Numerical Uncertainty and Its Implications

Copyright © 2014 António F. Rodrigues, Nuno O. Martins. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accordance of the Creative Commons Attribution License all Copyrights © 2014 are reserved for SCIRP and the owner of the intellectual property António F. Rodrigues, Nuno O. Martins. All Copyright © 2014 are guarded by law and by SCIRP as a guardian.

Optimal scheduling with uncertainty in the numerical data on the basis of a stability analysis

Scheduling, job shop, uncertainty, mixed (disjunctive) graph, stability analysis

Quantifying the effects of temperature and concentration on variable-density flow in numerical modeling of groundwater systems: Implications for predictive uncertainty and data collection

Groundwater systems of different densities are often mathematically modeled to understand and predict environmental behavior such as seawater intrusion or submarine groundwater discharge. Additional data collection may be justified if it will cost-effectively aid in reducing the uncertainty of a model's prediction. The collection of salinity, as well as, temperature data could aid in reducing predictive uncertainty in a variable-density model. However, before numerical models can be created, rigorous testing of the modeling code needs to be completed. This research documents the benchmark testing of a new modeling code, SEAWAT Version 4. The benchmark problems include various combinations of density-dependent flow resulting from variations in concentration and temperature. The verified code, SEAWAT, was then applied to two different hydrological analyses to explore the capacity of a variable-density model to guide data collection. ^ The first analysis tested a linear method to guide data collection by quantifying the contribution of different data types and locations toward reducing predictive uncertainty in a nonlinear variable-density flow and transport model. The relative contributions of temperature and concentration measurements, at different locations within a simulated carbonate platform, for predicting movement of the saltwater interface were assessed. Results from the method showed that concentration data had greater worth than temperature data in reducing predictive uncertainty in this case. Results also indicated that a linear method could be used to quantify data worth in a nonlinear model. ^ The second hydrological analysis utilized a model to identify the transient response of the salinity, temperature, age, and amount of submarine groundwater discharge to changes in tidal ocean stage, seasonal temperature variations, and different types of geology. The model was compared to multiple kinds of data to (1) calibrate and verify the model, and (2) explore the potential for the model to be used to guide the collection of data using techniques such as electromagnetic resistivity, thermal imagery, and seepage meters. Results indicated that the model can be used to give insight to submarine groundwater discharge and be used to guide data collection. ^

From local to global importance measures of uncertainty propagation in composite structures

The influence of uncertainties of input parameters on output response of composite structures is investigated in this paper. In particular, the effects of deviations in mechanical properties, ply angles, ply thickness and on applied loads are studied. The uncertainty propagation and the importance measure of input parameters are analysed using three different approaches: a first-order local method, a Global Sensitivity Analysis (GSA) supported by a variance-based method and an extension of local variance to estimate the global variance over the domain of inputs. Sample results are shown for a shell composite laminated structure built with different composite systems including multi-materials. The importance measures of input parameters on structural response based on numerical results are established and discussed as a function of the anisotropy of composite materials. Needs for global variance methods are discussed by comparing the results obtained from different proposed methodologies. The objective of this paper is to contribute for the use of GSA techniques together with low expensive local importance measures.

Uncertainty propagation in inverse reliability-based design of composite structures

An approach for the analysis of uncertainty propagation in reliability-based design optimization of composite laminate structures is presented. Using the Uniform Design Method (UDM), a set of design points is generated over a domain centered on the mean reference values of the random variables. A methodology based on inverse optimal design of composite structures to achieve a specified reliability level is proposed, and the corresponding maximum load is outlined as a function of ply angle. Using the generated UDM design points as input/output patterns, an Artificial Neural Network (ANN) is developed based on an evolutionary learning process. Then, a Monte Carlo simulation using ANN development is performed to simulate the behavior of the critical Tsai number, structural reliability index, and their relative sensitivities as a function of the ply angle of laminates. The results are generated for uniformly distributed random variables on a domain centered on mean values. The statistical analysis of the results enables the study of the variability of the reliability index and its sensitivity relative to the ply angle. Numerical examples showing the utility of the approach for robust design of angle-ply laminates are presented.

