UNIVERSITY OF MINNESOTA

This is to certify that I have examined this copy of a master’s thesis by

XIAOLIANG WANG

And have found that it is complete and satisfactory in all respects,

and that any and all revisions required by the final

examining committee have been made.

__________________________________________________________

Name of Faculty Advisor

__________________________________________________________

Signature of Faculty Advisor

__________________________________________________________

Date

GRADUATE SCHOOL

OPTICAL PARTICLE COUNTER (OPC) MEASUREMENTS

AND

PULSE HEIGHT ANALYSIS (PHA) DATA INVERSION

A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL

OF THE UNIVERSITY OF MINNESOTA BY

XIAOLIANG WANG

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

JUNE 2002

i

Acknowledgements

I am deeply indebted to my advisor Prof. Peter H. McMurry, whose stimulating

suggestions and encouragement helped me throughout my research and preparation of

this thesis. It is hard to believe that he looked at this thesis closely for more than three

times, offering suggestions for improvement from English style, grammars to contents,

although he is extremely busy. He is so knowledgeable in aerosol science and technology.

I am very fortunate to obtain advice from such a versatile scientist and excellent mentor

during the course of my graduate studies.

I would like to sincerely thank Dr. Hiromu Sakurai, who has contributed a lot of

helpful ideas, knowledge and time to this project.

Thanks to Kueng Shan Woo, Jongsup Park, Kihong Park, Qian Shi, and all other

PTL faculty and students who have helped me in either experiments or data analysis.

Finally, this thesis is dedicated to my family, from where I obtain ceaselessly and

tirelessly encouragement and support.

Funding for this work was provided by the United States Environmental

Protection Agency as a part of the Supersite Program through a subcontract from

Washington University in St. Louis.

ii

Abstract

Two types of Optical particle counters (OPCs) were used in this study: PMS

Lasair 1002 and Climet Spectro.3. The table data provided by the instruments usually

report “optical equivalent” sizes, and the resolutions are low. To obtain particle mobility

sizes directly and to obtain higher size resolution, a multichannel analyzer (MCA) was

connected to the OPC analog output to record OPC’s voltage responses to particles (pulse

height distribution).

Kernel functions of monodisperse particles were needed to convert pulse height

distributions to size distributions. According to my calibrations of the Lasair-MCA

system with PSL, DOS and diesel exhaust particles, kernel functions were found to fit

lognormal distributions very well. Therefore, two parameters: peak voltage and geometric

standard deviation were needed to define a kernel function.

• For spherical particles, remarkable agreements between the calibrated peak responses

of PSL and DOS and Mie theoretical responses were achieved. Hence the theoretical

peak voltages of spherical particles with arbitrary refractive index could be calculated.

PSL were found to be more nearly monodisperse than aerosols classified by the DMA,

and the pulse height distributions of DMA classified aerosols could be predicted

using the width of PSL responses and DMA transfer functions. Therefore, the

geometric standard deviations of PSL kernel functions were used for kernel functions

of other particles.

• For non-spherical particles, the shape played an important role in both OPC-MCA

peak voltages and standard deviations. The kernel functions of these particles were

determined by calibration.

Twomey algorithm and its modified version (STWOM) were adapted to invert the

pulse height distribution. Numerical experiments demonstrated that this algorithm could

invert pulse height distributions to size distributions accurately and quickly. The

inversion helped to realize the strength of the OPC-PHA technique: much higher

resolution and more accurate sizing than the table data. However, the uncertainties of the

iii

inverted distribution both ends of the size range were large due to the large uncertainties

of counting efficiencies of these sizes.

A Lasair and a Climet were used in the atmospheric aerosol measurement in

Metropolitan St. Louis (IL-MO). Particles of 450nm were found to be externally mixed

and have quite different optical properties. In this work, refractive indices of brighter

particles obtained from hourly calibration were used for data analysis. OPC distributions

obtained from different methods were compared to SMPS distributions. The original

OPC table data were found to match SMPS data best, but the table data corrected by

refractive index and inverted distributions were systematically higher than SMPS

concentrations. A Lasair was also used in the diesel exhaust mass distribution

measurement. A discrepancy of ±200% was found between the inverted OPC and SMPS

size and mass distributions. The reason of the discrepancies between the SMPS and the

Lasair is still under investigation.

iv

Table of Contents Acknowledgements.............................................................................................................. i Abstract ............................................................................................................................... ii Table of Contents............................................................................................................... iv Chapter 1: Introduction ....................................................................................................... 1

1.1 Principles of Optical Particle Counter (OPC)................................................. 1 1.2 Introduction of OPC PHA Data Inversion ...................................................... 4 1.3 Thesis Content ................................................................................................ 6

Chapter 2: Optical Particle Counter Calibration................................................................. 8 2.1 Introduction of OPC Calibration..................................................................... 8 2.2 Experiment Apparatus .................................................................................... 9 2.3 Data Acquisition Software............................................................................ 23 2.4 Lasair Calibration results .............................................................................. 23 2.5 Climet Calibration Results............................................................................ 48 2.6 Summary ....................................................................................................... 50

Chapter 3: Adaptation of the STWOM Method for OPC Pulse Height Analysis Data Inversion ........................................................................................................................... 52

3.1 The Twomey and STWOM Nonlinear Iterative Inversion Algorithms........ 52 3.2 OPC Pulse Height Analysis Data Inversion Codes....................................... 54 3.3 Some Details of the OPC PHA Inversion Codes and Numerical Experiments ....................................................................................................................... 56 3.4 Discussions of Twomey Inversion................................................................ 64 3.5 Summary ....................................................................................................... 67

Chapter 4: Atmospheric Aerosol and Diesel Exhaust Measurements .............................. 68 4.1 St. Louis Size Distribution Measurement ..................................................... 68 4.2 Diesel Exhaust Measurements ...................................................................... 92 4.3 Summary ....................................................................................................... 94

Chapter 5: Conclusions and Suggestions for Future Work............................................... 96 5.1 Conclusions................................................................................................... 96 5.2 Recommendations for Future Work.............................................................. 97

References:...................................................................................................................... 101 Appendix A: OPC Calibration Results ........................................................................... 104

A.1 South Pole Lasair High Gain ...................................................................... 104 A.2 South Pole Lasair Low Gain ....................................................................... 108 A.3 St. Louis Lasair Low Gain .......................................................................... 110 A.4 Climet Low Gain......................................................................................... 113

Appendix B: Codes for the Twomey Inversion Package................................................ 114 Appendix C: Codes for the Lasair and Climet Response Calculations .......................... 140 Appendix D: Codes for Lasair and Climet Table Data Refractive Index Corrections ... 147

1

Chapter 1: Introduction

1.1 Principles of Optical Particle Counter (OPC)

The scattering of light by homogenous spherical particles is well-defined by three

scattering regimes according to the optical size parameter α, which is given by:

λπα d

= (1.1)

where

d = particle geometric diameter

λ = wavelength.

The three scattering regimes are [1], [2]:

• Rayleigh scattering: (α<0.15)

• Lorenz-Mie scattering: 0.15≤ α≤15 (approximate)

• Geometric scattering: α>15 (approximate)

In our current atmospheric research, particle sizes are comparable to wavelength.

Therefore, scattering occurs in the Lorenz-Mie regime. Since light scattering theory can

provide exact results for homogenous spherical particles, it forms the basis for building

sensitive and accurate particle measuring instruments [1]. Single optical particle counters

(OPC) count and size aerosol particles by measuring the light that is scattered when

individual particles pass through a light beam [3]. Figure 1.1 is a schematic diagram of a

generic forward scattering optical particle counter. It illustrates the steps required to

convert raw voltage pulse data to particle size distributions [4].

2

Figure 1.1 Optical particle counter and data process steps [4]

In the OPC shown above, a narrow stream of aerosol particles surrounded by

filtered sheath air flows through a scattering volume into which an illuminating beam of

light is tightly focused. Only one particle is illuminated at a time. The photo-detector

collects the scattered light in a defined angular range and generates a voltage pulse that is

proportional to the amount of light collected. Then the signal processor amplifies the

pulses and classifies them into several discrete voltage bins (this is usually done by a

built-in pulse height analyzer (PHA) or a multichannel analyzer (MCA)) to form a pulse

height distribution. A set of comparators compare the pulse height distribution to the

threshold voltages determined by calibration, and the pulse height distribution is finally

converted to particle size distribution and reported as tabulated data. Given the aerosol

flow rate, we can measure the aerosol concentration by counting the number of the

scattering events per unit time [1], [5].

Optical particle counters avoid physical contact with particles and provide real-

time measurement. However, these advantages are offset by three major shortcomings.

First, the built-in pulse height analyzer (PHA) board only sizes particles into a few

channels (e.g. 8 channels for the PMS Lasair 1002, 16 channels for the Climet Spectro .3).

This configuration doesn’t take full advantage of the inherent resolutions provided by

these instruments. Second, the threshold voltage of each size bin is usually determined by

Calibrated Response

Pulse Height Distribution

Size Distribution

3

polystyrene latex (PSL) calibrations. PSL is spherical and has a refractive index of 1.590

(at 0.589µm wavelength). Particles measured with OPCs often have shapes and refractive

indices that are quite different from those of PSL. Therefore, if we use the preset

threshold voltages, the size information will be inaccurate. For example, if we are

measuring di-octyl sebacate (DOS) spherical particles, whose refractive index is 1.448 (at

0.589µm wavelength), the Lasair voltage response to DOS particles of a given size will

be smaller than that to PSL of the same size. This means that the OPC tends to

underestimate the DOS sizes. Third, in an ideal OPC, all particles with the same size

would be classified into the same channel. However, real OPCs produce a distribution of

pulse heights when sampling monodisperse particles. Therefore, there is not a unique

relationship between the pulse height and particle size, as is suggested in Figure 1.1. In

order to extract the maximum amount of information from measured pulse height

distributions, it is necessary to take into account what is known about the response of the

OPC to real, complex particles.

To overcome these three problems, we connected an external Multichannel

Analyzer (MCA) (produced by EG&G ORTEC) to the OPC analog voltage output

instead of using the internal MCA board. We refer this as an Optical Particle Counter –

Pulse Height Analysis (OPC-PHA) System. The external MCA has 2048 channels, which

significantly improves the particle size resolution. Furthermore, we used a differential

mobility analyzer (DMA) to produce monodisperse calibration aerosols from the

measured aerosols (e.g., atmospheric and diesel exhaust particles) as well as calibration

standards such as PSL and DOS. These provided us accurate information on the OPC’s

response to measured aerosols. I developed an “inversion” algorithm to convert pulse

height distributions measured by the OPC-PHA system to aerosol size distributions. This

inversion algorithm utilized “kernel functions”, which define the probability that a

particle of a given size will be counted in a given MCA channel. Kernel functions were

obtained by calibration with monodisperse particles. Rather than assuming that all

particles in a given channel were produced by particles of the same size, the inversion

algorithm determined the contribution of particles in various sizes to the number of

counts in each MCA channel. For example, suppose that 100 particles were counted in

4

channel 500. Kernel functions say that 10% of these particles are 200nm – 300nm, 80%

are 300nm – 400nm, and 10% are 400nm – 500nm. Then the 100 particles should be

allotted to these three size intervals according to their percentages.

1.2 Introduction of OPC PHA Data Inversion

As described in the previous section, the objective of OPC PHA data inversion is

to unravel the true size distribution from the pulse height distribution recorded by the

multi-channel analyzer (MCA). Mathematically, our object is to solve the following

Fredholm integral equation of the first kind for the aerosol size distribution, )( pDf , at

each channel:

∫∞

=0

)()( pppii dDDfDKy , i = 1, 2 , … , m, (1.2)

where

iy = number of pulses counted by the ith MCA channel

Dp = particle diameter

)( pi DK = the probability that a particle with size Dp will be counted by MCA

channel i (kernel function), 1)(0 ≤≤ pi DK

)( pDf = particle size distribution function

m = total number of MCA channels.

The physical meaning of this equation can be interpreted in this way: pp dDDf )(

is the number of particles in the size range [ pD , pD + pdD ]. And pppi dDDfDK )()( is the

number of particles in [ pD , pD + pdD ] that will be classified into channel i. If all possible

sizes are integrated, we can get the total number of particles counted in the ith MCA

channel, i.e. iy .

The most common approach of solving equation 1.2 is to evaluate )( pDf at a

discrete set of diameters, pjD . The number of sizes is called “resolution”. Equation 1.2

can be reduced into the following discrete sum form:

5

∑∑==

=∆≈n

jjpjpi

n

jpjjpjpii DfDADDfDKy

11)()()()( , i = 1, 2 , … , m, (1.3)

where

n = resolution of data inversion

pjD = particle diameter where the size distribution is to be calculated

pjjpijpi DDKDA ∆= )()( . (1.4)

This is a system of m equations with n unknowns ( )(jpDf ). It can be rewritten in matrix

notation as:

)( pDAfy = (1.5)

where y is a m×1 vector, A is a m×n matrix, and )( pDf is a n×1 vector.

A straightforward solution of )( pDf can be obtained by simple matrix inversion:

yADf p1)( −= . (1.6)

Unfortunately, there are several limitations preventing us from solving these equations in

this way. First, if m(channel number) > n (resolution), this is an overdetermined problem,

and 1−A does not exist. Second, if m(channel number) < n (resolution), this is an

underdetermined problem, and again 1−A does not exist. Furthermore, the solution to this

problem is not unique. Third, even when m(channel number) = n (resolution), the matrix

A is nearly singular and ill conditioned for many aerosol measurements [6]. Therefore, 1−A is very large or does not exist at all.

A variety of inversion methods have been derived to solve this problem. A

comprehensive review was given by Milind Kandlikar [6]. Among these methods, the

programs developed by Crump and Seinfeld, INVERSE and CINVERSE, were reported

to be able to give good results for impactor and optical particle counter data. But

INVERSE often gives negative values in the tail of the inverted size distribution, and

CINVERSE is difficult to automate [7]. The MICRON package developed by

Wolfenbarger and Seinfeld has been successfully used in inverting the Ultrafine

Condensation Nucleus Counter pulse height distributions [8]. This code is very long and

6

difficult to understand, so it is not easy to be modified or adapted for inversions of data

from other instruments. Furthermore, it requires substantial computational resources.

In this study, we have chosen Twomey’s non-linear iterative algorithm [9] and its

modified version STWOM [7] to invert the OPC pulse height distribution data. Our work

shows that these algorithms can give good result and they are relatively simple to use.

1.3 Thesis Content

The objective of this work is to develop a software package that:

(1) generates kernel functions pertinent to the refractive index of measured particles;

(2) inverts measured OPC pulse height data with their kernel functions to obtain

mobility size distributions.

An outline of this thesis work is shown in Figure 1.2.

Refractive index ofmeasured aerosols

OPC responsecalculation

Kernel functions oflaboratory aerosols

Kernel functions ofmeasured aerosols

Measured pulseheight distribution

Data Inversionprogram

Inverted sizedistribution

Figure 1.2 Overview of the research in this thesis

Three optical particle counters were used in this work. One PMS Lasair 1002 and

one Climet Spectro .3 were used in measurements of atmospheric aerosol size

distributions in the St. Louis Supersite Program. Another PMS Lasair 1002 was used in

measuring diesel engine exhaust and laboratory generated aerosol mass distributions. The

detailed description of these two instruments is given in Chapter 2. OPC calibrations are

7

essential for checking the performance of instruments, determining OPC’s response to

particles of different sizes and refractive indices, and eventually obtaining good kernel

functions and inverted size distributions. Chapter 2 presents the OPC calibration

experiment setup and results. The calculation of kernel functions is discussed in detail.

OPC theoretical responses are also calculated according to Mie theory and compared to

the measured responses.

Chapter 3 is devoted to describing the Twomey and STWOM non-linear data

inversion method. A number of numerical experiments have been performed to evaluate

the performance of this inversion algorithm for OPC PHA data.

In chapter 4, the inversion package is applied to atmospheric and diesel exhaust

aerosol measurements. Conclusions are presented in Chapter 5.

8

Chapter 2: Optical Particle Counter Calibration

2.1 Introduction of OPC Calibration

The objective of calibrating the OPC is to obtain the instrument responses to

monodisperse particles. We refer to these response functions as kernel functions. The

kernel functions can help us to understand the OPC’s responses to particles of different

sizes, refractive index, and shape. As was explained in Chapter 1, kernel functions are

required to obtain size distributions by inverting raw pulse height distribution data. In this

chapter, the Lasair’s response to monodisperse polystyrene latex (PSL), di-octyl sebacate

(DOS), sodium chloride (NaCl), and diesel soot particles is discussed in detail. Some

calibration results for the Climet are also presented.

Some of the properties of the particles are listed in Table 2.1 [1].

Table 2.1 Properties of measured particles

PSL DOS NaCl Diesel soot

Shape Spherical Spherical Cubic Chain agglomerates

Refractive index

(λ=589nm) 1.590 1.448 1.544 (1.96-0.66i)

Density (g/cm3) 1.05 0.915 2.20 0.3 ~ 1.1 ①

In the first part of this chapter, I present the instruments and the experiment setup

used for calibration. Then the calibration results are presented and discussed. (Detailed

calibration results for each instrument are listed in Appendix A.) The theoretical OPC

responses are calculated and compared to measurements. After that, the Lasair counting

efficiency is discussed. Finally, I discuss the calculation of kernel functions for

homogenous spherical particles with arbitrary refractive indices.

① Data measured by Kihong Park

9

2.2 Experiment Apparatus

The OPC calibration experiment system can be divided into two subsystems, a

monodisperse particle generation system that generates the monodisperse particles, and

an Optical Particle Counter – Pulse Height Analysis (OPC-PHA) data acquisition system

that measures the kernel functions. The entire system is shown in Figure 2.1.

atomizer

dry, cleancompress air

DMAdi

ffus

ion

drye

r

neutralizer

filter

excess flow

sheath flowHEPA

C.O

excess flow

make up flow

HEPAamplifierfilter

0

0

0

0

0

0

0

0

0

0

u2u3u4u5un

u1

x2

x1 * / *

u4u4

u2u3u4u5un

u1

x2

x1 * / *

u4u4

to vacuumto vacuum

Lasair

Climet

CNC

MCAPC

H.V. Power Supply

liquid trap

voltagedivider

Vin1

V in2

Vout1

Vout1

Vout2

Vout2

Lab-PC-1200

Symbols:

Critical OrificeBall ValveLaminar Flowmeter

qa

qc

qm

qs

Figure 2.1 OPC calibration experiment setup

2.2.1 Monodisperse Particle Generation System

The laboratory monodisperse particle generation system used in this experiment is

a very typical system that has been widely used in the Particle Technology Laboratory for

many years [10], [11], [12], [13].

In this system, particles were generated by atomizing solutions or suspensions. In

my experiments, deionized water was used to atomize PSL or NaCl particles. Typical

10

concentrations were 5 drops of 1.5 % of PSL in 250cc DI water, and 0.1% (by weight) of

NaCl. DOS was dissolved in isopropyl alcohol to make a 0.1% (by volume) solution.

Compressed air was passed through a dryer and a filter before it enters the atomizer. The

pressure of compressed air at the entrance to the atomizer was controlled at around 30 psi

by a pressure regulator (These parts are not shown in Figure 2.1).

Because the Lasair, Climet and CNC needed only part of the aerosol flow

provided by the atomizer, the excess flow was directed through a filter into the room air.

A liquid trap was used downstream of the atomizer to collect big droplets. This reduced

the amount of water that must be collect by the diffusion dryer.

The droplets coming out the atomizer contained a mixture of the solvent and

solute. To get pure solute particles, a diffusion dryer filled with silica gel was used to

absorb the water from the PSL and NaCl droplets. To remove the isopropyl alcohol from

the DOS solution, the diffusion dryer was filled with activated carbon.

In some cases, the concentration of the particles was so high that it exceeded the

upper limit that could be counted by the OPC. When this occurred, multiple particles

could be simultaneously present in the scattering volume, and the MCA dead time was

high, causing large errors in sizing and concentration. Dilution, which was achieved by

filtering a fraction of the aerosol flow, was then used to reduce the concentration.

The particles produced by the atomizer had an unknown distribution of charges.

A Po-210 neutralizer was used to ensure that particles entering the DMA had the

Boltzmann equilibrium charge distribution.

The Differential Mobility Analyzer (DMA) was the core instrument used to

generate monodisperse aerosols used for calibration. The DMA selects particles

according to the electrical mobility Zp , which is defined as the ratio of electrostatic drift

velocity to the magnitude of electric field [1]:

p

c

DqC

EZp

πη3v

== (2.1)

where

v = particle velocity

E = electric field strength

11

q = particle’s charge

Cc = Cunningham slip correction factor

η = air viscosity

pD = particle diameter

As shown in Figure 2.1, the DMA analyzing region consists of a center rod that

can be maintained at a known voltage and a grounded outer housing. Both clean sheath

air and aerosol flow enter near the top of the DMA. The aerosol flows through a thin

annular region near the inner wall of the DMA housing. Charged particles move across

the sheath flow to the center rod due to the electrical force. Particles having a narrow

range of electrical mobilities will reach the sampling slit near the bottom of the DMA

analyzing region. This range is given by ZpZp ∆±* , where *Zp is the centroid mobility,

and Zp∆ is half width of the mobility range of the extracted particles. These parameters

can be expressed as [11], [14]:

Vqq

Zp mc

Λ+

=π4

* (2.2)

Vqq

Zp sa

Λ+

=∆π4

(2.3)

)/ln( abL

=Λ (2.4)

where

a = outer radius of the center rod

b = inner radius of the housing

L = distance between the mid-planes of the DMA entrance and exit slits

aq = aerosol (polydisperse) flow rate

cq = clean (sheath) air flow rate

mq = main (excess) outlet flow rate

sq = sampling (monodisperse) flow rate

V = center rod voltage

12

Note that particles of different mobilities can be selected by varying the voltage applied

to the center rod.

The resolution of the DMA is defined as the relative half-width, which is

mc

sap

qqqq

ZpZ

++

=∆

* . (2.5)

Since the aerosol coming out of the DMA is not monodisperse, the DMA broadening

effect is defined by the DMA transfer function Ω, which is the probability that an aerosol

particle of electrical mobility Zp entering the DMA will leave the DMA via the

monodisperse aerosol outlet. Figure 2.2 shows the DMA transfer function [14], [15].

Figure 2.2 Theoretical DMA transfer function [15]

If aq = sq , cq = mq , the transfer function shown in Figure 2.2 can be simplified to the

following form [15]:

+∞≤≤∆+

∆+≤≤+∆+∆−

≤≤∆−−∆−∆

∆−≤≤∞−

=∆Ω

ppp

pppppppp

pppppppp

ppp

ppp

ZZZ

ZZZZZZZZ

ZZZZZZZZ

ZZZ

ZZZ

*

***

***

*

*

0

)1/(/

)1/(/

0

),,( . (2.6)

13

In my experiment, the GMWDMA was used [16]. This instrument had

dimensions of: L = 44.348cm, a = 0.943cm, b = 1.927cm. A critical orifice was used to

control the sheath air flow rate. I used two methods to ensure that the DMA did not leak.

First, I reduced the vacuum inside the DMA column to 600 mmHg and closed all the

valves. My criterion for a “leak free” column was that the pressure did not drop more

than 5 mmHg in a 30-minnute period [17]. Second, I set the DMA voltage to zero,

balanced the aerosol and sheath flow, and monitored the outlet aerosol flow using a TSI

Condensation Nucleus Counter (CNC 3760). No particles would be detected by the CNC

if there were no leak.

In this experiment, DMA flow rates were regulated such that sa qq = , and

cm qq = . Therefore, *Zp was only a function of V and cq (see Equation 2.2). The high

voltage supply operated over the range from 0V to 10000V, and the sheath flow rate

cq could be varied to obtain particles in a desired size range. Different sizes of critical

orifices were used to control the sheath air flow rate. In order to get good resolution, the

aerosol flow rates were almost always set to 101 of the sheath air flow rate. Under this

condition, 101

* =∆

ZpZ p . Furthermore, because all particles I measured in these experiments

were bigger than 100nm, the diffusion broadening of particle size distributions was not

significant. However, I found that the OPC pulse height distribution produced by DMA-

generated particles were significantly wider than would be produced by truly

“monodisperse” particles. This effect needed to be accounted for when obtaining kernel

functions. This will be discussed in detail later in this Chapter.

The DMA center rod voltage was supplied by a Bertan Model 205A-10R high

voltage power supply. Usually, the voltage indicated on the front panel is not exactly

equal to voltage applied. I used a mulitmeter and a high voltage probe to calibrate the

voltage supply.

After leaving the DMA, the “monodisperse aerosol” flow was mixed with filtered

make up air before it was sampled by particle measuring instruments.

14

2.2.2 OPC-PHA Data Acquisition System

The OPC data acquisition system consists of a PMS Lasair 1002, a Climet

SPECTRO .3 and two multichannel analyzers. In our system, the OPC’s responses to

particles were recorded both by the OPCs themselves and by the MCAs. A TSI CNC

3760 sampled the aerosol in parallel with the OPCs to independently measure the total

particle concentration. The OPC’s counting efficiency for monodisperse particles could

be calculated by dividing the MCA concentration by the CNC concentration.

