Electro-Optical Aerosol Phase Function Database 

PFNDAT2005

by Richard C. Shirkey and David H. Tofsted 

ARL-TR-3684 November 2005 

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Army Research Laboratory 

White Sands Missile Range, NM 88002-5501 

ARL-TR-3684 November 2005 

Electro-Optical Aerosol Phase Function Database 

PFNDAT2005

Richard C. Shirkey and David H. Tofsted 

Computational Information Sciences Directorate 

Approved for public release; distribution is unlimited. 
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Electro-Optical Aerosol Phase Function Database PFNDAT2005 

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U.S. Army Research Laboratory 

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Battlefield Environment Division (ATTN: AMSRD-ARL-CI-EE) 

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14. ABSTRACT 

The Phase Function Database (PFNDAT) 2005 consists of a series of wavelength dependent phase functions, single scattering albedos, 

extinction coefficients, and asymmetry parameters for ten naturally occurring and four manmade aerosols along with brief descriptions of the 

scattering parameters, concentrations, and aerosol size distribution characteristics. The naturally occurring aerosols consist of maritime, 

urban, rural, tropospheric, fog, rain, snow, and dust aerosols; a wind-lofted desert aerosol; and the Navy aerosol model. The manmade 

aerosols consist of dust produced from high-explosive munitions; white phosphorus; fog oil; and hexachloroethane smokes. Many of the 

models are functions of relative humidity, wind speed, and other parameters. The database includes information at wavelengths from 0.35 to 

40.0 μm, dependent on the availability of index of refraction data for each scattering species. This version of PFNDAT has added a 

tropospheric model, a wind speed dependent desert aerosol model, the Navy aerosol model, expanded the fog distributions (now including 

moderate advection and heavy radiation) and a completely revised snow phase function. 

15. SUBJECT TERMS

aerosols, scattering, phase function, transmission 

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19a. NAME OF RESPONSIBLE PERSON

Richard C. Shirkey 

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iii 

Contents 

List of Figures iv

List of Tables v

Preface vii

Executive Summary ix

1. Overview 1

2. Introduction 1

2.1 The Phase Function and Associated Aerosol Parameters 1 

2.1.1 Extinctions and Albedos 3 

2.1.2 Phase Function 4 

2.1.3 Asymmetry Parameter 5 

2.2 Particle Size Distribution Models 5 

2.2.1 Gamma Distribution 6 

2.2.2 Modified Gamma Distribution 6 

2.2.3 Lognormal Distribution 7 

3. Aerosol Models 7

3.1 Maritime, Urban, Rural, and Tropospheric Models 7 

3.2 Fog Models 9 

3.3 The Navy Aerosol Model 10 

3.3.1 Examination of the NAM Phase Functions 13 

3.4 Rain and Snow Models 17 

3.4.1 Rain Model 17 

3.4.2 Snow Model 18 

3.5 Dust Aerosol Models 25 

3.5.1 Dust Model 25 

3.5.2 HE Dust Model 27 

3.5.3 Vehicular Dust 27 

3.5.4 Desert Model 27 
iv

3.6 Fog Oil, White Phosphorus, and Hexachloroethane Smoke Aerosol Models 28 

4. The PFNDAT2005 Database 30

References 33

Appendix: A FORTRAN90 Program for Interrogating the Phase Function Files 37

Acronyms and Abbreviations 43

Distribution List 44

List of Figures 

Figure 1. NAM phase functions at 0.55μm and 99% RH as a function of AMP, current (U), 

and average (U ) wind speed................................................................................................... 14 

Figure 2. NAM phase functions at 0.55μm, 99% RH, current wind speed (U) of 12 m/s, and 

AMPs of 1, 5, and 10 as a function of the average wind speed U . ........................................ 15 

Figure 3. NAM phase functions at 0.55μm, 99% RH, current wind speed (U) of 12 m/s, and 

average wind speeds (U ) of 3 and 8 m/s as a function of AMP............................................. 15 

Figure 4. Snow phase function measurements taken by Winchester at 0.6238 μm. ................... 19 

Figure 5. Phase function comparison of the 4-term HG (solid black) with the Mie (solid 

blue) at 0.6238 μm. Winchester’s measurements are represented by the solid diamond 

shaped symbol (black). ........................................................................................................... 21 

Figure 6. Peak phase function comparison (0o to 8o) of the 4-term HG (solid black) with the 

Mie (solid blue) at 0.6238 μm. Winchester’s measurements are represented by the solid 

diamond shaped symbol (black). ............................................................................................ 21 
v

List of Tables 

Table 1. Aerosol types contained in the PFNDAT2005 and their meteorological parameter 

range.......................................................................................................................................... 2 

Table 2. Mode radii (rg), standard deviation (σg), and number densities (N) as functions of 

relative humidity for the small and large modes of the maritime haze aerosol model. ............ 8 

Table 3. Mode radii (rg), standard deviation (σg), and number densities (N) as functions of 

relative humidity for the small and large modes of the urban haze aerosol model. ................. 8 

Table 4. Mode radii (rg), standard deviation (σg), and number densities (N) as functions of 

relative humidity for the small and large modes of the rural haze aerosol model. ................... 8 

Table 5. Mode radii (rg), standard deviation (σg), and number densities (N) as functions of 

relative humidity for the tropospheric aerosol model............................................................... 9 

Table 6. Hygroscopic aerosol mass concentration (μg/m3) as a function of relative humidity 

for small mode, large mode, and total mass concentration for maritime, urban, rural, and 

tropospheric aerosols. ............................................................................................................... 9 

Table 7. Mode radius (rc),β, γ, A, B, N, and liquid water content for the fog aerosols. .............. 10 

Table 8. The D and E parameters and their humidity range for the NAM aerosol 

components. ............................................................................................................................ 11 

Table 9. Values selected for AMP, U (m/s), and U (m/s) for all wavelengths and relative 

humidities................................................................................................................................ 16 

Table 10. Visibilities (km) for the selected NAM aerosols as a function of AMP, average and 

current wind speed (m/s), and relative humidity. ................................................................... 16 

Table 11. Concentrations (μg/m3) and number densities (cm-3) for the selected NAM 

aerosols as a function of AMP, average and current wind speed (m/s), and relative 

humidity. ................................................................................................................................. 16 

Table 12. Precipitation rates, number densities, and equivalent liquid water for the rain 

model....................................................................................................................................... 18 

Table 13. Coefficients for the 4-term HG phase function. .......................................................... 20 

Table 14. Independent parameters for the two snow types.......................................................... 23 

Table 15. Dependent parameters for the two snow types (visibility = 3 km).............................. 23 

Table 16. Modified visibility criteria for snowfall intensitya (based on temperature and day 

or night)................................................................................................................................... 24 

Table 17. Snow visibility (km) vs. snowfall intensity. ................................................................ 25 

Table 18. Values of lognormal particle size distributions parameters for the dust and high 

explosive (HE) dust models: bulk density, number density, mass loading, mode radius, 

and geometric standard deviation. .......................................................................................... 26 

Table 19. Particle size distribution parameters for desert aerosol components: bulk density, 

number density, mass loading, mode radius, and geometric standard deviation.................... 28 
vi

Table 20. Representative parameters of inventory smokes at various relative humidities.......... 29 

Table 21. Comparison of theoretical and experimentally measured mass extinction 

coefficients at 50% RH for various smoke aerosols. .............................................................. 30 

Table 22. Aerosol ID as a function of aerosol type and RH........................................................ 31 

Table 23. Structure of an aerosol phase function data file........................................................... 32 
vii 

Preface 

The 2005 edition of the electro-optical aerosol Phase Function Database (PFNDAT2005) 

consists of a series of wavelength dependent phase functions, single scattering albedos, 

extinction coefficients, and other relevant properties for ten naturally occurring and four 

manmade aerosols characteristic of the near surface battlespace. These phase functions are 

useful in simulating the near surface atmospheric propagation conditions for target acquisition 

and scattering studies where typical scattering species are required. The naturally occurring 

aerosols consist of maritime, urban, rural, tropospheric, fog, rain, snow, and dust aerosols; a 

wind-lofted desert aerosol; and the Navy aerosol model of the marine boundary layer. The 

manmade aerosols consist of dust produced from high-explosive munitions and white 

phosphorus; fog oil; and hexachloroethane smokes. Many of the models are functions of relative 

humidity, wind speed, and other parameters, resulting in a total of 69 aerosol variations. Users 

may also create new composite (mixed) aerosols (e.g., rain plus haze) by making linear 

combinations of aerosol statistics. 

The database includes information at wavelengths from 0.35 to 40.0 μm, dependent on the 

availability of index of refraction data for each scattering species. This report contains brief 

descriptions of the scattering parameters and the aerosol size distribution characteristics used to 

generate the phase functions, albedos, extinction coefficients, and asymmetry parameters. This 

version of PFNDAT has added a tropospheric model, expanded the fog distributions (now 

including moderate advection and heavy radiation), a desert aerosol model that is a function of 

wind speed, the Navy aerosol model, and a completely revised snow phase function that includes 

“dry” and “wet” snow. The phase functions are provided on CD along with a short computer 

program that may be used to interrogate them. 
viii 

INTENTIONALLY LEFT BLANK.
ix 

Executive Summary 

In order to acquire targets on the dirty battlefield, we must know how much “signal” is lost as it 

propagates through the atmosphere from the target to the sensor, and how much “noise” is added 

back in through atmospheric scattering processes. These effects are often expressed as a loss of 

target contrast. Calculating this quantity requires information about the atmospheric constituents 

and how they interact with radiation of various wavelengths passing through the atmosphere. 

The aerosol Phase Function Database (PFNDAT) is a repository of aerosol scattering data. The 

PFNDAT database provides the necessary information to characterize a wide variety of battleinduced and natural aerosols. Included in this collection are datasets describing four main 

classes of hazes (rural, maritime, urban, and tropospheric) at varying relative humidities; two 

types of fog (radiation and advection); rain at three precipitation rates (drizzle, moderate, and 

heavy); and two classes of snow, “dry” and “wet”. Dusts are treated under four categories: 

light, heavy, wind-lofted desert, and battlefield-induced high explosive munition. Inventory 

smokes include hygroscopic white phosphorus at three relative humidities, hexachloroethane, 

and fog oil. Finally, in the natural aerosol arena, the Navy aerosol model has been added to 

provide up-to-date modeling of the marine environment. 

Providing scattering information in a database format allows aerosol scattering species to be used 

within larger radiative transfer models. The resulting propagation characteristics can readily be 

used within radiative transfer codes under a variety of conditions. 

The PFNDAT database provides an accurate description of aerosol scattering properties for a 

wide range of particulates to be encountered on the dirty battlefield. This upgraded version, 

PFNDAT2005, provides significantly more resolution, detail, and versatility of aerosol species of 

interest to Army systems developers and radiative transfer specialists than previous editions 

have. 
x 

INTENTIONALLY LEFT BLANK.
1 

1. Overview 

Determining the amount of radiation transmitted through the atmosphere requires information on 

how the atmospheric constituents interact with that radiation. The aerosol Phase Function 

Database (PFNDAT) has been constructed to provide a significant part of this information. 

PFNDAT characterizes a wide variety of battle-induced and natural aerosols, including four 

classes of hazes (rural, maritime, urban and tropospheric) at varying relative humidity levels; two 

types of fog (radiation and advection), each of which has a moderate and heavy aerosol loading; 

rain at three precipitation rates (drizzle, moderate, and heavy); two classes of snow (“dry” and 

“wet”); and four dust categories (light and heavy aerosol loading, wind-lofted desert, and dust 

raised by high energy munitions). It also uses the Navy aerosol model for two air mass 

parameters, two average wind speeds, one current wind speed, and four relative humidities. 

Inventory smokes include white phosphorus (WP) at three relative humidities, hexachloroethane 

(HC), and fog oil. 

This version, PFNDAT2005, corrects minor errors in the previous versions and adds 

2 distributions for fog, 4 distributions for desert aerosols, 8 tropospheric haze distributions, 

16 variations of the Navy aerosol model, and 2 snow phase functions. Where data permits, we 

have extended the wavelength range of these aerosols. This allows the PFNDAT2005 results to 

be used in other radiative transfer applications, such as solar loading and energy balance 

calculations. We have also included the concentration density and asymmetry parameters for 

these aerosols and have reformatted them for easier computational use. 

2. Introduction 

2.1 The Phase Function and Associated Aerosol Parameters 

The propagation of electro-magnetic energy within the Earth’s atmosphere depends on the 

wavelength of the radiation and on the nature of the medium being traversed. This medium 

consists of various molecular species and aerosol particles. These atmospheric particles, or 

aerosols, are ubiquitous in nature and frequently are the determining factor in the amount of 

radiation received at a sensor. At all wavelengths aerosols generally scatter radiation out of the 

line-of-sight (LOS) into various angles. The probability of scattering per particle encounter is 

given by the single scattering albedo (π0). In contradistinction to π0, the angular probability 

distribution of scattering radiation, according to a direction relative to the incident radiation (i.e., 

the probability that an incident photon will be scattered into a particular angle θ relative to its 

initial direction of propagation), is given by the “phase function.” 

