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NASA NTRS1993

Confidence bounds on structural reliability

Different approaches for quantifying physical, statistical, and model uncertainties associated with the distribution parameters which are aimed at determining structural reliability are described. Confidence intervals on the distribution parameters of the input random variables are estimated using four algorithms to evaluate uncertainty of the response. Design intervals are evaluated using either Monte Carlo simulation or an iterative approach. A first order approach can be used to compute a first approximation of the design interval, but its accuracy is not satisfactory. The regression approach which combines the iterative approach with Monte Carlo simulation is capable of providing good results if the performance function can be accurately represented using regression analysis. It is concluded that the design interval-based approach seems to be quite general and takes into account distribution and model uncertainties.

Mehta, S. R.↗

DOE OSTI2025

Explainable multi-fidelity Bayesian neural network for distribution system state estimation

Distribution System State Estimation (DSSE) is frequently constrained by limited real-time measurements, the uncertainties introduced by distributed energy resources, and the presence of bad data. To address them, this paper proposes an enhanced Multi-Fidelity Bayesian Neural Network (MFBNN) DSSE approach. A low-fidelity layer based on a Deep Neural Network (DNN) is first pre-trained on pseudo-measurement data to learn fundamental state features. Subsequently, a high-fidelity Bayesian Neural Network (BNN) layer leverages limited but high-quality real-time measurements to refine these features, thereby achieving accurate DSSE. Additionally, the deep SHapley Additive exPlanation (SHAP) is developed to quantify the influence of measurement data on DSSE through dual perspectives of global feature importance and local nodal contributions, establishing a hierarchical explainability framework for machine learning-based DSSE. Comparative studies conducted on the IEEE 13-bus system and a real-world 2135-node system from Dominion Energy demonstrate that the proposed method excels in estimation accuracy, even under situations of high noise levels, bad data, and missing data. Further comparisons with Weighted Least Squares (WLS) and other machine learning-based DSSE approaches verify that the proposed framework offers higher accuracy, improved interpretability, and enhanced robustness.

Bad data↗

NASA NTRS2018

HZETRN Radiation Transport Validation Using Balloon-Based Experimental Data

The deterministic radiation transport code HZETRN (High charge (Z) and Energy TRaNsport) was developed by NASA to study the effects of cosmic radiation on astronauts and instrumentation shielded by various materials. This work presents an analysis of computed differential flux from HZETRN compared with measurement data from three balloon-based experiments over a range of atmospheric depths, particle types, and energies. Model uncertainties were quantified using an interval-based validation metric that takes into account measurement uncertainty both in the flux and the energy at which it was measured. Average uncertainty metrics were computed for the entire dataset as well as subsets of the measurements (by experiment, particle type, energy, etc.) to reveal any specific trends of systematic over- or under-prediction by HZETRN. The distribution of individual model uncertainties was also investigated to study the range and dispersion of errors beyond just single scalar and interval metrics. The differential fluxes from HZETRN were generally well-correlated with balloon-based measurements; the median relative model difference across the entire dataset was determined to be 30%. The distribution of model uncertainties, however, revealed that the range of errors was relatively broad, with approximately 30% of the uncertainties exceeding ± 40%. The distribution also indicated that HZETRN systematically under-predicts the measurement dataset as a whole, with approximately 80% of the relative uncertainties having negative values. Instances of systematic bias for subsets of the data were also observed, including a significant underestimation of alpha particles and protons for energies below 2.5 GeV/u. Muons were found to be systematically over-predicted at atmospheric depths deeper than 50 g/cm(sup 2) but under-predicted for shallower depths. Furthermore, a systematic under-prediction of alpha particles and protons was observed below the geomagnetic cutoff, suggesting that improvements to the light ion production cross sections in HZETRN should be investigated.

