Rosenblueth's Point Estimate Method Research Articles

Investigation into Mining Economic Evaluation Approaches Based on the Rosenblueth Point Estimate Method

Conducting technical and economic evaluations is important for mining investment and mining operation decision-making. Traditional economic evaluation methods rarely address the issue of evaluation reliability and usually require complex calculations to obtain the optimal solution. In this study, the Rosenblueth point estimate method for reliability evaluation of engineering project schemes is introduced. Combined with the cash flow method for economic evaluation of mines, the Rosenblueth point estimate method for evaluating the reliability of mining economy is established. Based on the technical and economic index of the case mine, taking the ore grade as a sensitivity indicator, empirical research on established models and methods was carried out. The results of the economic reliability evaluation and the variation rules obtained using the Rosenblueth point estimate method model were basically consistent with the actual production and operation rules of mining enterprise. The similar results also proved that the proposed model has good applicability and reliability for mining economic evaluation. Using the proposed RPEM economic reliability model, the economic reliability of a certain iron mine in Liaoning Province was calculated to be 99.95, which was a huge improvement compared with the traditional evaluation method. Additionally, the calculation process of the proposed model for economic reliability evaluation is simple and the accuracy is controllable. The economic reliability of the project can be calculated based on changes in sensitivity indicators, and the value range of sensitivity indicators can also be calculated through the required reliability. The obtained results and the proposed evaluation model provide a decision-making basis for mining investment projects and operation management.

Parametric uncertainty analysis on hydrodynamic coefficients in groundwater numerical models using Monte Carlo method and RPEM

Groundwater resources are the only source of water in many arid and semi-arid regions. It is important to manage these resources to have a sustainable development. However, there are many factors influencing the accuracy of the results in groundwater modeling. In this research, the uncertainty of two important groundwater model parameters (hydraulic conductivity and specific yield) were considered as the main sources of uncertainty in estimating water level in an unconfined aquifer, in Iran. For this purpose, a simple method called Rosenblueth Point Estimate Method (RPEM) was used to assess groundwater modeling parametric uncertainty, and its performance was compared with Monte Carlo method as a very complicated and time-consuming method. According to calibrated values of hydraulic conductivity and specific yield, several uncertainty intervals were considered to analyze uncertainty. The results showed that the optimum interval for hydraulic conductivity was 40% increase–30% decrease of the calibrated values in both Monte Carlo and RPEM methods. This interval for specific yield was 200% increase–90% decrease of the calibrated values. RPEM showed better performance using the evaluating indices in comparison with Monte Carlo method for both hydraulic conductivity and specific yield with 43% and 17% higher index values, respectively. These results can be used in groundwater management and future prediction of groundwater level.

A point-estimate based method for soil amplification estimation using high resolution model under uncertainty of stratum boundary geometry

This study provides a practical technique for stochastically estimating soil amplification under uncertainty of the geometry of stratum boundary. To evaluate the possible values of the non-linear soil response under ambiguity, a nonlinear dynamic analysis with a high resolution 3D-FEM model is preferable, although the computations still represent a rigorous task. This study modifies Rosenblueth's point estimate method (PEM), which is greatly advantageous from the viewpoint of computational time, and demonstrates that the proposed method can provide good estimations with a very small number of calculations compared to the Monte-Carlo method. To evaluate the ground response stochastically, the preferable probability density function (PDF) is also examined. Furthermore, the estimation accuracy of the exceedance probability and the limitations of the proposed method are also discussed.

Probabilistic analysis of the active earth pressure on retaining wall for c-f soil backfill under seismic loading conditions

En este artículo se describe una metodología basada en el método de estimación puntual de Rosenblueth para el análisis del empuje activo desarrollado en un muro de retención con relleno cohesivo-friccionante bajo condiciones de carga sísmica. El principio básico de esta metodología es usar dos estimaciones puntales, i.e., la desviación estándar y el valor medio, para examinar una variable en el análisis de seguridad. Es posible mostrar que aumentando el valor del coeficiente de aceleración sísmica horizontal, el factor de seguridad por volteo decrece y la probabilidad de falla aumenta, especialmente para coeficientes mayores que 0.2. Por otro lado, es observado que el valor medio del factor de seguridad crece cuando aumenta el coeficiente de aceleración sísmica vertical, sin embargo la probabilidad de falla se mantiene prácticamente igual para el valor del factor de seguridad considerado como crítico (1.15).

