First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.

Uncertainty in simulating wheat yields under climate change

Projections of climate change impacts on crop yields are inherently uncertain1. Uncertainty is often quantified when projecting future greenhouse gas emissions and their influence on climate2. However, multi-model uncertainty analysis of crop responses to climate change is rare because systematic and objective comparisons among process-based crop simulation models1, 3 are difficult4. Here we present the largest standardized model intercomparison for climate change impacts so far. We found that individual crop models are able to simulate measured wheat grain yields accurately under a range of environments, particularly if the input information is sufficient. However, simulated climate change impacts vary across models owing to differences in model structures and parameter values. A greater proportion of the uncertainty in climate change impact projections was due to variations among crop models than to variations among downscaled general circulation models. Uncertainties in simulated impacts increased with CO2 concentrations and associated warming. These impact uncertainties can be reduced by improving temperature and CO2 relationships in models and better quantified through use of multi-model ensembles. Less uncertainty in describing how climate change may affect agricultural productivity will aid adaptation strategy development andpolicymaking.

Visualizing probabilistic flood forecast information: expert preferences and perceptions of best practice in uncertainty communication

The aim of this article is to improve the communication of the probabilistic flood forecasts generated by hydrological ensemble prediction systems (HEPS) by understanding perceptions of different methods of visualizing probabilistic forecast information. This study focuses on interexpert communication and accounts for differences in visualization requirements based on the information content necessary for individual users. The perceptions of the expert group addressed in this study are important because they are the designers and primary users of existing HEPS. Nevertheless, they have sometimes resisted the release of uncertainty information to the general public because of doubts about whether it can be successfully communicated in ways that would be readily understood to nonexperts. In this article, we explore the strengths and weaknesses of existing HEPS visualization methods and thereby formulate some wider recommendations about the best practice for HEPS visualization and communication. We suggest that specific training on probabilistic forecasting would foster use of probabilistic forecasts with a wider range of applications. The result of a case study exercise showed that there is no overarching agreement between experts on how to display probabilistic forecasts and what they consider the essential information that should accompany plots and diagrams. In this article, we propose a list of minimum properties that, if consistently displayed with probabilistic forecasts, would make the products more easily understandable. Copyright © 2012 John Wiley & Sons, Ltd.

Quantifying uncertainty in remotely sensed land cover maps

Remotely sensed land cover maps are increasingly used as inputs into environmental simulation models whose outputs inform decisions and policy-making. Risks associated with these decisions are dependent on model output uncertainty, which is in turn affected by the uncertainty of land cover inputs. This article presents a method of quantifying the uncertainty that results from potential mis-classification in remotely sensed land cover maps. In addition to quantifying uncertainty in the classification of individual pixels in the map, we also address the important case where land cover maps have been upscaled to a coarser grid to suit the users’ needs and are reported as proportions of land cover type. The approach is Bayesian and incorporates several layers of modelling but is straightforward to implement. First, we incorporate data in the confusion matrix derived from an independent field survey, and discuss the appropriate way to model such data. Second, we account for spatial correlation in the true land cover map, using the remotely sensed map as a prior. Third, spatial correlation in the mis-classification characteristics is induced by modelling their variance. The result is that we are able to simulate posterior means and variances for individual sites and the entire map using a simple Monte Carlo algorithm. The method is applied to the Land Cover Map 2000 for the region of England and Wales, a map used as an input into a current dynamic carbon flux model.

Government policy under price uncertainty: a source of volatility in illegal immigration

In this paper we provide an alternative explanation for why illegal immigration can exhibit substantial fluctuation. We develop a model economy in which migrants make decisions in the face of uncertain border enforcement and lump-sum transfers from the host country. The uncertainty is extrinsic in nature, a sunspot, and arises as a result of ambiguity regarding the commodity price of money. Migrants are restricted from participating in state-contingent insurance markets in the host country, whereas host country natives are not. Volatility in migration flows stems from two distinct sources: the tension between transfers inducing migration and enforcement discouraging it and secondly the existence of a sunspot. Finally, we examine the impact of a change in tax/transfer policies by the government on migration.