Table 2.2 shows some of the main specifications of the two optical particle

counters. More information about the instruments can be found in Lasair User’s Guide to

Operate [18] and Lasair Technical Service Manual [19], Spectro.3 Laser Particle

Spectrometer Operation Manual [20], and the web sites of the two manufacturers:

http://www.pmeasuring.com/, http://www.climet.com/.

Table 2.2 Some specifications of Lasair 1002[19] and Climet Spectro .3 [20]

PMS Lasair 1002 Climet Spectro. .3

Flow rate 0.002 CFM (0.057 LPM) 0.035 CFM (1.0 LPM)

Max. concentration 50,000,000/ft3 28,000,000/ ft3

Optical design Wide angle 90˚ collecting optics Elliptical Mirror

Laser source HeNe, 633nm 50mW laser Diode, 780nm②

Analog output 0 ~ -10V 0 ~ +2.9V

Computer interface RS-232 and RS-485 RS-232 and RS-485

2.2.2.1 PMS Lasair 1002

The Lasair 1002 is produced by Particle Measuring Systems. Figure 2.3 and

Figure 2.4 show the optical system and flow system diagrams of the Lasair 1002 [18].

② Data from personal communication with Randy Grater (Technical Service Manager of Climet Instruments)

15

Figure 2.3 Optical system of Lasair [18]

Figure 2.4 Flow system for Lasair 1002 [18]

The operation of the Lasair is similar to that for the generic OPC we discussed in

Chapter 1. The source of illumination is a 633nm 10-milliwatt HeNe laser. As a particle

passes through the sample cavity, it is illuminated by the laser beam and scatters light.

The main signal processing steps are illustrated in Figure 2.5 and discussed below [19]:

• Photodector board: The photodetector senses the scattered light and produces a

current pulse. This pulse is proportional to amount of the scattered light, and contains

size and refractive index information about the particle. Then this current pulse is

converted to a negative voltage pulse. The preamplifiers amplify the signal into

several gain stages according to different amplification factors and send it to the

internal pulse height analysis (PHA) board.

16

ScatteredLight

Analog Output

Laser ReferenceVoltage

Photodectorboard

Photodector

Preamplifiers

Amplifiers

Comparators

Signal Pulse

Amplified signal inup to 4 gains

External PHABoard

Digital Board

Size information in8 channels

Screen/Printer

Table data RS-232/RS-485Table Data

File

Pulse HeightDistribution

Internal PHAboard

Figure 2.5 Lasair data process flow chart

• Internal PHA: The internal PHA board amplifies the signal from the detector board

again. Then the signal goes on in two separate routines. One is sent to the rear panel

I/O as a 0 to -10VDC analog output, which can be connected to an external MCA to

record the pulse height distribution. The other routine goes to the comparators, where

the signal is compared to preset threshold voltages for the eight channels and assigned

to the appropriate size bin. This information is sent to the digital board to create the

17

table data. Table 2.3 shows the particle size channels of the Lasair 1002 provided by

the manufacturer. The threshold voltages for the eight channels are based on

calibrations done with monodisperse polystyrene latex spheres (PSL). The voltage vs.

size curve provided by PMS is shown in Figure 2.6 [19]. Thresholds are

automatically adjusted to account for the changes of laser reference voltage (LRV) by

voltage dividers shown in Figure 2.7. This is done by setting

Threshold = LRV×R2/(R1+R2).

Table 2.3 Size channels for Lasair 1002 table data

High gain Low gain

Channel 1 2 3 4 5 6 7 8

Size (µm) 0.1–0.2 0.2-0.3 0.3-0.4 0.4-0.5 0.5-0.7 0.7-1.0 1.0-2.0 >2.0

Figure 2.6 Voltage vs Size Interval Curve – Lasair 1001 and 1002 [19]

High gainLow gain

18

Lasair referencevoltage input

Threshold voltage outputto internal MCA

R1 R2

Figure 2.7 Voltage divider to set the threshold of each size bin

• Digital Board: The digital board controls the data in and out of the Lasair. It processes

data from the internal PHA board and outputs it either as the Lasair screen display

(table data) or a printed hard copy. It can also read from or write to RS-232/ RS-485

serial ports. In my LabVIEW program, I used RS-232 serial communication to

control sampling and save the table data as a file in the computer.

I used two Lasair 1002’s in my work. Serial number 38107 was used in St Louis

Supersite aerosol measurements. For this Lasair, only the low gain was calibrated and

used. Serial number 14705 was used at the South Pole during December 2000. I used this

instrument to study diesel exhaust aerosols and laboratory-generated aerosols. Both the

high gain and the low gain of this Lasair were calibrated and used. In this thesis, these

two instruments are referred to as the St Louis (STL) Lasair and South Pole (SP) Lasair,

respectively.

2.2.2.2 Climet SPECTRO .3

Climet Spectro .3 is produced by Climet Instruments Company. The operation

principle of Climet is quite similar to the Lasair, but as shown in Table 2.2, there are four

main differences between the Climet and the Lasair. First, the flow rate of the Climet is

about twenty times higher than the Lasair. This enables the Climet to collect more

particles than the Lasair during the same sample period. Second, the Climet covers a

wider size range than the Lasair. It can detect particles as big as 10µm. Third, the Climet

uses an elliptical mirror instead of mangin mirrors to focus the scattered light to the

detector. (This will be discussed in more detail later in this Chapter.) Finally, as with the

Lasair, the Climet also has both table data and analog DC voltage outputs. But the Climet

19

table data have 16 channels in 3 separate gains, as shown in Table 2.4. The analog

voltage output is 0 ~ +2.9VDC [20].

Table 2.4 Size channels for Climet SPECTRO .3 table data [20]

Digital③ High gain

Channel 0 1 2 3 4 5 6 7

Size (µm) 0.3–0.4 0.4-0.5 0.5-0.63 0.63-0.8 0.8-1.0 1.0-1.3 1.3-1.6 1.6-2.0

Low gain

Channel 8 9 10 11 12 13 14 15

Size (µm) 2.0–2.5 2.5-3.2 3.2-4.0 4.0-5.0 5.0-6.3 6.3-8.0 8.0-10 >10.0

2.2.2.3 Multichannel Analyzer (MCA)

The multichannel analyzer consists a Multichannel Buffer (MCB) card and a

personal computer. The MCB takes the Lasair or Climet analog voltage output as its

input, and classifies voltage pulses into different channels. The computer is used to

control instruments and to display and record measurements.

The MCB used in our experiments is the TRUMP-2K Multichannel Buffer Card

produced by EG&G ORTEC. This card has a resolution of 2048 channels. The inputs to

the card are voltage pulses in the range from 0 to 10V. However, the manufacturer

reserved channel 2001 to 2048 to improve the linearity performance and the data in this

area is not valid. Therefore, we can only use data from channel 0 to 2000④.

For proper performance of the MCA, two things should be addressed: dead time

and lower level discriminator (LLD). The MCB is not able to count signals during the

time required for ADC conversion and data transfer. This is called dead time. When the

concentration is very high, the possibility of losing pulse counts increases, which yields

incorrect particle concentration data. In our experiments, the concentrations of particles

were controlled so that the MCA dead time is less than 8%. The Lower Level

③ The amplification factor of the digital gain is 5 times higher than the high gain. The signal from the digital gain is applied as a digital pulse, rather than as an analog pulse, to the comparators. This information was not used in the PHA analysis of this work. ④ From personnel communication with Joe Lassater , a technician in Ametec, Inc, (Joe.Lassater@ametec-online.com)

20

Discriminator (LLD) adjustment is used to prevent small noise pulses from being

counted. If the noise is counted, the dead time will increase tremendously, and the

recorded pulse height distribution will include data from both noise and particles. The

manufacturer (ORTEC) generally set the LLD to 75 mV [21], which corresponds to

channel 15. In order to adjust the LLD to the noise level of the OPCs, I put a filter at the

OPC inlet so that no particles were entering the OPC. The lowest MCA channel at which

noise was detected was identified. A safety factor of about 10 channels was added to this

lowest channel to set the LLD. In contrast to LLD, there is a upper level discriminator

(ULD) which sets the highest amplitude pulse that will be stored in MCA. The ULD was

set to one channel less than the maximum channel as required by the manufacture.

In my experiments, I assumed that the voltage response changed linearly with the

channel number. Because the lower end of channel 1 corresponded to 0 V, and the upper

end of channel 2048 corresponded to 10V, the upper voltage limit for channel i was:

204810 iVVi ×= . (2.7)

2.2.2.4 Inverting and Non-inverting Amplifiers

The Lasair analog outputs are voltages from 0 to –10V, and the MCB input

voltage range is 0 to +10V. Therefore, I built an inverter to enable the MCB to detect the

Lasair output signals. At the same time, in order to increase the resolution in a selected

range of particle sizes, I sometimes amplified the Lasair output signal. For example, for

the particle size distribution measurements in St Louis, we wanted the Lasair to cover the

size range of 0.3µm to 1.0µm. The threshold voltages of these two sizes were about

0.171V and 3.937V (Figure 2.6), respectively. We amplified the Lasair output pulse by a

factor of 2.5. Hence the adjusted voltage range was from 0.428V to 9.843V. This

significantly improved the resolution over the size range of interest. However, the analog

outputs of the Climet were voltage pulses in the range from 0 to +2.9V, I used a non-

inverting amplifier to amplify the signal to increase the resolution. The amplification

factor was 4.282, which enabled the Climet-PHA data to cover the size range of 0.4µm to

1.3µm.

21

Typical inverting and non-inverting amplifiers are shown in Figure 2.8 and Figure

2.9, respectively.

+15V

-15V

R1

R2

Input ( from Lasair )

output ( to MCA )OP 27G

+

-

Figure 2.8 Inverting Amplifier used with Lasair

(Values for R1 and R2 are given in Table 2.5)

+15V

-15V

R1

R2

Input ( from Climet )

output ( to MCA )OP 37G

+

-

Figure 2.9 Non-inverting Amplifier used with Climet

(Values for R1 and R2 are given in Table 2.5)⑤

In order to obtain the correct amplification factor and maintain the shape of the

signal, the amplifiers should have appropriate slew rates (defined as the voltage change

rate per unit time). The signal durations of the Lasair and the Climet are about 20µsec

and 4µsec, respectively. This means the amplifier of the Climet should be faster than that

⑤ The amplifier used with this Climet was originally OP27G. Later we found this amplifier was too slow that it did not provide the performance we desired. So we replaced it with a faster amplifier OP37G.

22

of the Lasair. OP 27G and OP 37G have slew rates of 2.8V/µsec and 17V/µsec,

respectively. Oscilloscope tests showed that these two amplifies worked very well for the

Lasair and the Climet. For the inverting amplifier in Figure 2.8, the amplification factor is

R2/R1. The amplification factor of the non-inverting amplifier in Figure 2.9 is 1+R2/R1.

The amplifier settings for the two Lasairs and the Climet of my experiment are listed in

table 2.5.

Table 2.5 Amplifier parameters

R1⑥ (Ω) R2 (Ω) Amplification

factor

PHA size range

(PSL) (µm)

South Pole Lasair

(high gain) 21.49K(22K) 27.59K(27K) 1.284 0.1 – 0.2

South Pole Lasair

(low gain) 9.8K (10K) 27.74K(27K) 2.831 0.3 – 1.0

St Louis Lasair

(low gain)⑦ 201.9 (200) 461.0 (470) 2.283 0.3 – 1.0

St Louis Climet

(high gain) 9.77K (10K) 32.07K (33K) 4.282 0.4 – 1.3

2.2.2.5 Condensation Nucleus Counter and Lab-PC-1200 Data Acquisition

Card

In these experiments, a CNC 3760 was used to measure the total concentration of

the monodisperse particles, which was then used to calculate the OPC counting

efficiency. As shown in Figure 2.1, the CNC, Lasair and Climet sampled the calibration

aerosol in parallel downstream of the DMA. In order to make sure that these three

instruments sample aerosols of the same concentration, the aerosol flows and the make up

flow must be very well mixed. To achieve this, the flow path between the mixing point

⑥ Values in parenthesis are the nominal values. ⑦ The amplifiers for the St. Louis Lasair worked well, but the resistor values were too small. Usually, the higher the input impedance, the better the op amp performance. On the other hand, too high resistor will suffer from Johnson noise. Therefore, resistors on an order of several kΩ are suggested for future work.

23

and the sampling point was extended to about 2 meters and an orifice (not shown in

Figure 2.1) was added between the tubes to help mixing.

Lab-PC-1200 is a data acquisition card manufactured by National Instruments.

This card provides a counter to record the CNC counts. The digital output of CNC 3760

is a 15V square pulse, but the Lab-PC-1200 can only take 0~10V input. Therefore, a

voltage divider was used between the CNC and Lab-PC-1200 to reduce the CNC output

to the amplitude acceptable to the Lab-PC-1200.

2.3 Data Acquisition Software

I wrote LabVIEW programs “Lasair_calib.vi” and “Climet_calib.vi” to control

the instruments, do measurements and record data. When the programs start, they send

commands to the serial ports that control the OPCs to set the sampling parameters, such

as the sample interval, and sample mode (continuous or not). Then they order the OPCs

to start sampling. At the same time, the program sends one command to the counter to

start the CNC 3760 counting, and another to the MCB card to start measurements with

the PHA. At the completion of the sampling interval, both the OPC table data and PHA

data are stored on the computer hard disk.

2.4 Lasair Calibration results

2.4.1 PSL Kernel Functions

As indicated earlier, the manufacturer of the Lasair (Particle Measuring Systems

Inc.) uses polystyrene latex (PSL) to calibrate the Lasair. They did not report complete

kernel functions. Instead, they provided the average Lasair voltage responses

corresponding to the peaks in the pulse height distributions of several selected PSL sizes

(Figure 2.6). These values were used to set the threshold voltages for size bins to create

table data. PSL spheres have standard deviations of about 2%. The size range is so

narrow that the dispersion in size can be neglected. Therefore, the measured PSL kernel

functions were deemed as the true PSL kernel functions in our work. Kernel functions of

particles generated by the DMA (DOS, NaCl, diesel soot, etc) can be estimated from the

PSL kernels by Mie response calculation (homogenous, spherical particles) or by

24

calibration (non-spherical particles). The PSL kernel functions were also used to check

the performance of the particle measuring system, and to study the effect of refractive

index on kernel functions.

The response of the Lasair to 404nm PSL monodisperse particles recorded by the

MCA (pulse height distribution) is shown in Figure 2.10.

0

20

40

60

80

100

120

140

0 500 1000 1500 2000

MCA channel number

coun

ts

Figure 2.10 Pulse height distribution of 404nm PSL (Lasair low gain)

In order to obtain size distributions by inverting measured pulse height

distributions, we need to fit mathematical functions to the measured kernels. These

functions can then be interpolated or extrapolated to provide estimates of kernel functions

for particle sizes for which no measurements are available. The procedure that I used to

obtain generic kernel functions is as follows:

• First, only the peak corresponding to the desired size was kept in analysis. Peaks of

doublets, triplets, etc. (which appear more commonly in DOS calibrations) were

deleted.

• Second, normalized pulse height distributions were obtained by dividing the number

of counts in each channel by the total number of the counts in the main peak.

• Third, the channel numbers were converted to voltages (pulse height) by assuming

that the channel numbers were linearly proportional to voltages (Equation 2.7).

During the sampling, the Lasair reference voltage (LRV) varied from 6.5 to 9.0V. All

pulse heights were normalized to a LRV of 10V to enable comparisons of

25

measurements obtained at different LRV levels. The conversion from channel i to the

upper limit voltage Vi used Equation 2.8:

LRViVi

102048

×= (2.8)

• Finally, the measured kernel functions were fit to lognormal distributions according

to the following equations [1]:

∑∑=

i

iig C

VCV

lnln (2.9)

21

2

1)ln(ln

ln

−

−=

∑∑

i

giig C

VVCσ (2.10)

−−= 2

2

)(ln2)ln(ln

expln2

1

g

g

g

VV

VdVdC

σσπ (2.11)

where

gV = count median voltage

gσ = geometric standard deviation

iV = voltage corresponding to the upper limit of channel i

iC = normalized counts in channel i.

C = normalized counts distribution (kernel function).

The most frequent pulse height voltage (mode) pV was calculated by Equation 2.12.

)lnexp( 2ggp VV σ−×= . (2.12)

Both the measured and fitted kernel functions for the 404nm PSL data in Figure 2.10 are

shown in Figure 2.11. Note that the lognormal curve fits the measurements very well.

26

0

1

2

3

4

5

1.5 1.7 1.9 2.1

Pulse Height (V)

dC/d

V

measured kernelfitted kernel

Figure 2.11 Measured and fitted kernel function of 404nm PSL

All of the PSL kernel functions for calibrated sizes were obtained by the method

described above. Figure 2.12 shows the measured and fitted PSL kernels in the size

range from 305nm to 1099nm for the South Pole Lasair low gain. Note that the lognormal

distribution fits the PSL kernels quite well for most sizes.

0.1

1

10

0.0 2.0 4.0 6.0 8.0 10.0 12.0Pulse Height (V)

dC/d

V

305nm

404nm482nm

505nm595nm

672nm653nm

701nm720nm 845nm

913nm

1099nm

Figure 2.12 Measured and fitted PSL kernel function (SP Lasair low gain)

If we take the peak of each pulse height distribution (the fitted Vp from Equation

2.12), we can draw a graph of peak voltage vs. particle mobility size, which is shown in

Figure 2.13.

27

0

2

4

6

8

10

12

200 400 600 800 1000 1200

Dp (nm)

Peak

Vol

tage

(V)

PSL MeasuredPMS Provided

Figure 2.13 Peak voltage versus size from my measurement

and from calibration data provided by PMS (SP Lasair low gain)

As we can see from this plot, the peak voltage (Vp) increases monotonically with particle

diameter, except for the data point of 672nm. Also shown in Figure 2.13 are some peak

voltages calculated from the PMS calibration data (Figure 2.6). They were obtained by

multiplying the PMS calibration data by the amplification factor of the external inverting

amplifier. Note that these data fit my calibration very well. Figure 2.14 shows a plot of

geometric standard deviations (σg) of PSL pulse height distributions. A straight line was

fitted to these points, and the fitted line equation was used to calculate standard

deviations of all sizes. I found that when particle diameter exceeded 2µm, the

extrapolated standard deviation was very close to 1 (see Figure A.3.2 in Appendix A).

Since we did not have PSL calibration data above 2µm, the standard deviation of 2µm

PSL was used for particles bigger than 2µm.

28

y = -3E-05x + 1.0691R2 = 0.8666

1.03

1.04

1.05

1.06

400 500 600 700 800 900 1000 1100

Dp (nm)

σ g

Figure 2.14 Geometric standard deviations of fitted PSL kernel functions

(SP Lasair low gain)

2.4.2 DOS Kernel Functions and DMA Broadening Effect

The response of the Lasair to DMA selected “monodisperse” 404nm DOS

particles is shown in Figure 2.15. Note that there are two peaks in Figure 2.15: a main

peak at channel 136, and a minor peak around channel 456. I believe that the minor peak

was produced by “doublets”. The doublets have the same electrical mobility as the singly

charged particles, but they are doubly charged, and are therefore larger.

0

50

100

150

200

250

300

350

0 500 1000 1500 2000

MCA channel number

Cou

nts

Figure 2.15 Pulse height distribution of 404nm DOS (Lasair low gain)

29

According to Equation 2.1,

p

c

DqC

EVZp

πη3== (2.1)

and 21 ZpZp = , 212 qq =

2

22

1

11

33 p

c

p

c

DCq

DCq

πηπη=⇒

1

122

2

c

pcp C

DCD

××=⇒ (2.13)

where the subscripts 1 and 2 represent singly and double charged particles, respectively.

In this case, nmDp 4041 = ; From Equation 2.13 it follows that nmDp 7052 = . On the

other hand, my calibration showed that the peak of 701nm DOS pulse height distribution

appears at channel 437. This confirms that the particles in the minor peak were doublets.

In Chapter 3, this pulse height distribution is inverted to obtain the size distribution.

Again, we will see these two peaks in the size distribution. Since we can calculate the

size of doublets precisely, the peak of doublets can be considered as a calibration data

point [22]. However, in this study, only the main peak was used to calculate the kernel

function, and peaks of doublets, triplets etc. were deleted.

Both the fitted and the measured pulse height distributions for the 404nm DOS

data in Figure 2.15 are shown in Figure 2.16. Note that the lognormal curve also fits the

“monodisperse” DOS measurement data very well.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.4 0.6 0.8 1.0 1.2 1.4

Pulse Height (V)

dC/d

V

measured kernelfitted kernel

30

Figure 2.16 Measured and fitted pulse height distribution of 404nm DOS

However, the pulse height distribution shown in Figure 2.16 is not the true kernel

function for 404nm DOS particles because of the DMA broadening effect. As was

indicated in Section 2.2.1, particles coming out of DMA had a mobility range of

ZpZp ∆±* (in most of my experiments, 101

* =∆

ZpZ p ). This mobility range corresponds to a

diameter range of 375.6nm to 438.5nm. This range is much wider than that of 404nm

PSL, which according to the manufacturer is 400nm to 408nm. Figure 2.17 compares the

measured 404nm PSL and DOS pulse height distributions. We can see that the measured

DOS pulse height distribution is much wider than the measured PSL kernel function.

0

2

4

6

8

10

0.5 1.0 1.5 2.0 2.5

Pulse Height (V)

dC/d

V

measured 404nm PSLmeasured 404nm DOSscaled 404nm DOS

Figure 2.17 Pulse height distributions for 404nm PSL and DOS. The measured 404nm

PSL and DOS were obtained directly from measurement. The scaled 404nm DOS curve

was obtained by scaling the 404nm PSL kernel. The scaling method is discussed below.

Theoretically, the true DOS kernel functions can be solved through Equation 1.2

[8],

∫∞

=0

)()( pppii dDDfDKy , i = 1, 2 , … , m, (1.2)

where iy represents measured pulse height distribution of DMA selected “monodisperse”

DOS particles, )( pi DK are true kernel functions, and )( pDf is the aerosol distribution

exiting the DMA. If we assume that the DMA inlet particle concentration over the narrow

31

range of ZpZp ∆±* is constant, then pp dDDf )( becomes the DMA transfer function

(Equation 2.6). Equation 1.2 can be changed to the matrix form as shown below:

1

2

1

1

111

1

2

1

)(

)(

)(

)()(

)()(

×××

Ω

Ω

Ω

=

npn

p

p

nmpnmpm

pnp

mm D

D

D

DKDK

DKDK

Y

YY

M

L

MOM

L

M (2.14)

However, because the number of unknowns ( )( pi DK ) generally exceeds the number of

equations, it is not possible to solve this matrix to get the true kernel functions. Instead, I

scaled the PSL kernel function to obtain the true DOS kernel function. The scaling factor

was the ratio of the Mie response to DOS and to PSL of the same size (The Mie response

calculation is discussed later in this chapter). The scaled 404nm DOS kernel is also

shown in Figure 2.17. It was obtained by multiplying the x value (pulse height) of PSL

kernel by the scaling factor, while dividing the y value (dC/dV) by the same scaling

factor. If the scaled kernels are true kernels, then I can solve Equation 2.14 to get Yi,

which is the pulse height distribution of DMA selected 404nm DOS particles. The

measured and calculated pulse height distributions are shown in Figure 2.18. Note that

the two curves are pretty close. The small peak shift is due to the small difference

between the measured and calculated peak voltages of 404nm DOS and PSL particles.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.5 1.0 1.5 2.0 2.5Pulse Height (V)

dC/d

V

measured 404nm DOS

calculated 404nm DOS

Figure 2.18 Measured and calculated pulse height distributions

32

of DMA selected 404nm DOS

In conclusion, the kernel functions scaled from PSL kernel functions are good

estimates of true DOS kernel functions. I assumed this was also true for other spherical

particles in my work. For non-spherical particles such as NaCl and diesel exhaust aerosol,

no Mie response calculation result was available. And the shape factors affected the

Lasair responses a lot. Therefore, the kernel functions of these aerosols were determined

from calibration.

2.4.3 Effect of Refractive Index on Lasair Response

Because the Lasair was calibrated with polystyrene latex (PSL), the “optical

equivalent size” [23] provided by the Lasair internal MCA (table data) corresponds to the

size of a PSL sphere that would scatter the same amount of light as the measured particle.

The intensity of scattered light tends to decrease with decreasing size and refractive index.

Therefore, if the refractive index of the measured particle is smaller than that of PSL, the

light scattering diameter provided by the Lasair table data will be smaller than the true

particle size. This underestimation of particle size by different optical particle counters

has been addressed previously [12], [24].

In our work, we were not using the Lasair table data to get the particle size

distribution. Instead, before carrying out measurements, we calibrated the Lasair using

mobility-classified particles selected from the measured aerosol. Then we used these

calibrated responses to obtain particle size distributions. This method allowed us to get

the size distribution, without being affected by refractive index.

Figure 2.19 shows the responses of the Lasair to PSL (m = 1.59) and DOS (m =

1.448). Note that the peak voltages of PSL are systematically higher than those of DOS

of the same size in this size range.