James E Warner↗

NASA NTRS2014

Past Changes in the Vertical Distribution of Ozone Part 1: Measurement Techniques, Uncertainties and Availability

Peak stratospheric chlorofluorocarbon (CFC) and other ozone depleting substance (ODS) concentrations were reached in the mid- to late 1990s. Detection and attribution of the expected recovery of the stratospheric ozone layer in an atmosphere with reduced ODSs as well as efforts to understand the evolution of stratospheric ozone in the presence of increasing greenhouse gases are key current research topics. These require a critical examination of the ozone changes with an accurate knowledge of the spatial (geographical and vertical) and temporal ozone response. For such an examination, it is vital that the quality of the measurements used be as high as possible and measurement uncertainties well quantified. In preparation for the 2014 United Nations Environment Programme (UNEP)/World Meteorological Organization (WMO) Scientific Assessment of Ozone Depletion, the SPARC/IO3C/IGACO-O3/NDACC (SI2N) Initiative was designed to study and document changes in the global ozone profile distribution. This requires assessing long-term ozone profile data sets in regards to measurement stability and uncertainty characteristics. The ultimate goal is to establish suitability for estimating long-term ozone trends to contribute to ozone recovery studies. Some of the data sets have been improved as part of this initiative with updated versions now available. This summary presents an overview of stratospheric ozone profile measurement data sets (ground and satellite based) available for ozone recovery studies. Here we document measurement techniques, spatial and temporal coverage, vertical resolution, native units and measurement uncertainties. In addition, the latest data versions are briefly described (including data version updates as well as detailing multiple retrievals when available for a given satellite instrument). Archive location information for each data set is also given.

stratospheric ozone layer measurement↗

NASA NTRSDate not supplied

Uncertainty Quantification of Inducer Natural Frequency using Conditional Assessment of Modeling and Modal Testing of Simpler Systems

The low pressure fuel pump inducer of the new Space Launch System RS25 core stage engine operates in a highly complex environment that substantially affects its modal characteristics. Some of the more important effects are fluid-added mass resulting from operation within a light liquid (Hydrogen), and the magnification of this effect due to tight tip clearance. Since higher order cavitation has been identified as a significant harmonic driver, knowledge of the natural frequency of potentially excitable modes is critical for safe operation, but this frequency cannot be measured during the severe operational environment. A comprehensive testing and analysis program has therefore been performed over the last four years to identify the nominal value and uncertainty of the frequency by modeling and testing four simpler structures which share some of the characteristics of the operational inducer. This testing was used to assess and adjust modeling techniques and excellent correlation was achieved. Identification of the uncertainty in the inducer frequency itself was still problematic, however. This difficulty led to an investigation of Bayesian uncertainty quantification techniques, and to the application of the relatively simple technique of Multi-Variate Normal conditional distributions to calculate the inducer natural frequency uncertainty. Assumptions on prior distributions of uncertainty of the fluid-added mass and tip clearance effect are initially applied to models of each of the simple structures and the inducer itself, and these uncertainties are propagated to generate natural frequencies using design of experiments. Simple response surfaces are then created from this data in order to calculate a Covariance Matrix relating all of these natural frequencies. Finally, the results from modal test of the simple structures are considered to be observations and used to calculate the conditional variance of the desired inducer frequencies. As this method is less rigorous than more complicated Bayesian methods reported in the literature, a conservative factor is applied to the result, but the resulting uncertainty is still significantly less than originally estimated and will greatly assist certification of the inducer for use in the engine.

Uncertainty Quantification↗

NASA NTRSDate not supplied

Uncertainty Quantification of Inducer Eigenvalues using Conditional Assessment of Models and Modal Test of Simpler Systems

Uncertainty Quantification↗

NASA NTRS1979

Impact melting early in lunar history

The total amount of impact melt produced during early lunar history is examined in light of theoretically and experimentally determined relations between crater diameter (D) and impact melt volume. The time dependence of the melt production is given by the time dependent impact rate as derived from cratering statistics for two different crater-size classes. Results show that small scale cratering (D less than or equal to 30 km) leads to melt volumes which fit selected observations specifying the amount of impact melt contained in the lunar regolith and in craters with diameters less than 10 km. Larger craters (D greater than 30 km) are capable of forming the abundant impact melt breccias found on the lunar surface. The group of large craters (D greater than 30 km) produces nearly 10 times as much impact melt as all the smaller craters, and thus, the large impacts dominate the modification of the lunar surface. A contradiction between the distribution of radiometric rock ages and a model of exponentially decreasing cratering rate going back to 4.5 b.y. is reflected in uncertainty in the distribution of impact melt as a function of time on the moon.

