single particle light scattering

C O P Y R I G H T © 2 0 0 9 D R O P L E T M E A S U R E M E N T T E C H N O L O G I E S ,

I N C .

Data Analysis

User’s Guide

Chapter I:

Single Particle

Light Scattering

DOC-0222, Rev A

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Boulder, CO 80301-5727 USA

Data Analysis User’s Guide – Chapter I: Single Particle Light Scattering

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C O N T E N T S

Foreword ........................................................................................ 4

1.0 Single Particle Light Scattering ................................................... 5

1.1 Theory ........................................................................................ 5

1.1.1 Particle Size Determination .......................................................... 5

1.1.2 Sample Volume Determination ...................................................... 11

1.2 Uncertainties and Limitations ........................................................... 13

1.2.1 Sizing Errors ............................................................................ 14

1.2.2 Counting and Concentration Errors ................................................. 16

1.2.3 Contamination from Drop and Crystal Shattering ................................ 17

1.3 Data Analysis ............................................................................... 21

1.3.1 Size Distributions ...................................................................... 21

1.3.2 Derivation of Bulk Parameters ...................................................... 29

1.3.2.1 Total Number Concentration ................................................. 31

1.3.2.2 Extinction Coefficient and Visibility ........................................ 31

1.3.2.3 Liquid Water Content ......................................................... 32

1.3.2.4 Mean, Median Volume and Effective Diameter ............................ 33

1.3.3 Derivation of Refractive Index ...................................................... 34

1.3.4 Derivation of Asphericity ............................................................ 37

1.3.5 Evaluation of Cloud Microstructure ................................................ 40

1.4 Further Reading on Single Particle Scattering ......................................... 42

L i s t o f T a b l e s

Table 1.1: CAS Setup Table (Forward Scattering Only) ......................................... 23

Table 1.2: CAS Setup Table with and without Combined Channels ........................... 24

Table 1.3: Cloud Properties, Hydrometeor Characteristics, and Environmental Impact .. 29

Table 1.4: Hydrometeor Properties ............................................................... 30

Table 1.5: Sample Table of Forward and Backward Scattering Cross Sections .............. 36

Table 1.6: Sample Table of Aspect Ratios vs. Forward to Backscatter ....................... 38


Form DOC-0222 4 © 2009 DROPLET MEASUREMENT TECHNOLOGIES, INC.

Foreword

The analysis of cloud particle measurements requires a certain degree of understanding

of the basic principles of operation of the sensors and, perhaps more importantly, an

appreciation of the limitations and uncertainties that are associated with the

measurement techniques. This user‟s guide is designed to introduce the reader to the

fundamental concepts that Droplet Measurement Technologies (DMT) employs to measure

cloud particles, the inherent uncertainties and limitations associated with these

measurement techniques and suggestions on how the measurement results can be

evaluated and analyzed within these constraints.

This manual is not a substitute for the many publications that have been written on the

subject of particle measurements and methodologies for data analysis. Throughout this

guide we will extract from many of such publications with extensive, up-to-date

reference lists at the end of each chapter. As newer publications become available this

guide will be updated accordingly with addendums that describe new analysis techniques,

correction algorithms or new technology that decreases the uncertainties or improves

resolution or response.

This guide is separated into three general chapters related to the technology that is

employed in the DMT instruments: 1) single particle light scattering, 2) single particle

imaging and 3) thermal analysis of liquid water. Each chapter contains a simplified

description of the measurement theory, a discussion of the uncertainties and limitations

inherent in the technique, derivation of the most commonly used atmospheric

parameters and some examples of typical applications. At the end of each chapter is a

list of references and suggested reading.

Most of the graphs throughout the manual were generated by the DMT analysis program,

“PAPI”, that runs on the Wavemetrics Inc. software package IGOR. PAPI is an open source

set of routines for ingesting data files written by DMT‟s Particle Acquisition and Display

System (PADS). In various parts of the chapters, examples will be given to show how PAPI

can help users analyze and display data. There is a separate User‟s Guide for PAPI, but

the examples contained in this manual do not require an extensive knowledge of IGOR or

PAPI to run on the user‟s own data set.



1.0 Single Particle Light Scattering

1.1 Theory

1.1.1 Particle Size Determination

The operating principle of instruments that detect particle by light scattering is based on

the concept that the intensity of scattered light is proportional to the particle size and

can be predicted theoretically if the shape and refractive index of the particle is known

as well as the wavelength of the incident light. As illustrated in Fig. 1.1, scattering is the

result of the interaction of the electromagnetic wave of incident light with the molecular

structure of a particle that subsequently produces electromagnetic emissions with

components of reflection, refraction and diffraction.

Figure 1.1

The composite of these three components is a field of scattered light whose intensity

varies as a function of angle around the particle. If the particle is spherical and of

homogeneous composition, the scattered intensity is symmetric around the axis parallel

with the incident wave but varies in intensity from 0 to 180º, where 0º is the most

forward scattering and 180º is directly backward. Figure 1.2 shows an example of the

angular pattern of scattering. This angular dependency of the scattering around a

Reflection, Q1

Refraction, Q2

Diffraction, Q3

Q2



spherical particle can be calculated using the equations that were developed by Mie

(1908) for a specific diameter, refractive index and incident wavelength.

This theory is applied in optical particle counters (OPCs) by collecting scattered light

from particles that pass through a light beam of controlled intensity and wavelength and

converting the photons to an electrical signal whose amplitude can be subsequently

related back to the size of the particle.

Figure 1.2

The property of a particle to interact with light is usually described by its scattering cross

section, σ. This is the product of the scattering efficiency, θ, and cross sectional area,

(π/4)D2, where D is the particle diameter. Figure 1.3 shows the scattering cross section

as a function of angle for particles with three different diameters and an incident

wavelength of 680 nm. In order to determine the amount of light scattered in any

direction, it is only necessary to multiply the scattering cross section by the intensity of

the incident light. Alternatively, if we have an optical system that collects light over a

range of angles and we measure the intensity of scattered light collected from a particle,

we can determine the particle size from the information shown in Figure 1.3 by

integrating the scattering function over the range of angles used in the instrument.

The collection angles used in the DMT single particle scattering instrument, i.e. the cloud

and aerosol spectrometer (CAS), the cloud droplet probe (CDP) and the fog monitor, are

from 3.5º to 12º, as shown on Fig. 1.3 (the backscattering angles used in the CAS are also

shown). The near forward angles are used because, as can be seen in Fig. 1.3, the largest

percentage of light scattered from a particle is in the forward direction. This is generally

true when the size of the particle is large with respect to the incident wavelength.



Figure 1.3

The DMT light scattering spectrometers use an optical configuration similar to the block

diagram shown in Fig. 1.4 that is the conceptual design for the CAS with polarization

option (discussed in section 1.3). A diode laser is used as the source for coherent,

monochromatic light that is focused and transmitted across the airstream that carries the

particles. The beam falls on a light absorbing circular „dump‟ spot that blocks the beam

from the detectors. The forward scattered light is collected by the optics and focused

onto two detectors via a beam splitter (the purpose of this arrangement is discussed

below in Section 1.1.2). The acceptance angles are determined by the diameter of dump

spot and the aperture of the primary collection lens.

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

Angle

1E -010

1E -009

1E -008

1E -007

1E -006

Sc

att

erin

g C

ro

ss

Se

cti

on

(c

m-2

)

5 um

15 um

25 um

CA

S a

nd

CD

P F

orw

ard CA

S B

ack



Figure 1.4

The detectors convert the collected photons into an electrical current that is

subsequently conditioned and digitized into a signal whose peak amplitude is proportional

to the scattered light of the particle as it passes through the beam. To derive the size of

the particle, two steps are needed: 1) the amplitude of the digitized signal must be

converted into a light scattering cross section and 2) the light scattering cross section

must be linked to the diameter of the particle that produced it.

The first step is executed through the calibration of the system using particles of known

scattering cross section. Polystyrene Latex Spheres (PSL) with a refractive index of 1.58

and crown glass beads with refractive index of 1.51 (at the wavelength of 680 nm used in

DMT instruments) are used for this purpose. These particles are commercially available

and come in a range of sizes. Each set of beads is also relatively monodispersed with

standard deviations about the mean size of only a few percent.

In order to obtain the scale factor between digitized volts and scattering cross section,

the calibration particles are passed through the system and an average, peak voltage, V0,

is measured. The scale factor is S = I0/V0, where I0 is the scattering cross section for a

calibration particle with diameter, D, and refractive index, m, for an optical detection

system with an incident wavelength of λ, and collection angles of θ0-θ1. This scattering

cross section is calculated using Mie theory and can be computed with programs such as

described by Bohren and Huffman (1983).