33

0

2

4

6

8

10

12

300 400 500 600 700 800 900 1000 1100

mobility diameter (nm)

Peak

vol

tage

(V)

PSL peak voltageDOS peak voltage

Figure 2.19 Peak voltages of PSL and DOS

The data in Figure 2.19 can be used to determine the equivalent optical scattering

diameters of the DOS, which are shown in Figure 2.20. The ratio of the DOS optical

equivalent diameter to its mobility equivalent diameter (i.e., true diameter) is shown in

Figure 2.21. Note that this ratio varies with particle diameter, and ranges from 77% to

90% for the range of sizes investigated.

300

500

700

900

1100

300 400 500 600 700 800 900 1000 1100

mobility diameter (nm)

equi

vale

nt o

ptic

al d

iam

eter

(nm

)

Figure 2.20 DOS equivalent optical scattering diameter vs. mobility diameter

34

0.76

0.8

0.84

0.88

0.92

400 500 600 700 800 900 1000 1100

mobility diameter (nm)

diam

eter

rat

io

Figure 2.21 DOS Diameter ratio vs. mobility diameter

Information on the sensitivity of Lasair response to different refractive indices can

also provide us some insight into the physical or chemical properties of the measured

particles. Figure 2.22 compares the Lasair responses to mobility classified diesel soot

aerosols at different engine loads. We found that the peak voltages of particle emitted at

75% engine load were always higher than those at lower engine loads. Kihong Park has

shown that at low load, diesel particles probably contain more oil, and they are somewhat

more compact. As engine load increases, the diesel soot particles of a given mobility

become more highly agglomerated and particles are mostly composted of carbon. A

detailed study of the reason that particles formed at high engine loads scattered more light

is beyond the scope of this thesis.

35

0

1

2

3

4

5

130 160 190 220 250 280 310

mobility diameter (nm)

peak

resp

onse

(V)

10% load50% load75% load

Figure 2.22 Lasair response to mobility-classified diesel engine emissions at

different engine loads

2.4.4 Lasair Theoretical Response Calculation

2.4.4.1 Lasair Scattering Geometry

The light collecting optics of the Lasair 1002 consists of two mangin mirrors

mounted at the right angle to the laser beam [19]. Figure 2.23 illustrates the scattering

geometry of the Lasair. Light scattered by the particle is collected by the mirrors in the

cone with semi angle β from 18˚ to 53˚. The theoretical scattering intensity can be

calculated in two steps. First, integrate the light scattered in the cone with semi angle 53˚,

and then subtract the light in the smaller cone with semi angle 18˚.

36

Figure 2.23 Light scattering geometry of Lasair 1002

2.4.4.2 Optical Particle Counter Response Calculation Theory

The scattering of light by homogenous spheres is based on Mie theory, which has

been well defined and used extensively [11], [25], [26], [3]. The response of a single

optical particle counter is proportional to the rate at which the scattered electromagnetic

energy enters the collecting optics. It is a function of instrument properties (optical

design, source of illumination, and electronics), and particle properties (size, refractive

index, shape, orientation of non-spherical particles relative to the illuminating beam) [4].

The OPC response can be calculated with Equation 2.15 if we assume that all of the

scattered light for given values of scattering angle θ and azimuth angle φ enters the

detector [3].

λϕθθλλ dddrPfIIR sin)()()( 2||∫∫∫ += ⊥ (2.15)

where

⊥I = scattered irradiance for the vertically polarized incident light

||I = scattered irradiance for the horizontally polarized incident light

)(λf = wavelength distribution of the incident light

37

)(λP = wavelength-dependent response of the OPC detector

λ = incident light wave length

θ = scattering angle

ϕ = azimuth angle

For the Lasair, the incident laser is coherent and unpolarized. The wavelength is

fixed at 633nm, so the integral over wavelength will be omitted and replaced with an

instrument-dependent constant. This constant can be determined empirically by

calibrating the Lasair with particles having known size and refractive index. From Figure

2.20, we also know that the collecting optics is external to the laser cavity, and they are

mounted normal to the laser beam. Taking all these factors into consideration, Equation

2.15 can be reduced to Equation 2.16 as follows [3].

λϕθθλλπλ

dddPfSSI

R sin)()(24

22

21

2

20

+

= ∫∫∫

θθθβη

βηdSSC sin)(2

22

1 Ψ+= ∫+

− (2.16)

where

C = λλλπλ

dPfI

)()(4 2

20∫ (2.17)

1S , 2S = infinite series that relate the scattered and incident electric field

β = collecting aperture semi angle (18˚ to 53˚ for Lasair 1002, Figure 2.23)

η = the angle between the incident light and the axis of collecting aperture

(90˚ for Lasair 1002)

)(θΨ =

=Ψ →

− −°=−

θβθ

ηθηθβ η

sincoscos)(

sinsincoscoscoscos 1901 . (2.18)

In this work, S1 and S2 were calculated by the computer program BHMIE, which

was given by Bohren and Hoffman [27]. The Lasair responses were calculated by a

Fortran program, which was originally developed by W.W. Szymanski and S. Palm.

Several modifications were made to adapt this program to the Lasair. The Fortran codes

are listed in Appendix C.

38

2.4.4.3 Lasair Response Calculation Results

We assumed that PSL and DOS particles were homogenous spheres so that Mie

theory could be used to calculate scattering intensities. Figure 2.24 to Figure 2.26 are the

calculation results for both high and low gain of the two Lasairs. Also shown are the

measured responses. As stated in the previous subsection, there was an instrument-

dependent factor between the calculated response and the measured response. The factor

k was obtained using the least squares approach to minimize the function )(kg , which

was the difference between the calculated and measured responses at the same diameter:

( )∑=

−=m

iii xkykg

1

2)( (2.19)

where

iy = calculated response at ith size

ix = measured response at ith size

m = number of sizes measured

The function )(kg reached its minimum when

( ) ( )∑∑==

−=⇒−∂∂

=m

iii

m

iii xkyxky

k 11

2 00 . (2.20)

Consequently,

∑∑==

=m

ii

m

ii yxk

11/ (2.21)

Figures 2.24 to Figure 2.26 show that the measured particle responses match the

calculated ones very well.

39

0

2

4

6

8

10

12

14

80 120 160 200 240Dp (nm)

Res

pons

e (V

)

PSL measuredPSL (given by PMS)PSL CalculatedDOS MeasuredDOS Calculated

Figure 2.24 Measured and calculated responses (SP Lasair, high gain)

0.1

1

10

100

200 400 600 800 1000 1200

Dp (nm)

Resp

onse

(V)

PSL MeasuredPSL (given by PMS)PSL CalculatedDOS MeasuredDOS Calculated

Figure 2.25 Measured and calculated responses (SP Lasair, low gain)

40

0.1

1

10

100

300 500 700 900 1100 1300 1500 1700 1900 2100

Dp (nm)

Resp

onse

(V)

PSL MeasuredPSL (given by PMS)PSL CalculatedDOS MeasuredDOS Calculated

Figure 2.26 Measured and calculated responses (STL Lasair, low gain)

2.4.5 Lasair Counting Efficiencies

When particle size approaches the lower detection limit, the Lasair counting

efficiency drops. This is because the Lasair uses comparators to eliminate pulses whose

magnitudes are less than those produced by 0.1µm PSL particles. Similarly, the MCA has

a lower level discriminator (LLD) to avoid counting noise signals and an upper lever

discriminator (ULD) to eliminate large signals. In order to obtain the true size distribution

measured by OPCs, it is necessary to account for these size-dependent counting

efficiencies.

In my experiments, a TSI 3760 CPC was used to sample DMA classified

monodisperse aerosols in parallel with the Lasair. Because the CPC has a lower detection

limit of 0.014µm, which is well below that of the Lasair (0.1µm for table data), the

concentration measured by the CPC can be regarded as the true concentration. The Lasair

counting efficiency was then obtained by dividing the Lasair concentration by the CPC

concentration. Figure 2.27 is an example of counting efficiency measurements.

41

Lasair counting efficiency for PSL

0

50

100

0 100 200 300 400 500 600 700 800mobility diameter (nm)

coun

ting

effic

ienc

y (%

)

MCATable

Figure 2.27 Comparison of PSL counting efficiencies obtained

from the Lasair table and the MCA data

Note that table data has a higher counting efficiency than the MCA data at smaller sizes.

This is because the MCA data only covers part of the low gain while the table data

records both low gain and high gain signals.

OPC counting efficiency is a function of refractive index. Particles having smaller

refractive indices scatter less light, therefore, counting efficiencies are higher for particles

with higher refractive indices. Figure 2.28 compares Lasair-MCA counting efficiencies of

PSL (n=1.590) and DOS (n=1.448). We can see that PSL has smaller detectable size than

DOS.

42

Lasair-MCA counting efficiencies for PSL and DOS

0

50

100

100 300 500 700 900 1100mobility diameter (nm)

coun

ting

effic

ienc

y %

PSLDOS

Figure 2.28 Comparison of Lasair-MCA counting efficiencies for PSL and DOS

There are three problems in measuring the true monodisperse counting

efficiencies:

• The first one is related to multiple charged particles. For example, as we can see from

Figure 2.28, the counting efficiency for 263nm DOS is around 0. But double charged

“doublets” will also pass through the DMA. They have a diameter of 437nm and the

counting efficiency of these particles is near 100%. Therefore, when we measure the

counting efficiency for 263nm DOS, we need to subtract the doublet concentration

from the total concentrations measured with the CPC and Lasair. This can easily be

done with the Lasair-PHA data because the doubly charged particles are clearly

separated from singly charged particles. It is difficult to make this correction for the

table data, because the table data resolution is too low.

• To correct counting efficiency for Lasair-PHA data, we need to know counting

efficiencies for each size. In this case, PSL works very well because of its narrow size

range. However, the counting efficiency for DMA classified ‘monodisperse’ particles

is not truly the counting efficiency for that size because counting efficiencies can vary

substantially over the range of sizes selected by the DMA. We need to deconvolute

the pulse height distribution to obtain the true kernel and the true counting efficiency.

On the other hand, the resolution of table data is so low that it is very difficult to

make counting efficiency corrections. For example, the table data counting

43

efficiencies in the size range of 100nm to 200nm are: 12% for 102nm, 84% for

150nm and 93% for 199nm PSL. Clearly, it would be necessary to have better size

resolution in this range to properly account for detection efficiencies. Therefore, it is

not clear how to correct for counting efficiencies for the table data in the 100-200nm

size bin.

• The third problem is that the Lasair counting efficiency is a function of laser

reference voltage (LRV). When the LRV drops, the minimum detectable size

increases. This problem can be solved by the technique discussed below.

I have identified two approaches for counting efficiency corrections for Lasair-

PHA size distribution measurements. One is to incorporate counting efficiencies with

kernel functions when inverting the PHA data. In this case, the kernel functions indicate

the probability that a particle of a given size will be detected in a given MCA channel and

the total probability that particles will be detected is less than 1. The other way is to

obtain the size distribution of detected particles first, and then divide the OPC counting

efficiencies [8]. In this case the kernel functions indicate the probability that detected

particles of a given size will be detected in a given MCA channel, and the total

probability for detecting “detected” particles equals 1. In my case, counting efficiencies

can be easily included in kernel functions by setting kernels below the LLD and above

the ULD to 0. Therefore, the sum of kernels will be less than or equal to 1, and this sum

is equal to the counting efficiency. Figure 2.29 shows the measured and modeled

counting efficiencies for PSL. We can see that they are in reasonable agreement.

44

Measured and modeled Lasair-MCA counting efficiencies for PSL

0

20

40

60

80

100

120

100 200 300 400 500 600 700 800mobility diameter (nm)

coun

ting

effic

ienc

y %

measuredmodeled

Figure 2.29 Measured and modeled Lasair-MCA counting efficiencies for PSL

As introduced earlier, the DMA classified DOS particles are not truly

monodisperse particles. Therefore, the measured DOS counting efficiency is not the true

counting efficiency for that size. However, as shown in Figure 2.18, I can model the

DMA classified DOS “kernel function” using truly monodisperse kernel functions and

DMA transfer function. Ideally, I can model the measured counting efficiency by

summing the DMA classified “kernel function” over the channel range of LLD to ULD.

Figure 2.30 shows the counting efficiencies of measured and modeled DMA classified

DOS particles. Again, we see they match very well, with discrepancies of less than 10%.

However, as shown in Figure 2.31, the measured and modeled counting efficiencies for

diesel soot do not agree as well. The maximum discrepancy is about a factor of 1.7. I

suspect this discrepancy is due to the complex shapes of diesel exhaust particles, which

make it difficult to predict kernel functions very well.

45

Measured and modeled Lasair-MCA counting efficiencies for DOS

0

20

40

60

80

100

120

100 300 500 700 900 1100mobility diameter (nm)

coun

ting

effic

ienc

y %

measured

modeled

Figure 2.30 Measured and modeled Lasair-MCA counting efficiencies for DOS

Measured and modeled Lasair-MCA counting efficiencies for diesel soot

0

20

40

60

80

100

120

50 100 150 200 250 300 350

mobility diameter (nm)

coun

ting

effic

ienc

y (%

)

measuredmodeled

Figure 2.31 Measured and modeled Lasair-MCA counting efficiencies

for diesel exhaust aerosols

As I mentioned earlier, counting efficiency is a function of laser reference voltage

(LRV). The modeled counting efficiencies for 263nm PSL at different LRV are shown in

Figure 2.32. Because kernel functions are always adjusted to the LRV at each

measurement in data inversion, this LRV dependence is automatically accounted for in

kernel functions.

46

PSL (263nm) counting efficiency vs. Laser Reference Voltage

0

50

100

6 7 8 9 10

Laser Reference Voltage (V)

coun

ting

effic

ienc

y (%

)

Figure 2.32 Lasair-MCA counting efficiencies for 263nm PSL at different LRV

2.4.6 Lasair Kernel Functions for Particles with Arbitrary Refractive

Index

As indicated in Chapter 1, one of the objectives of this work is to generate kernel

functions pertinent to the refractive index of measured particles. My calibration of the

Lasair using PSL, DOS, NaCl, and diesel soot particles has shown that the kernel

functions are described very well by lognormal distributions. Therefore only two sets of

parameters are needed to create kernel functions: count median voltage and geometric

standard deviation (Equation 2.9 to 2.11).

For homogenous spherical particles, if the refractive index is given, then the peak

voltage response of the Lasair to each size of these particles can be calculated according

to Mie theory as discussed in the previous subsection, and the median voltage can be

obtained from the peak voltage by Equation (2.12). However, when the refractive index is

not known, we must obtain refractive index first. In this work, the refractive indices of

atmospheric aerosols in the St Louis measurement were obtained with two different

methods. One was using Bill Dick’s model for calculating refractive indices from

measurement of aerosol composition [28]. The other method was using hourly DMA

selected 450nm aerosol calibration data to obtain the peak voltage. By comparing this

peak voltage to the responses of 450nm particle with different refractive indices, we

47

could determine the refractive index of atmospheric aerosol. This will be discussed in

detail in Chapter 4. Because the DMA mobility-classified aerosols were not truly

monodisperse due to the DMA broadening effect, the kernel function shapes of these

aerosols were assumed to be the same with PSL kernel functions in this work. Therefore,

the geometric standard deviations of calibrated PSL kernel functions were used for these

aerosols. For SP Lasair low gain, the fitted line in Figure 2.14 was used to calculate

standard deviations of aerosols measured by the low gain of this Lasair. A similar curve

was used for the STL Lasair low gain (Figure A.3.2 in Appendix A). For SP Lasair high

gain, only three sizes of PSL were available to do calibration (102nm, 152nm, 199nm).

The standard deviation of the 152nm PSL was used as the average standard deviation of

all particles. After we obtain both count median voltages and standard deviations, we can

plug them into Equation 2.11 to calculate kernel functions for spherical particles with

arbitrary refractive index.

For non-spherical particles Mie theory doesn’t apply, hence calibrated median

voltages must be used instead. For example, as shown in Figure 2.33, a logarithmic curve

was fit to the calibrated median voltages of 50% engine load diesel soot, and the fitted

equation was used to calculate the median voltages for all sizes. For agglomerate particles

like diesel exhaust, the orientation of particles in the viewing volume affected the Lasair

response greatly. Therefore, kernel functions of these particles were inherently wider than

spherical particles, and PSL standard deviations were no longer good estimations of these

particles. The calibrated DMA classified kernel functions were used.

ln(Vg) for diesel soot (50% engine load)

y = 3.0644Ln(x) - 15.785R2 = 0.9928

-1

0

1

2

100 150 200 250 300 350

Dp (nm)

lnV

48

Figure 2.33 Median voltages of Lasair response to 50% engine load diesel soot

2.5 Climet Calibration Results

Since we were using the Climet high gain in St Louis, only the high gain had been

carefully calibrated by PSL. The result of these calibrations is shown in Figure 2.34.

0

0.005

0.01

0.015

0.02

0.025

0 500 1000 1500 2000

MCA channel

norm

aliz

ed c

ount

s

404nm

482nm505nm

595nm653nm 701nm

845nm913nm 1099nm

Figure 2.34 Climet high gain calibration with PSL

Unfortunately, we found that it was difficult to fit one distribution to the Climet

kernel functions. Therefore much less effort was made in my work to invert the Climet

PHA data.

As shown in Figure 2.35, the light scattering geometry of the Climet is much

more complicated than that of the Lasair (Figure 2.23). The mirror is elliptical, and the

equation of the inside surface is:

000.163.068.0 2

2

2

2

=+YX (2.22)

49

Figure 2.35 Climet ellipsoidal mirror (by Climet Instrument Company)

Similar to the Lasair mangin mirrors, the Climet mirror axis is 90˚ (η) to the laser beam,

and the mirror covers a semi-angle (angle AF1D) of 112.1˚ (β). Since the Off Axis

Collection Aperture calculation given in [3] is valid when η> β, the mirror is divided into

two parts: BCDEF and ABFG. The theoretical Mie response can be calculated in the

following seven steps:

• Step1: calculate response of BCDEF (the same with BAHGF) with semi-angle of

89.9˚ (off axis) I1.

• Step2: calculate response of AHG with semi-angle of 67.9˚ (off axis) I2.

• Step3: calculate response of hole CDE with semi-angle of 22.5˚ (off axis) I3.

50

• Step4: calculate the response of particle holes with semi-angle of 18.7˚ (approximate)

(off axis) I4.

• Step5: calculate the response of forward laser beam hole particle hole with scattering

angle of 0 to 15.8˚ (approximate) (on axis) I5.

• Step6: calculate the response of backward laser beam hole particle hole with

scattering angle of 164.2 to 180˚ (approximate) (on axis) I6.

• Step7: calculate the total response:

654321 22 IIIIIII −−×−−−×= (2.23)

The Fortran codes are listed in Appendix C. The calculated responses and the

measured median voltages for PSL are shown in Figure 2.36. Also shown are data

provided by the manufacturer (after amplification correction) [20].

0

1

10

100

0 500 1000 1500 2000Dp (nm)

Res

pons

e (V

)

theoreticalmeasuredClimet provided

Figure 2.36 Calculated and measured Climet response to PSL

Note that our measurements match the Climet provided data very well. The measured

responses to 845nm and 913nm are about 18% lower than the theoretical values. For

particles above 1µm, the data provided by Climet are higher than the theoretical values.

2.6 Summary

Calibration is one of the most important steps in OPC measurements. It quantifies

the OPC’s responses to particles with different physical and chemical properties. When

kernel functions corresponding to the particles being measured are used in the PHA data

51

inversion, the true particle sizes instead of the PSL equivalent optical scattering sizes are

obtained.

In this chapter, I focused on answering the question of how to calculate Lasair

kernel functions for particles with given refractive indices. OPC calibration with DMA

classified “monodisperse” particles were discussed in detail. We found that

“monodisperse” particles selected by a DMA were not truly monodisperse due to the

width of the mobility distribution of particles that were classified by the DMA. Because

PSL spheres were more nearly monodisperse than aerosols selected by a DMA from a

polydisperse input aerosol, the standard deviations of the OPC responses obtained from

PSL calibration were used for generating kernel functions of other aerosols. I also

presented the theoretical response calculations of the two OPCs, and found that they

matched the calibration results very well. This provided us the possibility to calculate the

OPC’s responses (peak voltages) to spherical particles of arbitrary refractive index. For

the Lasair, lognormal distributions fitted kernel functions very well. So we could use the

PSL standard deviations and the calculated response to generate kernel functions for

particles of arbitrary refractive index in a reasonable size range. However, we haven’t

found a good analytical model to describe the kernel functions for the Climet, and

therefore less effort was spent on the Climet kernel function study.

52

Chapter 3: Adaptation of the STWOM Method for OPC Pulse Height Analysis Data Inversion

3.1 The Twomey and STWOM Nonlinear Iterative Inversion

Algorithms

As discussed in Chapter 1, the objective of OPC pulse height analysis (PHA) data

inversion is to solve Equations 1.3 for )( pDf .

∑∑==

=∆≈n

jjpjpi

n

jpjjpjpii DfDADDfDKy

11)()()()( , i = 1, 2 , … , m. (1.3)

Twomey’s non-linear iterative method and its modified version STWOM were used in

this work. Twomey’s algorithm begins with an initial guess, )()0(pjDf , based on the data

obtained from measurements. Then the initial guess distribution is iteratively refined by

multiplying small weighting factors. These multiplicative factors are related to the kernel

functions as shown below:

njmiDfDKaDf pjk

pjiipjk KK ,2,1,,2,1),()]()1(1[)( )()1( ==×−+=+ (3.1)

where ia is the ratio of the measured counts to the calculated counts in channel i, which

is given by:

∑=

∆= n

jpjjp

kjpi

ii

DDfDK

ya

1

)( )()(. (3.2)

In equations (3.1) and (3.2), the superscripts k+1 and k represent the new and old trial

solutions.

Because 0≥ia , and 1)(0 ≤≤ pji DK , )()1(pj

k Df + will be positive as long as the

initial guess is positive. This means that the algorithm gives positively constrained results.

Another advantage of this algorithm is that it does not limit the number of points, pjD , at

which the solutions are calculated.

53

Gregory R. Markowski developed STWOM [7], a modified version of the

Twomey routine. The main proposes of the modifications are to find smoother solutions

and to reduce computation time. The two main modifications are:

1. Before the kth trial solution )()(pj

k Df is put into the Twomey routine, it is

smoothed by applying the following three point average:

4

)(2

)(4

)()( )1(

)()()1(

)()( +− ++= jp

kpj

kjp

k

pjk DfDfDf

Df . (3.3)

For points at both ends, smoothing is done by:

4

)(4

)(3)( 1

)(0

)(

0)( p

kp

k

pk DfDf

Df +×

= (3.4)

4)(

4)(3

)( )1()()(

)( −+×

= npk

pnk

pnk DfDf

Df . (3.5)

This will reduce the roughness in the solution.

2. A stopping point is added when the calculated counts yit are close enough to the

measurements yi. This stopping criterion SIGMA is defined as:

[ ]2

1/)(1 ∑

=

−=m

ii

tii Eyy

mSIGMA (3.6)

where Ei are error tolerances which typically are the experimental standard

deviation or uncertainties in yi. The Twomey routine is exited when SIGMA < 1.

This reduces the computation time. The flow diagram of STWOM is shown in

figure 3.1 [7].

Twomey first applied his method to the inversion of filter measurements [9].

Markowski used STWOM to invert low pressure Berner cascade impactor and Electrical

Aerosol Analyzer (EAA) data [7]. Both Twomey’s routine and STWOM have been

widely used in aerosol measurement data analysis.

54

Calculate initial guess

Smooth trialsolution

Is SIGMA>limit?

Stop

STOW M Start

Twomey Routine Start

Read in trial solutionN(k)(Dpj)

Return

Yes

No

∑=

∆

= n

jpjjp

k

jpi

ii

DDfDK

ya

1

)( )()(

)()]()1(1[)( )()1(pj

kpjiipj

k DfDKaDf ×−+=+

Calculate next trial soluiton

njmi KK ,2,1,,2,1 ==

)()0(pjDfUse Twomey routine

to find the initial trialsolution

Calculate next trialsolution with Twomeyroutine

)()1(pj

k Df +

)()1(pjDf

)()(pj

k Df

)()1(pj

k Df +

Figure 3.1 Flow diagram of STWOM algorithm [7]

3.2 OPC Pulse Height Analysis Data Inversion Codes

The codes I used in inverting OPC PHA data were originally written by Hwa-Chi

Wang based on the Twomey inversion routine [29]. Modifications had been made by W.

Winklmayr (1987-1988) and some features of the STWOM were applied. This program

was especially tested for the inverting data from the Berner Impactor [30]. I made many

modifications to adapt this code to invert the OPC PHA data. The codes are listed in

Appendix B.

This package consists 6 Fortran files and a parameter data file:

55

• main.for: This is the main program that controls the execution of the flow chart

shown in figure 3.2. It provides the users with options of conducting “ numerical

experiments” or carrying out inversion of actual data.

• kernel.for: This routine is different for spherical and non-spherical particles. For

spherical particles, it calls Mie response routine to calculate theoretical responses.

