Linear Propagation Of Uncertainties Research Articles

Estimation of Aerosol Layer Height from OLCI Measurements in the O2A-Absorption Band over Oceans

The aerosol layer height (ALH) is an important parameter that characterizes aerosol interaction with the environment. An estimation of the vertical distribution of aerosol is necessary for studies of those interactions, their effect on radiance and for aerosol transport models. ALH can be retrieved from satellite-based radiance measurements within the oxygen absorption band between 760 and 770 nm (O2A band). The oxygen absorption is reduced when light is scattered by an elevated aerosol layer. The Ocean and Land Colour Imager (OLCI) has three bands within the oxygen absorption band. We show a congruent sensitivity study with respect to ALH for dust and smoke cases over oceans. Furthermore, we developed a retrieval of the ALH for those cases and an uncertainty estimation by applying linear uncertainty propagation and a bootstrap method. The sensitivity study and the uncertainty estimation are based on radiative transfer simulations. The impact of ALH, aerosol optical thickness (AOT), the surface roughness (wind speed) and the central wavelength on the top of atmosphere (TOA) radiance is discussed. The OLCI bands are sufficiently sensitive to ALH for cases with AOTs larger than 0.5 under the assumption of a known aerosol type. With an accurate spectral characterization of the OLCI O2A bands better than 0.1 nm, ALH can be retrieved with an uncertainty of a few hundred meters. The retrieval of ALH was applied successfully on an OLCI dust and smoke scene. The found ALH is similar to parallel measurements by the Tropospheric Monitoring Instrument (TROPOMI). OLCI’s high spatial resolution and coverage allow a detailed overview of the vertical aerosol distribution over oceans.

Transcranial ultrasound simulation with uncertainty estimation.

Transcranial ultrasound simulations are increasingly used to predict in situ exposure parameters for ultrasound therapies in the brain. However, there can be considerable uncertainty in estimating the acoustic medium properties of the skull and brain from computed tomography (CT) images. This paper shows how the resulting uncertainty in the simulated acoustic field can be predicted in a computationally efficient way using linear uncertainty propagation. Results for a representative transcranial simulation using a focused bowl transducer at 500 kHz show good agreement with unbiased uncertainty estimates obtained using Monte Carlo.

A novel linear uncertainty propagation method for nonlinear dynamics with interval process

Interval process is a preferable model for time-varying uncertainty propagation of dynamic systems when only the range of uncertainties can be obtained. However, for nonlinear systems, except for Monte Carlo (MC) simulation, there are still few efficient uncertainty propagation methods under the interval process model. This paper develops a non-intrusive and semi-analytical uncertainty propagation method, named the “convex model linearization method (CMLM),” by constructing a linearization formulation of a nonlinear system in a non-probabilistic sense. First, the criterion to evaluate the difference between the original system and the linearization formulation is derived, represented by the discrepancy of middle point, radius and correlations of response. By minimizing these three parameters, the coefficients of linear equations will be optimized to obtain the linearization formulation of the original system. Then, analytical equations are built to calculate uncertainty response under the interval process, without time-consuming analysis of the original system. To further improve the efficiency of the linearization process, Chebyshev polynomial is introduced to approximate the nonlinear dynamic analysis. Two numerical examples of duffing oscillators and vehicle rides are set to test the proposed CMLM. Compared to the MC method, with comparable uncertainty response precision, the CMLM just needs 1–10% times of dynamic analyses of the nonlinear system. Furthermore, a practical launch vehicle ascent trajectory problem with black-box dynamics is solved by, respectively, the CMLM and MC method. The results verify the capacity of the CMLM to deal with black-box problems and show that the CMLM performs better in terms of accuracy, efficiency and robustness.

Optimal experimental design of physical property measurements for optimal chemical process simulations

Chemical process simulations depend on physical properties, which are usually available through property models with parameters estimated from experiments. The required experimental effort can be reduced using the method of Optimal Experimental Design (OED). OED reduces the number of experiments by minimising the expected uncertainty of the estimated parameters. In chemical engineering, however, the main purpose of an experiment is usually not to determine property parameters with minimum uncertainty but to simulate processes accurately. Therefore, this paper presents the OED of physical property measurements resulting in the most accurate process simulations: c-optimal experimental design (c-OED). c-OED aims to minimise the uncertainty of the process simulation results, which is estimated by linear uncertainty propagation from uncertain property parameters through the process model. We use c-OED to design liquid-liquid equilibrium and diffusion experiments minimising thermodynamic and economic performance metrics of three solvent-based processes. In all three case studies, the c-optimal design substantially reduces the experimental effort for the same simulation accuracy compared to state-of-the-art OED that neglects the process. Our findings are confirmed by a Monte-Carlo simulation of the designed experiments. Furthermore, we assess the limits of c-OED for highly nonlinear process models. Thus, the work shows how c-OED can successfully reduce experimental effort required for accurate process simulations by tailoring experimental designs to the process model.

