Download Probabilistic seismic hazard analysis

MASTERS IN EARTHQUAKE ENGINEERING
AND ENGINEERING SEISMOLOGY
UNIVERSITY OF PATRAS, GREECE
PROBABILISTIC SEISMIC HAZARD ANALYSIS –
AN INTRODUCTION TO THEORETICAL BASIS AND
APPLIED METHODOLOGY
A Dissertation Submitted in Partial
Fulfillment of the Requirements for the Master Degree in
EARTHQUAKE ENGINEERING
by
JESSICA GODINHO
Supervisor: Dr APOSTOLOS PAPAGEORGIOU
January, 2007
The dissertation entitled “Probabilistic seismic hazard analysis – an introduction to theoretical
basis and applied methodology”, by Jessica Godinho, has been approved in partial fulfilment
of the requirements for the Master Degree in Earthquake Engineering.
Dr. Apostolos Papageorgiou …… …
………
i
ABSTRACT
Probabilistic seismic hazard analysis (PSHA) is a method used to evaluate seismic hazard by
computing the probability of a specified level of ground motion being exceeded at a site or area
of interest. This article presents the basic methodologies of PSHA in an attempt to provide a
clear and concise introduction into the theoretical basis and application of PSHA in engineering
practice today. Various models commonly applied in PSHA are introduced and discussed,
including methods for describing source geometry, source seismicity, and ground motion
attenuation. Common assumptions made in PSHA, specifically for magnitude and temporal
distributions, are also presented and analyzed, providing insight into the validity of each
assumption for different source and site scenarios. Additionally, the mathematical framework for
hazard computations is presented in a closed analytical form providing an efficient tool for
engineers and decision-makers to easily extract information for assessing seismic risk.
i
TABLE OF CONTENTS
Page
ABSTRACT……………………………………………………………………………………..i
TABLE OF CONTENTS...........................................................................................................................ii
LIST OF FIGURES…………………………………………………………………………….v
LIST OF TABLES……………...………………………………………………………...……vii
1. INTRODUCTION……......…………………………………………………………………1
1.1 Deterministic vs. Probabilistic………………………………………………………1
1.2 Organization………...…....…………………………………………………………1
2. SOURCE GEOMETRY…………………………………………………………………….2
2.1 Types of Sources……………………………………………………………………2
2.1.1 Areal Sources….…..................………………………………………………...2
2.1.2 Fault Sources……………………………....…………………………………2
2.2 Estimating Fault Dimensions………………………...……………………………...2
2.3 Spatial Uncertainty………………………………………………………………….3
3. SOURCE SEISMICITY.........................................................................................................................3
3.1 Magnitude Scales......................................................................................................................3
3.1.1 Surface, Body & Local Magnitude Scales..................................................................4
3.1.2 Moment Magnitude......................................................................................................4
3.2 Magnitude Recurrence Relations...........................................................................................5
3.2.1 Magnitude Distributions..............................................................................................5
a) Truncated Exponential Model...............................................................................5
b) Characteristic Earthquake Model..........................................................................6
c) Composite Model.....................................................................................................7
3.3 Empirical Scaling Relations.....................................................................................................8
3.4 Activity Rates............................................................................................................................9
3.4.1 Historical Seismicity.....................................................................................................9
3.4.2 Geological Information............................................................................................10
4. EARTHQUAKE OCCURRENCES WITH TIME........................................................................11
4.1 Memory-less Model...............................................................................................................12
4.2 Models w/ Memory...............................................................................................................15
ii
4.2.1 Renewal Models..........................................................................................................15
a) Lognormal...............................................................................................................15
b) Brownian Passage Time........................................................................................17
c) Weibull & Gamma................................................................................................20
4.2.2 Markov & Semi-Markov Models..............................................................................22
4.2.3 Slip Predictable Models..............................................................................................23
4.2.4 Time Predictable Models...........................................................................................25
4.2.5 Combined Slip & Time Predictable Models...........................................................26
5. ESTIMATING GROUND MOTIONS...........................................................................................26
5.1 Ground Motion Parameters.................................................................................................26
5.1.1 Amplitude.....................................................................................................................26
5.1.2 Frequency Content.....................................................................................................27
5.1.3 Duration.......................................................................................................................28
5.2 Empirical Ground Motion Equations.................................................................................28
5.2.1 Factors Influencing Attenuation...............................................................................31
a) Tectonic Regime...................................................................................................32
b) Focal Mechanism..................................................................................................33
c) Hanging Wall.........................................................................................................33
d) Site Conditions......................................................................................................34
e) Near-Fault..............................................................................................................35
5.3 Stochastic Ground Motion Methods..................................................................................35
5.3.1 Ground Motion Spectra.............................................................................................35
a) Source Effects.......................................................................................................36
b) Path Effects............................................................................................................37
c) Site Effects.............................................................................................................39
d) Type of Ground Motion.....................................................................................39
5.3.2 Obtaining Ground Motions......................................................................................40
a) Simulating Time Series.........................................................................................40
b) Response Spectra..................................................................................................41
6. Hazard Curves........................................................................................................................................41
6.1 Deaggregation of Hazard......................................................................................................43
iii
7. Uncertainty..............................................................................................................................................44
7.1 Aleatory Variability................................................................................................................44
7.2 Epistemic Uncertainty...........................................................................................................45
7.2.1 Logic Trees...................................................................................................................45
8. Conclusions.............................................................................................................................................46
iv
LIST OF FIGURES
Page
Figure 3-1. Magnitude probability density function of truncated exponential model........................6
Figure 3-2. Magnitude probability density function of truncated normal model................................7
Figure 3-3. Magnitude probability density function of composite characteristic and exponential
model.................................................................................................................................................8
Figure 4-1. Probability density function of earthquake occurrence for exponential distribution
model...............................................................................................................................................13
Figure 4-2. Hazard rate of earthquake occurrence for exponential distribution model (Poisson
assumption).....................................................................................................................................14
Figure 4-3. Probability density function of earthquake occurrence for lognormal distribution
model...............................................................................................................................................16
Figure 4-4. Hazard rate of earthquake occurrence for lognormal distribution model.....................17
Figure 4-5. Probability density function of earthquake occurrence for BPT distribution
model...............................................................................................................................................18
Figure 4-6. Hazard rate of earthquake occurrence for BPT distribution model...............................19
Figure 4-7. Example of load state paths for a Brownian relaxation oscillator...................................19
Figure 4-8. Probability density function of earthquake occurrence for Weibull distribution
model...............................................................................................................................................20
Figure 4-9. Probability density function of earthquake occurrence for Gamma distribution
model...............................................................................................................................................21
Figure 4-10. Hazard rate of earthquake occurrence for Weibull distribution model........................21
Figure 4-11. Hazard rate of earthquake occurrence for Gamma distribution model.......................22
Figure 4-12. Schematic representation of the trajectory of a semi-Markov process.........................22
Figure 4-13. Slip predictable model: (a) Time history of stress accumulation and release;
(b) Relationship between coseismic slip and time between seismic events; and (c) Sample
path for the corresponding Markov renewal process..............................................................24
Figure 4-13.Comparison between Poisson model and slip-predictable model for probabilities of
at least one event M>7.5and M>8.0 in the next 100 yrs as a function of the gap t1..........24
Figure 4-14. Time predictable model. (a) Time history of stress accumulation and release;
(b) Relationship between coseismic slip and time between seismic events; (c) Sample path
v
for the corresponding Markov renewal process.......................................................................25
Figure 5-1. Example of response spectrum.............................................................................................28
Figure 5-2. Source-to-site distance measures for ground motion attenuation models for (a)
vertical faults, and (b) dipping faults...........................................................................................31
Figure 5-3. Attenuation relation for (a) peak spectral acceleration at 0.1 sec and (b) peak ground
acceleration using Campbell & Bozorgnia (2003) model........................................................32
Figure 5-4. Peak spectral acceleration (5% damping) using Campbell & Bozorgnia (2003) ground
motion attenuation model showing effects of faulting mechanism. Evaluated using
M=7.0, rsesis=10km, firm soil........................................................................................................33
Figure 5-5. Peak spectral acceleration (5% damping) using Campbell & Bozorgnia (2003) ground
motion attenuation model showing effects of site conditions. Evaluated using M=7.0,
rsesis=10km, strike-slip fault...........................................................................................................34
Figure 5-6. Ground motion amplitude spectra using models in Table 5-1 for M=7.5 &
M=4.5..............................................................................................................................................36
Figure 5-7. Observed attenuation of ground motions with distance in eastern North America and
effect of geometrical spreading & whole path attenuation using model applied by
Atkinson & Boore (1995).............................................................................................................38
Figure 5-8. Observed duration from earthquakes in eastern North America and duration function
applied by Atkinson & Boore (1995) in ground motion simulation model for eastern
North America...............................................................................................................................38
Figure 5-9. Combined effect of site amplification and path-independent diminution.....................39
Figure 5-10. Simulation of a time series using stochastic methods.....................................................40
Figure 6-1. Example of individual source hazard curves for spectral period of 2 sec......................43
Figure 6-2. Example of deaggregation of hazard for spectral period of 2 sec at ground motion
level of 0.5g.....................................................................................................................................44
Figure 7-1. Example of a logic tree used in a PSHA............................................................................45
vi
LIST OF TABLES
Page
Table 3-1. Examples of magnitude-area scaling relations.......................................................................8
Table 5-1. Ground motion amplitude spectra relations: (a) Spectra shape factor relations for
various models and (b) corresponding corner frequencies and moment ratios...................29
Table 5-2. (a) Spectra shape factor relations for various models and (b) corresponding corner
frequencies and moment ratios....................................................................................................37
vii
1
INTRODUCTION
The assessment of risk due to an earthquake can be characterized as consisting of three
components: 1) the event (how, when, where), 2) the resulting ground motion (amplitude,
duration, frequency), and 3) the effect on the structure (e.g. forces, deformations). First
introduced by Cornell in 1968, probabilistic seismic hazard analysis (PSHA) involves the first two
components, providing a method to quantitatively represent the relationship between potential
seismic sources, associated ground motion parameters, and respective probabilities of occurrence.