The geometric standard deviations are those obtained from PSL calibration. For non-

spherical particles, it reads in fitted kernel function parameters for sizes used for

calibration. Then the kernel functions of all the sizes to be evaluated (Dpj) are

calculated.

• Scatter.for: This subroutine calculates the Mie responses of the Lasair to spherical

homogenous particles with given refractive indices. The peak voltage is used in

kernel function calculation.

• Ln04.for: This subroutine is used to generate ideal “data” for unimodal or bimodal

lognormal distribution for testing the performance of the inversion routine. Users are

asked to provide mean diameters, standard deviations and the maximum dn/dlogDp’s

of two lognormal distributions. This routine calls the routine “kernel.for” and

calculates the number of counts in each channel (synthetic pulse height distribution).

• Poisson.for: This code is used to generate random error according to Poisson

possibility, which is used to test the sensitivity of the data inversion routine to error

and to test minimum count requirements in each channel.

• Ti04.for: This is the core routine for STWOM data inversion.

• TI.PAR: This data file contains control parameters of the Twomey routine, such as

the iteration steps, error range, smooth steps, resolutions, etc. These values should be

adjusted by the user to adapt the routine for different inversions.

The flow chart of this package is shown in Figure 3.2. More detailed descriptions of these

codes can be found within the codes, which are listed in the Appendix B of this thesis.

56

Stop

Start

NumericalExperiment?

(main.for) Generate idealdistribution(Ln04.for )

Calculate Kernelfunctions

(kernel.for)

Yes

Read from input file thecounts in all the channels

(Ti04.for)

Add random error to the input data?

(Ti04.for)

No

Generaterandom error(Poisson.for)

Yes

STOWMinversion(Ti04.for)

Calculate Kernelfunctions

(kernel.for)

Calculate OPC Mieresponse

(Lasair.for)

Calculate OPC Mieresponse

(Lasair.for)

No

Twomey routinecontrol parameters

(Ti.PAR)

Figure 3.2 Structure of the OPC data inversion package

3.3 Some Details of the OPC PHA Inversion Codes and Numerical

Experiments

In this section, we will discuss some important questions such as initial guess,

reliable reconstruction range, and effect of random error. I have carried out various

numerical experiments to determine the proper values of control parameters and to

evaluate the quality of the inverted size distributions.

57

In this work, bimodal lognormal distributions were used to carry out “numerical

experiments” to evaluate the performance of the STWOM routine for OPC PHA data

inversion. These bimodal distributions are sum of two lognormal distributions, as shown

in equation 3.7.

−−+

−−= 2

2

22

221

21

1 )(log2)log(log

exp)(log2

)log(logexp

log)(

g

gpj

g

gpj

p

pj DDM

DDM

DdDdn

σσ (3.7)

where

1M , 2M = amplitudes of the two lognormal distributions

1gD , 2gD = geometric mean diameters of the two lognormal distributions

1gσ , 2gσ = geometric standard deviation of the two lognormal distributions

In my package, these bimodal lognormal distributions were generated by the program

(Ln04.for). The synthetic pulse height distributions iy for these bimodal lognormal

distributions were also calculated by this code using Equation 1.3. In the content of this

chapter, the parameters input by the user are given in the form of

(M1, Dg1, σg1; M2, Dg2, σg2). (3.8)

If either M1 or M2 is zero, the distribution becomes a unimodal lognormal distribution.

The kernel functions used in the numerical experiments presented in this chapter

were diesel soot kernels (50% engine load) as were discussed in the previous chapter.

3.3.1 Initial Guess Distribution

A good initial guess distribution is very important for getting the correct result

with the Twomey and STWOM algorithms. A bad initial guess not only costs a lot of

computer time, but also usually gives unrealistic solutions. In order to give a good initial

guess, we should consider both the measured pulse height distribution and the kernel

functions.

In a perfect instrument, particles of a given material and size would produce the

same amplitude OPC response, which would appear in a single MCA channel. Although

particles of a given size are actually detected in a range of MCA channels, the “ perfect

58

instrument” model provides us some clues for selecting a good first guess distribution.

Actually, this is the traditional method of pulse height distribution data inversion.

If we assume that each MCA channel corresponds to a single particle size, then

the pulse height distribution can be converted to size distribution directly by comparing

the voltage of each channel to the theoretical response curve (spherical particles) or to the

calibration data (non-spherical particles). Unfortunately, size distributions obtained in

this way are usually pretty rough because the measured pulse height distributions are

usually not smooth.

To get a smoother initial guess, I did some modification to the method discussed

above. If we assume that the pulse height distribution of monodisperse particles with size

Dp is lognormal, then 68.26% of theses particles will fall within the response voltage

range defined by

)logexp(log gσµ ± , i.e. ),/( gσµσµ × (3.9)

where

µ = median voltage of the monodisperse lognormal kernel function

σg = geometric standard deviation.

Although particles of other sizes will also fall into this range, particles with size Dp are

the dominant contributors. From Equation 1.3, the number of pulses counted by the ith

channel is:

∑=

∆≈n

jpjjpjpii DDfDKy

1)()( .

The initial guess for the distribution function for particles with jth size is

))(()(1

∑ ∑∑=

≈∆i

n

jjpi

iipjjp DKyDDf (3.10)

where i ’s are the channels between the lower and higher limits corresponding to the

voltage range of ),/( gg σµσµ × . Actually, this voltage range can be adjusted to adapt to

different instruments and measured aerosols. If it is set as narrow as the range of one

MCA channel, this becomes the traditional method I mentioned earlier.

This method indicates that a good initial guess can only be obtained for the size

range in which the peak voltages of particles fall between channel LLD and 2048. If a

59

significant fraction of particles lie outside of this range, error in the distribution function

will be quite large due to large uncertainties in counting efficiencies.

One numerical example of the true distribution, initial guess, and the inverted

results are shown in Figure 3.3. We can see that the initial guess deviates from the

original distribution quite a bit, but the inverted result is in good agreement with the true

distribution. To reduce the roughness in the initial guess for true measurements, this

initial guess is smoothed five times using Equations 3.3~3.5 before it is put into the

Twomey routine.

Test initial gess (100,150,1.5;100,300,1.2)

0

20

40

60

80

100

120

140

0 100 200 300 400 500 600

Dp (nm)

dn/d

logD

p

originalguessinverted

Figure 3.3 Test initial guess

3.3.2 Reliable Reconstruction Range (RRR) [6]

As discussed in the last section, good inverted distribution can be obtained for

particles whose peak responses fall between channel LLD and 2048 if the measured pulse

height distributions contain no error. According to my calibration, LLD (channel 45) and

channel 2048 of the Lasair high gain correspond to the diesel soot size range of 105nm to

366nm (at 10V LRV). Therefore, size distribution can only be achieved for particles that

fall within this size range. The original and inverted distributions in Figure 3.3 have

shown this restriction.

The relative error, which is defined as,

60

%100Re ×−

=valuetrue

valuetruevalueinvertederrorlative (3.12)

is plotted in Figure 3.4 for the inversion in Figure 3.3. Note that in the size range of 96nm

to 393nm the relative errors are within ±10%. This is close to the range I predicted

(105nm to 366nm). Also shown in Figure 3.4 is the predicted counting efficiency for each

size. Note that it is the sharp drops of counting efficiency outside of the RRR that cause

big errors in the inverted result.

-50

0

50

100

150

0 100 200 300 400 500 600

Dp (nm)

coun

ting

effic

ienc

y&

rela

tive

erro

r (%

)

relative errorcounting efficiency

Figure 3.4 Relative error and counting efficiency for the inversion in Figure 3.3

In summary, reliable reconstruction range can be predicted by comparing channel

LLD and channel 2048 voltages to the calibrated or calculated peak voltages. These

ranges can also be determined from numerical experiments.

3.3.3 Effect of Random Error

To test the effect of measurement uncertainty on the inverted size distributions,

and to find out the minimum number of counts required for good inversion results, a

Poisson random number generator [31] was constructed to simulate the effect of counting

statistical errors on pulse height distributions. I assumed that counts collected in each

channel obeyed the Poisson probability distribution, and took the counts of the original

61

pulse height distribution as the mean value. The Poisson random number generated a

simulated pulse height distribution according to Poisson distribution.

Figure 3.5 and 3.6 are two examples of random error effect tests. The left figures

show the original and simulated pulse height distributions. The right plots show the

hypothetical distribution, the inverted distributions with and without random error.

Figure 3.5 Pulse height distribution and inverted size distribution when counts in

channels are low (10000,150,1.5; 10000,300,1.2)

Figure 3.6 Pulse height distribution and inverted size distribution when counts in

channel are high (100000,150,1.5; 100000,300,1.2)

Note that when counts in each channel are low (Figure 3.5), the counting statistical errors

cause big oscillations in the inverted data. However, Figure 3.6 shows that when counts

exceed 50 per channel, the inverted distribution is in good agreement with the true

distribution. In our atmospheric aerosol measurements, particles counted in each channel

0

5

10

15

20

25

30

35

0 500 1000 1500 2000

MCA channel number

coun

ts

originalpoisson estimate

0

4000

8000

12000

16000

0 200 400 600Dp (nm)

dn/d

logD

p

originalno errorpoisson

0

50

100

150

200

250

0 500 1000 1500 2000

MCA channel number

coun

ts

originalpossion estimate

0

40000

80000

120000

160000

0 200 400 600Dp (nm)

dn/d

logD

p

originalno errorpoisson

62

usually fell below 50 for the Lasair, which caused the inverted data to oscillate. To

reduce the uncertainty, we can combine channels so that the number of counts per

grouped channel is 50 or more. Of course, this will reduce the resolution as a trade-off.

The other choice is to increase sampling time to get enough counts, at the expense of

losing some time resolution.

3.3.4 Examples of Bimodal Distribution Inversion

Similar to the experiments done by Winklmayr, [30], this series of numerical

experiments was carried out to test the code’s ability to recover bimodal lognormal

distributions. In my diesel engine emission measurements, the particle size range was

(100nm, 400nm), and the maximum number of counts in pulse height distributions

ranged from 103 to 104. The parameters in these numerical experiments were selected to

be representative of those observed in the diesel measurements. One mode (d1 = 140nm,

σg1 = 1.5) is held constant, while the values of d2 for the other mode are varied from

160nm, 200nm, 250nm, to 300nm (keeping σg2 = 1.2). The amplitudes of dn/dlogDp of

both modes are held fixed at 5105× . The results are shown in Figure 3.7. These figures

show that this package is able to recover bimodal distributions very well except for very

small or very large particles.

The 404nm DOS calibration shown Figure 2.15 (Chapter 2) is an example of a

true measured bimodal distribution. The calculated result indicates that the two peaks are

404nm and 705nm, respectively. The inverted size distribution of this bimodal pulse

height distribution is shown in Figure 3.8. We can see that the inversion recovers the two

peaks of singlets and doublets exactly. In the next chapter, more tests are described in

which the inversion algorithm is applied to the measurements of diesel exhaust, and

atmospheric aerosol size distributions.

63

0

200000

400000

600000

800000

1000000

0 100 200 300 400 500 600

Dp (nm)

dn/d

logD

p

true

inverted

0

200000

400000

600000

800000

1000000

0 100 200 300 400 500 600

Dp (nm)

dn/d

logD

p

true

inverted

Figure 3.7.1 d2=160nm Figure 3.7.2 d2=200nm

Figure 3.7.3 d2=250nm Figure 3.7.4 d2=300nm

0

50000

100000

150000

200000

250000

300000

350000

300 400 500 600 700 800 900

Dp (nm)

dn/d

logD

p

Figure 3.8 Inverted size distribution for 404nm DOS calibration data

0

200000

400000

600000

800000

0 100 200 300 400 500 600

Dp (nm)

dn/d

logD

p

true inverted

0

200000

400000

600000

800000

0 100 200 300 400 500 600

Dp (nm)

dn/d

logD

p

true inverted

64

3.4 Discussions of Twomey Inversion

The numerical experiments show that Twomey inversion can recover size

distributions from pulse height distributions very well. In this section, I will compare size

distributions obtained from Twomey inversion to other methods.

The straightforward way of obtaining size distribution is to use the Lasair table

data. Figures 3.9(a) to 3.9(c) compare Lasair table data and Twomey inversion for DMA

selected “monodisperse” PSL, diesel soot aerosols and DOS. From these charts, we see

apparently that Twomey inversion has two advantages compared to the Lasair table data:

much higher resolution, and independence of refractive index. In contrast, the table data

reports “optical equivalent” sizes. Because diesel soot is not spherical and absorbs light,

and because DOS has a lower refractive index than PSL, the optical equivalent sizes of

diesel soot and DOS are lower than their mobility sizes. Therefore, table data

underestimate true sizes of these particles as was shown earlier in Figure 2.19 to 2.21.

However, since we know the refractive index of DOS, and we can calculate the

true size corresponding to each optical equivalent size according to theoretical response

of DOS and PSL, it is straightforward to determine the true DOS size limits for bins of

the table data. The “corrected table” data, as well as the Twomey inversion data, are

shown in Figure 3.9(d). From this chart, we can see that the corrected table provides more

precise size information than the original table, but the resolution is even lower because

size bins become wider. Figure 3.10 shows the Lasair high gain responses to PSL and

diesel soot. Note that 140nm and 300nm diesel soot particles are optically equivalent to

115nm and 170nm PSL respectively. But it is difficult to find the mobility sizes of diesel

soot with 100nm and 200nm PSL equivalent sizes. Hence it’s very difficult to do

refractive index corrections for the Lasair table data for diesel soot measurements.

65

3.9 (a) 404 nm PSL 3.9 (b) 200nm diesel soot

3.9 (c) 404nm DOS 3.9 (d) 404nm DOS

Figure 3.9 Comparison of Lasair table data and Twomey inversion

66

Lasair response to PSL and diesel soot (high gain)

0

1

2

3

4

5

100 150 200 250 300

mobility diameter (nm)

Resp

onse

(V)

PSL

diesel soot

Figure 3.10 Comparison of Lasair high gain response to PSL and diesel soot

Traditionally, people obtain size distributions from pulse height distributions by

simply converting response voltages to sizes according to calibrations. This is a special

case of the Twomey inversion initial guess as was discussed in section 3.3.1. The major

problem of this method is that it does not take the kernel broadening into consideration.

Figure 3.11 compares the traditional method and Twomey inversion for PSL and diesel

soot. We find that results are pretty close, except that Twomey inversion is smoother and

narrower.

Figure 3.11 Comparison of traditional method and Twomey inversion

Comparison of Twomey Inversion and traditional method (PSL 404nm)

0

100000

200000

300000

300 400 500 600Dp (nm)

dn/d

logD

p

Traditional methodTwomey inversion

Comparison of Twomey Inversion and traditional method (diesel 200nm)

0

50000

100000

150000

200000

0 100 200 300 400 500

Dp (nm)

dn/d

logD

p

traditional method

Twomey invertion

67

3.5 Summary

In this chapter, the Twomey and STWOM algorithms were introduced with

particular reference to the structure of my OPC PHA data inversion codes. Some

important issues such as initial guess, good size range and random error were discussed

in detail. From numerical experiments, we can see that if all parameters are set properly,

this algorithm can unravel the size distribution from the pulse height distribution very

well. I also compared Twomey inversion with Lasair table data and traditional inversion

method. The results showed that Twomey inversion was much more precise than the

original table data. If the table data is corrected by refractive index, it can give correct

size information, but the resolution is still low. I found that the traditional method could

also do a good job in obtaining size distributions, but the Twomey method was smoother

and more precise.

In the next chapter, I will provide a more detailed discussion of the performance

of this algorithm for inverting OPC measurements of diesel soot and atmospheric

aerosols.

68

Chapter 4: Atmospheric Aerosol and Diesel Exhaust Measurements

We used OPCs to measured diesel engine exhaust size and mass distributions and

St. Louis atmospheric aerosol size distributions. I have discussed the OPC calibration

with these aerosols in Chapter 2, and the data inversion algorithm in Chapter 3. In this

chapter, I will discuss the application of my inversion algorithm to the analysis of these

data.

4.1 St. Louis Size Distribution Measurement

4.1.1 Introduction of St. Louis Measurement

The objective of St. Louis-Midwest Fine Particle Supersite Project is to provide

physical and chemical measurements needed by the health effects, atmospheric science

and regulatory communities. Several groups have been conducting measurements of

particle size distributions, mass concentrations and compositions continuously at the

Supersite in Metropolitan St. Louis (IL-MO) since April 2001 [32].

Our research group (University of Minnesota) is using a particle size distribution

(PSD) measurement system and an integrated moments monitoring (IMM) system [33] to

provide accurate and fast measurements of aerosol size distributions. The PSD system

consists of a Nano-SMPS (TSI Nano-DMA column + TSI 3025A UCPC), a Regular

SMPS (UMN PTL DMA column (long) + TSI 3760 CPC), a PMS Lasair 1002, and a

Climet Spectro .3 (http://www.menet.umn.edu/~hiromu/). For the two OPCs, both the

table data and the pulse height data are recorded. A schematic diagram of the PSD system

is shown in Figure 4.1. The size ranges covered by the four instruments are listed in

Table 4.1.

69

Figure 4.1 Schematic diagram of the PSD system (drawn by Dr. Hiromu Sakurai)

Table 4.1 Size ranges covered by the PSD instruments

Instrument Size range PHA size range

Nano SMPS 3nm ~ 45nm

Regular SMPS 30nm ~ 400nm

Lasair 1002 100nm ~ 2000nm 300nm ~ 1000nm

Climet Spectro .3 300nm ~ 10000nm 400nm ~ 1300nm

The PSD system operates in two modes: calibration mode and measurement

mode. During the first 10 minutes of each hour, the PSD system is set to the calibration

mode. The two OPCs there sample DMA-classified 450nm “monodisperse” atmospheric

particles. For the remaining 50 minutes of each hour, the system operates in the

measurement mode. A complete size distribution is measured every 5 minutes. There are

several proposes of dedicating 10 minutes of each hour to OPC calibrations. First, these

measurements enable us to check the performance of the OPCs. It would be apparent

from these measurements if the DMA were to leak or something were wrong with the

70

OPCs. Second, since the calibration size is fixed at 450nm, the pulse height distribution is

a function of atmospheric aerosol properties such as refractive index and particle shape.

We can obtain much useful information about particle properties by analyzing this hourly

calibration data, which will be discussed in the next section in detail.

Because the refractive index of water (1.333@ 0.589µm [1]) is much lower than

other atmospheric aerosol constituents, the optical properties of atmospheric aerosol

depend strongly on water content [28]. Particle water content depends on the relative

humidity (RH). In the PSD system, we minimize the effect of relative humidity changes

of the sampled air to the OPC size distributions by controlling the RH in the sampling

line. A RH conditioner was designed to fix the RH at 40% in the instruments’ inlet line

[34].

In Situ measurements of aerosol chemical properties were conducted by other

groups: PM2.5 sulfate was measured by Harvard University using HSPH continuous

monitor with SO2 detector; organic carbon (OC) and elemental carbon (EC) was

measured by University of Wisconsin-Madison with the continuous EC/OC analyzer

manufactured by Sunset Laboratoeies. Other chemical species such as nitrate,

ammonium, etc. were also measured, but the data are not yet available.

4.1.2 Refractive Index Calculation and Modeling

In the St. Louis measurements, both table data and PHA data are recorded for

both OPCs. In all cases, true refractive index of measured aerosols is required to obtain

exact size distributions. The table data report the optical equivalent size distributions

based on PSL calibrations. To obtain mobility size distributions, we need to use the true

refractive index of the measured aerosol to convert the optical equivalent sizes to true

sizes, as was shown in Figure 3.9 (d). For the Lasair PHA data, kernel functions are

strongly dependent on atmospheric aerosol refractive indices. Therefore, kernel functions

corresponding to the true refractive index should be used to invert the PHA data.

The true refractive index can be obtained in two ways: from our hourly OPCs

calibration data and from the chemical species mass concentration data measured by

other groups in this project. Dick [28] has shown how to obtain refractive index from

measured compositions.

71

4.1.2.1 Refractive Index Calculation Using Hourly Calibration Data

Refractive index can be obtained from calibration data by assuming 450nm

atmospheric particles are homogenous, non-light absorbing spheres. The OPC responses

to 450nm particles as a function of refractive index are obtained using Mie theory, as was

discussed earlier. This response-refractive index relationship is shown in Figure 4.2.

From the St Louis hourly calibration PHA data, I can obtain the 450nm atmospheric

aerosol peak response. By comparing this response to the curve in Figure 4.2, the

refractive index of the 450nm aerosol sampled at that time is obtained.

y = 4.8868x - 5.8351R2 = 0.9995

0.5

1.0

1.5

2.0

1.30 1.35 1.40 1.45 1.50 1.55 1.60

refractive index

resp

onse

(V)

Figure 4.2 Theoretical Lasair responses to 450nm particles

with different refractive indices

An example of the Lasair hourly calibration PHA data of 450nm St. Louis

atmospheric aerosol is shown Figure 4.3. In this chart, there is a distinct peak around

1.37V. There are also significant particle counts below 0.5V. If it is not cut by the MCA

lower limit discriminator (LLD), there will be a secondary peak in this lower channel

range. These results are typical of our hourly calibration data, which indicates that 450nm

particles often contain external mixtures of particles with varied compositions. Also their

shapes may be different. In this thesis, I name particles that produce the small and large

pulses “dark” particles and “bright” particles, respectively.

72

0

2

4

6

8

0 0.5 1 1.5 2 2.5 3

Response Voltage (V)

dn/d

V (#

/cc)

Figure 4.3 Lasair pulse height distributions of 450nm

St. Louis atmospheric aerosol calibration

For the bright particles, since the peak response voltage is 1.37V, we can read

from Figure 4.2 that the refractive index is about 1.48. But for the dark particles, if we

still assume they are spherical and non-light absorbing, the refractive index will be much

lower than water, which is not realistic for materials likely to be found in atmospheric

aerosols. Therefore, our assumption that particles are non-absorbing spheres does not

work for these particles.

Since the dark particles scatter much less light, we suspect that they are made up

of elemental carbon (EC) and they are not spherical. I measured 450nm diesel soot

emitted by the John Deere Engine in the Center for Diesel Research Facilities in our

department. The Lasair PHA distribution for both the 450nm diesel soot and St. Louis

calibration data are shown in Figure 4.4. The calibration data is the sum of 48 hours’

calibration data for July 8th and 9th, 2001.

bright particles

dark particles

73

1

10

100

1000

0 1 2 3

Response Voltage (V)

Cou

nts

450nm diesel sootSt. Louis calibration

Figure 4.4 Comparison of 450nm diesel soot particles and St. Louis

hourly calibration data

For the data shown in Figure 4.4, the counting efficiency of diesel soot particles is only

5%, while the counting efficiency for the dark particles is about 60%⑧. This means that

the Lasair response to diesel soot is lower than that to the dark particles calibrated in St.

Louis. However, there is no reason that these two aerosols should have the same optical

properties. Even if the dark particles were originally soot particles, they had experienced

transformations in the atmosphere before they were measured. These transformations

could affect both physical and chemical properties of the particles.

In addition, we compared the dark particle number concentration to the elemental

carbon mass concentration to see whether they are correlated. The EC concentration was

measured with the second-generation continuous EC/OC analyzer developed by Sunset

Laboratory and operated by Schauer and coworkers at the University of Wisconsin-

Madison. Figure 4.5 shows one week’s comparison data.

⑧ Assume Lasair table data records all 450nm particles, and the counting efficiency for bright particles is 100%, then dark particle counting efficiency = dark particle counts in MCA/(total counts in Lasair table data-bright particle counts in MCA)

74

Dark particle number concentration

0.0

0.5

1.0

1.5

2.0

9/9/01 9/10/01 9/11/01 9/12/01 9/13/01 9/14/01 9/15/01 9/16/01

Date

Conc

entra

tion

[#/c

c]

Figure 4.5 (a) Dark particle number concentration

Elemental carbon mass concentration

0.0

1.0

2.0

3.0

4.0

5.0

9/9/01 9/10/01 9/11/01 9/12/01 9/13/01 9/14/01 9/15/01 9/16/01

Date

Mas

s co

nc (µ

g/m

3)

Figure 4.5 (b) Elemental carbon mass concentration

Note that dark particle concentrations are clearly correlated to elemental carbon

concentrations. This gives us clear evidence that the dark particles are associated with EC.

Dark soot particles are not spherical, and it is impractical to calculate their Mie responses.

Therefore, we cannot calculate the refractive index of these particles with the method we

used for bright particles. To simplify the analysis, I assume that all particles behave like

bright particles. Effects of external mixing will be discussed later.

75

As discussed earlier, it is straightforward to determine the refractive index of

bright particles by comparing the calibrated median response of the bright peak to Figure

4.2. Because the number of 450nm particles counted during a 10 minutes calibration

period may be insufficient to accurately determine the location of the voltage peak,

sometimes we need group several hours’ calibration data to accurately determine the peak

location.

4.1.2.2 Refractive Index Modeling Using Chemical Species Mass

Concentrations

In the St Louis project, the chemical species mass concentrations are measured by

other groups. Particle water content can be determined from atmosphere RH, RH in the

sampling line and particle composition. Therefore, we can estimate refractive index of

atmospheric aerosol using these chemical data and Bill Dick’s refractive index model

[28].

