First-order Second-moment Analysis Research Articles

Uncertainty Quantification for Aircraft Noise Emission Simulation: Methods and Limitations

Implementing measures for the reduction of aircraft noise impact, such as optimization of flight paths and aircraft, requires sophisticated simulation capabilities. These tools have to incorporate simulation of aircraft noise generation at the source (i.e., emission) and account for prevailing sound propagation effects to ultimately predict noise levels as received on the ground. Obviously, understanding the associated uncertainties is crucial when aiming at a reliable and meaningful assessment. It also becomes essential when comparing different technologies, mixing experimental and numerical data, or using simulation methods of different fidelity (e.g., semi-empirical and first-order methods). This research focuses on quantifying uncertainties in the first step in aircraft noise simulation (i.e., the prediction of the emission situation). The quantified uncertainties reflect imperfections of models for different aircraft noise sources and variability of model input parameters at different operating conditions. A general framework is presented, which also accounts for limited knowledge of the underlying data distributions, and quantitative comparisons of different uncertainty methods are provided. In particular, the first-order second-moment analysis is compared to higher-order polynomial chaos methods, and the advantages and disadvantages of the different methods are discussed.

Sensitivity and uncertainty estimation of cement paste properties to microstructural characteristics using FOSM method

The spatial distribution of microstructures in cement paste affects its mechanical and thermal properties. To evaluate the material properties of cement pastes statistically, sensitivity analysis using the first-order second-moment (FOSM) method is performed. Two random variables, namely porosity and pore continuity, are selected as input variables, and the stiffness and thermal conductivity of cement paste are selected as output variables. The statistical distributions of the input variables are obtained from X-ray micro-computed tomography (CT) images of 64 specimens and responses are estimated by using virtual specimens reconstructed from the data of the real specimens. The uncertainty of the properties when considering two random input variables agrees well with the simulation results based on the micro-CT images of real specimens. It is demonstrated that FOSM analysis with a microstructure reconstruction process can be used as a framework for estimating the sensitivity and uncertainty of material properties as long as the output responses are monotonic and approximately linear to the input variables.

Reliability of sight distance at stop-control intersections

The American Association of State Highway and Transportation Officials presents guidelines for intersection sight distance (ISD) based on extreme values of the component design variables such as speed and time gap. The guidelines for a stopped vehicle at a stop-control intersection on a minor road assume that the component design variables are deterministic. This paper presents a reliability method that considers the design variables as random. The method uses the means and variances of the probability distributions of the random variables and accounts for their correlations. A safety margin is defined as the difference between available and required ISD. Relationships for the mean and standard deviation of the safety margin are developed based on first-order second-moment analysis. The proposed method is useful for ISD design of a new intersection or redesign of an existing intersection for a desired reliability level. For single-unit trucks, the obstruction clearances from the minor- and major-road pavement edges are approximately 10% higher than those for passenger cars. The proposed method should be of interest to highway engineers involved in road safety and management.

Reliability Analysis of Minimum Pedestrian Green Interval for Traffic Signals

AbstractThe current method of computing the minimum pedestrian green interval for intersection signal timing assumes that the component variables are deterministic. This paper presents a probabilistic method in which the pedestrian start-up time and walking speed are random variables. To establish pedestrian characteristics, data were collected at 14 intersections in downtown, suburban, and tourist areas. The method is based on a safety margin that is defined as the difference between the supplied and demanded green intervals, where the demanded green interval is a random variable. Relationships for the mean and standard deviation of the safety margin of the demanded green interval are developed on the basis of the first-order second-moment analysis. A closed-form solution for the minimum supplied green interval is then derived as a function of the relevant variables, including the vehicular intergreen interval and its component variables. A procedure for establishing the walk and the flashing “don’t walk...

Uncertainty and sensitivity analysis techniques for hydrologic modeling

Formal uncertainty and sensitivity analysis techniques enable hydrologic modelers to quantify the range of likely outcomes, likelihood of each outcome and an assessment of key contributors to output uncertainty. Such information is an improvement over standard deterministic point estimates for making engineering decisions under uncertainty. This paper provides an overview of various uncertainty analysis techniques that permit mapping model input uncertainty into uncertainty in model predictions. These include Monte Carlo simulation, first-order second-moment analysis, point estimate method, logic tree analysis and first-order reliability method. Also presented is an overview of sensitivity analysis techniques that permit identification of those parameters that control the uncertainty in model predictions. These include stepwise regression, mutual information (entropy) analysis and classification tree analysis. Two case studies are presented to demonstrate the practical applicability of these techniques. The paper also discusses a systematic framework for carrying out uncertainty and sensitivity analyses.