Hydrologic modeling and uncertainty analysis of an ungauged watershed using mapwindow-swat

Dissertation submitted in partial fulfillment of the requirements for the Degree of Master of Science in Geospatial Technologies

Dark Clouds or Silver Linings? Knightian Uncertainty and Climate Change

This paper examines the impact of Knightian uncertainty upon optimal climate policy through the prism of a continuous-time real option modelling framework. We analytically determine optimal intertemporal climate policies under ambiguous assessments of climate damages. Additionally, numerical simulations are provided to illustrate the properties of the model. The results indicate that increasing Knightian uncertainty accelerates climate policy, i.e. policy makers become more reluctant to postpone the timing of climate policies into the future.

Local and global error models to improve uncertainty quantification

In groundwater applications, Monte Carlo methods are employed to model the uncertainty on geological parameters. However, their brute-force application becomes computationally prohibitive for highly detailed geological descriptions, complex physical processes, and a large number of realizations. The Distance Kernel Method (DKM) overcomes this issue by clustering the realizations in a multidimensional space based on the flow responses obtained by means of an approximate (computationally cheaper) model; then, the uncertainty is estimated from the exact responses that are computed only for one representative realization per cluster (the medoid). Usually, DKM is employed to decrease the size of the sample of realizations that are considered to estimate the uncertainty. We propose to use the information from the approximate responses for uncertainty quantification. The subset of exact solutions provided by DKM is then employed to construct an error model and correct the potential bias of the approximate model. Two error models are devised that both employ the difference between approximate and exact medoid solutions, but differ in the way medoid errors are interpolated to correct the whole set of realizations. The Local Error Model rests upon the clustering defined by DKM and can be seen as a natural way to account for intra-cluster variability; the Global Error Model employs a linear interpolation of all medoid errors regardless of the cluster to which the single realization belongs. These error models are evaluated for an idealized pollution problem in which the uncertainty of the breakthrough curve needs to be estimated. For this numerical test case, we demonstrate that the error models improve the uncertainty quantification provided by the DKM algorithm and are effective in correcting the bias of the estimate computed solely from the MsFV results. The framework presented here is not specific to the methods considered and can be applied to other combinations of approximate models and techniques to select a subset of realizations

Numerical modeling of the effect of variation of boundary conditions on vadose zone hydraulic properties

An accurate estimation of hydraulic fluxes in the vadose zone is essential for the prediction of water, nutrient and contaminant transport in natural systems. The objective of this study was to simulate the effect of variation of boundary conditions on the estimation of hydraulic properties (i.e. water content, effective unsaturated hydraulic conductivity and hydraulic flux) in a one-dimensional unsaturated flow model domain. Unsaturated one-dimensional vertical water flow was simulated in a pure phase clay loam profile and in clay loam interlayered with silt loam distributed according to the third iteration of the Cantor Bar fractal object Simulations were performed using the numerical model Hydrus 1D. The upper and lower pressure heads were varied around average values of -55 cm for the near-saturation range. This resulted in combinations for the upper and lower constant head boundary conditions, respectively, of -50 and -60 cm, -40 and -70 cm, -30 and -80 cm, -20 and -90 cm, and -10 and -100 cm. For the drier range the average head between the upper and lower boundary conditions was set to -550 cm, resulting in the combinations -500 and -600 cm, -400 and -700 cm, -300 and -800 cm, -200 and -900 cm, and -100 and -1,000 cm, for upper and lower boundary conditions, respectively. There was an increase in water contents, fluxes and hydraulic conductivities with the increase in head difference between boundary conditions. Variation in boundary conditions in the pure phase and interlayered one-dimensional profiles caused significant deviations in fluxes, water contents and hydraulic conductivities compared to the simplest case (a head difference between the upper and lower constant head boundaries of 10 cm in the wetter range and 100 cm in the drier range).

