DEVELOPMENT OF A JOINT SEISMIC HAZARD CURVE FOR MULTIPLE SITE SEISMIC HAZARD

Transcription

1 DEVELOPMENT OF A JOINT SEISMIC HAZARD CURVE FOR MULTIPLE SITE SEISMIC HAZARD by DARYN R HOBBS B.S., University of Colorado Boulder, 2013 M.S., University of Colorado Boulder, 2013 A Master s Report submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Department of Civil, Environmental, and Architectural Engineering 2013

2 This Master s Report entitled: Development of a Joint Seismic Hazard Curve for Multiple Site Seismic Hazard written by Daryn Hobbs has been approved for the Department of Civil, Environmental, and Architectural Engineering Professor Keith Porter, Ph.D., P.E. Professor Ross Corotis, Ph.D., P.E. Professor Abbie Liel, Ph.D., P.E. Date The final copy of this Master s Report has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline.

3 Hobbs, Daryn R (M.S., Civil Engineering) Development of a Joint Seismic Hazard Curve for Multiple Site Seismic Hazard Master s Report directed by Associate Research Professor Keith Porter The state of the art in probabilistic seismic hazard analysis was examined and these concepts, along with recent work in spatial ground motion correlation, were extended to develop the notion of a joint seismic hazard curve: a 3-dimensional surface that shows the probability of exceeding a ground motion intensity measure at two sites within the same time period. Some elementary earthquake characterization concepts were reviewed, i.e. local magnitude, moment magnitude, etc., and different intensity measures were considered including macroseismic intensity and instrumental intensity. Ultimately peak ground acceleration, a measure of instrumental intensity, was chosen for use in the joint seismic hazard curve although the same methodology could be used for spectral acceleration or other measures of instrumental intensity. A seismic hazard curve was created using data from the OpenSHA event set calculator, the theorem of total probability, and the assumption of Poisson arrivals for earthquake occurrence. A methodology was created for the creation of a joint seismic hazard curve for two sites with two separate underlying assumptions: 1) Sites have conditionally independent ground motions and 2) Sites have correlated ground motions. It was found that the correlated ground motion assumption should not be used for sites farther than 10 km apart. i

4 TABLE OF CONTENTS 1. INTRODUCTION Problem Statement Objectives Organization of Report CHARACTERIZING EARTHQUAKES Magnitude Macroseismic Intensity Instrumental Intensity THE DEVELOPMENT OF SEISMIC HAZARD Early Descriptions of Hazard Evolution of PSHA Earthquake Rupture Forecasts Ground Motion Prediction Equations Probabilistic Calculation SPATIAL GROUND MOTION CORRELATION Site Seismic Hazard and Separation Distance Relationships for Correlation Coefficients Use of Semivariograms in Ground Motion Correlation JOINT SEISMIC HAZARD CURVES Conditionally Independent Joint Seismic Hazard Curve Correlated Joint Seismic Hazard Curve CONCLUSIONS REFERENCES ii

5 LIST OF FIGURES Figure 1 Seismic hazard curves Figure 2 Test sites in San Francisco Bay Area, CA Figure 3 Test sites in San Fernando Valley, CA Figure 4 Test sites in Memphis, TN Figure 5 - Comparison of Correlation Coefficients Figure 6 Sample Semivariogram Figure 7 Joint Seismic Hazard Curve, Same Site Figure 8 - Joint Seismic Hazard Curve, Separate Sites iii

6 1. INTRODUCTION 1.1. PROBLEM STATEMENT The concept of probabilistic seismic hazard dates back to the 1960s and has since been continuously modified and refined (Field E. H., 2005). One of the products of probabilistic seismic hazard analysis (PSHA) is a seismic hazard curve that characterizes seismic hazard in terms of a ground motion intensity measure (IM) Modified Mercalli Intensity (MMI), peak ground acceleration (PGA), damped elastic spectral acceleration response (Sa) (henceforth all mention of Sa assumes 5% damping), peak ground velocity (PGV), etc. for a single site in a finite period of time, e.g. one year. The curve shows the probability of exceeding any probable value of an IM in that time period. This curve is useful when planning for future seismic risk and, in particular, it is used in the USGS National Seismic Hazard Mapping Program. This information can then be transferred to engineers and policy makers for use in the development of engineering standards and codes. To better understand the seismic hazard at two or more sites the concept of a joint seismic hazard curve is proposed. The joint curve considers the probability of exceeding an IM at a site A and a site B at least once during the same earthquake in a specified period of time. A joint seismic hazard curve could be particularly useful in managing a group of properties that vitally operate together to perform critical functions for an organization or city. For example, consider two vehicular bridges that serve as the 1

7 only means of egress from an isolated community susceptible to earthquakes. Knowing the probability that the ground motion will exceed a certain value at each site at the same time would be valuable when designing or retrofitting the structures, planning emergency evacuation routes, or for large earthquake simulations such as ShakeOut. A joint seismic hazard curve would provide a systems approach to understanding the seismic vulnerability of the community s major transportation infrastructure and could also be extended for use in quantifying the risk of other natural hazards OBJECTIVES This report first explores the framework necessary to create a seismic hazard curve. It works within this framework to formulate the steps required to create a joint seismic hazard curve for two sites under the assumption that their ground motions are independent of each other. An attempt is then made to improve this assumption by incorporating spatial ground motion correlation ORGANIZATION OF REPORT This section introduces the problem and lays out the objectives of the report. Section 2 provides a review of the methods used to characterize the size of an earthquake and the resultant ground shaking. A detailed explanation of the PSHA process is covered in Section 3, ending with a seismic hazard curve created by the author. Section 4 covers recent work that has been completed on spatial ground motion correlation and introduces empirical expressions for the correlation of the 2