Combining concepts of probability theory with seismological models of source and ground
motion characteristics, PSHA computes how often a specified level of ground motion will be
exceeded at the site of interest. Presenting information in a closed analytical form (typically as an
annual rate of exceedance or return period), PSHA provides seismic hazard computations that
can easily be transformed into evaluations of seismic risk. This provides engineers and decisionmakers a useful tool in assessing the relationship between seismic resistance and potential loss in
a structure. Additionally, PSHA allows for a clearer understanding of seismic hazard itself, in
particular, insight into the relationship between different source and site characteristics and
resulting ground motion parameters, an understanding that plays a pivotal role in determining
appropriate design ground motions.
1.1
DETERMINISTIC VS. PROBABILISTIC
Prior to PSHA, most hazard assessments were completed using a deterministic approach,
considering individual scenarios of magnitude and location for each source, often with the
“worst-case” scenario, or largest magnitude/closest source-to-site distance used to evaluate the
design ground motion (Abrahamson, 2006). PSHA can be viewed as the assessment of an infinite
number of deterministic hazard analyses, with the hazard being integrated over all potential
earthquake sources for all possible scenarios of magnitude and distance. Further, by assigning
probability distributions to source and ground motion characteristics, a reasonable ground
motion, at some accepted level of probability of occurrence, can be chosen for design. This
allows for a more intelligent and economic design in comparison with the often overly
conservative, deterministic, “worst-case” scenario approach.
1.2
ORGANIZATION
The remainder of this article presents the basic steps in computing seismic hazard using PSHA.
Section 2 discusses concepts of source identification and spatial characterization. Section 3 begins
by introducing different methods of magnitude scaling and continues with a discussion of
earthquake recurrence relations and the definition of magnitude distribution functions and source
activity rates. Two approaches to modeling of earthquake occurrence with time are presented in
1
Section 4, with each assumption defining the earthquake process as either having a “memory” or
“memory-less” nature. Section 5 describes important ground motion parameters and introduces
both empirical and stochastic methods for generating ground motion estimates. Section 6
presents the final mathematical framework of the hazard computation and the introduction of
hazard curves. Further, the process of dividing the hazard into its relative contributions of
magnitude and distance, known as deaggregation of hazard, is discussed. Lastly, Section 7 deals
with uncertainty, identifying the two types present in hazard computations and introducing
appropriate methods for handling each within the PSHA.
2
SOURCE GEOMETRY
The characterization of seismic sources describes the rate at which earthquakes of given
magnitude and dimensions occur at a given location. The first step of the source characterization
involves identifying potential sources and modeling their geometric parameters. This includes
defining the source type, estimating the source dimensions, and finally assigning a distribution to
the uncertainty in earthquake location within the source.
2.1
TYPES OF SOURCES
2.1.1
Areal Sources
Due to insufficient geological data of known faults, seismic sources were initially modeled as areal
source zones based on historical seismicity data. Generally, these seismic zones were assumed to
have uniform source properties in both time and space. Although today most hazard analyses are
completed using a fault source characterization, areal sources are still used to model seismicity in
regions with unknown fault locations. In addition, areal sources can be used to model
“background zones” of seismic regions to account for any earthquakes that may occur off
identified faults (Abrahamson, 2006).
2.1.2
Fault Sources
As more geological data became available, locations of faults were able to be identified and more
accurately defined. Although originally only modeled as linear sources, most fault source models
now have multi-planar features and ruptures are assumed to be distributed over the entire fault
plane (Abrahamson, 2006).
2.2
ESTIMATING RUPTURE DIMENSIONS
Estimates of fault rupture dimensions typically involve one of two methods, estimating the
dimensions based directly on the size of the fault rupture plane or by basing the estimate on the
size of the aftershock zone (Wells & Coppersmith, 1994; Henry & Das, 2001). Rupture
dimensions can also be estimated from the corner frequency fc of the source spectra which is
2
obtained from ground motion recordings (Molnar et al, 1973; Beresnev, 2002).
Estimating fault rupture dimensions directly requires the measurement of the length (L) of the
fault expression on the free surface and the estimation of the seismogenic zone (corresponding to
the width, W). This estimation can be difficult, particularly in the case when there is no surface
rupture or the rupture is sub-ocean. Further, it is important when estimating the fault rupture
length to distinguish between primary and secondary source rupture. Primary source rupture is
related directly to the tectonic rupture, that is, the fault rupture plane intersecting the ground
surface. Secondary rupture corresponds to fractures that are formed from causes associated with
the initial rupture such as ground shaking, landslides, or ruptures from earthquakes triggered on
nearby faults (Wells & Coppersmith, 1994).
The second method of estimating fault dimensions determines the subsurface rupture length as
indicated by the spatial pattern of aftershocks. The rupture width can also be determined through
this method or can be estimated as the seismogenic depth as described above. Although studies
have shown that this method is relatively reliable, there are inevitably factors that contribute to its
uncertainty. Studies have shown that the aftershock zone expands as a function of time;
therefore, depending on the time at which the zone is interpreted, the rupture dimensions will
change (Henry & Das, 2001). If the time period after the main event is relatively small,
commonly taken as one day, studies have shown that the aftershock area still provide good
estimates of rupture dimensions (Henry & Das, 2001).
2.3
SPATIAL UNCERTAINTY OF SOURCES
Possible locations of earthquakes are usually assumed to be uniformly distributed along the fault
strike. This assumption has been supported by studies mapping hypocenter locations for strike
and dip-faults (Henry & Das, 2001). Although additional studies have suggested that hypocenters
of large earthquakes associated with subduction faults tend to be located towards the ends of
ruptures, the error in assuming a random and therefore uniformly distributed hypocenter location
is small ( Henry & Das, 2001).
3
SOURCE SEISMICITY
Once the geometry of a seismic source is defined the next step is to estimate the distribution of
all possible size earthquakes that can occur within the source dimensions. This involves defining
a consistent measure of magnitude scale and representing the source seismicity through a
magnitude recurrence relation as defined by a magnitude distribution function and activity rate.
3.1
MAGNITUDE SCALES
When dealing with the definition of seismicity it is important to pay attention to the terminology
3
and parameters used. For instance, there are several different ways to express the magnitude of
an earthquake, the most common using surface, body, local and moment magnitude scales.
3.1.1 Surface, Body, & Local Magnitude Scales
In the past, many earthquake magnitudes were determined using scales based on the
measurement of seismic wave amplitudes at a selected period. Surface wave magnitude, MS, is
measured using the amplitude of surface Raleigh waves at a period of 20 sec (Gutenberg &
Richter, 1936). Body wave magnitude, mb, often used for deep earthquakes in which the surface
waves are too small to measure, is related to the amplitude of the first few cycles of P-waves and
is measured at a period of 1 sec (Gutenberg, 1945). The local magnitude, ML developed by
Richter to measure shallow, local earthquakes in Southern California is also measured at a period
of around 1 sec. Magnitude scales that are measured in this period range are more frequently used
and are often regarded as better measures of seismic damage. This is because most common
structures have a natural period which lies in the neighborhood of 1 sec.
The scales described above are not directly related to physical parameters of the earthquake
source itself but rather are related to its associated ground shaking characteristics. Because
ground shaking characteristics do not increase at the same rate as that of the total energy released
during an earthquake, saturation of the scales occurs for large sized earthquakes. This is indicated
by the magnitude scale becoming less sensitive to the size of the earthquake as the size of the
earthquake increases (around 6-7 for body and local magnitude scales and 8 for surface) (Kramer,
1996).
3.1.3 Moment Magnitude – (Kanamori 1977, Hanks & Kanamori 1979)
An alternative to the scales mentioned above, and the most widely used scale today, is the
moment magnitude, MW. The moment magnitude, as defined by Hanks & Kanamor (1979), is
related to the total amount of energy released during an earthquake, as expressed through the
seismic moment, M0 (Aki, 1966) which is the most fundamental physical parameter of a seismic
source that expresses the size of an earthquake.
