Atmospheric Factors Influencing Daily Sulfate Concentrations in Chicago Air

Abstract

Confidence interval construction for central tendency is a problem of practical consequence for those who must analyze air contaminant data. Determination of compliance with relevant ambient air quality criteria and assessment of associated health risks depend upon quantifying the uncertainty of estimated mean pollutant concentrations. The bootstrap is a resampling technique that has been steadily gaining popularity and acceptance during the past several years. A potentially powerful application of the bootstrap is the construction of confidence intervals for any parameter of any underlying distribution. Properties of bootstrap confidence intervals were determined for samples generated from lognormal, gamma, and Weibull distributions. Bootstrap t intervals, while having smaller coverage errors than Student's t or other bootstrap methods, under-cover for small samples from skewed distributions. Therefore, we caution against using the bootstrap to construct confidence intervals for the mean without first considering the effects of sample size and skew. When sample sizes are small, one might consider using the median as an estimate of central tendency. Confidence intervals for the median are easy to construct and do not under-cover. Data collected by the Northeast States for Coordinated Air Use Management (NESCAUM) are used to illustrate application of the methods discussed.     

Atmospheric lead concentration distribution in Northern Taiwan

Atmospheric lead concentrations were measured randomly, approximately once per week, at five traffic sites in northern Taiwan from September 1994 to May 1995. Three types of theoretical distributions, lognormal, Weibull and gamma were selected to fit the frequency distribution of the measured lead concentration. Four goodness-of-fit criteria were used to judge which theoretical distribution is the most appropriate to represent the frequency distributions of atmospheric lead.The results show that atmospheric lead concentrations in total suspended particulates fit the lognormal distribution reasonably well in northern Taiwan. The intervals of fitted theoretical cumulative frequency distributions (CFDs) can successfully contain the measured data when the population mean is estimated with a 95% confidence interval. In addition, atmospheric lead concentration exceeding a critical concentration is also predicted from the fitted theoretical CFDs.     

Estimating the frequency distributions of particulate matter and their metal elements in a temple

Air quality inside Asian temples is typically poor because of the burning of incense. This study measured and analyzed concentrations of fine (PM2.5) and coarse (PM2.5-10) particulate matter and their metal elements inside a temple in central Taiwan. Experimental results showed that the concentrations of metals Cd, Ni, Pb, and Cr inside the temple were higher than those at rural, suburban, urban, and industrial areas in other studies. Three theoretical parent distributions (lognormal, Weibull, and gamma) were used to fit the measured data. The lognormal distribution was the most appropriate distribution for representing frequency distributions of PM10, PM2.5, and their metal elements. Furthermore, the central limit theorem, H-statistic-based scheme, and parametric and nonparametric bootstrap methods were used to estimate confidence intervals for mean pollutant concentrations. The estimated upper confidence limits (UCLs) of means between different methods were very consistent, because the sample coefficient of variation (CV) was < 1. When the sample CV was > 1, the UCL based on H-statistical method tended to overestimate the UCLs when compared with other methods. Confidence intervals for pollutant concentrations at different percentiles were evaluated using parametric and nonparametric bootstrap methods. The probabilities of pollutants exceeding a critical concentration were also calculated.     

Estimating the Frequency Distributions of Particulate Matter and Their Metal Elements in a Temple

Abstract

Air quality inside Asian temples is typically poor because of the burning of incense. This study measured and analyzed concentrations of fine (PM_2.5) and coarse (PM_2.5–10) particulate matter and their metal elements inside a temple in central Taiwan. Experimental results showed that the concentrations of metals Cd, Ni, Pb, and Cr inside the temple were higher than those at rural, suburban, urban, and industrial areas in other studies. Three theoretical parent distributions (lognormal, Weibull, and gamma) were used to fit the measured data. The lognormal distribution was the most appropriate distribution for representing frequency distributions of PM₁₀, PM_2.5, and their metal elements.

Furthermore, the central limit theorem, H-statistic-based scheme, and parametric and nonparametric bootstrap methods were used to estimate confidence intervals for mean pollutant concentrations. The estimated upper confidence limits (UCLs) of means between different methods were very consistent, because the sample coefficient of variation (CV) was <1. When the sample CV was >1, the UCL based on H-statistical method tended to overestimate the UCLs when compared with other methods. Confidence intervals for pollutant concentrations at different percentiles were evaluated using parametric and nonparametric bootstrap methods. The probabilities of pollutants exceeding a critical concentration were also calculated.     