Methodology for quantitative landslide risk analysis in residential projects

This paper provides a quantitative assessment for the hazard, the vulnerability and the risk associated to slope instability as a tool for urban planners and policy makers. Analysis methodologies and overall numerical procedures are presented in detail. In order to assess a hazard, which is expressed as a temporal probability, FOSM technique was used along with Rosenblueth's point estimate method. Reliability index (β) was used as a standard measure to compare the results assessed with other information presented in published literature for a number of geotechnical projects. Concerning the analysis of vulnerability, a new approach was proposed by combining local methodology for seismic vulnerability designed for buildings in Medellin with generic analysis methodology for vulnerability of people when exposed to landslides. The risk was assessed by a simple mathematical crossing between hazard and vulnerability. A database of 120 residential projects located on natural slopes in the city of Medellin (Colombia) was used for the analysis. The results were presented as FN charts relating the calculated frequency of landslides to the number of potential life or economic losses.

Probabilistic modeling of rainfall-induced shallow landslide using a point-estimate method

The assessment of rainfall-induced shallow landslides using a physical-based model requires accounting for the uncertainty of soil parameters that cannot be accurately quantified. In this study, the modified Iverson model was integrated with the Rosenblueth point-estimate method to probabilistically model rainfall-induced shallow landslides, while simultaneously considering the uncertainty and correlation of soil parameters. Using various soil parameters, hillslope conditions, and hydrological conditions, the applicability of the Rosenblueth point-estimate method was compared with that of the Monte Carlo simulation method. The simulated results indicated that the Rosenblueth point-estimate method can accurately and efficiently assess the probability of rainfall-induced shallow landslides. A lognormal distribution of the safety factor can be assumed when the soil parameters are lognormally distributed random variables. Accounting for the correlation of soil parameters decreases the standard deviation of the safety factor, but does not affect the mean of the safety factor. Neglecting the correlation of soil parameters could substantially misestimate the probability of rainfall-induced shallow landslides.

Reliability based approach to tunnel lining design using a modified point estimate method

This paper presents a new reliability-based design approach to evaluate the performance of a composite tunnel lining using a modified Rosenblueth Point Estimate Method (PEM), First Order Reliability Method (FORM), Monte Carlo sampling method and finite element analysis. The modified PEM involves the selection of additional evaluation points to provide a more complete assessment of system performance across the range of statistically significant values. To demonstrate this approach, the support performance at the Yacambú-Quibor tunnel in Venezuela was assessed. The general rockmass characteristics and the details of the composite lining system are presented for a segment along the tunnel. Numerical analyses were completed using finite element modeling to determine the behaviour of the lining over the range of possible rockmass and in situ stress conditions. The results of these analyses were then used to determine the reliability index (β) and probability of failure (pf) for a given liner section.

Application of Neural Network for Prediction of Unmeasured Mode Shape in Damage Detection

The major problem in the vibration-based damage detection field is still a limited number of sensors and the existence of uncertainties. In this paper, a new approach combines a multi-stage ANN model and statistical method to detect damage based on the limited number of sensors with consideration of uncertainties. The first stage of the ANN is used to predict the unmeasured mode shapes data based on limited measured modal data. The second stage ANN is devoted to predicting the damage location and severity using the complete modal data from the first-stage ANN. To incorporate the uncertainties in modal data, Gaussian noise is applied to the input variables and the probability of damage existence is calculated using Rosenblueth's point estimate method. The feasibility of the proposed method is demonstrated using an analytical model of a continuous two-span reinforced concrete slab. The application of a multi-stage ANN showed results having a high potential of overcoming the issue of using a limited number of sensors in structural health monitoring.

Assessing the impact of uncertainty on flood risk estimates with reliability analysis using 1-D and 2-D hydraulic models

