* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Probabilistic seismic hazard analysis
Survey
Document related concepts
Casualties of the 2010 Haiti earthquake wikipedia , lookup
Kashiwazaki-Kariwa Nuclear Power Plant wikipedia , lookup
2010 Canterbury earthquake wikipedia , lookup
2008 Sichuan earthquake wikipedia , lookup
1570 Ferrara earthquake wikipedia , lookup
2009–18 Oklahoma earthquake swarms wikipedia , lookup
1992 Cape Mendocino earthquakes wikipedia , lookup
1880 Luzon earthquakes wikipedia , lookup
April 2015 Nepal earthquake wikipedia , lookup
Earthquake engineering wikipedia , lookup
Seismic retrofit wikipedia , lookup
Transcript
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 REFERENCES Abrahamson, N.A., (2006). Notes on Probabilistic Seismic Hazard Analysis – An Overview. Rose School, Pavia, Italy. Abrahamson, N.A., and J.J. Bommer (2005).Opinion Paper: Probability and Uncertainty in Seismic Hazard Analysis,” Earthquake Spectra, Vol. 21, No. 2, pp. 1-5. Abrahamson, N.A., and K.M. Shedlock. (1997). Overview – Estimate of Ground Motions. Seism. Res. Let., Vol. 68, No. 1. Abrahamson, N.A., and P.G. Somerville. (1996). Effects of the Hanging Wall and Footwall on Ground Motions Recorded During Northridge Earthquake. Bull. Seism. Soc. Am., Vol.86, pp. S93-S99. Abrahamson, N.A., and W.J. Silva. (1997). Empirical Response Spectral Attenuation Relations for Shallow Crustal Earthquakes. Seism. Res. Let., Vol. 68, No. 1. Aki, K. (1972). Earthquake Mechanism. Tectonophysics, Vol. 13, pp. 423-446. Atkinson, G.M., and D.M. Boore. (1997). Earthquake Ground Motion Prediction Equations for Eastern North America. Bull. Seism. Soc. Am., Vol.93, No.1, pp. 314-331. Bazzurro, P., and A.C. Cornell (1999). Disaggregation of Seismic Hazard. Bull. Seism. Soc. Am., Vol. 89, No. 2, pp.501-520. Beresnev, I.A. (2002). Source Parameters Observable from the Corner Frequency of Earthquake Spectra. Bull. Seism. Soc. Am., Vol. 92, No. 5, pp. 2047-2048. Bolt, B.A. (1969). Duration of Strong Motion. Proc. of 4th Conf. EQ Eng., pp. 1304-1315. Santiago, Chile. Bommer, J.J., and A. Martinez-Pereira (1999). The Effective Duration of Earthquake Strong Motion. J. Earthquake Eng. Vol. 3, No. 2, pp. 127-172. Boore, D.M. (2003). Simulation of Ground Motion Using the Stochastic Method. Pure Appl. Geophys. Vol. 160. pp. 635-676. Campbell, K.W., and Y. Bozorgnia (2003). Updated Near-Source Ground Motion (Attenuation) Relations for the Horizontal and Vertical Components of Peak Ground Acceleration and 46 Acceleration Response Spectra. Bull. Seism. Soc. Am., Vol.93, No.1, pp. 314-331. Cartwright, D.E. and M.S. Longuet-Higgins (1956). "The Statistical Distribution of the Maxima of a Random Function," Proc. of the Royal Society of London, Series A (Mathematics and Physical Science), Vol. 237, pp. 212-232. Cornell, C.A. (1968). Engineering Seismic Risk Analysis. Bull. Seism. Soc. Am., Vol. 58, pp. 1583-1606. Cornell, C.A., and S. Winterstein (1988). Temporal and Magnitude Dependence in Earthquake Recurrence Models. Bull. Seism. Soc. Am., Vol. 78, No. 4, pp. 1522-1537. Ellsworth, W.L., et al (1999). A Physically-Based Earthquake Recurrence Model for Estimation of Long Term Earthquake Probability. Workshop on Earthquake Recurrence: State of the Art and Directions for the Future. Instituto Nazionale de Geofisica, Rome, Italy. Feburary 22-25. Gutenberg, B. (1945). Amplitudes of P, PP, & S and Magnitudes of Shallow Earthquakes. Bull. Seism. Soc. Am., Vol. 35, pp. 57-69. Gutenberg, B. (1945). Amplitudes of Surface Waves and Magnitudes of Shallow Earthquakes. Bull. Seism. Soc. Am., Vol. 35, pp. 3-12. Gutenberg, B. and C.F. Richter (1956). Earthquake Magnitude, Intensity, Energy and Acceleration. Bull. Seism. Soc. Am., Vol. 