Figure 1.5 shows the scattering cross section as a function of diameter and the optical

configuration of the CAS and CDP. The dashed line is the scattering cross section for

crown glass beads and the solid line is the equivalent was water droplets.

Sizing Detector

Backward Scatter

Detector (S-pol)

Laser Diode

(polarized)

Qualifying

Detector

Backward Scatter

Detector (No-pol)

Polarized Filter



Figure 1.5

With this scale factor, the signal measured from each particle is converted to a diameter

by multiplying the peak voltage by the scale factor to obtain the scattering cross section

and then going to a chart like Figure 1.5, or a lookup table stored in the data acquisition

system, to find the diameter of the particle that has the calculated scattering cross

section.

A pause here is needed to emphasize a very important point: single particle optical

scattering spectrometers do not measure particle size; they measure optical

scattering cross section. Hence, in order to determine particle size, the refractive

index of the particle that scattered the light must be known or assumed and it must

also be assumed that the particle was spherical.

From Figure 1.5 it is seen that particles with the same size but different refractive

indices may have substantially different scattering cross sections. Put a different way,

0 5 10 15 20 25 30 35 40 45 50

Particle Diameter ( m)

0.00E+000

2.50E-007

5.00E-007

7.50E-007

1.00E-006

1.25E-006

1.50E-006

1.75E-006

2.00E-006

2.25E-006

2.50E-006

2.75E-006

3.00E-006

3.25E-006

3.50E-006

3.75E-006

4.00E-006S

catt

eri

ng

Cro

ss S

ecti

on

(cm

2) Water

Glass


Form DOC-0222 1 0 © 2009 DROPLET MEASUREMENT TECHNOLOGIES, INC.

depending on the assumed refractive index, the scattering cross section derived from the

measurement will lead to different particle diameters. For example, in Fig. 1.5, a 25 μm

water droplet and 30 μm glass calibration particle have the same scattering cross section.

To further complicate the issue of sizing, for the same refractive index, two particles

with different sizes can sometimes have nearly identical cross sections. For example, as

seen in Fig. 1.6, 5 and 7 μm water droplets have almost the same cross section. This

measurement limitation is elaborated further in section 1.2.1.

Figure 1.6

The conversion from signal voltage to particle size is further complicated in the CAS, the

OPC that covers the size range from 0.5 to 50 μm. The scattering cross sections of water

droplets in this size range cover almost six orders of magnitude. The linear amplifiers

used in the CAS, and their associated A/D converters, can only measure over two orders

of magnitude. Hence, the CAS uses three sets of electronics, each with different

amplification, to measures this range of diameters. This requires that the calibration is

done with particles whose sizes will fall in each of the gain stages and three scale factors

are derived to relate the digitized volts to scattering cross section.

The DMT spectrometers can transmit two types of information: digitized counts from

individual particles, also known as particle-by-particle (PBP) and frequency distributions

of particle sizes. In the PBP case, the counts range from 0 to 3072 for the CAS and 0-4095

for the CDP. These can be subsequently scaled to scattering cross sections and converted

0 1 2 3 4 5 6 7 8 9 10 11 12

Particle Diameter ( m)

0.00E+000

5.00E-008

1.00E-007

1.50E-007

2.00E-007

2.50E-007

3.00E-007

Scatt

eri

ng

Cro

ss S

ecti

on

(cm

2) Water

Glass



to a size as discussed previously. Section 1.3 describes in greater detail the utility of PBP

and how additional information about particle properties can be obtained from these

data.

The frequency distribution information comes in the form of a table of values that is

generated by the spectrometer and transmitted at fixed intervals (usually once per

second) to the data acquisition system. The table consists of a preselected number of

channels (bins), usually 30, where each bin represents a voltage range given as A/D

counts. The scattering intensity of each particle that is measured within the sample

period is digitized and represents a single event that is added to the bin whose A/D

counts (scattering intensity interval) encompasses that of the measured particle.

The method for selecting the channel intensity thresholds is described in Section 1.3.

Suffice to say at this point that these are typically set to represent evenly spaced size

intervals of particles with predetermined refractive index, such as water droplets. Since

these thresholds are scattering cross sections, not particle diameters, they can be

derived for whatever refractive index is assumed, i.e. if aerosols like ammonium

sulfate are being measured, the size thresholds will be different than if measuring

water but the scattering cross sections remain the same.

1.1.2 Sample Volume Determination

In addition to deriving the size from the scattered light, the particles counted and sized

in a selected time period must be converted to a number concentration, i.e. number of

particles per unit volume or mass of air. This requires a definition of the sample volume.

The sample volume is the amount of air that passes through the sensitive area of the

laser beam during the sample period. This is defined as:

Sample Volume = (Sample Area)(airspeed)(sample time) (1.1)

In the open-flow CAS, CDP and Fog monitor, particles pass randomly through various

sections of the beam. Given that the laser beam does not have uniform intensity across

its cross section, and because it is focused so that it is most intense in the center of the

sample tube or between the arms of these instruments, a technique is implemented to

only accept and convert to a size those particles that pass through the most intense

section of the laser.

The sensitive area of the laser beam is defined with a combination of optical and

electronic techniques that selects a depth-of field (DOF) that is a certain length along the

beam, and an effective beam diameter (EBD) that is a portion of the beam width. This is

accomplished using two detectors as is shown above in Fig. 1.4. The scattered signal is

divided with a beam splitter into two components that are measured independently by



sizing and qualifier detectors. The qualifier detector has an optical mask that is in the

shape of a rectangle. As shown in Figure 1.7, this mask allows a full image of a particle

that is in the beam to appear at the detector only when the particle passes within a

certain distance of the center chord through the beam. When the particle passes through

the center of focus, the image is distinct and represents the true size of the particle. As

the particle moves away from the center of focus, the image seen at the detector

becomes larger and fuzzier and the complete image will no longer be within the edges of

the optical mask.

As a result, as compared with the detector that has no optical mask, the measured signal

will be smaller. This is illustrated in Figure 1.8 by the dashed line that represents the

voltage signal from the qualifying detector and the solid curve is the signal from the

sizer. The qualifier signal is larger than the sizing signal when a particle is in focus

because the beam splitter divides the scattered light in such a way that the qualifier

receives more light than the sizer, typically 67% for the qualifier, 33% for the sizer (the

CDP uses a 50:50 split in signal), when a particle is in focus. This is done in order to keep

the gains of the two detection systems the same while providing a good discrimination

between the qualifier and signal when particles are in and out of focus.

Figure 1.7

Out of focus

Depth of Field

Beam

Diameter

In focus

Center of Focus



Figure 1.8

As shown in Fig. 1.7, the shape of the sample area is not actually rectangular but is more

like an oblate spheroid due to the inherent nature of the optical definition of the depth

of field. The sample area is mapped at DMT using a droplet stream moved with a micro-

positioner.

1.2 Uncertainties and Limitations

The uncertainties and limitations of the light scattering probes are associated with the

derivation of size from light scattering, with the determination of the sample volume,

with issues of counting statistics and coincidence and with contamination from particles

produced from shattering of water drops and ice crystals on the inlet (CAS) or arms (CDP)

of the sensors.

The errors in counting and sizing propagate into parameters that are derived from these

measurements, i.e. number, area and mass concentrations and size parameters like the

median volume diameter and effective radius. The limitations and the associated

uncertainties are described in the following sections and the estimated uncertainties in

derived parameters are elucidated in section 1.3.

Sizing Signal

Qualifier Signal

Out of Focus

Qualifier Signal

In Focus



1.2.1 Sizing Errors

The sizing accuracy is affected by several factors. The lasers used in the probes have a

Gaussian intensity distribution, i.e. the center of the beam is the most intense with

exponentially decreasing intensity towards the edge of the beam. The optical mask on

the qualifying detector has a dimension such that only particles that pass through the

beam where the intensity is within 15% of the maximum intensity will be accepted. This

means that for a given particle size, the measured scattering intensity has an uncertainty

of approximately ±5%, assuming that on average, particles pass through the portion of the

beam with approximately 90% of the maximum intensity and that the system is calibrated

under that assumption. The uncertainty in the sizing, however, may be more or less than

±5% depending on where on the scattering cross section curve the size is being derived. In

general, the uncertainty is larger at larger sizes where the slope of the scattering cross

section as a function of size is less steep than at smaller sizes. Mis-sizing will also occur

as a result of particles coincident in the beam which will be detected as a single particle

but sized somewhat larger. This type of sizing error will usually be negligible and, at

present, no attempt is made to correct the data for this event.

