A critical examination of seismic response uncertainty analysis in earthquake engineering

The last decade of performance‐based earthquake engineering (PBEE) research has seen a rapidly increasing emphasis placed on the explicit quantification of uncertainties. This paper examines uncertainty consideration in input ground‐motion and numerical seismic response analyses as part of PBEE, with particular attention given to the physical consistency and completeness of uncertainty consideration. It is argued that the use of the commonly adopted incremental dynamic analysis leads to a biased representation of the seismic intensity and that when considering the number of ground motions to be used in seismic response analyses, attention should be given to both reducing parameter estimation uncertainty and also limiting ground‐motion selection bias. Research into uncertainties in system‐specific numerical seismic response analysis models to date has been largely restricted to the consideration of ‘low‐level’ constitutive model parameter uncertainties. However, ‘high‐level’ constitutive model and model methodology uncertainties are likely significant and therefore represent a key research area in the coming years. It is also argued that the common omission of high‐level seismic response analysis modelling uncertainties leads to a fallacy that ground‐motion uncertainty is more significant than numerical modelling uncertainty. The author's opinion of the role of uncertainty analysis in PBEE is also presented. Copyright © 2013 John Wiley & Sons, Ltd.     

Seismic reliability assessment and the nonergodicity in the modelling parameter uncertainties

Modelling uncertainty can significantly affect the structural seismic reliability assessment. However, the limit state excursion due to this type of uncertainty may not be described by a Poisson process as it lacks renewal properties with the occurrence of each earthquake event. Furthermore, considering uncertainties related to ground motion representation by employing recorded ground motions together with modelling uncertainties is not a trivial task. Robust fragility assessment, proposed previously by the authors, employs the structural response to recorded ground motion as data in order to update prescribed seismic fragility models. Robust fragility can be extremely efficient for considering also the structural modelling uncertainties by creating a dataset of one-to-one assignments of structural model realizations and as-recorded ground motions. This can reduce the computational effort by more than 1 order of magnitude. However, it should be kept in mind that the fragility concept itself is based on the underlying assumption of Poisson-type renewal. Using the concept of updated robust reliability, considering both the uncertainty in ground motion representation based on as-recorded ground motion and non ergodic modelling uncertainties, the error introduced through structural reliability assessment by using the robust fragility is quantified. It is shown through specific application to an existing RC frame that this error is quite small when the product of the time interval and the standard deviation of failure rate is small and is on the conservative side.     

Accounting for source location errors in the bayesian analysis of seismicity and seismic hazard

A method is presented for incorporating the uncertainties associated with hypocentral locations in the formulation of probabilistic models of the time and space distributions of the activity of potential seismic sources, as well as of the resulting seismic hazard functions at sites in their vicinity. For this purpose, a bayesian framework of analysis is adopted, where the probabilistic models considered are assumed to have known forms and uncertain parameters, the distribution of the latter being the result of an a priori assessment and its updating through the incorporation of the direct statistical information, including the uncertainty associated with the relations between the actual hypocentral locations and the reported data. This uncertainty is incorporated in the evaluation of the likelihood function of the parameters to be estimated for a given sample of recorded locations. For the purpose of illustration, the method proposed is applied to the modelling of the seismic sources near a site close to the southern coast of Mexico. The results of two alternate algorithms for the incorporation of location uncertainties are compared with those arising from neglecting those uncertainties. One of them makes use of Monte Carlo simulation, while the other is based on a closed-form analytical integration following the introduction of some simplifying assumptions. For the particular case studied, accounting for location uncertainties gives place to significant changes in the probabilistic models of the seismic sources. Deviations of the same order of magnitude can be ascribed to differences in the mathematical and/or numerical tools used in the uncertainty analysis. The resulting variability of the seismic hazard at the site of interest is less pronounced than that affecting the estimates of activity of individual seismic sources.     

Model uncertainty analysis by variance decomposition

Errors and uncertainties in hydrological, hydraulic and environmental models are often substantial. In good modelling practice, they are quantified in order to supply decision-makers with important additional information on model limitations and sources of uncertainty. Several uncertainty analysis methods exist, often with various underlying assumptions. One of these methods is based on variance decomposition. The method allows splitting the variance of the total error in the model results (as estimated after comparing model results with observations) in its major contributing uncertainty sources. This paper discusses an advanced version of that method where error distributions for rainfall, other inputs and parameters are propagated in the model and the “rest” uncertainties considered as model structural errors for different parts of the model. By expert knowledge, the iid assumption that is often made in model error analysis is addressed upfront. The method also addresses the problems of heteroscedasticity and serial dependence of the errors involved. The method has been applied by the author to modelling applications of sewer water quantity and quality, river water quality and river flooding.     

