Insights on Numerical Damping Formulations Gained from Calibrating Two-Dimensional Ground Response Analyses at Downhole Array Sites

TL;DR

This study evaluates different numerical damping formulations in 2D ground response analyses to improve seismic wave attenuation modeling, finding that Rayleigh Mass damping with inflated damping ratios offers better accuracy and efficiency.

Contribution

It introduces a site-specific calibration approach for damping in 2D GRAs and demonstrates the superiority of Rayleigh Mass damping over traditional methods.

Findings

Inflated damping ratios improve mode representation in 2D GRAs.

Rayleigh Mass damping closely matches empirical transfer functions at most sites.

Frequency-dependent damping formulations outperform frequency-independent approaches.

Abstract

Accurately modeling seismic wave attenuation is critical for ground response analyses (GRAs), which aim to replicate local site effects in ground motions. However, theoretical transfer functions (TTFs) from GRAs often overestimate empirical transfer functions (ETFs) when the small-strain damping ratio ($D_{\text{min}}$) is set equal to laboratory measurements. Prior studies addressed this by inflating $D_{\text{min}}$ in one-dimensional (1D) GRAs to account for apparent damping mechanisms such as diffraction and mode conversions that cannot be captured in 1D. Although this approach improved fundamental-mode predictions, it often overdamped higher modes. This study explores more direct modeling of apparent damping using two-dimensional (2D) GRAs at four downhole array sites: Delaney Park (DPDA), I-15 (I15DA), Treasure Island (TIDA), and Garner Valley (GVDA). At each site, three numerical damping formulations, Full Rayleigh, Maxwell, and Rayleigh Mass, were implemented using both conventional $D_{\text{min}}$ and an inflated $D_{\text{min}}$ ($m \times D_{\text{min}}$) obtained from site-specific calibration. Results show that the appropriate $D_{\text{min}}$ multiplier ($m$) correlates with the site's velocity contrast. Using inflated $D_{\text{min}}$, Full Rayleigh and Maxwell damping systematically overdamped higher modes, with Maxwell damping also shifting modal peaks. In contrast, Rayleigh Mass damping consistently achieved the closest match to ETFs at three of the four sites while offering faster computational performance. These findings demonstrate that inflated $D_{\text{min}}$ can represent unmodeled attenuation in 2D GRAs, particularly at sites with low velocity contrast, and that frequency-dependent formulations such as Rayleigh Mass damping can more accurately predict site response than traditional frequency-independent approaches.