The multi-valued nature of the scattering function, and its sensitivity to refractive index

and shape, discussed in section 1.1.1, presents the largest uncertainty in deriving particle

size. As shown in Fig. 1.9, the actual, atmospheric size distribution may have a smooth,

Gaussian-like shape (solid, blue curve) but because of the multi-valued scattering

function, the measured size distribution will have anomalous dips as shown by the solid

black curve.

By having multiple bins in the regions of ambiguity, the spectra can be sensibly smoothed

by combining the counts in channels that adjoin the delinquent sections, i.e. where

fluctuations in the spectra are counter to our understanding of physical processes. For

example, in clouds, although bi-modal size distributions are frequently found and we

understand this process as a result of entrainment and mixing, we do not expect to have

abrupt, anomalous dips in the size spectrum as seen in Fig. 1.9. In this figure, channels

have been combined to produce an adjusted size spectrum with smoother shape, albeit

with less resolution.



Figure 1.9

Another sizing uncertainty implied in the principles of operation is that these probes

cannot discriminate between water and ice particles. Ice particles pass through the

sample volume of these probes in random orientation and the measured sizes will depend

upon the orientation and the shape of the crystals. Ice particles outside the nominal

sample area of these probes will sometimes be sized and counted if they fall partially

within the sample volume. In general, aspherical particles will be undersized; however, if

assumptions are made about the shape, corrections can be applied (Borrmann et al.,

2000).

2 4 6 8 10 12 14 16 18 20

Particle Diameter (um)

0

50

100

150

200

Co

nc

en

tra

tio

n

Measurements with Original Bins

Measurements after grouping bins



The uncertainty in deriving particle diameter depends, in part, on the diameter of the

particle, its shape (if aspherical) and its refractive index (if different than what is

assumed when relating its scattering cross section to a size). On average, the estimated

error in diameter is ±20%. This error can be much larger in certain size ranges and if the

particle shape differs greatly from spherical.

1.2.2 Counting and Concentration Errors

Counting errors will occur when there are high concentrations and two or more particles

are coincident in the beam volume. Within the nominal depth of field of 1.5 mm,

qualifier width of 0.10 mm and beam diameter of 2.0 mm, the volume within which two

or more particles could be coincident and counted as only one is approximately 0.3 mm3.

Assuming that particles in the atmosphere are uniformly, randomly distributed, with their

spacing described by Poisson statistics, the probability of more than one particle in the

beam volume is less than 1% until particle concentrations exceed 1000 cm-3.

The determination of number, surface (extinction), or mass (water content)

concentrations is sensitive not only to the counting accuracy but to the uncertainty in

defining the volume of air within which particles are counted. As previously described in

section 1.1.2, the sample volume is the product of the effective beam diameter, depth of

field, air speed and sample period. The spectrometers have been designed to allow air to

pass isokinetically through the laser beam so that the air is neither accelerated nor

decelerated with respect to environmental air. This does not account for mounting

location, however, where the aircraft fuselage or wings may cause airflow distortion (for

additional information see further reading, Appendix I).

The cross-sectional area of the beam that sweeps out the sample volume is the product

of the effective beam diameter and depth of field, both determined by the optical

magnification of the collection optics and the geometry of the optical mask that is placed

in front of the qualifying detector (Fig. 1.7). The form of this cross-sectional area is not a

precise rectangle but has its maximum dimension in the center of focus, then tapers

slightly towards the edges of the DOF, each side of the center of focus. This produces a

shape that is like a prolate spheroid and presents a challenge for a precise measurement.

As previously stated, the sample area has been mapped using a droplet stream moved

with a micro-positioner. The area measured this way has been validated by producing

monodispersed particles, e.g. PSL or glass beads, in very clean air, free of any

background particles, and measuring the concentration of these particles with a

condensation nucleus (CN) counter sampling air after it has passed through the CAS and

CDP. The CN counter is a high precision particle counter with a well-defined sample

volume. The sample area of the DMT spectrometers was estimated by comparing with the

CN measurements and the uncertainty is approximately ±20%.



1.2.3 Contamination from Drop and Crystal Shattering

It has been recognized since 1985 (Gardiner and Hallett, 1985) that the inlets and arms

on particle probes were a potential obstruction to the airflow and provided surfaces

where water droplet and ice crystals can impact and produce secondary fragments, some

of which arrive in the measurement sample volume.

More recent studies (McFarquhar et al. 2007; Heymsfield, 2007) have shown correlations

between anomalously high concentrations of small particles (< 20 μm), measured with

light scattering probes, and the ice water content in ice crystals larger than 100 μm. This

is indirect evidence that large ice crystals are shattering and producing fragments in the

measurement volume of the instrument. Figure 1.10 is another way to look at the

problem whereby comparisons are made of the measurements with the CAS and CIP

(cloud imaging probe). The CIP is also susceptible to crystal shattering on its arms but the

measurements shown in this figure have been corrected for this problem (see section

2.2.3). The CIP in this example is the gray-scale imaging probe with 15 μm resolution so

that it has an overlap in sizes with the CAS from 15-45 μm. What is shown in Fig. 1.10 is

that when the ice crystals are small and in low concentrations, there is a good agreement

between the CAS and CIP measurements in the overlapping size range. As the ice crystals

become larger and in higher concentration, the discrepancy increases, indicating that the

CAS measures what are presumably false particles, i.e. those being produced through

fragmentation.

This generation of false particles is a major source of error in the CAS measurements.

This problem is under investigation and solutions are being assessed to resolve this

problem. Flow simulations have been conducted to assess the magnitude of the problem

and to develop algorithms that correct those measurements that have already been

made.

As shown in Fig. 1.11, flow field analyses with the commercial software, Fluent, are used

to compute the three dimensional pressure, temperature and velocity fields around the

DMT instruments, in this case the CAS. These flow fields, dependent on ambient pressure

and temperature, are then used as the boundary conditions for trajectory analysis of

particles embedded in these fields. Figure 1.12 shows an example of particle trajectories

computed from the flow fields. What is important to note is that there is a very small

subset of particle trajectories that eventually result in a particle measured in the CAS

sample volume. The analysis shows that only those particles whose trajectories result in a

reverse course after breaking or bouncing will produce a path that eventually arrives in

the probe sample volume.



Figure 1.10

Figure 1.11

T = 200 K

P = 100 mb

AOA = 0o

Elasticity = 0.2-1.0

Exit Angles = All possible

Fragment diameters = 5-50 μm

Fragment density = 0.2 to 0.9 g cm-3

Flow modeling with FLUENT

Unknowns



Figure 1.12

On average, this is less than 1% of the particles that strike any segment of the inlet. On

the other hand, because of the surface area of the inlet, approximately 5 cm2 for the

original CAS (the new design uses a much thinner inlet), each second the CAS sweeps out

a volume of 1000 cm2 at an airspeed of 200 m-1, the velocity of the NASA WB57F. If we

assume that ice crystals break into evenly sized spheres of 15 μm when they strike the

inlet, we can calculate the potential number of fragments that would be measured by the

CAS. Figure. 1.13 shows the relationship between concentration of particles produced

from fragmentation and ice crystal volume, using measurements from three projects

when the CAS was flown on the WB57F. The individual points are the actual

measurements from the CAS and CIP and the solid lines are predictions based upon the

flow model predictions for the fraction of 15 μm particles that would be produced and

arrive in the sample volume. The inlet of the CAS also has a shroud whose purpose is to

reduce sensitivity to changes in flow angle due to aircraft attitude. As seen in Figure

1.13, (black curves), this shroud introduces an additional surface where particles will

shatter. What is clear from the figure is that the slope of the measurements is very

Particle Diameter

10 m, =0.9

30 m, =0.9

45 m, =0.9

10 m, =0.5

30 m, =0.5

45 m, =0.5



similar to the predicted slope and that the shroud is an additional source of crystal

fragments.

The fragments are embedded in the field of real particles and represent a bias that needs

to be removed. During the CRAVE experiment, the shroud was removed and the inlet

made much thinner. This clearly decreased the number of fragments, as seen by the

comparison with predicted concentrations (top panel, Fig. 1.13), but at higher ice crystal

concentrations there remains a clear fragmentation signature.

Figure 1.13

Corrections algorithms are under development and will be implemented to remove

some fraction of the erroneous particles in the CAS measurements from previous

projects. The technique uses the CIP measurements as a benchmark against which to

compare those from the CAS and estimate the number of fragments being produced as a

function of size.