Quantification of variability and uncertainty in lawn and garden equipment NOx and total hydrocarbon emission factors

Variability refers to real differences in emissions among multiple emission sources at any given time or over time for any individual emission source. Variability in emissions can be attributed to variation in fuel or feedstock composition, ambient temperature, design, maintenance, or operation. Uncertainty refers to lack of knowledge regarding the true value of emissions. Sources of uncertainty include small sample sizes, bias or imprecision in measurements, nonrepresentativeness, or lack of data. Quantitative methods for characterizing both variability and uncertainty are demonstrated and applied to case studies of emission factors for lawn and garden (L&G) equipment engines. Variability was quantified using empirical and parametric distributions. Bootstrap simulation was used to characterize confidence intervals for the fitted distributions. The 95% confidence intervals for the mean grams per brake horsepower/hour (g/hp-hr) emission factors for two-stroke engine total hydrocarbon (THC) and NOx emissions were from -30 to +41% and from -45 to +75%, respectively. The confidence intervals for four-stroke engines were from -33 to +46% for THCs and from -27 to +35% for NOx. These quantitative measures of uncertainty convey information regarding the quality of the emission factors and serve as a basis for calculation of uncertainty in emission inventories (EIs).     

Quantification of Variability and Uncertainty in Lawn and Garden Equipment NOx and Total Hydrocarbon Emission Factors

Abstract

Variability refers to real differences in emissions among multiple emission sources at any given time or over time for any individual emission source. Variability in emissions can be attributed to variation in fuel or feedstock composition, ambient temperature, design, maintenance, or operation. Uncertainty refers to lack of knowledge regarding the true value of emissions. Sources of uncertainty include small sample sizes, bias or imprecision in measurements, nonrepresentativeness, or lack of data. Quantitative methods for characterizing both variability and uncertainty are demonstrated and applied to case studies of emission factors for lawn and garden (L&G) equipment engines. Variability was quantified using empirical and parametric distributions. Bootstrap simulation was used to characterize confidence intervals for the fitted distributions. The 95% confidence intervals for the mean grams per brake horsepower/hour (g/hp-hr) emission factors for two-stroke engine total hydrocarbon (THC) and NO_x emissions were from -30 to +41% and from -45 to +75%, respectively. The confidence intervals for four-stroke engines were from -33 to +46% for THCs and from -27 to +35% for NO_x. These quantitative measures of uncertainty convey information regarding the quality of the emission factors and serve as a basis for calculation of uncertainty in emission inventories (Els).     

Comparisons of statistical characteristic of air pollutants in Taiwan by frequency distribution

The lognormal, Weibull, and type V Pearson distributions were selected to fit the concentration frequency distributions of particulate matter with an aerodynamic diameter of < or = 10 microm (PM10) and SO2 in the Taiwan area. Air quality data from three stations, Hsin-Chu, Shalu, and Gain-Jin, were fitted with three distributions and compared with the measured data. The parameters of unimodal and bimodal fitted distributions were obtained by the methods of maximum likelihood and nonlinear least squares, respectively. Moreover, the root mean square error (RMSE), index of agreement (d), and Kolmogorov-Smirnov (K-S) test were used as criteria to judge the goodness-of-fit of these three distributions. These results show that the frequency distributions of PM10 concentration at the Hsin-Chu and Shalu stations are unimodal, but the distribution at Gain-Jin is bimodal. The distribution type of PM10 concentration varied greatly in different areas and could be influenced by local meteorological conditions. For SO2 concentration distribution, the distributions were all unimodal. The results also show that the lognormal distribution is the more appropriate to represent the PM10 distribution, while the Weibull and lognormal distributions are more suitable to represent the SO2 distribution. Moreover, the days exceeding the air quality standard (AQS) (PM10 > 125 microg/ m3) for the Hsin-Chu, Shalu, and Gain-Jin stations in the coming year are successfully predicted by the theoretic distributions.     

A physical explanation of the lognormality of pollutant concentrations

Investigators in different environmental fields have reported that the concentrations of various measured substances have frequency distributions that are lognormal, or nearly so. That is, when the logarithms of the observed concentrations are plotted as a frequency distribution, the resulting distribution is approximately normal, or Gaussian, over much of the observed range. Examples include radionuclides in soil, pollutants in ambient air, indoor air quality, trace metals in streams, metals in biological tissue, calcium in human remains. The ubiquity of the lognormal distribution in environmental processes is surprising and has not been adequately explained, since common processes in nature (for example, computation of the mean and the analysis of error) usually give rise to distributions that are normal rather than lognormal. This paper takes the first step toward explaining why lognormal distributions can arise naturally from certain physical processes that are analogous to those found in the environment. In this paper, these processes are treated mathematically, and the results are illustrated in a laboratory beaker experiment that is simulated on the computer.     