Benchmarking observational uncertainties for hydrology: rainfall,river discharge and water quality

This review and commentary sets out the need for authoritative and concise information on the expected error distributions and magnitudes in observational data. We discuss the necessary components of a benchmark of dominant data uncertainties and the recent developments in hydrology which increase the need for such guidance. We initiate the creation of a catalogue of accessible information on characteristics of data uncertainty for the key hydrological variables of rainfall, river discharge and water quality (suspended solids, phosphorus and nitrogen). This includes demonstration of how uncertainties can be quantified, summarizing current knowledge and the standard quantitative results available. In particular, synthesis of results from multiple studies allows conclusions to be drawn on factors which control the magnitude of data uncertainty and hence improves provision of prior guidance on those uncertainties. Rainfall uncertainties were found to be driven by spatial scale, whereas river discharge uncertainty was dominated by flow condition and gauging method. Water quality variables presented a more complex picture with many component errors. For all variables, it was easy to find examples where relative error magnitudes exceeded 40%. We consider how data uncertainties impact on the interpretation of catchment dynamics, model regionalization and model evaluation. In closing the review, we make recommendations for future research priorities in quantifying data uncertainty and highlight the need for an improved ‘culture of engagement’ with observational uncertainties. Copyright © 2012 John Wiley & Sons, Ltd.     

Influence of variable uncertainties in seismic tomography models on constraining mantle viscosity from geoid observations

The radial viscosity structure of the Earth is explored on the basis of the geoid observations. The variations of uncertainty in seismic tomography models are accounted for when finding the radial viscosity structure. The new methodology we propose attempts to fit more closely those features of the geoid that are better constrained by tomography models and avoids to fit those features that are poorly constrained. This approach is particularly important because the error of geoid predictions caused by uncertainties in seismic tomography models is overwhelmingly larger than the noise in the geoid measurements. The synthetic tests indicate that the viscosity structures obtained by disregarding the uncertainty variations in seismic tomography models can be biased depending on the geoid spectral band and on the ‘input’ seismic tomography model. When the uncertainty variations in seismic models are considered in the inversion process, results do not indicate a viscosity in the transition zone lower than in the upper mantle. A robust feature found with the new method is a viscosity in the upper mantle two orders of magnitude smaller than in the lower mantle. The error covariance of seismic tomography models is critical for the method we suggest. A covariance matrix rigorously derived by seismologists should help to even more reliably infer the viscosity structure and relation between anomalies in density and seismic velocities from surface observations such as the geoid, and thus lead to a better knowledge of the Earth interior.     

Using experimental data to reduce the single-building sigma of fragility curves: case study of the BRD tower in Bucharest, Romania

The lack of knowledge concerning modelling existing buildings leads to significant variability in fragility curves for single or grouped existing buildings. This study aims to investigate the uncertainties of fragility curves, with special consideration of the single-building sigma. Experimental data and simplified models are applied to the BRD tower in Bucharest, Romania, a RC building with permanent instrumentation. A three-step methodology is applied： （1） adjustment of a linear MDOF model for experimental modal analysis using a Timoshenko beam model and based on Anderson＇s criteria, （2） computation of the structure＇s response to a large set of accelerograms simulated by SIMQKE software, considering twelve ground motion parameters as intensity measurements （IM）, and （3） construction of the fragility curves by comparing numerical interstory drift with the threshold criteria provided by the Hazus methodology for the slight damage state. By introducing experimental data into the model, uncertainty is reduced to 0.02 considering Sd ） as seismic intensity IM and uncertainty related to the model is assessed at 0.03. These values must be compared with the total uncertainty value of around 0.7 provided by the Hazus methodology.     