8 ground motion at two sites as a function of distance. Section 5 explains the steps taken to create a conditionally independent joint seismic hazard curve and a correlated joint seismic hazard curve. 3

9 2. CHARACTERIZING EARTHQUAKES 2.1. MAGNITUDE Several methods have been developed to quantify the amount of energy released by an earthquake. Richter s definition of magnitude is one of the most wellknown magnitude scales and was first developed in His scale uses data describing ground motion obtained from a Wood-Anderson seismograph; specifically, the maximum amplitude from the horizontal ground motion measurements taken 100 km away from the epicenter is used (Bullen & Bolt, 1985). His equation is shown below: Equation 1: Bullen & Bolt (1985) where A ( m) is the maximum amplitude traced on a seismogram located at a (1) distance (km) from the epicenter and A0 is an empirical correction factor that accounts for the distance the measurement was taken from the epicenter. The magnitude found using this method is now referred to as the local magnitude, ML. Instrumental limitations require this method to use data taken far away from a very large earthquake s source and hence use data that are less accurate in determining energy levels (Housner & Jennings, 1982). Further, Richter s scale is only appropriate for shallow earthquakes (Bullen & Bolt, 1985). Refinement by Richter and Gutenberg resulted in a modified Richter scale. Empirical tables were developed to calculate magnitudes for significantly deeper earthquakes and two notions of magnitude were developed using measurements 4

10 from surface waves, Ms, and body waves, Mb. They determined that the two measures were related and developed a relationship to calculate the energy contained in these waves (Kanamori, 1978). This relationship is an approximation as the scales only use surface and body waves with specific periods and do not account for the entire spectrum of waves. Error exists when using this method to estimate the energy of very large earthquakes that produce significant long-period waves (Kanamori, 1978). With advances in seismometer technology came advances in using longperiod wave data to describe the energy produced by very large earthquakes. The notion of seismic moment, M0, was developed by Aki (1966) and can be calculated with Equation 2: Equation 2: Aki (1966) where (dyn/cm 2 ) is the rigidity of the material surrounding the fault, D (cm) is the average displacement discontinuity along the rupture plane, and S (cm 2 ) is the surface area of the fault s rupture. The seismic moment can then be used to calculate the elastic strain energy released in an earthquake, shown in Equation 3: (2) Equation 3: Kanamori (1977) where is the stress drop in the fault. Kanamori (1977) found that for large earthquakes (Ms > 6.5) the stress drop is almost constant (see Equation 5). Therefore by using the relationship for energy given in Equation 4, Equation 5 and (3) 5

11 the energy-magnitude relationship developed by Gutenberg and Richter, shown in Equation 6, the moment magnitude, Mw, can be calculated per Equation 7: Equations 4: Kanamori (1977) (4) (5) (6) (7) Equations 5-7: Kanamori (1978) where M0 is in dyn-cm, H is the energy lost due to friction, and W is the wave energy. Equation 5 neglects the heat loss during the fault rupture (Kanamori, 1977). Moment magnitude is the standard scale used today largely due to its lack of saturation at large magnitudes MACROSEISMIC INTENSITY An earthquake s effects on buildings, infrastructure, and people cannot always be directly related to the amount of energy released, as described by its magnitude. Intensity scales have been developed to better describe the relative shaking intensity an earthquake produces and the subsequent damage. The macroseismic parameters include those that can be described in layman s terms and are a result of the general nature of the ground acceleration and the complex action of the seismic waves. Since the above parameters are not easily quantified in a holistic sense, a qualitative basis is used for intensity scales. These scales describe macroseismic intensity and conglomerate the effects of the parameters mentioned above to a simplified intensity level. The most prominent scale in North America is 6

12 the Modified Mercalli Intensity Scale (MMI). An earthquake s intensity is measured on this scale by how damaging it was to different building types, how people reacted to it, what the effects were on the geography of the surrounding area, and other qualitative descriptions. The MMI scale contains discrete levels ranging from I to XII. For example, at level I no ground shaking is felt, at level VI the earthquake is felt by everyone but there is no significant damage, at level IX heavy damage is caused with poorly designed structures collapsing or experiencing severe damage, and at level XII severe damage is incurred to all structures, surface waves can be seen on the ground, and large ground masses are displaced or altered (Bullen & Bolt, 1985). An intensity scale such as this can be used to map out regions that experienced similar intensities which are separated by lines called isoseismal curves (Bullen & Bolt, 1985). As waves travel farther from the source of the earthquake, energy is dissipated and intensity decreases. Isoseismal lines look similar to a topographical map with the largest intensity regions near the epicenter INSTRUMENTAL INTENSITY Instrumental intensity measures especially PGA and Sa are commonly used to quantify ground shaking at specific locations during a particular size of earthquake. In structural design, a nominal, maximum force is desired and this must respect the dynamic nature of earthquake loads in combination with the corresponding dynamic behavior of the structure. The maximum ground acceleration, and resultant structural response at a given site is of interest here, 7

