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

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.

💡 Research Summary

This paper addresses the persistent discrepancy between theoretical transfer functions (TTFs) derived from ground response analyses (GRAs) and empirical transfer functions (ETFs) measured at downhole array sites. While previous work demonstrated that inflating the small‑strain damping ratio (Dₘᵢₙ) in one‑dimensional (1‑D) GRAs can compensate for unmodeled attenuation mechanisms such as diffraction and mode conversion, the approach often over‑damps higher modes. To investigate whether a more direct representation of apparent damping is feasible in two‑dimensional (2‑D) analyses, the authors performed calibrated 2‑D GRAs at four well‑instrumented sites: Delaney Park (DPDA), I‑15 (I15DA), Treasure Island (TIDA), and Garner Valley (GVDA). For each site three numerical damping formulations were implemented: Full Rayleigh, Maxwell, and Rayleigh Mass. Simulations were first run using the conventional laboratory‑measured Dₘᵢₙ, which reproduced the known over‑prediction of TTFs, especially at higher frequencies. A site‑specific multiplier (m) was then applied to Dₘᵢₙ (i.e., Dₘᵢₙ → m·Dₘᵢₙ) to calibrate the models against the observed ETFs. The calibrated multiplier correlated strongly with the velocity contrast between the shallow and deep layers; sites with low velocity contrast required only modest inflation, whereas high‑contrast sites needed larger multipliers. When the inflated Dₘᵢₙ was used, Full Rayleigh and Maxwell damping both systematically over‑damped higher modes, with Maxwell also shifting modal peaks toward lower frequencies. In contrast, Rayleigh Mass damping, which introduces frequency‑dependent damping coefficients, matched the amplitude and frequency of both fundamental and higher modes across three of the four sites. Moreover, Rayleigh Mass required fewer matrix operations, delivering roughly 30 % faster computational times than the other two formulations. The study concludes that (1) inflating Dₘᵢₙ can effectively represent unmodeled attenuation in 2‑D GRAs, particularly at sites with modest velocity contrast, and (2) frequency‑dependent damping formulations such as Rayleigh Mass provide a more accurate and efficient means of reproducing site‑specific response than traditional frequency‑independent approaches. These findings offer practical guidance for selecting damping models and calibrating damping ratios in advanced seismic site‑response simulations.