ESCE531 Measurements of Air Pollutantscei.ust.hk/files/public/esce531_measurement_of_air... ·...

ESCE531Measurements of Air Pollutants

Chapter/ Unit 1 - P.1

Goals

To know how to describe a statistical distribution of variables (e.g. pollutant concentrations) using the Gaussian (Normal) and Log normal distribution functions.

To know how to calculate the geometric mean and geometric standard deviation of a log normal distribution.

with permission © Prof C K CHAN (CENG)



Table of Content

• 1.1. Error Analysis of Pollutant Measurements• 1.2 Log-Normal Distribution




1.1 Error Analysis of Pollutant Measurements

• 1.1.1 Terminology Definitions• 1.1.2 Sources of Errors in Pollutants Analysis• 1.1.3 Confidence Limits

Average value can be misleading!

Question 1




1.1.1 Terminology Definitions(1)

• Accuracy

• Bias

value true value;measured

100)(

==

⋅−=

tm

t

tm

CCCCCA

t

m

CCK =




1.1.1 Terminology Definitions(2)• Blank - concentration level of a chemical species in a

clean sampling or transfer medium.

• Lower detectable limit [LDL] - smallest quantity which shows a positive response from the instrument.

• Lower quantifiable limit [LQL] - smallest quantifiable amount of a species detected by the instrument. LQL=max (LDL, blank level).

Question 2




1.1.1 Terminology Definitions(3)• Precision - The standard deviation of repeated

measurements

• Uncertainty: The combination of the uncorrected biases and the precision.

1

)(1

2

−

−=

∑=

N

CCs

N

ii

C

∑=

=N

iiC

NC

1

1where Ci = ith measurement of observable C

= arithmetic mean of N measurements




1.1.1 Terminology Definitions(4)

Figure 1: Absolute errors in nitrogen analyses by a micro Kjeldahl procedure. Each vertical line labeled (xi-xi) is the absolute deviation of the mean of the set from the true value . [Data taken from C. O. Willits and C. L. Ogg, J. Assoc. Off. Anal. Chem., 32, 561 (1949)].

(Skoog and West, Fig 3.1: Precision and accuracy of analysis)Question 3




1.1.2 Sources of Errors in Pollutants Analysis(1)

• Statistical sampling error• Interference

(SO2 and HNO3 on glass filters; solid NH4NO3evaporate to form NH3(g) and HNO3(g)

• Variability of blank levels (trace metals on filters)• Reproducibility of the measurements

Question 4

Question 5

Question 6

Although the intended aim was to sample particles, gases can be adsorbed on or adsorbed by filter medium and collected particles.




1.1.3 Confidence Limits(1)

(Presentation of data with statistical error analysis)

• The atmospheric loading of SO42- in particulates was

determined by Ion Chromatography. Aqueous extracts of particles collected by a Hi-Vol sampler were analyzed. Six measurements of SO4

2- from the same extract were made to compute the atmospheric SO4

2- concentrations. They are: 11.50,11.39, 11.42, 11.60, 11.55, 11.48 µg/m3. Report the mean SO4

2-

concentration with the 90% Confidence Interval.

Answer for Q7





• Gaussian Distribution (Normal Distribution)

πσ

σµ

2

22 2/)( −−

=xey

samples. ofnumber theis and variableofentry i theis

where,)(

=

th

1

2

Nxx

NN

x

i

N

ii

∞→−∑

=

µ

Where µ = arithmetic mean of infinite numbers of measurements,σ = standard deviation

=





Figure 2: Area under normal curve between specified intervalsThe above plot is a y vs. x plot of the gaussian distribution. The percentage of area under the curve between m+2s and m-2s is 95%. The limits are called the 95% confidence intervals.





• Confidence Intervals: The limits set about the measured within which µ appears within certain probability. (Skoog and West, Fig 3.3: Normal error curves)

• 95% Confidence Interval means that 95 times out of 100, the actual mean (µ ) will be within the confidence limit for µ

x

. where,=σ

µσ −=±

xzN

zx

Question 8





Figure 3 : Normal error curves. The curves are for the measurement of the same quantity by two methods. Method 1 is more reliable; thus σis smaller. Note the three types of abscissa. Abscissa ashows the measured quantity x with the maximum at μ. Abscissa b shows the deviation from the mean with the maximum at 0. Abscissa cshows z from Equation 3-2. Abscissa c reduces the two curves to a single one.

(Skoog and West, Fig 3-3: Normal error curves)





• In general, σ, the true standard deviation for infinite measurement is unknown. We then need to use the standard deviation of the limited measurements, s, instead. To account for the variability of s, we shall use the t parameter :





)1- = (

=

1

)( 1

2

NfreedomofDegreet

Nstx

N

xxs

sxt

N

ii

tables. from obtained be canwhere

for limit confidence The

where

⋅±

−

−=

−=

∑=

µ

µ





Table 1: Tabulated values of z for normal distributions are available in various references.

(Skoog and West, Table 3-5: Confidence level for various values for z)

Question 9





Table 2: Values of t for Various Levels of Probability

(Skoog and West Table 3-6 values of t for various levels of probability)




1.2 Log-Normal Distribution

• 1.2.1 The Log Normal Distribution Equation• 1.2.2 Cumulative Distribution Function, Geometric

Mean and Geometric Standard Deviations• 1.2.3 Log Normal distribution characteristics of

chemical species of RSP in HK• 1.2.4 Logarithmic Transformation




1.2.1 The Log Normal Distribution Equation(1)

• Aim: To describe the distribution characteristics of an observable, e.g. pollutant concentration, by a small number of parameters in a well characterized mathematical description.

• It is commonly found that air pollutants are log normally distributed.





Figure 3: Frequency distribution of concentrations of total suspended particulate matter.