Using numerical classification of profiles based on Vis-NIR spectra to distinguish soils from the Piracicaba Region, Brazil

Considering that information from soil reflectance spectra is underutilized in soil classification, this paper aimed to evaluate the relationship of soil physical, chemical properties and their spectra, to identify spectral patterns for soil classes, evaluate the use of numerical classification of profiles combined with spectral data for soil classification. We studied 20 soil profiles from the municipality of Piracicaba, State of São Paulo, Brazil, which were morphologically described and classified up to the 3rd category level of the Brazilian Soil Classification System (SiBCS). Subsequently, soil samples were collected from pedogenetic horizons and subjected to soil particle size and chemical analyses. Their Vis-NIR spectra were measured, followed by principal component analysis. Pearson's linear correlation coefficients were determined among the four principal components and the following soil properties: pH, organic matter, P, K, Ca, Mg, Al, CEC, base saturation, and Al saturation. We also carried out interpretation of the first three principal components and their relationships with soil classes defined by SiBCS. In addition, numerical classification of the profiles based on the OSACA algorithm was performed using spectral data as a basis. We determined the Normalized Mutual Information (NMI) and Uncertainty Coefficient (U). These coefficients represent the similarity between the numerical classification and the soil classes from SiBCS. Pearson's correlation coefficients were significant for the principal components when compared to sand, clay, Al content and soil color. Visual analysis of the principal component scores showed differences in the spectral behavior of the soil classes, mainly among Argissolos and the others soils. The NMI and U similarity coefficients showed values of 0.74 and 0.64, respectively, suggesting good similarity between the numerical and SiBCS classes. For example, numerical classification correctly distinguished Argissolos from Latossolos and Nitossolos. However, this mathematical technique was not able to distinguish Latossolos from Nitossolos Vermelho férricos, but the Cambissolos were well differentiated from other soil classes. The numerical technique proved to be effective and applicable to the soil classification process.

Impacts of upland open drains upon runoff generation: a numerical assessment of catchment-scale impacts

Shallow upland drains, grips, have been hypothesized as responsible for increased downstream flow magnitudes. Observations provide counterfactual evidence, often relating to the difficulty of inferring conclusions from statistical correlation and paired catchment comparisons, and the complexity of designing field experiments to test grip impacts at the catchment scale. Drainage should provide drier antecedent moisture conditions, providing more storage at the start of an event; however, grips have higher flow velocities than overland flow, thus potentially delivering flow more rapidly to the drainage network. We develop and apply a model for assessing the impacts of grips on flow hydrographs. The model was calibrated on the gripped case, and then the gripped case was compared with the intact case by removing all grips. This comparison showed that even given parameter uncertainty, the intact case had significantly higher flood peaks and lower baseflows, mirroring field observations of the hydrological response of intact peat. The simulations suggest that this is because delivery effects may not translate into catchment-scale impacts for three reasons. First, in our case, the proportions of flow path lengths that were hillslope were not changed significantly by gripping. Second, the structure of the grip network as compared with the structure of the drainage basin mitigated against grip-related increases in the concentration of runoff in the drainage network, although it did marginally reduce the mean timing of that concentration at the catchment outlet. Third, the effect of the latter upon downstream flow magnitudes can only be assessed by reference to the peak timing of other tributary basins, emphasizing that drain effects are both relative and scale dependent. However, given the importance of hillslope flow paths, we show that if upland drainage causes significant changes in surface roughness on hillslopes, then critical and important feedbacks may impact upon the speed of hydrological response. Copyright (c) 2012 John Wiley & Sons, Ltd.