Probabilistic Response of a Bladed Disk With Uncertain Geometry

AbstractGeometric uncertainties in the blade manufacturing process have important consequences in terms of dynamical properties of bladed disks. In this paper, we address the problem of modeling a full bladed disk composed by blades having uncertain geometry. The geometric imperfection of the blades is represented and analyzed according to a procedure previously presented by the authors, based on the principal component analysis (PCA) and the mesh morphing. The dynamical model of the full disk is constructed following the component mode synthesis (CMS) approach. The blade geometry is represented using a probabilistic model constructed from an experimental dataset. The effect of the geometric uncertainties is assessed using a linear uncertainty propagation approach, leading to a procedure that is fast enough to be embedded into a Monte Carlo simulation (MCS) loop.

Prediction of Heat Transfer in a Jet Cooled Aircraft Engine Compressor Cone Based on Statistical Methods

The paper presents the setup and analysis of an experimental study on heat transfer of a jet cooled compressor rear cone with adjacent conical housing. The main goal of the paper is to describe the systematic derivation of empirical correlations for global Nusselt numbers to be used in the design process of a jet engine secondary air system. Based on the relevant similarity parameters obtained from literature, operating points are deduced leading to a full factorial design experiment to identify all effects and interactions. The varied similarity parameters are the circumferential Reynolds number, the non-dimensional mass flow, the non-dimensional spacing between rotor and stator, and the jet incidence angle. The range of the varied similarity parameters covers engine oriented operating conditions and is therefore suitable to predict Nusselt numbers in the actual engine component. In order to estimate measurement uncertainties, a simplified model of the test specimen, consisting of a convectively cooled flat plate, has been derived. Uncertainties of the measured quantities and derived properties are discussed by means of a linear propagation of uncertainties. A sensitivity study shows the effects of the input parameters and their interactions on the global Nusselt number. Subsequently, an empirical correlation for the global Nusselt numbers is derived using a multivariate non-linear regression. The quality of the empirical correlation is assessed by means of statistical hypotheses and by a comparison between measured and predicted data. The predicted values show excellent agreement with experimental data. In a wide range, accuracies of 15% can be reached when predicting global Nusselt numbers. Furthermore, the results of the sensitivity study show that pre-swirled cooling air does not have a positive effect on heat transfer.

Uncertainty propagation in the N-body problem using Dromo elements

Reliable and efficient uncertainty propagation is crucial for the task of monitoring possible impacts of Near Earth Asteroids with our planet. It is well known that a switch of the primary body can greatly reduce the numerical truncation error in the case of planetary flybys. In the present work, the advantages of performing a primary body switch in the uncertainty propagation problem are explored. For this purpose, first we present the linear uncertainty propagation using Dromo formulation, which has been shown in previous works to have a satisfactory performance when propagating the orbit uncertainty of Near Earth Asteroids. Next, we introduce the concept of primary body switch for the uncertainty propagation problem. The algorithm is based on the following procedure. We sample the initial orbit uncertainty distribution and linearly propagate the samples, considering the N-body gravitational influence. A primary body switch, which is a nonlinear mapping, is performed for all the samples at a threshold distance from the approaching planet, the Earth. The orbit of the samples is then linearly propagated with respect to the Earth using the Dromo formulation until the threshold distance is reached again. Finally, the propagation center is changed back to the heliocentric frame and the linear propagation continues. We apply the proposed method to an extensive set of asteroids that approach the Earth. Results suggest that the average error of the linear propagation can be reduced up to a factor of 30 when compared to a purely heliocentric linear propagation using Dromo elements.

A posteriori uncertainty quantification of PIV-based pressure data

A methodology for a posteriori uncertainty quantification of pressure data retrieved from particle image velocimetry (PIV) is proposed. It relies upon the Bayesian framework, where the posterior distribution (probability distribution of the true velocity, given the PIV measurements) is obtained from the prior distribution (prior knowledge of properties of the velocity field, e.g., divergence-free) and the statistical model of PIV measurement uncertainty. Once the posterior covariance matrix of the velocity is known, it is propagated through the discretized Poisson equation for pressure. Numerical assessment of the proposed method on a steady Lamb–Oseen vortex shows excellent agreement with Monte Carlo simulations, while linear uncertainty propagation underestimates the uncertainty in the pressure by up to 30 %. The method is finally applied to an experimental test case of a turbulent boundary layer in air, obtained using time-resolved tomographic PIV. Simultaneously with the PIV measurements, microphone measurements were carried out at the wall. The pressure reconstructed from the tomographic PIV data is compared to the microphone measurements. Realizing that the uncertainty of the latter is significantly smaller than the PIV-based pressure, this allows us to obtain an estimate for the true error of the former. The comparison between true error and estimated uncertainty demonstrates the accuracy of the uncertainty estimates on the pressure. In addition, enforcing the divergence-free constraint is found to result in a significantly more accurate reconstructed pressure field. The estimated uncertainty confirms this result.