The phase function only provides information applicable to a single scattering event; in multiple 

scattering calculations, it is used repetitively. To calculate the phase function, the aerosol’s 

composition, number density, size distribution, and refractive index at the wavelength under 

consideration must be known; often the relative humidity must also be taken into account. 

Further, a given aerosol species frequently is composed of weighted sums of more than one 
2

particle type, as in the case of various dusts. In this document, we discuss the aerosol 

composition, number density, and size distribution components. Since refractive indices for 

the aerosol models may be found in the references contained in the aerosol model sections, 

they are not presented here. In view of the fact that many of the refractive indices and extinction 

coefficients, along with plots of the phase functions, were presented in previous versions of 

this database (1, 2), we have included those documents on the distribution CD. The aerosol 

types and the range of meteorological parameters at which they were computed are presented 

in table 1. 

Table 1. Aerosol types contained in the PFNDAT2005 and their 

meteorological parameter range. 

Aerosol Type Relative Humidity 

(%) 

Maritime 0 50 70 80 90 95 98 99 

Urban 0 50 70 80 90 95 98 99 

Rural 0 50 70 80 90 95 98 99 

Fog (heavy advection) N/A 

Fog (moderate radiation) N/A 

Rain (drizzle) N/A 

Rain (widespread) N/A 

Rain (thunderstorm N/A 

Snow (dry) – 

Snow (wet) – 

Fog (moderate advection) N/A 

Fog (heavy radiation) N/A 

Desert aerosol (U=0) – 

Desert aerosol (U=10) – 

Desert aerosol (U=20) – 

Desert aerosol (U=30) – 

Tropospheric 0 50 70 80 90 95 98 99 

Dust (light loading) – 

Dust (heavy loading) – 

High explosive dust – 

WP smoke 17 50 90 

Fog oil 50 

HC smoke 85 

Navy aerosol 

(AMP=3 & 8; U =3 & 8; U=12) 

50 80 95 99 

NOTE: AMP is the air mass parameter, U is the average wind speed 

(m/s), and U is the current wind speed (m/s). 

Extinction coefficients, albedos, asymmetry factors, aerosol concentrations, and number 

densities are included in the phase function data files as functions of relative humidity and 
3

wavelength. Their values may be examined using the “aerosol_info” program located in the 

appendix and on the distribution CD. The user is cautioned that when interpolating the 

extinction within wavelength bands (0.35–0.75 μm, 3.0–5.0 μm, 8.0–40.0 μm) care must be 

taken: peaks or valleys in the extinction coefficient may easily be missed due to coarse 

wavelength resolution. 

When performing these calculations, the following standard assumptions were made: 

• The particles are assumed to be spaced far enough apart that radiation scattered by one 

particle does not affect how radiation is scattered from another particle. Therefore, each 

scattering event is independent. 

• A Mie scattering code is assumed valid for predicting the behavior of each scattering 

species. The usage of a Mie scattering method assumes that the particles can be 

approximated as spherical in shape. This assumption is often made even for non-spherical 

particles, because in most cases the orientation of the particles is random since no external 

influences, such as strong magnetic fields or hydrodynamic forces, are present. Orientation 

averaging then produces nearly the same result as the assumption of sphericity. 

• The scattering properties of a given type of particulate distribution can be represented by a 

weighted integral over the particle size distribution. This approach depends on the 

assumption of independent scattering, as discussed above. 

• Polarization effects in both the incident radiation and the scattered fields have been 

ignored. Hence, the azimuthal and polar field components arising from the Mie scattering 

analysis have been merged into a single scattered flux statistic. 

• The particulate size distribution is homogeneous over the volume concerned. 

2.1.1 Extinctions and Albedos 

Having determined the scattering properties of various aerosol species, one can then use these 

quantities to accurately calculate the scattering and absorption of radiation at a given wavelength 

as it passes through the atmosphere. As alluded to above, the relevant information needed to 

determine these radiation results includes the angular scattering probability distribution (P), the 

volume extinction coefficient (κ) (in length-1), and the single scattering albedo (π0). The volume 

extinction coefficient κ, which determines the attenuation of the incident radiation is composed 

of two parts: 

• A scattering coefficient κs that describes the radiation scattered out of the LOS without a 

change in wavelength 

• The absorption coefficient κa that describes the amount of radiation along the LOS that is 

converted into other forms of energy or that undergoes a change in wavelength 

These two quantities are related to κ and π0 by 

π0 = κs / (κs + κa); κs =π0 κ; κa = (1-π0) κ. (1) 

The albedo π0 represents the probability that interacting radiation will be scattered rather than 

absorbed: for pure scattering, π0 = 1, and for total absorption, π0 = 0. Making the assumption 
4

that the obscurant material properties are essentially uniform, we can relate the extinction 

coefficient to the transmission (T) by the Beer-Lambert law for monochromatic radiation, 

 T = e-κL

, (2) 

where L is the path length under consideration. 

Frequently in practical problems dealing with smokes, dust, and other aerosols that may vary 

dynamically over time and space, it is observed that the volume extinction coefficient tends to 

fall into ranges of about 0.001–100 km-1. Under these conditions the “mass” extinction 

coefficient, denoted here as α in units of m2 g

-1, is typically used, which is in contradistinction to 

the “volume” extinction coefficient κ, which has units of km-1. Using the mass extinction 

coefficient, the transmission can be expressed as 

 T = e-α CL, (3) 

where the term concentration length (CL) is used for the total mass density integrated over the 

LOS. The concentration C is typically measured in g m-3 and the path length L in units such that 

either α CL or κ L is dimensionless. In support of this application, the aerosol concentration of 

each species is computed and included in the database. The extinction coefficient can be related 

to the concentrations by combining equations 2 and 3. 

Since smoke densities are frequently a function of time, a method for reassessing the extinction 

coefficient at different times is useful. This reassessment may be accomplished by comparing 

equations 2 and 3, and noting that 

α C = κ, (4) 

 = N s, (5) 

where s is the extinction cross section per particle (unit area) and N is the particle number 

density in units of particles/unit volume. Equating equations 5 and 6, we find for the volume 

extinction coefficient: 

κ = N s. (6) 

Thus, the volume extinction coefficient κ can be scaled as a function of time, if N (or the 

quantity α C) is known as a function of time. 

2.1.2 Phase Function 

Transmittance is important, because it quantifies when a received signal will be below some 

operational threshold of an electro-optical device, thereby directly affecting target acquisition. 

But transmittance is not the only quantity that determines the energy that is detected. Just as 

extinction removes energy along a path, multiple scattering can return some of that energy into 

the LOS. Radiation from other external sources will also scatter into the LOS, affecting image 

contrast and/or impacting the system’s dynamic range handling capabilities. The aerosol phase 

function is an essential ingredient in determining the amount of radiation scattered by the 

aerosol. 
5

The angular scattering distribution (the phase function) gives the directional distribution of 

radiation scattered by the aerosol under consideration: the phase function is proportional to the 

probability that incoming radiation is scattered through a scattering angle θ into an element of 

solid angle dΩ. For incident unpolarized radiation, the phase function, as used here, is 

normalized as 

∫

4π

 P(θ) dΩ = 1.0. (7) 

2.1.3 Asymmetry Parameter 

Simple approximations to the phase function are frequently sought for calculations where many 

scatterings have occurred and/or the optical depth is sufficiently large. One such approximation 

is the Henyey-Greenstein (HG) phase function (3): 

, (1 2 cos( ) )

1

4

1 ( , ) 2 3 / 2

2

g g

g PHG g − +− = π θ

θ (8) 

where g is called the asymmetry factor and is a relative measure of forward or backward 

scattering. This phase function is particularly appealing as it is dependent only on the single 

parameter g. For isotropic scattering, g is 0; for highly peaked phase functions, as is common at 

visible wavelengths, g approaches its maximum value of 1. The asymmetry formulation also 

admits to a negative solution, where the phase function would be peaked in the reverse direction 

(180°). The asymmetry factor is defined as the average cosine of the scattering angle weighted 

by the phase function: 

 g = < cosθ > = ∫

4π

 cosθ P(θ) dΩ / ∫

4π

 P(θ) dΩ. (9) 

We have included values of g for all aerosols. 

2.2 Particle Size Distribution Models 

Aerosols are frequently a combination of more than one type; each type may have its own 

relative density distribution and refractive index properties. For example, typical dusts are 

assumed to be composed of various component quantities of quartz, montmorillonite, and 

ammonium sulfate particles. Each particle type has a particulate bulk density ρ in units of 

g cm-3 or as a dimensionless specific gravity. As mentioned above, each species may be 

characterized by a given mass concentration, C (g/cm3

), which represents the weight of lofted 

material mixed within a unit volume of air. From a meteorological standpoint, for aerosols 

comprised primarily of water, C is often referred to by the term “liquid water content.” The term 

“mass concentration” is used to describe both effects in this text. 

The quantities ρ and C are related by the particle size distribution, denoted by n(r), where r is the 

radius of a spherical particle. The total number of particles per unit volume is denoted by the 

number density N, which is related to the particle size distribution through 

 N = ∫

0

∞

 n(r) dr, (10) 
6

where n(r) has units of particles per cm3- μm. 

We may now write the mass concentration as a function of the particle size distribution, 

 C = 4/3 π∫0

∞

n(r) r3 ρ dr, (11) 

where (4/3)πr

3

ρ is the mass of a particle of radius r. The total mass concentration of a mixed 

aerosol is assumed to be a weighted sum of individual constituent parts. 

The three aerosol particle size distributions (gamma, modified gamma, and lognormal) used in 

determining the phase functions are presented in more detail in sections 2.2.1–2.2.3. 

2.2.1 Gamma Distribution 

The gamma distribution, used by the rain and snow models, is given by the equation 

 n(r) = N0 rm exp(-Λr), (12) 

where N0 is the y intercept, Λ is the slope, m is the order of the gamma size distribution, and r is 

the radius. 

Taking m = 0, we find the number density to be 

 N = N0/Λ. (13) 

The mass concentration for this distribution can be expressed as 

 C = 4πρN0Γ(4)/(3Λ4

) = 8πρN0/Λ4. (14) 

2.2.2 Modified Gamma Distribution 

The modified gamma (MG) distribution, used by the fog models, is given by the equation 

 n(r) = rc rβ exp [-(β/γ) (r/rc)

γ

], (15) 

where rc is the mode radius, and β and γ are fit coefficients of the distribution. By integrating 

equation 15, we find the number density 

 N = A B-(β-1)/γ Γ[(β+1)/γ] γ

-1. (16) 

The mass concentration for this distribution can be expressed as 

[( 1)/ ]

[( 4)/ ]

3

4

3 /

3

β γ

γ

β

β γ πρ γ

Γ + ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

Γ + C = rc N . (17) 
7

2.2.3 Lognormal Distribution 

The lognormal distribution, used by the smoke, dust, rural, urban, maritime, tropospheric, desert, 

and the Navy aerosol models, is given by the equation 

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡ − − ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡ = ∑=

2

2

10 10

1 2

(log log )

exp 2 ln(10) ( )

i

gi

i

i

k

i

r r

r

N

n r

π σ σ

, (18) 

where rgi is the distribution geometric mean radius (or mode radius) in μm, σi is the width of the 

distribution measured in log10 space (2, 4), and Ni is the number density of the ith component 

with rgi. We note in passing that some Mie codes, in particular AGAUS, the code used for this 

work, use natural instead of common logs to specify this distribution. The mass concentration 

equation for this distribution is given by 

 Ci = 4/3 πρNi rgi

3

 exp [9/2(ln(10)σi )2]. (19) 

3. Aerosol Models 

3.1 Maritime, Urban, Rural, and Tropospheric Models 

The parameters for the maritime, urban, rural, and tropospheric aerosol models have been taken 

from Shettle and Fenn (4); they are bimodal lognormal (equation 18 with k = 2), with the mode 

radius varying as a function of relative humidity. The rural aerosol model consists of small and 

large rural distributions with correspondingly different indices of refraction. Similarly, the urban 

aerosol model consists of small and large urban distributions. The maritime aerosol model 

consists of the small rural (continental) distribution along with a large particle oceanic 

distribution. The tropospheric model is identical to the rural model, but without the large particle 

component of that size distribution. The indices of refraction for the individual aerosols are 

listed in Models for the Aerosols of the Lower Atmosphere and the Effects of Humidity 

Variations on Their Optical Properties (4). 

