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Some Useful Formulae for Aerosol Size

Distributions and Optical Properties

R.G. Grainger

9 June 2017

1

Some Useful Formulae for Aerosol Size Distributions and Optical Properties

Acknowledgements

I would like to acknowledge helpful comments that improved this manuscriptfrom Dwaipayan Deb.

iii


Contents

1 Describing Aerosol Size 1

1.1 Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Moments, Mean, Mode, Median and Standard Deviation . . . . . . 11.3 Geometric Mean and Standard Deviation . . . . . . . . . . . . . . 21.4 Effective radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Area, Volume and Mass Distributions . . . . . . . . . . . . . . . . 3

2 Normal and Logarithmic Normal Distributions 4

2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Properties of the Lognormal Distribution . . . . . . . . . . . . . . 6

2.2.1 Normal versus Lognormal . . . . . . . . . . . . . . . . . . . 62.2.2 Median and Mode . . . . . . . . . . . . . . . . . . . . . . . 72.2.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.4 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Log-Normal Distributions of Area and Volume . . . . . . . . . . . 92.4 A Different Basis State . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Other Distributions 13

3.1 Gamma Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Modified Gamma Distribution . . . . . . . . . . . . . . . . . . . . . 143.3 Inverse Modified Gamma Distribution . . . . . . . . . . . . . . . . 153.4 Regularized Power Law . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Modelling the Evolution of an Aerosol Size Distribution 18

5 Optical Properties 19

5.1 Volume Absorption, Scattering and Extinction Coefficients . . . . . 195.2 Back Scatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.3 Phase Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.4 Single Scatter Albedo . . . . . . . . . . . . . . . . . . . . . . . . . 205.5 Asymmetry Parameter . . . . . . . . . . . . . . . . . . . . . . . . . 205.6 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.7 Formulae for Practical Use . . . . . . . . . . . . . . . . . . . . . . . 215.8 Cloud Liquid Water Path . . . . . . . . . . . . . . . . . . . . . . . 235.9 Aerosol Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

v

Note that this is a working draft. Comments that will be excluded fromthe final text are indicated by XXX .


1 Describing Aerosol Size

A complete description of an ensemble of particles would describe the com-position and geometry of each particle. Such an approach for atmosphericaerosols whose concentrations can be ∼ 10, 000 particles per cm3 is imprac-ticable in most cases. The simplest alternate approach is to use a statisticaldescription of the aerosol. This is assisted by the fact that small liquiddrops adopt a spherical shape so that for a chemically homogeneous aerosolthe problem becomes one of representing the number distribution of particleradii. The size distribution can be represented in tabular form but it is usualto adopt an analytic functional. The success of this approach hinges uponthe selection of of an appropriate size distribution function that approximatesthe actual distribution. There is no a priori reason for assuming this can bedone.

1.1 Size Distribution

The distribution of particle sizes can be represented by a differential radiusnumber density distribution, n(r) which represents the number of particleswith radii between r and r + dr per unit volume, i.e.

N(r) =∫ r+dr

rn(r) dr. (1)

Hence

n(r) =dN(r)

dr. (2)

The total number of particles per unit volume, N0, is then given by

N0 =∫

∞

0n(r) dr. (3)

1.2 Moments, Mean, Mode, Median and Standard De-viation

The moments of a size distribution are simple metrics to express the shapeof the distribution. The ith moment mi of n(r) is defined

1

mi =1

N0

∫

∞

0(r − c)in(r) dr (4)

1Many texts omit the normalising term before the integral as for a probability distri-bution function N0 = 1.

1


where c is some constant. The raw moment is the moment where c = 0. Themoments about the mean are called central moments.

Generally a size distribution is characterised by its centre and by itsspread. The centre of a distribution can be represented by the

mean, µ0 defined by

µ0 =

∫

∞

0 rn(r) dr∫

∞

0 n(r) dr=

1

N0

∫

∞

0rn(r) dr. (5)

The mean is the first raw moment of a size distribution.

mode The mode is the peak (maximum value) of a size distribution.

median The median is the ”middle” value of a data set, i.e. 50 % of particlesare smaller than the median (and so 50 % of particles are larger).

while the spread is captured through the variance, σ20, defined

σ20 =

1

N0

∫

∞

0(r − µ0)

2n(r) dr. (6)

The variance is the second central moment of the size distribution. Finally,the standard deviation, σ0, of a distribution is the square root of the variance.