13 and these are quantified by PGA and Sa. It is important to note that Housner and Jennings (1982) suggest that using a single number can be problematic, It is inherently impossible to describe a complex phenomenon by a single number, and a great deal of information is inevitably lost when this is attempted. The nature of the ground motion time history is also important; the effect of a given acceleration can be different if it is short and abrupt than if it is long and powerful. To calculate spectral acceleration, response spectra are created based on historic earthquake data to account for the various damping characteristics and modal frequencies that a structure may have under possible earthquake scenarios. Response spectra are created using a time history of ground acceleration, velocity, or displacement recorded during a single earthquake at a specific site. These data, typically acceleration data, are then used as input for structural models that calculate the dynamic response of a single degree-of-freedom elastic oscillator with a certain natural period of vibration and damping ratio (most often 5%). The calculations are repeated for many different natural periods and the maximum relative displacement, relative velocity, and total acceleration is recorded for each period. This is used to create a spectrum of the expected response to the specified ground motion at a site. A response spectrum shows how an earthquake affects structures oscillating at different frequencies and can produce spectral accelerations, velocities, and displacements. This entire process can be repeated using a different damping ratio each time, typically ranging from 0% to 10% damping. 8

14 Response spectra are useful for retroactively determining how a structure would respond to an earthquake at a given site, but a more involved approach must be used for design purposes. Since it is nearly impossible to predict the precise ground motion of an impending earthquake, even with historical data, design spectra must consider the probability an earthquake will occur at all possible locations that could affect the site, as well as the ground motion attenuation between the earthquake s location and the site of interest. Further, to be inserted into a design standard or code, the design spectra must be simplified and made more conservative to be applicable on a regional basis. ASCE 7-10, for example, offers a simplified design spectrum that has a constant design acceleration section around Sa(0.2 sec, 5%) and a long period shape that represents the average of many records. For these reasons, understanding site seismic hazard allows a more refined analysis to be performed than what is required in a simplified building code. 9

15 3. THE DEVELOPMENT OF SEISMIC HAZARD 3.1. EARLY DESCRIPTIONS OF HAZARD In the early years of modern seismology, late nineteenth and early twentieth centuries, the concept of seismic hazard was ill-defined and poorly understood. It was during this time that seismologists postulated that when the ground slips and cracks along faults, earthquakes occur and not vice versa (Reitherman, 2012). Discovering this relationship was an essential starting point in defining and calculating seismic hazard. This crucial concept also provides a hint at the inherent intricacies involved in determining hazard since fault sections and other potential earthquake sources must be considered on a source-by-source basis. Advancement can be seen in the post-earthquake publications of the 1906 San Francisco Earthquake and Fire. Serving as a major learning experience for engineers and seismologists of the time, the analysis of the earthquake s destruction resulted in many conclusions that would assist in the later developments of hazard quantification. Some of these conclusions include the amplifying effect of soft and alluvial soils, the notion of how complex ground motion attenuation can be, and the ways in which a fault slip can propagate (Gilbert et al., 1907). Engineers realized that these parameters were significant contributors to how damaging the San Francisco Earthquake was to specific sites but were unable to develop distinct relationships between them to be useful for assessing site seismic hazard in the future. The conclusion that the damage at a site caused by an earthquake is a 10

16 function of the distance from the earthquake source and the type of ground the site is on was vital in the development of seismic hazard. One of the early methods for quantifying seismic hazard at a specific site was developed by Cornell in He attempted to provide the method for integrating the individual influences of potential earthquake sources, near and far, more active or less, into the probability distribution of maximum annual intensity. (Cornell, 1968). This is regarded as the first development of PSHA written in English. One of the results of this approach is the number of expected earthquakes in one year that will exceed a given level of MMI. More generally, this method can produce the probability distribution of the greatest MMI that will occur at a site throughout some interval of time (Cornell, 1968). Aside from MMI, Cornell demonstrated that other variables of interest could be calculated if a relationship existed between that variable and an earthquake s magnitude and focal distance. Thus, in a similar fashion, calculations could be performed to create probability distributions for instrumental intensity measures such as PGA, Sa, and PGV. Cornell s approach is an application of the theorem of total probability, integrating over the occurrence frequency of nearby possible earthquake ruptures and the resulting probabilistic ground motion at the site of interest, to arrive at an exceedance rate (events per year, for example) of various levels of IM. If one assumes that earthquakes can be treated as a Poisson process, one can then estimate the probability that any specified value of IM will be exceeded during any period of interest. 11