2
M = log( M 0 ) − 10.7
3
(3-1)
The seismic moment, being the product of the rigidity μ of the earth, the area A of the surface
that slips and the average slip u can be related to the elastic strain energy that is released by the
earthquake source / fault. Specifically,
M 0 = μ Au
where
(3-2)
μ = Shear modulus of crust (3 x 1011 dyne / cm)
A = Area of fault rupture
4
u = Average displacement (slip) over rupture surface
By using relations (3-1) and (3-2) above, the moment magnitude can be expressed directly as a
function of the source’s physical parameters (Hanks & Kanamori, 1979):
2
2
2
M W = log( A) + log(u ) + log( μ ) − 10.7
3
3
3
(3-3)
3.2 MAGNITUDE RECURRENCE RELATIONS
Once the geometry of the source is defined and an appropriate magnitude scale chosen, the
seismicity of a source can be expressed through a magnitude recurrence relation. Describing the
average rate at which earthquakes with magnitudes greater than or equal to a specified magnitude,
M, occur on a source, magnitude recurrence relations are characterized by a source’s activity rate
and magnitude distribution function. As shown below, a recurrence relation is computed by
integrating the magnitude distribution density function and scaling by the activity rate:
λM = vM
M max
min
∫
f m (m)dm
(3-4)
m= M
3.2.1 Magnitude Distribution
Randomness in the number of relative number of large, moderate and small magnitude
earthquakes that will occur on a given source can be defined through a probability density
function. Two types of models are typically used to represent magnitude distributions, the
truncated exponential model & characteristic earthquake model. Although the exponential model
works well for large regions in which the hazard is not controlled by one particular fault, studies
have shown that the characteristic model is more appropriate for characterizing individual fault
sources (Youngs & Coppersmith, 2000). Some models take advantage of a combined magnitude
distribution, using the truncated exponential model for the distribution of small-to-moderate
earthquakes and the characteristic model for large magnitude events. Differences in resulting
hazard between these two models is a function of fault-to-site distance and acceleration level and
therefore is specific to each analysis (Youngs & Coppersmith, 2000).
3.2.1.1 Truncated Exponential Model
The truncated exponential model is based on Gutenberg-Richter magnitude recurrence relation
(Gutenberg-Richter, 1956). The Gutenberg-Richter recurrence relation is expressed as:
log λm = a − bm
(3-5)
The a-value represents the activity rate of the source which represents the absolute rate of
earthquake occurrence with magnitudes greater than zero. Based on empirical estimates, the bvalue is related to the relative likelihood of earthquakes with different magnitudes and typically
takes a value between 0.8-1.0. The truncated exponential model is often rewritten in the following
5
form of a standard recurrence relation:
λm = ν 0 ⋅ exp(− β m)
where
(3-6)
ν 0 = 10a and
β = b ln(10) 2.3b
As can be seen, the earthquake magnitudes are exponentially distributed. This implies that the
mean recurrence rate for small magnitude earthquakes is greater than that of large magnitude
earthquakes.
Although the standard Gutenberg-Richter recurrence relation can be applied to an infinite range
of magnitudes, it is common to apply bounds at minimum and maximum magnitude values. This
is because seismic sources are usually associated with a capacity to produce some maximum
magnitude Mmax and for engineering purposes earthquakes of very small magnitudes that do not
cause damage to structures are not of interest (Abrahamson, 2006). The methods developed for
determining the values of the minimum and maximum magnitude values are discussed in the next
section. The corresponding probability density function using these minimum and maximum
values is expressed below in its bounded form and is displayed in Figure 3-1:
f m ( m) =
β ⋅ e− β ( m− M
min )
1 − e − β ( M max − M min )
(3-7)
2
1.8
Probability Density Function, fm(m)
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
5
5.5
6
6.5
7
Magnitude
7.5
8
8.5
9
Figure 3-1 – Magnitude probability distribution function of truncated exponential model
3.2.1 Characteristic Earthquake Models
First developed by Schwartz and Coppersmith (1985), characteristic earthquake models are based
6
on the assumption that individual faults tend to generate same size (within ½ magnitude unit of
each other), or “characteristic” earthquakes. Prior to the 1980’s the magnitude associated with the
characteristic earthquake was based on the assumption that some fraction of the total fault length
would rupture (e.g. ¼-½) (Abrahamson, 2006). Now, however, the fault is typically defined into
segments that serve as boundaries of rupture dimensions. This allows for the consideration of
single segment ruptures in addition to multi-segment ruptures, also known as “cascading”
(Abrahamson, 2006).
One form of the characteristic earthquake model, also known as “maximum magnitude” model,
does not account for small-to-moderate sized earthquake occurrences along the fault. This is
based on the assumption that all seismic energy is released through characteristic earthquakes
(Abrahamson, 2006). Expressed as a truncated normal distribution, this model allows for a
narrow range of magnitudes for the characteristic earthquake and is given by the magnitude
density function shown in Figure 3-2:
5
4.5
Probability Density Function, fm(m)
4
3.5
3
2.5
2
1.5
1
0.5
0
6
6.5
7
7.5
Magnitude
8
8.5
9
Figure 3-2 – Magnitude probability density function of truncated normal model
3.2.2 Composite Model
Past studies have used a combination of the truncated exponential and characteristic model to
accommodate the distribution of both small and large magnitude earthquakes (Youngs &
Coppersmith, 1985). This allows for the modeling of characteristic earthquake behavior without
the exclusion of other magnitude events. Characterized by an exponential distribution until some
magnitude, m’ and a uniform distribution of a given width (often 0.5 magnitude units) centered
on the mean characteristic magnitude, the magnitude density function for this model is shown in
Figure 3-3. In addition to limits placed on minimum and maximum magnitudes, this model
7
requires an additional constraint to define the relative amplitudes of the two distributions. Based
on empirical data, this constraint represents the relative amount of seismic moment released
through smaller magnitude events and characteristic earthquakes (Youngs & Coppersmith, 1985).
1
Probability Density Function, fm(m)
10
0
10
-1
10
-2
10
5
5.5
6
6.5
Magnitude
7
7.5
8
Figure 3-3 – Magnitude probability density function of composite characteristic and
exponential model
3.3 EMPIRICAL SCALING RELATIONS
Magnitude distribution models, such as those discussed above, are typically bounded between
some minimum and maximum magnitude values. Minimum magnitudes represent the minimum
level of energy release that is expected to cause damage to structures and are often taken as five
(Abrahamson, 2006). Maximum magnitudes are related to fault dimensions and stress drop, a
parameter describing the distribution of seismic moment release in space and time. Studies have
shown that observed stress drop is relatively constant among different tectonic regions and most
empirical scaling relations are developed based on this assumption, allowing for relations to be
developed that are functions only of rupture area (Kanamori & Anderson, 1975). Examples of
scaling relations between rupture dimensions and magnitude for various models are shown in
Table 3-1.
Wells and Coppersmith (1994)
All fault types
Wells and Coppersmith (1994)
Strike-slip
M = 0.98 log(A) + 4.01
M = 1.02 log(A) + 3.98
8
Wells and Coppersmith (1994)
Reverse
Ellsworth (2001)
Strike-slip for A>500 km2
Somerville et al (1999)
All fault types
M = 0.90 log(A) + 4.33
M = log(A) + 4.2
M = log(A) + 3.95
Table 3-1 – Examples of magnitude-area scaling relations
3.4 ACTIVITY RATES
While magnitude distribution models provide the relative rate of earthquakes at various
magnitudes, in order to completely represent source seismicity through a recurrence relation the
absolute rate of earthquakes above a minimum magnitude, known as the activity rate, is needed.
There are two approaches to determining the activity rate of a seismic source, either through
historical seismicity or through geological data.
(a) Historical Seismicity
In seismically active regions in which there is significant historical data available it is possible to
estimate activity rates based on information recorded in earthquake catalogs. This method is
primarily used in conjunction with the truncated exponential distribution model to estimate
activity rates for small to moderate earthquake occurrences where most seismicity data is
available. Fitting the exponential distribution model to historical data, seismicity parameters such
as the activity rate and the b-value in Guternberg-Richter’s recurrence relation can be computed
using a regression analysis such as the maximum likelihood method. A detailed description of
how this method is applied in addition to the benefits of the maximum likelihood method over
other regression tools such as the least-squares method can be found in McGwire (2001).
When relying on earthquake catalogs to provide data on earthquake occurrence it is important to
recognize that the accuracy of the estimated activity rate is dependent on the reliability of the
catalog. Therefore, data in historical catalogs must be carefully assessed to be both complete and
appropriate for use. All dependent events, such as aftershocks and foreshocks should not be
considered (Abrahamson, 2006). This is because probability models used in the analysis typically
assume that all events are independent, and including these events would violate that assumption.
Additionally, since all earthquake occurrences may not have been reported in the catalog,
particularly small magnitude events, the completeness of the data used in the analysis must be
assessed. This can be evaluated through methods such as that developed by Stepp (1973) which
examines the stationary nature of the activity rate. If the catalog is assessed as sufficiently
complete the b-value and activity rate can then be calculated using the max likelihood method.
(b) Geological Information (i.e. slip rate)
While the historical data method is appropriate for estimates of activity rates corresponding to
9
the truncated exponential model, using geological information, specifically slip rate, can be used
to estimate activity rates for other earthquake models such as the characteristic earthquake model
(Youngs & Coppersmith, 1985). This method has the benefit of being applicable to seismic areas
where there is little recorded data on earthquake occurrence and further provides recurrence
information that spans several seismic cycles of large magnitude earthquakes, allowing for a
better estimate of the average earthquake frequency (Youngs & Coppersmith, 1985).
While the accuracy of the activity rate estimate based on historical data is dependent on the
reliability of the historical catalog, using geological data requires a reliable estimate of the fault
slip rate. In the estimation of this parameter several assumptions are typically made (Youngs &
Coppersmith, 1985):
1) All observed slip is considered as seismic slip, that is, slip as an effect of creep is not
recognized unless explicitly defined.
2) The slip rate represents an average value and therefore short term fluctuations are not
considered. Further, this average value is assumed to be applicable to the future time period of
interest.
3) All surface measurements are assumed to represent slip rates at seismogenic depths and along
the entire length of the fault.
The activity rate is computed by balancing the long term accumulation of seismic moment with
its long term release. Based on Aki’s definition of seismic moment, the rate of moment build up
can be defined as (Aki, 1979):
dM 0
du
= μA
= μ Au
dt
dt
(3-8)
where u = slip rate (cm/yr), A = fault rupture area, and μ = shear modulus.