Lange, M. A.↗

NASA NTRS2019

Assessing Uncertainties of a Geophysical Approach to Estimate Surface Fine Particulate Matter Distributions from Satellite-Observed Aerosol Optical Depth

Health impact analyses are increasingly tapping the broad spatial coverage of satellite aerosol optical depth (AOD) products to estimate human exposure to fine particulate matter (PM2.5). We use a forward geophysical approach to derive ground-level PM2.5 distributions from satellite AOD at 1 km2(exp) resolution for 2011 over the northeastern US by applying relationships between surface PM2.5 and column AOD (calculated offline from speciated mass distributions) from a regional air quality model (CMAQ; 12×12 km2(exp) horizontal resolution). Seasonal average satellite-derived PM2.5 reveals more spatial detail and best captures observed surface PM2.5 levels during summer. At the daily scale, however, satellite-derived PM2.5 is not only subject to measurement uncertainties from satellite instruments, but more importantly to uncertainties in the relationship between surface PM2.5 and column AOD. Using 11 ground-based AOD measurements within 10 km of surface PM2.5 monitors, we show that uncertainties in modeled PM2.5∕AOD can explain more than 70 % of the spatial and temporal variance in the total uncertainty in daily satellite-derived PM2.5 evaluated at PM2.5 monitors. This finding implies that a successful geophysical approach to deriving daily PM2.5 from satellite AOD requires model skill at capturing day-to-day variations in PM2.5∕AOD relationships. Overall, we estimate that uncertainties in the modeled PM2.5∕AOD lead to an error of 11 µg m−3(exp) in daily satellite-derived PM2.5, and uncertainties in satellite AOD lead to an error of 8 µg m−3(exp). Using multi-platform ground, airborne, and radiosonde measurements, we show that uncertainties of modeled PM2.5∕AOD are mainly driven by model uncertainties in aerosol column mass and speciation, while model representation of relative humidity and aerosol vertical profile shape contributes some systematic biases. The parameterization of aerosol optical properties, which determines the mass extinction efficiency, also contributes to random uncertainty, with the size distribution being the largest source of uncertainty and hygroscopicity of inorganic salt the second largest. Future efforts to reduce uncertainty in geophysical approaches to derive surface PM2.5 from satellite AOD would thus benefit from improving model representation of aerosol vertical distribution and aerosol optical properties, to narrow uncertainty in satellite-derived PM2.5.

aerosol optical depth↗

NASA NTRS1993

Some effects of dust on photometry of high-z galaxies: Confounding the effects of evolution

Photometric observations of very distant galaxies--e.g., color vs. z or magnitude vs. z, have been used over the past decade or so in investigations into the evolution of the stellar component. Numerous studies have predicted significant color variations as a result of evolution, in addition to the shifting of different rest wavelengths into the band of observation. Although there is significant scatter, the data can be fit with relatively straightforward, plausible models for galaxian evolution. In very few cases are the effects of dust extinction included in the models. This is due in a large part to the uncertainty about the distribution and optical properties of the grains, and even whether or not they are present in significant numbers in some types of galaxies such as ellipticals. It is likely that the effects of dust on broadband observations are the greatest uncertainty in studies of very distant galaxies. We use a detailed Monte Carlo radiative transfer model within a spherical geometry for different star/dust distributions to examine the effects of dust on the broadband colors of galaxies as a function of redshift. The model fully accounts for absorption and angular redistribution in scattering. In this summary, we consider only the effects on color vs. redshift for three simple geometries each with the same total dust optical depth. Elsewhere at this conference, Capuano, Thronson, & Witt consider other effects of altering the relative dust/star distribution.