Real-Time Demand Estimation and Confidence Limit Analysis for Water Distribution Systems

A real-time estimation of water distribution system state variables such as nodal pressures and chlorine concentrations can lead to savings in time and money and provide better customer service. While a good knowledge of nodal demands is prerequisite for pressure and water quality prediction, little effort has been placed in real-time demand estimation. This study presents a real-time demand estimation method using field measurement provided by supervisory control and data acquisition systems. For real-time demand estimation, a recursive state estimator based on weighted least-squares scheme and Kalman filter are applied. Furthermore, based on estimated demands, real-time nodal pressures and chlorine concentrations are predicted. The uncertainties in demand estimates and predicted state variables are quantified in terms of confidence limits. The approximate methods such as first-order second-moment analysis and Latin hypercube sampling are used for uncertainty quantification and verified by Monte Carlo simulation. Application to a real network with synthetically generated data gives good demand estimations and reliable predictions of nodal pressure and chlorine concentration. Alternative measurement data sets are compared to assess the value of measurement types for demand estimation. With the defined measurement error magnitudes, pipe flow data are significantly more important than pressure head measurements in estimating demands with a high degree of confidence.

Probabilistic assessment for seismic performance of port structures

Past experience has shown that ports are often susceptible to severe damage during earthquakes. From field damage data of 1995 Kobe earthquake, it is observed that the seismic behavior of port structures shows significant variability. In this study, a 2D numerical model, representing PC1 berth located in Port Island, Kobe and damaged in the 1995 Kobe earthquake, has been developed and used to simulate seismic behavior. It has been found that the uncertainties in the friction angle and the shear modulus of reclaimed soil contribute most to the variability of the residual horizontal displacement (RHD) response of the quay wall of port structures. To investigate the propagation of uncertainties of soil–structure system to the quay wall, a tornado diagram and a first-order second-moment analysis are used. Uncertainty of ground motions has also been investigated. Based on the results, design considerations have been provided.

Reliability of nailed timber connections with gusset plates

First order second moment (FOSM) analysis techniques, recently employed in the development of limit states design codes allow the engineer/designer to take into account the variations and random nature of the material properties and load effects present in the design of structural timber. A generated non-linear equation is used in conjunction with basic design mechanics to model the moment–rotation behaviour of dowel type (nailed) timber connections. The connections examined incorporated a series of nailed gusset beam to column joints with varying number of nails, subjected to a series of permanent and variable loading conditions. Joint reliability is evaluated by imposing an end rotation and calculating the reliability index under imposed deformation levels. The parameters of the equation considered as random variables were nail diameter, timber density, connection geometry and the applied loading condition. The statistics of the random variables considered are modelled using the normal, lognormal and extreme value type-1 (Gumbel) distributions. Results show that irrespective of the magnitude of variation of the other parameters, nail diameter significantly influences the moment-capacity of the connection, especially for smaller diameters. Use of probabilistic analysis techniques in design is intended to provide improved insight into system behaviour, the influence of different random variables on system performance and the interaction between different system components. This paper demonstrates how the incorporation of FOSM analysis with a non-linear moment–rotation model provides a more rational description of connection behaviour without the need to resort to more traditional Monte Carlo simulation, which can be both costly and time consuming.

MODEL ERROR PROPAGATION AND DATA COLLECTION DESIGN. AN APPLICATION IN WATER QUALITY MODELLING

The scope of this paper is to analyse the error propagation structure of a simple two compartment water quality model, and to describe the effect on model predictions from uncertainties in initial conditions and forcing inflow. Following a brief review of linear dynamic systems, a variance estimation model for the water quality model is described. The methodology employed in relating model prediction uncertainty and data collection design, is based on systems or ordinary differential equations combined with first-order second-moment analysis (FOSMA). In the paper some important characteristics of uncertain water quality systems are discussed, namely the steady-state concentration and system time scale. Applying FOSMA, the variance expressions of the system characteristics lead to some specific suggestions in the practical design of water quality models and the relation to data collection accuracy. Further it was found that the variance estimator for the steady-state concentration, provided that the system matrix is deterministic, is expressed as a weighted sum of the variances of the initial concentrations and the inflow transport. The weights are functions of the system time scales and thus related to the model parameters.

