Characterising geotechnical model uncertainty by hybrid Markov Chain Monte Carlo simulation

Geotechnical models are usually associated with considerable amounts of model uncertainty. In this study, the model uncertainty of a geotechnical model is characterised through a systematic comparison between model predictions and past performance data. During such a comparison, model input parameters (such as soil properties) may also be uncertain, and the observed performance may be subjected to measurement errors. To consider these uncertainties, the model uncertainty parameters, uncertain model input parameters and actual performance variables are modelled as random variables, and their distributions are updated simultaneously using Bayes’ theorem. When the number of variables to update is large, solving the Bayesian updating problem is computationally challenging. A hybrid Markov Chain Monte Carlo simulation is employed in this paper to decompose the high-dimensional Bayesian updating problem into a series of updating problems in lower dimensions. To increase the efficiency of the Markov chain, the model uncertainty is first characterised with a first order second moment method approximately, and the knowledge learned from the approximate solution is then used to design key parameters in the Markov chain. Two examples are used to illustrate the proposed methodology for model uncertainty characterisation, with insights, discussions, and comparison with previous methods.     

Experimental investigation of contaminant migration in filled fracture

Experiment equipments involved in the single tube fracture and double tube fracture models are designed to research the characteristics of groundwater flow and solute transport in filled fracture. During the experiment, the state of groundwater flow can be characterized as linear flow, and satisfies Darcy’s law. Therefore, based on the pipe flow of hydraulics and Darcy’s law, the flow rate and water flow velocity can be calculated. Also, dispersion parameters were calculated with the fitting of observed data and analytical solution in the single tube fracture model. Furthermore, effects of some factors on solute transport are involved in the double tube fracture model, and length of branch fracture, particles’ diameter and flow rate in water inlet have been discussed. Results show that the arrival time of concentration peak value in the single tube fracture model is faster than that in the double tube fracture model, and two concentration peak values exist in the double tube fracture model. Arrival time of concentration peak value is faster with the increase of branch fracture length. Furthermore, if the branch fracture is longer, arrival time of the first concentration peak value is faster, while arrival time of the second concentration peak value is slower, relative to short branch fracture.     

Sensitivity and uncertainty analyses for performance assessment modeling

Sensitivity and uncertainty analyses methods for computer models are being applied in performance assessment modeling in the geologic high-level radioactive-waste repository program. The models used in performance assessment tend to be complex physical/chemical models with large numbers of input variables. There are two basic approaches to sensitivity and uncertainty analyses: deterministic and statistical. The deterministic approach to sensitivity analysis involves numerical calculation or employs the adjoint form of a partial differential equation to compute partial derivatives; the uncertainty analysis is based on Taylor series expansions of the input variables propagated through the model to compute means and variances of the output variable. The statistical approach to sensitivity analysis involves a response surface approximation to the model with the sensitivity coefficients calculated from the response surface parameters; the uncertainty analysis is based on simulation. The methods each have strengths and weaknesses.     

The effects of database parameter uncertainty on uranium(VI) equilibrium calculations

The propagation of database parameter uncertainty has been assessed for aqueous and mineral equilibrium calculations of uranium by Monte Carlo and quasi-Monte Carlo simulations in simple inorganic solution compositions. The concentration output distributions of individual chemical species varies greatly depending on the solution composition modelled. The relative uncertainty for a particular species is generally reduced in regions of solution composition for which it is predicted to be dominant, due to the asymptotic behaviour imposed by the mass balance constraint where the species concentration approaches the total element concentration. The relative uncertainties of minor species, in regions where another species comprising one or several of the same components is predicted to be dominant with a high probability, also appear to be reduced slightly. Composition regions where two or several species are equally important tend to produce elevated uncertainties for related minor species, although the uncertainties of the major species themselves tend to be reduced. The non-linear behaviour of the equilibrium systems can lead to asymmetric or bimodal output distributions; this is particularly evident close to equivalence points or solubility boundaries. Relatively conservative estimates of input uncertainty can result in considerable output uncertainty due to both the complexity of uranium solution chemistry and the system interdependencies. The results of this study suggest that for some modelling scenarios, “classical” speciation calculations based on mean value estimates of the thermodynamic values may result in predictions of a relatively low probability compared to an approach that considers the effects of uncertainty propagation.     