Bill Dick estimated the complex refractive index of particles made up of

ammoniated sulfate, organic carbon, elemental carbon and water using volume-averaged

complex indices of individual species [28]:

∑∑=

i

ii

vnv

n (4.1)

∑

∑=i

ii

vkv

k (4.2)

where

n, k = real and imaginary parts of the mixture refractive index

vi = volume or volume concentration of the ith species

ni, ki = real and imaginary parts of the refractive index of ith species

To estimate the refractive index of atmospheric aerosols in St. Louis, I made the

following assumptions:

1. Particles are composed of organic carbon, elemental carbon, sulfate and water. These

are chemical species measured in St. Louis. Nitrate data is only available for selected

periods, and it is not included in this modeling. Nitrate concentrations tend to be

76

higher in the winter, and it would be important to include nitrate when estimating

refractive index during the winter. The dry species properties used in Bill Dick’s

study are listed in Table 4.2. Since refractive indices are wavelength dependent, and

the values listed in Table 4.2 are valid for λ = 488nm, we need to adjust refractive

indices to Lasair wavelength (633nm) based on molar refractions [28]. Furthermore,

these data were based on Dick’s measurements in the Southeastern Aerosol and

Visibility Study (SEAVS) in 1995. The aerosol properties measured in St. Louis are

different from that study. More assumptions are used to best estimate St. Louis

aerosol properties.

Table 4.2 Dry species properties used in Dick’s refractive index modeling [28]

Species Mass CCF⑨ Density (g/cm3) n⑩ k

OC 2.1 1.4 1.46 0.0

EC 1.0 1.9 1.93 0.66

H2SO4 1.021 1.841 1.425 0.0

NH4HSO4 1.198 1.805 1.494 0.0

(NH4)3H(SO4)2 1.287 1.787 1.520 0.0

(NH4)2SO4 1.376 1.769 1.542 0.0

H2O 1.0 1.0 1.337 0.0

NH4NO311 1.290 1.725 1.55412 0.0

2. OC is not hygroscopic at any RH. Dr. Dick found that OC was mildly hygroscopic in

the Southeastern Aerosol and Visibility Study. However, he also found that OC

absorbed significantly less water than sulfate on a volume basis, as is shown in Figure

4.6. Therefore, assuming that OC does not absorb water is not a very bad assumption.

Since organic species are very complicated, we still use Dick’s mass CCF value of

2.1 and density value of 1.4 g/cm3. But the refractive index of OC is adjusted to fit

best with our measured refractive index. This will be discussed later in this chapter.

⑨ The component conversion factor (CCF) is the multiplicative factor that converts OC to organics or sulfate to a salt of ammonium and sulfate. ⑩ These are refractive index values at 488nm wavelength. 11 NH4NO3 was not used in Bill Dick’s model. These data were provided by (Tang, I, 1996). 12 at wavelength = 0.58µm

77

Figure 4.6 Comparison of hydration curves for several chemical species [28]

3. Elemental carbon does not appear in the bright particles. As discussed in the previous

section, EC is an important species in dark particles. Here I assume that all EC exists

in those dark particles.

4. Sulfates are present as ammonium sulfate [(NH4)2SO4]. Theoretically, sulfate will be

present as a mixture of sulfuric acid, ammonium bisulfate and ammonium sulfate, etc.,

depending on the ammonium-to-sulfate molar ratio. Ammonium data was measured

in the St. Louis study, but they were not available to me yet. To simplify calculations,

I assume all sulfate species are (NH4)2SO4. The mass CCF of SO4 to (NH4)2SO4 is

1.376, and the refractive index is 1.534 when λ = 633nm.

5. According to the above assumptions, only (NH4)2SO4 is hygroscopic. Therefore,

water volume can be calculated from (NH4)2SO4 water uptake properties. According

to Tang et al. [35], (NH4)2SO4 deliquesces at 80% relative humidity (RH) when the

RH is increased for dry crystalline particles and crystallizes (“effloresces”) at about

37% RH when the RH is decreased. Figure 4.7 shows the phase change of (NH4)2SO4

as a function of RH, where m0 is the mass of particles at 0% RH. Bill Dick fitted the

anhydrous (NH4)2SO4 solute mass fraction as a function of water activity to a fifth-

order polynomial [28]:

78

5432 700.3748.11856.147400.90894.272646.4% wwwww aaaaawt −+−+−= (4.3)

where

wt% = weight percent of (NH4)2SO4 in the solution

aw = water activity (approximate equals to RH), 0.392≤ aw ≤1.000

The fitted curves of both particle growth and evaporation are shown in Figure 4.8. I

compared the atmosphere RH and the RH in the sampling line during each

measurement. If the outside RH was lower than the sampling line RH, the equilibrium

curve was used to calculate the weight percent of (NH4)2SO4. Otherwise, the

metastable curve was used.

Now we have all parameters needed for refractive index calculation from

chemical species data except the OC refractive index. Using the method discussed in the

next section, I decided to use 1.483 as the OC refractive index.

Figure 4.7 Phase change of (NH4)2SO4 as a function of RH [35], illustrating the

hysteresis that occurs when dry particles are humidified or wet particles are dehumidified

equilibrium curve

metastable curve

deliquescence point

efflorescence point

79

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

relative humidity

wei

ght p

erce

nt o

f (NH

4)2S

O4

Figure 4.8 (NH4)2SO4 mass fraction as a function of RH

To summarize the discussion above, the way I model the real component of the

refractive index of atmospheric aerosol measured in St. Louis is:

OHSONHOC

OHOHSONHSONHOCOC

i

ii

vvvnvnvnv

vnv

n2424

22424424

)(

)()(

++

×+×+×==

∑∑ (4.4)

where

OCOCOCOC CCFmv ρ/×= (4.5)

42444424 )()( / SONHSOSOSONH CCFmv ρ×= (4.6)

OHSONHSONH

SOSOOH wt

wtCCFm

v2424

424

44

2/)%1(

% )()(

ρ−××

= . (4.7)

The corrected dry chemical species properties used in this study are given in

Table 4.3.

Table 4.3 Dry species properties used in refractive index modeling of St. Louis

atmospheric aerosols

Species Mass CCF Density (g/cm3) n13 k

OC 2.1 1.4 1.483 0.0

(NH4)2SO4 1.376 1.769 1.534 0.0

13 These are refractive index values at 633nm wavelength.

Increasing RH (equilibrium)

Decreasing RH(metastable)

80

H2O 1.0 1.0 1.332 0.0

4.1.2.3 Results of Refractive Index Calculation and Modeling

As mentioned in the previous section, properties of OC in atmospheric aerosols

vary with time and location, and we have no way of calculating OC refractive index

precisely [28]. In this work, I calculate the OC refractive index by plotting the measured

refractive index based on hourly calibration as a function of OC/ SO4 mass ratio. Ideally,

according to our model, when particles are dry (RH below crystallization point of

(NH4)2SO4, i.e.39.2%) and OC/ SO4 mass ratios are large, the measured refractive index

will approach the OC refractive index. Figure 4.9 shows the measured refractive index in

September 2001 as a function of OC/ SO4 mass ratio. From this chart, we see that dry

particles converge towards a refractive index value around 1.48 as OC/ SO4 mass ratio

increases.

Measured refractive index in 0901

1.351.371.391.411.431.451.471.491.511.53

0.1 1 10 100

OC/SO4 mass ratio

refr

activ

e in

dex

RH 0-39.2RH 39.2-50RH 50-100

Figure 4.9 Measured refractive index as a function of OC/SO4 mass ratio and RH

By using the assumed OC refractive index value of 1.483 in Equation 4.4, we can

calculate the atmospheric aerosol refractive index during the same period using the

chemical species mass concentration data. The result is shown in Figure 4.10. Figure 4.11

shows a scatter plot comparison of measured and modeled refractive index in the same

81

period, and Figure 4.12 shows the difference between measured and modeled refractive

index values.

modeled refractive index in 0901

1.381.4

1.421.441.461.481.5

1.521.54

0.1 1 10 100

OC/SO4 mass ratio

refra

ctiv

e in

dex

RH 0-39.2RH 39.2-50RH 50-100

Figure 4.10 Modeled refractive index as a function of OC/SO4 mass ratio and RH

(assuming OC refractive index = 1.483)

1.35

1.4

1.45

1.5

1.55

1.35 1.4 1.45 1.5 1.55measured refractive index

mod

eled

ref

ract

ive

inde

x

RH 0-39.2RH 39.2-50RH 50-100

Figure 4.11 Scatter plot comparison of modeled and measured refractive index for

September 2001 (assuming OC refractive index = 1.483)

82

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.1 1 10 100

OC/SO4 mass ratio

mea

sure

d n

- mod

eled

n

RH 0-39.2RH 39.2-50RH 50-100

Figure 4.12 Difference between measured and modeled refractive index as a function of

OC/SO4 mass ratio and RH for September 2001 (assuming OC refractive index = 1.483)

Note that in Figure 4.12, the difference between the measured and modeled refractive

index is about ±0.06. When RH is high (50-100%), modeled refractive indices are higher

than measured values. This may be because OC is also hygroscopic, or because

ammonium-to-sulfate ratios are not equal to 2, as was assumed. However, we have no

way to justify other assumptions based on the information that was measured in St. Louis.

Since there are many mysteries regarding particle’s chemical properties, we have

more confidence in the measured refractive index than the calculated value. Furthermore,

the refractive index obtained from hourly calibration is what the Lasair really sees. Hence

measured refractive indices are used in our data analysis. Figure 4.13 shows the

September atmospheric aerosol refractive indices obtained by Lasair hourly calibration in

St. Louis. And Figure 4.14 shows the dependence of the measured refractive index on

relative humidity.

83

Measured refractive Index (real part) 09/2001

1.341.361.381.4

1.421.441.461.481.5

1.521.54

9/1/01 9/8/01 9/15/01 9/22/01 9/29/01 10/6/01

Date

Refr

activ

e In

dex

Figure 4.13 Measured refractive indices in St. Louis in September, 2001

Measured refractive Index vs. RH in 0901

1.351.371.391.411.431.451.471.491.511.53

10 20 30 40 50 60 70 80 90

RH(%)

refra

ctiv

e in

dex

Figure 4.14 Measured refractive indices in St. Louis in September, 2001 as a

function of relative humidity

The uncertainty of the measured refractive index comes from three major factors:

• We only calibrated 450nm aerosol; the refractive indices of other sizes may be

different.

• For now, we can calculate the refractive indices of bright particles with reasonable

accuracy, but we know that dark particles are also present. Some error will be

84

introduced if we use the refractive index of the calibrated 450nm bright particles only

to obtain size distributions.

• When the calibrated particle concentration is very low, I grouped up to 24 hours’

calibration to obtain enough counts. If the number of grouped hours is very large, the

calculated refractive index is averaged over that many hours, not for that specific hour.

The worst condition happens when, even if I group 24 hours, insufficient counts were

obtained. Thus, time resolution was lost.

4.1.3 Lasair Size Distribution Analysis

As discussed in Chapter 3, four methods were used to obtain Lasair size

distributions: table data recorded by Lasair, table data corrected by refractive index,

traditional channel to diameter conversion of Lasair PHA data and Twomey inversion of

Lasair PHA data. Figure 4.15 shows the size distributions obtained using these four

methods for the same atmospheric aerosol measurement. Also shown in Figure 4.15 is the

SMPS size distribution measured at the same time.

7/7/2001 12:16:04 PM

85

Figure 4.15 Comparison of SMPS and Lasair size distribution for St. Louis measurement

First thing to notice in Figure 4.15 is that the Lasair distributions obtained by the

four methods are pretty close in their common size range. But we need have a close look

at each of them.

The original table data is obtained directly from the Lasair without adjusting for

refractive index. The size bins were decided by the manufacturer according to PSL

calibrations. Hence these sizes are PSL optical equivalent sizes. Because the refractive

indices of atmospheric particles differ from those of PSL, we should do refractive index

correction to this original table data to obtain mobility size distributions.

The corrected table data takes the refractive index difference into consideration.

Size limits of the table bins are changed to mobility size according to theoretical response

of PSL and atmospheric aerosol with refractive index calculated from hourly calibrations.

That is:

mobility equivalent size = correction factor ×PSL optical equivalent size,

where the correction factors depend on size and refractive index. Typically, these

correction factors are between 1.0 and 1.5. There are four major problems related to the

corrected table data.

• First, because the table data has only 8 size bins, the resolution is very low.

• Second, it is difficult to do refractive index corrections for big size bins due to Lasair

multi-valued responses. Figure 4.16 shows the response curves for both PSL (n=1.59)

and atmospheric aerosols (n = 1.506). We can see that when particles are smaller than

1.1µm, both PSL and atmospheric aerosol responses increase monotonically, and PSL

responses are always higher than the measured aerosol. But for particles above

1.1µm, oscillations occur in both response curves. In this size range, the two curves

are intertwined and particles with different sizes can produce the same response. This

makes it difficult to find the mobility size for 2µm optical equivalent size. The

uncertainty in estimating 2µm mobility size leads to ambiguities in the last two size

bins of the corrected table data.

86

• Third, only one atmospheric aerosol refractive index based on calibrated 450nm bight

particles was used for this correction. We did not account for the known presence of

dark particles.

• Fourth, it is difficult to do counting efficiency corrections for the Lasair table data.

0

5

10

15

20

0 500 1000 1500 2000

Dp (nm)

Lasa

ir re

spon

se (V

)

n=1.455n=1.59

Figure 4.16 Lasair theoretical responses to PSL and atmospheric aerosol

The advantage of Lasair table data is that it includes both high and low gain

information, while the MCA data discussed below only covers part of the low gain size

range.

The inverted Lasair-PHA data has three advantages. First, it inherently takes the

refractive index and counting efficiencies into consideration by employing kernel

functions. Therefore, it directly provides mobility sizes and true size distributions.

Second, it has a higher resolution. For the curve shown in Figure 4.15, the resolution is

100 size bins. Third, it considers the broadening response of OPC-PHA system to

monodisperse particles, and uses kernel functions to invert the pulse height distribution.

Ideally, this gives a more precise size distribution. However, the inverted distribution also

has three major problems:

• First, as we can see from Figure 4.15, the inverted size distribution does not do a

good job for very small and very large particles. Part of the reason for this was

discussed in section 3.3.2. I did one more numerical experiment to investigate the

reliable reconstruction range. To simulate the measured pulse height distributions, I

87

assumed a lognormal distribution with parameters of (1E6, 250, 1.5) and inverted the

synthetic pulse height distribution. Figures 4.17 and 4.18 show the inverted result, the

relative reconstruction error and the modeled counting efficiency. From these charts,

we found that the reliable reconstruction range (error = ±10%) corresponded to the

size range where the counting efficiency was around 1.0. For this case the good size

range is about 300nm to 1300nm.

• Second, when counts in each channel are too few, the measurement uncertainty is

large. This problem has also been addressed in Chapter 3. One solution is to group

several measurements to obtain enough counts at the expense of losing time

resolution. The other solution is to group data from several MCA channels to obtain

enough counts at the expense of losing size resolution.

• Third, as listed in Table 4.1, the designed Lasair-PHA data size range (PSL

equivalent) is from 300nm to 1000nm. When the refractive index is lower than that of

PSL and the Laser reference voltage goes below than 10V, the PHA size range will

change somewhat. However, the MCA data covers only a portion of the size range

covered by the low gain table data. Therefore, the MCA data truncate the range of

sizes included in the inverted distribution.

Figure 4.17 Original and inverted lognormal distributions

88

Figure 4.18 Relative error and counting efficiency for the inversion in Figure 4.17

The traditional method assumes that particles of a given size will all be classified

into the same channel. Thus we can invert the PHA data by converting MCA channel

numbers to particle sizes based on Mie response calculations. Particles in each channel

are deemed to have the same size. The main problem with this method is that it doesn’t

take the OPC kernel functions into consideration. Also, when oscillations in the Mie

response occur, the size distribution becomes invalid because dlogDp’s are no longer

correct.

By comparing the SMPS data and Lasair data in Figure 4.15, surprisingly, we find

that the original Lasair table data matches the SMPS data best. The table data corrected

by refractive index also approaches the SMPS data at the smaller size end, but it deviates

from the SMPS data trend as sizes become larger. The size distribution obtained by

Twomey inversion wanders between the original and correct table data in the size range

from 0.4µm to 1.3µm with some fine structure. But it is about 3 times higher than the

SMPS data between 0.3µm and 0.4µm. At present, we don’t know the reason for this

discrepancy. Three possible reasons are:

1. We do not fully understand the refractive index of atmospheric aerosols. As presented

earlier, we observed dark particles and bright particles in the St. Louis hourly

calibration of 450nm particles, which suggested that some particles were externally

89

mixed. However, in our size distribution analysis, we only considered the refractive

index of bright particles. In this case, the dark particles are considered smaller than

their true mobility sizes. This causes overestimation of smaller particles and

underestimation of bigger particles. This would cause Lasair data to be systematically

higher than SMPS data in the size range they overlap.

2. There is some systematic error in flow rates of the Lasair and SMPS.

3. The SMPS data does not give correct size distribution at the big size end due to

uncertainties in multiple charge correction. We know that the Lasair table data does

not correctly correspond to the mobility sizes, but it matches the SMPS data very

well. This suggests that there could be some problems with the SMPS data.

Based on our comparison of SMPS and Lasair data obtained using different

methods, we decided to use the refractive index-corrected table data at this time. The

reasons are:

• Compared to the inverted PHA data, the table data covers a wider size range;

• Although the corrected table data has a larger discrepancy with the SMPS data than

the original table data, we cannot ignore the refractive index difference between

atmospheric aerosol and PSL.

4.1.4 Climet Size Distribution

Compared to the Lasair, I did much less work on Climet PHA data analysis

because the amplifier between the Climet and the MCA was improperly designed. The

amplifier was too slow and therefore did not provide linear amplification factors for

particles in the size range of interest to us. We were unable to establish a good fit for

Climet kernel functions. Therefore, for this analysis, Climet table data is used.

Similar to the Lasair table data, the Climet table data should also be corrected by

refractive index. Figure 4.19 shows the calculated and calibrated Climet responses to both

PSL and DOS. We can see that the measured responses follow the theoretical curve well

up to 1µm. However, there is a big bump in the theoretical curve between 1µm and 2µm,

and the responses provided by the manufacturer are much higher in this region.

Furthermore, for particles larger than 1µm, there are oscillations in the response curves

that lead to ambiguities in the refractive index correction. Since the refractive indices for

90

big particles could be quite different from small ones, and because we did not measure

optical properties of large particles, we only did refractive index correction for Climet

table data below 1µm. Figure 4.20 shows the SMPS, original and corrected Lasair and

Climet table data for the same measurement shown in Figure 4.15.

0

10

20

30

40

50

0 500 1000 1500 2000 2500 3000 3500 4000

Dp (nm)

Clim

et r

espo

nse

(V)

PSL theoreticalDOS theoreticalPSL measuredPSL provided by Climet

Figure 4.19 Calculated and measured responses of Climet to PSL and DOS

Figure 4.20 Comparison of size distributions measured by Climet, SMPS and Lasair

7/7/2001 12:16:04 PM

91

Note that the Climet table data without refractive index correction match SMPS data very

well. However, we know that we should do refractive index correction, therefore, the

corrected Climet table data is used. Since the measured counting efficiencies are 35% for

0.3µm PSL, and 100% for 0.4µm PSL, which are close to nominal values provided by the

manufacturer, no counting efficiency correction is needed.

4.1.5 Summary of OPC Measurements in St. Louis

Two optical particle counters (Lasair and Climet) were used in the St. Louis

atmospheric aerosol measurement project. Since the OPCs were calibrated with PSL

spheres, and atmospheric aerosols have different refractive indices from PSL, we need to

use the true refractive indices to obtain mobility size distributions.

It’s a great challenge to obtain precise values for the refractive indices of

atmospheric aerosols because of the chemical complexity of these particles. I tried two

methods: one was based on OPCs hourly calibration for 450nm aerosols, and the other

was using chemical mass concentration and Bill Dick’s volume-averaging refractive

index model [28]. Comparison of results from these two methods showed that the

agreement was only fair (±0.06 refractive index discrepancies). The main reason was that

many properties of chemical species such as OC were highly uncertain. Therefore, we

used the refractive index calculated from hourly calibration in our data analysis. However,

we found that some particles were externally mixed, and they had totally different optical

properties. In this work, we only used the refractive index of those bright particles in

calculating size distribution. This led to inherent uncertainty in the inverted distribution.

OPC size distributions obtained in different ways were compared to SMPS

distributions. We found that the original table data provided by the instruments matched

the SMPS data best among all of these methods. We know, however, that the table data

provide PSL optical equivalent sizes instead of mobility sizes. Therefore, it is surprising

that the data do not agree better after they are corrected for refractive index. The

discrepancy between the corrected table data, Twomey inversion of Lasair data and the

SMPS data may be also caused by external mixing. Since we did not see any clear

advantage of Twomey inversion and because the corrected table data covered a wider

size range, we decided to use the corrected table data for both the Lasair and Climet.

92

4.2 Diesel Exhaust Measurements

In this experiment, Kihong Park and I measured number and mass distributions

for diesel exhaust aerosols. The experiment was carried out in the Center for Diesel

Research Facilities in our department. All OPC measurements were done on the John

Deere 4045 Engine with EPA (400ppmS) fuel. Figure 4.21 shows the schematic diagram

of this experiment.

Diluted diesel exhaust

Lasair

SMPS /DMA-APM

MOUDI Figure 4.21 Schematic diagram of diesel exhaust measurement setup

Before the diesel exhaust reached the sampling instruments, the Variable

Residence Time Dilution System (VRTDS) [36] was employed to simulate diesel exhaust

mixing into atmosphere. The VRTDS had two separate dilution stages with well-defined

dilution factors.

Particle size distributions were measured by the optical particle counter (OPC)

and Scanning Mobility Particle Sizer (SMPS). The SMPS covered the size range from

20nm to 500nm. Because the aerosol concentration above 300nm was low, only the

Lasair high gain was used. The Lasair PHA data covered 110nm to 350nm, which was

dependent on refractive index and shape of particles and laser reference voltage (LRV).

Using the DMA-APM technique developed by Professor McMurry et al. recently [37],

Kihong Park measured masses of individual particles as a function of mobility diameter.

The mass distribution was obtained by multiplying the number distribution by the

measured size-dependent particle mass. Besides the OPC and SMPS, a Micro Orifice

Uniform Deposit Impactor (MOUDI) was connected in parallel to measure mass

distributions directly as a function of aerodynamic diameter. McMurry et al. [37] showed

that the DMA-APM could be used to estimate the relationship between mobility diameter

93

and aerodynamic diameter. This information facilitates a direct comparison of mass

distributions obtained using the three different instruments.

As mentioned earlier, diesel soot particles are chain agglomerates and light

absorbing. They have totally different optical behaviors from PSL and DOS. The

calibration results and kernel functions of these monodisperse diesel particles have been

presented in Chapter 2. Figure 4.22 compares the inverted Lasair number concentration

with that measured by SMPS. Note that the discrepancies between these two curves are

±200%. This discrepancy in number concentration leads to even larger discrepancies in

mass distributions. Figure 4.23 compares the mass distributions measured with SMPS,

OPC and MOUDI. Note that the SMPS and MOUDI concentration matches reasonably

well, but the OPC mass concentration differs from the other two by a factor of 2. So far I

haven’t figured out the reason for this discrepancy.

Diesel engine number concentration

0

5000

10000

15000

20000

25000

10 100 1000mobility diameter (nm)

dn/d

logD

p (#

/cc)

SMPSOPC

Figure 4.22 Diesel exhaust number concentration measured by SMPS and Lasair

02/22/2002

94

Diesel engine mass concentration

0

2000

4000

6000

8000

10000

12000

10 100 1000

m obility diam e te r (nm )

dm

/dlo

gD

p (

ug

/m3 )

MOUDI data (mobility s ize)SMPSOPC

Figure 4.23 Diesel exhaust mass concentration measured by SMPS, Lasair and MOUDI

4.3 Summary

This chapter focuses on the application of optical particle counters (OPCs) for

atmospheric and diesel exhaust aerosol measurements.

Two methods were used to estimate the refractive indices of atmospheric aerosols.

One was based on OPC hourly calibration and Mie response calculation; the other

involved calculating the refractive index from the measured chemical species mass

concentration using the volume-averaging model developed by Bill Dick. From the OPC

hourly calibrations, we found that particles with the same mobility size had different

optical properties, suggesting that they were externally mixed. More study showed that

the abundance of dark particles was well correlated with EC mass concentrations.

Because these particles were not spherical, it was difficult to analyze their optical

behaviors. Therefore, I estimated the refractive indices of bright particles in this work.

The refractive index model used organic carbon, sulfate and water mass concentrations.

Sulfate was assumed to be present as ammonium sulfate, which was assumed to be the

only hygroscopic species. The refractive indices obtained from these two methods had

discrepancies of ±0.06. In pervious work using multiangle light scattering, Dick found

that measured and calculated refractive indices agreed to within ±0.02. I believe the

major reason for the large discrepancy between measured and calculated refractive

02/22/2002

95

indices in this thesis is that properties of chemical species such as OC were not

adequately characterized. Therefore, I used the refractive index of bright particles

calculated from hourly calibrations in data inversion.