Simple uncertainty frameworks for selecting weighting schemes and interpreting multimodel ensemble climate change experiments

Future climate change projections are often derived from ensembles of simulations from multiple global circulation models using heuristic weighting schemes. This study provides a more rigorous justification for this by introducing a nested family of three simple analysis of variance frameworks. Statistical frameworks are essential in order to quantify the uncertainty associated with the estimate of the mean climate change response. The most general framework yields the “one model, one vote” weighting scheme often used in climate projection. However, a simpler additive framework is found to be preferable when the climate change response is not strongly model dependent. In such situations, the weighted multimodel mean may be interpreted as an estimate of the actual climate response, even in the presence of shared model biases. Statistical significance tests are derived to choose the most appropriate framework for specific multimodel ensemble data. The framework assumptions are explicit and can be checked using simple tests and graphical techniques. The frameworks can be used to test for evidence of nonzero climate response and to construct confidence intervals for the size of the response. The methodology is illustrated by application to North Atlantic storm track data from the Coupled Model Intercomparison Project phase 5 (CMIP5) multimodel ensemble. Despite large variations in the historical storm tracks, the cyclone frequency climate change response is not found to be model dependent over most of the region. This gives high confidence in the response estimates. Statistically significant decreases in cyclone frequency are found on the flanks of the North Atlantic storm track and in the Mediterranean basin.

Black carbon vertical profiles strongly affect its radiative forcing uncertainty

The impact of black carbon (BC) aerosols on the global radiation balance is not well constrained. Here twelve global aerosol models are used to show that at least 20% of the present uncertainty in modeled BC direct radiative forcing (RF) is due to diversity in the simulated vertical profile of BC mass. Results are from phases 1 and 2 of the global aerosol model intercomparison project (AeroCom). Additionally, a significant fraction of the variability is shown to come from high altitudes, as, globally, more than 40% of the total BC RF is exerted above 5 km. BC emission regions and areas with transported BC are found to have differing characteristics. These insights into the importance of the vertical profile of BC lead us to suggest that observational studies are needed to better characterize the global distribution of BC, including in the upper troposphere.

Understanding the WTA–WTP gap: Attitudes, feelings, uncertainty and personality

We present an experiment designed to study the psychological basis for the willingness to accept (WTA)–willingness to pay (WTP) gap. Specifically, we conduct a standard WTA–WTP economic experiment to replicate the gap and include in it five additional instruments to try to follow the psychological processes producing it. These instruments are designed to measure five psychological constructs we consider especially relevant: (1) attitudes, (2) feelings, (3) familiarity with the target good, (4) risk attitudes, and (5) personality. Our results provide important new insights into the psychological foundations of the WTA–WTP disparity, which can be used to organize some major previous results and cast serious doubts on the claim that the gap might be just a consequence of inappropriate experimental practice.

An environmental uncertainty-based diagnostic reference tool for evaluating the performance of supply chain value streams

This research has responded to the need for diagnostic reference tools explicitly linking the influence of environmental uncertainty and performance within the supply chain. Uncertainty is a key factor influencing performance and an important measure of the operating environment. We develop and demonstrate a novel reference methodology based on data envelopment analysis (DEA) for examining the performance of value streams within the supply chain with specific reference to the level of environmental uncertainty they face. In this paper, using real industrial data, 20 product supply value streams within the European automotive industry sector are evaluated. Two are found to be efficient. The peer reference groups for the underperforming value streams are identified and numerical improvement targets are derived. The paper demonstrates how DEA can be used to guide supply chain improvement efforts through role-model identification and target setting, in a way that recognises the multiple dimensions/outcomes of the supply chain process and the influence of its environmental conditions. We have facilitated the contextualisation of environmental uncertainty and its incorporation into a specific diagnostic reference tool.

Controls on inorganic nitrogen leaching from Finnish catchments assessed using a sensitivity and uncertainty analysis of the INCA-N model

The semi-distributed, dynamic INCA-N model was used to simulate the behaviour of dissolved inorganic nitrogen (DIN) in two Finnish research catchments. Parameter sensitivity and model structural uncertainty were analysed using generalized sensitivity analysis. The Mustajoki catchment is a forested upstream catchment, while the Savijoki catchment represents intensively cultivated lowlands. In general, there were more influential parameters in Savijoki than Mustajoki. Model results were sensitive to N-transformation rates, vegetation dynamics, and soil and river hydrology. Values of the sensitive parameters were based on long-term measurements covering both warm and cold years. The highest measured DIN concentrations fell between minimum and maximum values estimated during the uncertainty analysis. The lowest measured concentrations fell outside these bounds, suggesting that some retention processes may be missing from the current model structure. The lowest concentrations occurred mainly during low flow periods; so effects on total loads were small.