The number densities (rounded) for each mode of the maritime, urban, rural, and tropospheric 

distribution types along with the mode radii (rg), variance (σg), and related liquid water content 

information are provided in tables 2–5 for the user who may wish to change the visibilities. The 

geometric mean variance, σg used herein, is related to σ by 

σ = log10(σg). (20) 

The hygroscopic aerosol mass concentration data is contained in table 6 and has been derived 

using equation 19. The extinction coefficients were generated for a meteorological visibility of 

5.0 km; the number densities correspond to those found in tables 8–10 in reference 4. 
8

Table 2. Mode radii (rg), standard deviation (σg), and number densities (N) as functions of relative humidity for 

the small and large modes of the maritime haze aerosol model.

 Relative Humidity 

(%) 

Parameter 0 50 70 80 90 95 98 99 

rg (S) (μm) 0.02700 0.02748 0.02846 0.03274 0.03884 0.04238 0.04751 0.05215 

σg (S) 2.239 2.239 2.239 2.239 2.239 2.239 2.239 2.239 

N (S) (cm-3) 38251 35129 27757 13902 9697 6976 4360 2948 

rg (L) (μm) 0.1600 0.1711 0.2041 0.3180 0.3803 0.4606 0.6024 0.7505 

σg (L) 2.512 2.512 2.512 2.512 2.512 2.512 2.512 2.512 

N (L) (cm-3) 386.4 354.8 280.4 140.4 98.0 70.5 44.0 29.8 

NOTE: S = small and L = large. 

Table 3. Mode radii (rg), standard deviation (σg), and number densities (N) as functions of relative humidity for 

the small and large modes of the urban haze aerosol model. 

Relative Humidity 

(%) 

Parameter 0 50 70 80 90 95 98 99 

rg (S) (μm) 0.02500 0.02563 0.02911 0.03514 0.04187 0.04904 0.05996 0.06847 

σg (S) 2.239 2.239 2.239 2.239 2.239 2.239 2.239 2.239 

N (S) (cm-3) 87204 83354 64829 42776 27693 18217 10516 7286 

rg (L) (μm) 0.4000 0.4113 0.4777 0.5805 0.7061 0.8634 1.1691 1.4858 

σg (L) 2.512 2.512 2.512 2.512 2.512 2.512 2.512 2.512 

N (L) (cm-3) 10.9 10.4 8.1 5.4 3.5 2.3 1.3 0.9 

NOTE: S = small and L = large. 

Table 4. Mode radii (rg), standard deviation (σg), and number densities (N) as functions of relative humidity for 

the small and large modes of the rural haze aerosol model.

 Relative Humidity 

(%) 

Parameter 0 50 70 80 90 95 98 99 

rg (S) (μm) 0.02700 0.02748 0.02846 0.03274 0.03884 0.04238 0.04751 0.05215 

σg (S) 2.239 2.239 2.239 2.239 2.239 2.239 2.239 2.239 

N (S) (cm-3) 79076 76305 70804 51674 33895 27052 19290 14761 

rg (L) (μm) 0.4300 0.4377 0.4571 0.5477 0.6462 0.7078 0.9728 1.1755 

σg (L) 2.512 2.512 2.512 2.512 2.512 2.512 2.512 2.512 

N (L) (cm-3) 9.9 9.5 8.9 6.4 4.2 3.4 2.4 1.9 

NOTE: S = small and L = large. 
9

Table 5. Mode radii (rg), standard deviation (σg), and number densities (N) as functions of relative humidity 

for the tropospheric aerosol model. 

Relative Humidity 

(%) 

Parameter 0 50 70 80 90 95 98 99 

rg (μm) 0.02700 0.02748 0.02846 0.03274 0.03884 0.04238 0.04751 0.05215 

σg 2.239 2.239 2.239 2.239 2.239 2.239 2.239 2.239 

N (cm-3) 87180 84113 78115 56956 37152 29502 21673 16802 

Table 6. Hygroscopic aerosol mass concentration (μg/m3) as a function of relative humidity for small mode, large 

mode, and total mass concentration for maritime, urban, rural, and tropospheric aerosols. 

Relative Humidity 

Aerosol Type (%) 

0 50 70 80 90 95 98 99 

Maritime 

Small 58.7 56.8 49.9 38.0 44.3 41.4 36.4 32.6 

Large 301.7 338.7 454.4 860.5 1027.3 1313.0 1833.2 2400.9 

Total 360.3 395.5 504.3 898.5 1071.6 1354.4 1869.7 2433.5 

Urban 

Small 106.2 109.4 124.6 144.7 158.4 167.5 176.7 182.3 

Large 133.0 137.9 168.3 201.3 234.8 282.1 395.8 561.7 

Total 239.2 247.3 292.9 346.0 393.3 449.6 572.5 744.9 

Rural 

Small 121.3 123.4 127.2 141.3 154.8 160.5 161.2 163.2 

Large 150.0 151.8 162.0 200.4 216.0 229.8 421.1 588.9 

Total 271.3 275.3 289.2 341.7 370.8 390.3 582.3 751.4 

Tropospheric 

Total 133.7 136.0 140.3 155.8 169.7 175.0 181.2 185.7 

3.2 Fog Models 

Data from Shettle and Fenn (4) were also used for the fog models. In previous versions of 

PFNDAT, these models included only the heavy advection and moderate radiation. For this 

version, we have added two additional cases: moderate advection and heavy radiation. These 

models use the MG distribution (equation 15); the input parameters are listed in table 7 along 

with the mass concentrations (liquid water content) and number densities. According to Shettle 

and Fenn (4), these fogs are typical of radiation fogs and advection fogs, although they are also 

characteristic of developing and mature fogs, respectively. 
10

Table 7. Mode radius (rc),β, γ, A, B, N, and liquid water content for the fog aerosols. 

Fog Type 

Parameters 

Heavy Advection Moderate 

Advection Heavy Radiation Moderate Radiation 

rc (μm) 10 8 4 2 

β 3 3 6 6 

γ 1 1 1 1 

A (particles/cm3-cm) .027 .06692 2.37305 607.5 

B (cm-1) .3 .375 1.5 3.0 

N (particles/cm-3) 20 20 100 200 

Liquid water content 

(μg/m3

) 372,300 190,636 62,553 15,640 

3.3 The Navy Aerosol Model 

The Navy Aerosol Model (NAM) was developed in the 1980s (5) to describe the marine aerosol 

and its optical and infrared (IR) propagation properties for the atmosphere’s marine boundary 

layer (MBL) using local meteorological parameters at shipboard level (nominally, 10 m). The 

model is a function of relative humidity, wind speed (24-hr average and current), wavelength, 

and, implicitly, visibility. This model has no vertical structure, although such structure is being 

investigated (6, 7). When the model is used in the MBL at levels away from the surface, it is 

assumed that the MBL is extremely well mixed and that the size distribution of the aerosol 

within this layer is constant from the sea surface to the top of the MBL. 

The NAM is a four-component lognormal aerosol model, expressed as 

{ [ ] }2

0

4

1

exp ln( /( )i

i

i C r f r

f

A

dr

dN = ∑ − =

, (21) 

where dN/dr is the number of particles per cm3

 per micrometer; i is the component index; r is 

particle radius in micrometers; r0 is the mode radius; C = 1/(2 σ

2

), where σ = 1/ 2 (therefore, 

C=1) is the standard deviation of the size distributions; Ai’s are parameters fitted to the 

meteorological conditions and particle concentrations of the NAM component; and f is the effect 

of relative humidity given by the ratio of the particle radius at the ambient relative humidity to 

the particle radius at the standard relative humidity of 80% (7). The first and second components 

of equation 21 have the same mode radius, but are different classes of aerosols. The first class is 

a nonsoluble aerosol present only at high Air Mass Parameters (AMPs), discussed below, and the 

second class is water soluble. Equation 21 can be related to equation 18 by noting that the 

geometric mean radius rg used in equation 18 is equivalent to the median radius in NAM and is 

related to their mode radius by rg = rmedian = rmode exp(σ

2

), where the NAM form of the lognormal 

distribution has absorbed the r dependence into the exponential by modifying the Ai coefficients 

(7). Plots of the NAM number density as a function of radius may be found in Optical 

Properties of the Marine Aerosol as Predicted by the Navy Aerosol Model (5). 
11

The original NAM used the hygroscopic growth formulization of Fitzgerald (8) for changes to 

the particle mode radii. Gerber (9) later extended Fitzgerald’s relationship to particle sizes 

greater than 0.1 μm, relative humidities greater than 98%, and for varying properties of the 

different aerosol components. In Gerber’s formulization, used in this work, 

 f = [(D – rh)/(E (1 – rh))]1/3, (22) 

where rh is the relative humidity expressed in fractional form and f is the “growth factor” used 

for the hygroscopic growth of the particles in a humid environment. The NAM growth rule is 

designed around a baseline value such that f = 1.0 at 80% relative humidity for all aerosol 

types. The D’s and E’s in equation 22 take on the values in table 8 for the various NAM 

aerosol components. 

Table 8. The D and E parameters and their humidity range for the NAM aerosol components. 

NAM Component a Aerosol Material Mode Radii 

(μm) D E Humidity Range 

(%) 

1 Dust 0.03 NA NA NA 

2 B1 0.03 1.17 1.87 < 99 

3 Sea salt plus water 0.24 1.83 5.13 < 99.9 

4 Sea salt plus water 2.0 1.97 5.83 < 99.99 

 aas used in this work. 

The first component of the NAM is Shettle and Fenn’s (4) dust-like aerosol, assumed to be 

nonsoluble; the second component is Volz’s (10) B1 aerosol; and the third and fourth 

components are sea-salt water droplets and wave-generated sea water droplets. The mode radii 

for these components may be found in table 8. The first two components are considered to be 

background aerosols that are related to the AMP, which (as will be discussed below) is itself a 

function of relative humidity and transit time from point of origin. The third component is 

comprised of aerosols that have been produced by earlier high wind conditions that exhibit a 

relatively long residence time and, thus, are a function of the previous 24-hour average of the 

wind speed. The fourth component, comprised of the largest aerosols, is considered to be related 

to the current wind speed and the current white water phenomenon. A more complete 

description of the components may be found in references 5 and 7. 

The number densities of the various aerosol components as a function of the AMP may be found 

using the following (7, 11): 

when AMP ≤ 5, 

 N 1 = 0 cm-3

 N 2 = 136.55 AMP2

 cm-3

when AMP > 5, 

 N 1 = 0.3 x 136.55 AMP2

 cm-3

 N 2 = 0.7 x 136.55 AMP2

 cm-3
12

and for all AMP, 

 N 3 = 0.5462 x MAX(5.866 x (U - 2.2), 0.5) cm-3

 N 4 = 4.5518 x 10(0.06 U - 2.8) cm-3

where U is the 24-hour average wind speed (m/s) and U is the current wind speed (m/s). 

The NAM refractive indices (11) are modified by Hänel’s volume interpolation method (12): 

 m = mw + (mo + mw) (f 3

(0)/f 3(rh)), (23) 

where mw is the refractive index of water and mo is the refractive index of the dry aerosol. 

We now have all the parameters necessary to determine the aerosol optical properties with the 

exception of the AMP, which describes the general condition of the air mass. An AMP of 1 

indicates a pure air mass without contaminants and an AMP of 30 is the worse case situation that 

might be found downwind of a large industrial complex (7). There are four methods for 

determining the AMP parameter: 1) empirical judgment, 2) measurements related to MBL (i.e., 

the atmospheric radon content), 3) the elapsed time for the current air mass to reach the point of 

observation at sea from a distant land mass, or 4) analysis using the current wind speed, average 

wind speed, relative humidity, and visibility (7). Since none of these methods are easily 

obtainable for the typical user, we have instead initially taken values for the AMP and average 

wind speeds from the Target Acquisition Weapons Software TAWS (13). Available values in 

TAWS for AMP are 1, 5, or 10 and the available values for the average wind speeds are 3, 8, or 

15 m/s. To complete the necessary input values, we have chosen current wind speeds of 0, 5, 12, 

and 20 m/s, and relative humidities of 50%, 80%, 95%, and 99%. Phase functions and extinction 

coefficients were calculated using these values at 32 wavelengths resulting in 4608 possible 

combinations. Since there are no existing published phase functions for NAM, we have 

compared the extinction coefficients generated using NAM6 (11) and AGAUS (14) for 

verification purposes. At 0.55 μm, the average error, using NAM6 as the “correct” value, was 

less than 1%: maximum errors, greater than 2% but less than 2.5%, were found at the 4 values of 

current wind speed (0, 5, 12, and 20 m/s) at a relative humidity of 50%, an AMP of 10, and an 

average wind speed of 15 m/s. Over all wavelengths, relative humidities, AMPs, U ’s, and U’s, 

the average error was 2.8%. 
13

3.3.1 Examination of the NAM Phase Functions 

In the discussion that follows, we implicitly assume that the technique for calculating scattering 

effects will use a truncation method for the peak of the phase function, such as the deltaEddington or small-angle approximation, that allows us to discount the forward peak values 

(0° to ~10°). We have examined the 144 phase function files to determine if there are 

similarities over the various AMPs, U ’s, and U’s that would allow us to reduce this number. We 

made visual comparisons by plotting the phase functions at a wavelength of 0.55 μm with a 

relative humidity of 99%. Here, the size parameter, 2πr/λ (where λ is the wavelength), is 

relatively large and the scattering from aerosol particles is significant. These phase functions are 

presented in figure 1. An examination of figure 1 shows little variation in the phase function 

with current wind speed. Therefore, we opted to use only one current wind speed, arbitrarily 

chosen to be 12 m/s. To look for other possible reductions, a similar examination was performed 

holding the current wind speed constant at 12 m/s. Again, as may be seen in figure 2, visual 

inspection shows that there is little difference between 8 and 15 m/s for the average wind speed. 