Uncertainty analysis in bed load transport prediction of gravel bed rivers by ANN and ANFIS

River flow is a complex dynamic system of hydraulic and sediment transport. Bed load transport have a dynamic nature in gravel bed rivers and because of the complexity of the phenomenon include uncertainties in predictions. In the present paper, two methods based on the Artificial neural networks (ANN) and adaptive neuro-fuzzy inference system (ANFIS) are developed by using 360 data points. Totally, 21 different combination of input parameters are used for predicting bed load transport in gravel bed rivers. In order to acquire reliable data subsets of training and testing, subset selection of maximum dissimilarity (SSMD) method, rather than classical trial and error method, is used in finding randomly manipulation of these subsets. Furthermore, uncertainty analysis of ANN and ANFIS models are determined using Monte Carlo simulation. Two uncertainty indices of d factor and 95% prediction uncertainty and uncertainty bounds in comparison with observed values show that these models have relatively large uncertainties in bed load predictions and using of them in practical problems requires considerable effort on training and developing processes. Results indicated that ANFIS and ANN are suitable models for predicting bed load transport; but there are many uncertainties in determination of bed load transport by ANFIS and ANN, especially for high sediment loads. Based on the predictions and confidence intervals, the superiority of ANFIS to those of ANN is proved.     

Assessing spatial uncertainty in mapping soil erodibility factor using geostatistical stochastic simulation

Soil erosion is one of most widespread process of degradation. The erodibility of a soil is a measure of its susceptibility to erosion and depends on many soil properties. Soil erodibility factor varies greatly over space and is commonly estimated using the revised universal soil loss equation. Neglecting information about estimation uncertainty may lead to improper decision-making. One geostatistical approach to spatial analysis is sequential Gaussian simulation, which draws alternative, equally probable, joint realizations of a regionalised variable. Differences between the realizations provide a measure of spatial uncertainty and allow us to carry out an error analysis. The objective of this paper was to assess the model output error of soil erodibility resulting from the uncertainties in the input attributes (texture and organic matter). The study area covers about 30 km² (Calabria, southern Italy). Topsoil samples were collected at 175 locations within the study area in 2006 and the main chemical and physical soil properties were determined. As soil textural size fractions are compositional data, the additive-logratio (alr) transformation was used to remove the non-negativity and constant-sum constraints on compositional variables. A Monte Carlo analysis was performed, which consisted of drawing a large number (500) of identically distributed input attributes from the multivariable joint probability distribution function. We incorporated spatial cross-correlation information through joint sequential Gaussian simulation, because model inputs were spatially correlated. The erodibility model was then estimated for each set of the 500 joint realisations of the input variables and the ensemble of the model outputs was used to infer the erodibility probability distribution function. This approach has also allowed for delineating the areas characterised by greater uncertainty and then to suggest efficient supplementary sampling strategies for further improving the precision of K value predictions.     

Probabilistic analysis of groundwater and radionuclide transport model from near surface disposal facilities

As per the regulatory requirements controlling the disposal of radioactive waste, the performance of waste disposal facilities needs to be assessed quantitatively using predictive models. This estimates the potential impact of disposal on the environment and public health. Near Surface Disposal Facilities (NSDFs), constructed to contain the low level radioactive waste are considered to model the radionuclide migration from the system to the geo-sphere. The radiation dose experienced by an individual through drinking water pathway is the endpoint of assessment of the model. A three dimensional groundwater contaminant transport model with a decaying source is modelled numerically to determine the radiation dose for short-lived and long-lived radionuclides. The consideration of uncertainties constitutes an intrinsic part of modelling. The uncertain input parameters include porosity, longitudinal dispersivity, transverse dispersivity, diffusion coefficient and distribution coefficient. The uncertainty propagation and quantification is carried out using collocation based stochastic response surface method (CSRSM). To run the simulations for the huge set of input, a code is developed using built-in python interface in the numerical model. The results are processed further to obtain the sensitive parameters affecting the output concentrations. Further, the probability of radiation dose exceeding permissible value is estimated by subset simulation.     