A Physical Explanation of the Lognormality of Pollutant Concentrations

Investigators In different environmental fields have reported that the concentrations of various measured substances have frequency distributions that are lognormal, or nearly so. That is, when the logarithms of the observed concentrations are plotted as a frequency distribution, the resulting distribution is approximately normal, or Gaussian, over much of the observed range. Examples include radionuclides in soil, pollutants in ambient air, Indoor air quality, trace metals In streams, metals in biological tissue, calcium In human remains. The ubiquity of the lognormal distribution in environmental processes is surprising and has not been adequately explained, since common processes in nature (for example, computation of the mean and the analysis of error) usually give rise to distributions that are normal rather than lognormal. This paper takes the first step toward explaining why lognormal distributions can arise naturally from certain physical processes that are analogous to those found in the environment. In this paper, these processes are treated mathematically, and the results are illustrated in a laboratory beaker experiment that Is simulated on the computer.     

Characterization of distribution of pesticide residues in crop units

The characteristic features of distribution of pesticide residues in crop units and single sample increments were studied based on more than 19,000 residue concentrations measured in root vegetables, leafy vegetables, small-, medium- and large-size fruits representing 20 different crops and 46 pesticides. Log-normal, gamma and Weibull distributions were found to provide the best fit for the relative frequency distributions of individual residue data sets. The overall best fit was provided by lognormal distribution. The relative standard deviation of residues (CV) in various crops ranged from 15–170%. The 100–120 residue values being in one data set was too small to identify potential effects of various factors such as the chemical and physical properties of pesticides and the nature of crops. Therefore, the average of CV values, obtained from individual data sets, were calculated and considered to be the best estimate for the likely variability of unit crop residues for treated field (CV = 0.8) and market samples (CV = 1.1), respectively. The larger variation of residues in market samples was attributed to the potential mixing of lots and varying proportion of non-detects. The expectable average variability of residues in composited samples can be calculated from the typical values taking into account the sample size.     

Confidence limits for air quality model evaluations,as estimated by bootstrap and jackknife resampling methods

Air quality models are used to make decisions regarding the construction of industrial plants, the types of fuel that will be burnt and the types of pollution control devices that will be used. It is important to know the uncertainties that are associated with these model predictions. Standard analytical methods found in elementary statistics textbooks for estimating uncertainties are generally not applicable since the distributions of performance measures related to air quality concentrations are not easily transformed to a Gaussian shape. This paper suggests several possible resampling procedures that can be used to calculate uncertainties or confidence limits on air quality model performance. In these resampling methods, many new data sets are drawn from the original data set using an empirical set of rules. A few alternate forms of the socalled bootstrap and jackknife resampling procedures are tested using a concocted data set with a Gaussian parent distributions, with the result that the jackknife is the most efficient procedure to apply, although its confidence bounds are slightly overestimated. The resampling procedures are then applied to predictions by seven air quality models for the Carpinteria coastal dispersion experiment. Confidence intervals on the fractional mean bias and the normalized mean square error are calculated for each model and for differences between models. It is concluded that these uncertainties are sometimes so large for data sets consisting of about 20 elements that it cannot be stated with 95% confidence that the performance measure for the ‘best’ model is significantly different from that for another model.     

Simultaneous Total Oxidant and Chemiluminescent Ozone Measurements in Ambient Air

The purpose of this project was to investigate the relationship of ambient air quality measurements between two analytical methods, referred to as the total oxidant method and the chemiluminescent method. These two well documented analytical methods were run simultaneously, side by side, at a site on the Houston ship channel. They were calibrated daily. The hourly averages were analyzed by regression techniques and the confidence intervals were calculated for the regression lines. Confidence intervals for point estimates were also calculated. These methods were used with all data sets with values greater than 10 parts per billion and again with values greater than 30 parts per billion. A regression line was also calculated for a second set of data for the preceding year. These data were generated before a chromium triox-ide scrubber was installed to eliminate possible chemical interferences with the Kl method.

The results show that in general the chemiluminescent ozone method tends to produce values as much as two times higher than the simultaneous total oxidant values. In one set of data collected an 80 ppb chemiluminescent ozone value predicted a value of 43.9 ppb total oxidant with a 95% confidence interval of 7.7 to 80.4 ppb. In the second set of data an 80 ppb chemiluminescent ozone value predicted a value of 78 ppb total oxidant with a 95% confidence interval of 0.4 to 156 ppb. Other statistical analyses confirmed that either measurement was a very poor predictor of the other.     

