Linear Site Response Model Applicable to Levees in Sacramento-San Joaquin Delta Region of California

Introduction

The Sacramento-San Joaquin Delta, herein referred to as the Delta, is located in California’s Central Valley, roughly 50 km east of the San Francisco Bay area. Roughly 700,000 acres of land within the Delta are protected by more than 1,770 km of levees, which also serve as a conduit for approximately two-thirds of the State’s drinking water supply. Vital transportation, utility, and communication lifelines cross the patchwork of subsided islands, which are protected by the levees.

Before the lands were reclaimed, the area was a great tidal freshwater marsh. As the wetland vegetation died, the plant matter was only able to partially decay before the material became submerged and/or overgrown leaving organic deposits known as peat. The thickness of these organic deposits vary across the Delta, reaching up to 15 m in some areas. Many of the Delta levees were constructed atop these peaty deposits, as illustrated schematically in Figure 1.

In 2009 the Delta Risk Management Strategy (DRMS) conducted a probabilistic seismic hazard analysis (PSHA) for the Delta and concluded that seismic events pose a significant hazard to the Delta levees; this conclusion has since been reinforced by recent research efforts (Wong et al. 2021). The peaty soils affect risk in two ways: (1) seismic demand as a result of site amplification of ground motions and (2) ground motion deformation potential associated with permanent shear and/or volumetric deformations. Sites with peat may be characterized as having 30-m time-averaged shear wave velocities, V_S30, in the range of 100 to 200 m/s, which is softer than the lower limit for NGA-West2 ergodic site response models. We suspect these models to poorly capture the true behavior of such soft soil sites, therefore this work is concerned with investigating the (linear) site response of soft peaty organic soil sites in the Delta region.

Data Selection

We assembled a ground motion database that significantly expands upon the NGA-West2 database for California events and sites. From this larger database we extracted 526 ground motions recorded by 45 seismic instruments located within the Delta jurisdictional boundaries corresponding to 68 events with a wide range of source-to-site azimuths (shown in Figure 2). We only consider motions produced by events with moment magnitudes M>4 to avoid difficulties that may be encountered in the analysis of site terms using smaller magnitude data. Additionally, magnitude-distance cutoffs suggested by (Boore et al. 2014; BSSA14) were considered during record selection, horizontal components are combined to median-component (RotD50) intensity measures, and a lowest usable frequencies are enforced.

Characterization of Peaty Delta Sites

Site-specific V_S30 were computed at sites with a measured V_S profile obtained from a V_S profile database (Ahdi et al. 2018). Proxy-based models were used to assign V_S30 values at sites without measurements. At sites where peaty deposits are not suspected to be encountered we assigned V_S30 by combining the geologic- and topographic-based proxy value (Thompson 2018) (2/3 weight) and terrain-based proxy value (Yong et al. 2012) (1/3 weight). These models are not well calibrated for organic soils, therefore we developed a regional proxy model based on peat thickness and site condition (free-field or levee) for sites where peat is expected to be encountered:

\begin{equation} \ln{(\bar{V}_{S30})} = \mu_{\ln{V_{S30}}} + C(t_p -\mu_{t_p}) \pm \epsilon_n \sigma \label{(1)} \end{equation}

where μ_lnVS30 and μ_tp represent the mean lnV_S30 value (in m/s) and peat thickness (in m) for the given site condition, respectively; t_p represents the thickness of peat (in m); C quantifies the correlation between lnV_S30 and t_p; σ represents the model standard deviation; and ε_n is the fractional number of standard deviations of a single predicted value away from the mean.

Table 1. V_S30-peat thickness proxy model coefficients.

Methodology

Our aim is to evaluate site response in the Bay-Delta region from empirical data, and then use the observed site response effects to investigate its dependence on readily available site parameters (e.g., V_S30, peat thickness, etc.). We apply procedures for evaluating non-ergodic site response effects from ground motion recordings (Stewart et al. 2017), which utilize residuals analyses. We begin by computing total residuals, R_ij, between the observed ground motion recorded at site i from event j and a prediction from an ergodic ground motion model (e.g., BSSA14). Through the use of mixed-effects analyses, we partition R_ij into a fixed-effect (c_k which represents the overall model bias) and various random-effects: (1) an event term, η_Ei, which contains systematic source effects; (2) a site term, η_Sj, which contains systematic site effects; and (3) the remaining residual, ε_ij, which contains any remaining unattributed effects.

\begin{equation} R_{ij} = c_k + \eta_{Ei} + \eta_{Sj} + \epsilon_{ij} \label{(2)} \end{equation}

Non-ergodic site response is computed using η_Sj. To minimize bias for each site we use only data from sites with more than four observations and whose η_Sj standard errors, SE_ηSj, are below a threshold value. This period-dependent threshold was determined as the mean SE_ηSj plus two standard deviations computed from 755 sites which have recorded 10 or more ground motions in our expanded database. Out of the 45 sites in the Delta, 33 satisfy this selection criterion with a corrosponding V_S30 range from about 100 to 400 m/s.

