Wavelet Packet-Based Diffusion Model for Ground Motion Generation with Multi-Conditional Energy and Spectral Matching

TL;DR

The paper tackles the challenge of generating ground motions that satisfy both target response spectra and explicit temporal-energy evolution constraints, addressing nonstationarity and damage potential in nonlinear analyses. It introduces a multi-conditional diffusion model operating in a wavelet packet domain, leveraging the Elucidating Diffusion Model with a second-order Heun sampler and a Transformer-based conditional encoder with cross-attention to integrate diverse conditioning signals. The approach yields spectrum-compatible motions with realistic energy evolution and enables uncertainty quantification through conditional diversity sampling, validated on the NGA-West2 database. This framework offers a scalable path toward parameterized, site-aware ground-motion synthesis for performance-based seismic engineering and can be extended with additional physics-based constraints and conditioning variables.

Abstract

Temporal energy distribution strongly affects nonlinear structural response and cumulative damage. We propose a multi-conditional diffusion framework for ground motion synthesis that simultaneously matches temporal energy evolution and target response spectra. Wavelet packet decomposition provides the signal representation and enables direct waveform reconstruction via orthogonal filter banks. A Transformer-based conditional encoder with cross-attention integrates heterogeneous conditions, including spectral ordinates, Arias intensity, temporal parameters, and Husid curves. The framework adopts the Elucidating Diffusion Model (EDM) with second-order Heun sampling to improve inference efficiency without sacrificing quality. Tests on the NGA-West2 database show that explicit temporal-energy constraints markedly improve control of energy onset and significant duration while preserving spectrum matching and maintaining stable diversity sampling. The framework yields spectrum-compatible motions with realistic energy evolution and supports uncertainty quantification via conditional diversity sampling.

Figures (9)

• Figure 1: Multi-conditional diffusion framework for constrained ground motion generation: (a) wavelet packet decomposition and diffusion in coefficient space; (b) conditional encoder fuses vector conditions ($S_a$, $H(t)$) and scalar IMs ($I_A$, $T_5$, $D_{5-95}$, $E_{th}$), and injects them into a U-Net denoiser via cross-attention; (c) EDM sampling (Heun solver) from Gaussian noise followed by inverse wavelet packet reconstruction to obtain time-domain waveforms.
• Figure 2: Comparison between STFT+Griffin-Lim and wavelet packet reconstruction. (a) STFT spectrogram with maximum reconstruction error $\varepsilon_{\mathrm{max}} = 6.14 \times 10^{1}$; (b) Wavelet packet coefficients with $\varepsilon_{\mathrm{max}} = 1.78 \times 10^{-14}$; (c) original waveform (black) compared with STFT+Griffin-Lim reconstruction (blue dashed) and wavelet packet reconstruction (red dashed); (d) STFT+Griffin-Lim reconstruction error; (e) Wavelet packet reconstruction error.
• Figure 3: Comparison of scalar intensity parameter control precision between Case 1 and Case 2.
• Figure 4: Comparison of generated waveforms and observed waveforms for 8 representative samples from Case 1 and Case 2.
• Figure 5: Distribution histograms of accuracy metrics for 15,000 generated samples (150 conditions $\times$ 100 samples). Upper row: response spectrum $R^2$ (left) and RMSE (right) distributions. Lower row: Husid curve $R^2$ (left) and RMSE (right) distributions.