⚡ electrical engineering

Direct Damping-Dependent Vertical Ground Motion Model for Stiff Soils in Active Crustal Regions

This paper proposes and validates a direct ground-motion prediction model for estimating vertical pseudo-acceleration response spectra across a range of damping ratios (1–20%) on stiff soils in active crustal regions, demonstrating that while it yields comparable variability to indirect scaling methods, it produces distinct median estimates that vary with earthquake magnitude.

Original authors: Héctor Dávalos, Jorge Ruiz-García, David Aguilar

Published 2026-08-20

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CC BY 4.0

Original authors: Héctor Dávalos, Jorge Ruiz-García, David Aguilar

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Earthquakes do not just shake the ground side to side; they also push and pull it up and down. While the horizontal jolts are often the most visible, the vertical motion can be just as destructive, particularly for heavy structures like bridges and tall buildings. When the ground thrusts upward, it can suddenly increase the weight a building column must support, potentially causing it to buckle or fail in ways that standard safety designs might not anticipate. To keep people safe, engineers need to know exactly how strong these vertical shakes will be at different locations. For decades, the standard way to predict this shaking has relied on a single, fixed assumption: that the buildings and materials involved absorb energy at a specific, uniform rate. In reality, however, different materials and structures absorb that energy at very different rates, meaning the standard predictions can sometimes miss the mark.

A team of researchers from Mexico has developed a new way to forecast these vertical ground motions that accounts for these differences. Instead of relying on a single, rigid rule, their model adjusts its predictions based on how much energy a specific structure is likely to absorb. They analyzed thousands of recorded earthquake events, focusing on areas with firm, stiff soil—a type of ground common in many major cities. By studying the data from these events, they created a tool that can estimate the intensity of vertical shaking for a wide range of structural behaviors, from very rigid systems to those that are more flexible. This approach allows engineers to get a more precise picture of the forces a building might face, rather than guessing based on a one-size-fits-all assumption.

The researchers began by gathering a massive collection of data: 2,408 separate records of vertical ground motion from earthquakes with magnitudes greater than 5.5. These recordings came from sites where the ground was stiff, with a specific measure of soil stiffness falling between 180 and 360 meters per second. This type of soil is significant because it is the default classification used in many building codes when specific local data is missing. The team looked at how the ground moved over a range of time intervals, from very rapid shakes lasting 0.1 seconds to slower movements lasting up to 2.0 seconds. Crucially, they did not just look at the standard 5% energy absorption rate used in most existing models. Instead, they calculated the shaking intensity for six different levels of energy absorption, ranging from 1% to 20%. This allowed them to see how the predicted shaking changes when the structure behaves differently.

Using this extensive dataset, the team built a new prediction model that directly links the size of the earthquake and the distance from the fault to the expected shaking intensity, while simultaneously factoring in the energy absorption rate. They tested this new model against the traditional method, which involves a much longer, multi-step process. The old way requires engineers to first predict the horizontal shaking, then apply a ratio to guess the vertical shaking, and finally adjust that result using a separate formula for energy absorption. The researchers found that their new, direct method produced results that were comparable in terms of variability but often different in the actual numbers predicted. For moderate-sized earthquakes, their model predicted shaking that was significantly lower than the traditional method, while for very large earthquakes, it predicted shaking that was notably higher.

The study suggests that this direct approach offers a more practical and consistent way to estimate vertical shaking, especially for structures that do not behave like the standard 5% assumption. By avoiding the need to chain together multiple different models and guess at how they might interact, the new method reduces the uncertainty that comes from combining separate predictions. The researchers found that their model works well for earthquakes ranging from magnitude 5 to 8 and for distances up to 200 kilometers. They concluded that by accounting for the specific way different structures absorb energy, engineers can move away from rough approximations and toward a more accurate understanding of the vertical forces that threaten our buildings during an earthquake. This refinement does not just add a layer of complexity; it provides a clearer, more direct path to safety in regions where the ground shakes from below.