17 Cornell s analytical approach relies heavily on its assumption that earthquakes be modeled as Poisson arrivals. It assumes that earthquakes occur independently of each other in time. As is pointed out in his paper, this assumption does not consider aftershocks and is not consistent with elastic rebound theory. In order for his method to be simple, it also considers that there is an equally likely chance for an earthquake to occur everywhere along a fault. Cornell s method provided an analytical model of determining seismic hazard suitable for hand calculations but a more detailed model including more refined empirical attenuation relationships was necessary to increase its applicability EVOLUTION OF PSHA Advances in PSHA have resulted in refined methods that address a breadth of the variables that affect earthquake occurrence and subsequent ground motion. Modern PSHA consists of three parts: one or more earthquake rupture forecasts (each with an associated weight or Bayesian probability), one or more ground motion prediction equations (GMPEs, again each with an associated weight or Bayesian probability) and an application of the theorem of total probability similar to that done by Cornell (1968) but with a logic tree to account for different earthquake rupture forecasts and GMPEs (Field E. H., 2005). In the past three decades there have been major advancements in earthquake rupture forecasts and GMPEs which have provided a better grasp on seismic hazard, especially in the Western United States. 12

18 EARTHQUAKE RUPTURE FORECASTS As an attempt to address all factors contributing to earthquake occurrence, some recent seismic earthquake rupture forecasts have become lengthy and complex. These models use a brute-force method that exhaustively accounts for all possible earthquake sources and all estimated sources of uncertainty. As time passes with today s level of seismic recording technology, the amount of historical data that researchers have available to them increases. These data serve as a baseline for models to build off and a means to judge accuracy. In recent years, major contributions to rupture forecasts have come from the development of the Uniform California Earthquake Rupture Forecast (UCERF), now in its second version developed in 2008 and referred to as UCERF 2. This forecast addresses the first steps required in PSHA: to calculate a probability that a given magnitude earthquake will occur in a region in a defined time span. UCERF 2 consists of four sets of models used in this order: fault models, deformation models, earthquake rate models, and then probability models. The fault models break all of California s known faults into sections and assign each section a set of parameters of interest such as geometry, average slip rate, and aseismicity factor (Field et al., 2009). These sections are created and assembled based on various theories and this results in two models: FM2.1 and FM2.2. Next, deformation models use the fault models to calculate a slip rate for each fault section. Six deformation models, DM2.1 through 2.3 for FM2.1 and DM2.4 through 2.6 for FM2.2, were created to reduce epistemic uncertainty; the 13

19 main discrepancy being how to distribute the slip rates among neighboring faults. Thirdly, Earthquake rate models take the above information and determine the rate at which all possible damaging earthquakes could occur on each fault section. The models created here are dependent on the type of source, type-a, -B, or -C, the global slip rate between the North American and Pacific Plates, and the long-term and short-term nature of how energy is released through earthquakes. Type-A, -B, and -C sources are classified by the amount of information known about the fault or region. Enough is known about type-a sources that permanent rupture boundaries (segment endpoints) can be hypothesized and a stress-renewal recurrence model can be applied (Field et al., 2009). Type-B sources include faults that have slip-rate estimates but there are not enough historical data to create stress-renewal probabilities. Type-C sources include area sources where there is not enough information to assign slip to distinct faults. One fault model reduces the rate at which earthquakes occur in order to produce larger events while another does the opposite. A segmented model only allows earthquakes to occur along predefined segments of fault sections, disallowing fault-to-fault ruptures and an unsegmented model allows for fault ruptures to jump between segments. Finally, probability models describe how the earthquakes could occur throughout a given period of time. The probability models can be grouped into two categories: time independent and time dependent (Field et al., 2009). One of the defining characteristics of UCERF 2 is its means of managing epistemic uncertainties by way of a logic tree. Each of the previously mentioned sets 14

20 of models represents the seismological community s best guess as to how the phenomenon should be modeled and with which variables. At this time, it is not known which model perfectly predicts the phenomenon of interest. To manage these uncertainties the logic tree considers the leading models and weights them according to the UCERF 2 authors degree of belief in each model. However, this tree has 480 branches that have varying results and for this reason efforts have been made to trim the tree and use only the uncertainties that contribute the most (Porter et al., 2012). The full logic tree is somewhat computationally intensive and in the end, an average value is used for the final probability of earthquake occurrence (Field et al., 2009) GROUND MOTION PREDICTION EQUATIONS Earthquake rupture forecasts contain the information needed to quantify the seismicity of a region and the next step in PSHA is to use GMPEs to determine how strongly an earthquake will shake a particular site within that region. As used previously by Cornell, the two main input variables among the early GMPEs are earthquake magnitude and source-to-site distance. Early attenuation calculations were theoretically based and followed physicsbased attenuation law. They used the exponential decay function with the following independent variables: wave frequency, travel path, velocity, and an attenuation constant which was dependent on geologic conditions (Trifunac & Brady, 1975). Trifunac and Brady also mention the use of finite element analysis but conclude that this method is not ideal because it is too computationally expensive and there 15

21 are a lack of available data required to create a precise geophysical ground model. There was a trend in the 1980s and 1990s of researchers not only working on mathematically based analytical models, but also producing empirical ones. Different approaches were taken in curve-fitting and multiple equations have been developed with this methodology. Similar to the physics-based attenuation law, the functional form of most regression-based equations is exponential; one reason being that the definition of moment magnitude is exponential (Boore & Joyner, 1982). These equations started out with separate terms for magnitude, distance, and site characteristics. Regression coefficients are on most or all terms. Equations 8-10 are examples of early GMPEs with variable names modified for ease of comparison: Equation 8: Joyner & Boore (1982) Equation 9: Campbell (1988) (8) (9) (10) Equation 10: Boore Joyner, & Fumal (1993) where in all equations Y is the IM of interest: peak ground acceleration, velocity, displacement, or spectral acceleration, M is moment magnitude, and r is some measure of source-to-site distance. In Equation 8, S accounts for site soil conditions and is binary: 0 for rock and 1 for soil, and P represents the uncertainty in the prediction. In Equation 9, s is a function of fault type, directivity, soil type, building size, and building embedment. In Equation 10, GB and GC represent site classification and are binary: GB=1 for site class B and GC=1 for site class C and both are zero otherwise, r and e are variables that account for the variability 16