Using the current wind speed of 12 m/s and average wind speeds of 3 and 8 m/s, this process was 

again repeated for AMPs of 1, 5, and 10. These results are shown in figure 3 where we find little 

difference between AMPs of 5 and 10. In figure 1, the NAM phase functions at 0.55μm and 

99% relative humidity (RH) are presented as a function of AMP, current (U), and average (U ) 

wind speed. 
14

Figure 1. NAM phase functions at 0.55μm and 99% RH as a function of AMP, current (U), and average 

(U ) wind speed. 

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

0 45 90 135 180

Angle (degrees)

0

5

12

20

U (m/s)

RH = 99% 

Amp = 1

U = 3 m/s 

NAM Phase Function

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

0 45 90 135 180

Angle (degrees)

0

5

12

20

RH = 99%

Amp = 1

U = 8 m/s

U (m/s)

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

0 45 90 135 180

Angle (degrees)

0

5

12

20

RH = 99%

Amp = 1

U = 15 m/s

U (m/s)

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

0 45 90 135 180

Angle (degrees)

0

5

12

20

RH = 99%

Amp = 5

U = 3 m/s

U (m/s)

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

0 45 90 135 180

Angle (degrees)

0

5

12

20

U (m/s)

RH = 99%

Amp = 5

U = 8 m/s

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

0 45 90 135 180

Angle (degrees)

0

5

12

20

RH = 99%

Amp = 5

U = 15 m/s

U (m/s)

NAM Phase Function

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

0 45 90 135 180

Angle (degrees)

0

5

12

20

RH = 99%

Amp = 10

U = 3 m/s

U (m/s)

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

0 45 90 135 180

Angle (degrees)

0

5

12

20

RH = 99%

Amp = 10

U = 8 m/s

U (m/s)

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

0 45 90 135 180

Angle (degrees)

0

5

12

20

U (m/s)

RH = 99%

Amp = 10

U = 15 m/s
15

Figure 2. NAM phase functions at 0.55μm, 99% RH, current wind speed (U) of 12 m/s, and AMPs of 1, 5, 

and 10 as a function of the average wind speed U . 

Figure 3. NAM phase functions at 0.55μm, 99% RH, current wind speed (U) of 12 m/s, and average wind 

speeds (U ) of 3 and 8 m/s as a function of AMP. 

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

0 45 90 135 180

Angle (degrees)

3

8

15

Ubar 

( /)

RH = 99% 

Amp = 1

U = 12 m/s 

NAM Phase Function

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

0 45 90 135 180

Angle (degrees)

3

8

15

RH = 99%

Amp = 5

U = 12 m/s

Ubar (m/s)

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

0 45 90 135 180

Angle (degrees)

3

8

15

RH = 99%

Amp = 10

U = 12 m/s

Ubar (m/s)

NAM Phase Function

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

0 45 90 135 180

Angle (degrees)

1

5

10

RH = 99%

Ubar = 3

U = 12 m/s

AMP

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

0 45 90 135 180

Angle (degrees)

1

5

10

RH = 99%

Ubar = 8

U = 12 m/s

AMP
16

Table 9 shows the values we settled on for AMP, U , and U at all wavelengths and relative 

humidities. This reduces the total number of phase functions from 4608 to 512 and the number 

of files from 144 to 16. Table 10 presents the meteorological visibilities associated with the 

selected NAM extinction coefficients; table 11 presents the concentrations and number densities 

for the selected AMP, U , U, and relative humidities. 

Table 9. Values selected for AMP, U (m/s), and U (m/s) for all wavelengths and relative humidities. 

AMP Average Wind Speed, U Current Wind Speed, U 

Range Value Range Value Range Value 

0 ≤ AMP < 4 3 0 ≤ U < 8 3 

4 ≤ AMP ≤ 10 8 8 ≤ U ≤ 15 8 

0 ≤ U ≤ 20 12 

Table 10. Visibilities (km) for the selected NAM aerosols as a function of AMP, average and current wind 

speed (m/s), and relative humidity. 

RH Aerosol IDa

 AMP U U 

50% 80% 90% 95% 

58–61 3 3 12 139 97 41 12 

62–65 3 8 12 61 41 18 6.2 

66–69 8 3 12 22 18 9.5 3.0 

70–73 8 8 12 18 14 7.4 2.4 

a for details of the aerosol ID ranges see table 22. 

Table 11. Concentrations (μg/m3

) and number densities (cm-3) for the selected NAM aerosols as a function of 

AMP, average and current wind speed (m/s), and relative humidity. 

Aerosol ID AMP U U RH Concentration 

(μg/m3

) 

Number Density 

(cm-3) 

58 3 3 12 50% 2.498 1231.60 

59 " " " 80% 3.774 1231.60 

60 " " " 90% 10.15 1231.60 

61 " " " 95% 44.18 1231.60 

62 3 8 12 50% 3.988 1247.62 

63 " " " 80% 6.038 1247.62 

64 " " " 90% 16.28 1247.62 

65 " " " 95% 70.92 1247.62 

66 8 3 12 50% 4.198 8741.60 

67 " " " 80% 5.732 8741.60 

68 " " " 90% 13.40 8741.60 

69 " " " 95% 54.33 8741.60 

70 8 8 12 50% 5.688 8757.62 

71 " " " 80% 7.996 8757.62 

72 " " " 90% 19.53 8757.62 

73 " " " 95% 81.07 8757.62 
17

3.4 Rain and Snow Models 

The phase functions generated for rain and snow particle distributions are complicated by the 

large size parameters (2πr/λ) encountered. Because the size of these particles is very large, the 

amount of forward scatter due to diffraction can be extreme. According to Hodkinson and 

Greenleaves (15), when the airborne particles of an aerosol species are larger than a few 

wavelengths of the radiation being transmitted and a range of particle sizes or wavelengths 

exists, the combined single-scattering characteristics may be approximated by a combination of 

Fraunhofer diffraction, geometrical transmission, and reflection. While studying light scattering 

by irregular particles larger than the wavelength (such as snow), Hodkinson (16) found that, 

although the diffraction patterns of individual irregular particles vary greatly with shape, the 

resultant forward diffraction lobe for an ensemble of nonspherical particles with random 

orientations would be similar to an ensemble of spheres with cross-sectional areas equal to the 

particles’ areas. Because the phase functions for typical rain and snow size distributions have 

sharp peaks in the forward direction, a set of angles concentrated in the forward direction were 

used for these phase functions. Otherwise, difficulties arise in the phase function interpolation 

processes. Both the rain and snow models use the gamma distribution (equation 12) with m = 0. 

3.4.1 Rain Model 

The most widely used analytical parameterization for raindrop size distribution is the gamma 

distribution, which has been found to realistically describe averaged raindrop size distributions in 

many parts of the world. Marshall and Palmer (17) proposed a simple negative exponential 

parameterization for the raindrop size distribution, 

 n(r) = N0 exp(-Λr) = 1.6 x 10-5 exp(-8.2x10-3 R-0.21 r) cm-3 μm-1, (24) 

as a fit to filter paper measurements of raindrop size spectra for rain rates (R) between 1 and 

23 mm h-1. Their parameterization has been found to remain a realistic representation of 

averaged raindrop size distributions for much higher rain rates. Uijlenhoet and Stricker (18), as 

the result of an analytical derivation based on a theoretical parameterization for the raindrop size 

distribution, found that 

 n(r) = N0 exp(-8.46x10-3 R-0.214 r). (25) 

They noted that the power law dependence between rainfall related variables is far from 

coincidental. Rather, it is a direct consequence of the fact that rainfall is not a continuous 

process, as is often assumed, but a discrete process consisting of individual raindrops with 

different sizes and fall speeds. The actual shape of the raindrop size distribution is determined 

by the relative magnitude of the competing microphysical processes, which lead to growth 

(coalescence, condensation) or decay (breakup, evaporation) of the raindrops as they fall to the 

ground (19). However, the measurement of small raindrops are close to the detection limit of 

many raindrop sampling devices and, moreover, they are more sensitive to the effects of wind, 

turbulence, and splash than large raindrops. Also for increasing rain rates, the proportion of 

large raindrops increases, and consequently, the proportion of small drops decreases. 

Phase functions for rain were generated using a Marshall-Palmer particle size distribution with 

rain rates of 1, 5, and 10 mm/h, corresponding to rain types of drizzle, widespread, and 

thunderstorm, respectively. The number densities and liquid water content are given in table 12. 
18

The Marshall-Palmer distribution yields somewhat larger particles at higher rain rates, causing 

the forward direction lobe of the phase function to be narrower and sharper. The phase functions 

given are considered to be reasonable for most rain conditions. Extinction for other rain rates 

may be calculated using the following (20), 

κ = 0.51 R0.63, km-1 (drizzle), (26) 

κ = 0.36 R0.63, km-1 (widespread), (27) 

κ = 0.16R0.63, km-1 (thunderstorms), (28) 

where R is in mm h-1. 

Table 12. Precipitation rates, number densities, and equivalent liquid water for the rain model. 

Precipitation Type 

Parameters 

Rain Drizzle Rain 

Widespread 

Rain 

Thunderstorm 

Rain rate (mm/hr) 1.0 5.0 10.0 

Number density (particles/cm3) .001951 .00274 .00316 

Liquid water content (μg/m3) 89,000 344,000 616,000 

3.4.2 Snow Model 

The exact nature of snow size distributions is complicated by the size and shape of snow crystals, 

which are influenced by the temperature and humidity conditions that exist during their 

formation and growth. The basic crystal can also be modified by the attachment of super-cooled 

drops in a process called riming. These processes result in a wide variation in snow types with a 

subsequent large variation in associated visibilities. Because snow particle size distributions 

may vary greatly for a given snow rate, it is difficult to make generalizations about the scattering 

characteristics of a typical snow. Compounding this difficulty is the nature of the behavior of the 

refractive indices of ice, which are temperature, humidity, and wavelength dependent. 

Additionally no theory presently exists to calculate exactly the extinction coefficient for complex 

shapes such as snow crystals. Since determination of snow phase functions, which are highly 

dependent upon the size distribution chosen, and determination of visibilities, which are not 

dependent upon the size distribution but are highly dependent upon the snow type, we have 

divided the following discussion into two sections: phase function determination and visibility 

determination. 

Phase Function Determination 

Calculations for the snow phase function and its associated scattering parameters use a gamma 

distribution (21) (c.f., equation 12). The particle number density, precipitation rate, and 

equivalent liquid water content values, derived from equations presented in the “Visibility 

Determination” section, are given in tables 14 and 15. It should be noted that while the 

extinction coefficient is dependent on the number of particles, the phase function per se is not. 

For the snow size distribution used here, the size parameters in the visible range are large 

(>20,000) and Mie calculations are therefore suspect. In the mid- and far-IR the size parameter 
19

is considerably smaller and Mie calculations, while still suspect, become more reasonable. 

Therefore, we have chosen to use Mie generated phase functions only in the mid- to far-IR. A 

hybrid approach comprised of fits to measured data combined with Mie calculations was used in 

the visible and near-IR. 

To model the scattering properties at visible and near-IR wavelengths, we used Winchester’s 

measured phase functions (22, 23), covering a series of 5 snowfalls at a wavelength of 0.6238 

μm (see figure 4). His measurements, which include propagation through large aggregates of 

snow particles, covered scattering angles from 10° to 170° and were normalized to a value of 1.0 

at 10°. Scattering in this range, which is outside the forward peak region, essentially comprises 

geometric optic reflections and refractions within the particles. These geometric effects are 

relatively wavelength independent due to the large size parameters involved. 

Figure 4. Snow phase function measurements taken by Winchester at 0.6238 μm. 