1.0E+002 1.0E+003 1.0E+004 1.0E+005 1.0E+006 1.0E+007 1.0E+008

volume ( m3 cm-3)

1E-002

1E-001

1E+000

1E+001

1E+002

1E+003

1E+004

1E+005

Nu

mb

er

co

nc

(l-1

)

Measurements - 15 m

Inlet Breakup - 15 m

Inlet + shroud Breakup - 15 m

CRYSTAL/FACE

1E-002

1E-001

1E+000

1E+001

1E+002

1E+003

1E+004

1E+005

Nu

mb

er

co

nc

(l-1

)

MidCix

volume ( m3 cm-3)1E-002

1E-001

1E+000

1E+001

1E+002

1E+003

1E+004

1E+005

Nu

mb

er

co

nc

(l-1

) CRAVE



The CAS inlet is also being redesigned to further limit the problem of crystal

fragmentation and contamination of the measurements. This manual will be revised at a

later date when the correction algorithms have been validated and a methodology

developed that can be implemented by users to correct measurements prior to the date

when a new CAS inlet is implemented.

1.3 Data Analysis

1.3.1 Size Distributions

The most commonly used analysis technique for evaluating particle measurements is the

display of particle concentration as a function of size. The concentrations can be

number, water content, surface area or any other extrinsic property derived with respect

to the volume or mass of air. Prior to constructing such size histograms, however, the

concentrations must be binned according to size. It may be recalled that the scattering

probes transmit a frequency histogram with a predefined number of bins and the

thresholds of the bins are predefined in terms of the A/D counts that are related to the

scattering intensity. It then remains to associate these counts to an optical diameter

(NOTE: the term optical diameter is used here instead of geometric diameter because

of the method being used to determine the size).

The channels thresholds are typically determined prior to a measurement campaign and

reflect calibrations with particles of known size, as discussed in Section 1.1.1, and are

adjusted for the expected particle composition, e.g. water or haze droplets. It is highly

recommendable, however, to recalculate the sizes associated with the A/D counts after

post-project calibrations. It is also sometimes of interest to see how the shape of the size

distribution might change if the particles measured were some refractive index other

than the one assumed when the channels were defined.

As a demonstration of how this is done, we take a CAS as the target instrument since it is

the more complicated to set up, due to the three gain stages. The CAS used in this

example has been set up to measure nominally between 0.3 and 20 μm.

Step I

The scale factors for the three gain stages are computed from the measurements with

calibration particles. In this example, 0.65 μm PSL particles were used to calibrate the

high gain (channels 1-10), 2 μm PSL particles calibrated the mid-gain (channels 11-20)

and 15 μm glass beads calibrated the low gain. These three calibrations set the scale

factors, Shigh, Smid and Slow. Caution! Although the counts in the CAS are listed in the

setup tables ranging from 0 to 3072 over the 30 channels, each gain stage is actually

represented by A/D counts from 0-1023. If the 0.65 μm particles peak in channel 8



with A/D counts of 600, then the scale factor for this stage is Ihigh/600, where Ihigh is

the scattering cross section for a 0.65 μm PSL particle. If the 2 μm particles peak in

channel 16, with A/D counts of 1400 as the channel threshold, the scale factor, Smid

for this stage is Imid/(1400-1023), where Imid is the scattering cross section for 2 μm

PSL. Likewise, the low gain average A/D count must have 2047 subtracted before

calculating the scale factor.

Step II – 1st option

The A/D counts are converted to scattering cross sections using the scale factor. In this

case, the counts have already been set and the goal is to associate a diameter with the

counts. The counts are converted to a scattering cross section by multiplying the

threshold counts of each channel for the scale factor associated with the appropriate

gain stage. Caution! As in the case of calculating the scale factors, the mid gain counts

must have 1023 subtracted and the low gain counts must have 2047 subtracted

before applying the scale factor.

Step II – 2nd option

In this case, the goal is to determine the threshold counts for the channels based upon

preselected sizes and refractive index for the particles. Here, we go to the table of

scattering cross sections as a function of size and refractive index, select the 30 sizes

that we wish to associated with the 30 channels, and extract the scattering cross sections

for each of the sizes. These values are converted to counts by dividing by the scale

factor (recall that the scale factor has units of cm2/counts). The subsequent counts will

all range between 0 and 1023, so the user must decide whether to add 1023 or 2047

based upon the range of sizes and which gain stage will contain each size range.

Step III – 1st option

Each of the scattering cross sections must be associated with a size by using the table of

scattering cross sections as a function of size and refractive index. For a given refractive

index, several sizes may have similar scattering cross sections. This requires that an

average is calculated of all the possible sizes and used to associate the counts with a

size.

Step III – 2nd option

There is no third step for the option of preselecting sizes and channel counts.

Table 1.1 shows how the 30 channels were set up and the sizes associated with the A/D

counts, assuming refractive indices of 1.33 (water) and 1.44 (sulfuric acid). The effect of

changing the composition of the particles is to slightly increase the range of sizes from

0.41 to 36 μm to 0.37 to 40 μm. In general, increasing the refractive index will increase



the range of sizes because the slope of the scattering cross section, for this range of

collection angles and incident wavelength, decreases as refractive index increases.

Adding a light absorbing component via the imaginary part of the refractive index has a

similar impact.

Table 1.1: CAS Setup Table (Forward Scattering Only)

CH# Forward

Counts

Forward

Intensity

Forward Size

(water, m=1.33)

Forward Size

(m = 1.48)

1 60 6.357E-011 0.41 0.37

2 90 9.536E-011 0.44 0.4

3 131 1.388E-010 0.47 0.43

4 186 1.971E-010 0.5 0.46

5 259 2.744E-010 0.53 0.49

6 352 3.73E-010 0.56 0.51

7 470 4.98E-010 0.59 0.54

8 618 6.548E-010 0.62 0.56

9 800 8.477E-010 0.65 0.59

10 1024 1.085E-009 0.68 0.62

11 1084 3.018E-009 0.83 0.78

12 1124 5.03E-009 0.93 0.88

13 1159 6.791E-009 0.99 0.97

14 1224 1.006E-008 1.11 1.31

15 1309 1.434E-008 1.21 1.91

16 1414 1.962E-008 1.81 2.01

17 1549 2.641E-008 2.51 2.11

18 1684 3.32E-008 2.61 2.15

19 1849 4.15E-008 2.72 2.21

20 2024 5.03E-008 2.8 2.31

21 2109 1.156E-007 5.6 5.71

22 2136 1.668E-007 8 8.91

23 2183 2.559E-007 10.2 11.51

24 2268 4.17E-007 15 17.21

25 2373 6.16E-007 18 20.81

26 2438 7.392E-007 20.5 23.61

27 2538 9.287E-007 23.5 25.91

28 2693 1.223E-006 26.8 30.91

29 2848 1.516E-006 31.4 34.01

30 3072 1.941E-006 35.8 40.21



Once the size thresholds are determined, the concentrations can be displayed in

graphical form by creating a plot of the concentrations as a function of particle size.

Figures 1.14 and 1.15 illustrate a size histogram of number concentration as a function of

particle diameter, displayed in a number of ways. Table 1.2 lists the channel definitions

assuming water droplets and for the original 30 channels. Also shown in this table is a

new size definition where channels have been combined to encompass those regions

where there are multiple values of size that have the same scattering cross sections. In

Fig. 1.14 the same information has been displayed but using the original and a

redistribution of the information in the size channels and by normalizing by the

logarithm, base 10, of the width of each size bin (also listed in Table 1.2).

Table 1.2: CAS Setup Table with and without Combined Channels

Channel Original

Size

Log10

Width

New Size Log10 Width

1 0.60 0.08 0.5 - 5.0 1.00

2 0.70 0.07 5.0 -10.0 0.30

3 0.75 0.03 10.0 - 15.0 0.18

4 0.80 0.03 15.0 - 20.0 0.12

5 0.90 0.05 20.0 - 25.0 0.10

6 0.95 0.02 25.0 - 30.0 0.08

7 1.0 0.02 30.0 - 35.0 0.07

8 1.1 0.04 35.0 - 40.0 0.06

9 1.2 0.04 35.0 - 45.0 0.05

10 1.25 0.02 45.0 - 50.0 0.05

11 1.5 0.08

12 2.0 0.12

13 2.5 0.10

14 3.0 0.08

15 3.5 0.07

16 4.0 0.06

17 5.0 0.10

18 6.5 0.11

19 7.0 0.03

20 8.0 0.06

21 10.0 0.10

22 12.5 0.10

23 15.0 0.08

24 20.0 0.12

25 25.0 0.10

26 30.0 0.08

27 35.0 0.07



28 40.0 0.06

29 45.0 0.05

30 50.0 0.05

The black, solid line is the average number concentration, as a function of the 30 sizes

defined in Table 1.2, for all cirrus clouds measured on May 2, 2004 by the CAS on the

NASA WB57F during the MidCix project. The roughness or irregularities are a result of the

multi-valued nature of the scattering cross sections. The irregularities are also a result of

non-uniform intervals in the sizes, i.e. in the small sizes some of the intervals are 0.05

μm wide, whereas others are 0.1 or 0.5. Normalization by the width of an interval, i.e.

dividing the concentration by the width of the size intervals decreases the irregularity, as

shown by the black, dashed line in Fig. 1.14. In this case the base 10 logarithm of the size

interval has been used (log10(di+1/di) for the reason that is discussed below.

