to the VS30 value, the VS profiles that go with the specific VS30 should also
be provided. That is, the VS30 scaling in the GMM has an implied VS profile, such as those shown
in Figure 7.8. Currently, most GMMs do not provide the VS profile that is implied by the site
categories used (7.8 or other categories). This has lead to a common misunderstanding by users
of GMMs: they think that because they enter VS30 into the GMM, it must be a key physical
parameter for the site response, but it is really just an index that points to a VS profile. In the
next few years, we will begin to see VS profiles provided as part of the GMM to make it clear that
it is the VS profile that controls the site amplification and not the VS30 value itself.
Other site parameters, such as the depth to rock and the site period have been used in GMMs.
These parameters provide additional information about the VS profile. A full VS profile for the
GMM site class, as discussed above, would cover these additional site parameters.
A common misuse of the VS30 scaling in GMMs is to use this as a replacement for a site-specific
profile that does not fit the profile used to develop the VS30 scaling in the GMM. If the site-specific
VS profile does not match the profile implicit in the GMM, a site response analysis should be
performed.
7.5.8
Other Site Parameters
The VS30 site parameter is a single parameter that is commonly used to classify sites. As mentioned
earlier, it is just a proxy for the entire VS profile. It does a reasonable job of distinguishing the
site amplification between different velocity profiles, but one limitation of this parameter is that
it does not distinguish between shallow soil sites and deep soil sites. To address this limitation, a
second site parameter is being introduced to some ground-motion models: the depth to a specified
VS value. In the NGA models, two depths are considered in the various models: the depth to
VS = 1 km/s (Z1.0 ) and the depth to VS = 2.5 km/s (Z2.5 ). Other site parameters have been
studied, such as site period. So far, there has not been a single site parameter that works better
than VS30 , but including parameters in addition to VS30 can improve the model. These additional
parameters help to better define the full VS profile.
7.6
Finite-Fault Effects and Saturation
In GMMs, there is a concept of magnitude saturation at short distances. Most GMMs use some
measure of the shortest distance from the site to the rupture. This is like assuming the the
earthquake is a point source at this closest location. For large magnitudes, we know that the
rupture has a finite dimension (Area(M) model) so that the energy is not all concentrated at
the closest point. To address this issue, GMMs use an effective distance rather than the closest
distance. In eq 7.4, the effective distance is the R + C6 f1 (M ) term. In the point-source model,
the effective distance is used as the point-source distance to make the point-source model work for
large magnitudes. While large earthquakes are not point sources, we can make the point-source
model produce reasonable results for large magnitude earthquakes by using an effective distance.
At sites very close to large magnitude earthquakes, the median ground motion tends to reach
7.8
Spring 2023
a limit, called magnitude saturation. The idea is that once the maximum rupture width has been
reached, the larger magnitude earthquakes become longer, but the additional length is far from
the site (see Figure 7.10). Full magnitude saturation implies that there is no magnitude scaling for
short-period ground motions at RRU P = 0 distance for magnitudes above some threshold (typically
M6.5 for crustal earthquakes). The magnitude saturation can be imposed on the GMM through
either the c6 T (M ) term or through the magnitude dependence of the geometrical spreading term:
c5 M . The conditions for full saturation using the magnitude dependent geometrical spreading is
shown below. First, for short periods, there is no need for the quadratic magnitude scaling term
(we are above the corner frequency). So setting c3 = 0, T (M ) = 1, and RRU P = 0, then the
magnitude terms in eq 7.4 reduce to M (c4 ln(c6 ) + c2 )). To have no magnitude scaling, the terms
multiplied by M must be zero, so c4 = −c2 /ln(c6 ). This condition is called full saturation. An
example of this type of saturation for M > 6.5 is shown in Figure 7.11. A similar constraint can
be developed using c5 = 0 and T (M ) = a1 exp(a2 M ).
7.9
Spring 2023
Notation
Repi
Rhypo
RJB
RRU P
RX
RY
RY 0
Table 7.1: Commonly used distance metrics for GMMs
Distance metric
Epicentral distance (horizontal distance)
Hypocentral distance (slant distance)
Closest horizontal distance to the vertical projection of the rupture
(called the “Joyner-Boore” distance)
Closest distance to the rupture surface (slant distance)
Horizontal distance to the top edge of the rupture measured
perpendicular to the fault strike
Horizontal distance from the center of the top edge of the
rupture measured parallel to the fault strike
Horizontal distance from the closest end of the top edge of the rupture
measured parallel to the fault strike
7.10
Spring 2023
Figure 7.1: Example of spectral shapes for different tectonic regions.
Figure 7.2: Example of aleatory variability for different tectonic regions.
7.11
Spring 2023
FW side
HW side
RJB
RJB=0
0
Depth Z(km)
RRUP
ZTOR
RRUP
5
DIP
10
15
-20
-15
-10
-5
0
Rx (km)
5
10
15
20
Figure 7.3: Example of distance, dip, and HW definitions. The blue line is the rupture. The Rx
is measured from the vertical projection of the top of the rupture. The R − y is measured into the
page.
7.12
Spring 2023
60
40
0
20
-40
0
-20
-20
-40
-60
20
-80
0
-100
-20
-10
0
10
20
30
40
50
Figure 7.4: Example of Rx computed for a bending fault using GC2.
7.13
Spring 2023
Figure 7.5: Example of Rx computed using GC2 for fault with changes in strike over short distances.
7.14
Spring 2023
!
Figure 7.6: Example of correlation of log VS30 and the site amplification for short and moderate
period response spectral values (PGA, T=0.1s, T=0.2s, and T=0.5s)
7.15
Spring 2023
!
Figure 7.7: Example of correlation of log VS30 and the site amplification for long-period response
spectral values (T=1s, T=2s, T=3s, and T=5s)
7.16
Spring 2023
!
Figure 7.8: Example of range of VS profiles for sites in California with VS30 close to 270 m/s.
(From Kamai et al, 2015)
7.17
Spring 2023
!
Figure 7.9: Example of range of VS profiles for sites in Japan with VS30 close to 270 m/s. (From
Kamai et al, 2015)
7.18
Spring 2023
!
Figure 7.10: For sites located close to a large rupture, only the part of the rupture close to the site
(within about 2 rupture widths) contributes to the peak amplitude of the ground motion. (From
Baltay and Hank, 2014)
7.19
Spring 2023
!
Figure 7.11: Example of saturation in GMMs. This example is for the Campbell and Bozorgnia
(2008) GMM for peak acceleration on a soft-rock site condition.
7.20
Spring 2023