1.3 Geometric Mean and Standard Deviation

The geometric mean µg of a set of n numbers {r1, r2, . . . , rn} is

µg = n√r1 × r2 × . . .× rn (7)

The geometric mean can also be written

µg = exp[

ln(

n√r1 × r2 × . . .× rn

)]

= exp

(

∑ni=1 ln rin

)

(8)

The geometric standard deviation σg or S is defined

σg = S = exp

√

∑ni=1 (ln ri − lnµg)

2

n

(9)

that is, the geometric standard deviation is the exponential of σ, the standarddeviation of ln r. So ln(σg) = ln(S) = σ.

2


1.4 Effective radius

The effective radius (or area-weighted mean radius) of an aerosol distribution,re, is defined as the ratio of the third moment of the drop size distributionto the second moment, i.e.

re =m3

m2

=

∞∫

0r3 n(r)dr

∞∫

0r2 n(r)dr

. (10)

The usefulness of effective radius comes from the fact that energy removedfrom a light beam by a particle is proportional to the particle’s area (pro-vided the radius of the particle is similar to or larger than larger than thewavelength of the incident light).

1.5 Area, Volume and Mass Distributions

Analogous to the description of the distribution of particle number with ra-dius it is also possible to describe particle area, volume or mass with equiv-alent expressions to Equations 1-3.

The distribution of particle area can be represented by a differential areadensity distribution, a(r) which represents the area of particles whose radiilie between r and r + dr per unit volume, i.e.

A(r) =∫ r+dr

ra(r) dr. (11)

Hence

a(r) =dA

dr. (12)

For spherical particles

a(r) =dA

dN

dN

dr= 4πr2n(r). (13)

The total particle area per unit volume, A0, is then given by

A0 =∫

∞

0a(r) dr. (14)

The distribution of particle volume can be represented by a differentialvolume density distribution, a(r) which represents the volume contained inparticles whose radii lie between r and r + dr per unit volume, i.e.

V (r) =∫ r+dr

rv(r) dr. (15)

3


Hence

v(r) =dV

dr. (16)


v(r) =dV

dN

dN

dr=

4

3πr3n(r). (17)

The total particle area per unit volume, V0, is given by

V0 =∫

∞

0v(r) dr. (18)

The distribution of mass can be represented by a differential mass densitydistribution, m(r) which represents the mass contained in particles with radiibetween r and r + dr per unit volume, i.e.

M(r) =∫ r+dr

rm(r) dr. (19)

Hence

m(r) =dM

dr. (20)


m(r) =dM

dN

dN

dr=

4

3πr3ρn(r), (21)

where ρ is the density of the aerosol material. The total particle mass perunit volume, M0, is

M0 =∫

∞

0m(r) dr. (22)

2 Normal and Logarithmic Normal Distribu-

tions

2.1 Definitions

One particle distribution to consider adopting is the normal distribution

n(r) =N0√2π

1

σ0

exp

[

−(r − µ0)2

2σ20

]

, (23)

4


where µ0 is the mean and σ0 is the standard deviation of the distribution.The size of particles in an aerosol generally covers several orders of magni-tude. As a result the normal distribution fit of measured particle sizes oftenhas a very large standard deviation. Another drawback of the normal distri-bution is that it allows negative radii. Aerosol distributions are much betterrepresented by a normal distribution of the logarithm of the particle radii.Letting l = ln(r) we have

nl(l) =dN(l)

dl=

N0√2π

1

σexp

[

−(l − µ)2

2σ2

]

(24)

where the mean, µ, and the standard deviation, σ, of l = ln(r) are defined

µ =

∫

∞

−∞lnl(l) dl

∫

∞

−∞nl(l) dl

=1

N0

∫

∞

−∞

lnl(l) dl (25)

σ2 =1

N0

∫

∞

−∞

(l − µ)2nl(l) dl (26)

It is common for the lognormal distribution to be expressed in terms of theradius. Noting that

dl

dr=

1

r(27)

then in terms of radius rather than log radius we have

n(r) =dN(r)

dr=

dN(l)

dl

dl

dr= nl(l)

dl

dr=

N0√2π

1

σ

1

rexp

[

−(ln r − µ)2

2σ2

]

(28)

It is also common to express the spread of the distribution using thegeometric standard deviation, S. From this definition S must be greater orequal to one otherwise the log-normal standard deviation is negative. WhenS is one the distribution is monodisperse. Typical aerosol distributions haveS values in the range 1.5 - 2.0.