22 within each earthquake record and between earthquakes, respectively. All other variables are regression coefficients and can be found in tables provided by the appropriate authors. Site class is defined by VS30 where for site class A VS30 is greater than 750 m/s, for site class B VS30 is less than 750 m/s and greater than 360 m/s, for site class C VS30 is less than 360 m/s and greater than 180 m/s, and for site class D VS30 is less than 180 m/s. Equation 10 is not applicable for site class D. The above equations use different measures of source-to-site distance. This includes hypocentral distance, epicentral distance, and various distances to the rupture surface. Some variables, including distance, are magnitude dependent. Equation 9 uses either surface wave magnitude or local magnitude but Equations 8 and 10 use moment magnitude. In later research, moment magnitude prevails and is prominent throughout all modern equations. Also seen in Equations 8-10 are differing methods of site classification. Equation 9 uses a function, s, to account for site classification, faulting parameters and even building attributes while Equations 8 and 10 use simple binary switches that turn regression coefficients on or off. In 1997, Boore, Joyner and Fumal as well as Abrahamson and Silva refined previous regression models and used more sophisticated site classification variables. Their models are shown in brief in Equations 11 and 12: (11) ( ) ( ) (12) Equation 11: Boore Joyner, & Fumal (1997), Equation 12: Abrahamson & Silva (1997) 17

23 where in both equations M is moment magnitude. In Equation 11 Vs is the average shear-wave velocity in the top 30 m of earth, r is a function of epicentral distance and a fictitious depth, and the bi s and VA are determined via regression. In Equation 12 the fi s represent functions of the enclosed variables and each function contains its own regression coefficients, rrup is the closest distance to the rupture plane, F accounts for fault type, HW accounts for the hanging wall effect, S accounts for site class, and pgarock is the expected peak ground acceleration on rock, and is used to estimate non-linear soil response. Over time, as GMPEs used terms for more source, site, and path variables and the data used in the regression became more numerous and included larger magnitude earthquakes, the applicability of the equations increased. This is true for both magnitude and distance. For example, the most recent equations initiated by the Next Generation Attenuation (NGA) Program can be applied to earthquakes with magnitudes between 4 and 8.5 and at distances from km whereas an early GMPE, i.e. Boore, Joyner, and Fumal (1993), could only be applied to magnitudes between 5 and 7.7 and at distances below 100 km. Two of the NGA equations are shown below: ( ) ( ) Equation 13: Boore & Atkinson (2008) ( ) ( ) ( ) (13) (14) Equation 14: Abrahamson & Silva (2008) where in Equation 13 FM, FD, and FS are magnitude, distance, and site amplification functions, respectively, Rjb is the Joyner-Boore distance which is the 18

24 closest horizontal distance to the surface projection of the fault, ε is the fractional number of standard deviations of a single predicted value of ln(y) away from the mean of ln(y), and σ τ represents the inter- and intra-event aleatory uncertainty (Boore & Atkinson, 2008). In Equation 14 the fi s are functions of the enclosed variables, FRV, FNM, FAS, and FHW, are flags for reverse faulting, normal faulting, aftershocks, and hanging wall effects, respectively, PGA1100 is the median peak acceleration for VS30 = 1100 m/s, Rx is the horizontal distance from the top edge of rupture measured perpendicular to the fault strike, δ is the fault dip angle, ZTOR is the depth to top of fault rupture, and Z1.0 is the depth to VS=1.0 km/s (Abrahamson & Silva, 2008). The NGA equations have an increased range of applicability over the previous GMPEs but not without added complexity. Site classification was quantified using a continuous variable, Vs30, and the type of faulting was described in more detail. One may have to go through as many as six equations to calculate the soil depth term, f10. However, the NGA equation authors suggest an increase in applicability over previous GMPEs and even extrapolated beyond earthquake magnitudes used in the regression analysis. The availability of strong-motion data obtained close to the source (R < 20 km) of large earthquakes (M > ~7.0) has limited GMPEs (Joyner & Boore, 1988) but as more data become available over time, it is likely that new equations have potential for higher accuracy and wider applicability. 19

