Validation of stochastic ground motion model modification by comparison to seismic demand of recorded ground motions

Abstract

An important consideration for the adoption of stochastic ground motion models in performance-based earthquake engineering applications is that the probability distribution of target intensity measures from the developed suites of time-histories is compatible with the prescribed hazard at the site and structure of interest. The authors have recently developed a computationally efficient framework to modify existing stochastic ground motion models to facilitate such a compatibility. This paper extends this effort through a validation study by comparing the seismic demand of recorded ground motions to the demand of stochastic ground motion models established through the proposed modification. Suites of recorded and stochastic ground motions, whose spectral acceleration statistics match the mean and variance of target spectra within a period range of interest, are utilized as input to perform response history analysis of inelastic single-degree-of-freedom (SDoF) case-study systems. SDoF systems with peak-oriented hysteretic behavior, strain hardening, and (potentially) degrading characteristics, experiencing different degree of inelastic response, are considered. Response is evaluated using the peak inelastic displacement and the hysteretic energy given by the work of the SDoF restoring force as engineering demand parameters (EDPs). The resultant EDP distributions are compared to assess the effect of (and validate) the proposed modification. It is shown that the proposed modification of stochastic ground motion models can provide results that are similar to these from recorded ground motion suites, improving any (in some cases large) discrepancies that exist for the initial, unmodified stochastic ground motion model.

Appendix 1: Details for the stochastic ground motion modification framework

The statistical characterization p_g(ln(S_a(T_i))|μ,Σ) for IM S_a(T_i) through the stochastic ground motion model can be obtained by propagating the uncertainty originating from the predictive relationship p(θ|μ,Σ) for the model parameter vector θ and the stochastic sequence involved in the model description, denoted w herein. Following Tsioulou et al. (2018b) the statistical characterization for the log IM, ln(S_a(T_i)), is approximated as Gaussian with mean and variance:

$$\ln (\bar{S}_{a}^{g} (T_{i} ,{\varvec{\upmu}},{\varvec{\Sigma}})) = \int {\int {\ln (S_{a}^{g} (T_{i} ,{\varvec{\uptheta}},{\mathbf{w}}))} } p({\varvec{\uptheta}}|{\varvec{\upmu}},{\varvec{\Sigma}})p({\mathbf{w}})d{\varvec{\uptheta}}d{\mathbf{w}}$$

(2)

$$(\sigma^{g} (T_{i} ,{\varvec{\upmu}},{\varvec{\Sigma}}))^{2} = \int {\int {\left[ {\ln (S_{a} (T_{i} ,{\varvec{\uptheta}},{\mathbf{w}})) - \ln (\bar{S}_{a}^{g} (T_{i} ,{\varvec{\upmu}},{\varvec{\Sigma}}))} \right]} }^{2} p({\varvec{\uptheta}}|{\varvec{\upmu}},{\varvec{\Sigma}})p({\mathbf{w}})d{\varvec{\uptheta}}d{\mathbf{w}}$$

(3)

where p(w) is the probability distribution for the stochastic sequence w and $S_{a}^{g} (T_{i} ,{\varvec{\uptheta}},{\mathbf{w}})$ denotes the estimate for S_a(T_i) established through the stochastic ground motion model for specific values of the model parameter vector θ and a specific white-noise sequence w [i.e. for a specific ground motion time-history provided through the model]. Equations (2) and (3) ultimately represent the uncertainty propagation (from θ and w) to estimate the probability distribution for S_a(T_i), in this case approximated as a lognormal distribution.

The hazard compatible modification of the probability model for θ, ultimately of the parametric description defined through μ and Σ, is formulated as a multi-objective optimization problem

$$\left[ {{\varvec{\upmu}},{\varvec{\Sigma}}} \right]^{*} = \arg \hbox{min} \{ F_{1} ({\varvec{\upmu}},{\varvec{\Sigma}}|{\mathbf{z}}),F_{2} ({\varvec{\upmu}},{\varvec{\Sigma}}|{\mathbf{z}})\}$$

(4)

As discussed in Sect. 2, the first objective F₁ is the discrepancy between the target and predicted hazard probabilistic descriptions, quantified through the relative entropy as (Tsioulou et al. 2018b)

$$F_{1} ({\varvec{\upmu}},{\varvec{\Sigma}}|{\mathbf{z}}) = \frac{1}{{\sum\nolimits_{i = 1}^{{n_{y} }} \gamma_{i} }}\sum\limits_{i = 1}^{{n_{y} }} \gamma_{i} \int_{\mathbb{R}} {p_{g} (\ln (S_{a} (T_{i} ))|{\varvec{\upmu}},{\varvec{\Sigma}})\log \left[ {\frac{{p_{g} (\ln (S_{a} (T_{i} ))|{\varvec{\upmu}},{\varvec{\Sigma}})}}{{p_{t} (\ln (S_{a} (T_{1} ))|{\mathbf{z}})}}} \right]} d\ln (S_{a} (T_{i} ))$$