Combining those channels whose size thresholds encompass regions where the Mie

scattering function is multi-valued (illustrated in Fig. 1.6) further removes the

irregularities in the size distribution as shown by the solid red line in Fig. 1.14. Some of

the detail in the size distribution, of course, is also removed by combining the channels.

As Table 1.2 shows, the newly defined size thresholds also are not uniform in their

intervals so that the concentrations need to be normalized by the width of each size

channel. In addition, in order to compare size distributions, be they from different

instruments with different size definitions or from the same instrument with different

size intervals, normalization is mandatory. Figure 1.14 shows such a comparison and

illustrates how the subsequent shape of the distributions can look quite different

dependent on how the information is presented.



Figure 1.14

The size distributions in Fig. 1.14 are displayed on a log-log plot in order to show the

details of size distribution. It is often better, however, to display the concentrations on a

linear-log plot, as seen in Fig. 1.15. The reason is that from the graphical perspective,

when the concentrations are normalized by the base 10 logarithm of the size interval and

when the abscissa is plotted as a base 10 logarithmic scale, the area under the curve

between two diameters, di and di+1, is directly proportional to the total concentration of

particles within this size range. This allows by quick inspection an estimate of where the

majority of particles are found, e.g., in this case, between 5 and 20 μm.

Likewise, by computing the extinction and water mass as a function of size (described in

Section 1.3.2), the graphical representation of these size distributions represents the

measurements so that we can see which size of particles contribute the most to these

two derived parameters that are important for assessing impacts on climate and on

precipitation or hydrological impacts, respectively. This is illustrated in Figs. 1.16 and

1.17.

0.01

0.1

1

10

100

Nu

mb

er

Co

nce

ntr

ati

on

5 6 7 8 9

12 3 4 5 6 7 8 9

102 3 4 5

Diameter ( m)

All channels All channels normalized Combined channels Combined channels normalized



Figure 1.15

100

80

60

40

20

0

Nu

mb

er

Co

nce

ntr

ati

on

5 6 7 8 9

12 3 4 5 6 7 8 9

102 3 4 5

Diameter ( m)

All channels normalized Combined channels normalized



Figure 1.16

Figure 1.17

40

30

20

10

0

Exti

ncti

on

(m

-1m

-1)

5 6 7 8 9

12 3 4 5 6 7 8 9

102 3 4 5

Diameter ( m)

All channels Combined channels

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

Liq

uid

Wate

r C

on

ten

t (g

m-3

m-1

)

5 6 7 8 9

12 3 4 5 6 7 8 9

102 3 4 5

Diameter ( m)

All channels Combined channels



1.3.2 Derivation of Bulk Parameters

The particle distributions measured by the particle spectrometers can provide estimates

of many parameters that are relevant to atmospheric processes. Tables 1.3 and 1.4

summarize these parameters and the equations that relate the parameters to the size

distributions. Some of the more commonly derived parameters are described in sections

1.3.2.1 – 1.3.2.5.

The uncertainties that are estimated in the following sections use error propagation

whereas the error in a derived parameter is estimated from the errors in counting, sizing

and sample volume discussed in section 1.2. In general, the uncertainty, σ2, in a derived

parameter, P, is equal to

σP2 = σx1

2 + σx22 + …. + σxn

2 (1.2)

where x1 … xn are the n independent variables and σx12 + σx2

2 + …. + σxn2 are the

estimated uncertainties in these variables. This methodology is often referred to as the

„root sum squared‟ approach to estimating uncertainties from propagated errors.

Table 1.3: Cloud Properties, Hydrometeor Characteristics, and Environmental Impact

Cloud Property Hydrometeor

Characteristic

Environmental Impact

Albedo Number and Surface Area

Phase function and

Extinction

Effective Radius

Climate

Lifetime Number and mass

Fall Velocity

Climate, Weather, and

Hydrological Cycles

Spatial Distribution Number, surface area and

mass

Fall Velocity


Hydrological Cycles

Precipitation Efficiency

and Rain rate

Number and mass

Fall Velocity


Hydrological Cycles

Chemical Processing

Efficiency

Number, Surface Area and

Mass Concentration

Fall Velocity

Climate and Hydrological

Cycles



Table 1.4: Hydrometeor Properties

Number concentration (cm-3) N =

m

i

ic

1

Surface area concentration ( m2cm-3) S =

2

1

i

m

i

iidcs

Mass concentration (g m-3) M =

3

1

6 i

m

i

iiidcs

Fall Velocity (cm s-1) Vt = f(di, g, a, CD)

Phase Function P , = drrcrrF

r

r

)(),,,(2

2

1

Extinction (m-1) Be = drrcrrQ

r

r

e)(),,(

2

2

1

Effective Radius ( m)

Re = m

i

ii

m

i

ii

dc

dc

1

2

1

3

4

3

m = number of size categories

ci = number concentration of hydrometeors in size category, i.

di = average diameter of size category, i.

ri = average radius of size category, i.

si = shape factor of the hydrometeor of size category, i, to account for asphericity

(this factor depends upon the intrinsic property being defined)

i = density of the hydrometeor in size category, i.

a = air density

g = gravitational acceleration

CD = drag coefficient

F = Angular scattering intensity efficiency

Qe = Extinction efficiency

= hydrometeor refractive index

= wavelength of incident light

= Angle of light scattered from the hydrometeor



1.3.2.1 Total Number Concentration

Particle number concentration is defined as the number of particles per unit volume and

is also the 0th moment of the size distribution. In the case of the light scattering probes

with size ranges from 0.1 to 50, the conventional units are in the number of particles per

cubic centimeter. The method of calculating the total concentration is

SV

n

C

m

i

i

T

1 (1.3)

where

CT = total concentration units = # cm-3

ni = number of particles accumulated in channel i

m = total number of size channels

SV = sample volume as defined in equation 1.1

The estimated uncertainty in number concentration, ignoring coincidence, is ±20%, due

to the uncertainties in sample volume determination.

1.3.2.2 Extinction Coefficient and Visibility

The extinction coefficient is proportional to the geometric cross section of particles and

for spherical particles over the size range from r1 to r2 is defined as

m

i

iieedcdQB

1

2),,(

4 (1.4)

where

Qe = Extinction efficiency

= hydrometeor refractive index

= wavelength of incident light

ci = number concentration of hydrometeors in size category, i.

di = average diameter of size category, i.

For cloud droplets and visible wavelengths, Qe is approximately 2 and the extinction

coefficient is just twice the optical cross section. In equation 2.3 the units of σe are in

inverse length. The extinction coefficient is normally expressed in inverse meters,



kilometers or mega-meters, depending on whether it is being computed for aerosols or

clouds.

Visibility is inversely proportional to the extinction and is calculated via the Koschmeider

equation, 3.91/σe.

The uncertainty in extinction is:

σe = (σθ2 + σc

2 + 2σd2)1/2 = ((20%)2 + (20%)2 + 2(20%)2)1/2 = ±40% (1.5)

where

σθ = uncertainty in the extinction efficiency

σc = uncertainty in the number concentration

σd = uncertainty in the diameter

Here we multiply the uncertainty in the diameter by two because the diameter is squared

in equation 1.4.

1.3.2.3 Liquid Water Content

The liquid water content, W, is the summation of the individual particle masses per unit

sample volume,

m

i

ii

wdcW

1

3

6 (1.6)

where ρw = density of water.

Note, given that W is usually expressed in g m-3, the value of ρw is 1.0e6 g m-3 and ci

and di are converted to units of # m-3 and meters, respectively.

The uncertainty in W is:

σw = (σc2 + 3σd

2)1/2 = ((20%)2 + 3(20%)2)1/2 = ±40% (1.7)

where

σc = uncertainty in the number concentration

σd = uncertainty in the diameter

Here we multiply the uncertainty in the diameter by three because the diameter is

squared in equation 1.6. Note: this uncertainty can be much larger if the particles are

aspherical and the density is different than that of water.