On the numerical integration of rigid body nonlinear dynamics in presence of parameters singularities

One of the main complexities in the simulation of the nonlinear dynamics of rigid bodies consists in describing properly the finite rotations that they may undergo. It is well known that, to avoid singularities in the representation of the SO(3) rotation group, at least four parameters must be used. However, it is computationally expensive to use a four-parameters representation since, as only three of the parameters are independent, one needs to introduce constraint equations in the model, leading to differential-algebraic equations instead of ordinary differential ones. Three-parameter representations are numerically more efficient. Therefore, the objective of this paper is to evaluate numerically the influence of the parametrization and its singularities on the simulation of the dynamics of a rigid body. This is done through the analysis of a heavy top with a fixed point, using two three-parameter systems, Euler's angles and rotation vector. Theoretical results were used to guide the numerical simulation and to assure that all possible cases were analyzed. The two parametrizations were compared using several integrators. The results show that Euler's angles lead to faster integration compared to the rotation vector. An Euler's angles singular case, where representation approaches a theoretical singular point, was analyzed in detail. It is shown that on the contrary of what may be expected, 1) the numerical integration is very efficient, even more than for any other case, and 2) in spite of the uncertainty on the Euler's angles themselves, the body motion is well represented.

Quantifying uncertainty in the simulation of crop-climate processes

The impacts of climate change on crop productivity are often assessed using simulations from a numerical climate model as an input to a crop simulation model. The precision of these predictions reflects the uncertainty in both models. We examined how uncertainty in a climate (HadAM3) and crop General Large-Area Model (GLAM) for annual crops model affects the mean and standard deviation of crop yield simulations in present and doubled carbon dioxide (CO2) climates by perturbation of parameters in each model. The climate sensitivity parameter (λ, the equilibrium response of global mean surface temperature to doubled CO2) was used to define the control climate. Observed 1966–1989 mean yields of groundnut (Arachis hypogaea L.) in India were simulated well by the crop model using the control climate and climates with values of λ near the control value. The simulations were used to measure the contribution to uncertainty of key crop and climate model parameters. The standard deviation of yield was more affected by perturbation of climate parameters than crop model parameters in both the present-day and doubled CO2 climates. Climate uncertainty was higher in the doubled CO2 climate than in the present-day climate. Crop transpiration efficiency was key to crop model uncertainty in both present-day and doubled CO2 climates. The response of crop development to mean temperature contributed little uncertainty in the present-day simulations but was among the largest contributors under doubled CO2. The ensemble methods used here to quantify physical and biological uncertainty offer a method to improve model estimates of the impacts of climate change.

Quantification of physical and biological uncertainty in the simulation of the yield of a tropical crop using present-day and doubled CO2 climates

The impacts of climate change on crop productivity are often assessed using simulations from a numerical climate model as an input to a crop simulation model. The precision of these predictions reflects the uncertainty in both models. We examined how uncertainty in a climate (HadAM3) and crop General Large-Area Model (GLAM) for annual crops model affects the mean and standard deviation of crop yield simulations in present and doubled carbon dioxide (CO2) climates by perturbation of parameters in each model. The climate sensitivity parameter (lambda, the equilibrium response of global mean surface temperature to doubled CO2) was used to define the control climate. Observed 1966-1989 mean yields of groundnut (Arachis hypogaea L.) in India were simulated well by the crop model using the control climate and climates with values of lambda near the control value. The simulations were used to measure the contribution to uncertainty of key crop and climate model parameters. The standard deviation of yield was more affected by perturbation of climate parameters than crop model parameters in both the present-day and doubled CO2 climates. Climate uncertainty was higher in the doubled CO2 climate than in the present-day climate. Crop transpiration efficiency was key to crop model uncertainty in both present-day and doubled CO2 climates. The response of crop development to mean temperature contributed little uncertainty in the present-day simulations but was among the largest contributors under doubled CO2. The ensemble methods used here to quantify physical and biological uncertainty offer a method to improve model estimates of the impacts of climate change.