Technical Summary: Direct Damping-Dependent Vertical Ground Motion Model for Stiff Soils in Active Crustal Regions

Problem Statement
While ground-motion prediction models (GMMs) are routinely developed for 5%-damped vertical acceleration response spectra ( $S_{aV}$ ), many structural and nonstructural systems possess damping ratios that deviate significantly from this conventional value. Current methodologies for estimating $S_{aV}$ for non-standard damping ratios typically rely on indirect approaches. These involve a multi-step process: first estimating horizontal spectral acceleration ( $S_{aH}$ ), converting it to vertical using a vertical-to-horizontal ( $V/H$ ) ratio model, and finally applying a vertical damping scaling factor (DSF). This indirect method introduces significant complexity, requiring the combination of three distinct models and, critically, the knowledge of correlation coefficients between the residuals of these random variables ( $S_{aH}$ , $V/H$ , and $DSF_v$ ). As noted in the literature, these correlation coefficients are often unavailable or limited in applicability, leading to the propagation of uncertainty and potential epistemic bias in the final estimates.

Methodology
To address these limitations, the authors propose a direct GMM for estimating $S_{aV}$ across a range of damping ratios for shallow crustal earthquakes recorded on stiff soil sites (NEHRP Site Class D, $V_{S30}$ between 180 and 360 m/s).

Database: The model utilizes a dataset of 2,408 vertical ground-motion records from the NGA-West2 database. The records correspond to events with moment magnitudes ( $M_w$ ) greater than 5.5 and Joyner–Boore distances ( $R_{jb}$ ) shorter than 300 km.
Spectral Computation: Elastic response spectra were computed for 17 vibration periods ranging from 0.1 s to 2.0 s. Calculations were performed for six critical damping ratios: 1%, 2%, 5%, 10%, 15%, and 20%.
Functional Form: The model employs a mixed-effects regression approach to estimate the natural logarithm of $S_{aV}$ . The functional form is a simplified version of the Bozorgnia and Campbell (2016b) model, dependent on $M_w$ and $R_{jb}$ . It comprises an event term ( $F_E$ ) and a path term ( $F_P$ ), with coefficients ( $b_1$ through $b_6$ ) derived independently for each period and damping ratio.
Predictor Variables: Preliminary analyses indicated that including additional variables such as faulting mechanism, hypocentral depth, or depth to the 2.5 km/s horizon yielded only minor reductions in variability (e.g., 1.8% for faulting mechanism). Consequently, the authors opted for a simpler functional form to reduce parametric uncertainty.
Variability: The model explicitly quantifies within-event ( $\phi$ ) and between-event ( $\tau$ ) variability, as well as total logarithmic standard deviation ( $\sigma$ ).

Key Results

Model Performance: Residual analyses (within-event vs. $R_{jb}$ and between-event vs. $M_w$ ) demonstrated no significant trends, suggesting the functional form adequately captures the scaling of $S_{aV}$ with magnitude and distance.
Variability Trends: The logarithmic standard deviation is period-dependent. Lower damping ratios (1–2%) exhibit larger variabilities, particularly at periods between 0.03 s and 0.2 s. Higher damping ratios (10–20%) show more stable variabilities, with average $\sigma$ values ranging from 0.63 to 0.65.
Comparison with Indirect Approach: The proposed model (termed DRA26) was compared against an indirect approach combining the Bozorgnia and Campbell (2016b) GMM for 5% damping and the Rezaeian et al. (2014) DSF model (termed BC16+R14).
- Magnitude Dependence: For moderate magnitudes ( $M_w = 6.5$ ), the proposed model yields median $S_{aV}$ estimates approximately 31.4% lower than the indirect method at short distances (10 km). Conversely, for large magnitudes ( $M_w = 7.5$ ), the proposed model predicts values roughly 19.9% higher. These differences diminish as distance increases.
- Uncertainty: The proposed model offers a more consistent characterization of spectral variability because it avoids the need to assume correlations between $S_{aV}$ and $DSF_v$ or to approximate them using horizontal component correlations.

Significance and Claims
The paper posits that the direct prediction of damping-dependent vertical response spectra is a practical alternative to indirect damping-scaling procedures. The primary significance of the work lies in its ability to estimate $S_{aV}$ using a single functional form, thereby eliminating the need for intermediate $S_{aH}$ or $V/H$ models and the associated correlation coefficients that are often unavailable.

The authors claim that this direct approach reduces the propagation of uncertainty and avoids the epistemic uncertainty introduced by assuming independence or approximating correlations between random variables. The model is specifically recommended for estimating $S_{aV}$ from active crustal earthquakes with $M_w$ between 5 and 8, distances up to 200 km, and for NEHRP Site Class D sites. The study concludes that while both direct and indirect methods produce comparable estimates of spectral variability, the direct model provides a more robust framework for systems with effective damping ratios substantially different from the conventional 5% value.

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