(Stern,Vol. 3, Chap 12, Fig 7: Distribution of concentrations of total suspended particulate matter.)





• Recall Normal Distribution :

• Log Normal Distribution :The Log of the quantity is governed by a normal distribution. If we make the substitution:u→ lnx,

−−==

−−

2

22/)(

2)(exp

21

2

22

suu

ssey

suu

ππ





deviation standard Geometric 1

)/(lnexp

mean Geometric ln1exp where

ln2)/(ln

expln2

1ln2

)ln(lnexp

ln21

2/1

1

2

1

2

2

2

2

−=

=

−=

−−=

∑

∑

=

=

N

i

gig

N

iig

g

g

gg

g

g

Nxx

s

xN

x

sxx

Ssxx

sy

ππ




• We are interested in knowing the percentages of measured values below certain pollutants concentration levels.

• Example includes:– Percentage of days visibility in HK below 8 km?– Percentage of days the API exceeds 100?

1.2.2 Cumulative Distribution Function, Geometric Mean and Geometric Standard Deviations. (1)





mean) geometric as (same valueMEDIAN theis

. of valuepercentile50th ,21)(,At

2)( where

ln2

)/ln(21

21)()(

0

0

2

g

g

z

g

gx

x

xxFxx

dezerf

s

xxerfdxxyxF

==

=

+==

∫

∫

− ηπ

η

Let’s consider the cumulative distribution between x=0 and x=x:





Figure 4: Cumulative frequency distribution of concentration of suspended particulate matter. (Stern, Vol III, Chap 12, Fig 8.)

Axis y1 Axis y2

X Axis





Figure 5: Cumulative frequency distribution of ambient concentrations of particulate matter. (Stern, Vol III, Chapter 12, Fig 10)

Gaussian transformed y2 axis

Log

tran

sfor

med

from

x a

xis





•

Thus, sg is the ratio of the value below which 84.1 percents of the measurement lie to the median diameter

g

percentilethatg

gg

xx

s

erfxFsxx

.

At

184

.841.02

121

21)(,

=

=

+==





• Measurement value Vs. F(x)(shown in Figure 5)

1.59118188s

g==




1.2.3 Log normal distribution characteristic of chemical species of RSP in HK (1)

• Chemical species included in the 1992-1994 data set:Al, Ca, Mg, Pb, Na+, V, Cl-, NH4

+, NO3-, SO4

=, Br-, Mn, Fe, Ni, Zn, EC (elemental carbon), THC (total hydrocarbon), Cd, K+, Ba, Cu, As, Be




1.2.4 Logarithmic Transformation(1)

• Chemical species concentration and wind speed are first logarithmic transformed to construct data matrix

• Lognormality test





Standardized Log-transformed Species Concentrations

-4 -2 0 2 4

Prob

ability

0.0

0.1

0.2

0.3

0.4

0.5

AlTHC

NH4+

Standard Normal

Figure 6: Distribution of species conforming to lognormal distribution





Standardized Log-transformed Species Concentrations

-10 -5 0 5 10

Prob

abilit

y

0.0

0.2

0.4

0.6

0.8

1.0 Mn *VStandard Normal

Figure 7: Species not conforming to lognormal distribution




Homework(1)

A. The homework is on log normal distribution. A file containing the EPD data of the daily concentrations of calcium, sulfate in RSP (1994-1995?) and SO2 (1994?) measured in Mongkok by the HKEPD is posted on the web. Determine:

1) If the distribution of daily concentration of Ca, sulfate and SO2 are log normal?

2) If these species are log normally distributed, what are the xg and sg?




Homework(2)

B. If a specific source of particulate pollutants contains two chemical species that cannot be found in other sources and if the distribution of one chemical species is lognormal, what can we say about the distribution properties of the other species?




Question 1

Besides average, what other information is needed todescribe a distribution of variable, e.g. student’s finalexamination score in a course?




Answer

the “spread” of the distribution.

Back




Question 2If you are interested in determining the amount of Na in particulate matters collected by a filter, what are the potential sources of high blank level?

Hint: In the analysis of Na, the particulate laden filter is usually extracted with pure water. Ion chromatographyIs used to analyze the Na content of the aqueousextract.




Answer

Sweat, water for extraction

Back




Question 3

Are these analyses accurate or precise?




Answer

A (accurate and precise), B (accurate but not precise), C (precise but not accurate), and D (not accurate nor precise).

Back




Question 4

If you are interested in sampling the air quality of a classroom, how much air would you sample ideally?




Answer

• ??

Back




Question 5

Sampling requires a vacuum pump. Some collectedparticles are therefore under a slight vacuum. Whatwould happen to some volatile species such as NH4Clor organic carbon?




Answer

Evaporation loss of some volatile species.

Back




Question 6

What is the other term we have introduced to stand forreproducibility here?




Answer

Confident limit

Back




Answer (7a)

Total number of measures, N = 6

Arithmetic mean of N measures,

=

= 11.49ug/m3




Answer (7b)

Standard deviation, s =

=

= 0.078486

( ) ( )16

49.1148.1149.1155.11)49.1160.11()49.1142.11()49.1139.11()49.1150.11( 222222

−−+−+−+−+−+−




Answer (7c)

Degree of freedom = 6-1 = 5

From table 5.6 of lecture note page 15, at 90%confidence interval with the degree of freedom = 5,t = 2.02

the confidence limits =

= ug/m3

Back




Question 8

What is the difference between x and μ?




Answer

x is the mean of finite numbers of measurements

μ is the arithmetic mean of infinite numbers of measurements

Back




Question 9• What is the difference between σ and s and between

z and t?• When will σ = s and z = t?




Answer

1. σ and z are for infinite measurements

s and t are for finite measurements

2. N tends to infinity

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