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Fourier Amplitude Spectrum Calculation and Smoothing Operation

One of the most important factors to consider when evaluating ground motions is frequency content. The most common measure of frequency content is the Fourier amplitude spectrum, which indicates how the amplitude of the ground motion is distributed across different frequencies. Calculation of the spectrum requires a transformation of the ground motion from the time domain to the frequency domain. This transformation is called a Fourier transform. In RSSeismic, the transformation is completed using a Fast Fourier Transform (FFT). The resulting Fourier spectrum is then used to calculate the Fourier amplitude spectrum using the following equations:

Equation 1Eqn. 1
Equation 2Eqn. 2

where fi is the i-th frequency, n is the number of points in the FFT, |F|i is the Fourier amplitude at the i-th frequency, and Ci is the i-th amplitude and phase (in complex number representation) of the FFT. The maximum frequency that can be contained in the motion is dictated by the motion’s time step. This maximum frequency is called the Nyquest frequency and is calculated using the following equation:

Equation 3Eqn. 3

RSSeismic can also smooth the calculated Fourier amplitude spectrum to make interpretation easier by providing a clearer view of the overall frequency content. RSSeismic uses a triangle smoother in log space (also called a log-triangle smoother). The smoothing routine in RSSeismic uses a sliding triangular smoothing window in log-space and is adapted from a routine developed by David Boore. The weights assigned to each point are based on the log distance from the point of interest. Currently, the maximum smoothing width is set to 0.2. At each frequency of the spectrum the weights of the smoothing window are calculated as follows:

  • For frequencies below the current frequency:
Equation 4Eqn. 4
  • For the current frequency:
Wi = 1Eqn. 5
  • For frequencies above the current frequency:
Equation 6Eqn. 6

where the upper and lower bound indices are determined using the desired window width and index of the current frequency.

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