The Precision Associated with the Sampling Frequencies of Total Particulate at Indianapolis,Indiana

The frequency distribution of total suspended particulate matter for Indianapolis, Ind., was examined in order to determine the precision associated with any given sampling scheme. By assuming a basic log_e-normal distribution, a theoretical set of confidence intervals about the geometric mean was derived for random sampling. Verification of the log_e-normal distribution was made for particulate matter in Indianapolis. Application of the derived confidence intervals revealed that for a 30-day period 20 samples must be taken to ensure that the 90% confidence interval will be within 10% of the geometric mean. Analysis of the records for 19 sampling locations within Indianapolis revealed that only 2 sites possessed sufficient data to allow monthly climatological evaluation over the period 1968-1970.     

On the Evaluation of Predictions from a Gaussian Plume Model

The performance of a CRSTER equivalent Gaussian plume model (CEQM) is examined using data from the EPRI Plume Model Validation study at the Klncaid, Illinois site. Four-way comparisons are made on the ordered statistics or the cumulative frequency distribution (CFD) of maximum hourly observed and predicted concentrations. Using the uniform random distribution and the lognormal random distribution as simple predictive schemes without any physical context, it Is found that the CEQM predicts a concentration CFD which matches the observed CFD significantly closer than the CFD predicted by the uniform random distribution. The two-parameter lognormal random distribution predicts the concentration CFD better than the CEQM over all concentration ranges; however, the CEQM fits the upper range of the concentration distribution better than the lognormal random distribution,, despite the fact that the predictions are generated using dispersion conditions entirely different from those of the observations. The nature of this ergodicity of distribution is probed by exercising CEQM using randomized input based on the observed frequency distributions of the Input parameters instead of feeding the hour-by-hour model input matched by time into CEQM as is customarily done. The exercise of the model by uncoupling the time linkage in model Input has no systematic effect on the predicted cumulative frequency distribution of concentrations. Only at the highest concentration range (99.5% or higher) do the two sets of predictions begin to diverge.     

Statistics of extreme values of air quality—a simulation study

Some of the existing National Ambient Air Quality Standards require the use of an extreme observed concentration in a year to determine compliance. Since observed extreme values tend to be not reliable, different statistical approaches for determining the extreme values have been used or suggested. However, none of these approaches properly take into account the effects of an underlying trend and the serial correlation of the air quality time series. By means of a time series simulation, these effects can be considered concurrently in estimating the extreme values.This paper reports the results of such a simulation for determining the statistics of the seven highest values (rank m = 1–7, m = 1 representing the highest value) using actual air quality data that contain both trends and autocorrelations. The result of this simulation shows that for a high-pollution season of 122 days, the commonly used asymptotic distributions overestimate the maximum (m = 1) values and underestimate their uncertainties. As one moves from m = 1 to m = 7, the over- and underestimations by the asymptotic distributions worsen (compared to the simulation result). These findings in logarithmic space are further enhanced when they are converted back to concentration space. The simulation using the oxidant data for Azusa, California further shows that the uncertainties associated with the estimates Of the extreme values are typically 20% of the values for m = 1 and 10% of the values for m = 7. Compared to the observed data, which is a single series for each year, the result based on the popular lognormal distribution consistently overpredicts the extreme values, by about 40% for the maximum values and about 20% for the seventh highest values.Our results illustrate the difficulty of estimating the extreme values of air quality time series with accuracy and confidence. However, the accuracy and confidence of the estimates improve as the rank moves away from the extreme. This result calls for the need for using a less extreme value in setting a sensible air quality standard. Of course, such a standard can be set without sacrificing its stringency.     

Frequency distribution of dust concentration in Barbados as a function of averaging time

Two sets of daily-averaged dust concentration data from Barbados, West Indies, have been analyzed to determine the affect of averaging time, ranging from 1 to 7 days, on the dust concentration frequency distribution. On each of the time scales examined, the frequency distribution is characterized as a bimodal lognormal distribution. The major effects of increasing the averaging are a major reduction in the percentage of the samples represented by the lower of the two modes and a significant increase in the geometric mean concentration of that mode. Consequently, predictions of the distributions on a shorter time scale are likely to substantially underestimate the frequency of low concentration samples.     