Linear and Monte Carlo uncertainty analysis for subsurface contaminant transport simulation

Transport simulation can be used to predict subsurface contaminant distributions and the effectiveness of site remediation technologies. Because of the uncertainty inherent in subsurface transport prediction, an integral part of predictive modeling is uncertainty analysis. We have investigated the uncertainty of simulated trichloroethylene (TCE) concentrations due to parameter uncertainty and variation in conceptual model. The transport simulation is performed with T2VOC, a three‐dimensional integral finite difference code for three‐phase (gas, aqueous, non‐aqueous phase liquid), three‐component (air, water, volatile organic compound), nonisothermal subsurface flow. Our modeling is based on an actual site and considers a three‐dimensional system with a thick vadose zone (25 m) into which 35,000 kg of TCE was disposed of in surface trenches over 10 years. Subsurface transport involves TCE moving in the vadose and saturated zones from the source trench toward a nearby residence and includes the processes of advection, diffusion, and adsorption over extended distances and timescales. Uncertainties in the calculated concentrations due to variance in the major transport parameters are quantified using the inverse modeling code ITOUGH2. ITOUGH2 calculates uncertainty in the T2VOC simulation by both first‐order, second‐moment (FOSM) and Monte Carlo analyses. Significant uncertainty in simulated TCE concentrations at a site of potential human exposure is observed owing to uncertainty in permeability, porosity, diffusivity, chemical solubility, and adsorption within a single conceptual model. For the case study considered the linear FOSM analysis generally captures the uncertainty range calculated by the more accurate Monte Carlo method. Calculations also show that significant output uncertainty is introduced by conceptual model variation. Because human exposure and health risk depend strongly on contaminant concentrations, risk assessment and remediation selection based on transport simulation is meaningful only if the analysis includes quantitative estimates of transport simulation uncertainty.

An efficient formulation for limit state function gradient calculation

In the first-order second moment (FOSM) reliability analysis, the gradients of the limit state function with respect to the basic random variables need to be calculated. For complex structures, these calculations can often be performed with the use of the probabilistic finite element (PFE) routines. However, for practical problems with a large number of basic random variables, the computation of the gradients is usually very expensive. This is particularly true in the FOSM analysis since the computation of the gradient vector is required to be repeated at each iteration step in the optimization algorithm. A new formulation, which reduces CPU time in computing the gradient vector without affecting the accuracy, is developed in this paper. For a simple example, direct comparison of the new technique with other methods that it is able to significantly reduce the CPU time and storage space requirements.

Canadian highway bridge evaluation: load and resistance factors

Consistent load and resistance factors are developed for a range of target values of the reliability index, β, following first-order second-moment analysis techniques for use in the evaluation of highway bridges. Dead load factors are established for steel girders, concrete girders, concrete bridge decks, and wearing surfaces, taking into account the statistical variations of weights and the range of load fractions as determined from field measurements. Live load factors are established for four categories of live loads: NP — non-permit traffic that are permitted by legislation; PM — permit, multiple trip, bulk haul, divisible loads; PS — permit, single trip, unsupervised, mixed with non-permit traffic; and PC — permit, controlled, supervised extremely heavy loads with escort. These live load factors are based on field surveys of truck weights, in Alberta and elsewhere. The event curves for NP, PS, and PM traffic have been used to determine the maximum annual truck, as the period of evaluation was chosen as 1 year based on a life-safety criterion-related to the consequences of failure. Because PC traffic is so rare, it was dealt with on an event basis. Impact data of others were analyzed to determine the appropriate bias coefficients and coefficients of variation. Uncertainties in the transverse distribution of both dead and live loads were also considered.Resistance factors are based on statistical data reported in the literature and take into account the variation in material properties, member size, and the resistance formulations. Key words: dead and live load factors, resistance factors, impact, maximum annual, traffic categories, transverse distribution, weight fractions.

First-order second-moment analysis of the buckling of shells with random imperfections

First-order second-momentum analysis is given for nominally circular cylindrical shells possessing axisymmetric as well as general nonsymmetric random imperfections. Results of measurements of initial imperfection Fourier coefficients are used to construct their second-order statistical properties. The MIUTAM computer code is applied to determine buckling loads. Results of reliability calculations are compared with those delivered by the Monte Carlo method and are found to be in good agreement.