25 PROBABILISTIC CALCULATION The last step in PSHA is to combine earthquake rupture forecasts and GMPEs mentioned above, calculating the level of hazard on a site-specific level. The earthquake rupture forecasts provide the rate at which earthquakes of various, discrete magnitudes occur throughout time. A GMPE takes certain characteristics of each of these earthquakes magnitude, distance from site, fault type, etc. along with the site conditions to calculate the probability distribution for the desired IM: PGA, Sa, PGV, etc. For engineering design purposes, it is the probability that an IM will be exceeded that is of greatest interest. The rate that a ground motion will be exceeded in time span T is calculated by multiplying the rate,, at which any earthquake i will occur by the probability that an IM will exceed a certain value given that earthquake occurs, then summing this product over all possible earthquakes that can cause substantial ground shaking at the site, as shown in Equation 15: (15) where Ei represents any one of N earthquakes that occurs with rate, IMA is the intensity measure at site A, and IM1 is some arbitrary intensity measure. It is important to note that one should avoid including earthquakes with a source-to-site distance greater than the applicable distance of the GMPE being used and with a magnitude outside of the GMPE s applicable range. 20

26 To arrive at a probability of exceedance, it is convenient to assume that earthquakes occur as Poisson events. Equation 15 is used in Equation 16, to calculate the probability of at least one occurrence of IM1 being exceeded, resulting in Equation 17. (16) ( ) (17) This process can be repeated to create a seismic hazard curve for any site in a region sufficiently described by an earthquake rupture forecast and with applicable GMPEs. An example of a seismic hazard curve was created using the steps described above and can be seen in Figure 1. The Event Set Calculator from OpenSHA was used to collect the required data and perform the calculations needed to find the effective earthquake sources and their respective rates as well as the probability distributions of the ground shaking at the site; both terms are shown in Equation 15. UCERF 2 was used to identify earthquake sources and their respective rates of occurrence at varying magnitudes and the Boore & Atkinson (2008) GMPE was used to calculate mean PGA at sites in the San Francsico Bay Area. Then Equation 10 was used to calculate the probabilities of exceedance for various PGA levels. 21

27 Figure 1 Seismic hazard curves shown for two sites 7.2 km apart. Site B is base of the new San Francisco-Oakland Bay Bridge tower ( N, W) and Site A is 7.2 km northeast of Site B ( N, W). 22

28 4. SPATIAL GROUND MOTION CORRELATION There are situations where one must quantify the ground shaking at multiple, closely spaced sites instead of studying ground shaking on an individual site basis. In doing so it is important to consider spatial ground motion correlation, or the tendency of two sites separated in space to have similar ground shaking under the same earthquake. For example, in a series of earthquakes two sites may have PGAs that both tend to be above or below the mean PGA, calculated with a GMPE, in each earthquake. Physically, this correlation can be attributed to the similar travel paths of seismic waves as well as both sites being a similar distance from discontinuities along the fault if the fault rupture is very long compared to the source-to-site distance (Park et al., 2007). Including ground motion correlation at multiple sites is of considerable importance in the risk assessment of a portfolio of properties in close proximity (Park et al., 2007; Jayaram & Baker, 2009). Park et al. (2007) note that by accounting for the correlation of ground motion between multiple sites, portfolio loss estimates become more accurate; if spatial ground motion correlation is not accounted for then the occurrence frequency of very strong ground motion at both sites tends to be underestimated and the occurrence frequency of weaker ground motion at both sites tends to be overestimated. Further, utilizing correlation can be of particular interest to increase the accuracy of loss estimates and ground shaking maps that are produced immediately after an earthquake occurs less than 20 minutes after an event (David Wald, personal communication, March 15, 2013) 23

29 such as the USGS PAGER and ShakeMaps programs (Park et al., 2007). These prompt post-earthquake reports are especially helpful in an emergency response context where response crews need to be dispatched to the most severely affected areas first SITE SEISMIC HAZARD AND SEPARATION DISTANCE The author performed an investigation studying how the seismic hazard between two sites varies with distance for three regions: San Francisco Bay Area, CA; San Fernando Valley, CA; and Memphis, TN. The purpose of this investigation was to gain a sense of the distance scale for which seismic hazard differs significantly between two sites as a function of separation distance. Might sites have similar seismic hazard if they are 100 km apart? 10 km? 1 km? A main site of interest was chosen and several sites surrounding the main site located along two azimuths and at various distances. The azimuths were chosen to be parallel and perpendicular to faults immediately surrounding the site. See Figures 2-4. Figure 2 Test sites in San Francisco Bay Area, CA. Faults shown in red. 24

30 Figure 3 Test sites in San Fernando Valley, CA. Faults shown in red. Figure 4 Test sites in Memphis, TN. Then a seismic hazard curve was created for each site using the USGS Java Ground Motion Parameter Calculator. This tool uses USGS data from 2002 to calculate seismic hazard assuming site conditions are on the NEHRP Site Class B-C boundary where Vs30 = 760 m/s. Specific probabilities of exceedance (PE) were examined for comparison including 50% in 10 years, 10% in 50 years, 5% in 50 years, and 2% in 50 years. 25

31 The results, shown in Table 1, show that for sites chosen perpendicular to the surrounding faults, as distance from the main site increases, similarity to the PGA of the main site decreases at a faster rate than for sites chosen parallel to the surrounding faults. For example when the new Bay Bridge tower in the San Francisco Bay Area was the main site, a second site along a parallel azimuth to nearby faults could be as far as 40 km away from the main site before PGA (2% PE in 50 years) differed by more than 10% from the main site while a second site only 4 km away along a perpendicular azimuth differed by more than 10%. It was concluded that on a particular site basis there are factors other than separation distance that affect the similarity of one site s seismic hazard to another s. These include directivity in relation to faults and soil conditions. Although they were not considered in this particular investigation, it can be expected that site soil conditions can significantly increase the difference in PGA between two sites considering all modern GMPEs account for soil conditions. Table 1 - Results for 10% PE in 50 Years Location Directionality Separation Distance (km) Difference in PGA from Main Site San Francisco Bay Area San Fernando Valley Nearby Faults 7.2 1% Nearby Faults % Nearby Faults % Nearby Faults % 26