46, pp. 105-145. Hanks, T.C., and H. Kanamori. (1979). A Moment Magnitude Scale. J. Geophys. Res., Vol. 84, pp. 2345-2350. Henry, C. and S. Das (2001). Aftershock Zones of Large Shallow Earthquakes: Fault Dimensions, Aftershock Area Expansion and Scaling Relations. Geophys. J. Int., Vol. 147, pp. 272-293. Kanamori, H. and D.L. Anderson (1975). Theoretical Basis of Some Empirical Relations in Seismology. Bull. Seism. Soc. Am., Vol. 65, No. 5, pp. 1073-1095. Kiremidjian, A., and T. Anagnos (1984). Stochastic Slip-Predictable Model for Earthquake Occurrences. Bull. Seism. Soc. Am., Vol. 74, No. 2, pp. 739-755. Kiremidjian, A., and T. Anagnos (1984). Stochastic Time-Predictable Model for Earthquake Occurrences. Bull. Seism. Soc. Am., Vol. 74, No.6, pp. 2593-2611. Kramer, S.L. (1996). Geotechnical Earthquake Engineering. New Jersey: Prentice Hill. 47 Matthews, M.V., W.L. Ellsworth and P.A. Reasenberg (2002). A Brownian Model for Recurrent Earthquakes. Bull. Seism. Soc. Am., Vol. 92, No. 6, pp. 2233-2250. McGwire, R. (2004). Seismic Hazard and Risk Analysis. Earthquake Engineering Research Institute, MNO-10. McGwire, R.K. (2001). Deterministic vs. Probabilistic Earthquake Hazards and Risks. Solid Dynamics and EQ Eng., Vol. 21, pp.377-384. Molnar, P., B.E. Tucker and J.N. Brune (1973). Corner Frequencies of P and S Waves and Models of Earthquake Sources. Bull. Seism. Soc. Am., Vol. 63, No. 6, pp. 2091-2104. Patwardhan, A.S., R.B. Kulkarni, and D. Tocher. (1980). A Semi-Markov Model for Characterizing Recurrence of Great Earthquakes. Bull. Seism. Soc. Am., Vol. 70, pp. 323-347. Peterson, M., et al (2005). Time Independent and Time Dependent Seismic Hazard for State of CA. CA Seismic Hazard Paper, November 7. Reid, H.F. (1911). The Elastic-Rebound Theory of Earthquakes. Berkley, CA. University of California Press. Richter, C.F. (1935). An Instrumental Earthquake Magnitude Scale. Bull. Seism. Soc. Am., Vol. 25, pp. 1-32. Sadigh, K., C.Y. Chang, J.A. Egan, F. Makdisi, and R.R. Youngs (1997). Attenuation Relations for Shallow Crustal Earthquakes Based on California Strong Motion Data. Seism. Res. Letters. Vol. 68, No. 1, pp. 180-189. Shaw, B.E. (1997). Model Quakes in the Two-Dimensional Wave Equation. J. Geophys. Res. Vol.102, No.27, pp. 367-377. Shimazaki, K. and T. Nakata (1980). Time Predictable Recurrence for Larger Earthquakes. Geophys. Res. Letters 86, pp. 279-282. Somerville, P.G., N.F. Smith, R.W. Graves, and N.W. Abrahamson. (1997). Modification of Empirical Strong Ground Motion Attenuation Relations to Include Amplitude and Duration Effects of Rupture Directivity. Seism. Res. Let., Vol. 68, No. 1. pp. 199-222. Stepp, J.C. (1973). Analysis of Completeness of the Earthquake Sample in the Puget Sound Area. Contributions to Seismic Zoning: U.S. National Oceanic and Atmosphere Administration Technical Report ERL 267-ESL30, pp. 16-28. Stewart, J.P., S. Chiou, J.D. Bray, R.W. Graves, P.G. Somerville, and N.A. Abrahamson.(2001). 48 Ground Motion Evaluation Procedures for Performance Based Design. Peer Report 2001/09. Peer Center, College of Engineering, UC Berkley. Toro, G.R., N.A. Abrahamson, and J.F. Schneider (1997). Model of Strong Ground Motions from Earthquakes in Central and Eastern North America: Best Estimates and Uncertainty. Seism. Res. Letters, Vol. 68, No. 1, pp. 41-57. Wells, D.L., and K.J. Coppersmith (1994). New Empirical Relationships Among Magnitude, Rupture Length, Width, Area & Surface Displacement. Bull. Seism. Soc. Am., Vol. 84, No. 4, pp. 974-1002. Youngs, R.R. and K.J. Coppersmith (1985). Implications of Fault Slip Rates and Earthquake Recurrence Models to Probabilistic Seismic Hazard Estimates. Bull. Seism. Soc. Am., Vol. 75, No.4, pp. 939-964. 49