1.3.2.4 Mean, Median Volume and Effective Diameter

The mean diameter is the arithmetric average of all the particle sizes, also known as the

1st moment of the size distribution, and is calculated by

T

m

i

ii

C

dc

D1 (1.8)

The uncertainty in the mean diameter is approximately the same as the uncertainty in

diameter, ±20%, even though in (1.8) there is the concentration parameter also in the

equation. Given that concentration appears in the numerator and denominator of (1.8),

on average the uncertainty in this variable cancels and it is only the uncertainty in

determining the size that dominates.

The median volume diameter is the size of droplet, below which 50% of the total water

volume resides. This is computed as follows.

Step 1: Compute the liquid water content (2.4)

Step 2: Beginning at the first size channel and calculate the accumulated mass, Sn= w1+

w2 + ... wn, where w1 is the mass of water in channel 1 and wn is the channel where the

accumulated mass > 0.5w, i.e. greater than or equal to 50% of the total LWC.

Step 3: Compute the median volume diameter, Dmvd, by interpolating linearly between

the channels that bracket where the accumulated mass exceeded the total LWC

Dmvd = Dn-1 + (0.5 – Sn-1/Sn)*(Dn-Dn-1) (1.9)

where

Sn = the sum of water masses in channels 1 to n, up until S> 0.5w

Sn-1 = the sum of water masses in channels 1 to n-1

w = total liquid water content

Dn = the diameter of the size channel when S = Sn

Dn-1 = the diameter of the size channel when S = Sn-1

The effective radius is a parameter used by climate modelers to describe the optical

properties of cloud hydrometeors and also in retrievals of cloud properties from satellite

measurements. It is not so much a physical dimension of the particle population as a way

to parameterize the optical properties of this population as a function of the liquid water

content and the optical cross section of the particles. It has been expressed many ways

but is commonly seen in the literature (McFarquhar and Heymsfield, 1998) as



m

i

iw

e

r

wr

1

24

3 (1.10)

As discussed by McFarquhar and Heymsfield (1998), this definition is only meaningful for

water clouds. The definition becomes more complicated in ice clouds and the

denominator of 2.7 should be replaced with a cross sectional area that represents the

average population of non-spherical particles.

Given that operationally the effective radius is also defined as the ratio of the water

content to the extinction, although there is some correlation between the uncertainties

in these two parameters, a conservative estimate of the uncertainty in effective radius is

σRE = (σw2 + σe

2)1/2 = ((40%)2 + (40%)2)1/2 = ±57% (1.11)

1.3.3 Derivation of Refractive Index

The refractive index is derived using a look-up table that has the theoretical values for

the forward and backward scattering cross sections as a function of refractive index. As

shown for three refractive indices, in Fig. 1.18, there are various pairs of forward and

backscattering cross sections that are unique to specific refractive indices. Table 1.5 also

lists these pairs as a function of particle diameter and refractive index. In the PBP file for

the forward and non-polarized scattering data, the information is listed as counts. These

data must be converted to equivalent scattering cross section values as discussed above

in section 1.3.1.

To derive the refractive index of a particle, the measured forward and backscattering

counts must be converted to the corresponding scattering cross sections using the

calibration scale factors discussed in section 1.1.1 and 1.3.1. The second step is to search

the table for all occurrences of the forward scattering value. Since an exact match will

not necessarily be found, a range of values around the measured value can be prescribed

that is plus and minus some percentage of the measured value. For each of the forward

scattering values found in the table, compare the corresponding backscatter values with

the one measured. Again, a plus or minus range should be prescribed prior to the search.

When a pair of forward and backward scattering cross sections is found that match those

that were measured within the specified error for acceptance, this will indicate the most

probable refractive index for that particle.



In many cases there may not be a unique answer, i.e., as seen in Figure 1.18, there are

overlapping regions in the graph where particles with different refractive indices will

have similar forward to backward relationships. In those cases where more than one

match is found, the investigator can choose to discard the information or keep all the

matches but in a separate analysis category that indicates the possibility of more than a

single refractive index. If the particle population is assumed to be composed of

approximately the same composition, then for an ensemble of particles of many sizes,

those sizes that have a unique forward to backward relationship will help decide the

correct refractive index for those particles whose size is associated with more than one

forward to backward scattering pair.

Note that the refractive index can be derived only if the particles are assumed spherical.

In addition, the sample shown in Table 1.5 does not take into account any light

absorption by the particles, i.e. the complex refractive indices only include the real

component. Tables can also be calculated to include imaginary components but this

increases the complexity of the analysis as well as the resulting uncertainty.

Figure 1.18

1E-009 1E-008 1E-007 1E-006

Forward Scattering Cross Section (cm2)

1E-011

1E-010

1E-009

1E-008

1E-007

Bac

k S

ca

tteri

ng

Cro

ss S

ec

tio

n (

cm

2)

