DSHA

Classic Deterministic Seismic Hazard Assessment

The calculation of the seismic hazard was based on assuming here the occurrence of a large earthquake of a certain magnitude at a specific location that affects the site where the motion assessment is made. For its computation, this code follows a series of steps (Reiter, 1990; Krinitzsky, 1995; Kramer, 1996; Campbell, 2005) that allow the deterministic derivation of the seismic hazard with a zone-based method:

(i) A catalogue construction of NFseismogenic sources {SSj}, which may affect the study area, such an inventory compiles their geometry, depth, and geographic location.

(ii) Characterisation of the seismic potential of each source SSj that is capable of a significant ground motion at the site for the characteristic or maximum credible earthquake MCEj and the prevailing focal mechanism (normal, thrust, or strike-slip) of its seismicity.

(iii) For the seismic ground motion intensity parameter Y, by which the hazard is to be characterised, select the set {fi} of the empirical Ground-Motion Prediction Equations (GMPEs) or attenuation laws Yi=fi(M,R,fik,siY), with their corresponding parameters fikthat include possible dependence on focal mechanism, shear type or S-wave velocity, and the random uncertainties siY for each i-th prediction equation.

(iv) Arbitrarily select the desired probability of exceedance PRexed=Pr[Y>=ymax/M,R], for a seismic scenario given by the (M,R) pair.

(v) Calculate the p-th percentile equivalent to the probability of exceedance:

PCPr= Pr[Y<yp/M,R]=1- Pr[Y>=yp/M,R]

(vi) Calculate the standard normal random variable Z01, of mean 0 and variance 1, that matches the percentile PCPr.

(vii) For a site P located at the geographic coordinate position (x,y,h) with latitude, longitude and hypsometry (height or depth above mean sea level).

(viii) For each j-th seismic source SSj (j=1,.... NF), assuming that the worst-case scenario (M,R, focal mechanism) is set, defined as the occurrence of an earthquake of magnitude M=MCEj from a point of the j-th seismogenic source at the shortest possible distance R = Rmin =min{d(P,SSj)}, based on the distance to be handled (RJB, Rep, Rhip, etc.) in the attenuation function (Figure 4).

(viii.1) Calculate the mean ground motion (50th percentile or GMPE treated as deterministic) of the seismic parameter Yi50,jat the site, with each i-th GMPE: Yi50,j = fi(M,R,fik).

(viii.2) Calculate the p-th percentile of the seismic parameter at the site with the i-th equation of motion:

log Yip,j = log Yi50,j + Z01σiY

(viii.3) Obtain the on-site seismic ground shaking parameter Yp,j on site P produced by each j-th earthquake source.

(ix) Deterministic evaluation of the th percentile of the seismic hazard Yp(x,y,h) at the site located at (x,y,h) as the largest parameter of the intensity of ground shake obtained from each seismogenic source: Yp(x,y,h) = max{Yp,j}

(x) Write Yp(x,y,h) output to change to a new location of the P site to assess.

(xi) Repetition of this process at numerous locations spread over a geographical region allows mapping of the hazard assessment over the geographical study area.

The previous hazard estimation is based on the classical deterministic method; however, it is not statistically accurate considering that the intensity of the seismic motion Yp, obtained for a minimum p-percentile, turns out to be the maximum value of a set of independent random variables with a standard distribution function. Since in each GMPE the log Y value is distributed as a normal random variable of mean logY with standard deviation s (Kramer, 1996), then, since Yp(x,y,h) = max{Yp,j}, both Yp and Yp,jare random variables. Thus, if the p-th percentile Yp is calculated as follows:

in accordance with the classical scheme, with a single dominant seismic source, and where F denotes the cumulative density function of the standard normal distribution (i.e., mean m = 0, and variance = 1). Now, considering the distribution of Yp,NSas the extreme value of a set of values (Coles, 2001; Ang and Tang, 2007), the above approximation is computed as:

This implies incorporation of the effect of the remaining NS seismic sources in the hazard percentile estimation in the hazard assessment. The estimation is more complicated than in the case of a single source, as it is now necessary to solve the nonlinear equation:

To assess the hazard for a specific exceedance probability, considering the scenario in which all seismic sources contribute (maximum Y distribution) and with a logic tree approach a subroutine has been included in the program that solves the nonlinear equation lineal by means of a numerical Bisection method, also known as halving, binary search, dichotomy or Bolzano (Atkinson, 1991; Süli and Mayers, 2003). Since this process must be repeated at each site where the hazard is estimated and, in this case, for each GMPE, the convergence characteristics of this algorithm allow it to be repeated for a total of 100 iterations, which are sufficient to achieve an admissible error |Xo - X*| in the root X* solution of the equation (Burden et al., 2015).

Cited Refs.

Ang, A. H. and Tang, W. H.: Probability Concepts in Engineering Planning: Emphasis on Applications to Civil and Environmental Engineering, John Wiley and Sons, 2007.

Atkinson, G. M., Bommer, J. J., and Abrahamson, N. A.: Alternative Approaches to Modeling Epistemic Uncertainty in Ground Motions in Probabilistic Seismic‐Hazard Analysis, Seismological Research Letters, 85, 1141–1144, https://doi.org/10.1785/0220140120, 2014.

Burden, R. L., Faires, J. D., and Burden, A. M.: Numerical Analysis, Cengage Learning, 918 pp., 2015.

Campbell, K.: Overview of Seismic hazard Approaches with Emphasis on the Management of Uncertainties.“, in: 2nd ICTP Workshop on Earthquake Engineering for Nuclear Facilities: Uncertainties in Seismic Hazard”, Trieste, Italy, 14–25, 2005.

Coles, S.: An Introduction to Statistical Modeling of Extreme Values, Springer, London, https://doi.org/10.1007/978-1-4471-3675-0, 2001.

Kramer, S. L.: Geotechnical Earthquake Engineering, Prentice-Hall Civil Engineering and Engineering Mechanics Series, Prentice Hall, Upper Saddle River, 653, 1996.

Krinitzsky, E. L.: Deterministic versus probabilistic seismic hazard analysis for critical structures, Engineering Geology, 40, 1–7, https://doi.org/10.1016/0013-7952(95)00031-3, 1995.

Reiter, L.: Earthquake Hazard Analysis, Issues and Insights, Columbia U, Press, New York, NY, 1990.

Süli, E. and Mayers, D. F.: An Introduction to Numerical Analysis, Cambridge University Press, 440 pp., 2003.