An Improved Hilbert Spectral Representation Method for Synthesizing Spatially Correlated Earthquake Ground Motions and Its Error Assessment

This paper is an extension of the random amplitude-based improved Hilbert spectral representation method (IHSRM) that the authors developed previously for the simulation of spatially correlated earthquake ground motions (SCEGMs) possessing the nonstationary characteristics of the natural earthquake record. In fact, depending on the fundamental types (random phase method and random amplitude method) and matrix decomposition methods (Cholesky decomposition, root decomposition, and eigendecomposition), the IHSRM possesses various types. To evaluate the influence of different types of this method on the statistic errors, i.e., bias errors and stochastic errors, an error assessment for this method was conducted. First, the random phase-based IHSRM was derived, and its reliability was proven by theoretical deduction. Unified formulas were given for random phase- and random amplitude-based IHSRMs, respectively. Then, the closed-form solutions of statistic errors of simulated seismic motions were derived. The validness of the proposed closed-form solutions was proven by comparing the closed-form solutions with estimated values. At last, the stochastic errors of covariance (i.e., variance and cross-covariance) for different types of IHSRMs were compared, and the results showed that (1) the proposed IHSRM is not ergodic; (2) the random amplitude-based IHSRMs possessed higher stochastic errors of covariance than the random phase-based IHSRMs; and (3) the value of the stochastic error of covariance for the random phase-based IHSRM is dependent on the matrix decomposition method, while that for the random amplitude-based one is not.