The integrated effects of climate and hydrologic uncertainty on future flood risk assessments

This work examines future flood risk within the context of integrated climate and hydrologic modelling uncertainty. The research questions investigated are (1) whether hydrologic uncertainties are a significant source of uncertainty relative to other sources such as climate variability and change and (2) whether a statistical characterization of uncertainty from a lumped, conceptual hydrologic model is sufficient to account for hydrologic uncertainties in the modelling process. To investigate these questions, an ensemble of climate simulations are propagated through hydrologic models and then through a reservoir simulation model to delimit the range of flood protection under a wide array of climate conditions. Uncertainty in mean climate changes and internal climate variability are framed using a risk‐based methodology and are explored using a stochastic weather generator. To account for hydrologic uncertainty, two hydrologic models are considered, a conceptual, lumped parameter model and a distributed, physically based model. In the conceptual model, parameter and residual error uncertainties are quantified and propagated through the analysis using a Bayesian modelling framework. The approach is demonstrated in a case study for the Coralville Dam on the Iowa River, where recent, intense flooding has raised questions about potential impacts of climate change on flood protection adequacy. Results indicate that the uncertainty surrounding future flood risk from hydrologic modelling and internal climate variability can be of the same order of magnitude as climate change. Furthermore, statistical uncertainty in the conceptual hydrological model can capture the primary structural differences that emerge in flood damage estimates between the two hydrologic models. Copyright © 2014 John Wiley & Sons, Ltd.     

Uncertainty Analysis of Strong-Motion and Seismic Hazard

In this article we present the modelling of uncertainty in strong-motion studies for engineering applications, particularly for the assessment of earthquake hazard. We examine and quantify the sources of uncertainty in the basic variables involved in ground motion estimation equations, including those associated with the seismological parameters, which we derive from a considerable number of strong-motion records. Models derived from regression analysis result in ground motion equations with uncertain parameters, which are directly related to the selected basic variables thus providing an uncertainty measure for the derivative variable. These uncertainties are exemplified and quantified. An alternative approach is presented which is based on theoretical modelling defining a functional relationship on a set of independent basic variables. Uncertainty in the derivative variable is then readily obtained when the uncertainties of the basic variables have been defined. In order to simplify the presentation, only the case of shallow strike-slip earthquakes is presented. We conclude that the uncertainty is approximately the same as given by the residuals typical for regression modelling. This implies that uncertainty in ground motion modelling cannot be reduced below certain limits, which is in accordance with findings reported in the literature. Finally we discuss the implications of the presented methodology in hazard analyses, which is sensitive to the truncation of the internal error term, commonly given as an integral part of ground motion estimation equations. The presented methodology does not suffer from this shortcoming; it does not require truncation of the error term and yields realistic hazard estimates. This revised version was published online in July 2006 with corrections to the Cover Date.     

Assessing uncertainties in urban drainage models

The current state of knowledge regarding uncertainties in urban drainage models is poor. This is in part due to the lack of clarity in the way model uncertainty analyses are conducted and how the results are presented and used. There is a need for a common terminology and a conceptual framework for describing and estimating uncertainties in urban drainage models. Practical tools for the assessment of model uncertainties for a range of urban drainage models are also required to be developed. This paper, produced by the International Working Group on Data and Models, which works under the IWA/IAHR Joint Committee on Urban Drainage, is a contribution to the development of a harmonised framework for defining and assessing uncertainties in the field of urban drainage modelling. The sources of uncertainties in urban drainage models and their links are initially mapped out. This is followed by an evaluation of each source, including a discussion of its definition and an evaluation of methods that could be used to assess its overall importance. Finally, an approach for a Global Assessment of Modelling Uncertainties (GAMU) is proposed, which presents a new framework for mapping and quantifying sources of uncertainty in urban drainage models.     

Error distribution modelling of satellite soil moisture measurements for hydrological applications

Satellite‐based soil moisture data accuracies are of important concerns by hydrologists because they could significantly influence hydrological modelling uncertainty. Without proper quantification of their uncertainties, it is difficult to optimize the hydrological modelling system and make robust decisions. Currently, the satellite soil moisture data uncertainty has been limited to summary statistics with the validations mainly from the in situ measurements. This study attempts to build the first error distribution model with additional higher‐order uncertainty modelling for satellite soil moisture observations. The methodology is demonstrated by a case study using the Soil Moisture and Ocean Salinity satellite soil moisture observations. The validation is based on soil moisture estimates from hydrological modelling, which is more relevant to the intended data use than the in situ measurements. Four probability distributions have been explored to find suitable error distribution curves using the statistical tests and bootstrapping resampling technique. General extreme value is identified as the most suitable one among all the curves. The error distribution model is still in its infant stage, which ignores spatial and temporal correlations, and nonstationarity. Further improvements should be carried out by the hydrological community by expanding the methodology to a wide range of satellite soil moisture data using different hydrological models. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite Difference Simulations of Seismic Wave Propagation for the 2007 Mw 6.6 Niigata-ken Chuetsu-Oki Earthquake: Validity of Models and Reliable Input Ground Motion in the Near-Field