Thronson, H. A., Jr.↗

NASA NTRS2005

Managing Lunar and Mars Mission Radiation Risks: Cancer Risks, Uncertainties, and Shielding Effectiveness - Part 1

This document addresses calculations of probability distribution functions (PDFs) representing uncertainties in projecting fatal cancer risk from galactic cosmic rays (GCR) and solar particle events (SPEs). PDFs are used to test the effectiveness of potential radiation shielding approaches. Monte-Carlo techniques are used to propagate uncertainties in risk coefficients determined from epidemiology data, dose and dose-rate reduction factors, quality factors, and physics models of radiation environments. Competing mortality risks and functional correlations in radiation quality factor uncertainties are treated in the calculations. The cancer risk uncertainty is about four-fold for lunar and Mars mission risk projections. For short-stay lunar missins (<180 d), SPEs present the most significant risk, but one effectively mitigated by shielding. For long-duration (>180 d) lunar or Mars missions, GCR risks may exceed radiation risk limits. While shielding materials are marginally effective in reducing GCR cancer risks because of the penetrating nature of GCR and secondary radiation produced in tissue by relativisitc particles, polyethylene or carbon composite shielding cannot be shown to significantly reduce risk compared to aluminum shielding. Therefore, improving our knowledge of space radiobiology to narrow uncertainties that lead to wide PDFs is the best approach to ensure radiation protection goals are met for space exploration.

Cucinotta, Francis A.↗

DOE OSTI2024

Measurements of jet cross-section ratios in 13 TeV proton-proton collisions with ATLAS

Measurements of jet cross-section ratios between inclusive bins of jet multiplicity are performed in 140 fb -1 of proton-proton collisions with √s = 13 TeV center-of-mass energy, recorded with the ATLAS detector at CERN’s Large Hadron Collider. These ratios are constructed from double-differential cross-section measurements that are made in bins of jet multiplicity and other observables that are sensitive the energy scale and angular distribution of radiation due to the strong interaction in the final state. Additionally, the scalar sum of the two leading jets’ transverse momenta is measured triple differentially, in bins of the third jet’s transverse momentum and of jet multiplicity. These measurements are unfolded to account for acceptance and detector-related effects. The measured distributions are used to construct ratios of the inclusive jet-multiplicity bins, which have been shown to be sensitive to the strong coupling Formula Presented while being less sensitive than other observables to systematic uncertainties and parton distribution functions. The measured distributions are compared with state-of-the-art QCD calculations, including next-to-next-to-leading-order predictions for two- and three-jet events. These predictions are generally found to model the data well and perform best in bins with a modest requirement on the third jet’s transverse momentum. Significant differences between data and Monte Carlo predictions are observed in events with large rapidity gaps and invariant masses of the leading jet pair. Studies leading to reduced jet energy scale uncertainties significantly improve the precision of this work and are documented herein.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2026

Dynamic Transmission Line Switching Amid Wildfire-Prone Weather Under Decision-Dependent Uncertainty

During dry and windy seasons, environmental conditions significantly increase the risk of wildfires, exposing power grids to disruptions caused by transmission line failures. Wildfire propagation exacerbates grid vulnerability, potentially leading to prolonged power outages. To address this challenge, we propose a multistage optimization model that dynamically adjusts transmission grid topology in response to wildfire propagation, aiming to develop an optimal response policy. By accounting for decision-dependent uncertainty, where line survival probabilities depend on usage, we employ distributionally robust optimization to model uncertainty in line survival distributions. We adapt the stochastic nested decomposition algorithm and derive a deterministic upper bound for its finite convergence. To enhance computational efficiency, we exploit the Lagrangian dual problem structure for a faster generation of Lagrangian cuts. Using realistic data from the California transmission grid, we demonstrate the superior performance of dynamic response policies against two-stage alternatives through a comprehensive case study. In addition, after solving the multistage formulation, we construct easy-to-implement policies that significantly reduce computational burden while maintaining good performance in real-time deployment. History: Accepted by Russell Bent, Area Editor for Network Optimization: Algorithms and Applications. Funding: This work was supported by the U.S. Department of Energy, Office of Electricity [Grant DE-AC02-05CH11231]. The work of R. Jiang was supported in part by the U.S. National Science Foundation, Division of Electrical, Communications and Cyber Systems [Grant ECCS-1845980] and the U.S. Air Force Office of Scientific Research [Grant FA9550-23-1-0323]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2025.1210 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2025.1210 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Estrada-Garcia, Juan-Alberto↗

NASA NTRS1998

Cormack Research Project: Glasgow University

The aim of this project was to investigate and improve upon existing methods of analysing data from COMITEL on the Gamma Ray Observatory for neutrons emitted during solar flares. In particular, a strategy for placing confidence intervals on neutron energy distributions, due to uncertainties on the response matrix has been developed. We have also been able to demonstrate the superior performance of one of a range of possible statistical regularization strategies. A method of generating likely models of neutron energy distributions has also been developed as a tool to this end. The project involved solving an inverse problem with noise being added to the data in various ways. To achieve this pre-existing C code was used to run Fortran subroutines which performed statistical regularization on the data.