Conditional Latin Hypercube Simulation of (Log)Gaussian Random Fields

In earth and environmental sciences applications, uncertainty analysis regarding the outputs of models whose parameters are spatially varying (or spatially distributed) is often performed in a Monte Carlo framework. In this context, alternative realizations of the spatial distribution of model inputs, typically conditioned to reproduce attribute values at locations where measurements are obtained, are generated via geostatistical simulation using simple random (SR) sampling. The environmental model under consideration is then evaluated using each of these realizations as a plausible input, in order to construct a distribution of plausible model outputs for uncertainty analysis purposes. In hydrogeological investigations, for example, conditional simulations of saturated hydraulic conductivity are used as input to physically-based simulators of flow and transport to evaluate the associated uncertainty in the spatial distribution of solute concentration. Realistic uncertainty analysis via SR sampling, however, requires a large number of simulated attribute realizations for the model inputs in order to yield a representative distribution of model outputs; this often hinders the application of uncertainty analysis due to the computational expense of evaluating complex environmental models. Stratified sampling methods, including variants of Latin hypercube sampling, constitute more efficient sampling aternatives, often resulting in a more representative distribution of model outputs (e.g., solute concentration) with fewer model input realizations (e.g., hydraulic conductivity), thus reducing the computational cost of uncertainty analysis. The application of stratified and Latin hypercube sampling in a geostatistical simulation context, however, is not widespread, and, apart from a few exceptions, has been limited to the unconditional simulation case. This paper proposes methodological modifications for adopting existing methods for stratified sampling (including Latin hypercube sampling), employed to date in an unconditional geostatistical simulation context, for the purpose of efficient conditional simulation of Gaussian random fields. The proposed conditional simulation methods are compared to traditional geostatistical simulation, based on SR sampling, in the context of a hydrogeological flow and transport model via a synthetic case study. The results indicate that stratified sampling methods (including Latin hypercube sampling) are more efficient than SR, overall reproducing to a similar extent statistics of the conductivity (and subsequently concentration) fields, yet with smaller sampling variability. These findings suggest that the proposed efficient conditional sampling methods could contribute to the wider application of uncertainty analysis in spatially distributed environmental models using geostatistical simulation.     

Simultaneous Estimation of Geologic and Reservoir State Variables Within an Ensemble-Based Multiple-Point Statistic Framework

Assessment of uncertainty due to inadequate data and imperfect geological knowledge is an essential aspect of the subsurface model building process. In this work, a novel methodology for characterizing complex geological structures is presented that integrates dynamic data. The procedure results in the assessment of uncertainty associated with the predictions of flow and transport. The methodology is an extension of a previously developed pattern search-based inverse method that models the spatial variation in flow parameters by searching for patterns in an ensemble of reservoir models. More specifically, the pattern-searching algorithm is extended in two directions: (1) state values (such as piezometric head) and parameters (such as conductivities) are simultaneously and sequentially estimated, which implies that real-time assimilation of dynamic data is possible as in ensemble filtering approaches; and (2) both the estimated parameter and state variables are considered when pattern searching is implemented. The new scheme results in two main advantages—better characterization of parameters, especially for delineating small scale features, and an ensemble of head states that can be used to update the parameter field using the dynamic data at the next instant, without running expensive flow simulations. An efficient algorithm for pattern search is developed, which works with a flexible search radius and can be optimized for the estimation of either large- or small-scale structures. Synthetic examples are employed to demonstrate the effectiveness and robustness of the proposed approach.     

基于节理不确定性的可靠度分析

边坡稳定性一直是边坡安全的重点研究对象,针对边坡评价中常见的不确定性因素,可靠度分析是值得利用的方法。为评价某节理发育的岩质岸坡稳定性,通过有限元计算软件,结合现场勘探测绘数据,建立以边坡节理强度参数c、φ为输入变量,安全系数为输出变量的点估计（PEM）计算概率模型,计算结果表明:节理发育对该边坡变形具有明显控制作用;边坡整体可靠性较好,破坏概率极低。最后,通过蒙托卡罗法对可靠度结果进行验证,结果表明两种方法的计算结果不存在显著性差异。研究结果表明节理对岩质边坡稳定具有良好的敏感性,基于节理不确定性的点估计法分析边坡可靠度是一种有效的方法。     

Equilibrium modeling of U(VI) speciation in high carbonate groundwaters: Model error and propagation of uncertainty

Remediation of U-contaminated sites relies upon thermodynamic speciation calculations to predict U(VI) movement in the subsurface. However, reliability and applicability of geochemical speciation and reactive transport models may be limited by determinate (model) errors and random (uncertainty) errors in the equilibrium speciation calculations. This study examines propagated uncertainty in two types of subsurface calculations: I. Dissolved U(VI) speciation based on measured analytical constraints and solution phase equilibria and II. Overall U(VI) speciation which combined the dissolved phase equilibria with previously published adsorption reactions. Three levels of uncertainty, instrumental uncertainty, temporal variation and spatial variation across a site, were investigated using first-derivative sensitivity calculations and Monte Carlo simulations. Dissolved speciation calculations were robust, with minimal amplification of uncertainty and normal output distributions. The most critical analytical constraints in the dissolved system are pH, DIC, total U and total Ca, with some effect from dissolved . When considering adsorption equilibria, calculations were robust with respect to adsorbed U(VI) concentration prediction, but bimodal distributions of dissolved U(VI) concentrations were observed in simulations with background levels of total U(VI) and higher (spatial and temporal variability) estimates of input uncertainty. Consequently, sorption model predictions of dissolved U(VI) may not be robust with respect these higher levels of uncertainty.     