Finite difference simulations of seismic wave propagation are performed in the Niigata area, Japan, for the 2007 Mw 6.6 Niigata-ken Chuetsu-Oki earthquake at low frequencies. We test three 3D structural models built independently in various studies. First aftershock simulations are carried out. The model based on 3D tomography yields correct body waves in the near field, but later phases are imperfectly reproduced due to the lack of shallow sediment layers; other models based on various 1D/2D profiles and geological interpretation provide good site responses but generate seismic phases that may be shifted from those actually observed. Next, for the mainshock simulations, we adopt two different finite source models that differ in the near-field ground motion, especially above the fault plane (but under the sea) and then along the coastline. Each model is found to be calibrated differently for the given stations. For engineering purposes, the variations observed in simulated ground motion are significant, but for seismological purposes, additional parameter calibrations would be possible for such a complex 3D case.     

Markov chain Monte Carlo algorithms for target-oriented and interval-oriented amplitude versus angle inversions with non-parametric priors and non-linear forward modellings

In geophysical inverse problems, the posterior model can be analytically assessed only in case of linear forward operators, Gaussian, Gaussian mixture, or generalized Gaussian prior models, continuous model properties, and Gaussian-distributed noise contaminating the observed data. For this reason, one of the major challenges of seismic inversion is to derive reliable uncertainty appraisals in cases of complex prior models, non-linear forward operators and mixed discrete-continuous model parameters. We present two amplitude versus angle inversion strategies for the joint estimation of elastic properties and litho-fluid facies from pre-stack seismic data in case of non-parametric mixture prior distributions and non-linear forward modellings. The first strategy is a two-dimensional target-oriented inversion that inverts the amplitude versus angle responses of the target reflections by adopting the single-interface full Zoeppritz equations. The second is an interval-oriented approach that inverts the pre-stack seismic responses along a given time interval using a one-dimensional convolutional forward modelling still based on the Zoeppritz equations. In both approaches, the model vector includes the facies sequence and the elastic properties of P-wave velocity, S-wave velocity and density. The distribution of the elastic properties at each common-mid-point location (for the target-oriented approach) or at each time-sample position (for the time-interval approach) is assumed to be multimodal with as many modes as the number of litho-fluid facies considered. In this context, an analytical expression of the posterior model is no more available. For this reason, we adopt a Markov chain Monte Carlo algorithm to numerically evaluate the posterior uncertainties. With the aim of speeding up the convergence of the probabilistic sampling, we adopt a specific recipe that includes multiple chains, a parallel tempering strategy, a delayed rejection updating scheme and hybridizes the standard Metropolis–Hasting algorithm with the more advanced differential evolution Markov chain method. For the lack of available field seismic data, we validate the two implemented algorithms by inverting synthetic seismic data derived on the basis of realistic subsurface models and actual well log data. The two approaches are also benchmarked against two analytical inversion approaches that assume Gaussian-mixture-distributed elastic parameters. The final predictions and the convergence analysis of the two implemented methods proved that our approaches retrieve reliable estimations and accurate uncertainties quantifications with a reasonable computational effort.     

Incorporating structural uncertainty of hydrological models in likelihood functions via an ensemble range approach

ABSTRACT

Model ensembles are possibly the most powerful tool to assess uncertainties in runoff predictions stemming from inadequacies in model structure. But in many applications little knowledge is gained about the specific weaknesses of the individual models. Here we introduce the ensemble range approach (ERA). Compared to other ensemble techniques, ERA is primarily intended to facilitate hydrological reasoning about model structural uncertainty. This is attempted by separate modelling of data uncertainty and structural uncertainty with two different error density functions that are combined in one likelihood function. The width of the structural error density is in accordance with the range of runoff predictions calculated by a small model ensemble at each individual time step. Albeit not the only choice, this study is restricted on the use of a modified beta density to represent structural uncertainty. The performance of ERA is assessed in some synthetic and real data case studies. Ensembles of two structurally identical models are applied, made possible by estimating the parameters of ERA and both models simultaneously.