To emulate these measurements, we fit the data to the 10° to 170° range and extended it to 180° 

using a set of three HG analytic phase functions. We then added a fourth HG phase function to 

reflect the forward peak. 

The HG phase function is given by equation 8. The four term HG phase function can be written 

as 

, (1 2 cos( ) )

1 ( , , ) 2 3 / 2

2 4

1 i i

i

i

HG i

g g

g P g a a − +− = ∑= θ

θ (29) 

where the gi are asymmetry parameters and the ai are weighting coefficients: ∑=

=

4

1

1.0. iai 

The ai’s weight each HG term, which is appropriately peaked by the choice of gi. We assume 

that the forward peak of the phase function, which is due to the diffraction pattern at these sizes, 

contains one-half of the energy (24). While not strictly true, we find that integration of our Mie 

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

0 30 60 90 120 150 180

Angle (degrees)

Phase Function

15 Nov 1980

2 Feb 1983

3 Feb 1983

5 Feb 1983

9 Mar 1983
20

generated phase functions from 0o to 1o contain approximately 50% of the area under the phase 

function curve. Since the first HG term will represent the forward peak, we assigned a weight of 

0.5 to a1. Knowing a1, g1 can be determined by setting the phase function at θ = 0° equal to the 

Mie computed peak value at 0° ( AMie). Hence, 

HG AMie

g g

g P g = − +− = 2 3 / 2

2

(1 2 )

1

4

1 (0, )

π

. (30) 

Since g is required to be ≤ 1.0, we find the following equation for g1: 

Mie

Mie Mie

A

A A

g

π

π π

16

1 16 1 64

1

+ − + = . (31) 

This formulation also allows the user to generate a phase function with a supplied peak value. 

Additional guidance for the gi’s (i = 2-4) was found by noting that the second term should be 

peaked in the forward direction, the third term is isotropic, and the fourth term must be peaked in 

the backward direction. The ai’s were determined by fitting with the restriction that ∑=

=

4

2

0.5

i

ai . 

Table 13 displays our final values. Thus the visible snow phase functions all have the same 

functional curve from 10° to 180°, but have different peak values at 0°, reflecting differences in 

the diffraction patterns. Finally, these hybrid phase functions were renormalized to one. 

Table 13. Coefficients for the 4-term HG phase function. 

i gi ai

1 ~0.999 a 0.50 

2 0.860 0.45 

3 0.000 0.03 

4 -0.600 0.02 

a

variable: determined by using equation 31. 
21

Figure 5 presents the 4-term HG phase function, Winchester’s data, and our Mie calculations 

interpolated to 0.6238 μm. To more clearly see the peaked portion of figure 5, figure 6 presents 

an expanded segment from 0o to 8o. 

Figure 5. Phase function comparison of the 4-term HG (solid black) with the Mie (solid blue) at 

0.6238 μm. Winchester’s measurements are represented by the solid diamond shaped 

symbol (black). 

Figure 6. Peak phase function comparison (0o to 8o) of the 4-term HG (solid black) with the 

Mie (solid blue) at 0.6238 μm. Winchester’s measurements are represented by the 

solid diamond shaped symbol (black). 

Visibility Determination 

1. Relation to liquid equivalent snow rate: Current methods of estimating operational 

snowfall intensity are based on prevailing visibility. The U.S. National Weather Service 

estimates of snow “intensity” are defined as follows: light snow intensity is reported when 

the visibility is greater than or equal to 1.0 km, moderate snow intensity is reported when 

the visibility is less than 1.0 km but greater than 0.5 km, and heavy snow intensity occurs 

1.E-02

1.E+00

1.E+02

1.E+04

1.E+06

02468

Angle (degrees)

Phase Function

Measured

Mie

HG

1.E-04

1.E-02

1.E+00

1.E+02

1.E+04

1.E+06

0 30 60 90 120 150 180

Angle (degrees)

Phase Function

Measured

Mie

4 Term HG
22

when the visibility is less than or equal to 0.5 km. It has been shown (25) that this 

relationship frequently does not provide correct snowfall intensities, which is of particular 

concern to pilots. This visibility variation, leading to varying snowfall intensities, has been 

observed even though the liquid equivalent snowfall rate was constant. Liquid equivalent 

snowfall rate, hereafter referred to as “snowfall rate,” is expressed as light when the 

snowfall rate is less than 1 mm h-1, moderate when the snow rate is less than 2.5 mm hr-1

but greater than 1 mm h-1, and heavy when the snow rate is greater than 2.5 mm hr-1. 

Rasmussen, et al. (25) have investigated the relationship between visibility and snow 

intensity and found that snow rate estimates may be greatly improved by considering time 

of day and snowfall type. We synopsize their findings below. 

 There are many snow crystal types, their shape or habit being highly dependent upon the 

temperature and humidity in the cloud layer in which they are formed and whether or not 

these particles are being formed by deposition, aggregation, or riming. Riming is the 

freezing of super-cooled cloud droplets on snow crystals and is one of the fundamental 

snow growing mechanisms. Following the nomenclature of Rasmussen, we divide the 

snow into two general categories, wet and dry, which, excluding crystal types, covers 

almost all subsets of snow types. Generally, wet snow forms when the temperature is 

≥-1 °C and can be further defined as partially melted and/or rimed. Assuming that 

snowflakes can be approximated by spheres, the density of these two snow types can be 

found by 

ρdry = 0.17 D-1 g cm-3, (32a) 

 and 

ρwet = 0.724 D-1 g cm-3, (32b) 

 where D is the snowflake diameter in mm. Daytime meteorological visibility is calculated 

from 

 V = 3.912/κ cm, (33) 

 where V is the visibility and κ is the volume extinction coefficient. Using the gamma 

distribution for the snow particle size, it can be shown that the extinction coefficient, 

κ = π N0/Λ3

 cm2 cm-3, (34) 

 where N0 and Λ have been defined previously. Thus, 

 V = 1.245 x10-5 Λ3

/N0 = 1.245x10-5 Λ2/N km. (35) 

 The visibility can also be related to the mass median diameter, D0, by 

 V = 16.77/(D0

2

 N). (36) 

 
23

Therefore, knowing D0, N can be determined from the visibility. Now, for the snow size 

distribution used here, it can be shown that the (liquid equivalent) snowfall rate is 

 S = πρN0Vt/Λ4

 cm s-1, (37) 

 where Vt is the terminal snowflake velocity taken to be 100 cm s-1 for dry snow and 

200 cm s-1 for wet snow. Since the density of water is 1 g cm-3, the units of equation 38 

can also be written as g cm-2 s-1. Using equations 32, 36, and 37, we can now relate the 

snow rate to the visibility: 

 Sdry = 0.796 / V mm h-1 (38a) 

 or 

 Swet = 6.74 / V mm h-1, (38b) 

 where V is now in km. The wide range in visibility for a given snowfall rate is produced 

by two factors: 1) the smaller cross-sectional area of wet snowflake aggregates compared 

with that of dry snowflake aggregates of the same mass and 2) the higher terminal velocity 

of wet snowflake aggregates compared with that of the dry snowflake aggregates of the 

same mass (25). Thus for a snow rate of 1 mm h-1, the visibility can range from 0.8 km to 

6.7 km, over a factor of 8. Conversely, with a visibility of 3 km (the value used for our 

calculations), and subsequent “dry” and “wet” snow rates of 0.27 and 2.25 mm h-1, the 

snow intensity varies from light to moderate. This clearly points out the trouble using just 

observed visibility to determine snow intensity and/or snow rate without considering 

additional parameters, e.g. N, D0, ρ, or snow type. 

 We have determined a self-consistent set of parameters for the two snow types by taking 

the distribution slope, Λ, from Braham’s (21) table 4 (1026 EST on 21 Jan and 0855 EST 

on 14 Jan, his figures 3 and 4), and the terminal velocity (Vt) and the constant (C3) relating 

density to diameter from Rasmussen (25). These values are shown in table 14. Assuming 

a visibility of 3 km, we can then determine the dependent quantities, D0, κ, ρ, S, N0, and N 

from equations above. The dependent parameters, including the liquid water content, are 

presented in table 15. 

Table 14. Independent parameters for the two snow types. 

Snow Type Λ

(cm) 

Vt

(cm s-1) 

C3

(g cm-2) 

Dry 20.1 100 0.017 

Wet 17.7 200 0.072 

Table 15. Dependent parameters for the two snow types (visibility = 3 km). 

Snow 

Type 

Do 

(cm) 

κ

(km-1) 

ρ

(g cm-3) 

S 

(mm h-1) 

N0

(cm-3 cm-1) 

N 

(cm-3) 

Liquid Water Content 

(μg/m3

) 

Dry 0.183 1.30 0.0929 0.265 0.0337 1.68 x 10-3 60,300 

Wet 0.207 1.30 0.348 2.25 0.0230 1.30 x 10-3 256,000 
24

2. Day/night effect: Rasmussen also investigated the effect of nightfall on the estimation of 

snowfall rate by visibility. The American Meteorological Society (26) defines visibility as 

“the greatest distance in a given direction at which it is just possible to see and identify 

with the unaided eye 1) in the daytime, a prominent dark object against the sky at the 

horizon, and 2) at night, a known, preferably unfocused, moderately intense light source.” 

Daytime estimates of visibility are subjective evaluations of atmospheric attenuation of 

contrast, while nighttime estimates represent attempts to evaluate the attenuation of flux 

density. Rasmussen found that visibility-based estimates of snow intensity at night should 

be increased by one category. Thus, during the night, light snow intensity should be 

increased to moderate, and moderate snow intensity should be increased to heavy as 

presented below in table 16 (Rasmussen’s (25) table 6, reproduced with permission), and in 

a different format in table 17. To aid with visibility determination in wet or dry snow, 

these tables present snowfall intensity as a function not only of visibility, but also of 

temperature. 

Table 16. Modified visibility criteria for snowfall intensitya (based on temperature and day or night). 

Temperature Horizontal Visibility (Statute Miles and [km]) 

Condition 

°C °F 1⁄4 

[0.4] 

1⁄2 

[0.8] 

3⁄4 

[1.2] 

1 

[1.6] 

11⁄4 

[2.0] 

> 11⁄4 

[2.0] 

< -1 °C < 30 °F Heavy Moderate Light 

Snow 

Daylight 

(Light > 3 ft 

candles) ≥ -1 °C ≥ 30 °F Heavy Moderate 

Light 

< -1 °C < 30 °F Heavy Moderate Light 

Snow 

Nighttime 

(Light < 0.5 ft 

candles) ≥ -1 °C ≥ 30 °F Heavy Moderate Light 

NOTE: Rasmussen’s (25) table 6, reproduced with permission. 

a

 Light snowfall intensity is defined as less than 1 mm/hr equivalent liquid water precipitation, moderate 

snow as 1 mm/hr to 2.5 mm/hr, and heavy as greater than 2.5 mm/hr. 
25

Table 17. Snow visibility (km) vs. snowfall intensity. 

Temperature Snowfall Intensity 

Condition 

°C °F Heavy Moderate Light 

< -1 < 30 ≤ 0.4 0.8 ≥ 1.2 

Daylight 

≥ -1 ≥ 30 ≤ 0.8 1.2 ≥ 1.6 

< -1 < 30 ≤ 0.8 1.6 ≥ 2.0 

Nighttime 

≥ -1 ≥ 30 ≤ 1.6 1.6 – 2.0 > 2.0 

Visibility 

 (km) 

 We have calculated scattering values for two types of snow, dry and wet, using the 

refractive indices for ice taken from Warren (27). 

3.5 Dust Aerosol Models 

3.5.1 Dust Model 

Soil-derived aerosols are an important component of the total atmospheric aerosol content in 

certain geographic locations. Reported results of size distribution measurements for these 

aerosols vary widely. However, the general consensus is that the dust aerosols follow a bimodal 

lognormal distribution. Empirical data (28) fits this type of distribution well, and dust aerosols 

may be produced by a pulverization process in the soil. Epstein (29) has shown that such 

processes result in lognormal distributions. The bimodal distribution also provides a better fit, as 

empirical dust distributions appear to be characterized by more than one mode. Generally, the 

accumulation or small mode appears to be a characteristic of dust aerosols under all conditions, 

while the coarse or large mode is more a function of the parent soil size distribution (28). The 

latter component usually appears only under conditions of moderate to heavy aerosol dust 

loading. 

The parameters for light and heavy aerosol loading (table 18) were taken at various locations, 

predominantly in the southwestern United States. Analyses of the small particle mode (30) 

showed that the constituents were primarily ammonium sulfate, carbon, calcite, sodium nitrate, 

quartz, and montmorillonite for both distributions. The particles contained in the large mode 

were seen to settle quickly, in both light and moderate cases, as the wind speed diminished. To 

allow the distribution to be more representative of varying geographic locations (31), the 

accumulation mode was considered to be comprised of 80% quartz and 20% montmorillonite by 

mass for the heavy aerosol loading and 80% ammonium sulfate and 20% carbon by mass for the 

light aerosol loading case. 
26

Table 18. Values of lognormal particle size distributions parameters for the dust and high explosive (HE) dust 

models: bulk density, number density, mass loading, mode radius, and geometric standard deviation. 