We compute the observed linear site response for each site, f₁, as the sum of η_Sj and the linear site amplification from the ergodic site response model described by Seyhan and Stewart (2014; SS14), F_S. The dependence of f₁ on V_S30 is then examined and a linear site response model is developed using least-squares regression.

Functional Form

To this point, we have only investigated the dependence of site response in the region with V_S30. We adopt a mulit-linear functional form to capture the observed V_S30-scaling at soft soil sites which transitions to SS14 for stiffer soil conditions which are not typically encountered in the Delta:

\begin{equation} F_{lin}(V_{S30}) = \begin{cases} c_{low} \ln{(\frac{V_{S30}}{V_1})} + Δc \ln{(\frac{V_1}{V_{ref}})} + A, \text{ }\text{ }\text{ } V_{S30} \leq V_1 \\ Δc \ln{(\frac{V_{S30}}{V_{ref}})} + A, \quad\qquad\qquad\qquad V_1 < V_{S30} \leq 400 \text{ m/s} \\ m \log_{10}(V_{S30}) + b, \qquad\qquad\quad\quad\text{ }\text{ } 400 \text{ m/s} < V_{S30} \leq V_{ref} \\ c\ln{(\frac{V_{S30}}{V_{ref}})}, \quad\qquad\qquad\qquad\qquad\text{ }\text{ }\text{ } V_{ref} < V_{S30} \leq V_2 \\ c\ln{(\frac{V_2}{V_{ref}})}, \quad\qquad\qquad\qquad\qquad\text{ }\text{ }\text{ } V_2 \leq V_{S30} \end{cases} \label{(3)} \end{equation}

where V₁ and V₂ represent limiting velocites below- and above-which have different V_S30-scaling; V_ref is the velocity of the reference site condition (760 m/s); c_low, Δc, and c represent the V_S30-scaling for very soft soils (100 m/s< V_S30 < V₁), soft-moderate soils (V₁ < V_S30 < 400 m/s), and stiff soils (V_S30 > V₂), respectively; A is a fitting parameter used to ensure continuity between the first and second segments of the model; and coefficeints m and b are derived from regression results of the first two terms to facilitate the transition to SS14 at larger V_S30. All coefficients except for V_ref are period dependent.

Results

Coefficients in the first two terms of Eq. (3) are obtained by regressing f₁ on V_S30 for PGV, PGA and PSA at 105 oscillator periods (0.01 – 10 sec) and smoothing is performed across all periods to ensure smooth response spectra; the coefficients c and V₂ which appear in the last two terms are taken as c and V_c from SS14; and the coefficients in the third term are derived last. Figure 4 shows non-ergodic site responses for Bay-Delta sites, predictions of SS14, and predictions of the proposed local site response model for four oscillator periods.

Conclusions

• V₃₀-scaling predicted by SS14 does not accurately capture the site response observed in the Delta over a wide frequency range.
• There is a relatively large scatter in the data, however some trends are consistent:
1. Short periods (0.01-0.22 s): lower amplification than SS14.
2. Moderate periods (0.9-3 s): soft soils (V_S30 < 200 m/s) exhibit less V_S30-scaling than SS14.
3. Long periods (3-10 s): Delta sites amplify more than what SS14 predicts with soft soild having less V_S30-scaling.
• Implementation of our local Delta-specific linear site response model leads to improved ground motion predictions and a reduction of epistemic uncertainty, however more work is needed before the model can be used in engineering applications.
• The long-period site response observed from empirical data cannot be captured by ground response analyses, which has been the basis for site response estimation in prior work. This distinction is one example of the value of the present approach.
• Nonlinear site response effects in the region are expected to be significant, and are being evaluated using ground response analyses with regionally-customized nonlinear dynamic properties of the peat and other soils. The final site response will combine an empirical site amplification model with a simulation-based nonlinear model.

Future Work

• Our proposed model generally captures the observed site response, reducing uncertainty, however the functional form needs refinment which can be achieved using advanced regression methods (e.g., non-linear mixed-effects).
• Even with local V_S30-scaling, the linear site response model still has room for improvement. We intend to investigate the correlation between f₁ and additional site parameters (such as site fundamental period and peat thickness) to develop more sophisitcated model variants.
• To produce a comprehensive local site response model applicable to hazard level conditions (i.e., strong motions) we must develop a local non-linear model (in progress; discussed elsewhere).
• Lastly, we plan to run PSHA incorporating regional path (discussed elsewhere) and local site effects for the Delta region.

Acknowledgements

Funding for this study provided by California Department of Water Resources (DWR), Agreement 4600012415. We gratefully acknowledge this support. We want to particularly acknowledge Mike Driller (DWR), Tim Wheling (DWR), and Nick Novoa (DWR) for their assistance and support. Lastly, we want to acknowledge Ariya Balakrishnan (Division of Safety of Dams), Albert Kottke (Pacific Gas & Electric), Jamie Steidl (University of California, Santa Barbra and United States Geological Survey), and Ivan Wong (Lettis Consultants International) for their participation and guidance serving as advisory personnel.

References

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