Editor D. Koutsoyiannis; GUEST editor S. Weijs     

Seismic driven probabilistic classification of reservoir facies for static reservoir modelling: a case history in the Barents Sea

In this paper we present a case history of seismic reservoir characterization where we estimate the probability of facies from seismic data and simulate a set of reservoir models honouring seismically‐derived probabilistic information. In appraisal and development phases, seismic data have a key role in reservoir characterization and static reservoir modelling, as in most of the cases seismic data are the only information available far away from the wells. However seismic data do not provide any direct measurements of reservoir properties, which have then to be estimated as a solution of a joint inverse problem. For this reason, we show the application of a complete workflow for static reservoir modelling where seismic data are integrated to derive probability volumes of facies and reservoir properties to condition reservoir geostatistical simulations. The studied case is a clastic reservoir in the Barents Sea, where a complete data set of well logs from five wells and a set of partial‐stacked seismic data are available. The multi‐property workflow is based on seismic inversion, petrophysics and rock physics modelling. In particular, log‐facies are defined on the basis of sedimentological information, petrophysical properties and also their elastic response. The link between petrophysical and elastic attributes is preserved by introducing a rock‐physics model in the inversion methodology. Finally, the uncertainty in the reservoir model is represented by multiple geostatistical realizations. The main result of this workflow is a set of facies realizations and associated rock properties that honour, within a fixed tolerance, seismic and well log data and assess the uncertainty associated with reservoir modelling.     

Scale effects on modelling the seismic signature of gas: results from an outcrop analogue

Numerous examples of reservoir fields from continental and marine environments involve thin‐bedded geology, yet, the inter‐relationship between thin‐bedded geology, fluid flow and seismic wave propagation is poorly understood. In this paper, we explore the 4D seismic signature due to saturation changes of gas within thin layers, and address the challenge of identifying the relevant scales and properties, which correctly define the geology, fluid flow and seismic wave propagation in the field. Based on the study of an outcrop analogue for a thin‐bedded turbidite, we model the time‐lapse seismic response to fluid saturation changes for different levels of model scale, and explore discrepancies in quantitative seismic attributes caused by upscaling. Our model reflects the geological complexity associated with thin‐bedded turbidites, and its coupling to fluid flow, which in turn affects the gas saturation distribution in space, and its time‐lapse seismic imprint. Rock matrix and fluid properties are modelled after selected fields to reproduce representative field models with realistic impedance contrasts. In addition, seismic modelling includes multiples, in order to assess their contribution in seismic propagation through thin gas layers. Our results show that multiples could contribute significantly to the measured amplitudes in the case of thin‐bedded geology. This suggests that forward/inverse modelling involving the flow simulation and seismic domains used in time‐lapse seismic interpretation should account for thin layers, when these are present in the geological setting.     

地震危险性分析衰减不确定性校正中的主观不确定性

实际使用中的危险性分析模型均带有主观的性质，其与现实原型之间的差异包含两种性质不同的不确定性，即现实模型与理想模型之间的主观不确定性，和理想模型与现实之间的随机误差或客观不确定性。本文具体讨论对衰减规律作（客观）不确定性校正时主观不确定性的影响问题。文中分析了衰减不确定性校正的两种做法，即在危险性分析中直接校正和先忽略衰减不确定性，求得危险性曲线后总校正的做法，讨论了两者的等价性，并用半定量的方法     

Quantification of uncertainty in design water levels due to uncertain bed form roughness in the Dutch river Waal

Hydrodynamic river models are applied to design and evaluate measures for purposes such as safety against flooding. The modelling of river processes involves numerous uncertainties, resulting in uncertain model results. Knowledge of the type and magnitude of these uncertainties is crucial for a meaningful interpretation of the model results. Uncertainty in the hydraulic roughness due to bed forms is one of the main contributors to the uncertainty in the modelled water levels. The aim of this study was to quantify the uncertainty in the bed form roughness under design conditions and quantify the effect on the design water levels in the Dutch river Waal. Five roughness models that predict bed form roughness based on measured bed form and flow characteristics were extrapolated to design conditions. The results show that the 95% confidence interval of the predicted Nikuradse roughness values under design conditions ranges from 0.32 to 1.03 m. This uncertainty was propagated through the two‐dimensional hydrodynamic model, WAQUA, by means of a Monte Carlo simulation for an idealized schematization of the Dutch river Waal. The uncertain bed form roughness results in an uncertainty in the design water levels, with a 95% confidence interval of 0.53 m, which is significant for Dutch river management practice. The uncertainty in the bed form roughness was mainly caused by a lack of knowledge about the physical process of bed form evolution that causes roughness. An improved estimation of bed form roughness can significantly reduce the uncertainty in the design water levels. Copyright © 2012 John Wiley & Sons, Ltd.