Dust Type Light Loading Heavy Loading HE Dust 

Mode Small Large Small Large Small Large 

Composition Ammonium sulfate Quartz Montmorillonite Quartz 

Bulk density (g/cm3) 1.769 2.32 2.5 2.32 2.5 2.5 

Number density (cm-3) 1988 3.79 39.62 0.1128 200 0.07 

Mass loading (μg/m3) 16 40 1000 10000 15930 48680 

rg (μm) 0.05 0.5 0.5 15 0.5 22.5 

σg 2.0 2.0 2.25 1.6 2.6 1.87 

Composition Carbon Quartz 

Bulk density (g/cm3) 1.8 2.32 

Number density (cm-3) 488.5 1218.6 

Mass loading (μg/m3) 4 4000 

rg (μm) 0.05 0.5 

σg 2.0 1.6 

The refractive indices for quartz were interpolated from Weinman and Peterson (32) for 0.55 and 

1.06 μm; from Jennings and Gillespie (33) for 3.0–5.0 μm; and from Spitzer and Kleinman (34) 

for 8.0–12.0μm. The refractive indices for ammonium sulfate were interpolated from the work 

of Toon et al. (35); and the refractive indices of carbon were interpolated from the work of 

Gillespie and Goedecke (36). The refractive indices of montmorillonite at 0.55 and 1.06 μm 

were interpolated from the work of Egan and Hilgeman (37); for wavelengths greater than 

4.5 μm, values were interpolated from the work of Toon et al. (38). The errors introduced by the 

interpolation are probably small due to the close proximity of the wavelengths used here and in 

the aforementioned references. However, for the 3.0–4.5 μm band, the refractive indices had to 

be interpolated between 2.6 and 5.0 μm. Because montmorillonite is a clay material with water 

chemically bonded in its crystal lattice structure, the refractive indices and derived quantities 

should be used with caution in this wavelength band (3.0–4.5 μm). The heavy loading dust type 

reflects very large mode radius constituents associated with high wind speeds. The light loading 

would be the case normally considered. Mixing the heavy and light cases should simulate 

intermediate condition dust cases. Because quartz is an optically positive uniaxial crystal (39), 

the scattering problem is divided into two parts. Two-thirds of the scattering material is treated 

using the ordinary indices of refraction. The remaining third of the material is treated using the 

extraordinary indices. It has been pointed out to the authors that the scattering properties at 

0.55 μm for this phase function appear to be anomalous at higher wind speeds for backscatter 

angles greater than ~164° (c.f., figure A-8b in reference 1). This appears to be a function of the 

quartz component of the aerosol (40, 41) being increasingly lofted as the wind speed increases. 
27

3.5.2 HE Dust Model 

The HE dust model was generated using the empirical results of field tests (42) taken at 

Huntsville, AL, and Orogrande, NM. The results were empirically fitted to a bimodal lognormal 

curve with the parameters presented in table 18. The refractive indices were taken from the work 

of Ivlev and Popova (43). The refractive indices are synthetic spectra chosen because no 

consistent set of measurements covers the wavelength range in PFNDAT. A comparison of the 

synthesized spectrum with the measurements at the wavelengths available shows good 

agreement (30). 

3.5.3 Vehicular Dust 

The heavy, light, and HE dust models can, in some circumstances, be used for vehicular dust. 

The following is provided as guidance. The heavy and HE dust modes can be used for dust 

being raised by the vehicle; however, the coarse mode of these distributions will settle out in 

about 20 s, in which case the light dust model would be appropriate. The user is cautioned that 

soil types vary considerably and these models may not be representative of local soils. Refer to 

The Infrared & Electro-Optical Systems Handbook (44) for further information. 

3.5.4 Desert Model 

The characteristics of the desert aerosol model are detailed in Longtin et al. (45). This model 

arises from measurements taken in North Africa and the southwest United States between 1977 

and 1985. The measurements included size distributions, composition, and total mass loading. 

The desert air mass is composed of three components; each component has a different lognormal 

particle size distribution and indices of refraction. The components represent carbonaceous 

particles, water soluble particles, and sand. Two types of sand contribute to the sand component: 

50% of the sand particles are pure quartz, and 50% are quartz contaminated with 10% hematite. 

The amount of carbonaceous and water soluble particles are constant (once scaled by the 

visibility), the sand component increases with wind speed. This model does not have humidity 

dependence. 

While Longtin’s data formed the baseline of these calculations, the Longtin data shows identical 

values for indices of refraction of pure quartz for the ordinary and extraordinary rays in both the 

visible and near-infrared bands. More up-to-date information on quartz refractive indices was 

available in Hoock and Sutherland (44). The Hoock and Sutherland data showed distinctions in 

these values throughout the visible to far-IR bands and were used here as they were based on 

later, more accurate measurements. For the carbonaceous aerosol, we used the Longtin 

refractive indices, but we note that data from vegetative fires in Hoock and Sutherland show 

significant differences; however, these data are associated with aerosols extracted close to the 

source of emission and thus are battlefield aerosols in origin. Therefore the carbonaceous 

aerosols included here represent long-range atmospheric constituents with sources outside the 

immediate area. The input parameters for the six distributions are presented in table 19. We 

used equation 19 to recover the number density based on known concentration values C. The 

data for the carbonaceous and water soluble aerosols (i.e., ammonium sulfate) are the same for 

each wind speed case, while the sand statistics vary. In our case six aerosol components were 

necessary to accommodate the carbonaceous, water soluble, and four sand constituents: two 

each types of sand―pure quartz and 10% hematite contaminated quartz and two types of 

ray―ordinary (2/3rds) and extraordinary (1/3rd) with relative weightings as indicated. As a 
28

combined effect, we thus divided the total sand loading into 2/6ths ordinary ray pure quartz, 

1/6th extraordinary ray pure quartz, 2/6ths ordinary ray 10% hematite contaminated quartz, and 

1/6th extraordinary ray hematite contaminated quartz. Results obtained for extinction 

coefficients differ somewhat from those obtained by Longtin et al., primarily at shorter 

wavelengths due to the differing refractive indices. 

Table 19. Particle size distribution parameters for desert aerosol components: bulk density, number density, mass 

loading, mode radius, and geometric standard deviation. 

Aerosol Component Type Carbonaceous Water Soluble Sand 

Wind speed (m/s) NA NA 0 10 20 30 

Bulk density (g/cm3), ρ 2.00 1.77 2.65 2.65 2.65 2.65 

Geometric mean (μm),rg 0.0118 0.0285 6.24 7.76 9.28 10.80 

Width parameter, σg 2.00 2.24 1.89 2.14 2.42 2.74 

Number density (cm-3), N 367.9 3650.6 0.002459 0.01494 0.07137 0.31613 

Mass loading (μg/m3),C 0.044 11.69 41.07 1048.1 21277.1 427587.8 

3.6 Fog Oil, White Phosphorus, and Hexachloroethane Smoke Aerosol Models 

The phase functions for inventory smokes can be calculated almost exactly because the particles 

are nearly spherical. Discrepancies between theory and measurement can be attributed to 

uncertainties in the particle size spectrum or complex refractive indices. Experiments (33) have 

shown that the particle size spectrum is closely approximated by a lognormal distribution. 

Reference to the mass loading or mass concentration (C) of the particulate material (equation 11) 

rather than the number density is conventional in smoke applications. The extinction 

calculations have thus been performed for a constant aerosol loading factor of 1g/m3 to permit 

scaling of results to arbitrary concentrations during simulations. The magnitude of C has no 

effect on the phase function or the mass extinction coefficient α, and only linearly scales the 

volume extinction coefficient κ (c.f., equation 4). Table 20 lists the parameters considered 

representative of inventory smokes and includes the mass median diameter (MMD), often used 

in the literature in place of rg. MMD is related to rg and σg through the relation 

 ln(MMD) = ln 2rg + 3ln2

σg, (39) 

where rg and σg are listed in table 20. This table also shows that the mass concentration C was 

arbitrarily set to 106 μg/m3

. 
29

Table 20. Representative parameters of inventory smokes at various relative humidities. 

Aerosol Species WP Fog Oil HC 

RH (%) 17 50 90 50 85 

Geometric mean (μm), rg 0.241 0.269 0.365 0.190 0.422 

Width parameter, σg 1.450 1.450 1.450 1.800 1.450 

Bulk density (g/cm3), ρ 1.617 1.443 1.178 0.890 1.220 

Mass loading (μg/m3), C 106

 106 106 106 106

MMD 0.729 0.814 1.104 0.575 1.338 

Table 20 also lists the particle spectrum parameters appropriate for fog oil dissemination by 

current military generators designed to produce particles most efficient for obscuration at visible 

wavelengths (46). Other experimental generators may produce larger particles. WP and HC 

have parameters listed for specific values of relative humidity. Hygroscopic growth has been 

modeled for these conditions by semi-empirical relations (47, 48). Other evidence (49) shows 

that at high humidities (greater than 75% RH) a bimodal particle size spectrum may be expected 

that would be most pronounced for WP smoke. 

The final considerations are the real and imaginary refractive indices. Hoock and Sutherland’s 

(44) real and imaginary refractive indices for WP and fog oil were used. Reliable experimental 

measurements would be preferred, but such measurements are usually impossible because of the 

complex reaction products formed in producing smoke (50). The values for selected 

wavelengths from the visible through the IR for HC were taken from Weast and Astle (50). 

They derived coefficients based on laboratory measurements performed on the major constituent 

of HC, zinc chloride (ZnCl2). 

Fog oil smoke is not considered hygroscopic, so only a single dataset is used. In the visible, the 

imaginary index for fog oil is so small as to be beyond instrumental sensitivity; it can be 

considered negligible for most applications. This small value for the imaginary index leads to a 

single-scattering albedo of nearly unity, implying that extinction is entirely due to scattering. 

Table 21 compares the average mass extinction coefficients (α) computed using the AGAUS 

Mie code (14), versus laboratory experimental results (47–50) for several spectral bands of 

interest. Since the measured results represent band averages, a typical cloud thickness of 

0.01 km was assumed and results were computed by averaging the computed transmission 

through 1 g/m3

 density aerosols. The table shows that all the comparisons are reasonable. 

Disparities are no larger than those found among various experiments throughout the above cited 

literature. Comparison of results for WP in the 8–12 μm region is sometimes taken as evidence 

that secondary reaction products are significant for WP smokes (51). The disparate results for 

fog oil at visible wavelengths may be due to the use of a single wavelength for the modeled 
30

results. Differences in the longwave IR band results may reflect different assumptions regarding 

sources. It is probably significant that the experimental data were obtained by the vapor 

condensation method rather than by pyrotechnic dissemination. 

Table 21. Comparison of theoretical and experimentally measured mass extinction 

coefficients at 50% RH for various smoke aerosols. 

WP Fog Oil HC λ 

(μm) Model Experiment Model Experiment Model Experiment

visiblea 4.282 3.940 5.367 7.730 3.227 4.579 

1.06 1.963 1.410 3.737 3.500 2.601 2.040 

3–5 0.284 0.290 0.262 0.270 0.193 0.190 

8–12 0.284 0.366 0.021 0.014 0.068 0.052 

NOTE: All mass extinction coefficients are given in m2/g. a 0.55 μm for the model; 0.4–0.7 μm for the experiment. 

4. The PFNDAT2005 Database 

There are 14 different aerosol classes contained in the database. These aerosols are functions of 

other parameters. The subtypes within a given aerosol class (advective, radiation fog, etc.) and 

RH, etc., result in 69 different aerosol variations. Each aerosol has a numerical ID associated 

with it (see table 22). 

The database is comprised of a series of ASCII files, one for each of the aerosol identifiers (the 

numerical ID) listed in table 22. With the exception of the snow aerosol at visible and near-IR 

wavelengths, these phase functions and associated information were generated with AGAUS 

(14). Information relevant to the entire dataset has been placed on the first line, i.e. the phase 

function ID, the number of angles, the number of wavelength datasets, the aerosol concentration 

(g m-3), the total number density (particles/cm3

), and the RH (%), formatted as 

(3(I3,1X),2(E12.6,1X),F6.2): this is a change from previous versions of PFNDAT. Where 

values were either not applicable or unknown, a value of -99.99 has been used as a placeholder. 