= 1.33

= 1.44

= 1.60



Table 1.5: Sample Table of Forward and Backward Scattering Cross Sections

Size

fwdrf1.3

3

bckrf1.3

3

fwdrf1.4

0

bckrf1.4

0

fwdrf1.4

4

bckrf1.4

4

fwdrf1.4

8

bckrf1.4

8

1 7.00E-09 3.01E-11 7.49E-09 2.85E-11 7.29E-09 4.12E-11 6.96E-09 8.79E-11

1.01 7.25E-09 2.65E-11 7.72E-09 2.79E-11 7.56E-09 4.86E-11 7.33E-09 1.10E-10

1.11 1.07E-08 3.05E-11 1.11E-08 8.72E-11 9.86E-09 1.10E-10 8.06E-09 1.27E-10

1.21 1.41E-08 4.33E-11 1.27E-08 5.51E-11 1.15E-08 9.86E-11 9.48E-09 1.87E-10

1.31 1.87E-08 2.27E-11 1.44E-08 9.12E-11 1.04E-08 1.62E-10 7.05E-09 2.96E-10

1.41 2.09E-08 5.33E-11 1.55E-08 8.62E-11 1.11E-08 1.03E-10 6.41E-09 1.64E-10

1.51 2.47E-08 2.43E-11 1.29E-08 1.10E-10 7.84E-09 3.14E-10 5.93E-09 4.89E-10

1.61 2.45E-08 6.28E-11 1.27E-08 1.01E-10 6.54E-09 1.23E-10 4.43E-09 2.63E-10

1.71 2.47E-08 6.51E-11 8.85E-09 2.05E-10 6.75E-09 3.48E-10 7.94E-09 3.60E-10

1.81 2.30E-08 1.08E-10 7.33E-09 1.87E-10 6.74E-09 3.21E-10 1.41E-08 5.82E-10

1.91 1.85E-08 1.57E-10 8.27E-09 3.06E-10 1.20E-08 3.07E-10 2.05E-08 5.19E-10

2.01 1.65E-08 2.24E-10 9.11E-09 4.02E-10 2.07E-08 5.52E-10 3.22E-08 4.43E-10

2.11 1.11E-08 3.16E-10 1.56E-08 4.52E-10 2.87E-08 4.76E-10 4.36E-08 1.16E-09

2.21 1.01E-08 3.99E-10 2.61E-08 7.42E-10 4.13E-08 6.48E-10 4.93E-08 6.27E-10

2.31 1.05E-08 5.53E-10 3.44E-08 7.04E-10 5.36E-08 1.13E-09 5.26E-08 7.46E-10

2.41 1.23E-08 6.25E-10 4.90E-08 7.33E-10 5.88E-08 9.74E-10 5.46E-08 1.77E-09

2.51 2.05E-08 7.02E-10 5.91E-08 1.29E-09 6.28E-08 7.01E-10 4.65E-08 1.03E-09

2.61 2.82E-08 9.77E-10 6.87E-08 1.04E-09 6.16E-08 1.63E-09 3.68E-08 9.28E-10

2.71 4.13E-08 8.41E-10 7.40E-08 1.06E-09 5.53E-08 1.43E-09 3.01E-08 2.56E-09

2.81 5.51E-08 1.31E-09 7.28E-08 1.19E-09 4.54E-08 8.47E-10 2.19E-08 1.75E-09

2.91 6.68E-08 1.05E-09 6.88E-08 1.72E-09 3.56E-08 1.59E-09 1.88E-08 8.69E-10

3.01 8.07E-08 1.22E-09 6.16E-08 1.67E-09 2.79E-08 2.45E-09 2.57E-08 4.24E-09

3.11 8.75E-08 1.47E-09 4.73E-08 1.40E-09 2.31E-08 9.66E-10 3.53E-08 2.02E-09

3.21 9.44E-08 1.42E-09 3.95E-08 1.07E-09 2.33E-08 1.57E-09 4.88E-08 1.80E-09

3.31 9.47E-08 1.73E-09 3.12E-08 2.39E-09 3.02E-08 2.44E-09 6.41E-08 5.41E-09

3.41 9.16E-08 1.36E-09 2.69E-08 2.08E-09 4.46E-08 8.69E-10 7.66E-08 3.06E-09

3.51 8.68E-08 1.63E-09 2.88E-08 7.22E-10 5.61E-08 2.31E-09 8.48E-08 1.83E-09

3.61 7.76E-08 1.79E-09 3.27E-08 1.91E-09 6.97E-08 1.36E-09 8.53E-08 2.71E-09

3.71 6.84E-08 1.78E-09 4.62E-08 3.60E-09 8.35E-08 1.43E-09 8.08E-08 3.21E-09

3.81 6.05E-08 2.10E-09 5.94E-08 8.53E-10 8.88E-08 2.37E-09 7.02E-08 2.63E-09

3.91 5.08E-08 1.11E-09 6.87E-08 1.71E-09 9.06E-08 1.66E-09 6.36E-08 3.01E-09

4.01 5.03E-08 2.16E-09 8.06E-08 1.48E-09 8.81E-08 1.93E-09 5.11E-08 3.40E-09

4.11 4.46E-08 1.91E-09 8.84E-08 1.55E-09 8.08E-08 2.41E-09 4.81E-08 3.00E-09

4.21 5.25E-08 1.61E-09 9.40E-08 1.57E-09 7.37E-08 2.16E-09 5.13E-08 3.84E-09

4.31 5.28E-08 1.23E-09 9.44E-08 1.84E-09 7.04E-08 2.51E-09 4.99E-08 3.73E-09

4.41 5.85E-08 9.05E-10 9.28E-08 3.62E-09 5.97E-08 2.87E-09 6.02E-08 3.10E-09

4.51 6.27E-08 2.10E-09 8.71E-08 2.13E-09 6.41E-08 2.40E-09 6.49E-08 4.80E-09

4.61 6.62E-08 2.12E-09 8.89E-08 2.17E-09 6.01E-08 3.25E-09 7.43E-08 3.13E-09

4.71 8.03E-08 1.08E-09 8.18E-08 2.43E-09 6.36E-08 2.71E-09 7.73E-08 3.64E-09

4.81 8.16E-08 9.20E-10 7.99E-08 2.64E-09 7.05E-08 3.00E-09 8.02E-08 4.60E-09

4.91 8.87E-08 9.73E-10 8.29E-08 2.69E-09 6.79E-08 3.18E-09 8.46E-08 4.19E-09

5.01 8.51E-08 2.83E-09 7.56E-08 2.66E-09 7.54E-08 3.39E-09 7.86E-08 5.63E-09



1.3.4 Derivation of Asphericity

The particle shape is evaluated on a particle by particle basis using the same approach as

deriving refractive index, except in this case the refractive index is assumed constant and

the relationship between the forward to backscatter cross sections changes as a result of

shape deviation, i.e. asphericity. There are two approaches. Both approaches require

converting the counts to their respective scattering cross sections. Similar to the tables

for refractive index, a similar table has been constructed for the relationship between

the forward and backscattering as a function of the aspect ratio, where an aspect ratio of

unity designates a sphere, an aspect ratio smaller than one is an oblate spheroid (disk

shaped) and greater than one is a prolate spheroid (football shaped). Table 1.6I shows

the ratio of forward to backscattering for particles with radius between 1 – 10 μm with

the refractive index of ice (1.31) and a range of aspect ratios.

A slightly different methodology is used than with the estimation of refractive index.

First, the particle effective diameter is calculated from the forward scattering cross

section, where the effective diameter is found by assuming the refractive index of the

particle then finding the size in the Mie lookup table that corresponds to the measured

forward scattering signal. Then the ratio is calculated between the measured forward

and backscattering cross sections, after converting the A/D counts to scattering. The

aspect ratio is found from the look up table by going to the column that represents the

size closest to the derived size and finding the ratio closest to the calculated ratio from

the measurements. The value in the first column that is associated with that row will be

the estimated aspect ratio.

An alternative approach is to search the entire table for the forward to back scattering

ratios that most closely match the ratio calculated from the measurements. As seen in

the examples in Table 1.6, there are multiple values that depend on the various pairs of

size versus aspect ratio. From all the matches, the one associated with the particle radius

that most closely matches the size derived from the forward scattering will be used to

estimate the aspect ratio.

An empirical approach can also be taken that takes advantage of all three signals, the

forward scattering, non-polarized and polarized backscattering (only in the CAS-POL). For

every particle, three ratios are calculated: 1) forward to non-polarized backscatter

(F2Bnon), 2) forward to polarized backscatter (F2Bpol) and 3) nonpolarized to polarized

backscatter ratio (noPOL2POL). Figure 1.19 shows averages of these three ratios, along

with their standard deviations (vertical bars), as a function of the average forward

scattering value for water droplets, quartz and dust particles. From these comparisons,

we can see that there are some particle sizes, represented by the forward scattering

value, where there is a much greater difference between the ratios for the three types of

particles than for other sizes. This figure also shows that the forward scattering values



where the greatest separation is seen as a function of particle type is not the same for

each of the three ratios. This means that the three ratios can be used, in different

combinations with respect to the forward scattering values to differentiate between

different shapes.

Table 1.6: Sample Table of Aspect Ratios vs. Forward to Backscatter

Particle Radius

Aspect

Ratio 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

0.5 307 850 290 506 390 0 0 0 0 0

0.6 338 602 221 462 286 314 352 308 265 204

0.7 310 265 153 303 214 319 256 245 180 164

0.8 216 186 108 163 150 273 164 156 127 116

0.9 164 215 74 118 105 173 98 80 75 73

1 151 248 64 73 68 165 57 82 61 95

1.1 159 201 72 90 117 139 87 76 68 57

1.2 180 165 83 103 98 169 101 97 79 74

1.3 207 176 87 115 108 165 146 116 100 95

1.4 238 215 91 122 117 152 209 158 134 123

1.5 271 269 101 136 150 154 226 171 146 134

1.6 310 344 120 165 183 134 180 184 196 139

1.7 354 439 149 190 214 135 164 185 193 199

1.8 404 538 202 227 256 147 150 166 143 159

1.9 455 635 282 278 292 153 0 0 0 0



Figure 1.19

800 1000 1200 1400 1600 1800

Forward Scattering Counts

12

16

20

Fo

rwa

rd t

o N

on

-Po

l B

ac

ks

ca

tte

r Water

Quartz

Dust

2

4

6

8

10

Fo

rward

to

Po

l B

ac

ks

ca

tte

r

0

10

20

30

40

50

No

n-P

ol to

Po

lari

za

tio

n B

ack

scatt

er



1.3.5 Evaluation of Cloud Microstructure

Measurements of the fine-scale structure of clouds can give us useful information about

the physical processes that underlie cloud evolution. Although the fundamental processes

of nucleation, condensation and coalescence are well understood conceptually, when

taken in the context of real clouds in the atmosphere, their relative interactions are

much more complex. For example, entrainment and mixing, turbulence, and

inhomogeneities in supersaturation will produce changes in the cloud‟s microstructure

that will lead to changes in the rate at which the new droplets are formed, and others

are removed or grow more rapidly than conventional theory predicts.

The light scattering probes offer the opportunity to evaluate the cloud microstructure

because they analyze individual particles. This feature has been exploited since the mid-

1980‟s to examine how droplets are spatially distributed in cloud (Baumgardner, 1986;

Paluch and Baumgardner, 1989; Baker, 1992; Baumgardner et al., 1993; Malinowski et

al., 1994). All of these studies investigated the inhomogeneity of cloud droplet

distributions by analyzing the particle-by-particle measurements from light scattering

instruments.

The basic premise is that droplets should be distributed uniformly and randomly in a well-

mixed cloud. This means that when the spatial distribution is measured, i.e. the distance

between individual droplets, their spacing can be described with a Poisson probability

distribution, as shown in Fig. 1.20. In this figure, at the top is illustrated that the for the

CAS, used in this example, the time between particles are recorded, the interarrival

times. These are used to generate a frequency distribution as shown in the graph of

frequency of events versus arrival time. The frequency has been normalized here by the

total number of events. In this way of representing the data, for a large number of

events, a frequency distribution is an approximation to a probability distribution.

The probability that an arrival time, Δt, will be longer than some time, t, is given by

P(Δt > t) = e-nt 1.12

where n is the particle rate,

n = (Concentration)(Sample Area)(Air Velocity) 1.13

and the sample area is that of the CAS. Hence, we can predict the probability distribution

from the concentration measurements, derived from equation (1.3).