Using a scaling relation to define the characteristic magnitude of the fault in consideration,
MW = log(A) + 4 (general form)
(3-9)
a moment-magnitude relation can be used to express the amount of moment released by an
individual characteristic.
MW =
2
log10 ( M 0 ) − 10.7
3
log10 ( M 0 ) = 1.5M W + 16.05
(3-10a)
(3-10b)
The total rate of moment release can be expressed as the product of the moment release per
characteristic earthquake and the rate of earthquake occurrence.
10
M0
⋅ vM = M 0Released
EQ
(3-11)
Equating the rate of moment release with the rate of moment build-up allows for the direct
computation of the activity rate.
M 0Release = M 0Build-up
(3-12a)
M0
⋅ vM = μ Au
EQ
(3-12b)
νM =
M0
μ Au
=
M 0 / EQ M 0 / EQ
(3-13)
The above example of the application of this method is expressed in its simplified form assuming
the case in which only one size of earthquake occurs on the fault. A more generalized approach
can be applied using an arbitrary form of the magnitude probability density function and a mean
moment per earthquake (Abrahamson, 2006):
max
⎡M ⎤
Mean ⎢ 0 ⎥ = ∫ 101.5 M +16.05 ⋅ f m ( M )dM
⎣ EQ ⎦ M min
M
(3-14)
where f m ( M ) =Magnitude Distribution PDF
Note that the computed activity rate is now defined as the rate of earthquake occurrence above a
specified minimum magnitude, Mmin and is expressed as
νM
4
min
=
μ Au
Mean [ M 0 / EQ ]
(3-15)
EARTHQUAKE OCCURRENCES WITH TIME
Once the recurrence rate of an earthquake of a given magnitude is computed, the next step is to
convert it into a probability of earthquake occurrence. This requires an assumption regarding
earthquake occurrence with time, specifically whether the process of earthquake occurrence
follows a “memory or memory-less” pattern. In order to assess the appropriateness of either
assumption it is important to understand the physical process of earthquake occurrence which
can be described by the theory of elastic rebound, first introduced by Reid (1911). According to
11
this theory, the occurrence of earthquakes is a product of the successive build-up and release of
strain energy in the rock adjacent to faults. The build-up of strain energy is a result of the relative
movement of the earth’s plates which cause shear stresses to increase on fault planes that serve as
boundaries between the plates. When the shear stresses reach the shear strength of the
surrounding rock, the rock fails and the accumulated strain energy is released. If the adjacent rock
is relatively weak and ductile, the little strain energy that is able to build-up will be slowly released
aseismically. In the case of strong/brittle rock the failure will occur rapidly, creating a sudden
release of energy in the form of an earthquake (Kramer, 1996).
4.1 MEMORY- LESS MODEL
Most probabilistic seismic hazard analyses are based on the assumption that the earthquake
process is memory-less, that is, there is no memory of the time, size and location of preceding
events. This implies that the probability of an earthquake occurring in a given year does not
depend on the elapsed time since the previous earthquake.
This assumption is typically made by defining the occurrence of earthquakes as a Poisson process
characterized by an exponential distribution of earthquake recurrence intervals.
ft (t ) = λM e− λM t
t
t
0
0
Ft (t ) = ∫ ft (t )dt = ∫ λM e − λM t = 1 − e
− λM t
(Probability Density Function)
(4-1)
(Cumulative Distribution Function)
(4-2)
where λM is the recurrence rate and t is the intermit time between events.
1
0.9
Probability Density Function, ft(t)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
Time, t
6
7
8
9
10
12
Figure 4-1 – Probability density function of earthquake occurrence for exponential
distribution model
Applying Bayes’ probability theorem, the probability of an earthquake occurrence within t years
from the previous event can be expressed as:
T0 + t
P ⎡⎣T0 + t T0 ⎤⎦ =
P [T0 + t , T0 ]
P [T0 ]
=
∫
f t (t )dt
T =T0
∞
∫
=
f t (t )dt
FT (T0 + t ) − F (T0 )
1 − FT (T0 )
(4-3)
T =T0
where T0 is the elapsed time since the previous earthquake and t is intermit time between events
as previously defined. Evaluating the probability expression using the cumulative distribution
function associated with the Poisson assumption:
P ⎡⎣T0 + t T0 ⎤⎦ =
1/ − e − λM (T0 +t ) − 1 + e − λM T0 e − λM T0 (1 − e− λM t )
= 1 − e−λM t
=
− λM T0
− λM T0
1−1+e
e
(4-4)
Notice that the elapsed time since the last earthquake, T0, is no longer present in the probability
expression, demonstrating the model’s “memory-less” nature. This is also observed through the
exponential distribution’s hazard function, generally defined as:
ht (t ) =
f t (t )
1 − Ft (t )
(4-5)
In reliability theory, the hazard function describes the instantaneous “failure rate” at any point of
time. In the case of renewal-time distributions, it describes the instantaneous occurrence rate of
an earthquake at any point in time and is useful in examining the time dependence of the
assumed distribution. Specifically the following parameters are of interest (Matthews et al, 2002):
1) Asymptotic hazard rate.
h∞ lim h(t )
t →∞
2) Likelihood of immediate occurrence after an event.
h(0)
3) General shape of hazard function
Shown below, the hazard rate of the exponential distribution is constant, confirming the
stationary nature of the Poisson assumption. Further, the hazard rate at t=0 is a non-zero value,
indicating that there is a possibility of an earthquake occurrence immediately following the
13
preceding event. This is not in accordance with elastic rebound principles and characteristic
earthquake theory which imply that if the local accumulated strain of a fault section is released in
an earthquake h(0) should be zero.
1
1
Hazard Rate, ht(t)
1
1
1
1
1
0
0.5
1
1.5
2
2.5
Time, t
3
3.5
4
4.5
5
Figure 4-2 – Hazard rate of earthquake occurrence for exponential distribution model
(Poisson assumption)
Many past studies have examined the accuracy of the Poisson assumption in its use in seismic
hazard analyses and have concluded that in most practical cases its application is in fact
appropriate (Cornell & Winterstein, 1988). However, these studies have also shown that this is
not the case in which the seismic hazard is controlled by a single source with an elapsed time
since the last event greater than the average intermit time or for sources exhibiting strong
characteristic earthquake behavior (Cornell & Winterstein, 1988). In these instances, it is more
appropriate to use models that can capture the broader process of earthquake occurrence that
includes memory of prior events in the assessment of future earthquake occurrence rates.
4.2
MODELS W/ MEMORY
4.2.1 Renewal Models
A common way to represent earthquake occurrence with time which includes memory of
previous events is through renewal models in which the occurrence of large earthquakes is
assumed to have some periodicity. Unlike the Poisson model in which earthquake recurrence
intervals are assumed to be exponentially distributed, renewal models apply different
distributions, allowing for the probability of occurrence, Pc to increase with elapsed time since the
previous event (Cornell & Winterstein, 1988). Typical distributions of earthquake recurrence
intervals include - a) Lognormal, b) Brownian Time Passage, c) Weibull and d) Gamma.
14
Most renewal model distributions are characterized by two statistical parameters, the mean and
covariance. The covariance represents the measure of periodicity of the earthquake recurrence
intervals, with low values indicating a very periodic process while a value approaching 1.0
indicates a Poisson characterization (VT = 1.0). In practice, typical values of VT in practice range
from 0.4-0.6 (Cornell & Winterstein, 1988). The mean represents the average intermit time
between events and is typically estimated using the inverse of the mean activity rate of the source.
(a) Lognormal
One of the most common distributions of earthquake recurrence intervals used in practice is the
lognormal distribution:
f ln (t ) =
⎛ −(ln t − ln μ ) 2 ⎞
exp ⎜
⎟
2σ ln t 2
2πσ ln t t
⎝
⎠
1
(4-6)
1
0.9
Probability Density Function, ft(t)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
Time, t
6
7
8
9
10
Figure 4-3 – Probability density function of earthquake occurrence for lognormal
distribution model
It is important to note that the lognormal is characterized by median, μ and standard deviation,
σlnt rather than more commonly used mean and covariance. These parameters can be related
through the following expressions:
15
μ=
T
⎛σ 2 ⎞
exp ⎜ ln t ⎟
⎝ 2 ⎠
(4-7)
σ ln t = ln(1 + VT ) 2
(4-8)
Unlike the exponential model, the lognormal distribution allows for a steady increase in the
hazard rate to a finite maximum near the mean recurrence time and then decreases asymptotically
to a level in which the conditional probability of the occurrence of an event becomes time
independent. While these distribution properties provide a more accurate representation of the
time dependent nature of the earthquake occurrence process than that under the Poisson
assumption, the lognormal distribution does have its shortcomings, particularly when assessing
values of elapsed time that exceed twice the mean recurrence interval. As can be seen in the plot,
at these values the hazard rate begins to decrease rather quickly, considered by some as a
violation of the basic concept of the renewal model. Further, the asymptotic value of the hazard
function is zero, implying that if significant time has elapsed since the previous event, the
probability of a future earthquake eventually reduces to zero. While some claim that this
disqualifies the lognormal distribution as a reliable model, others assert that when a characteristic
event is long overdue, the accumulated stress and moment associated with that event may have
been dissipated by alternative seismic or aseismic mechanisms (Matthews et al, 2002). This fault
behavior is defined by “transient characteristic failure modes” in which the pattern of recurring
characteristic events eventually diminishes (Shaw, 1997).