Skinner, Susan↗

DOE OSTI2024

Bayesian mixture model approach to quantifying the empirical nuclear saturation point

The equation of state (EOS) in the limit of infinite symmetric nuclear matter exhibits an equilibrium density, $n_0 \approx 0.16 \, \mathrm{fm}^{-3}$, at which the pressure vanishes and the energy per particle attains its minimum, $E_0 \approx -16 \, \mathrm{MeV}$. Although not directly measurable, the nuclear saturation point $(n_0,E_0)$ can be extrapolated by density functional theory (DFT), providing tight constraints for microscopic interactions derived from chiral effective field theory (EFT). However, when considering several DFT predictions for $(n_0,E_0)$ from Skyrme and Relativistic Mean Field (RMF) models together, a discrepancy between these model classes emerges at high confidence levels that each model prediction's uncertainty cannot explain. How can we leverage these DFT constraints to rigorously benchmark nuclear saturation properties of chiral interactions? To address this question, we present a Bayesian mixture model that combines multiple DFT predictions for $(n_0,E_0)$ using an efficient conjugate prior approach. The inferred posterior distribution for the saturation point's mean and covariance matrix follows a Normal-inverse-Wishart class, resulting in posterior predictives in the form of correlated, bivariate $t$-distributions. The DFT uncertainty reports are then used to mix these posteriors using an ordinary Monte Carlo approach. At the 95\% credibility level, we estimate $n_0 \approx 0.157 \pm 0.010 \, \mathrm{fm}^{-3}$ and $E_0 \approx -15.97 \pm 0.40 \, \mathrm{MeV}$ for the marginal (univariate) $t$-distributions. Combined with chiral EFT calculations of the pure neutron matter EOS, we obtain bivariate normal distributions for the nuclear symmetry energy and its slope parameter evaluated at $n_0$: $S_v \approx 32.0 \pm 1.1 \, \mathrm{MeV}$ and $L\approx 52.6\pm 8.1 \, \mathrm{MeV}$ (95\%), respectively. Furthermore, our Bayesian framework is publicly available, so practitioners can readily use and extend our results.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS1988

Advanced probabilistic methods for quantifying the effects of various uncertainties in structural response

The effects of actual variations, also called uncertainties, in geometry and material properties on the structural response of a space shuttle main engine turbopump blade are evaluated. A normal distribution was assumed to represent the uncertainties statistically. Uncertainties were assumed to be totally random, partially correlated, and fully correlated. The magnitude of these uncertainties were represented in terms of mean and variance. Blade responses, recorded in terms of displacements, natural frequencies, and maximum stress, was evaluated and plotted in the form of probabilistic distributions under combined uncertainties. These distributions provide an estimate of the range of magnitudes of the response and probability of occurrence of a given response. Most importantly, these distributions provide the information needed to estimate quantitatively the risk in a structural design.

Nagpal, Vinod K.↗

NASA NTRS1990

Stratosphere chemistry in a 2-D model with residual circulation

The objective of this research was to examine the effects of chemical perturbations on the stratosphere using models which can incorporate fully interactive radiative, chemical, and dynamical responses, in the context of a zonally averaged model. Model runs for the unperturbed, chlorine-perturbed and simultaneously chlorine-and CO2-perturbed cases were completed using the JPL-87 chemical kinetics data. The base case was analyzed and submitted for publication. The perturbed cases show substantial sensitivity of the predicted column ozone depletion to the perturbations affecting lower stratosphere temperature, but less to far dynamical perturbations. The column ozone distribution changed substantially when the kinetics data was changed. This implies a greater-than-expected uncertainty in predicted latitude distributions of ozone depletion, due to uncertainty about the accuracy and completeness of the chemical kinetics data set.