Abstract. This paper addresses the use of reliability techniques such as Rosenblueth's Point-Estimate Method (PEM) as a practical alternative to more precise Monte Carlo approaches to get estimates of the mean and variance of uncertain flood parameters water depth and velocity. These parameters define the flood severity, which is a concept used for decision-making in the context of flood risk assessment. The method proposed is particularly useful when the degree of complexity of the hydraulic models makes Monte Carlo inapplicable in terms of computing time, but when a measure of the variability of these parameters is still needed. The capacity of PEM, which is a special case of numerical quadrature based on orthogonal polynomials, to evaluate the first two moments of performance functions such as the water depth and velocity is demonstrated in the case of a single river reach using a 1-D HEC-RAS model. It is shown that in some cases, using a simple variable transformation, statistical distributions of both water depth and velocity approximate the lognormal. As this distribution is fully defined by its mean and variance, PEM can be used to define the full probability distribution function of these flood parameters and so allowing for probability estimations of flood severity. Then, an application of the method to the same river reach using a 2-D Shallow Water Equations (SWE) model is performed. Flood maps of mean and standard deviation of water depth and velocity are obtained, and uncertainty in the extension of flooded areas with different severity levels is assessed. It is recognized, though, that whenever application of Monte Carlo method is practically feasible, it is a preferred approach.

RosenPoint: A Microsoft Excel-based program for the Rosenblueth point estimate method and an application in slope stability analysis

The Rosenblueth point estimate method is one of the probabilistic analyses in estimating failure probability of a system, such as a slope. The essence of the approach is to use two point estimates, mean value±standard deviation, to present a variable in safety evaluation. The simple and straightforward framework leads to its wide application, but as a system governed by n variables (n is large), mass computations (2n repetitions in calculation) are required during the analysis. This prevents the possibility of hand computation using the approach, and a proper computing tool is needed under this situation. In this study, a Microsoft Excel-based program, RosenPoint, was developed for the Rosenblueth approach, and the program developments, descriptions and modifications are given in detail. The program is successfully demonstrated by computing the failure probability of an infinite slope under earthquake condition with a deterministic factor of safety (FOS) equal to 1.77. As the critical FOS is equal to 1.4, the slope that is considered stable by a conventional analysis is found associated with a substantial failure probability around 20%. Since the current version of RosenPoint is designed for estimating slope failure probability, the program needs modification as it is used for other tasks. Owing to the separated programming structure in RosenPoint, the subroutine governing FOS algorithms only needs to be replaced or recompiled as modification is needed. In addition, the capacity of the current RosenPoint is limited to 19 variables due to the dimension constraint of Excel spreadsheets (=220 rows). However, the capacity can be easily improved with sacrificing output completeness. This program modification is also described in this paper.

Uncertainty and Reliability Analysis Applied to Slope Stability: A Case Study From Sungun Copper Mine

Nowadays, there are many new methods for slope stability analysis; including probabilistic methods assessing geotechnical uncertainties to develop safety factors. In this paper, a reliability index analysis for the Sungun copper mine slope stability is evaluated based on three methods of uncertainties consisting Taylor series method, Rosenblueth point estimate method and Monte-Carlo simulation method. Sungun copper mine will be one of the Iran’s biggest mines with final pit’s height of 700 meters. For this study two of its main slopes were assessed, one dipping to the NE (030) and the other to the SE (140). Probability density function of cohesion and angle of friction for the slopes were developed using limit equilibrium methods. These shear strengths were then used to determine the probability density function of safety factor and reliability index using the probabilistic methods. Results of the probabilistic analysis indicate that with ascending values of the uncertainties the reliability index decreases. Furthermore, it was determined that with the Monte Carlo simulation the seed number used has little effect on the reliability index of the safety factor especially with seed numbers in excess of 1200. Variations in the overall reliability index of safety factor were observed between the two slopes and this difference is explained by the differences in complexities of the geology within the cross-section.

Influence of Uncertain Source Parameters on Strong Ground Motion Simulation with the Empirical Green's Function Method

A combined stochastic and Green's function approach was developed to simulate strong ground motions in Southwest Western Australia (SWWA) in a previous study. Although it was demonstrated that adopting the source parameters derived from other regions yielded reasonable simulation of ground motions in SWWA as compared with a few available strong motion records, the effect of source parameter variations on simulated ground motions was not known. This article performs a statistical study of the effects of random fluctuations of the seismic source parameters on simulated strong ground motions. The uncertain source parameters, i.e., stress drop ratio, rupture velocity, and rise time corresponding to the empirical source models are assumed to be the respective mean value of the parameter and normally distributed with an assumed coefficient of variation. The Rosenblueth's point estimate method [Rosenblueth, 1981] is used to calculate the statistics of the simulate ground motion parameters corresponding to different magnitudes and epicentral distances. The accuracy of the Rosenblueth's point estimate method in estimating the mean and standard deviation of ground motion PGA, PGV, and response spectrum is proven by simulating the ground motions from an ML6.0 and epicentral distance 100 km event with both the Rosenblueth's point estimate method and the Monte Carlo simulation method. A sensitivity analysis is preformed to investigate the effect of random fluctuations of each source parameters on strong ground motion simulation. A coefficient of variation model for ground motion parameters is developed based on the simulated data as a function of the variations of the three source parameters and earthquake magnitude, which can be used in probabilistic predictions of earthquake ground motions with uncertain source parameters.