Consensus and uncertainty: using forecast probabilities of output declines

A number of studies have addressed the relationship between intra-personal uncertainty and inter-personal disagreement about the future values of economic variables such as output growth and inflation using the SPF. By making use of the SPF respondents' probability forecasts of declines in output, we are able to construct a quarterly series of output growth uncertainty to supplement the annual series that are often used in such analyses. We also consider the relationship between disagreement and uncertainty for probability forecasts of declines in output.

Forecasting US output growth with non-linear models in the presence of data uncertainty

We consider the impact of data revisions on the forecast performance of a SETAR regime-switching model of U.S. output growth. The impact of data uncertainty in real-time forecasting will affect a model's forecast performance via the effect on the model parameter estimates as well as via the forecast being conditioned on data measured with error. We find that benchmark revisions do affect the performance of the non-linear model of the growth rate, and that the performance relative to a linear comparator deteriorates in real-time compared to a pseudo out-of-sample forecasting exercise.

Direct soil moisture controls of future global soil carbon changes: An important source of uncertainty

The nature of the climate–carbon cycle feedback depends critically on the response of soil carbon to climate, including changes in moisture. However, soil moisture–carbon feedback responses have not been investigated thoroughly. Uncertainty in the response of soil carbon to soil moisture changes could arise from uncertainty in the relationship between soil moisture and heterotrophic respiration. We used twelve soil moisture–respiration functions (SMRFs) with a soil carbon model (RothC) and data from a coupled climate–carbon cycle general circulation model to investigate the impact of direct heterotrophic respiration dependence on soil moisture on the climate carbon cycle feedback. Global changes in soil moisture acted to oppose temperature‐driven decreases in soil carbon and hence tended to increase soil carbon storage. We found considerable uncertainty in soil carbon changes due to the response of soil respiration to soil moisture. The use of different SMRFs resulted in both large losses and small gains in future global soil carbon stocks, whether considering all climate forcings or only moisture changes. Regionally, the greatest range in soil carbon changes across SMRFs was found where the largest soil carbon changes occurred. Further research is needed to constrain the soil moisture–respiration relationship and thus reduce uncertainty in climate–carbon cycle feedbacks. There may also be considerable uncertainty in the regional responses of soil carbon to soil moisture changes since climate model predictions of regional soil moisture changes are less coherent than temperature changes.

Quantifying errors due to frequency changes and target location uncertainty for radar refractivity retrievals

Radar refractivity retrievals can capture near-surface humidity changes, but noisy phase changes of the ground clutter returns limit the accuracy for both klystron- and magnetron-based systems. Observations with a C-band (5.6 cm) magnetron weather radar indicate that the correction for phase changes introduced by local oscillator frequency changes leads to refractivity errors no larger than 0.25 N units: equivalent to a relative humidity change of only 0.25% at 20°C. Requested stable local oscillator (STALO) frequency changes were accurate to 0.002 ppm based on laboratory measurements. More serious are the random phase change errors introduced when targets are not at the range-gate center and there are changes in the transmitter frequency (ΔfTx) or the refractivity (ΔN). Observations at C band with a 2-μs pulse show an additional 66° of phase change noise for a ΔfTx of 190 kHz (34 ppm); this allows the effect due to ΔN to be predicted. Even at S band with klystron transmitters, significant phase change noise should occur when a large ΔN develops relative to the reference period [e.g., ~55° when ΔN = 60 for the Next Generation Weather Radar (NEXRAD) radars]. At shorter wavelengths (e.g., C and X band) and with magnetron transmitters in particular, refractivity retrievals relative to an earlier reference period are even more difficult, and operational retrievals may be restricted to changes over shorter (e.g., hourly) periods of time. Target location errors can be reduced by using a shorter pulse or identified by a new technique making alternate measurements at two closely spaced frequencies, which could even be achieved with a dual–pulse repetition frequency (PRF) operation of a magnetron transmitter.