Figure 1.20

The dashed line shown in Fig. 1.20 shows the predicted distribution based on the number

concentration for an example of measurements in a wave cloud. The solid line shows the

distribution of arrival times that was actually measured. The differences between the

predicted and measured provide information about the underlying processes that are

changing the spatial distribution from one that is well-mixed to that where the droplets

are organized in a different arrangement than predicted from Poisson statistics.

It is beyond the scope of this document to describe the possible mechanisms by which

droplet ensembles do not behave a expected, but some of the possible processes could

be entrainment that mixes cloud-free air with the cloud, small scale turbulence,

electrification and droplet charging, or activation of new droplets or coalescence.

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

Arr

iva

l F

req

uen

cy

600500400300200100

Interarrival Time ( s)

Upwind Measured Upwind Predicted

Δt1 Δt2 Δt3 Δt4 Δt5 Δt6

Predicted for randomly uniform distribution

and measured averageconcentration, Cm

Steeper measured

slopes signifies cluster

of particles with higher

concentration.

Cm = Σn/(Volume)

Cderived Frequency

distribution

slope



1.4 Further Reading on Single Particle Scattering

Baker, B., 1992: Turbulent entrainment and mixing in clouds: A new observational

approach, J. Atmos. Sci., 49, 387-404.

Baumgardner, D., 1983: An analysis and comparison of five water droplet measuring

instruments. J. Appl. Meteor., 22, 891-910.

Baumgardner, D., J.W. Strapp, and J.E. Dye, 1985: Evaluation of the forward scattering

spectrometer probe Part II: Corrections for coincidence and dead-time losses.,

J.Atmos. and Oceanic Tech., 2, 626-632.

Baumgardner, D., 1986: A new technique for the study of cloud microstructure. J.

Oceanic and Atmos. Tech., 3, 340-343.

Baumgardner, D., W.A. Cooper, and J.E. Dye, 1990: Optical and electronic limitations of

the forward scattering spectrometer probe. Liquid particle Size Measurement

Techniques: 2nd Volume, ASTM STP 1083, E. Dan Hirleman, W.D. Bachalo, and Philip

G. Felton, Eds. American Society for Testing and Materials, Philadelphia, 1990,

115-127.

Baumgardner, D., J.E. Dye, and B.W. Gandrud, 1989: Calibration of the forward

scattering spectrometer probe used on the ER-2 during the airborne Antarctic

ozone experiment. J. Geophys. Res., 94, 16,475-16,480.

Baumgardner, D. and M. Spowart, 1990: Evaluation of the forward scattering

spectrometer probe. Part III: Time response and laser inhomogeneity limitations. J.

Atmos. Oceanic Tech., 7, 666-672.

Baumgardner, D., J.E. Dye, R.G. Knollenberg, and B.W. Gandrud,1992: Interpretation of

measurements made by the FSSP-300X during the Airborne Arctic Stratospheric

Expedition, J. Geophys. Res., 97, 8035-8046.

Baumgardner, D., B. Baker, and K. Weaver 1993: A technique for the measurement of

cloud structure on centimeter scales, J. Atmos. Oceanic Tech., 10, 557-565.

Baumgardner, D., J.E. Dye, B. Gandrud, K. Barr, K. Kelly, K.R. Chan, 1996: Refractive

indices of aerosols in the upper troposphere and lower stratosphere, Geophys. Res.

Lett, 23, 749-752.

Baumgardner, D., H. Jonsson, W. Dawson, D. O‟Connor and R. Newton, 2001: The cloud,

aerosol and precipitation spectrometer (CAPS): A new instrument for cloud

investigations, Atmos. Res., 59-60, 251-264.

Baumgardner, D., J.F. Gayet, H. Gerber, A. Korolev, and C. Twohy, 2002: Clouds:

Measurement Techniques In-Situ, in the Encyclopaedia of Atmospheric Science,

Eds. J, Curry, J. Holton and J. Pyle, Academic Press, U.K., ISBN: 0-12-227090-8

Baumgardner, D., H. Chepfer, G.B. Raga, G.L. Kok, 2005: The Shapes of Very Small Cirrus

Particles Derived from In Situ Measurements, Geophys. Res. Lett.,32, L01806,

doi:10.1029/2004GL021300, 2005.

Bohren, C. F. and Huffman, D. R.: Absorption and Scattering of Light by Small Particles,

John Wiley, Hoboken, N.J., 1983.



Borrmann, S., Beiping Luot and Michael Mishchenko, 2000: Application of the T-matrix

method to the measurement of aspherical (ellipsoidal) particles with forward

scattering optical particle counters, ,J. Aerosol Sci. 31, 789-799.

Brenguier, J.L. and L. Amodei , 1989: Coincidence and Dead-Time Corrections for Particle

Counters. Part I: A General Mathematical Formalism, Journal of Atmospheric and

Oceanic Technology, 6,575–584.

Brenguier, J.L., 1989: Coincidence and Dead-Time Corrections for Particles Counters.

Part II: High Concentration Measurements with an FSSP, Journal of Atmospheric

and Oceanic Technology, 6, 585–598.

Brenguier, J.L., A.R. Rodi, G. Gordon, and P. Wechsler, 1993: Real-Time Detection of

Performance Degradation of the Forward-scattering Spectrometer Probe, Journal

of Atmospheric and Oceanic Technology, 10, 27–33.

Brenguier, J.L., D. Baumgardner, B. Baker, 1994: A review and discussion of processing

algorithms for FSSP concentration measurements, J. Atmos. Ocean. Tech.,11,

1409-1414.

Brooks, S.D., Toon, O.B., Tolbert, M.A., Baumgardner, D., Gandrud, B.E., Browell, E.,

Flentje, H., Wilson, J., 2004: Polar Stratospheric Clouds during SOLVE/THESEO:

Comparison of Lidar Observations with In-Situ Measurements, J. Geophys. Res.109,

D02212,doi: 10.1029/2003JD003463.

Chepfer, H., V. Noel, P. Minnis, D. Baumgardner, L. Nguyen, G. Raga, M.J. McGill, P.

Yang, 2005: Particle Habit In Tropical Ice Clouds During CRYSTAL-FACE:

Comparison of Two Remote Sensing Techniques With In Situ Observations, J.

Geophys. Res., VOL. 110, D16204, doi:10.1029/2004JD005455.

Cooper, W.A., 1988: Effects of coincidence on measurements with a forward scattering

spectrometer probe, Journal of Atmospheric and Oceanic Technology, 5, 823–832.

Dye, J.E. and D. Baumgardner, 1984: Evaluation of the forward scattering spectrometer

probe: I. Electronic and optical studies. J. Atmos. Ocean. Tech., 1, 329-344.

Field, P. R., R. Wood, P. R. A. Brown, P. H. Kaye, E. Hirst, R. Greenaway, and J. A.

Smith, 2003: Ice Particle Interarrival Times Measured with a Fast FSSP, Journal of

Atmospheric and Oceanic Technology, 20, 249–261.

Gardiner, B. A., and J. Hallett, 1985: Degradation of in-cloud forward scattering

spectrometer probe measurements in the presence of ice particles, J. Atmos.

Oceanic Technol., 2, 171– 180.

Heymsfield, A.J., 2007: On Measurements of Small Ice Particles in Clouds, Geophys. Res.

Lettr, 34, doi:10.1029/2007GL030951, 6 pp.

Knollenberg, R.G., 1981: Techniques for probe in cloud microstructure. Clouds, Their

Formation, Optical Properties, and Effects, P.V. Hobbs and A. Deepak, Eds.,

Academic Press, 495 pp.

McFarquhar, G. M. J. Um, Matt Freer, D. Baumgardner, G. L. Kok, and G. Mace, 2007:

Importance of small ice crystals to cirrus properties: Observations from the

Tropical Warm Pool International Cloud Experiment (TWP-ICE), Geophys. Res.

Lettr., 34, L13803, doi:10.1029/2007GL029865.

McFarquhar, G. M. and A.J. Heymsfield, 1998: The Definition and Significance of an

Effective Radius for Ice Clouds,J. Atmos. Sci.,55 , 2039-2052.



Malinowski, S.P., M.Y. Leclerc and D. Baumgardner, 1994: Fractal analysis of high

resolution cloud droplet measurements, J. Atmos. Sci., 51, 397-413.

Mie, G. (1908) Beitrage zur optik trüber medien speziell kolloidaler metallösungen. Ann.

Phys., 25, 377-445.

Paluch, I. and D. Baumgardner, 1989: Entrainment and fine scale mixing in a continental

convective cloud. J. Atmos. Sci., 46, 261-278.

Pontikis, C., E. Hicks, A. Rigaud, and D. Baumgardner, 1991: A method for validating FSSP

measurements using observational data, J. Atmos. Oceanic Tech.,8, 802-811.

single particle light scattering

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