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, ([email protected])
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
References: 1. Hinds, W.C., (1998) Aerosol Technology, Properties, Behavior, and Measurement of
Airborne Particles. 2nd Edition, Los Angeles, California: A Wiley-Interscience
Publication.
2. Lilienfeld, P., (2000) Light scattering fundamentals and optical measurements
(Aerosol and Particle Measurement Short Course (UMN, 2000))
3. McMurry, P.H., (1996) Optical Properties of Aerosol (ME 8613 class notes)
4. Szymanski, W.W., (2000) Optical Particle Counter (Aerosol and Particle
Measurement Short Course (UMN, 2000))
5. Reist, P.C., (1993) Aerosol Science and Technology: McGraw-Hill, Inc.
6. Kandlikar, M. and G. Ramachandran, (1999) "Inverse Methods for Analyzing
Aerosol Spectrometer Measurements: A Critical Review," J. Aerosol Sci., Vol. 30(No.
4): p. 413 - 437.
7. Markowski, G.R., (1987) "Improving Twomey's Algorithm for Inversion of Aerosol
Measurement Data," Aerosol Sci. Technology, 7: p. 127 -141.
8. Weber, R.J., et al., (1998) "Inversion of Ultrafine Condensation Nucleus Counter
Pulse Height Distribution To Obtain Nanoparticle (~3 - 10 nm) Size Distributions," J.
Aerosol Sci., Vol. 29(No. 5/6): p. 601-615.
9. Twomey, S., (1975) "Comparison of Constrained Linear Inversion and an Iterative
Nonlinear Algorithm Applied to the Indirect Estimate of Particle Size Distribution,"
Journal of Computational Physics, 18: p. 188-200.
10. Saros, M.T., (1995) Sensitivity of Pulse Heights to Pressure for Ultrafine Particle in
the Ultrafine Condensation Particle Counter (UCPC), M.S. Thesis, Dept. of
Mechanical Engineering, University of Minnesota
11. Dick, W.D., (1992) Size- and Composition - Dependent Response of the DAWN - A
Multiangle Single - Particle Optical Detector, M.S. Thesis, Dept. of Mechanical
Engineering, University of Minnesota
12. Hering, S.V. and P.H. McMurry, (1991) "Optical counter response to monodisperse
atmosphere aerosols," Atmospheric Environment, vol. 25A(No. 2): p. 463-468.
102
13. Gupta, A. and P.H. McMurry, (1989) "A Device for Generating Singly Charged
Particles in the 0.1 - 1.0 µm Diameter Range," Aerosol Science and Technology, 10: p.
451 - 462.
14. Knutson, E.O. and K.T. Whitby, (1975) "Aerosol Classification by electric mobility:
apparatus, theory, and applications," J. Aerosol Sci., Vol. 6: p. 443-451.
15. Rader, D.J. and P.H. McMurry, (1986) "Application of the Tandem Differential
Mobility Analyzer to Studies of Droplet Growth or Evaporation," J. Aerosol Sci., Vol.
17(No.5): p. 771 - 787.
16. McMurry, P.H., (1987) Operator's Manual for New University of Minnesota DMAs
(GMWDMA) (PTL report)
17. TSI, Model 3934 SMPS (Scanning Mobility Particle Sizer) Instruction Manual
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
22. Liu, B.Y.H., R.N. Berglund, and J.K. Agarwal, (1974) "Experimental Studies of
Optical Particle Counters," Atmospheric Environment, Vol. 8: p. 717-732.
23. Donovan, R.P., (2001) Contamination-Free Manufacturing for Semiconductors and
Other Precision Products: Marcel Dekker, Inc. 42-43.
24. Liu, Y. and P.H. Daum, (2000) "The Effect of Refractive Index on Size Distributions
and Light Scattering Coefficients Derived from Optical Particle Counters," J. Aerosol
Sci., Vol. 31(No. 8): p. 945-957.
25. Hulst, H.C.v.d., (1981) Light Scattering by Small Particles: Dover Publications, Inc.
26. Dick, W.D. and P.H. McMurry, (1994) "Size- and Composition-Dependent Response
of the DAWN-A Multiangle Single-Particle Optical Detector," Aerosol Sci.
Technology, (20): p. 345-362.
27. Bohren, C.F. and D.R. Huffman, (1983) Absorption and Scattering of Light by Small
Particles: John Wiley & Sons, New York.
103
28. Dick, W.D., (1998) Multiangle Light Scattering Techniques for Measuring Shape and
Refractive Index of Submicron Atmospheric Particles, Ph.D. Thesis, Mechanical
Engineering, University of Minnesota
29. Wang, H.-C. and W. John, (1988) "Characteristics of the Berner Impactor for
Sampling Inorganic Ions," Aerosol Sci. Technology, 8: p. 157-172.
30. Winklmayr, W., H.-C. Wang, and W. John, (1990) "Adaptation of the Twomey
Algorithm to the Inversion of Cascade Impactor Data," Aerosol Sci. Technology, 13:
p. 322-331.
31. Devroye, L., (1986) Non-Uniform Random Variate Generation: Springer-Verlag New
York Inc.
32. Turner, J., etc., (2000) St. Louis Supersite Project Description
33. Woo, K.-S., et al., (2001) "Use of continuous measurements of integral aerosol
parameters to estimate particle surface area," Aerosol Sci. Tech., 34: p. 1-9.
34. Woo, K.-S., (2002) Measurement of Atmospheric Aerosols: Size Distribution of
Nanoparticles, Estimation of Distribution Moments and Control of Relative Humidity,
Ph.D. Thesis, Department of Mechanical Engineering, University of Minnesota
35. Tang, I.N. and H.R. Munkelwitz, (1994) "Water activities, densities, and refractive
indices of aqueous sulfates and sodium nitrate droplets of atmospheric importance,"
Journal of Geophysical Research, 99(D9): p. 18801-18808.
36. Kittelson, D., (1998) "Engines and Nanoparticles: A Review," J. Aerosol Sci., (29): p.
574-588.
37. McMurry, P.H., et al., (2002) "The relationship between mass and mobility for
atmospheric particles: a new technique for measuring particle density," Aerosol Sci.
Tech., (36): p. 227-238.
38. Tang, I.N., (1996) "Chemical and size effects of hygroscopic aerosols on light
scattering coefficients," Journal of Geophysical research, Vol. 101(No. D14): p.
19,245-19,250.
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
125
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
126
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
127
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)
131
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)
132
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
133
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
134
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
136
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)
137
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? '
141
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,*)
142
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 )
143
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
144
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
145
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)
146
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)
147
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')
148
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
149
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
150
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.
Top Related