Generation of spectrum and energy-compatible (SEC) bi-directional ground motions via complex-valued wavelet transform

Modern seismic design codes mandate the simultaneous input of horizontal bi-directional ground motions in nonlinear time history analysis (NLTHA) to ensure comprehensive seismic assessments. This requirement highlights the critical importance of generating bi-directional ground motions that are compatible with the target maximum-direction spectrum (RotD100). However, accurate seismic response evaluations significantly depend on the selection of appropriate seismic inputs. For instance, the Arias intensity build-up process I_a(t), which quantifies the energy content of an accelerogram, has attracted increasing attention in performance-based seismic design. Consequently, it is necessary to propose a method capable of generating RotD100 response spectrum and energy-compatible (SEC) bi-directional ground motions. A novel approach is introduced to address this challenge, leveraging complex-valued wavelets derived from the impulse response of an under critically-damped oscillator. In contrast with previous methods that achieved RotD100 response spectral compatibility by separately manipulating two orthogonal components of bi-directional ground motions, the proposed complex-valued continuous wavelet transform (CWT) method represents the two horizontal components as the real and imaginary parts of complex numbers. The efficacy of the proposed algorithm is validated through the generation of SEC bi-directional ground motions using 40 pairs of seed records for two target RotD100 response spectra. Results suggest that, with the proposed algorithm, the modified bi-directional ground motions closely align with the target RotD100 response spectra and the I_a(t) build-up process mirrors that of the scaled records. Furthermore, the key attributes of the seed records, such as component-level spectral variability, nonlinear time history trace characteristics, and period-dependent polarity, are largely preserved. Additionally, the iterative modification of the I_a(t) build-up process leads to the improved displacement time history compared with the real-valued CWT-based method proposed by Montejo.