The Lasair size distributions obtained by four methods were discussed and

compared to the SMPS data. I found that the original OPC table data without any

correction matched SMPS data best. However, to obtain mobility size distributions, the

original table data should be corrected by refractive index. The Twomey inversion of

Lasair-PHA data provided some fine structure, but it had a bigger discrepancy relative to

the SMPS data. Compared to corrected table data, the advantages of Twomey inversion

discussed in Chapter 3 were not apparent here. The reason is still a mystery to me.

In the diesel experiment, both Lasair and SMPS were used to measure number

distributions. A DMA-APM system was used to measure particles mass. The mass

distribution was obtained by multiplying number distribution by the measured mass per

particle. A MOUDI was also used to measure mass distribution directly. We found that

the SMPS and MOUDI mass distributions matched reasonably well, but the discrepancy

between OPC and SMPS was as large as ±200%.

96

Chapter 5: Conclusions and Suggestions for Future Work

5.1 Conclusions

This thesis focuses on application of the optical particle counter-multichannel

analyzer (OPC-MCA) method for measurements of aerosol size distributions. The

content and conclusions of this work are summarized as follows:

1. Lasair kernel functions for spherical particles with arbitrary refractive index can

be predicted using Mie theory with reasonable accuracy. The shape of kernel

functions was found to be lognormal according to my calibration with

monodisperse PSL spheres. The variability in particle size for DMA classified

particles was found to lead to a significant broadening in the measured Lasair

kernel functions. Because PSL spheres are more nearly monodisperse than aerosol

classified by a DMA from a polydisperse input, standard deviations of PSL

kernel functions were used for kernel functions for particles with different

refractive indices. Theoretical calculations showed that Mie responses matched

calibrated perk voltages very well for PSL and DOS. Therefore, peak voltages of

kernel functions for spherical particles with arbitrary refractive index could be

obtained from Mie response calculations. Using these standard deviations and

peak voltages, kernel functions with lognormal shape can be obtained.

Because Mie theory does not apply for non-spherical particles, kernel functions of

such particles totally depend on calibration.

2. The STWOM algorithm was adapted to invert the pulse height distributions

measured by the OPC-MCA. Numerical experiments showed that this algorithm

could unravel size distributions from pulse height distributions very well. The

MCA has about 2000 channels, so it provides much higher resolution than the

OPC table data. Furthermore, the inversion method employs kernel functions and

corrects for the effect of refractive index automatically. The inverted result is the

mobility size distribution. However, the table data only reports PSL optical

97

equivalent size distributions. Refractive index corrections are required to obtain

mobility sizes.

3. A Fortran program was written to calculate kernel functions and to invert pulse

height distributions. This program was applied to analyze St. Louis atmospheric

aerosol measurements. Particles concentrations obtained by the STWOM

inversion were found to be about 3 times higher than those measured by the

SMPS. Big discrepancies between OPC-PHA and SMPS were also found when

this algorithm was applied to diesel exhaust measurements. I was unable to

identify the reason for these discrepancies.

5.2 Recommendations for Future Work

• Design logarithm amplifiers: The Lasair high gain covers 0.1 to 0.2µm, and low

gain covers 0.3 to 2.0µm as was shown in Figure 2.6. There is a gap between

these two gains, i.e., we cannot record pulse heights for particles in the size range

of 0.2 to 0.3µm with the current circuit design. Furthermore, because the size

range covered by the Lasair high gain is very narrow, and the Twomey inversion

does not give good results at both ends of the inverted size range, it is very

difficult to obtained good inversion results for high gain. One way to solve this

problem is to build a middle gain amplifier that connects the responses with both

high and low gains. However, this design would require three MCA cards to

record the whole size range. Alternatively, a logarithmic amplifier can be used.

Figure 5.1 shows the Mie response to PSL for the high gain of the South Pole

Lasair. Note that the response in the size range of 100nm-2000nm covers 5

decades. Because the MCA input voltage range is 0 to 10V, we need to take the

logarithm of the response. A detailed design of logarithmic amplifier needs to be

studied.

98

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

0 500 1000 1500 2000

Dp (nm)

Lasa

ir R

espo

nse

(V)

Figure 5.1 Mie responses to PSL for the South Pole Lasair high gain

• Evaluate SMPS-OPC-PHA technique: As stated earlier, there were large

discrepancies between the SMPS and OPC distributions. One way to investigate

this problem would be to set up an SMPS-OPC-PHA size distribution

measurement system. The OPC would be used instead of using a CPC as the

particle detector, and the MCA would be used to record the pulse height

distribution. If the measured aerosol consisted of spherical particles with known

refractive index, we could directly measure the multiply charged particle fractions

after the DMA and thereby test the SMPS inversion algorithm. We could also

compare the SMPS and Lasair size distribution taken in the same run to test

whether the Twomey inversion is working properly. Furthermore, if particles are

externally mixed, we can detect multiple peaks in the PHA data and measure the

size dependent ratios of these particles.

Several technical difficulties would need to be studied before this technique could

be implemented: First, since atmosphere aerosol concentrations for big sizes are

very low and sampling losses are high, the sampling time should be increased to

achieve statistical significance. Then the time resolution will be low. One solution

would be to choose an OPC with a flow rate higher than the Lasair 1002. Second,

external mixing and multiple charging may add complexities when interpreting

the MCA spectra. More work would be required to interpret the MCA pulse

99

height distribution. Third, this experiment requires the Lasair-MCA to cover a

wider size range, which would require a logarithmic amplifier as discussed earlier.

• Modify the PSD system: If the SMPS-Lasair-PHA system discussed above is

carefully evaluated and found to be work well, it could be used in atmospheric

field studies. In our present St. Louis PSD measuring system, the SMPS, Lasair,

and Climet were used to measure size distributions. Since our goal is to measure

size distributions based on mobility under certain temperature and RH, the best

way would be to use the SMPS to cover a size range as wide as possible. One of

the reasons that the SMPS was not used to measure larger sizes is that it is too

complicated and less accurate to do correction for multiple charges. But if the

SMPS-Lasair-PHA technique were to solve the problem of multiple charge

corrections, we could extend the SMPS to measure larger sizes. Actually, the

SMPS and Climet could cover the size range of the Lasair. If we were to add one

more SMPS system and use the Lasair as a detector, we could obtain more

additional information. First, we could still obtain size distributions from the

Lasair, though the time resolution may be lower. Second, it will provide

information about external mixing and particle refractive index with good size

and time resolution. Third, we could study particle water uptake properties by

changing the sampling RH.

• Study the temperature difference in and out of the Lasair: Since temperature

affects RH, and RH affects particle water content, the actual temperature inside

the Lasair is needed to calculate particle water content and refractive index.

Laboratory generated pure ammonium sulfate particles can be used to test

whether there is any significant optical property change at controlled temperature

and RH. Then we can tell whether there is significant temperature change inside

the Lasair.

• Change RH setting in atmospheric aerosol measurements: In the St. Louis

atmospheric aerosol measurements, we controlled our RH at 40%, which was

very close to the crystallization point of (NH4)2SO4 (37-40% [35]). Slight

fluctuation in the RH will cause large refractive index differences, which is not

100

ideal for OPC measurements. Table 5.1 lists the deliquescence RH (RHD) and

crystallization RH (RHC) of some important components of atmospheric aerosols

[38].

Table 5.1 Thermodynamic properties of several chemical species [38].

Chemical RHD (%) RHC (%)

NH4HSO4 40 20-0.05

(NH4)3H(SO4)2 69 44-35

(NH4)2SO4 80 40-37

NH4NO3 62 25-32

I suggest controlling RH at 50%, so that the water content of every chemical

species is well defined, and small fluctuations in RH will not cause big difference

in the particle refractive index.

• Do more studies on the Climet responses: In this work, only the Climet high

gain responses to PSL were studies, and no good analytical model had been found

to describe the Climet kernel functions. A careful study of Climet kernel functions

and the pulse height distribution inversion is needed.

101

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Airborne Particles. 2nd Edition, Los Angeles, California: A Wiley-Interscience

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13. Gupta, A. and P.H. McMurry, (1989) "A Device for Generating Singly Charged

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(GMWDMA) (PTL report)

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18. PMS, (1990) LASAIR User's Guide to Operations

19. PMS, (1991) LASAIR Technical Service Manual

20. Climet, (1999) Spectro.3 Laser Particle Spectrometer Operation Manual

21. ORTEC, E.G., (1998) TRUMP -8K/2K Multichannel Buffer Card Hardware Manual

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104

Appendix A: OPC Calibration Results This Appendix lists all calibration results for the two Lasairs and the Climet: peak

voltages, geometric standard deviations and counting efficiencies. The counting

efficiencies plotted here are the initial measured values. The effect of multiply charged

particles has not been taken into account.

A.1 South Pole Lasair High Gain

SP Lasair high gain responses

0

2

4

6

8

10

12

100 150 200 250 300

mobility diameter (nm)

peak

vol

tage

(V)

PSLDOSNaCldiesel 50% load

Figure A.1.1 SP Lasair high gain responses to different particles

105

pulse heigh distribution σg of SP Lasair high gain

1

1.2

1.4

1.6

1.8

100 150 200 250 300

mobility diameter (nm)

σg

PSLDOSNaCldiesel 50% load

Figure A.1.2 Geometric standard deviations of pulse height distributions

for the SP Lasair high gain

SP Lasair high gain peak responses for diesel soot

0

1

2

3

4

5

120 170 220 270 320

mobility diameter (nm)

peak

res

pons

e (V

) 10% load50% load75% load

Figure A.1.3 SP Lasair high gain responses to diesel soot at different engine loads

106

Pulse height distributions σg of diesel soot at different engine loads

1.4

1.5

1.6

1.7

120 170 220 270 320

mobility diameter

σ g

10% load50% load75% load

Figure A.1.4 Diesel soot geometric standard deviations at different engine

loads for the SP Lasair high gain

SP Lasair high gain counting efficiencies for PSL

0

20

40

60

80

100

100 120 140 160 180 200 220

mobility diameter (nm)

coun

ting

effic

ienc

y (%

)

MCATable

Figure A.1.5 SP Lasair high gain counting efficiencies for PSL

107

SP Lasair high gain counting efficiency for DOS

0

20

40

60

80

100

80 120 160 200 240

mobility diameter (nm)

coun

ting

effic

ienc

y (%

)

MCAtable

Figure A.1.6 SP Lasair high gain counting efficiencies for DOS

SP Lasair high gain counting efficiency for NaCl

0102030405060708090

100

80 120 160 200 240

mobility diameter (nm)

coun

ting

effic

ienc

y (%

)

MCAtable

Figure A.1.7 SP Lasair high gain counting efficiencies for NaCl

108

SP Lasair high gain counting efficiency for diesel soot at 50% engine load

0

20

40

60

80

100

120

80 120 160 200 240 280 320

mobility diameter (nm)

coun

ting

effic

ienc

y (%

)

MCAtable

Figure A.1.8 SP Lasair high gain counting efficiencies

for diesel soot at 50% engine load

A.2 South Pole Lasair Low Gain

SP Lasair low gain responses

0

2

4

6

8

10

12

300 600 900 1200

mobility diameter (nm)

peak

resp

onse

(V)

PSLDOSNaCl

Figure A.2.1 SP Lasair low gain responses to different particles

109

Pulse height distribution σg of SP Lasair low gain

1

1.1

1.2

1.3

0 200 400 600 800 1000 1200

mobility diameter (nm)

σg

PSLDOSNaCl

Figure A.2.2 Geometric standard deviations of pulse height distributions

for the SP Lasair low gain

SP Lasair low gain counting efficiencies for PSL

0

20

40

60

80

100

100 200 300 400 500 600 700 800

mobility diameter (nm)

coun

ting

effic

ienc

y (%

)

MCAtable

Figure A.2.3 SP Lasair low gain counting efficiencies for PSL

110

SP Lasair low gain counting efficiency for DOS

0

20

40

60

80

100

120

100 300 500 700 900 1100

mobility diameter (nm)

coun

ting

effic

ienc

y (%

)

MCAtable

Figure A.2.4 SP Lasair low gain counting efficiencies for DOS

SP Lasair low gain counting efficiency for NaCl

0

20

40

60

80

100

120

250 300 350 400 450 500 550

mobility diameter (nm)

coun

ting

effic

ienc

y (%

)

MCAtable

Figure A.2.5 SP Lasair low gain counting efficiencies for NaCl

A.3 St. Louis Lasair Low Gain

111

STL Lasair low gain responses

0

3

6

9

12

15

300 600 900 1200 1500 1800 2100

mobility diameter (nm)

peak

resp

onse

(V)

PSL (03/01)PSL(10/01)DOS(10/01)

Figure A.3.1 STL Lasair low gain responses

Pulse height distribution σg of STL Lasair low gain

1

1.1

1.2

0 500 1000 1500 2000

mobility diameter (nm)

σg

PSLDOS

Figure A.3.2 Geometric standard deviations of pulse height distributions

for the STL Lasair low gain

112

STL Lasair low gain counting efficiencies for PSL

0

20

40

60

80

100

120

200 300 400 500 600 700 800

mobility diameter (nm)

coun

ting

effic

ienc

y (%

)

MCAtable

Figure A.3.3 STL Lasair low gain counting efficiencies for PSL

STL Lasair low gain counting efficiency for DOS

0

20

40

60

80

100

120

200 400 600 800 1000 1200

mobility diameter (nm)

coun

ting

effic

ienc

y (%

)

MCAtable

Figure A.3.4 STL Lasair low gain counting efficiencies for DOS

113

A.4 Climet Low Gain

Median voltage of Climet high gain response to PSL

0

2

4

6

8

10

200 400 600 800 1000 1200

mobility diameter (nm)

med

ian

resp

onse

(V)

Figure A.4.1 Median voltage of Climet high gain response to PSL

0

0.005

0.01

0.015

0.02

0.025

0 500 1000 1500 2000

MCA channel

norm

aliz

ed c

ount

s

404nm

482nm505nm

595nm653nm 701nm

845nm913nm 1099nm

Figure A.4.2 Climet high gain kernel functions

114

Appendix B: Codes for the Twomey Inversion Package

This program is designed to invert Lasair pulse height distribution for spherical

particles using Twomey algorithms. All routines were written and compiled in the

Compaq Visual Fortran environment. The structure of this program and functions of

important routines have been shown in Chapter 3. More description can be found in the

codes. Because the inversion codes were original written for impactor data inversion,

some variable notations refer to impactors.

The non-spherical particle inversion case is not listed here. The user only needs to

change several lines in the subroutine “kernel”: instead of using Mie response

calculation, kernel functions are based on calibration.

Description of important variables:

calcount: calculated counts in each channel calibnum: calibrated points chcnt: inversion resolution difcount: stgcount(i)-calcount(i) dtrial: dN/dlogDp of each sizes error: assumed error in each channel fittol,abstol: tolerance of error fos(i,j):kernel of certain channel and size fosint: f(Dp)dDp gmtstd: geometric standard deviation of all calculated points gmtvlt: geometric mean voltage of all calculated points llmt, hlmt: low and high limit of diameter in consideration mcalibnum: maximum of calibrated sizes mchcnt: maximum resolution ratio: stgcount(i)/calcount(i)-1.0 refvolt: OPC reference voltage refvolt: OPC reference voltage response: scattering cross section stgcnt: total channel number stgcount: counts in each channel trial: counts of each sizes xinc: interval of diameters

Code list

115

program main c********************************************************************** c This is the entrance of this program. It controls the execution routine. If the user c chooses to do numerical experiment, the code calls "ln04" to generate synthesis pulse c height distribution for bimodal distributions, and then it calls "Ti04" to invert the c data. If the user chooses to invert measurement data, the code calls "Ti04" directly c and does Twomey inversion. c**********************************************************************

integer option write (6,1000) 1000 format(1x,'Please select your option: [1 or 2] :' + /1x, '1. Invert numerically generated data' + /1x, '2. Invert measurement data (default)') read (5,1010) option 1010 format (I4) if(option .eq. 1)then call ln04 endif call ti04 end subroutine ln04 c*********************************************************************** c This program generates synthetic pulse height distribution for lognormal c distributions. The user input (amplitude of dn/dlogDp, median diameter, and c geometric standard deviation) of two lognormal distributions (if unimodal c lognormal distribution needed, simply set the amplitude of the second modal c to 0. Then this code calls subroutine "kernel" to calculate kernel functions c and generate ideal pulse height distribution. The first row of the output c file is laser reference voltage (LRV) defined by the user. The original c lognormal distribution is stored in *.lgn, and the pulse height distribution is stored c in the user specified file. c***********************************************************************

implicit double precision (a-h,o-z) integer stgcnt,mchcnt,mcalibnum,lld integer chcnt,ilen parameter (stgcnt=2048) parameter (mchcnt=200)

parameter (mcalibnum =20) parameter (PI = 3.1415926) dimension fos(stgcnt,mchcnt),trial(mchcnt),dtrial(mchcnt) dimension stgcount(0:stgcnt),error(stgcnt) dimension dia(mchcnt) character*12 outfilename

116

character*1 ans, ans0 dimension d50(mcalibnum),gmtvlt1(mcalibnum),gmtstd1(mcalibnum)

dimension gmtvlt(mchcnt),gmtstd(mchcnt) common/d1/calibnum 10 write (6,1000) 1000 format(1x,'Enter resolution [max = 200] : ') read (5,1010)chcnt 1010 format (i3) stgcount(0) = 10.0 call kernel(fos,dia,stgcount(0),chcnt,xinc,gmtvlt,gmtstd,lld) do 20 j=1,12 outfilename(j:j) = ' ' 20 continue write (6,1040) 1040 format (1x,'Enter file name for output : [0=end] ') read(5,1050) outfilename 1050 format (a) if (outfilename(1:1) .eq. '0') goto 9999 ilen = 12 do 30 j = 1,12 if (outfilename(j:j) .eq. '.') ilen = j 30 continue write (6,1060) 1060 format(1x,'Enter zero setting [%] : ') read (5,1070)zerofrac zerofrac=zerofrac/100. 1070 format(f8.3) c parameters for test distribution: a1,d1,sigma1,a2,d2,sigma2 write (6,1080) 1080 format(1x,'Enter a1,d1,sigma1 : ')

read(5,*) a1,d1,sigma1 write(6,*) a1,d1,sigma1 write (6,1090) 1090 format(1x,'Enter a2,d2,sigma2 : ') read(5,*) a2,d2,sigma2 write(6,*) a2,d2,sigma2 c Calculate lognormal distributions xm1 = 0.0 xm2 = 0.0 xmtot=0.0 c ftn: dlog10(di) ftn = log10(xinc) do 40 j=1,chcnt

117

x1=-((log10(dia(j)/d1))**2) x2=-((log10(dia(j)/d2))**2) f1= x1/(2.0d0*log10(sigma1)**2) f2= x2/(2.0d0*log10(sigma2)**2) c these statements are included to avoid floating point underflow in the following c exponentiation if (f1 .lt. -40.0) then x3 = 0.0 else x3=a1*exp(f1) endif if (f2 .lt. -40.0) then x4 = 0.0 else x4=a2*exp(f2) endif xm1=xm1+x3*ftn xm2=xm2+x4*ftn trial(j)= (x3+x4)*ftn 40 continue xmtot=xm1+xm2 write(6,*) xmtot c calculated counts on channels do 50 i=1,stgcnt stgcount(i)=0.0 do 60 j=1,chcnt xvar = fos(i,j)*trial(j) stgcount(i) = stgcount(i)+xvar 60 continue 50 continue do 100 j = 1,chcnt dtrial(j)=trial(j)/ftn 100 continue c write stage data file open(unit=10,file=outfilename) write (6,1120)outfilename 1120 format (1x,'stagedata in : ',a12) do 1130 i=0,stgcnt write(10,1140) stgcount(i) 1140 format(10f18.9) 1130 continue close(10) c modify stage counts to percentage of adjecent channels c check from left to right do 110 l=1,stgcnt-1

118

if (stgcount(l+1) .lt. stgcount(l)*zerofrac) then stgcount(l+1) = stgcount (l)*zerofrac endif 110 continue c check again from right to left do 120 l=0,stgcnt-2 if (stgcount(stgcnt-l-1) .lt. stgcount(stgcnt-l)*zerofrac) then stgcount(stgcnt-l-1)=stgcount(stgcnt-l)*zerofrac endif 120 continue c write estimation file filename.ESM outfilename(ilen+1:ilen+1) = 'e' outfilename(ilen+2:ilen+2) = 's' outfilename(ilen+3:ilen+3) = 'm' open(unit= 7,file=outfilename,ERR=800) write (6,1150)outfilename 1150 format (1x,'esmdata in : ',a12) open (unit = 7,file=outfilename,status='new') do 140 i = 0,stgcnt write (7,1160) stgcount(i) 1160 format(9f9.4) 140 continue close (7) c write distribution file outfilename(ilen+1:ilen+1) = 'l' outfilename(ilen+2:ilen+2) = 'g' outfilename(ilen+3:ilen+3) = 'n' open(unit=11,file=outfilename,ERR=800) write (6,1170)outfilename 1170 format (1x,'logdata in : ',a12) write (11,1180) 1180 format ('dia dN dN/logDp') do 150 i = 1,chcnt write (11,1190) dia(i),trial(i),dtrial(i) 1190 format (3f12.4) 150 continue close (11) goto 9999 800 continue write (6,1200)outfilename 1200 format (1x,'Error opening : ',a12) 9999 continue end subroutine ti04

119

c **************************************************************** c This program was originally written by Hwa-Chi Wang based on Twomey's c inversion routine. Modifications were made by W. Winklmayr(1987-1988). It was c adapted to invert OPC-PHA data in my work. The algorithm used here was c described in [Wolfgang Winklmayr,et.al,1990]. c ****************************************************************

implicit double precision (a-h,o-z) integer stgcnt,mchcnt,mcalibnum parameter (stgcnt = 2048) parameter (mchcnt = 200) parameter (mcalibnum = 20) parameter (PI = 3.1415926) integer chcnt,debug,stgflg,txtflg,ich integer calibnum, ical, llmt, hlmt, lld integer ans0 double precision fos(stgcnt,mchcnt) double precision fittol(stgcnt),abstol(stgcnt) double precision calcount(stgcnt),fosint(stgcnt) double precision difcount(stgcnt),ratio(stgcnt) double precision guess(mchcnt),trial(mchcnt),dtrial(mchcnt) double precision dia(mchcnt),fossum(mchcnt) double precision debres(20,mchcnt),stgcount(0:stgcnt) double precision sqstddev(stgcnt),fosmax(stgcnt),error(stgcnt) double precision d50(mcalibnum),gmtvlt1(mcalibnum) double precision gmtstd1(mcalibnum) double precision gmtvlt(mchcnt),gmtstd(mchcnt) double precision totalsum, sum2000(mchcnt),count2000 double precision fossum2(stgcnt) double precision group(0:2048), gfos(2048,mchcnt) character*1 ans, ans1 character*12 datfilename,outfilename character*12 sdfilename,stagefilename character*12 exfilename,cdummy common/c1/ fos,stgcount,abstol,xexp,xlim,itl,isl,chcnt common/c2/ trial,ratio,difcount,calcount,sigma,fosmax common/d1/ calibnum common/d2/ tcount common/f1/ datfilename c read control parameters from TI.PAR c file TI.PAR holds important parameters for inversion routine. c this file must be in the default directory open (unit=8,file='TI.PAR',status='old',err=9999) write (6,*)'Reading : TI.PAR' c xexp : weighting factor for kernel functions c default value : 0.5

120

read (8,1000)cdummy,xexp write(6,1000)cdummy,xexp c xlim : if kernel functions dropp below this value c integration is stopped c default value : 0.01 read (8,1000)cdummy,xlim write(6,1000)cdummy,xlim c xend : if no error range for stage data is found c this value will be used as end condition c default value : 0.05 read (8,1000)cdummy,xend write(6,1000)cdummy,xend c sigstp: if improvment in sigma is less then specified c value after each twomey loop, iteration is stopped c default value : 1.05 read (8,1000)cdummy,sigstp write(6,1000)cdummy,sigstp c itl : integer count for consecutive twomey inversions. c end conditions will not be checked in this loop c default value : 10 read (8,1010)cdummy,itl write(6,1010)cdummy,itl c isl : integer count for calling inversion loop c end condition is checked every time within this loop c total iterations = itl*isl c default value : 8 read (8,1010)cdummy,isl write(6,1010)cdummy,isl c ismooth : integer flag for Markowsky smmothing c ismooth=0 :no smoothing; =1 :one smooth in each isl loop; default value : 0 read (8,1010)cdummy,ismooth write(6,1010)cdummy,ismooth c ifs : integer value for final smoothing c inverted distribution will be smoothed ifs times c default value : 2 read (8,1010)cdummy,ifs write(6,1010)cdummy,ifs c chcnt : resolution in particle diameter c default value : 100 read (8,1010)cdummy,chcnt write(6,1010)cdummy,chcnt c zero : percentage of mass of adjecent channel that c replaces a zero value ine th raw data c default value : 10. read (8,1000)cdummy,zero