Modeling tunneling-induced ground surface settlement development using a wavelet smooth relevance vector machine

Accurate prediction of ground surface settlement is necessary for effectively controlling the settlement that develops during tunneling. Many models have been established for this purpose by extracting the relationship between the settlement and the factors that influence it. However, most of the models focused on the maximum ground surface settlement and do not involve dynamic and real-time predictions. This paper investigated how tunneling-induced ground surface settlement developed using a smooth relevance vector machine with a wavelet kernel (wsRVM). Various factors that affect this settlement, including geometrical, geological and shield operational parameters were considered. The model was applied to earth pressure balance (EPB) shield-driven tunnels. The results indicate that the prediction model performs well and that the distribution of the predictions can provide a measure of the prediction uncertainty. Unlike conventional methods that requireadditional efforts to determine relevant model parameters, the proposed method can optimize the parameters in the training process. The results of the parametric study conducted show that the model performance can be improved by the optimization and that the method can serve as a simple tool for practitioners to use in estimating ground surface settlement development during tunneling.     

A neural network approach for attenuation relationships: An application using strong ground motion data from Turkey

This paper presents an application of neural network approach for the prediction of peak ground acceleration (PGA) using the strong motion data from Turkey, as a soft computing technique to remove uncertainties in attenuation equations. A training algorithm based on the Fletcher–Reeves conjugate gradient back-propagation was developed and employed for three sample sets of strong ground motion. The input variables in the constructed artificial neural network (ANN) model were the magnitude, the source-to-site distance and the site conditions, and the output was the PGA. The generalization capability of ANN algorithms was tested with the same training data. To demonstrate the authenticity of this approach, the network predictions were compared with the ones from regressions for the corresponding attenuation equations. The results indicated that the fitting between the predicted PGA values by the networks and the observed ones yielded high correlation coefficients (R²). In addition, comparisons of the correlations by the ANN and the regression method showed that the ANN approach performed better than the regression. Even though the developed ANN models suffered from optimal configuration about the generalization capability, they can be conservatively used to well understand the influence of input parameters for the PGA predictions.     

FRFI model application in groundwater non-point source pollution evaluation: a case study in the Luoyang Basin of North Henan province, China

The traditional non-point source (NPS) pollution models mainly focus on the flow path of NPS pollutants and attenuation during the flow. Extensive data set preparation and complex results analysis for these models are the most common problems encountered by the model user. In this study a new model, fuzzy-rough sets and fuzzy inference (FRFI), was introduced to evaluate groundwater NPS pollution. The proposed model involves two steps: the algorithm of fuzzy-rough sets attribute reduction (FRSAR) was applied to yield minimal decision rules from the fuzzy information system (FIS); the fuzzy inference technique was then used to forecast a groundwater synthesis pollution index based on the minimal decision rules. This model was applied in the Luoyang Basin, examining NPS pollution factors and hydrochemical variables data to validate the effectiveness of this model. The results indicate that it is only required to collect five NPS pollution factors or three hydrochemical variables; the groundwater synthesis pollution index can be predicted using the FRFI model. The prediction error is restricted to 2.9–6.1 % and 0.8–1.6 %, respectively. Therefore, the costs of computation and monitoring can be decreased, and the user is not required to prepare massive model parameters for the FRFI model. According to analyze the correlation between NPS pollution factors and hydrochemical variables, prevention measures are provided for treatment of the endemic disease and eutrophication. The FRFI model can be suitable for groundwater NPS pollution evaluation systems.     

Some applications of a backpropagation neural network in geo-engineering

A MATLAB based backpropagation neural network (BPNN) model has been developed. Two major geo-engineering applications, namely, earth slope movement and ground movement around tunnels, are identified. Data obtained from case studies are used to train and test the developed model and the ground movement is predicted with the help of input variables that have direct physical significance. A new approach is adopted by introducing an infiltration coefficient in the network architecture apart from antecedent rainfall, slope profile, groundwater level and strength parameters to predict the slope movement. The input variables for settlement around underground excavations are taken from literature. The neural network models demonstrate a promising result predicting fairly successfully the ground behavior in both cases. If input variables influencing output goals are clearly identified and if a decent number of quality data are available, backpropagation neural network can be successfully applied as mapping and prediction tools in geotechnical investigations.     

