EXTRACTION METHOD OF SINUSOIDAL WAVES FROM EARTHQUAKE GROUND MOTION TO SIMPLIFY THE MOTION

Abstract

 Simplification of earthquake ground motion using pulse waves has been studied to observe pulse-like ground motion. These simplified pulse waves have been utilized in studies of seismic isolated and tall buildings. Simplification of observed waveforms can also be utilized to identify a source model from a series of observed ground motions. Thus, establishing a simplification method is important. Most existing methods are based on extraction of waveforms by wavelet analysis. However, these methods have two main limitations: (i) they focus on non-stationary components; hence, stationary components are not extracted, and (ii) a wavelet for extraction does not have a well-defined period and amplitude, and does have a constant phase.
 First, the proposed method was formulated. In order to solve problem (i), the method extracts waveforms using multiple wavelets, as opposed to existing methods that use only one wavelet; for solving problem (ii), the wavelet is defined as the product of a sinusoidal waveform and a time-domain filter. The filter passes through a central region to clarify the definition of both period and amplitude. The shape of the filter is defined and can be changed using a non-negative integer, N. The proposed method selects and extracts optimum waveforms that minimize the sum of the squares of error using multiple wavelets that have arbitrary phase α and various N. When N has a small value, and α is a constant, the proposed method coincides with typical wavelet analysis. Then, the method extracts a local and non-stationary waveform. When N asymptotically approaches infinity, the method asymptotically approaches Fourier analysis. Then, the method extracts a repeat-cycled and stationary waveform. The method, therefore, can extract an appropriate waveform depending on the original waveform.
 The proposed method was applied on an acceleration waveform observed at Japan Railway Takatori station during the Kobe earthquake in 1995. Analytical cases 1, 2, and 3 used wavelets with N = 0, 1, and 2, respectively. Analytical case 4 used all wavelets with N less than 30. After extracting the optimum waveform, the remaining waveforms were repeatedly extracted 50 times. Comparison of the synthesized waveforms of the first 10 extracted waveforms for cases 1-4 showed that the original waveform was best reproduced by case 4. Among the proposed wavelets, residual displacement was observed only for N = 0. Only the first of the first 10 waveforms was extracted with N = 0. This implies that residual displacement primarily occurred in the region of the first extracted waveform. The transition of error against the extracted number shows that cases 1 and 2 were better than case 3 initially; however, case 3 was better than cases 1 and 2 when the number exceeded five. Case 4 was the best for any extracted number. The method was also applied to a frequency-swept sinusoidal waveform. The transition of error against the extracted number showed that cases 3, 2, 1 were good, in this order, for numbers less than 35. Wavelets with a small N value could not efficiently extract from a stationary waveform. Again, case 4 was the best for any extracted number. The advantage of case 4 over case 1 was larger for this waveform than that for the above earthquake waveforms. The frequencies of the extracted waveforms agreed well with those of the original waveform.
 The proposed method extracts essential waveforms from earthquake ground motion. Thus, this method can be utilized to resolve a waveform into various waveform components caused by different propagation paths. Moreover, because this method can reproduce a waveform with a small number of parameters, it can be applied as a compression technique for sound and image.