121

write(6,1000)cdummy,zero c txtflg : flag, if set headline will be included in output files. c default value : 1 read (8,1010)cdummy,txtflg write(6,1010)cdummy,txtflg c debug : integer flag. When set files with detailed information is created c default value : 0 read (8,1010)cdummy,debug write(6,1010)cdummy,debug 1000 format(a12,f8.4) 1010 format(a12,i8) close (8) 998 continue c INPUT SECTION stgflg = 1 c input data from file do 40 i=1,12 datfilename(i:i)=' ' outfilename(i:i)=' ' 40 continue 100 continue 110 write (6,1060) 1060 format (1x,'Enter filename of raw data [0 to exit] : ') read (5,1070) datfilename 1070 format (a) if (datfilename(1:1) .eq. '0') goto 9999 ilen =12 do 120 i=1,12 if (datfilename(i:i) .eq. '.') ilen = i 120 continue outfilename(1:ilen) = datfilename(1:ilen) outfilename(ilen+1:ilen+1)='s' outfilename(ilen+2:ilen+2)='t' outfilename(ilen+3:ilen+3)='g' open (unit =8,file = datfilename,status='old') do 130 i = 0,stgcnt read (8,*) stgcount(i) write (6,*) stgcount(i) 130 continue close (8) 140 continue c ADD random error to the counts in channels? write (6,1111) 1111 format(1x,'Please select Random Number option: [1 , 2 or 3] :' + /1x, '1. Percent random number (± n %)'

122

+ /1x, '2. Poisson random number' + /1x, '3. No random number') read (5,1112) ans0 1112 format (I4)

if (ans0 .eq. 1) then call rd04(error) do 88 i=1,stgcnt stgcount(i)=stgcount(i)*(1.0+error(i)/100.0) C stgcount(i)=stgcount(i)+sqrt(stgcount(i))*error(i)/100.0 c stgcount(i)=stgcount(i)-sqrt(stgcount(i)) 88 continue

elseif (ans0 .eq. 2) then call poisson(stgcount(0:stgcnt))

else goto 77 endif 77 continue vinc = 10.0*10.0/2048/(stgcount(0)) call kernel(fos,dia,stgcount(0),chcnt,xinc,gmtvlt,gmtstd,lld) c if standard deviation data are not available c the value of xend is assumed for all stages c except additional stages. do 250 i=1,stgcnt c sqstddev(i)=sqrt(stgcount(i)) + 0.001 sqstddev(i)=xend 250 continue c main loop * write (6,1330)outfilename 1330 format(1x,'creating : ',a12) if (stgflg .eq. 1) goto 2010 open (unit=11,file=outfilename) 2010 continue do 2020 i = 1,stgcnt c fit tolerance & absolute tolerance --absolute c uncertainty for the mass on stage i fittol(i) = sqstddev(i) abstol(i) = fittol(i)*stgcount(i) if (abstol(i) .lt. 0.001) abstol(i) = 0.001 2020 continue c Integrate kernel functions and search maximum do 290 i = 1,stgcnt fosint(i)=0.0 fosmax(i)=0.0 do 290 j = 1,chcnt if (fos(i,j) .gt. fosmax(i)) fosmax(i) = fos(i,j)

123

fosint(i)=fosint(i)+fos(i,j)*dlog(xinc)/dlog(1.0D1) fossum2(i)= fossum2(i)+fos(i,j) 290 continue do 295 j = 1,chcnt fossum(j) = 0.0 do 295 i = 1,stgcnt fossum(j) = fossum(j)+fos(i,j) 295 continue c initial guess distribution do 2030 i = 1,chcnt guess(i) = 0.0 2030 continue open (unit =8,file = 'limit.dat',status='unknown') do 2050 j=1,chcnt c if kernel sum very small, guess=0(for smaller sizes, cut by LLD) if(fossum(j) .le. 1.0e-5) then guess(j) = 0.0 else sum1 = 0 sum2 = 0 llmt = exp(gmtvlt(j)-1*gmtstd(j))/vinc; hlmt = exp(gmtvlt(j)+1*gmtstd(j))/vinc; write(8,*) dia(j),llmt,hlmt if (llmt .le.1) llmt = 1 if (llmt .ge.stgcnt) llmt = stgcnt if (hlmt .ge. stgcnt) hlmt = stgcnt do 2040 i=llmt,hlmt sum1 = sum1 + stgcount(i) c sum2 = sum2 + fosint(i) sum2 = sum2 + fossum2(i) 2040 continue if (hlmt .ge. stgcnt)then c 0.92 is an imperical factor if (((guess(j-1)/guess(j-2)) .ge. 0.92).and. # ((guess(j-1)/guess(j-2)) .le. 1.0).and. # (llmt.ge.stgcnt))then guess(j) = 0.92*guess(j-1) else guess(j) = 2*guess(j-1)-guess(j-2) endif else if (sum2 .le. 1e-12) then guess(j)=0.0 else guess(j)=sum1/sum2

endif

124

endif if(guess(j) .le. 0.0) guess(j) = .000001 2050 continue close (8) tcount=0. do 2060 i=1,stgcnt tcount=tcount+stgcount(i) 2060 continue tgues=0. do 2090 i=1,stgcnt do 2080 j=1,chcnt tgues=tgues+guess(j)*fos(i,j) 2080 continue 2090 continue do 2100 j=1,chcnt ftn = log10(xinc)

open (unit =8,file = 'initial.dat',status='unknown') c save first guess in debug array and initial.dat write(8,*) dia(j), guess(j),guess(j)/ftn debres(1,j) = guess(j) 2100 continue close(8) c group data if counts in each channel are too few, so that each channel has at c least 50 counts write (6,1001) 1001 format(1x,'Do you want to group data: [Y or N]?') read (5,1011) ans1 1011 format (A1) if((ans1 .eq. 'Y') .OR.(ans1 .eq. 'y') )then limit=50 do 13 i = 1,stgcnt

group(i)=0 13 continue i=lld+1 k=lld 11 k=k+1 22 i=i+1 if(i.le.stgcnt)then group(k)=group(k)+stgcount(i) do 14 ich=1, chcnt gfos(k,ich)=gfos(k,ich)+fos(i,ich) 14 continue if(group(k).le.limit)then goto 22 else

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goto 11 endif endif

do 133 i = 1,k stgcount(i)=group(i) do 133 ich=1,chcnt fos(i,ich)=gfos(i,ich) 133 continue do 134 i = k+1,stgcnt stgcount(i)=0 do 134 ich=1,chcnt fos(i,ich)=0 134 continue open (unit =8,file = 'group.dat',status='unknown') do 135 i=1,k write (8,*) group(i) 135 continue close(8) gstgcnt=k else gstgcnt=stgcnt endif c smooth linear interpolated guess to avoid peaks in inversion call smooth(guess,chcnt) call smooth(guess,chcnt) call smooth(guess,chcnt) call smooth(guess,chcnt) call smooth(guess,chcnt) tgues=0. do 2091 i=1,gstgcnt do 2081 j=1,chcnt tgues=tgues+guess(j)*fos(i,j) 2081 continue 2091 continue c normalize tgues to tcount c do 2101 j=1,chcnt c guess(j)=guess(j)*tcount/tgues c2101 continue write(6,1340) 1340 format(1x,'INITIAL GUESS COMPLETED') open (unit =8,file = 'guess.dat',status='unknown')

do 2110 j=1,chcnt trial(j)=guess(j) debres(2,j) = guess(j) write(8,*) dia(j),guess(j),guess(j)/ftn

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2110 continue close(8)

c first twomey inversion call twomey(gstgcnt) open (unit =8,file = 'first.dat',status='unknown') do 2122 i=1,chcnt write (8,*) trial(i) 2122 continue close(8) write (6,*) 'first call completed' c outer iteration loop isl = 30 do 2200 k=1,isl npass=(k+1)*itl n = 0 sigmat=sigma call twomey(gstgcnt) if ( ismooth .eq. 1) call smooth(trial,chcnt) do 2120 i=1,chcnt if (k .le. 6) debres(k+2,i)=trial(i) c debres(k+2,i)=trial(i) 2120 continue open (unit =8,file = 'out.dat',status='unknown') write(8,1350) npass,sigma 1350 format(1x,'Iterations ',i4,' Sigma ',f7.3) write(8,1360) (stgcount(i),i=1,stgcnt) write(8,1370) (calcount(i),i=1,stgcnt) 1360 format(1x,'SM : ',10f7.3) 1370 format(1x,'CM : ',10f7.3) write (8,1380)(ratio(i),i=1,stgcnt) 1380 format(1x,'RT : ',10f7.3) close(8) if (debug .eq. 1) then open (unit=10,file='debug.dat',status='unknown') write (6,*)'Saving debug information' write(10,1350) npass,sigma write(10,1360) (stgcount(i),i=1,gstgcnt) write(10,1370) (calcount(i),i=1,gstgcnt) write(10,1380) (ratio(i),i=1,gstgcnt) endif c skip stopping criterion in case sigma did not improve c stop iteration on this conditiom if (sigmat/sigma .lt. 1.0) goto 1385 if (sigmat/sigma .lt. sigstp) goto 2210 1385 continue

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if (sigma .lt. 0.005) goto 2210 c stop iteration on this conditiom c abs(yi/yij-1) <= error range for all channels, then stop nflag = 1 do 2001 i=1,gstgcnt if (abs(ratio(i)) .gt. fittol(i))then nflag = 0 endif 2001 continue if (nflag .eq. 1) goto 2210 2200 continue 2210 continue if (ifs .eq. 0) goto 2230 c apply final smoothing if specified do 2220 i=1,ifs 2220 call smooth(trial,chcnt) 2230 continue c save results in tables

ftn = log10(xinc) do 2240 i=1,chcnt debres(20,i)=trial(i) dtrial(i)=trial(i)/ftn 2240 continue open (unit =9,file = 'result.dat',status='unknown') write (9,1180) 1180 format ('dia dN dN/dlogDp') do 150 i = 1,chcnt write (9,1190) dia(i),trial(i),dtrial(i) 1190 format (3f16.3) 150 continue close(8) if (debug .eq. 1) write(10,1390) npass 1390 format('Total iterations = ',i3) 9999 continue end c--------------------------------------------------------------- c subroutine response c-------------------------------------------------------------- subroutine respon(gstgcnt) c This subroutine calculates correction ratio and sigma c for a trial distribution, called by Ti04 implicit double precision (a-h,o-z) integer stgcnt,mchcnt parameter (stgcnt = 2048) parameter (mchcnt =200)

128

double precision fos(stgcnt,mchcnt),stgcount(0:stgcnt) double precision trial(mchcnt),ratio(stgcnt),abstol(stgcnt) double precision calcount(stgcnt),difcount(stgcnt) double precision fosmax(stgcnt) integer itl,isl,chcnt common/c1/ fos,stgcount,abstol,xexp,xlim,itl,isl,chcnt common/c2/ trial,ratio,difcount,calcount,sigma,fosmax sigma=0.0D0 do 10 i=1,gstgcnt calcount(i)=0.0D0 do 20 j=1,chcnt c if (fos(i,j) .lt. xlim/1.0e6) goto 20 calcount(i)=calcount(i)+trial(j)*(fos(i,j)) 20 continue c ratio = yi/yij-1 if(calcount(i).lt.1.0e-30) then ratio(i) = -1 else ratio(i)=stgcount(i)/calcount(i)-1.0D0 endif difcount(i)=stgcount(i)-calcount(i) 10 continue do 30 i = 1,gstgcnt sigma=sigma+(difcount(i)/(abstol(i)))**2 30 continue sigma=dsqrt(sigma/gstgcnt) return end c------------------------------------------------------------ c subroutine Twomey c------------------------------------------------------------ subroutine twomey(gstgcnt) c This subroutine calculates new trial distribution c and calcount using Twomey algorithm, called by Ti04. implicit double precision (a-h,o-z) integer stgcnt,mchcnt parameter (stgcnt = 2048) c parameter (stgcnt = 2000) parameter (mchcnt =200) double precision fos(stgcnt,mchcnt),stgcount(0:stgcnt) double precision abstol(stgcnt) double precision trial(mchcnt),ratio(stgcnt) double precision calcount(stgcnt),difcount(stgcnt),fosmax(stgcnt) integer itl,isl,chcnt common/c1/ fos,stgcount,abstol,xexp,xlim,itl,isl,chcnt

129

common/c2/ trial,ratio,difcount,calcount,sigma,fosmax common/d2/ tcount call respon(gstgcnt) do 10 k=1,itl do 20 i=1,gstgcnt a=ratio(i) do 30 j=1,chcnt if (fos(i,j) .lt. xlim/1.0e10) goto 30 trial(j)=trial(j)*(1.0D0+a*fos(i,j)) 30 continue 20 continue call respon(gstgcnt) 10 continue return end c---------------------------------------------------------------- c subroutine smooth c---------------------------------------------------------------- subroutine smooth(trial,chcnt) c This subroutine applies smoothing to input function, called by Ti04. implicit double precision (a-h,o-z) integer mchcnt parameter (mchcnt =200) integer j,chcnt double precision trial(mchcnt) trial(1)=.75*trial(1)+.25*trial(2) do 10 j=2,chcnt-1 trial(j)=.25*trial(j-1)+.5*trial(j)+.25*trial(j+1) 10 continue trial(chcnt)=.75*trial(chcnt)+.25*trial(chcnt-1) return end c---------------------------------------------------------------- c input control file: TI.PAR

xexp : 0.5 xlim : 0.01 xend : 0.000005 sigstp : 1.01 itl : 10 isl : 8 ismooth : 1 ifs : 3 chcnt : 100 zero : 10. txtflg : 1

130

debug : 1

subroutine kernel(fos,dia,refvolt,chcnt,xinc,gmtvlt,gmtstd,lld) c ***************************************************************** c This subroutine is called by Ln04 and Ti04 to calculate kernel functions for c spherical particles with arbitrary refractive index. Kernel functions are lognormal c distributions, peak voltage is calculated from Mie theory, standard deviations c were calibrated using PSL c **************************************************************** implicit double precision (a-h,o-z) integer stgcnt,mchcnt,chcnt,lld parameter (stgcnt=2048) parameter (mchcnt=200) parameter (mcalibnum =20) parameter (PI = 3.1415926) double precision fos(stgcnt,mchcnt) double precision dia(mchcnt),fossum(mchcnt),kersum(mchcnt) double precision d50(mcalibnum),gmtvlt1(mcalibnum) double precision gmtstd1(mcalibnum), response(mchcnt) double precision gmtvlt(mchcnt),gmtstd(mchcnt) double precision diameter(mchcnt) common/d1/ calibnum write (6,*)'Calculating kernel parameters ' c inversion diameter range, need to be defined by the user dmin = 200 dmax = 2500

xinc = (dmax/dmin) ** (1./chcnt) chcnt=chcnt+1 do 20 ich = 1,chcnt dia(ich) = dmin * xinc**float(ich-1) diameter(ich)=dia(ich)/1000 20 continue c calcularte Mie response call scatter(response,diameter,chcnt) c factor is the scaling factor of the Mie response, determined by calibration factor=2.17E+09 do 21 ich = 1,chcnt gmtstd(ich)= -0.000025*dia(ich) + 0.066988 gmtvlt(ich)= log(factor*response(ich))+gmtstd(ich)**2 21 continue c normalized voltage interval (by reference voltage) vinc = 10.0*10.0/2048.0/refvolt open (unit=8,file='temp1.dat',status='unknown') do 11 j=1,chcnt write (8,*) dia(j),gmtvlt(j),gmtstd(j)

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11 continue close(8) c caliculate kernel functions fos(i,j), fos(i,j) is in percent, sum for each size is 1.

do 31 ich = 1,chcnt do 30 j = 1,stgcnt volt = vinc*j fos(j,ich) = 1/SQRT(2*PI)/volt/gmtstd(ich)*EXP(-1* # (log(volt)-gmtvlt(ich))**2/2/gmtstd(ich)**2)*vinc 30 continue 31 continue write (6,1050) 1050 format(1x,'Enter LLD of MCA : ') read (5,1060)lld 1060 format (i3) do 35 j = 1,lld do 36 ich = 1,chcnt fos(j,ich) = 0.0 36 continue 35 continue do 60 ich = 1,chcnt fossum(ich) = 0 do 60 j = 1,stgcnt fossum(ich)=fossum(ich)+fos(j,ich) 60 continue open(unit=12,file='kersum.dat',ERR=9999) do 61 j = 1,chcnt write (12,*) dia(j),fossum(j) 61 continue close(12) 9999 continue end

subroutine rd04(derror) c **************************************************************** c This is a routine to generate the random error in the format of c ∆yi =yi±(-1,1)*yi for the input counts in 2048 channels. User defines the c maximum error c ****************************************************************

implicit double precision (a-h,o-z) integer stgcnt real maxerror parameter (stgcnt = 2048) double precision derror(stgcnt) character*1 ans

write (6,1045)

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1045 format (1x,'Please set the maximum error [0~100] : ') read (5,1055) maxerror 1055 format(f12.0) call random_number(derror) open (unit=8,file='error.dat',status='unknown') do 11 j=1,stgcnt derror(j) = (2.0*derror(j)-1.0) * maxerror write (8,*) derror(j) 11 continue close(8) end subroutine poisson(x) c *****************************************************************

This subroutine simulates counts in each channel by Poisson possibility. C *****************************************************************

parameter (stgcnt = 2048) integer i character*12 datfilename double precision sum, possib, randnum,logp, stgcount(0:stgcnt) double precision x(0:stgcnt) common/f1/ datfilename open (unit =9,file = datfilename,status='unknown') do 130 i = 0,stgcnt read (9,*) stgcount(i) 130 continue close (9) open (unit =8,file = 'poissonrand.dat',status='unknown') do 10 i= 1,stgcnt if(stgcount(i).le.0.001) then c call random_number (randnum) c stgcount(i)=randnum stgcount(i)=0.0 endif ramda= nint(stgcount(i)) call random_number (randnum) x(i)=0 possib=exp(-ramda) logp=-ramda sum= possib do 20, while (randnum .gt. sum) x(i)=x(i)+1 logp=log(ramda/x(i))+logp possib=exp(logp) sum=sum+possib

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20 continue write(*,*) x(i) write(8,*) x(i) 10 continue close(8) x(0)=stgcount(0) end subroutine SCATTER(RESPONSE,DIAMETER,STEPNUM) C********************************************************************* C Program SCATTER is designed to calculate scattering cross section (cm2) vs. particle C diameter (um) for a sphere illuminated by a laser beam for various scattering C geometries. Formulae by W.W. Szymanski (1986), init. program by S. Palm (1986). C Corrections by A. Majerowicz (1986), update and changes by Szymanski and C Redermeier (1996). I used this code to calculate the Lasair 1002 response. C*********************************************************************

PARAMETER ( ndim = 15 ) PARAMETER ( MSTEP = 200 ) INTEGER STEPNUM DOUBLE PRECISION AUX(ndim),THETALO, THETAHI, REFMED DOUBLE PRECISION REFREAL, REFIMAG DOUBLE PRECISION PI,WAVELEN,DIALOW,DIASTEP,TOL,X DOUBLE PRECISION RADIUS,R,ETA,BETA,CHI,SUM DOUBLE PRECISION F,DIAMETER(MSTEP),FACT DOUBLE PRECISION BETADEG, RESPONSE(MSTEP),RES(2,MSTEP) INTEGER OPTION COMPLEX REFREL CHARACTER *80 FILENAME LOGICAL THERE COMMON /SCATPI/ PI COMMON/MIE/X,REFREL,BETA,ETA,CHI,OPTION DOUBLE PRECISION MIEFUN EXTERNAL MIEFUN PI=ACOS(-1.0) C Inputs REFMED=1.00028 WAVELEN=0.633 TOL=1E-4 WRITE (6,*) 'Refractive index of the particle, real part? ' READ (5,*) REFREAL WRITE (6,*) 'Imaginary part? ' READ (5,*) REFIMAG REFIMAG=0

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OPTION=1 ETADEG=90 CHIDEG=90 ETA=ETADEG*PI/180. CHI=CHIDEG*PI/180. BETADEG=53.0 I=1 3333 BETA=BETADEG*PI/180.0 THETALO=ETA-BETA THETAHI=ETA+BETA REFREL=CMPLX(REFREAL,REFIMAG)/REFMED F=(DIAHIGH-DIALOW)/(STEPNUM-1.D0) FACT = WAVELEN * WAVELEN * 1.E-8 / ( 2. * PI * PI ) DO 6000 J=1,STEPNUM c DIAMETER(J)=DIALOW+(J-1)*F X=PI*DIAMETER(J)*REFMED/WAVELEN CALL SINTGR (MIEFUN,THETALO,THETAHI,R,TOL,IERR,NMAX) IF ( IERR .EQ. 1 ) WRITE (6,*) 1'Accuracy not reached because of rounding errors for:' IF ( IERR .EQ. 2 ) WRITE (6,*) 1'Accuracy not reached because of parameter ndim too small for:' IF ( IERR .EQ. -1 ) WRITE (6,*) 1'Result too close to zero for:' RES(I,J) = R * FACT c WRITE (6,5500) DIAMETER,R 6000 CONTINUE IF(I==1)THEN I=2 BETADEG=18.0 GOTO 3333 ENDIF open (1,file='response.dat',status='unknown') WRITE (1,125) REFMED,REFREAL,REFIMAG WRITE (1,130) WAVELEN WRITE (1,140) WRITE (6,140) DO 6022 J=1,STEPNUM RESPONSE(J)=RES(1,J)-RES(2,J) WRITE (6,5500) DIAMETER(J)*1000,RESPONSE(J)*2.17E+09 WRITE (1,5500) DIAMETER(J)*1000,RESPONSE(J)*2.17E+09 6022 CONTINUE CLOSE(1) 4 FORMAT (A14,1P,G7.2E1) 5 FORMAT (A14,I5)

135

100 FORMAT (/,'SHPERE SCATTERING PROGRAM') 110 FORMAT ('SELECTED OPTION ',I1) 120 FORMAT(45A,//) 121 FORMAT(/,5X,'APERTURE=',1P,E10.4,/,5X,'INCLINATION=',E10.4 + ,/,5X,'AZIMUTH=',E10.4) 122 FORMAT(/,5X,'LOW ANGLE=',1P,E10.4,/,5X,'HIGH ANGLE=',E10.4) 125 FORMAT (5X,'REFMED = ',F8.4,3X,'REFRE =',E14.6,3X,'REFIM = ' + ,E14.6) 130 FORMAT (5X,'WAVELENGTH = ',F7.4) 140 FORMAT(//,5X,'Dp [nm] ',5X,'Flux ') 5500 FORMAT (5X,1P,E12.6,5X,E12.6) END c------------------------------------------------------------------------------------------- SUBROUTINE SINTGR (MIEFUN, A, B, S, EPS, IERR, NMAX) C Subroutine used to perform Simpsons integration of a function FUNC, that has to be C supplied by the caller. Mechanism is to pass the name of the function. A is the lower C bound limit, B upper bound limit, S the result of integration, EPS the error permitted to C integration (this value may be changed by the routine if the integral could not be C performed to a better accuracy than EPS), LINLOG is type logical and is .false. for C linear and .true. for logarithmic integration. C IERR is error parameter which is returned by the routine. C IERR = 0 ..... normal successful completion C IERR = -1 .... successful completion, but result too close to zero to reach accuracy C IERR = 1 ..... error reaching accuracy C IERR = 2 ..... Error presetting EPS is less or equal to zero C IERR = 3 ..... Integral cannot be evaluated on logarithmic basis, integral bounds less or C equal to zero C NMAX is the number of integration values used for the last integration step.