土壤环境质量支持向量机预测模型初探——以湖南长沙、株洲、湘潭地区为例

针对土壤环境质量时序连续监测数据缺乏的现状以及城市建设发展需要,笔者试图通过建立土壤环境质量影响因素预测模型,实现利用影响因素对土壤环境质量进行预测评估。基于支持向量机相对传统经验模型除了更适合样本少情况以外,还具有泛化力强、精确度较高的优势,尝试建立基于支持向量机的土壤环境质量预测模型。研究选择时序连续的9个土壤环境质量影响因素,如国内生产总值、有害废水、废气、固体废物产生量、人口总数、年降雨量、植被覆盖面积等作为土壤环境质量预测评价因子,以长沙、株洲、湘潭地区1986年和2003年的879个土壤样品的Cu、Pb、Zn、Cd、Co、Ni、Cr、Mn含量和17年的51个影响因素样本数据作为学习和预测检验样本,采用遗传算法优选并确定了高斯核函数参数(γ)、损失函数不敏感度(ε)以及惩罚因子参数值(C),它们分别为1.021、0.000416和1012。优化逼近了土壤环境质量与影响因素的关系隐函数,获得基于支持向量机的土壤环境质量预测模型,检验结果显示了模型的有效性。     

Using an approximation of the three-point Gauss–Hermite quadrature formula for model prediction and quantification of uncertainty

 Computer models are commonly used by regulators and managers to make predictions regarding groundwater flow and contaminant concentrations at various locations and times. However, the uncertainty associated with those predictions is often overlooked, despite the fact that an assessment of such uncertainty is critical in the formulation of policy decisions. One method of quantifying the uncertainty of model predictions, based on the collective uncertainties of the model parameter input values, is to use an approximation of the three-point Gauss–Hermite quadrature formula. The Gauss–Hermite approximation is a convenient substitute for simple Monte Carlo sampling, because it requires fewer model runs and provides an immediate sensitivity analysis of parameter main effects and two-way interactions. For example, a model with four parameters, each with its own associated uncertainty, needs to be run only 33 times to complete the Gauss–Hermite analysis. For an application to a contaminant-transport model, the Gauss–Hermite approximation compares well to the full method, with considerable savings in computing effort. By comparison, Latin hypercube sampling can be more flexible, but it is more complex to use in some circumstances and cannot as easily generate the detailed sensitivity analysis that the Gauss–Hermite approach offers. Received, October 1997 Revised, August 1998 Accepted, August 1998     

Reliability analysis of soil liquefaction based on standard penetration test

A review of probabilistic and deterministic liquefaction evaluation procedures reveals that there is a need for a comprehensive approach that accounts for different sources of uncertainty in liquefaction evaluations. For the same set of input parameters, different models provide different factors of safety and/or probabilities of liquefaction. To account for the different uncertainties, including both the model and measurement uncertainties, reliability analysis is necessary. This paper presents a review and comparative study of such reliability approaches that can be used to obtain the probability of liquefaction and the corresponding factor of safety. Using a simplified deterministic Seed method, this reliability analysis has been performed. The probability of liquefaction along with the corresponding factor of safety have been determined based on a first order second moment (FOSM) method, an advanced FOSM (Hasofer–Lind) reliability method, a point estimation method (PEM) and a Monte Carlo simulation (MCS) method. A combined method that uses both FOSM and PEM is presented and found to be simple and reliable for liquefaction analysis. Based on the FOSM reliability approach, the minimum safety factor value to be adopted for soil liquefaction analysis (depending on the variability of soil resistance, shear stress parameters and acceptable risk) has been studied and a new design safety factor based on a reliability approach is proposed.     

Modeling hydrologic responses of the Zwalm catchment using the REW approach: propagation of uncertainty in soil properties to model output

The research presented in this paper focuses on the application of a newly developed physically based watershed modeling approach, which is called representative elementary watershed approach. The study stressed the effects of uncertainty of input parameters on the watershed responses (i.e., simulated discharges). The approach was applied to the Zwalm catchment, which is an agriculture-dominated watershed with a drainage area of 114 km² located in East Flanders, Belgium. Uncertainty analysis of the model parameters is limited to the saturated hydraulic conductivity because of its high influence on the watershed hydrologic behavior and availability of the data. The assessment of output uncertainty is performed using the Monte Carlo method. The ensemble statistical watershed responses and their uncertainties are calculated and compared with measurements. The results show that the measured discharges fall within the 95% confidence interval of the modeled discharge. This provides the uncertainty bounds of the discharges that account for the uncertainty in saturated hydraulic conductivity. The methodology can be extended to address other uncertain parameters as far as the probability density function of the parameter is defined.