2
1.8
1.6
Hazard Rate, ht(t)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
Time, t
6
7
8
9
10
Figure 4-4 – Hazard rate of earthquake occurrence for lognormal distribution model
16
(b) Brownian Passage Time
Also known as the inverse Gaussian or Wald distribution, the Brownian Passage Time, “BPT”,
distribution is characterized by the mean recurrence interval, T and the a parameter representing
the aperiodicity, α (equivalent to the covariation, VT).
f (t ) =
⎡ (t − T ) 2 ⎤
exp
⎢ − 2α 2Tt ⎥
2πα 2t 3
⎣
⎦
T
(4-9)
1.4
1.2
Probability Density Function, ft(t)
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
Time, t
6
7
8
9
10
Figure 4-5 – Probability density function of earthquake occurrence for BPT distribution
model
Once again several noteworthy properties can be observed by examining the hazard rate function
of the distribution. As with the lognormal distribution, the probability of immediate earthquake
reoccurrence is zero, consistent with elastic rebound theory. Also similar to the lognormal
distribution, the hazard rate increases steadily to a finite maximum near the mean recurrence time
and then decreases asymptotically to a quasi-stationary level. In the BPT distribution however,
this level is a non-zero value that is a function of the model parameters (Matthews et al, 2002):
h∞ =
1
2T α 2
(4-10)
Ellsworth et al. (1999) assert that this asymptotic behavior distinguishes the BPT model as more
realistic than alternative models.
17
2.5
Hazard Rate, ht(t)
2
1.5
1
0.5
0
0
1
2
3
4
5
Time, t
6
7
8
9
10
Figure 4-6 – Hazard rate of earthquake occurrence for BPT distribution model
The BPT distribution is applied by Matthews et al. (2002) in the characterization of earthquake
occurrence through a Brownian relaxation oscillator, represented by the state variable Y(t).
Y (t ) = α t + ε (t )
(4-11)
Figure 4-7 – Example of load state paths for a Brownian relaxation oscillator. Taken from
Matthews et al (2002)
18
In this physically based model, the state variable builds up through a two component loading
scheme, a constant rate component, αt, and a random component known as Brownian motion,
ε(t). The occurrence of an event is defined when the state variable reaches a fixed threshold, Yf.
After the threshold has been reached, Y(t) returns to a fixed ground state, Y0 and the process
repeats itself. The modeling of earthquake occurrence through a Brownian relaxation oscillator is
an attempt to represent the mechanics of stress and strain accumulation and release through a
simple, physically-based, stochastic model. Further, the model was developed to allow for the
inclusion of external effects (such as stress-transfer effects from nearby earthquakes) through the
random component of Brownian motion in the state variable loading (Matthews et al, 2002).
(c) Weibull & Gamma
The Weibull and Gamma distributions have a similar general form and are both related to the
exponential density distribution, with parameters λ and k that are constants related to the mean
and variation of the distribution:
k⎛t ⎞
f t (t ) = ⎜ ⎟
λ⎝λ ⎠
ft (t ) =
t
( k −1)
k −1
e
⎛t⎞
−⎜ ⎟
⎝λ⎠
k
⎛t⎞
−⎜ ⎟
⎝λ⎠
e
Γ ( k )λ t
(Weibull)
(4-12)
(Gamma)
(4-13)
0.7
0.6
Probability Density Function, ft(t)
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
Time, t
6
7
8
9
10
Figure 4-8 – Probability density function of earthquake occurrence for Weibull
distribution model
19
0.8
0.7
Probability Density Function, ft(t)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
Time, t
6
7
8
9
10
Figure 4-9 – Probability density function of earthquake occurrence for Gamma
distribution model
In both models, the k-value defines the shape of the distribution and for earthquake recurrence
applications is typically taken as greater than 1.0. Gamma distributions with k > 1 have zero
hazard rate at time zero and increase to a finite asymptotic level that is always smaller than the
mean recurrence rate. Similarly, if k >1, the Weibull hazard rate function starts at zero, however it
does not reach a finite asymptotic level but rather increases to infinity.
4
3.5
3
Hazard Rate, ht(t)
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
Time, t
6
7
8
9
10
Figure 4-10 – Hazard rate of earthquake occurrence for Weibull distribution model
20
2
1.8
1.6
Hazard Rate, ht(t)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
Time, t
6
7
8
9
10
Figure 4-11 – Hazard rate of earthquake occurrence for Gamma distribution model
4.2.2 Markov & Semi - Markov Models
Many models of earthquake occurrence are based on stochastic processes that are characterized
by a Markov property. Processes that are defined as Markov are characterized by discrete states
and time intervals in which the successive state occupancy is governed by a transition probability
(Patwardhan et al, 1980). As defined by the Markov property, this transitional probability is
conditional only on the present state and is independent of the process’ state in the past:
P ( X n +1 = x X n = xn ,..., X 1 = x1 , X 0 = x0 ) = P ( X n +1 X n = xn )
(4-14)
Figure 4-12 – Schematic representation of the trajectory of a semi-Markov process. Taken
from Patwardhan et al (1980).
21
Earthquake occurrence models such as that developed by Patwardhan et al (1980) apply this basic
Markov property of one-step memory, defining the probability of a successive earthquake of
certain magnitude as dependent only on the magnitude of the previous earthquake. Further, the
Markov property can be built upon to expand the model into a “multi-step memory” process.
Referred to as a semi-Markov process, it includes a distribution of time intervals between
earthquake occurrences which is influenced by not only the magnitude of the previous
earthquake but also the magnitude of the successive. This dependency of the waiting time on the
successive earthquake magnitude represents the assumption that the time needed to buildup
sufficient strain to generate a certain magnitude earthquake will increase as the magnitude of
that earthquake increases (i.e. a magnitude 8 earthquake will require longer strain buildup than a
magnitude 7 earthquake). An illustration of the semi-Markov process is shown in Figure 4-12.
The application of Markov and semi-Markov properties in earthquake occurrence models allow
for the modeling of both the size and waiting time of successive earthquakes as dependent on
amount of strain energy released by the previous event (represented by the magnitude of the
previous). Patwardhan et al (1980) contend that this allows for a more accurate characterization
of the gradual accumulation of strain and its intermittent release and therefore provides a more
realistic representation of the physical process of earthquake occurrence than other developed
models.
4.2.3 Slip Predictable Model
One common property of many earthquake occurrence memory-models is the dependence of
future events on the time of the last occurrence. Based on the assumption that stress accumulates
at a constant rate for some period of time, the size of the successive earthquake, reflected by the
amount of stress released, is a function only of the time elapsed since the last event and is
independent of the previous earthquake’s size (Kiremidjian & Anagnos, 1984). This represents a
positive “forward” correlation between inter-arrival times (which are assumed to be randomly
distributed) and successive magnitudes and defines the basis of slip predictable models. A
schematic representation of the model developed by Kiremidjian & Anagnos (1984) that applies
slip-predictable and Markov properties is shown in Figure 4-13. As can be seen, the stress state
starts at some initial state, building-up at a constant rate until a random time, t, at which point all
accumulated stress is released in an earthquake of corresponding size (represented by the change
into a successive Markov state). In addition to correlating stress and magnitude, the size of an
event can also be characterized by coseismic slip, as shown in Figure 4-13b. This is considered by
some as a more appropriate parameter to describe earthquake size due to difficulties in assessing
the initial stress level and stress accumulation rate for a given fault zone (Kiremidjian & Anagnos,
1984).
22
Figure 4-13 – Slip-predictable model: (a) Time history of stress accumulation and release;
(b) Relationship between coseismic slip and time between seismic events; and (c)
Sample path for the corresponding Markov renewal process. Taken from Kiremidjian &
Anagnos (1984)
The slip-predictable model developed by Kiremidjian & Anagnos (1984) was applied to the
Middle America Trench in Oaxaca, Mexico assuming a Weibull distribution for inter-arrival
times. A comparison between the slip-predictable model and a Poisson model is illustrated in
Figure 4-14.
Figure 4-14 – Comparison between Poisson model and slip-predictable model for
probabilities of at least one event of M>7.5 and M>8.0 in the next 100 yrs as a function of
the gap t1. Dashed line – Poisson model. Solid line – slip-predictable model. Taken from
Kiremidjian & Anagnos (1984)
23
As can be seen, as the elapsed time, or “gap” increases beyond a specific range of time
(approximately 25 & 40 years for MW=7.5 & 8 respectively) the Poisson model becomes
increasingly un-conservative in comparison to the slip predictable. Thus, if the elapsed time since
an event is significantly large, the Poisson model can grossly underestimate the hazard resulting
from future occurrences. Further, at times immediately following an event, the Poisson model
may overestimate the hazard.
4.2.4 Time Predictable Model
While slip-predictable models use the time between events to estimate the size of the following
earthquake, an alternative model has been developed based on the assumption of timepredictable behavior. In time-predictable models the size of the last event provides information
on the occurrence time of next. This corresponds to a positive, backward correlation between
intermit times and magnitudes. Originally developed by Shimazaki and Nakata (1980), the timepredictable model is based on patterns in historical data that indicate the larger the previous
earthquake, the longer time until the next occurrence. Faults such as the San Andreas at Parkfield
have been shown to demonstrate this type of time-predictable behavior and serve as the basis for
validating corresponding models.
A schematic representation by Kiremidjian & Anagnos (1984), similar to that presented for the
slip-predictable model, is shown in Figure 4-15.
Figure 4-15 – Time-predictable model: (a) Time history of stress accumulation and
release; (b) Relationship between coseismic slip and time between seismic events; (c)
Sample path for the c. Taken from Kiremidjian & Anagnos (1984)
24
As previously assumed, the stress starts at some initial state and accumulates at a constant rate.
