A Short Note on Probabilistic Seismic Hazard Analysis

Although probabilistic seismic hazard analysis (PSHA) is the most widely used method to assess seismic hazard and risk for various aspects of public and financial policy, it contains a mathematical error in the formulations. This mathematical error results in difficulties in understanding and application of PSHA. A new approach is presented in this paper. Seismic hazards derived from the new approach are consistent with the inputs in temporal and spatial characteristics. The hazard curve derived from the new approach is similar to those derived from flood and wind hazard analyses and can be used in risk analysis in a similar way. Introduction Probabilistic seismic hazard analysis (PSHA) has become the most widely used method to assess seismic hazard and risk for various aspects of public and financial policy since it was introduced by Cornell (1968) more than three decades ago. For example, the U.S. Geological Survey used PSHA to develop the national seismic hazard maps (Algermissen and Perkins, 1976; Frankel and others, 1996, 2002). These maps are the basis for national seismic safety regulations and design standards, such as the NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (BSSC, 1998), the 2000 International Building Code (ICC, 2000), and the 2000 International Residential Code (ICC, 2000). The seismic design parameters for nuclear facilities, such as nuclear power plants, are also determined from PSHA (NRC, 1997). The use of PSHA has caused difficulty in selecting a hazard level (ground motion) or risk level (ground motion with a probability of exceedance in a period) for engineering designs and other policy applications, however. For example, the 2000 International Residential Code (IRC-2000), based on the 1996 USGS maps with 2 percent probability of exceedance (PE) in 50 years, gives a design peak ground acceleration (PGA) of about 0.6g for Paducah, Ky., higher than the design PGA for San Francisco (Wang and others, 2003; Malhotra, 2005). An extremely high ground motion (5.0g PGA or greater) would have to be considered for engineering design of the nuclear waste repository in Yucca Mountain, Nev., if PSHA is applied (Stepp and others, 2001; Bommer and others, 2004). The use of PSHA also has the results that “the seismic risk to life and property from code-designed buildings is very different in different parts of the country” (Malhotra, 2005). The difficulty in using PSHA for engineering designs and policy applications is not only caused by lack of understanding and lack of data on earthquakes, but also by the technical deficiencies of PSHA. It is well known that different practitioners could derive greatly different PSHA results (SSHAC, 1997). SSHAC (1997) concluded “that differences in PSHA results are due to procedural rather than technical differences.” In other words, “many of the major potential pitfalls in executing a PSHA are procedural rather than technical in character” (SSHAC, 1997). Technical problems may still be one of the main reasons for the large differences, however (Wang, 2005). They have resulted in: (1) unclear physical basis; (2) obscure uncertainty; and (3) difficulty in determining a correct choice (Wang and Ormsbee, 2005; Scherbaum and others, 2005; Wang, 2005, in press (a), (b)). PSHA clearly inherits some technical deficiencies (Wang and others, 2003, 2005; Scherbaum and others, 2005; Wang and Ormsbee, 2005; Wang, 2005, in press (a), (b)). In this short note, the formulations of current PSHA will be reviewed and the probable causes of those technical deficiencies will be discussed. A new approach will also be presented and discussed. PSHA PSHA was originally developed to derive a theoretical hazard curve (i.e., ground motion vs. return period) for engineering risk analysis in consideration of the uncertainty in the number, sizes, and locations of future earthquakes (Cornell, 1968). Later, Cornell (1971) extended his method to incorporate the possibility that ground motion at a site could be different (i.e., ground-motion uncertainty) for different earthquakes of the same magnitude at the same distance, because of differences in site conditions or source parameters. Cornell’s (1971) was coded into a FORTRAN algorithm by McGuire (1976) and became a standard PSHA (Frankel and others, 1996, 2002). Following McGuire’s (1995) formulation, annual probability of exceedance (γ) of a ground-motion amplitude y can be expressed as a triple integration over earthquake magnitude, epicentral distance, and ground-motion uncertainty as ∑ ∫∫∫ ≥ = j R M j dmdrd r m y Y P f r f m f v y ε ε ε γ ε ] , , | [ ) ( ) ( ) ( ) ( * , (1) where νj is the activity rate for seismic source j; fM (m), fR (r), and fε (ε) are earthquake magnitude, source-to-site distance, and ground-motion density functions, respectively; ε is ground motion uncertainty expressed in a standard deviation (logarithmic); and P[Y≥y|m,r, ε] is the conditional probability that Y exceeds y for a given m, r, and ε. Equation (1) is very complex and only computed through numerical approaches. Equation (1) is further complicated by the nonunique interpretations of seismological parameters, which are commonly treated by a logic-tree in PSHA (SSHAC, 1997; Stepp and others, 2001; Scherbaum and others, 2005). In order to better understand the basics of PSHA, a special case in which all the sources are characteristic will be discussed here. For the characteristic sources, we have c c M m if M m if M m f = ≠ = , 1 , 0 { ) ( , (2) c c R r if R r if R r f = ≠ = , 1 , 0 { ) ( , (3) and c c if if f ε ε ε ε ε ε = ≠ = , 1 , 0 { ) ( , (4) where MC is magnitude of the characteristic earthquake, RC is the shortest distance between the site and source, and εC is the ground-motion uncertainty for MC at the distance RC. Therefore, for characteristic sources, equation (1) becomes ] , , | [ 1 ) ( * C C C j j j R M y Y P T y ε γ ≥ =∑ , (5) where Tj is the average recurrence interval of the characteristic earthquake for source j. In current practice, the inverse of the annual probability of exceedance (1/γ), called return period (TP), is more often used and interpreted, as the ground motion (y) that will occur at least once in that return period (Frankel and others, 1996, 2002; Frankel, 2004). From equation (5), TP is equal to