32 4.2. RELATIONSHIPS FOR CORRELATION COEFFICIENTS Formal studies have been completed on spatial correlation of ground motions within a single earthquake in the past two decades. These start with the general formulation of a GMPE shown in Equation 18: (18) where is the median ground motion in an earthquake i at site j, and are the inter- and intra-event residuals respectively, and and are the standard deviations for the respective residuals. This general form is used by many recent authors who separate inter- and intra-event uncertainty. Next, a spatial correlation coefficient is introduced to quantify the correlation between sites and thus modify the term (Park et al., 2007). In essence this process adjusts the predicted ground motion parameter away from the mean to account for the correlation that exists between the site of interest and a nearby site. The correlation coefficient,, is defined in Equation 19: (19) where is the variance of the differences of the natural logarithm of the motion at two sites or the unexplained variance, and is the variance of the natural logarithm of the motion at a single site or the total variance. Multiple equations have been created to calculate using different data sets. These equations must satisfy the condition that at inter-site distances of 0 and, the correlation coefficient must be 1 and 0, respectively. Thus, most equations follow 27

33 the form of exponential decay (Abrahamson & Sykora, 1993; Boore et al., 2003; Park et al., 2007; Goda & Hong, 2008; and Goda & Atkinson, 2009) and this general form for PGA and SA(TN) is shown in Equation 20. Equations include independent variables that are frequency dependent indicating that correlation tends to increase for longer periods and decrease for shorter periods (Abrahamson & Sykora, 1993): ( ) Equation 20: Goda & Hong (2008) [ ( ) ] Equation 21: Goda & Atkinson (2009) Equation 22: Abrahamson & Sykora (1993) (20) (21) (22) from Boore & Atkinson (2008) (23) where,, c1, and c2 are constants, and is the separation distance between two sites in km. Equation 21 has been modified from its original form to extract a comparable correlation coefficient by assuming the value of to be the intraevent standard deviation calculated with the NGA GMPE from Boore & Atkinson (2008). Equations are examples of correlation coefficient relationships that are not frequency dependent and Park et al. (2007) examine the effects of simplifying Equation 20 to Equations 27 and 28: ( ) (24) ( ) (25) 28

34 ( ( )) Equations 24-26: Boore et al. (2003) (26) ( ) (27) Equations 27 & 28 : Park et al (28) Importantly restated, the equations above were established using different data sets e.g. Goda & Atkinson (2009) use data from the K-NET and KiK-net networks in Japan while Boore et al. (2003) use data only from the 1994 Northridge Earthquake; both studies examine PGA and Sa and their differences imply that the decay of spatial correlation can depend on geographic region and type of earthquake (Goda & Atkinson, 2009). It was not pertinent for this report to perform a detailed analysis comparing the equations shown above but a graphical comparison is provided in Figure 5 to show their variability. This report is most concerned with the existence of these equations and how they can be incorporated into joint seismic hazard. Although the above equations were developed for PGA and Sa, correlation coefficients also exist in terms of PGV. Wang & Takada (2005) perform a study on PGV correlation and adopt the same exponential decay expression shown in Equation 20. Their results show a separation distance range of km before ground motions are completely uncorrelated. 29

35 Correlation Coefficient, ρ Abrahamson & Sykora (1993) Boore et al. (2003) Park et al. (2007) (Eq. 26) Park et al. (2007) (Eq. 27) Goda & Hong (2008) Goda & Atkinson (2009) Separation Distance, Δ (km) Figure 5 - Comparison of Equations showing their variability and how correlation diminishes quickly with separation distance. The correlation coefficients drop off very quickly after a few kilometers of separation distance between sites, notably in Equations 20 and 28. Equations 20, 21, 27, and 28 show a general lack of correlation around Δ = 10 km in Figure 5. In these studies the correlation of ground motions in a single earthquake were of interest while in the author s investigation in Section 4.1, the similarity in overall seismic hazard as a function of separation distance was of interest. Although this difference exists, preventing the studies from being directly comparable to each other, it is important to note that the lack of correlation around Δ = 10 km corresponds well with the results of the investigation in Section

36 4.3. USE OF SEMIVARIOGRAMS IN GROUND MOTION CORRELATION Another way spatial correlation has been described in recent work is by use of semivariograms. This technique is typically used in geostatistics and provides a measure of how dissimilar two random variables can be in relation to distance; put simply, it quantifies the assumption that the closer two sites are, the more alike their ground motions will be in the same earthquake. The functional form is shown in Equation 29: { } (29) Equation 29: Jayaram & Baker (2009) where is the semivariogram function, is the spectral acceleration at site u, and is the spectral acceleration at a site a distance from site u. Jayaram and Baker (2009) show that the semivariograms created using PGA and Sa data from the 1994 Northridge and 1999 Chi-Chi Earthquakes are isotropic and hence, need not incorporate the directionality between sites. Two important parameters produced by a semivariogram are the range and sill. The range is the distance at which the ground motion is no longer correlated. The peak y-value of the semivariogram, realized at the range, is called the sill. If a sill exists then this implies there is a distance where ground motion is no longer correlated (Chiles & Delfiner, 2012). Jayaram & Baker (2009) show the relationship between a semivariogram and the correlation coefficient in Equation 30: (30) Equation 30: Jayaram & Baker (2009) 31