Incorporating Reliability into the Definition of the Margin of Safety in Total Maximum Daily Load Calculations

In the calculation of the total maximum daily load (TMDL), the margin of safety (MOS) is an attempt to account for the uncertainties related to the conceptual representation of the system and the inherently random nature of the processes under study. In general, the MOS is often the result of arbitrary or poorly supported decisions that have little to do with the reliability of the system. To improve the current practice in TMDL calculations, an enhanced reliability-based MOS is proposed here. The Modified Rosenblueth point estimate method is proposed in this paper as a valuable means to assess the uncertainty of a system, and, thus, its probability of failure. The Modified Rosenblueth method is a less computationally demanding uncertainty analysis technique, and can be easily coupled with deterministic hydraulic and water quality numerical models. A case study on arsenic concentrations in the Niagara River is discussed in this paper. The results of the uncertainty analysis are used to compute the effecti...

Evaluation of Probabilistic Point Estimate Methods in Uncertainty Analysis for Environmental Engineering Applications

This paper begins with a comprehensive review of various point estimate methods, with an emphasis on their differences and similarities in theory and application. The Rosenblueth probabilistic point estimate method is a computationally straightforward technique for the uncertainty analysis of engineering problems. It is capable of estimating a statistical moment of any order of a model output involving several stochastic variables that are correlated or uncorrelated, symmetric, or asymmetric. However, in multivariate problems with more than two stochastic variables involved, the Rosenblueth method is not able to provide a unique solution, rather an approximate solution to indeterminate problems. This is attributed to the fact that the number of unknowns to be solved is larger than the number of governing equations provided. An improved modified Rosenblueth point estimate method is proposed to circumvent the drawback of the nonunique solution of the original Rosenblueth method and to increase the computational efficiency in modeling. One example application on the particle terminal velocity computation is presented for illustration. A quantitative performance index is introduced to assess the performance of various point estimate methods. It is concluded in this study that the modified Rosenblueth method has a comparable performance to the Rosenblueth method and yet resolves the nonuniqueness problem in solutions.

Distinctive and fuzzy failure probability analysis of an anisotropic rock mass to explosion load

AbstractThis paper estimates failure probability of an anisotropic rock mass with random initial damage and random critical tensile strain to explosion loads. The initial random damage and critical tensile strain of the rock mass are estimated by approximate statistical methods. They are incorporated into the constitutive law of the anisotropic damage model. The statistical estimation of the rock‐mass response to underground explosion is evaluated by the Rosenblueth's point estimate method. A statistical anisotropic continuum damage model considering both the anisotropic initial damage and cumulative damage dependent on tensile strain is suggested. A beta distribution is proposed to represent the probabilistic distribution of the damage variable of the rock mass under blasting loads. Using the theory of reliability, failure probabilities are calculated according to different levels of failure criteria of the rock mass. A fuzzy definition is also proposed to describe the fuzzy nature of failure phenomenon of the rock mass. Based on the fuzzy random probabilistic theory, a model including both the effects of randomness of the rock‐mass properties and fuzziness of its failure criterion is proposed for the failure analysis of the rock mass. The suggested models are coded and linked with an available computer programe Autodyn3D as its user's subroutines. Using the Autodyn3D together with the suggested model, a series of field blasting tests are simulated. Numerical results of stress wave propagation are compared with field‐recorded data. The failure probabilities of the rock mass around the charge hole are estimated. Copyright © 2002 John Wiley & Sons, Ltd.