However, unlike the slip-predictable model, the accumulated stress in this model has no time
constraint, continuing to increase until it reaches a specified stress threshold. At this point, some
portion of the built-up stress is released through an earthquake whose size is proportional to the
change in stress level (Kiremidjian & Anagnos, 1984). Thus, the occurrence of the next
earthquake corresponds to the length of time needed to accumulate sufficient stress to reach its
threshold, triggering another event. While this property of time-predictable behavior allows for
forecasting of the time of the next earthquake, unlike the slip-predictable model, no information
regarding the size of the next earthquake is provided.
4.2.5 Combined Time-Slip Predictable Models
Some models take advantage of both time and slip predictable assumptions to create a model that
reflects both dependence on the previous earthquake magnitude and time of occurrence. This
creates a model in which successive magnitudes are no longer independent, and are related
through the elapsed time between events (Cornell & Winterstein, 1988).
5 ESTIMATING GROUND MOTION
Estimates of ground motion are used in the design of structures in one of two ways, either
implicitly through the use of building codes or explicitly through the site-specific design of
structures (Boore, 2003). Although recently there have been significant efforts to instrument
seismically active regions to gather more ground motion recordings, there are still not a sufficient
amount of data to allow for direct empirical estimation of design ground motions. As a result,
considerable work has been dedicated to develop relations to estimate ground motion parameters
critical for the design of structures based on seismic source characteristics (e.g. magnitude,
distance).
5.1
GROUND MOTION PARAMETERS
5.1.1 Amplitude
One of the most common parameters used in estimating ground motion amplitude is peak
horizontal acceleration. Although some relations have been developed for estimation of peak
vertical acceleration, it is not usually considered as the margins required under static gravity loads
are usually significant to resist seismic forces. Acceleration has been shown to demonstrate a
large proportion of high frequency content; therefore, peak ground accelerations are often used
in the design of very stiff structures. Less sensitive to high frequencies, peak ground velocity is
used to estimate ground motions for structures that are vulnerable to intermediate frequencies,
such as tall flexible structures. Another ground motion parameter, though not commonly used in
25
practice, is peak ground displacement which corresponds to the low frequency component of
ground motion.
5.1.2 Frequency Content
The dynamic response of a structure is very sensitive to the frequency at which it is loaded.
Therefore in addition to the estimation of amplitude it is necessary to have knowledge of the
frequency content which describes how the ground motion amplitude is distributed amongst
different frequencies. The frequency content can be expressed through different types of spectra
and their corresponding spectral parameters.
A Fourier spectrum, which is the product of performing a Fourier transform of a time series, is
represented by a plot of Fourier amplitude or phase versus frequency (Kramer, 2001). Once
plotted, the Fourier spectrum gives immediate indications as to the frequency content of the
ground motion. A narrow spectrum indicates a ground motion with a dominant frequency which
is characterized by a smooth, almost sinusoidal time history. On the contrary, a broad spectrum
represents a ground motion that includes a variety of frequencies and thus corresponds to a
jagged, irregular time history. As an alternative to employing a complete Fourier spectrum, the
frequency content can be represented by individual parameters, particularly through the
predominate period and bandwidth. The predominant period corresponds to the frequency in
which the maximum value of the Fourier amplitude spectrum is achieved whereas the bandwidth
is defined as the range of frequencies in which some level of Fourier amplitude is exceeded.
Another spectrum often used in describing frequency content is the power spectrum, which is a
plot of the power spectral density function of ground motion. If the ground motion is
characterized as a random process, the power spectrum allows for estimation of statistical
properties which are used in stochastic methods to develop ground motion estimates. The use of
power spectral density functions in stochastic ground motion equations is discussed later in this
section. Similar to Fourier spectra, power spectra can be defined through specific frequency
parameters. The central frequency is a measure of the frequency where the power spectral density
is concentrated. The dispersion of the power spectral density function about this central
frequency is defined through the shape factor.
Often used directly in structural design, a response spectrum describes the maximum response of
SDOF system with a specific level of viscous damping (e.g. 5%) as a function of natural
frequency. The response spectrum is often displayed on tripartite logarithm scale, providing peak
displacement, velocity and acceleration response on the same plot. This is possible since peak
acceleration is proportional to peak displacement and velocity through the natural frequency (ω2
and ω respectively). An example of a response spectrum is given below. As can be seen, the shape
of the spectrum indicates that peak values of displacement, velocity and acceleration occur at
different frequencies.
26
Figure 5-1 – Example of response spectrum
5.1.3 Duration
The duration of ground motion is critical to prevent damage caused by physical processes that
are sensitive to the number of load reversals, such as the degradation of stiffness and strength
and the buildup of pore water pressures which is associated to liquefaction. Related to the time
required to release accumulated strain energy, the duration of ground motion is positively
correlated to the length or area of rupture. Therefore, as the magnitude of an event increases,
indicating an increase in rupture dimensions, the duration of the resulting ground motion also
increases.
A common way to define duration is through the bracketed duration, defined by Bolt (1969) as
the time between the first and last exceedance of some value of threshold acceleration (typically
0.5g). Another parameter of duration often applied is the significant duration, which is a measure
of the time in which a specified amount of energy is dissipated. Duration can also be expressed
by the equivalent number of cycles of the ground motion, a parameter that is commonly used in
determining liquefaction potential (Stewart et al, 2001).
5.2
EMPIRICAL GROUND MOTION EQUATIONS
Characterizations of ground motions typically take the form of a probability distribution function
of a particular ground motion parameter (such as peak acceleration or response spectra) which is
defined by its statistical moments (e.g. median, standard deviation). These statistical moments are
determined using equations known as attenuation relations, which are derived through regression
analysis of empirical data and are based on magnitude, source-to-site distance, and other
27
seismological parameters. Examples of various models for ground motion attenuation in active
seismic zones are shown in Table 5-1.
Magnitude
Range
Distance
Range (km)
Distance
Measure
Site Parameters
Other Parameters
Atkinson & Boore
(1997)
5.5 – 7.5
0 - 100
rjb
30m-Vs
Fault type
Campbell (1997,
2000, 2001)
4.7 – 8.1
3 – 60
rseism
Soft rock, hard
rock, depth to rock
Abrahamson &
Silva (1997)
> 4.7
0 - 100
r
Soil/rock
Sadigh et al. (1997)
4.0 – 8.0
0 - 100
r
Soil/rock
Fault type,
Hanging-wall
ff
Fault type,
Hanging-wall
ff
Fault type
Idriss (1991, 1994)
4.6 – 7.4
1 - 100
r
Rock only
Fault type
Table 5-1 – Examples of attenuation models for horizontal spectral acceleration in active
fault zones
The general form of an attenuation relation can be expressed as:
1
c4
Ln(Y ) = c0 + c2 m + c3m + c5 ln(r ) + f ( F ) + f ( HW ) + f ( S ) + ε
2
3
(5-1)
4
where Y = Ground motion amplitude parameter (e.g. PHA)
c0, c1… c5 = Constants determined by regression analysis
m = Moment magnitude
r = Source to site distance
S = Factor accounting for local site conditions
F = Factor accounting for fault type (e.g. strike-slip, reverse)
HW = Factor accounting for hanging-wall effects
28
This general form relies on a number of assumptions that serve as the basis for most attenuation
relations (Stewart et al, 2001):
1) Uncertainty in ground motions – Attenuation relations define not only the mean ground
motion, Y, but additionally the uncertainty or variability, ε or σY, in ground motion amplitudes.
Typically, ground motion amplitudes are assumed to be lognormally distributed and thus the
mean and uncertainty are represented as Ln(Y) and σLnY respectively. This assumption however,
has been shown to break down for near-source ground motions and therefore other distributions
of uncertainty should be considered.
2) Magnitude dependence - Several magnitude scales such as moment magnitude are derived
using the logarithm of peak ground motion parameters. Therefore Ln(Y) is assumed to be
proportional to the magnitude, m of the event. This assumption however, has been shown to
break down for high frequency ground motion at large magnitudes due to saturation.
3) Radiation damping - The energy released by a seismic source during an earthquake is radiated
out through traveling body waves. As they travel away from the source, the wave amplitudes
reduce at a rate of 1/r (where r is the source-to-site distance), a phenomenon known as radiation
damping. The source-to-site distance, a key parameter is defining the attenuation of ground
motions, is a somewhat ambiguous term with several existing definitions. It is therefore critical to
clearly understand which distance parameter is appropriate for the specific model used. Some
examples of common definitions for r are shown in the figure below.
(a)
29
(b)
Figure 5-2 – Source-to-site distance measures for ground motion attenuation models for
(a) Vertical faults, and (b) Dipping faults
4) Factors influencing attenuation - Attenuation of ground motions can be significantly effected
by several factors related to source and site characteristics. Some of these factors are presented
and discussed in detail below. Further, where appropriate, their influence on the attenuation of
ground motions is illustrated and examined using a reference model.
Developed by Campbell & Bozorgnia (2003), the model applied in this section is a set of
mutually consistent near-source horizontal and vertical ground motion attenuation relations for
both peak ground acceleration and 5% damped pseudo-acceleration response spectra.
LnY = c1 + f1 ( M w ) + c4 ln
f 2 ( M w , rseis , S ) + f 3 ( F ) + f 4 ( S ) + f5 ( HW , F , M w , rseis ) + ε
(5-2)
As can be seen, this model shares a similar form to the general relation introduced above. In
addition to the standard magnitude and distance scaling characteristics, the model also includes
functions accounting for the type of faulting mechanism, near & far-source effects, local site soil
conditions and effects of the hanging wall. Further, the random error term, ε, has the property of
being defined either as a function of magnitude or a function of PGA. Examples of the model
are shown in Figure 5-3 for both peak ground acceleration and peak spectral acceleration at 0.1
sec.