37 γ( ) where is the experimental stationary semivariogram which is estimated from a sample of data and is an estimate of the ground motion correlation coefficient,. Equation 30 uses normalized intra-event residuals so that has a sill of 1 implying that there exists a separation distance where ground motions are uncorrelated. For more on the development of Equation 30 see Jayaram & Baker (2009). A sample semivariogram for visualization purposes is shown in Figure 6 using Equation 31 from Jayaram & Baker (2009). ( ) (31) Equation 31: Jayaram & Baker (2009) Separation Distance, Δ (km) Figure 6 Sample Semivariogram from Equation 31 Jayaram & Baker (2009) confirm the results of the frequency dependent correlation coefficients described in Section 4.2: that correlation tends to increase 32

38 with period. This is shown as an increasing range in the semivariograms calculated for the Northridge and Chi-Chi earthquakes when range was plotted against period. 33

39 5. JOINT SEISMIC HAZARD CURVES Let a joint seismic hazard curve refer to the combining of hazard curves for two sites into one 3-dimensional surface where the IM at Sites A and B are along the horizontal x- and y-axes and the probability of exceeding both of those IM s on the vertical z-axis. Rather than just understanding probabilistic seismic hazard at a single site, a joint seismic hazard curve provides a better understanding of hazard at two different sites that vitally act together CONDITIONALLY INDEPENDENT JOINT SEISMIC HAZARD CURVE It is proposed here that the form of a conditionally independent joint seismic hazard curve takes on a 3-D surface with IM at sites A and B along the x- and y- axes and PE on the vertical z-axis. The x-z plane where y=0 and y-z plane where x=0 show the independent hazard curves for each site separately. The calculation of the Probabilities of exceedance for the rest of the x-y plane, away from the axes, starts with the equation for the joint probability of two independent random variables shown in Equation 32: (32) where IMA and IMB are intensity measures at sites A and B, respectively and IM1 and IM2 are some arbitrary values to be exceeded. This assumes that IMA and IMB are independent random variables. Next Equation 32 is inserted in Equation 15 and followed through to Equation 17 to arrive at Equation 33: 34

40 (33) To develop the conditionally independent joint seismic hazard curve the same procedure described in Section was used except with Equation 33 in lieu of Equation 17. Two curves were created: Figure 7 shows perfect correlation, i.e. Site A and Site B are the same site, and Figure 8 is for two sites that are 7.2 km apart. Figure 7 Joint Seismic Hazard Curve showing PGA with 50-year probability of exceedance for the base of the new San Francisco-Oakland Bay Bridge tower ( N, W). Sites A and B are perfectly correlated, i.e. Site A=Site B. 35

41 Figure 8 - Joint Seismic Hazard Curve showing PGA with 50-year probability of exceedance. Site B is the base of the new San Francisco-Oakland Bay Bridge tower ( N, W) and Site A is 7.2 km northeast of Site B ( N, W). As stated previously, the curves, or surfaces rather, show the 2-dimensional hazard curves for sites A and B on the x-z plane where IMB=0 and y-z plane where IMA=0, respectively. Figure 7 shows that for a given IM at Site A, as the IM at Site B increases, the PE stays constant until IMA = IMB, and vice versa. Figure 8 does not show this same effect and the probability of exceeding the same IM at both sites tends to be less than in Figure 7. This can be attributed to the calculation of the joint probability using Equation CORRELATED JOINT SEISMIC HAZARD CURVE If ground motion correlation is accounted for, then the covariance of IMA and IMB conditioned on event Ei is no longer zero and they cannot be treated as conditionally independent events. In this case, Equation 34 must be used: (34) 36

42 where the exceedance probability of an IM at one site is influenced by the IM at another site. This conditional probability allows for treating correlation between the two sites in the following manner. Let and represent the expected value of the intensity measure at sites A and B, respectively. The exceedance probability of IMA (denoted by the second term on the right-hand side of Equation 34) is written out in Equation 35 using the cumulative distribution function. ( ) (35) Following the procedure used by Boore et al. (2003), the natural logarithm of IM at site A is assigned as the expected value of the natural logarithm at site B, shown in Equation 36: (36) Next the first term on the right-hand side of Equation 34 is defined in Equation 37. The numerator represents the conditional exceedance probability of IMA and IMB. In the procedure used by Boore et al. (2003), the exceedance probability of IMB depends on each distinct value of IMA and thus the numerator is necessarily an integral, rather than a discrete summation. Note that the second term in the integral uses the probability density function, not to be confused with the cumulative distribution function used in the first term. Since only the conditional exceedance probability of IMB given IMA is of interest, the exceedance probability of IMA must be divided out and exist in the denominator. 37