Following this are 65 discrete angles between 0o and 180o, formatted as (11(F6.2,1X)). For 

backward compatibility, the last angle is followed by a sentinel number of 999.99. Following the 

angles is a one-line preamble comprised of the wavelength (μm), the single scattering albedo, the 

total volume extinction coefficient (km-1), and the asymmetry parameter, formatted as 

(4(E12.6,1X)). Subsequent to this is the phase function, one value for each angle, at the 

wavelength under consideration, formatted as (6(E12.6,1X)). The total number of preambles and 

phase function sets in a given file corresponds to the “number of wavelength sets” read on the 

first line. The remainder of each file thus contains sets of aerosol scattering information (single 
31

scattering albedo, extinction, asymmetry, and phase function) for available wavelengths. The 

structure of these files is presented schematically in table 23. For most aerosols there will be 32 

such sets of data. For the smokes only 20 wavelength sets are provided since data were 

unavailable at wavelengths beyond 12 μm. There is a blank between all data items, thereby 

allowing unformatted reads if so desired. Phase functions 31–45, and 58–73 are new; they do 

not exist in previous versions. 

A FORTRAN90 computer program for interrogating all versions of the database is provided in 

the appendix and on the distribution CD. 

Table 22. Aerosol ID as a function of aerosol type and RH.

Aerosol ID RH 

(%) 

1) Maritime 0 

2) “ 50 

3) “ 70 

4) “ 80 

5) “ 90 

6) “ 95 

7) “ 98 

8) “ 99 

9) Urban 0 

10) “ 50 

11) “ 70 

12) “ 80 

13) “ 90 

14) “ 95 

15) “ 98 

16) “ 99 

17) Rural 0 

18) “ 50 

19) “ 70 

20) “ 80 

21) “ 90 

22) “ 95 

23) “ 98 

24) “ 99 

25) Fog (heavy advection) NA 

26) Fog (moderate radiation) NA 

28) Rain (widespread) NA 

29) Rain (thunderstorm) NA 

30) Snow (dry) NA 

31) Fog (moderate advection) 

32) Fog (heavy radiation) 

Aerosol ID RH 

(%) 

33) Desert aerosol (ws=0) 

34) Desert aerosol (ws=10) 

35) Desert aerosol (ws=20) 

36) Desert aerosol (ws=30) 

37) Tropospheric 0 

38) Tropospheric 50 

39) Tropospheric 70 

40) Tropospheric 80 

41) Tropospheric 90 

42) Tropospheric 95 

43) Tropospheric 98 

44) Tropospheric 99 

45) Snow (wet) 

46) reserved for future use 

47) reserved for future use 

48) reserved for future use 

49) reserved for future use 

50) Dust (light loading) NA 

51) Dust (heavy loading) NA 

52) HE dust NA 

53) WP smoke 17 

54) “ 50 

55) “ 90 

56) Fog Oil 50 

57) HC smoke 85 

58) NAM (AMP=3,U =3,U=12) 50 

59) “ 80 

60) “ 95 

61) “ 99 

62) NAM (AMP=3, U =8,U=12) 50 

63) “ 80 
32 

Table 22. Aerosol ID as a function of aerosol type and RH (continued).

Aerosol ID RH 

(%) 

64) “ 95 

65) “ 99 

66) NAM (AMP=8, U =3,U=12) 50 

67) “ 80 

68) “ 95 

Aerosol ID RH 

(%) 

69) “ 99 

70) NAM (AMP=8, U =8,U=12) 50 

71) “ 80 

72) “ 95 

73) “ 99 

Table 23. Structure of an aerosol phase function data file. 

PFN ID # θ’s # λ’s C N RH 

θ1 θ2 ... θ11

θ56 . ... θ65 999.99 

λ1 π01 κ1 g1

P(θ1,λ1,ID1) P(θ2,λ1,ID1) ... P(θ6,λ1,ID1)

P(θ61,λ1,ID1) ... P(θ65,λ1,ID1)

λ2 π02 κ2 g2

P(θ1,λ2,ID1) P(θ2,λ2,ID1) ... P(θ6,λ2,ID1)

P(θ61,λ2,ID1) ... P(θ65,λ2,ID1)

λmax π0max κmax gmax

P(θ1,λmax,ID1) P(θ2,λmax,ID1) ... P(θ6,λ1,ID1)

P(θ61,λmax,ID1) ... P(θ65,λmax,ID1)

NOTE: PFN ID = phase function identifier from table 22 

 # θ’s = total number of angles 

 # λ’s = total number of wavelengths for this ID 

 C = concentration (g m-3) 

 N = particle density (cm-3) 

 RH = relative humidity (%) 

 θi = discrete angles (degrees) 

 λj = discrete wavelengths (μm) 

 π0j = albedo for single scattering at wavelength j 

 κj = extinction coefficient (km-1) at wavelength j 

 gj = asymmetry parameter at wavelength j 

 P(θi,λj,ID1) = the value of the phase function at angle i, wavelength j, and aerosol type identifier ID 
33 

References 

1. Shirkey, R.C.; Sutherland, R.A.; Seagraves, M.A. EOSAEL 87 Volume 26 Aerosol Phase 

Function Data Base PFNDAT; ASL-TR-0221-26; U.S. Army Atmospheric Sciences 

Laboratory: White Sands Missile Range, NM, 1987. 

2. Tofsted, D.H.; Davis, B.T.; Wetmore, A.E.; Fitzgerrel, J.; Shirkey, R.C.; Sutherland, R.A. 

EOSAEL 92 Aerosol Phase Function Data Base PFNDAT; ARL-TR-273-9; U.S. Army 

Research Laboratory: White Sands Missile Range, 1997.

3. Henyey, L.C.; Greenstein, J.L. Diffuse radiation in the galaxy. Astrophys. J. 1941, 93, 

70–83.

4. Shettle E.P.; Fenn, R.W. Models for the Aerosols of the Lower Atmosphere and the Effects of 

Humidity Variations on Their Optical Properties; AFGL-TR-79-0214; Air Force Geophysics 

Laboratory: Hanscom Air Force Base, MA, 1979. 

5. Gathman, S.G. Optical Properties of the Marine Aerosol as Predicted by the Navy Aerosol 

Model. Opt. Eng. 1983, 22, 57-62. 

6. Gathman, S.G.; van Eijk, A.M.J.; Cohen, L.H. Characterizing Large Aerosols in the Lowest 

Level of the Marine Atmosphere, Proceedings of SPIE vol. 3433, Propagation and Imaging 

through the Atmosphere II; L.R. Bissonnette, Ed.; pp 41–52, 1998. 

7. Gathman, S.G.; Davidson, K.L. The Navy Oceanic Vertical Aerosol Model; NRaD Technical 

Report 1634; San Diego, December 1993. 

8. Fitzgerald, J.W. On the Growth of Aerosol Particles with Relative Humidity; NRL 

Memorandum Report 3847; 1978.

9. Gerber, H.E. Relative Humidity Parameterization of the Navy Aerosol Model (NAM); Navy 

Research Laboratory Report 8956; December 1985. 

10. Volz, F.E. Infrared Refractive Index of Atmospheric Aerosol Substrates. Applied Optics

1972, 11, 755–759. 

11. Zeisse, C. NAM6: Batch code for the Navy Aerosol Model; SPAWAR Systems Center 

Technical Report 1804; October 1999 (http://sunspot.spawar.navy.mil/2858/software/).

12. Hänel, G. The Properties of Atmospheric Aerosol Particles as Functions of Relative 

Humidity at Thermodynamic Equilibrium with the Surrounding Moist Air. Adv. Geophys. 

1976, 19, 73–188.

13. Shirkey, R.C.; Gouveia, M. Weather Impact Decision Aids: Software to Help Plan for 

Optimal Sensor and System Performance. Crosstalk, J. of Defense Software Engineering, 

2002, 15 (12), 17–21. 

14. Miller, A. Mie Code AGAUS 82; ASL-CR-83-0100-3; U.S. Army Atmospheric Sciences 

Laboratory: White Sands Missile Range, NM, 1983. 
34

15. Hodkinson, J.R.; Greenleaves, L. Computations of Light Scattering and Extinction by 

Spheres According to Diffraction and Geometrical Optics. J. of the Optical Society of 

America 1963, 53, 577–588. 

16. Hodkinson, J.R. “Light Scattering and Extinction by Irregular Particles Larger than the 

Wavelength” in Electromagnetic Scattering, M. Kerker, ed.; Macmillan: New York, 1963. 

17. Marshall, J.S.; Palmer, W.M. The Distribution of Raindrops with Size. J. Meteorology

1948, 5, 165–166.

18. Uijlenhoet, R.; Stricker, J.N.M. A consistent rainfall parameterization based on the 

exponential raindrop size distribution. J. of Hydrology, 1999, 218. 

19. Pruppacher, H.R.; Klett, J. D. Microphysics of Clouds and Precipitation; D. Reidel 

Publishing: Boston, MA, 1980. 

20. Quantitative Description of Obscuration Factors for Electro-Optical and Millimeter Wave 

Systems, Department of Defense Military Handbook, DOD-HDBK-178(ER), 1986. 

21. Braham, R.R., Jr. Snow Particle Size Spectra in Lake Effect Snows. J. of Applied 

Meteorology 1990, 29 (3). 

22. Winchester, L.W., Jr.; Jackovich, J.E.; Hanson, N.K. Phase Function Measurements at 

SNOW-TWO, SNOW-TWO Data Report, v1; U.S. Army Corps of Engineers Cold Regions 

Research & Engineering Laboratory Special Report 84-20; R. Jordan, ed., June 1984. 

23. Winchester, L.W., Jr.; Hanson, N.K.; Jackovich, J.E. Phase Function Measurements of Snow 

Crystals, Proceedings of the Snow Symposium III, v1, U.S. Army Corps of Engineers Cold 

Regions Research & Engineering Laboratory Special Report 83-31; Hanover, NH, 1983. 

24. van de Hulst, H.C. Light Scattering by Small Particles, Dover Publications: New York, NY, 

1981. 

25. Rasmussen, R.M.; Vivekanandan, J.; Cole, J. The Estimation of Snowfall Rate Using 

Visibility. J. of Applied Meteorology 1999, 38 (10). 

26. Glossary of Meteorology, T.S. Glickman, Ed.; American Meteorological Society: Boston, 

MA, 2000. 

27. Warren, S.G. Optical constants of ice from the ultraviolet to the microwave. Applied Optics

1984, 23 (8). 

28. Patterson, E.M.; Gillette, D.A. Commonalities in Measured Size Distributions for Aerosols 

Having a Soil-Derived Component. J. of Geophysics Research 1977, 82, 2074–2082. 

29. Epstein, B. The Mathematical Description of Certain Breakage Mechanisms Leading to the 

Logarithmic-Normal Distributio. J. of the Franklin Institute 1947, 244, 471–477. 

30. Jennings, S.G.; Pinnick, R.G.; Auvermann, H. J. Effects of Particulate Complex Refractive 

Index and Particle Size Distribution Variations on Atmospheric Extinction and Absorption 

for Visible Through Middle IR Wavelengths. Applied Optics 1978, 17, 3922–3928. 

31. Gillespie, J.B.; Lindberg, J.D. Seasonal and Geographic Variations in Imaginary Refractive 

Index of Atmospheric Particular Matter. Applied Optics 1992, 31, 2107–2112. 
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32. Weinman J.A.; Peterson, J.T. Optical Properties of Quartz Dust Particles at Infrared 

Wavelengths. J. of Geophysics Research 1969, 74, 6947–6952. 

33. Jennings, S.G.; Gillespie, J.B. Mie Theory Sensitivity Studies – The Effects of Aerosol 

Complex Refractive Index and Size Distribution Variations on Extinction and Absorption 

Coefficients; ASL-TR-0003; U.S. Army Atmospheric Sciences Laboratory: White Sands 

Missile Range, NM, 1978. 

34. Spitzer, W.G.; Kleinman, D.A. Infrared Lattice Bands of Quartz. Physical Review, 1961, 

121, 1324–1335. 

35. Toon, O.B.; Pollack, J.B.; Khare, B.N. Optical Constants of Several Atmospheric Aerosol 

Species, Ammonium Sulphate, Aluminum Oxide and Sodium Chloride. J. of Geophysics 

Research, 1976, 81, 5733–5748. 

36. Gillespie, J.B.; Goedecke, G. Refractive Indices of Powdered Materials Using Attenuated 

Total Reflectance Spectroscopy. Applied Optics 1989, 28, 3985–3992. 

37. Egan, W.G.; Hilgeman, T.W. Optical Properties of Inhomogeneous Materials; Academic 

Press: New York, NY, 1979. 

38. Toon, O.B.; Pollack, J.B.; Sagan, C. Physical Properties of the Particles Composing the 

Martian Dust Storm of 1971-1972. Icarus 1977, 30, 663–696. 