Guthrie, Paul D.↗

DOE OSTI2026

Conditional diffusion machine-learning framework for mapping valence electron distribution from convergent beam electron diffraction

Quantitative convergent beam electron diffraction (CBED) enables determination of aspherical valence electron distributions through refinement of low-order structure factors, which are highly sensitive to chemical bonding and charge density variations. However, conventional quantitative CBED (QCBED) requires solving a highly nonlinear inverse problem with many coupled parameters, and computationally intensive dynamical diffraction calculations, making it time-consuming and difficult to apply to complex systems. More broadly, reconstructing charge density and orbital electron distribution from diffraction data has long been a central challenge in both x-ray and electron crystallography. Here, in this study, we introduce an artificial-intelligence (AI)-based framework that replaces traditional refinement with a data-driven inverse solver. Using a large synthetic CBED dataset generated by Bloch-wave simulations, we train a conditional diffusion model to directly infer crystal structural parameters and multipole density formalism parameters, and hence valence electron distributions, from CBED patterns alone. By learning from forward simulations across realistic parameter space, the model effectively solves the inverse problem. Compared with direct regression approaches, the diffusion-based framework provides posterior parameter distributions for rigorous uncertainty quantification while preserving quantitative fidelity and reducing analysis time by orders of magnitude. By eliminating the need for external single-crystal x-ray diffraction data and complex nonlinear refinement, this approach enables practical, high-throughput, and in situ quantitative CBED, enabling real-time mapping of valence electron distributions and their correlation with functional responses in quantum and energy materials.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2024

Angular Distribution of Dimuons from Drell-Yan Production in p+Fe Interactions at 120 GeV Beam Energy

In the E906/SeaQuest Fermilab experiment, we report a measurement of the angular distributions by measuring the angular parameters $\lambda$, $\mu$, and $\nu$ of Drell-Yan dimuons produced using a 120 GeV proton beam incident on an iron target. The angular distribution in the naive Drell-Yan model does not show any $\cos2\phi$ dependency, where $\phi$ denotes the azimuthal angle of dimuons in the Collins-Soper frame. However, pion-induced Drell-Yan experiments, such as NA10 and E615, have observed a significant dependence on $\cos2\phi$. The Boer–Mulders function, a transverse momentum-dependent distribution function, represents the correlation between the transverse spin and the transverse momentum of the quark. A non-zero Boer-Mulders function or an improved higher-order Drell-Yan model considering QCD effects can produce a $\cos 2\phi$ modulation in the Drell-Yan angular distribution. To measure the angular distributions, we have used an event mixing method to construct the combin atorial background, which was then subtracted from the data to isolate the Drell-Yan signal. Following this, we corrected the detector, trigger, and reconstruction efficiencies using a doubly-iterative Bayesian Unfolding method. This iterative unfolding technique improves the response matrix based on the results of the previous unfolding step, ensuring robust convergence without exaggeration of uncertainties. The angular distributions of the dimuons were measured over the invariant mass range $5.0 < M_{\mu^+ \mu^-} < 8.0$ $GeV/c^2$, with dimuon transverse momentum $P_T < 2$ GeV/c and Feynman-x $-0.18 < x_F < 0.9$. The measured angular distributions are then compared with the QCD calculations for $p + \text{Fe}$ interactions, and proton-induced angular distribution measurements from other experiments. We have observed weak $\cos 2\phi$ modulations as a function of $P_T$. For $P_T > 1.0 \, \text{GeV}/c$, the predicted NNLO perturbative QCD value of $\nu$ is larger than what we have me asured at E906/SeaQuest. Moreover, we have not observed a strong dependence of $\nu$ on the kinematic variables, such as dimuon mass $M_{\mu^+ \mu^-}$ and Bjorken-$x$. The spin alignment of the virtual photon, $\lambda$, measured from the SeaQuest Drell-Yan $p+\text{Fe}$ data, is found to be strongly dependent on $P_T$, decreasing as $P_T$ increases. $\lambda$ also holds to the upper bound condition $\lambda < 1.0$ within the statistical uncertainty, showing a trend similar to that predicted by NNLO perturbative QCD. However, for $1.0 < P_T < 2.0 \, \text{GeV}/c$, the extracted $\lambda$ value from SeaQuest is smaller than that predicted by perturbative QCD at NNLO.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