30
(a)
(b)
Figure 5-3 – Attenuation relation for (a) peak spectral acceleration at 0.1 sec and (b) peak
ground acceleration using Campbell & Bozorgnia model. Taken from Campbell &
Bozorgnia (2003)
5.2.2 Factors Affecting Attenuation
(a) Tectonic Regime
One of the most fundamental factors affecting ground motion characteristics is the tectonic
region in which the seismic source is located. Typically attenuation relations are developed
31
independently for each respective region - active, subduction, and stable continental zones. A
large proportion of attenuation relations are developed for active tectonic regions due to the
significant amount of ground motion data available for these areas. For the case of stable
continental regions there is very little strong motion data available. Therefore, attenuation
relations for stable continental regions are typically based on simulated motions instead of
recorded data (Atkinson & Boore, 1995-1997b, Toro et al, 1997).
(b)
Focal Mechanism (Fault Type)
Studies have found that the faulting mechanism of a source influences ground motion
parameters, particularly amplitude and frequency content (Boore, 2003). Typically, strike slip
faults serve as the reference for attenuation relations and additional factors, either constant or
dependent on other seismic source parameters (e.g. distance, magnitude), are included to account
for effects of reverse, oblique, and normal faults. Some observations of fault-type effects include
higher mean ground motion amplitudes and a larger proportion of higher frequency content for
both reverse and thrust faults.
Figure 5-4 – Peak spectral acceleration (5% damping) using Campbell & Bozorgnia
ground motion attenuation model showing effects of faulting mechanism. Evaluated
using Mw = 7.0, rseis = 10km, firm soil. Taken from Campbell & Bozorgnia (2003)
(c)
Hanging Wall Effect
Studies have shown that there is a significant increase in ground motions for sites located over
the hanging wall of dipping faults (Somerville & Anderson, 1996). In the case of the 1994
Northridge earthquake, analyses have shown that this increase can be by as much as 50%
(Abrahamson & Silva, 1997). This effect is primarily a geometric effect since sites located on the
hanging wall are closer to a larger area of the source than the footwall sites.
32
(d)
Site Conditions
Effects of local site conditions can be represented in many forms, ranging from a simple constant
to more complex functions that attempt to characterize non-linearity in ground response. While
some models use a simple soil/rock soil classification (Abrahamson & Silva, 1997; Sadigh et al,
1997) others use more quantitative methods of classification, such as the 30m shear wave velocity
(Atkinson & Boore, 1997). In general, the standard error in attenuation relations are assumed to
be unaffected by site conditions.
(a)
(b)
Figure 5-5 – Peak spectral acceleration (5% damping) using Campbell & Bozorgnia
ground motion attenuation model showing effects of site conditions. Evaluated using Mw
= 7.0, rseis = 10km, strike-slip fault. Taken from Campbell & Bozorgnia (2003)
33
(e) Near-Fault Effects
The importance of near-fault effects on ground motion has been the study of many recent works
(Campbell & Bozorgnia, 2003). These studies have found that ground motion at near-source sites
(typically defined as within 20-60 km of fault rupture) are more sensitive to what is termed as
“rupture directivity” which primarily affects the duration and long period energy of a ground
motion. Forward directivity occurs when the rupture of a fault propagates towards the site and its
effects are principally in the horizontal direction normal to the fault rupture. Under this condition
ground motions are characterized by a shock wave effect as the wave-front arrives as a large pulse
of motion usually at the beginning of the record. This type of ground motion is associated with
large amplitudes at intermediate to long periods and a short duration. Oppositely, ground
motions affected by backward directivity (fault rupture propagating away from the site, as in the
case when a site is located at the epicenter of the energy release) are characterized by a relatively
long duration and low amplitude. In addition, permanent ground displacements occurring across
faults cause an effect termed as “fling step” which is associated with a unidirectional velocity
pulse that attempts to accommodate the slip in the fault parallel direction (Stewart et al, 2001). In
order to account for these effects on near-fault ground motions, many adjusted attenuation
relations have been developed, particularly for modified average horizontal spectra and increased
ratios between fault normal and fault parallel response spectra.
5.3 STOCHASTIC GROUND MOTION METHODS
If there is insufficient amount of ground motion recordings to develop empirically-based
equations it is possible to generate ground motions using stochastic methods to supplement
existing recordings. These methods are commonly used for ground motion estimation in stable
tectonic regions and for high frequency motions characterized by a large magnitude and short
source-to-site distance.
The standard stochastic method is based on the assumption that the far-field shear wave energy
generated by an earthquake source can be represented as a band-limited random process
(McGwire, 2001). Under this assumption ground motions can be represented as a band-limited,
finite duration, white Gaussian noise. A description of this method of representation, with
particular focus on the procedures outlined by Boore (2003) is described below.
5.3.1 Ground Motion Spectra
The first step in stochastically generating a ground motion is defining a spectrum of ground
motion amplitudes through a standard seismological model. The ground motion spectrum
accounts for physics of the earthquake process and wave propagation and is typically a function
of source-to-site distance (also referred to as path distance), magnitude, and local site conditions.
A general form of the ground motion amplitude spectrum can be expressed as:
Y ( M 0 , R, f ) = E ( M 0 , f ) ⋅ P ( R, f ) ⋅ G
( f )⋅ I( f )
N N
1
2
3
(5-13)
4
34
This general form breaks the total spectrum into relative contributions of the seismic source (E),
path (P), site (G) and type of motion (I). Details of the effects of these parameters on the ground
motion spectrum are discussed in detail below.
Figure 5-6 – Ground motion amplitude spectra using models in Table 5-1 for M=7.5 and
M=4.5. Taken from Boore (2003)
a)
Source effects
The shape and amplitude of a ground motion spectrum are functions of earthquake size and
therefore are directly related to source properties. The most common model to define this
relationship was developed by Aki in 1967 and is known as the “ω2 model”. Plots of the
amplitude spectrum computed using the “ω2 model” and other source-spectra models are shown
in Figure 5-6. The scaling of these spectra by magnitude is based on the dependence of the
corner frequency on seismic moment and can be expressed in general form as:
E (M 0 , f ) = C ⋅ M 0 ⋅ S (M 0 , f )
S (M0, f) is the displacement source spectrum (expressions for Sa and Sb for various models are
given in Tables 5-1a & b) and C is a constant that is based on the energy radiation pattern, free
surface amplification, source-to-site distance and local site shear wave velocity and density.
35
(a)
(b)
Table 5-2 – (a) Spectra shape factor relations for various models and (b) corresponding
corner frequencies and moment ratios. Taken from Boore (2003)
b) Path Effects
The function P(R, f) accounts for the geometric spreading and attenuation of ground motion
amplitudes. A decrease in peak amplitude with increasing ground motion duration is also
accounted for in P(R,f). This is because the duration of ground motion is directly influenced by
path effects, with a general increase of duration with distance which corresponds to the general
decrease in peak amplitude. Figures 5-7 & 5-8 show observed relations between amplitude,
duration and path distance in addition to path-dependent models commonly used in stochastic
ground motion methods.
36
Figure 5-7 – Observed attenuation of ground motions with distance in eastern North
America and effect of geometrical spreading and whole path attenuation using model
applied by Atkinson and Boore (1995) and Frankel et al. (1996) for . Taken from Boore
(2003)
Figure 5-8 – Observed duration from earthquakes in eastern North America and duration
function used by Atkinson and Boore (1995) in ground motion simulation model for
eastern North America. Taken from Boore (2003)
37
c)
Site Effects
Although site-specific parameters can be used in the G (f) function, site effects are most often
represented for general site conditions with the use of generic classifications of soil type.
Expressed below, the effects of site conditions are divided into two functions, an amplification
function, A(f) and a diminution function, D(f)
G ( f ) = A( f ) ⋅ D( f )
(5-14)
The amplification function, A(f), is usually defined relative to the source and is a function of
shear velocity versus depth. D(f) models the path independent loss of energy, or more
specifically, the loss of high frequency content in ground motions. An example of the combined
effect of amplification and diminution on ground motion amplitude is shown in Figure 5-9.
Figure 5-9 – Combined effect of site amplification and path-independent diminution.
Taken from Boore (2003)
d) Type of Ground Motion
The type of ground motion resulting from the stochastic simulation is controlled by the filter
function, I(f). The form of I(f) can be modified to produce different forms of ground motion
parameters such as displacement, velocity, acceleration or the response of oscillator (from which
response spectra can be derived).
5.3.2 Obtaining Ground Motions
Once the ground motion amplitude spectrum is defined, ground motions can be obtained using
one of two methods, time domain simulation or estimates of peak motions in the form of
response spectra.
38
(a)
Simulation of Time Series
In the simulation of a ground motion time series the first step involves the generation of white
noise for a specified duration (Figure 5-10a). Many definitions of strong- motion durations exist,
many of which are presented in a summary by Bommer and Martinez-Pereira (1999). Because
seismic motion is a non-stationary process, the white noise must be filtered by an appropriate
window (e.g. exponential, box or Jennings) (Figure 5-10b). Once filtered, the noise is transformed
into the frequency domain and is normalized by the square root of mean squared amplitude
spectra (Figure 5-10c). The final design spectrum is obtained by multiplying the normalized
spectrum by the ground motion spectrum (Figure 5-10d, 5-10e). This final spectrum is then
transformed back into the time domain for a final ground motion time series (Figure 5-10f). It
should be noted that it is not advisable to use individual time series in design – as the resulting
spectrum might not approximate the “target” spectrum accurately. Rather, the mean of individual
spectra for a number of simulations will ensure a resulting spectrum consistent with the “target”.