39. Born, M.; Wolf, E. Principles of Optics, 5th ed.; Pergamon Press: Oxford, UK, 1975. 

40. Liu, L.; Mishchenko, M.I.; Hovenier, J.W.; Volten, H.; Muñoz, O. Scattering matrix of 

quartz aerosols: comparison and synthesis of laboratory and Lorenz–Mie results. J. of 

Quantitative Spectroscopy & Radiative Transfer 2003, 79–80, 911–920. 

41. Haltrin, V.I.; Shybanov, E.B. Light scattering properties of quartz particles in seawater, 

Proceedings of the IEEE Geoscience and Remote Sensing Society 2000 Conference, 

Honolulu, HI, 2000, p1842–1844. 

42. Pinnick, R.G.; Fernandez, G.; Hinds, B.D. Explosion Dust Particle Size Measurements. 

Applied Optics 1983, 22, 95–102. 

43. Ivlev, L.S.; Popova, S.I. The Complex Refractive Indices of Substances in the AtmosphereAerosol Dispersed Phase. Atmospheric and Oceanic Physics 1973, 9, 587–591. 

44. Hoock, D.W.; Sutherland, R.A., “Obscuration Countermeasures,” chapter 6 of 

Countermeasure Systems, Volume 7, of The Infrared & Electro-Optical Systems Handbook; 

SPIE Optical Engineering Press: Bellingham, WA, 1996, pp 359–492. 

45. Longtin, D.R.; Shettle, E.P.; Hummel, J.R.; Pryce, J.D. A Wind Dependent Desert Aerosol 

Model: Radiative Properties; Air Force Geophysics Laboratory Technical Report, AFGLTR-88-0112; Hanscom Air Force Base, MA, 1988. 

46. Carlon, H.R. et al. Infrared Extinction Spectra of Some Cannon Liquid Aerosols. Applied 

Optics 1977, 1b, 1598–1605. 
36

47. Frickel, R.H.; Rubel, G.O.; Stuebing, E.W. Relative Humidity Dependence of the Infrared 

Extinction by Aerosol Clouds of Phosphoric Acid (UNCLASSIFIED), Proceedings of Smoke 

Symposium III (U), CONFIDENTIAL, PM Smoke/Obscurants, Aberdeen Proving Ground, 

MD, 1979. 

48. Rubel, G.O., Predicting the Droplet Size and Yield Factors of a Phosphorus Smoke as a 

Function of Droplet Composition and Ambient Relative Humidity Under Tactical Conditions; 

ARCSL-TR-78057; Chemical Systems Laboratory: Aberdeen Proving Ground, MD, 1978. 

49. Farmer, W. M., An Evaluation of Data Obtained During the H3S Test, Proceedings of Smoke 

Symposium IV, Adelphi, MD, 1980, p 49–86. 

50. Weast, R.C.; Astle, M.J., eds., CRC Handbook of Chemistry and Physics, 61st ed.; CRC 

Press: Boca Raton, FL, 1980-1981. 

51. Milham, M.E. et al. New Findings on the Nature of WP/RP Smokes; ARCSL-TR-77067; 

Chemical Systems Laboratory: Aberdeen Proving Ground, MD, 1977. 
37 

Appendix: A FORTRAN90 Program for Interrogating the Phase Function 

Files 

 program aerosol_info 

c Purpose: This interactive program will read any PFNDAT phase function file 

c at the requested wavelength (interpolation is not performed) and print out 

c the parameters listed below. PFNDAT 87/92 output values do not include 

c relative humidity*, particle concentration or number density. The 

c integral of the phase function is also computed. For additional 

c information see the references below. 

c Phase function normalization can be determined by integrating it 

c over all angles: 

c 

c 2pi pi 

c / / 

c | | P(cos(theta)) sin(theta) d(theta) d(phi) 

c / / 

c 0 0 

c 

c the asymmetry factor has an additional factor of cos(theta) 

 

c Usage: 

c The program is interactive: relevant wavelength and file name are requested. 

c The user can optionally have the phase function and associated angles 

c printed out. The program will attach the requested phase function file to 

c unit “udat”. Maximum input file name length is 9 characters. 

c Input: 

c Wavelength (um) [input value of 0 will examine all wavelengths in the file] 

c Name of phase function data file to be examined 

c Output: 

c angles (degrees) [optional] 

c phase function(s) at requested wavelength(s) [optional] 

c wavelength (microns) 

c single scattering albedo 

c extinction coefficient (km-1) 

c scattering coefficient (km-1) 

c absorption coefficient (km-1) 

c asymmetry parameter 

c integral of phase function 
38

c relative humidity(%) [*2005 only; for PFNDAT87/92 see tables in references] 

c particle concentration (g/m**3) [2005 only] 

c number density (particles/cc) [2005 only] 

c Language: FORTRAN 90 

c Parameters: 

c abextn: absorption coefficient (km-1) 

c angs: angles at which the phase function was computed (degrees) 

c ans: input determinor for p_switch 

c asymm: precomputed asymmetry factor 

c concentration: concentration of aerosol (g/m**3) 

c deg2rad: conversion factor 

c density: number density of aerosol (particles/cc) 

c extn: extinction coefficient (km-1) 

c file_name: input pfn file name 

c g: computed asymmetry factor 

c id: internal identification number of pfn file 

c id_check: numeric portion of pfn file name 

c num_angs: number of angles/values for pfn 

c num_waves: number of wavelengths contained in pfn file 

c p_switch: angle and phase function print switch 

c pfn: phase function value at related angle 

c pfnint: integral of phase function 

c pi: PI 

c rh: relative humidity (%) 

c scextn: scattering coefficient (km-1) 

c ssa: single scattering albedo 

c uin: input unit number 

c udat: unit number assigned to phase function file 

c uout: output unit number 

c w_switch: no valid wavelength indicator 

c wave: wavelength (micrometers) 

c wavein: selected wavelength to print (0 == all) 

c References: 

c Shirkey, R.C., and D.H. Tofsted, “Electro-Optical Aerosol Phase Function 

c Database DataBase, PFNDAT2005”, ARL-TR-3684, WSMR, NM. 

c 

c Tofsted, D., B. Davis, A. Wetmore, J. Fitzgerrel, R. Shirkey, 

c B. Sutherland, “EOSAEL92 Aerosol Phase Function Data Base PFNDAT,” 

c ARL-TR-273-9, June 1997, WSMR, NM. 

c 

c Shirkey, R.C., Sutherland, R.A. and M.A. Seagraves, EOSAEL 87, v26, 

c “Aerosol Phase Function Data Base PFNDAT”, ASL-TR-0221-26, October 1987, 

c WSMR, NM. 
39

 integer, parameter :: num_angs = 65 ! 65 angles 

 integer :: p_switch, w_switch, uin = 5, uout = 6, udat = 9 

 real, dimension(num_angs) :: angs, pfn 

 real :: wavein, wave, ssa, extn, asymm, scextn 

 character (len=9) :: file_name 

 character(len=1) ans 

 w_switch = 0 

 p_switch = 0 

 num_waves = 32 ! 32 wavelengths/file max 

 

 write (uout,*) ‘Input wavelength of interest (zero for all) ‘ 

 read (uin,*) wavein 

 write (uout,*)’Do you want the phase function printed out? (Y/N) ‘ 

 read (uin,*) ans 

 if (ans == ‘Y’ .or. ans == ‘y’) p_switch = 1 

 write (uout,*) ‘Input file name ‘ 

 read (uin,*) file_name ! name of pfn file to be examined 

 open (udat,file_name,status=‘old’,err=20) 

 read (udat,*) version ! 87 & 92 will have zero as first value 

 rewind (udat) ! 2005 has ID first value 

 write(uout,*) ‘ ‘ 

 if (version > 0.0) then 

 write(uout,*)’ ****** Results for PFNDAT 2005 ******’ 

 read (udat,*) id,num_ang2,num_wave2,concentration,density,rh 

 else 

 write(uout,*)’ ****** Results for PFNDAT 87/92 ******’ 

 endif 

 write (uout,*) ‘ ‘ 

 write (uout,*) ‘For file ‘,file_name 

 write (uout,*) ‘ ‘ 

 read (udat,*) (angs(i),i=1,num_angs) 

 do j = 1, num_waves ! # of wavelengths in a given file 

 if (version > 0.0) then ! 2005 

 read (udat,*,end=30) wave, ssa, extn, asymm ! different number of wavelengths 

 ! in different versions of PFNDAT 

 scextn = ssa * extn 

 read (udat,*) (pfn(i),i=1,num_angs) 

 else ! 87 or 92 

 read (udat,*,end=30) nang,id,wave,ssa,extn,scextn 
40

 read (udat,*) (pfn(i),i=1,num_angs) 

 endif 

 abextn = extn - scextn 

 if (p_switch == 1 .and. (wave==wavein .or. wavein==0.0)) then 

 if (j == 1) then 

 write (uout,*) ‘Angles follow’ 

 do k = 1,num_angs,11 

 if (k+10 < num_angs) then 

 write (uout,100) (angs(i),i=k,k+10) 

 else 

 write (uout,100) (angs(i),i=k,k+9) 

 endif 

 end do 

 write(uout,*) ‘ ‘ 

 endif 

 write(uout,*) ‘Phase function follows for wavelength = ‘, 

 + wave,’ um’ 

 do k = 1, num_angs,6 

 if (k+5 < num_angs) then 

 write (uout,101) (pfn(i),i=k,k+5) 

 else 

 write (uout,101) (pfn(i),i=k,k+4) 

 endif 

 end do 

 write (uout,*) ‘ ‘ 

 endif ! end optional print 

 call integrator(pfn,angs,pfnint,g,num_angs) 

 if (wave == wavein .or. wavein == 0.0) then 

 w_switch = 1 

 write(uout,*) ‘Associated information for ‘,file_name, 

 + ‘ at wavelength = ‘,wave,’ um’ 

 write (uout,*) ‘ Single scattering albedo = ‘,ssa 

 write (uout,*) ‘ Coefficients (km-1)’ 

 write (uout,*) ‘ extinction = ‘,extn 

 write (uout,*) ‘ scattering = ‘,scextn 

 write (uout,*) ‘ absorpton = ‘,abextn 

 if (version > 0.0) then 

 write (uout,*) ‘ Asymmetry = ‘,asymm ! 2005 precalculated 

 else 

 write (uout,*) ‘ Asymmetry = ‘,g ! on-the-fly; 87 & 92 

 endif 

 write (uout,*) ‘ Integral of phase function = ‘,pfnint 

 if (version > 0.0) then ! additional info for 2005 

 write (uout,*) ‘ Relative humidity = ‘,rh,’ %’ 
41

 write (uout,*) ‘ Concentration = ‘, 

 + concentration,’ g/m**3’ 

 write (uout,*) ‘ Density = ‘,density, 

 + ‘ particles/cc’ 

 endif 

 write (uout,*) ‘ ‘ 

 write (uout,*) ‘ ‘ 

 endif 

 end do ! wavelength loop 

 30 close (udat) 

 if (w_switch == 0) write (uout,*) ‘The input wavelength value of ‘ 

 + ,wavein,’ is invalid’ 

 stop 

 20 write(uout,*) ‘ File ‘,file_name,’ does not exist’ 

 100 format(1x,11(F6.2,1X)) 

 101 format(1x,6(E12.6,1X)) 

 end 

 

 subroutine integrator (pfn,angs,pfnint,g,num_angs) 

 real, dimension(num_angs) :: angs, pfn ! 65 angles max 

 pi = 3.14159 

 deg2rad = pi/180. 

c calculate normalization and asymmetry of phase function 

 pfnint = 0.0 

 g = 0.0 

 do i = 1,num_angs - 1 ! over zenith angle theta 

 pfnint = pfnint + .5*(pfn(i) + pfn(i+1))* ! zero to pi or zero to 180 

 1 abs(cos(angs(i+1)*deg2rad)-cos(angs(i)*deg2rad)) 

 g = g + .5*(cos(angs(i)*deg2rad)*pfn(i)+ 

 1 cos(angs(i+1)*deg2rad)*pfn(i+1))* 

 2 abs(cos(angs(i+1)*deg2rad)-cos(angs(i)*deg2rad)) 

 end do 

 

 pfnint = 2.* pi * pfnint ! over azimuthal angle phi 

 g = 2. * pi * g / pfnint ! zero to 2*pi or zero to 360 

 

 return 

 end 
42

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43 

Acronyms and Abbreviations 

ARL U.S. Army Research Laboratory 

AMP air mass parameter 

CL concentration length 

HC hexachloroethane 

HE high explosive 

IR infrared 

LOS line of sight 

MBL marine boundary layer 

MG modified gamma 

MMD mass median diameter 

NAM Navy Aerosol Model 

PFNDAT Phase Function Database 

RH relative humidity 

TAWS Target Acquisition Weapons Software 

WP white phosphorus 
44 

Distribution List 

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TOTAL 50 
46

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