(5)

where the integral corresponds to the relative entropy of the compared distributions and γ_i represent the weights prioritizing the match to different IM components (spectral accelerations at different structural periods), taken in this study all equal to 1. For the typical case, which is the one examined in this paper, that p_t(ln(S_a(T_i))|z) corresponds also to a Gaussian description for ln(S_a(T_i)) with mean $\ln (\bar{S}_{a} (T_{i} ,{\mathbf{z}}))$ and variance $\sigma^{2} (T_{i} ,{\mathbf{z}})$, F₁ in (6) has closed-form solution

$$\begin{aligned} F_{1} ({\varvec{\upmu}},{\varvec{\Sigma}}|{\mathbf{z}}) & = \frac{1}{{\sum\nolimits_{i = 1}^{{n_{y} }} \gamma_{i} }}\sum\limits_{i = 1}^{{n_{y} }} \gamma_{i} \\ & \quad \cdot \left\{ {\frac{{\left( {\ln (\bar{S}_{a}^{g} (T_{i} ,{\varvec{\upmu}},{\varvec{\Sigma}})) - \ln (\bar{S}_{a} (T_{i} ,{\mathbf{z}}))} \right)^{2} }}{{2\sigma^{2} (T_{i} ,{\mathbf{z}})}} + \frac{1}{2}\left[ {\frac{{(\sigma^{g} (T_{i} ,{\varvec{\upmu}},{\varvec{\Sigma}}))^{2} }}{{\sigma^{2} (T_{i} ,{\mathbf{z}})}} - 1 - \ln \left( {\frac{{(\sigma^{g} (T_{i} ,{\varvec{\upmu}},{\varvec{\Sigma}}))^{2} }}{{\sigma^{2} (T_{i} ,{\mathbf{z}})}}} \right)} \right]} \right\} \\ \end{aligned}$$

(6)

Objective F₂ is quantified by the entropy between the original probability model p(θ|μ_r(z),Σ_r), and the modified one, p(θ|μ,Σ)

$$\begin{aligned} F_{2} ({\varvec{\upmu}},{\varvec{\Sigma}}|{\mathbf{z}}) & = \int_{{\Re^{{n_{\theta } }} }} {p({\varvec{\uptheta}}|{\varvec{\upmu}},{\varvec{\Sigma}})\log \left[ {\frac{{p({\varvec{\uptheta}}|{\varvec{\upmu}},{\varvec{\Sigma}})}}{{p({\varvec{\uptheta}}|{\varvec{\upmu}}_{r} ({\mathbf{z}}),{\varvec{\Sigma}}_{r} )}}} \right]} d{\varvec{\uptheta}} \\ & = \frac{1}{2}\left[ {tr\left[ {{\mathbf{\varvec{\Sigma} \varvec{\Sigma} }}_{r}^{ - 1} } \right] + \left( {{\varvec{\upmu}}_{r} ({\mathbf{z}}) - {\varvec{\upmu}}} \right)^{T} {\varvec{\Sigma}}_{r}^{ - 1} \left( {{\varvec{\upmu}}_{r} ({\mathbf{z}}) - {\varvec{\upmu}}} \right) - n_{\theta } - \ln \left( {\det \left[ {{\mathbf{\varvec{\Sigma} \varvec{\Sigma} }}_{r}^{ - 1} } \right]} \right)} \right] \\ \end{aligned}$$

(7)

where tr[.] and det[.] stand for trace and determinant, respectively, and n_θ corresponds to the dimension of the θ vector.

The Pareto set corresponding to the multi-objective optimization problem of Eq. (4) can be identified through any standard numerical approach, for example through genetic algorithms or stochastic search, provided that the two objectives F₁ and F₂ can be efficiently estimated. For objective F₁ this represents a challenge, since its estimation involves the high-dimensional integrals described by Eqs. (2) and (3). A framework relying on surrogate modelling was established in Tsioulou et al. (2018b) to facilitate a computational efficient implementation. Foundation of the framework is the development of a surrogate model approximation for the conditional on θ statistics, with total statistics [corresponding ultimately to Eqs. (2) and (3)] calculated through Monte Carlo integration leveraging the computational efficiency of the aforementioned surrogate model. Though computational burden for development of the surrogate model is considerable, this is a one-time cost; once the surrogate model is developed it can be implemented to support an efficient identification of the Pareto front of Eq. (4) for any desired scenario z.

Beyond the ground motion modification framework that established hazard-compatibility, a simplified implementation also exists (Tsioulou et al. 2018a) that focuses only on compatibility for the mean IM, completely ignoring variability in the predictive relationships and ultimately representing p(θ|μ,Σ) simply by μ (i.e., assumes Σ = 0). This means that the average IM prediction in Eq. (2) simplifies to

$$\bar{S}_{a}^{g} (T_{i} ,{\varvec{\upmu}}) = \int {S_{a}^{g} (T_{i} ,{\varvec{\upmu}},{\mathbf{w}})p({\mathbf{w}})d{\mathbf{w}}}$$

(8)

since only source of variability is w, while the variance $\sigma^{g} (T_{i} ,{\varvec{\upmu}})$ cannot be directly controlled (only contributing factor is w) and therefore is ignored. The ground motion modification in this case focuses simply on matching the mean IM and corresponds to adjustment of objectives F₁ and F₂, respectively, to

$$\begin{aligned} F_{1} & = \frac{1}{{\sum\nolimits_{i = 1}^{{n_{y} }} \gamma_{i} }}\sum\limits_{i = 1}^{{n_{y} }} \gamma_{i} \frac{1}{2}\left( {\frac{{\bar{S}_{a}^{g} (T_{i} ,{\varvec{\upmu}}) - \bar{S}_{a} (T_{i} ,{\mathbf{z}})}}{{\bar{S}_{a} (T_{i} ,{\mathbf{z}})}}} \right)^{2} \\ F_{2} & = \frac{1}{2}\left( {{\varvec{\upmu}}_{r} ({\mathbf{z}}) - {\varvec{\upmu}}} \right)^{T} {\varvec{\Sigma}}_{r}^{ - 1} \left( {{\varvec{\upmu}}_{r} ({\mathbf{z}}) - {\varvec{\upmu}}} \right) \\ \end{aligned}$$

(9)

The identification of the Pareto front in this case is based on a similar surrogate modeling approximation and it is implemented with even higher computational efficiency (Tsioulou et al. 2018a), since estimation of F₁ involves no Monte Carlo integration step (no uncertainty to address with respect to θ).