4
3
3
2
2
1
Amplitude
Amplitude
1
0
0
-1
-1
-2
-2
-3
-3
-4
0
5
10
15
Time [sec]
20
25
-4
30
0
5
10
(a)
15
Time [sec]
20
25
(b)
0
10
4
10
-1
10
2
Fourier Amplitude
Fourier Amplitude
10
0
10
-2
10
-2
10
-4
10
-3
10
-2
10
-1
10
0
10
Frequency [Hz]
(c)
39
1
10
2
10
-1
10
0
10
1
10
Frequency [Hz]
(d)
2
10
30
10
10
20
15
8
10
10
6
10
Amplitude
Fourier Amplitude
5
4
10
2
10
0
-5
-10
0
10
-15
-2
10
-20
-4
10
-2
10
-1
10
0
1
10
2
10
10
Frequency [Hz]
(e)
3
10
-25
0
5
10
15
Time [sec]
20
25
30
(f)
Figure 5-10 – Simulation of a time series
(b) Response Spectrum
Rather than generating a ground motion time series, it is common to calculate a response
spectrum representing peak values of ground motion. This is achieved by using concepts of
random vibration theory which provide estimates of the ratio of peak motion (Ymax) to the RMS
motion (Yrms). This ratio, often referred to as the peak factor, is a function of natural frequency
and duration and is commonly determined using an expression developed by Cartwright and
Longuet-Higgons (1956). Using the amplitude spectrum to compute the spectral density function
of a linear oscillator, the RMS response of a single-degree-of-freedom linear oscillator with
natural frequency, fn, and damping ratio, ξ, can be computed. Applying this value with the peak
factor, yields the value of peak motion (Ymax) used to generate the response spectrum of the
ground motion.
6 HAZARD CURVES
Once ground motion estimates have been computed and characterized by distribution functions,
the final seismic hazard can be determined. This final step determines how often a specified level
of ground motion will be exceeded at the site of interest. This specified level can be the peak
ground acceleration or any ground motion parameter (e.g. duration, displacement). Taking the
form of an annual rate of exceedance or a return period, the resulting hazard consists of hazard
contributions from each independently defined source. The individual hazard of a single seismic
source can defined through the following expression:
∞
vi ( A > z ) = Ni ( M min ) ∫
M max
∫
r = 0 m = M min
f mi (m) f ri (r ) fε (ε ) P ( A > z m, r , ε )drdmd ε
(6-1)
40
This function integrates over the probability density functions for magnitude, source-to-site
distance, and ground motion. This allows for consideration of their variability to be included
explicitly in the analysis. Also included in the expression is the probability that the ground motion
exceeds the specified level z, for a magnitude M, distance r, and number of standard deviations ε.
It is in this term that the attenuation relations described above contribute to the hazard
calculation. For the given magnitude, distance and ε, the ground motion defined by the
attenuation relation determines whether this probability is either 0 or 1. In this way, the
probability term classifies which scenarios or combinations of magnitude, distance and standard
deviation, will produce ground motions greater than the specified level and which scenarios will
not.
Expression (6-1) defines the hazard function for the simple case of a point source which is
spatially defined only by the distance from the source to the site. In the case of fault sources, the
hazard function must be expanded to account for rupture dimensions (defined by either rupture
width and length or rupture width and area) and rupture locations (along strike and down dip).
Given these values the source-to-site distance can be calculated and the resulting hazard
determined using equation (6-2).
∞
vi ( A > z ) = N i ( M min )
∞
1
M max
1
∫ ∫ ∫ ∫ ∫
ε max
∫
W = 0 A= 0 x = 0 y = 0 m = M min ε =ε min
f mi (m) fWi (m, W ) f Ai (m, A) f locx ( x) flocy (m, y ) fε (ε )
i
i
P( A > z m, r ( x, y, A, W ), ε )dWdAdxdydm d ε
(6-2)
The hazard expressions introduced above only reflect the contribution of a single seismic source.
If multiple seismic sources are considered, it is necessary to sum the total hazard contributions
from each individual source.
v( A > z ) =
# Sources
∑
i =1
vi ( A > z )
(6-3)
Now that the total site hazard is quantified in the form of an annual rate of events or return
period, it is necessary to convert this value into a probability. This probability reflects the
likelihood that the ground motion will exceed the level z at least once during a specified time
interval and is determined by using one of the models discussed in Section 4 on earthquake
recurrence relations. Once the probability is computed the process can be repeated for different
levels of ground motion, resulting in a series of varying levels of ground motion and their
corresponding probabilities of exceedance. This data can be plotted to display a hazard curve for
the specific site that provides the designer with an excellent visual guide to the resulting seismic
hazard.
41
Figure 6-1 – Example of individual source hazard curves for spectral period of 2 sec.
Taken from McGwire (2001)
6.1 DEAGGREGATION OF HAZARD
While the expressions presented in the previous section represent the combined effect of all
magnitudes and distances on the resulting hazard it is often desirable to break down the hazard
into the relative contributions from each earthquake scenario. Referred to as deaggregation of
hazard, this procedure indicates what contributes the greatest to the hazard and is commonly
completed using a two-dimensional deaggregation into bins of different earthquake magnitudes
and source-to-site distances (Bazzuro & Cornell, 1999). This is achieved by integrating the
expression for total hazard and normalizing the result such that it sums to unity for all respective
scenarios of magnitude and distance. For simplicity, the expression below corresponds to
equation (6-1), in which the geometry of the source is presented in its generalized form as a point
source.
Deagg ( A > z, M 1 < M < M 2 , R1 < R < R2 ) =
# Sources
∑
i =1
∞
Ni ( M min ) ∫
M max
∫
r = 0 m = M min
f mi (m) f ri (r ) fε (ε ) P ( A > z m, r , ε )drdmd ε
(6-4)
v( A > z )
An example of the deaggregation of hazard corresponding to the total contribution from the
source hazard curves shown in Figure (6-1) is shown below. Note that it represents the
deaggregation of hazard for a specific spectral period and ground motion level.
42
Figure 6-2 – Example of deaggregation of hazard for spectral period of 2 sec at ground
motion level of 0.5 g. Taken from McGuire (2001)
7
UNCERTAINTY
An important aspect of a probabilistic seismic hazard analysis is the definition and treatment of
uncertainties. This involves identifying inherent variability in the earthquake process, defined as
aleatory variability, as well as considering uncertainty in the distribution models used in the
analysis, known as epistemic uncertainty. The distinction between these two types of
uncertainties is fundamental to understanding where uncertainty originates and further how it is
to be appropriately handled in hazard calculations.
7.1 ALEATORY VARIABILITY
Aleatory variability is defined as the innate randomness in a process. In discrete variables, this is
characterized by the probability of each possible value, while in continuous variables it is
characterized by probability density functions describing parameter distributions (e.g. magnitude
distributions). The aleatory variability in a hazard analysis is included directly in the calculations,
specifically through the standard deviation parameter, and thus it directly influences the resulting
hazard curve (Abrahamson & Bommer, 2005).
43
7.2 EPISTEMIC UNCERTAINTY
Epistemic uncertainty is often referred to as scientific uncertainty because it is a product of
limited data and knowledge. Unlike aleatory variability, as more information becomes available,
epistemic uncertainty can be reduced. Originating from parameters that are not random, but
rather have some correct, yet unknown value, epistemic uncertainty is characterized by the use of
alternative models (i.e. alternative probability density functions). Therefore, epistemic uncertainty
is not considered directly in the hazard calculations but rather is treated by developing alternative
models that yield respective alternative hazard curves (Abrahamson & Bommer, 2005).
7.2.1 Logic Trees
A common way to handle epistemic uncertainty is through the use of logic trees. As mentioned
above, epistemic uncertainty is considered by using different models for source characterization
or ground motion attenuation relations. With each combination of alternative models, the
resulting hazard is recomputed resulting in a collection of hazard curves. A logic tree provides a
method for effectively organizing and assessing the credibility of these models and their resulting
hazard curves.
An example of a logic tree is shown in Figure 7-1. As can be seen, it consists of a series of
branches that describe alternative models and/or parameter values with a set of weights located
at the tips. These weights represent the relative credibility of each model and must sum to unity
at each branch. Representing current scientific judgment on the merit of the alternative models,
the weights are based on data collected from analogous regions, simplified physical models, and
empirical observations.
Figure 7-1 – Example of a logic tree used in a PSHA
44
8 CONCLUSIONS
The concepts and methodologies presented in this article are intended to give an introduction to
the powerful tool of probabilistic seismic hazard analysis within engineering practice. PSHA
provides a consistent, quantitative method for assessing seismic hazard, yielding results that are
easily interpreted and can be readily applied to seismic risk computations. The mathematical
framework for the calculation of seismic hazard have been presented, along with the methods
and models used for quantifying hazard based on seismic source and ground motion
characteristics. Further, the relationship between different source and site characteristics and
resulting ground motion parameters was discussed. It is important to note that while the ideas
and models included in this article cover much of those commonly applied in PSHA, much
research has and continues to be completed in this area. Therefore, the work presented here is in
no way conclusive and intends to serve only as a brief introduction into the discipline of seismic
hazard assessment using PSHA.
45
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