Kriging metamodeling for seismic response distribution estimation

Abstract

This paper revisits the implementation of surrogate modeling (metamodeling) techniques within seismic risk assessment, for applications that the seismic hazard is described through stochastic ground motion models. Emphasis is placed on how to efficiently address the aleatoric uncertainty in the ground motions, stemming in this case from the stochastic sequence utilized within the excitation model. Previous work has accommodated this uncertainty by approximating the statistics of the engineering demand parameters (EDPs), something that required a large number of replication simulations (for different stochastic sequences) for each training point that was used to inform the metamodel calibration. Using kriging (Gaussian Process regression) as surrogate model, an alternative formulation is discussed here, aiming to minimize the replications for each training point. This is achieved by approximating directly the EDP distribution. It is shown that accommodating heteroscedastic behavior with respect to the aleatoric uncertainty is absolutely critical for achieving an accurate approximation, and two different approaches are presented for establishing this objective. The first approach adopts a stochastic kriging formulation, utilizing a small number of replications for judicially selected inputs, leveraging a secondary surrogate model over the latter inputs to address the heteroscedastic behavior. The second approach uses no replications, establishing a heteroscedastic nugget formulation to accommodate the EDP distribution estimation. A functional relationship is introduced between the nugget and the excitation intensity features to approximate the heteroscedastic behavior. This relationship is explicitly optimized during the metamodel calibration.

Addressing the different sources of excitation variability in seismic response distribution estimation using kriging metamodeling

“…, 18 since this approach has been shown to accommodate substantially higher computational benefits. Therefore, emphasis herein will be placed on the approximation of the EDP distribution considering the influence of the aleatoric variabilities in the seismic hazard description.…”

“…It is important to note that this lognormal assumption for the EDP distribution is not necessary for the proposed formulation. 18 Any other functional transformation (and resultant distribution) can be utilized for the definition of the metamodel output. The choice of a logarithmic transformation promotes consistency with earthquake engineering practice, as it common to assume that under the influence of aleatoric uncertainties the EDP distribution is lognormal.…”

“…Approaches to support computational efficiency of this estimation include: approximations that use static analysis, 7 efficient nonlinear response calculation, 8 advanced Monte Carlo methodologies, 9,10 reduced order modeling, 11 robust fragility procedures, 12 semi-analytical formulations based on random vibration theory, 13,14 or surrogate modeling techniques. [15][16][17][18] For applications that entail the detailed characterization of performance/risk using nonlinear response-history analysis (NLRHA) and complex, finite element structural models (FEMs), surrogate modeling techniques offer an attractive option for facilitating substantial computational savings in seismic risk/performance estimation. Surrogate models, also referenced as metamodels, establish a versatile datadriven approximation of the input (excitation and structural parameters) to output (EDPs) functional relationship using a database of structural simulations.…”

“…An alternative formulation was recently developed by the authors, 18 directly approximating the EDP distribution, instead of its statistics. This formulation was presented for applications that the seismic hazard is described through stochastic ground motion models, and uses kriging 29,30 as a metamodeling technique, leveraging its inherent probabilistic characteristics as a gaussian process (GP) emulator.…”

“…Section 3 reviews the framework for the stochastic kriging formulation with partial replications. 18 Section 4 presents in detail the different variants for the implementation of this framework to the problem formulation discussed in Section 2, examining alternative approaches for accommodating the different sources of ground motion variability. It also presents a validation framework that establishes consistent comparisons across these variants.…”