Percentile estimation using variable censored data

Much progress has been made in recent years to address the estimation of summary statistics, using data that are subject to censoring of results that fall below the limit of detection (LOD) for the measuring instrument. Truncated data methods (e.g., Tobit regression) and multiple-imputation are two approaches for analyzing data results that are below the LOD. To apply these methods requires an assumption about the underlying distribution of the data. Because the log-normal distribution has been shown to fit many data sets obtained from environmental measurements, the common practice is to assume that measurements of environmental factors can be described by log-normal distributions. This article describes methods for obtaining estimates of percentiles and their associated confidence intervals when the results are log-normal and a fraction of the results are below the LOD. We present limited simulations to demonstrate the bias of the proposed estimates and the coverage probability of their associated confidence intervals. Estimation methods are used to generate summary statistics for 2,3,7,8-tetrachloro dibenzo-p-dioxin (2,3,7,8-TCDD) using data from a 2001 background exposure study in which PCDDs/PCDFs/cPCBs in human blood serum were measured in a Louisiana population. Because the congener measurements used in this study were subject to variable LODs, we also present simulation results to demonstrate the effect of variable LODs on the multiple-imputation process.     

The Precision Associated with the Sampling Frequency of Log-Normally Distributed Air Pollutant Measurements

The frequency of air monitoring necessary to characterize an air pollutant for a given time period and area is an important problem. This paper deals with the precision of measuring an air pollutant concentration. Past research has shown that the distribution of many air pollutants can be described as log-normal. Using this result equations have been developed that predict the precision of the sample mean of the air pollutant as a function of: the frequency of sampling, the standard deviation of the logarithms of the air pollution measurements, and the level of confidence. An illustration is given to demonstrate their use. The equations are used to compare sampling plans. Tables are presented showing the precision associated with five sampling plans, for three geometric standard deviations, for three levels of confidence, and five periods of time over which the sampling plan is employed.

In an Appendix a mathematical development is presented showing the theoretical derivation of the equations. With these equations the precision of a sampling plan can be determined for any level of confidence or period of time. All that is needed is an estimate of the geometric standard deviation for the air pollution measurements.

Finally, the theoretical model is applied to air monitoring data that were collected at Roselawn School in Cincinnati, Ohio, between January 3, 1968, and April 1, 1968. The 90-day period was divided into three 30-day periods. All possible samples of size three were taken from each of the 30-day periods and their means and confidence intervals were calculated. The number of times the confidence intervals contained the true means was determined. The actual number of samples accepted as having contained the true mean, for the 80, 90, 95, and 9 9% level of confidence compared favorably with the theoretical. It is concluded that the model adequately described the behavior of air pollutants.     

Size distribution of large aerosol particles during AGASP-II: Absence of st. augustine eruptive particles in the Alaskan Arctic

Number distribution data for 0.1–45 μm diameter aerosol were obtained using optical counting and sizing probes flown over the Alaskan Arctic during the second Arctic Gas and Aerosol Sampling Program (AGASP-II), flights 201–203. Due to noise present in the lowest size channels of the optical probes, estimates of the H₂SO₄ component of Arctic haze were not attempted. Large particle (> 0.5 μm diameter) results are presented here. Large particle number and volume concentration were determined along with estimated mass, which was generally </ 0.1μg m⁻³. Lognormal fitting to > 0.3 μg m⁻³ mass loading sizedistributed aerosol data produced a means for comparing volume geometric median diameters (VGMD) for these higher-mass time intervals. These VGMDs showed that solid crustal particles previously observed during AGASP-II had VGMDs in the 1.2–1.6 μm range and that the shape of these fitted lognormal distributions was essentially constant. This result suggests very-long-range transport from a distant crustal source and, in conjunction with aerosol physical and chemical characterization data, argues against the presence of the Mt. Augustine eruptive particles during AGASP-II Alaskan Arctic sampling.     

Modeling distributions of air pollutant concentrations—II. Estimation of one and two parameter statistical distributions

In Part I of this series Taylor, Jakeman and Simpson (1986, Atmospheric Environment, 20, 1781–1789) examined the problem of identifying the appropriate distributional form for air pollution concentration data. In this paper we examine the parameter estimation problem. Monte Carlo simulation is used to compare methods for fitting statistical distributions to such data where the distributional form is known. Three methods are investigated for estimating the parameters of the lognormal distribution, two methods for the exponential distribution, three methods for the γ-distribution and four methods for the Weibull distribution. For all distributions and for each method we examine the accuracy with which the upper percentiles of the distribution are evaluated as it is these percentiles which are referred to by air quality standards. For each distribution a simple empirical model, which yields approximations to the relative root mean square error of the percentile estimates against sample size and parameter values, is developed and demonstrated. Thus for each distributional model an estimate of the relative error associated with evaluating high pollutant levels may be readily determined.