Numerical Analysis of Blast-Induced Stress Waves in a Rock Mass with Anisotropic Continuum Damage Models Part 2: Stochastic Approach

This paper reports the second part of the study carried out by the authors on the underground explosion-induced stress wave propagation and damage in a rock mass. In the accompanying paper reporting the first part of the study, equivalent material properties were used to model the effects of existing cracks and joints in the rock mass. The rock mass and its properties were treated as deterministic. In this paper, existing random cracks and joints are modeled as statistical initial damage of the rock mass. In numerical calculation, an anisotropic continuum damage model including both the statistical anisotropic initial damage and cumulative damage dependent on principal tensile strain and stochastic critical tensile strain is suggested to model rock mass behavior under explosion loads. The statistical estimation of stress wave propagation in the rock mass due to underground explosion is evaluated by Rosenblueth's point estimate method. The suggested models and statistical solution process are also programmed and linked to Autodyn3D as its user's subroutines. Numerical results are compared with the field test data and those presented in the accompanying paper obtained with equivalent material property approach.

Point-Estimate Method as Numerical Quadrature

Rosenblueth's point-estimate method for approximating the low-order moments of functions of random variables is widely used in geotechnical reliability analyses. It is a special case of numerical quadrature based on orthogonal polynomials. For normal variables, it corresponds to Gauss-Hermite quadrature, but Rosenblueth's procedure automatically generates the weights and abscissas of Gauss-Legendre and Gauss-Laguerre quadrature as well. Despite beliefs to the contrary, the method is not a form of Monte Carlo simulation or FOSM Taylor series expansion. The method is reasonably robust and can be satisfactorily accurate for a wide range of practical problems, although the computational requirements increase rapidly with the number of uncertain quantities. Caution should be exercised in applying the method when transformation of the uncertain quantities severely changes the form of the distributions or when moments of order greater than the two are involved. Examples with closed-form solutions serve to illustrate the use and accuracy of the method.

Dynamic response analysis of rock mass with stochastic properties subjected to explosive loads

Abstract This paper carries out probabilistic analysis of dynamic responses of rock mass under blast loads. Statistical analysis of the initial damage and strength constants of the rock mass is performed by using both the field and laboratory test data. The initial damage of the rock mass is found having the beta distribution, while the critical tensile strain has the normal distribution. These statistical properties are incorporated into the constitutive law and cumulative damage model for rock mass. The statistical estimation of stress wave propagation in the rock mass due to underground explosion is evaluated by Rosenblueth's point estimate method. In numerical calculation, an isotropic continuum damage model considering both the initial damage and the cumulative damage dependent on an equivalent tensile strain is suggested to model rock mass behaviour under blast loads. The suggested models are programmed and linked to an available computer program Autodyn2D through its user's subroutine capability. U...

Uncertainty Analysis by Point Estimate Methods Incorporating Marginal Distributions

The model performance of an engineering system is affected by many variables subject to uncer­ tainty. Point estimate (PE) methods are practical tools to assess the uncertainty features of a model involving multivariate stochastic parameters. Two PE methods have been developed for engineering applications. One is Rosenblueth's PE method, which preserves the first three moments of random variables and the other is Hare's PE method, which reduces the computations of Rosenblueth's method but only as appropriate for application to random variables with normal distributions. In this study, two algorithms are proposed to encompass the advantages of the two PE methods: computational practicality and the handling of mixture distributions. Through a numerical experiment, the proposed methods yielded more accurate estimations than those of Rosenblueth's method with about the same amount of computation as Hare's method. The two proposed methods were also applied to estimate statistical moments of a pier scouring model output to demonstrate their performance in an engineering application.

Calibration and Control for Estuary System under Uncertainty

A comprehensive stochastic management methodology is developed for a distributed-parameter estuary system (DPES) which is described by partial differential equations (PDEs). The proposed stochastic management methodology consists of uncertainty-based calibration (i.e. parameter estimation under uncertainty) and uncertainty-based control (i.e. determination of optimal freshwater inflows into the estuary under uncertainty). The technique for uncertainty-based calibration is based on Gauss–Newton minimization method and uncertainty analysis method such as Rosenblueth's point estimate method or Harr's point estimate method. The uncertainty-based calibration technique is used to estimate the optimal parameters such as Manning's roughness coefficient and dispersion coefficient in the PDEs while considering the uncertainties in the boundary conditions or initial conditions for the PDEs. The technique for uncertainty-based control is based on a real-time control method which is discrete-time stochastic linear quadratic feedback control. Real-time control is achieved by feedback control which uses salinities at test stations in the estuary as the state vector and controlled freshwater inflows as the control vector. The feedback control law determines the real-time control vector using observed states in a manner that best approximates desired states and controls (freshwater inflows). This new methodology for real-time control has been applied to the Lavaca–Tres Palacios Estuary in Texas for purposes of testing and illustration.