D-A-CH TAGUNG 2011 - SGEB

aggregated loss for a portfolio (widely located constructions of several types) and oncharacteristics of joint hazard and damage.In this study, we compared the effects of variations in the between-earthquake correlationand in the spatial (site-to-site) correlation on the seismic loss and damage estimations forextended objects (hypothetical portfolio) and critical elements (e.g., bridges) of a network.Based on the results of our previous work [29], the correlation distances vary from 0 km to 30km, and the between-earthquake correlation varies from 0.09 to 0.5, which is also within thereported values. A single event, a so-called “scenario” earthquake, was used as the source ofseismic influence. A set of hypothetical buildings representing a real building stock wasconstructed based on typical buildings in the Taiwan area.2 APPLICATION OF GROUND MOTION CORRELATION TO GROUND MOTIONMODELLINGFor the generation of the k-site random field of ground motion values that are spatiallycorrelated, it is necessary to generate a Gaussian vector of correlated standard normal variables(total residual term) X = [X 1 , X 2 , …., X k ] with a symmetric correlation matrix , or X ~N k (0,. The correlation matrix is defined as follows:1 k211112k 2... 1k 2k ... 1 , (6)where ijis the empirical correlation coefficient calculated for the sites separated by adistance .The procedure of the generation of k-site random field of ground motion errorvalues that are spatially correlated equals the generation of random variables X = [X 1 , X 2 , ….,X k ] with a correlation matrix from the k-dimensional Gaussian copula. Descriptions of theprocedure may be found in many sources (e.g., Refs. [12, 44]). The generation consists of thefollowing steps. First, a vector of independent standard normal variates U= [U 1 , U 2 , …., U k ]2with standard deviation, or U ~ N k (0, T, is generated. Then, a correlation matrix isconstructed and a Cholesky decomposition is applied to represent the correlation matrix asthe matrix product of matrix B and its transposition B T , i.e., S=B B T . The required vector X isobtained as X=BU. These X values are added to the median ground motion term ln Y toobtain a realisation of spatially correlated ground motions.jIn this study, we generated ground motion parameters (peak ground acceleration) from asingle scenario earthquake across a wide area (e.g., a city). Figure 2a shows the area of 22 km x18 km, which is divided into cells of 1 km x 1 km. Ground motion parameters were estimatedin the centres of the cells (marked by triangles) from an M 7.0 earthquake located near the area.The ground motion prediction equation for the calculation was used in the following form,which was recently proposed for Taiwan [29]:Ln PGA = -3.07+0.83M – 1.33ln[R+0.15exp(0.45M)]+0.0023R+σ T (7)where PGA is measured in units of g, and R is the hypocentral distance in km. The total2standard deviation for the earthquake was accepted as = 0.4 (total variance T= 0.16). Themedian expected PGA values, which were calculated using Equation 7, are shown in Figure 2b.ij7161