Investigation of Verification Artifacts in WR-03 Waveguides

This paper presents an investigation of verification artifacts in WR-03 waveguides. The waveguide verification artifacts include a cross-guide and a custom-made circular iris section. The investigation involves the transmission loss uncertainty analysis of the verification artifacts for vector network analyzer (VNA) waveguide systems operating at millimeter wavelengths. The measurement errors due to the dimensional tolerances and the flange misalignment are predicted by using a commercially available electromagnetic software package. The data analysis is carried out for complex-valued scattering parameters (S-parameters). The uncertainty due to different error sources is computed according to the Law of Propagation of Uncertainty. The real and imaginary data along with the associated uncertainties are converted to magnitude and phase representation via linear propagation of uncertainties. The experimental results are also compared with simulated results.

Uncertainties in power computations in a turbocharger test bench

A specific study of the uncertainties of turbine power output measured in turbocharger test benches is presented using the law of uncertainty propagation and the influence of the different terms that contribute to it is shown. Then, non-linear mixed integer mathematical programming algorithms used with the turbine power uncertainty expression become an essential tool to overcome the problem of selection new sensors to improve an existing test rig or to contribute to a new one. A method of optimisation is presented for two different scenarios: first, where the maximum cost is a constraint; second where the maximum uncertainty is a constraint and the total cost is minimised. When using a large transducers database, computational efforts may be reduced by solving the relaxed non-integer problem by means of sequential quadratic programming and then probing the ceilings and floors of the parameters to get an optimum approximation with low costs. A comparison between the linear uncertainty propagation model and Monte Carlo simulations is also presented, only showing benefits of the later method when computing high order statistical moments of the turbine power output probability distribution.

Size exclusion deep bed filtration: Experimental and modelling uncertainties

A detailed uncertainty analysis associated with carboxyl-modified latex particle capture in glass bead-formed porous media enabled verification of the two theoretical stochastic models for prediction of particle retention due to size exclusion. At the beginning of this analysis it is established that size exclusion is a dominant particle capture mechanism in the present study: calculated significant repulsive Derjaguin-Landau-Verwey-Overbeek potential between latex particles and glass beads is an indication of their mutual repulsion, thus, fulfilling the necessary condition for size exclusion. Applying linear uncertainty propagation method in the form of truncated Taylor's series expansion, combined standard uncertainties (CSUs) in normalised suspended particle concentrations are calculated using CSUs in experimentally determined parameters such as: an inlet volumetric flowrate of suspension, particle number in suspensions, particle concentrations in inlet and outlet streams, particle and pore throat size distributions. Weathering of glass beads in high alkaline solutions does not appreciably change particle size distribution, and, therefore, is not considered as an additional contributor to the weighted mean particle radius and corresponded weighted mean standard deviation. Weighted mean particle radius and LogNormal mean pore throat radius are characterised by the highest CSUs among all experimental parameters translating to high CSU in the jamming ratio factor (dimensionless particle size). Normalised suspended particle concentrations calculated via two theoretical models are characterised by higher CSUs than those for experimental data. The model accounting the fraction of inaccessible flow as a function of latex particle radius excellently predicts normalised suspended particle concentrations for the whole range of jamming ratios. The presented uncertainty analysis can be also used for comparison of intra- and inter-laboratory particle size exclusion data.

Metas.UncLib—a measurement uncertainty calculator for advanced problems

Metas.UncLib is a software library that facilitates the linear propagation of uncertainties through a measurement model. It is able to handle complex-valued and multivariate quantities and supports higher mathematics. It is therefore able to deal with advanced metrological problems that require, e.g., matrix manipulations. The software is optimized for short computation times and low memory use.

Efficient estimation of uncertainty in weakly non-linear algorithms for measurand reconstruction

Evaluation of measurement uncertainty is a key issue in measurement science since the practical value of any measurement result depends on how precise that uncertainty is assessed. This statement applies, in particular, to measurand reconstruction being a fundamental operation in any measuring system. Thus, the algorithms of measurand reconstruction should be provided with mechanisms for evaluating the uncertainty of the results of computation. If those algorithms are linear with respect to the data and all intermediate results of computation, and if they do not use the data or intermediate results more than once, then the linear propagation of uncertainties, executed step by step for each operation, yields a satisfactory solution to the problem of uncertainty analysis. Otherwise, a tendency for overestimating the uncertainty appears. An effective method for dealing with this problem in linear and weakly non-linear algorithms of measurand reconstruction is proposed in this paper. Its computational efficiency is demonstrated using weakly non-linear algorithms, viz., a non-recursive Cauchy filter and a recursive quadratic filter.