Zero-pad Effects on Conditional Simulation and Application of Spatially-varying Earthquake Motions

Realistic simulated ground motion are required for testing heath monitoring algorithms in connection with seismic excitations. To this end spatially-varying ground motions are important for long span structures such as bridges and pipelines. In conditional simulation of spatially-varying earthquake motions, a single earthquake motion time history at a site is known and a parametric coherency model is given. Spatially-varying earthquake motions at other sites are then simulated. To facilitate the use of the Fast Fourier Transform (FFT), dummy zero points must be added to the front and/or the end of the data. Investigating effects of zero- padis important for simulating ground motions to be used for testing health monitoring techniques. However, neither comparison of zero- paddedfiltered data with non-zero padfiltered data nor explanation for effects of zero padshave been reported. The zero- padeffects on FFT of the input motion and also on the simulated motions are examined in this paper. When the input motion is a recorded acceleration timehistory, the Fourier amplitude spectra of the zero- paddedinput and simulated accelerations have lower peaks and higher resolution compared to the non zero- padded. The two phenomena are due to the contribution of the added data points and the conservation of the power energy when the data length is increased, respectively. The effects of zero padon structural responses due to the simulated earthquake motions are also investigated by using a single degree of freedom model, where significant influence on the time history of structural responses at certain structural frequencies are observed. INTRODUCTION To facilitate the use of FFT in simulating earthquake motions for structural health monitoring (SHM) applications, dummy zero points must be added to the data, in the front and/or the end, as shown in Figure 1. At first, we consider a simple Nguyen Van Dinh and Biswajit Basu Post-doctoral Research Fellow, Professor, 1, 2 School of Engineering, Trinity College Dublin, Ireland. 6th European Workshop on Structural Health Monitoring Tu.3.D.3 Licence: http://creativecommons.org/licenses/by-nd/3.0 1 example of an 1-D array A1 [1 1 1 1], its FFT is [4 0 0 0]. If four more zeros are paddedat the end, A10 [1 1 1 1 0 0 0 0], the FFT becomes [4.0 1.0-2.4142i 0 1.0-0.4142i 0 1.0+0.4142i 0 1.0+2.4142i] Figure 1. Schematic chart of zero- paddeddata. The FFT of zero-added data A10 is different from that of the original A1. Because zero- paddingis necessary for the FFT algorithm, examining effects of zeropad is important. In this conditional simulator, the input data of acceleration, through its Fourier transform in frequency domain, is filtered by a coherency function and/or by a phase-variationfunction. The zero- padeffects on input data and also on simulated acceleration should be examined. Newland (1993) suggested a correction factor for correlation of two series due to zero- pads. Boore (2005, 2009) and Zerva (2009) analyzed these effects when processing raw earthquake records. However, neither comparison of zero- paddedfiltered data with non-zero padfiltered data nor explanation for effects of zero padshave been reported in the literature. SIMULATION OF SPATIALLY-VARYING EARTHQUAKE MOTIONS Having assumed that earthquake-induced motions are space-homogenous, the n Fourier coefficient ) ( k F of motion ak(t) at a station k can be evaluated by     r n jk wp jk n jk j k t t i n n F n F , exp ) ( ) ( ) (       (1) where, ) (n Fj , ) (n jk  , wp jk t and r n jk t ,  are, respectively, the n th Fourier coefficient of motion at the source station j, the n lagged coherency value, the time lag due to wave-passage effects and the n value of random arrival-time perturbation series. The motion at a discrete time tr = rt , r = 0, ..., (N – 1) is