PARAMETER ( JMAX = 14 ) PARAMETER ( maxdim = 2**JMAX + 2*JMAX + 3 ) INTEGER IERR LOGICAL LINLOG DOUBLE PRECISION MIEFUN DOUBLE PRECISION A, B, S, EPS DOUBLE PRECISION H, Y(maxdim), OLDS, X, ERR, AL, BL, FACTOR EXTERNAL MIEFUN LINLOG = .false. OLDS = -1.E-30 IF ( EPS .LE. 0.D0 ) THEN IERR = 3 RETURN ELSE IERR = 0 ENDIF

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IF ( LINLOG ) THEN IF ( A .LE. 0.D0 .OR. B .LE. 0.D0 ) THEN IERR = 3 RETURN ENDIF AL = DLOG(A) BL = DLOG(B) ELSE AL = A BL = B ENDIF DO 3 I = 4, JMAX NMAX = 2**I + 1 + 2 * (I+1) H = ( BL - AL ) / ( NMAX-1 ) X = A IF ( LINLOG ) THEN FACTOR = DEXP(H) DO 1 J = 1, NMAX Y(J) = MIEFUN (X) * X X = X * FACTOR 1 CONTINUE ELSE DO 2 J = 1, NMAX Y(J) = MIEFUN (X) X = X + H 2 CONTINUE ENDIF CALL SIMP (NMAX, H, Y, S) ERR = DABS(S-OLDS) IF ( ERR .LT. EPS * DABS(OLDS) ) RETURN OLDS = S 3 CONTINUE IERR = 1 IF ( DABS (S) .LT. EPS / NMAX ) IERR = -1 EPS = ERR RETURN END c------------------------------------------------------------------------------------------- SUBROUTINE SIMP (NDIM, H, Y, Z) C Subroutine used to calculate using Simpsons mechanism DOUBLE PRECISION Y, Z, H DIMENSION Y(NDIM) Z = Y (1) DO 1 I = 2, NDIM-2, 2 Z = Z + 4.D0 * Y(I) + 2.D0 * Y(I+1)

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1 CONTINUE Z = Z + 4.D0 * Y(NDIM-1) + Y(NDIM) Z = Z * H / 3.D0 RETURN END c------------------------------------------------------------------------------------------- DOUBLE PRECISION FUNCTION MIEFUN (THETA) C Function MIEFUN is used to preshape all function calls for subroutine QATR. DOUBLE PRECISION X,BETA,ETA,CHI,R DOUBLE PRECISION II,I1,I2,THETA,PSI,PHI1,PHI2,ALPHA1,ALPHA2 INTEGER OPTION COMPLEX REFREL,S1,S2 COMMON/MIE/X,REFREL,BETA,ETA,CHI,OPTION CALL BHMIE(X,REFREL,THETA,S1,S2) I1=CABS(S1)**2 I2=CABS(S2)**2 PSI=(COS(BETA)-COS(THETA)*COS(ETA))/ + (SIN(THETA)*SIN(ETA)) IF ( ABS(PSI) .GT. 1.D0 .AND. ABS(PSI) .LE. 1.001 ) + PSI = SIGN(1.,PSI) PSI=ACOS(PSI) PHI1=CHI-PSI PHI2=CHI+PSI ALPHA1=-0.25*(SIN(2.0*PHI2)-SIN(2.0*PHI1))+PSI ALPHA2=0.25*(SIN(2.0*PHI2)-SIN(2.0*PHI1))+PSI C R=(I1*ALPHA1+I2*ALPHA2)*SIN(THETA)*PSI R=(I1+I2)*SIN(THETA)*PSI MIEFUN = R RETURN END c------------------------------------------------------------------------------------------- SUBROUTINE BHMIE (X,REFREL,THETA,S1,S2) C Subroutine BHMIE calculates amplitude scattering matrix elements and efficiencies for C extinction, total scattering and backscattering for a given size parameter and relative C refractive index C DOUBLE PRECISION AMU,THETA,PI,TAU,PI0,PI1 COMPLEX D(3000),Y,REFREL,XI,XI0,XI1,AN,BN,S1,S2 DOUBLE PRECISION PSI0,PSI1,PSI,DN,DX DOUBLE PRECISION PIE,ANG,X INTEGER NANG C PIE is PI in all other routines, renamed by common. ! COMMON /SCATPI/ PIE DX=X

138

Y=X*REFREL C Series terminated after NSTOP terms XSTOP=X+4.*X**(1./3.)+2.0 NSTOP=XSTOP YMOD=CABS(Y) NMX=AMAX1(XSTOP,YMOD)+15 IF (NMX.GT.3000) THEN WRITE (6,*) 'ERROR IN SUBROUTINE BHMIE' WRITE (6,*) 'ARRAY D() IS NOT LARGE ENOUGH' WRITE (6,*) 'TRY DIMENSION OF GREATER THAN',NMX STOP ENDIF IF (THETA.GT.(PIE/2.0)) THEN AMU=COS(PIE-THETA) ELSE AMU=COS(THETA) ENDIF C Logarithmic derivative D(J) calculated by DOWNWARD C recurrence beginning with initial value 0.0+I*0.0 C at J = NMX D(NMX)=CMPLX(0.0,0.0) NN=NMX-1 DO 120 N=1,NN RN=NMX-N+1 120 D(NMX-N)=(RN/Y)-(1./(D(NMX-N+1)+RN/Y)) PI0=0.0 PI1=1.0 S1=CMPLX(0.0,0.0) S2=CMPLX(0.0,0.0) C Riccati-Bessel functions with real argument X calculated by upward recurrence PSI0=DCOS(DX) PSI1=DSIN(DX) CHI0=-SIN(X) CHI1=COS(X) APSI0=PSI0 APSI1=PSI1 XI0=CMPLX(APSI0,-CHI0) XI1=CMPLX(APSI1,-CHI1) N=1 200 DN=N RN=N FN=(2.*RN+1.)/(RN*(RN+1.)) PSI=(2.*DN-1.)*PSI1/DX-PSI0 APSI=PSI CHI=(2.*RN-1.)*CHI1/X-CHI0

139

XI=CMPLX(APSI,-CHI) AN=(D(N)/REFREL+RN/X)*APSI-APSI1 AN=AN/((D(N)/REFREL+RN/X)*XI-XI1) BN=(REFREL*D(N)+RN/X)*APSI-APSI1 BN=BN/((REFREL*D(N)+RN/X)*XI-XI1) TAU=RN*AMU*PI1-(RN+1.)*PI0 IF (THETA.GT.(PIE/2.0)) THEN T=(-1.)**N P=(-1.)**(N-1) S1=S1+FN*(AN*PI1*P+BN*TAU*T) S2=S2+FN*(AN*TAU*T+BN*PI1*P) ELSE S1=S1+FN*(AN*PI1+BN*TAU) S2=S2+FN*(AN*TAU+BN*PI1) ENDIF PSI0=PSI1 PSI1=PSI APSI1=PSI1 CHI0=CHI1 CHI1=CHI XI1=CMPLX(APSI1,-CHI1) N=N+1 RN=N PI=PI1 PI1=((2.*RN-1.)/(RN-1.))*AMU*PI1-RN*PI0/(RN-1.) PI0=PI IF(N-1-NSTOP) 200,300,300 300 RETURN END

140

Appendix C: Codes for the Lasair and Climet Response Calculations

Programs Lasair and Climet were special cases of the program “Scatter”, which

was designed to calculate scattering cross section (cm2) vs. particle diameter (um) for a

sphere illuminated by a laser beam for various scattering geometries by Szymanski, et al.

Except for the main routines “Lasair” and “Climet”, other subroutines (SINTGR, SIMP,

MIEFUN, BHMIE) are the same when calculating responses for these two instruments.

These subroutines have been listed in Appendix B.

*************************************************************

PROGRAM Lasair PARAMETER ( ndim = 15 ) PARAMETER ( MSTEP = 500 ) DOUBLE PRECISION AUX(ndim), THETALO, THETAHI

DOUBLE PRECISION REFMED, REFREAL,REFIMAG DOUBLE PRECISION PI,WAVELEN,DIALOW,DIASTEP,TOL,X DOUBLE PRECISION RADIUS,R,ETA,BETA,CHI,SUM DOUBLE PRECISION F,DIAMETER(MSTEP),STEPNUM,FACT DOUBLE PRECISION BETADEG, RESPONSE(MSTEP),RES(2,MSTEP) INTEGER OPTION COMPLEX REFREL CHARACTER *80 FILENAME LOGICAL THERE COMMON /SCATPI/ PI COMMON/MIE/X,REFREL,BETA,ETA,CHI,OPTION DOUBLE PRECISION MIEFUN EXTERNAL MIEFUN c sizes for SP Lasair low gain PSL c DATA STEPNUM,(DIAMETER(I),I=1,13)/13,0.263,0.305,0.404,0.482, c + 0.505,0.595,0.653,0.672,0.701,0.720,0.845,0.913,1.099/ PI=ACOS(-1.0) C Inputs REFMED=1.00028 WAVELEN=0.633 TOL=1E-6 WRITE (6,*) 'Refractive index of the particle, real part? ' READ (5,*) REFREAL WRITE (6,*) 'Imaginary part? '

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READ (5,*) REFIMAG 10 WRITE (6,*) 'Lower particle diameter [um]? ' READ (5,*) DIALOW IF (DIALOW.LE.0.D0) THEN WRITE (6,*) 'Particle diameter must be > 0' GOTO 10 ENDIF 20 WRITE (6,*) 'Upper particle diameter [um]? ' READ (5,*) DIAHIGH IF (DIAHIGH.LE.DIALOW) THEN WRITE (6,*) 'Particle range must be > 0' GOTO 20 ENDIF 30 WRITE (6,*) 'Number of particle diameter steps? ' READ (5,*) STEPNUM IF (STEPNUM .LE. 1) THEN WRITE (6,*) 'Number of steps must be > 1' GOTO 30 ENDIF OPTION=1 ETADEG=90 CHIDEG=90 ETA=ETADEG*PI/180. CHI=CHIDEG*PI/180. C LASAIR MIRROR ANGLE BETADEG=53.0 I=1 3333 BETA=BETADEG*PI/180.0 THETALO=ETA-BETA

THETAHI=ETA+BETA REFREL=CMPLX(REFREAL,REFIMAG)/REFMED F=(DIAHIGH-DIALOW)/(STEPNUM-1.D0) FACT = WAVELEN * WAVELEN * 1.E-8 / ( 2. * PI * PI ) DO 6000 J=1,STEPNUM DIAMETER(J)=DIALOW+(J-1)*F X=PI*DIAMETER(J)*REFMED/WAVELEN CALL SINTGR (MIEFUN,THETALO,THETAHI,R,TOL,IERR,NMAX) IF ( IERR .EQ. 1 ) WRITE (6,*) 1 'Accuracy not reached because of rounding errors for:' IF ( IERR .EQ. 2 ) WRITE (6,*)

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1 'Accuracy not reached because of parameter ndim too small for:' IF ( IERR .EQ. -1 ) WRITE (6,*) 1 'Result too close to zero for:' RES(I,J) = R * FACT 6000 CONTINUE IF(I==1)THEN I=2 C THE ANGLE OF THE HOLE IN THE LASAIR MIRROR BETADEG=18.0 GOTO 3333 ENDIF open (1,file='response.dat',status='unknown') WRITE (1,125) REFMED,REFREAL,REFIMAG WRITE (1,130) WAVELEN WRITE (1,140) WRITE (6,140) DO 6022 J=1,STEPNUM

RESPONSE(J)=RES(1,J)-RES(2,J) WRITE (6,5500) DIAMETER(J)*1000,RESPONSE(J) WRITE (1,5500) DIAMETER(J)*1000,RESPONSE(J) 6022 CONTINUE CLOSE(1) STOP 'Scatter - completed.' C Formats 4 FORMAT (A14,1P,G7.2E1) 5 FORMAT (A14,I5) 100 FORMAT (/,'SHPERE SCATTERING PROGRAM') 110 FORMAT ('SELECTED OPTION ',I1) 120 FORMAT(45A,//) 121 FORMAT(/,5X,'APERTURE=',1P,E10.4,/,5X,'INCLINATION=',E10.4 + ,/,5X,'AZIMUTH=',E10.4) 122 FORMAT(/,5X,'LOW ANGLE=',1P,E10.4,/,5X,'HIGH ANGLE=',E10.4) 125 FORMAT (5X,'REFMED = ',F8.4,3X,'REFRE =',E14.6,3X,'REFIM = ' + ,E14.6) 130 FORMAT (5X,'WAVELENGTH = ',F7.4) 140 FORMAT(//,5X,'Dp [nm] ',5X,'Flux ') 5500 FORMAT (5X,1P,E12.6,5X,E12.6) C End of main program C END ************************************************************************ PROGRAM CLIMET PARAMETER ( ndim = 15 ) PARAMETER ( MSTEP = 500 )

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DOUBLE PRECISION AUX(ndim), THETALO, THETAHI DOUBLE PRECISION REFMED, REFREAL,REFIMAG

DOUBLE PRECISION PI,WAVELEN,DIALOW,DIASTEP,TOL,X DOUBLE PRECISION RADIUS,R,ETA,BETA,CHI,SUM DOUBLE PRECISION F,DIAMETER(MSTEP),STEPNUM,FACT DOUBLE PRECISION BETADEG, RESPONSE(MSTEP),RES(6,MSTEP) INTEGER OPTION COMPLEX REFREL CHARACTER *80 FILENAME LOGICAL THERE COMMON /SCATPI/ PI COMMON/MIE/X,REFREL,BETA,ETA,CHI,OPTION DOUBLE PRECISION MIEFUN EXTERNAL MIEFUN c-------------------------------------------------------------------------- c sizes for Climet low gain DOS c DATA STEPNUM,(DIAMETER(I),I=1,11)/11,2.16,2.90,3.95,4.83, c +5.83,6.27,6.98,7.91,8.30,8.86,9.77/ c-------------------------------------------------------------------------- PI=ACOS(-1.0) C Inputs REFMED=1.00028 WAVELEN=0.78 TOL=1E-6 WRITE (6,*) 'Refractive index of the particle, real part? ' READ (5,*) REFREAL WRITE (6,*) 'Imaginary part? ' READ (5,*) REFIMAG 10 WRITE (6,*) 'Lower particle diameter [um]? ' READ (5,*) DIALOW IF (DIALOW.LE.0.D0) THEN WRITE (6,*) 'Particle diameter must be > 0' GOTO 10 ENDIF 20 WRITE (6,*) 'Upper particle diameter [um]? ' READ (5,*) DIAHIGH IF (DIAHIGH.LE.DIALOW) THEN WRITE (6,*) 'Particle range must be > 0' GOTO 20 ENDIF

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30 WRITE (6,*) 'Number of particle diameter steps? ' READ (5,*) STEPNUM IF (STEPNUM .LE. 1) THEN WRITE (6,*) 'Number of steps must be > 1' GOTO 30 ENDIF OPTION=1 ETADEG=90 CHIDEG=90 ETA=ETADEG*PI/180. CHI=CHIDEG*PI/180. BETADEG=89.99 I=1 3333 BETA=BETADEG*PI/180.0 THETALO=ETA-BETA THETAHI=ETA+BETA 6666 REFREL=CMPLX(REFREAL,REFIMAG)/REFMED F=(DIAHIGH-DIALOW)/(STEPNUM-1.D0) FACT = WAVELEN * WAVELEN * 1.E-8 / ( 2. * PI * PI ) DO 6000 J=1,STEPNUM DIAMETER(J)=DIALOW+(J-1)*F X=PI*DIAMETER(J)*REFMED/WAVELEN CALL SINTGR (MIEFUN,THETALO,THETAHI,R,TOL,IERR,NMAX) IF ( IERR .EQ. 1 ) WRITE (6,*) 1 'Accuracy not reached because of rounding errors for:' IF ( IERR .EQ. 2 ) WRITE (6,*) 1 'Accuracy not reached because of parameter ndim too small for:' IF ( IERR .EQ. -1 ) WRITE (6,*) 1 'Result too close to zero for:' RES(I,J) = R * FACT 6000 CONTINUE IF(I==1)THEN I=2 BETADEG=67.9 GOTO 3333 elseIF(I==2)THEN I=3 BETADEG=22.5 GOTO 3333 elseIF(I==3)THEN I=4 BETADEG=18.7

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GOTO 3333 elseIF(I==4)THEN I=5 option=2 THETALODEG=0 THETALO=THETALODEG*PI/180. THETAHIDEG=15.83 THETAHI=THETAHIDEG*PI/180.

GOTO 6666 elseIF(I==5)THEN

I=6 option=2 THETALODEG=164.17 THETALO=THETALODEG*PI/180. THETAHIDEG=180 THETAHI=THETAHIDEG*PI/180. GOTO 6666 ENDIF open (1,file='response.dat',status='unknown') WRITE (1,125) REFMED,REFREAL,REFIMAG WRITE (1,130) WAVELEN WRITE (1,140) WRITE (6,140)

DO 6022 J=1,STEPNUM RESPONSE(J)=2*RES(1,J)-RES(2,J)-RES(3,J)-2*RES(4,J) + -RES(5,J)-RES(6,J)

WRITE (6,5500) DIAMETER(J),RESPONSE(J) WRITE (1,5500) DIAMETER(J),RESPONSE(J) 6022 CONTINUE CLOSE(1) STOP 'Scatter - completed.' C Formats 4 FORMAT (A14,1P,G7.2E1) 5 FORMAT (A14,I5) 100 FORMAT (/,'SHPERE SCATTERING PROGRAM') 110 FORMAT ('SELECTED OPTION ',I1) 120 FORMAT(45A,//) 121 FORMAT(/,5X,'APERTURE=',1P,E10.4,/,5X,'INCLINATION=',E10.4 + ,/,5X,'AZIMUTH=',E10.4) 122 FORMAT(/,5X,'LOW ANGLE=',1P,E10.4,/,5X,'HIGH ANGLE=',E10.4) 125 FORMAT (5X,'REFMED = ',F8.4,3X,'REFRE =',E14.6,3X,'REFIM = ' + ,E14.6) 130 FORMAT (5X,'WAVELENGTH = ',F7.4) 140 FORMAT(//,5X,'Dp [nm] ',5X,'Flux ') 5500 FORMAT (5X,1P,E12.6,5X,E12.6)

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C End of main program END *********************************************************************** SUBROUTINE SINTGR SUBROUTINE SIMP (NDIM, H, Y, Z)

DOUBLE PRECISION FUNCTION MIEFUN (THETA) SUBROUTINE BHMIE (X,REFREL,THETA,S1,S2) (See Appendix B)

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Appendix D: Codes for Lasair and Climet Table Data Refractive Index Corrections

These programs were used for OPC table data size limits correction for refractive

index. The input refractive index file has several columns: the first columns are time in

the format of “ MM DD YY HH MM”, and the last column is the refractive index at that

time. Bin limits were corrected from 0.1µm to 1.0µm for the Lasair table data, and 0.3µm

to 0.8µm for Climet.

*************************************************************** program Table_Correction c This routine is for data I/O

character * 2 Month(12) /'01', '02', '03', '04', & '05', '06', '07', '08', & '09', '10', '11', '12'/ character * 2 Day(31) /'01', '02', '03', '04', '05', '06', & '07', '08', '09', '10', '11', '12', & '13', '14', '15', '16', '17', '18', & '19', '20', '21', '22', '23', '24', & '25', '26', '27', '28', '29', '30', '31'/ character * 2 Year(5) /'01', '02', '03', '04', '05'/ character * 2 Hour(24) /'00','01', '02', '03', '04', '05', '06', & '07', '08', '09', '10', '11', '12', '13', & '14', '15', '16', '17', '18', '19', '20', & '21', '22', '23'/ character * 2 Minute(60) /'00', '01', '02', '03', '04', '05', '06' & ,'07', '08', '09', '10', '11', '12', & '13', '14', '15', '16', '17', '18', & '19', '20', '21', '22', '23', '24', & '25', '26', '27', '28', '29', '30', '31', & '32', '33', '34', '35', '36', '37', '38', & '39', '40', '41', '42', '43', '44', '45', & '46', '47', '48', '49', '50', '51', '52', & '53', '54', '55', '56', '57', '58', '59'/ double precision refInd, corDia(8) open(11, file = 'SepRef.txt', status = 'old')

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open(2, file = 'SepDia.txt', access = 'append',status='unknown') DO WHILE (.NOT. EOF(11))

READ (11, *) monthN, idateN, iyearN, ihourN, minuteN, refInd call TableCorrect(refInd, corDia) write(*,*)Month(monthN) // '/' // Day(idateN) // '/' // & Year(iyearN) // ' ' // Hour(ihourN+1) // ':' // Minute(minuteN+1) write(2, 2000) Month(monthN) // '/' // Day(idateN) // '/' // & Year(iyearN) // ' ' // Hour(ihourN+1) // ':' // Minute(minuteN+1) & ,(corDia(i),I=1,8) END DO 1000 format(A8, 2x, A8, (F15.8, 1x)) 2000 format(A20, 2x, 8(f6.4,1x))

close(11) close(2) end ********************************************************************* subroutine LasairTableCorrect(Ref_Index,Dp_Corrected) c This program is designed to do Lasair table data correction for refractive index parameter (channel=8) parameter (mstepnum=200) double precision Dp_PSL(channel),Dp_Corrected(channel)

double precision Dp_Aerosol(mstepnum) double precision R_PSL(channel), R_Aerosol(mstepnum) double precision Ref_Index double precision dmin, dmax integer stepnum c Lasair table data channels DATA (Dp_PSL(I),I=1,8)/0.1,0.2,0.3,0.4,0.5,0.7,1.0,2.0/ c Theoretical response of Lasair to PSL DATA (R_PSL(I),I=1,8)/4.643661E-13,2.794850E-11,2.084067E-10, + 7.413423E-10,1.133376E-09,1.972216E-09, + 4.250987E-09,6.981118E-09/ dmin=0.1 dmax=2.5 stepnum=100 c xinc = (dmax/dmin) ** (1./stepnum) xinc = (dmax-dmin)/stepnum stepnum=stepnum+1 do 10 i = 1,stepnum c Dp_Aerosol(i)=dmin * xinc**float(i-1) Dp_Aerosol(i)=dmin + xinc*float(i-1) 10 continue

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call Lasair(R_Aerosol,Dp_Aerosol,stepnum,Ref_Index) c Look for the equivalent aerosol diameter for the PSL sizes do 20 i = 1,channel do 30 j = 1,stepnum-1

if(R_PSL(i).lt.R_Aerosol(1)) then Dp_Corrected(i)=Dp_Aerosol(1)+(R_PSL(i)-R_Aerosol(1))/ + (R_Aerosol(1)-R_Aerosol(2))*(Dp_Aerosol(1)-Dp_Aerosol(2)) elseif((R_Aerosol(j).le.R_PSL(i)).and. + (R_Aerosol(j+1).ge.R_PSL(i))) then Dp_Corrected(i)=Dp_Aerosol(j)+(R_PSL(i)-R_Aerosol(j))/ + (R_Aerosol(j+1)-R_Aerosol(j))*(Dp_Aerosol(j+1)-Dp_Aerosol(j))

elseif(R_PSL(i).gt.R_Aerosol(stepnum)) then Dp_Corrected(i)=Dp_Aerosol(stepnum-1)

+ +(R_PSL(i)-R_Aerosol(stepnum-1)) + /(R_Aerosol(stepnum)-R_Aerosol(stepnum-1)) + *(Dp_Aerosol(stepnum)-Dp_Aerosol(stepnum-1))

endif 30 continue 20 continue return end ********************************************************************* subroutine ClimetTableCorrect(Ref_Index,Dp_Corrected) c This program is designed to do Lasair table data correction for refractive index parameter (channel=16) parameter (mstepnum=200) double precision Dp_PSL(channel),Dp_Corrected(channel)

double precision Dp_Aerosol(mstepnum) double precision R_PSL(channel), R_Aerosol(mstepnum) double precision Ref_Index double precision dmin, dmax integer stepnum c Climet table data channels DATA (Dp_PSL(I),I=1,16)/0.3,0.4,0.5,0.63,0.8,1.0,1.3,1.6,2.0, + 2.5,3.2,4.0,5.0,6.3,8.0,10.0/

DATA (Dp_Corrected(I),I=6,16)/1.0,1.3,1.6,2.0, + 2.5,3.2,4.0,5.0,6.3,8.0,10.0/ c Theoretical response of Climet to PSL DATA (R_PSL(I),I=1,5)/4.897965E-10,2.235199E-09,6.322444E-09, + 1.567073E-08,3.008387E-08/ dmin=0.1 dmax=1.5

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stepnum=100 c xinc = (dmax/dmin) ** (1./stepnum) xinc = (dmax-dmin)/stepnum stepnum=stepnum+1 do 10 i = 1,stepnum c Dp_Aerosol(i)=dmin * xinc**float(i-1) Dp_Aerosol(i)=dmin + xinc*float(i-1) 10 continue call Climet (R_Aerosol,Dp_Aerosol,stepnum,Ref_Index) c Look for the equivalent aerosol diameter for the PSL sizes do 20 i = 1,channel do 30 j = 1,stepnum-1

if(R_PSL(i).lt.R_Aerosol(1)) then Dp_Corrected(i)=Dp_Aerosol(1)+(R_PSL(i)-R_Aerosol(1))/ + (R_Aerosol(1)-R_Aerosol(2))*(Dp_Aerosol(1)-Dp_Aerosol(2)) elseif((R_Aerosol(j).le.R_PSL(i)).and. + (R_Aerosol(j+1).ge.R_PSL(i))) then Dp_Corrected(i)=Dp_Aerosol(j)+(R_PSL(i)-R_Aerosol(j))/ + (R_Aerosol(j+1)-R_Aerosol(j))*(Dp_Aerosol(j+1)-Dp_Aerosol(j))

elseif(R_PSL(i).gt.R_Aerosol(stepnum)) then Dp_Corrected(i)=Dp_Aerosol(stepnum-1)

+ +(R_PSL(i)-R_Aerosol(stepnum-1)) + /(R_Aerosol(stepnum)-R_Aerosol(stepnum-1)) + *(Dp_Aerosol(stepnum)-Dp_Aerosol(stepnum-1))

endif 30 continue 20 continue return end ************************************************************************

subroutine Lasair(R_Aerosol,Dp_Aerosol,stepnum,Ref_Index) subroutine Climet(R_Aerosol,Dp_Aerosol,stepnum,Ref_Index)

Note: These two subroutines are pretty the same as the subroutine “scatter” in Appendix B, except one more parameter: Ref_Index in transfer. The other subroutines called by “scatter” are also used by these two routines.