The lognormal distribution appears in the atmospheric literature usingany of combination of rm or µ and σ or S with perhaps the commonest being

n(r) =N0√2π

1

ln(S)

1

rexp

[

−(ln r − ln rm)2

2 ln2(S)

]

(29)

Be particularly careful about σ and S whose definitions are sometimes re-versed!

5


Figure 1: Log-normal distribution with parameters N0 = 1, µ = −1 andσ = 0.4 plotted in log space (top panel) and linear space (bottom panel).Indicated are the distribution mode, median and mean in log and linear spacerespectively.

2.2 Properties of the Lognormal Distribution

2.2.1 Normal versus Lognormal

Figure 1 shows the particle number density per unit log(radius) where thedistribution is Gaussian. Also shown in the figure is the particle numberdensity distribution plotted per unit radius. In performing the transform thearea (ie. total number of particles) is conserved and the median, mean ormode in log space is the natural logarithm of the median radius, rm, in linearspace.

It is possible to show the geometric mean, rg, of the radius is the sameas the median in linear space, i.e.

rg = (r1 × r2 × . . .× rn)1

n (30)

⇒ ln(rg) = ln (r1 × r2 × . . .× rn)1

n =ln(r1) + ln(r2) + . . .+ ln(rn)

n(31)

6


which has the continuous form

ln(rg) =1

N0

∫

∞

−∞

lnl(l) dl = µ = ln(rm) (32)

⇔ rg = rm (33)

2.2.2 Median and Mode

It can be shown that the mode of the log-normal distribution, rM, is relatedto the median. Let

n(r) =N0√2π

1

ln(S)

1

rexp

[


2 ln2(S)

]

=A

rexp [B] (34)

where A =N0√2π

1

ln(S), B = −(ln r − ln rm)

2

2 ln2(S)and

dB

dr= −(ln r − ln rm)

ln2(S)r.

then

dn(r)

dr= −A

r2exp [B] +

A

rexp [B]

dB

dr(35)

= −A

r2exp [B]− A

r2exp [B]

(ln r − ln rm)

ln2(S)(36)

Setting the left hand side to zero so r becomes rM

0 = − A

r2Mexp [B]− A

r2Mexp [B]

(ln rM − ln rm)

ln2(S)(37)

−1 =(ln rM − ln rm)

ln2(S)(38)

ln rM = ln rm − ln2(S) = ln(rm)− σ2 (39)

2.2.3 Derivatives

The first and second derivatives of Equation 24 are

dnl

dN0

=nl

N0

(40)

dnl

dl= −(l − µ)

σ2nl (41)

dnl

dσ= nl

[

(l − µ)2

σ3− 1

σ

]

(42)

d2nl

dN20

= 0 (43)

7


d2nl

dl2= nl

[

(l − µ)2

σ4− l

σ2

]

(44)

d2nl

dσ2= nl

[

(l − µ)2

σ3− 1

σ

]2

− 3(l − µ)2

σ4+

1

σ2

(45)

If using Equation 29 the derivatives are :

dn

dN0

=n

N0

(46)

dn

dr= −n

r

[

1 +(ln r − ln rm)

ln2(S)

]

(47)

dn

dS=

n

S ln(S)

[

(ln r − ln rm)2

ln2(S)− 1

]

(48)

The second derivatives are:

d2n

dN20

= 0 (49)

d2n

dr2= n

[

2

r2+

3 ln r − 3 ln rm − 1

r2 ln2(S)+

(ln r − ln rm)2

r2 ln4(S)

]

(50)

d2n

dS2= n

[

(ln r − ln rm)4

S2 ln6(S)− 5

(ln r − ln rm)2

S2 ln4(S)− (ln r − ln rm)

2

S2 ln3(S)+

2

S2 ln2(S)+

1

S2 ln(S)

]

(51)

2.2.4 Moments

The i-th raw moment of a lognormal distribution is given by

mi = N0 exp

(

iµ+i2σ2

2

)

. (52)

The first three moments are

m1 =∫

∞

0rn(r) dr = N0 exp

(

µ+1

2σ2)

= N0rm exp(

1

2σ2)

≡ N0rm exp(

1

2ln2 S

)

m2 =∫

∞

0r2n(r) dr = N0 exp

(

2µ+ 2σ2)

= N0r2m exp

(

2σ2)

≡ N0r2m exp

(

2 ln2 S)

m3 =∫

∞

0r3n(r) dr = N0 exp

(

3µ+9

2σ2)

= N0r3m exp

(

9

2σ2)

≡ N0r3m exp

(

9

2ln2 S

)

The mean radius, the surface area density and the volume density of a log-normal distribution are given by

mean =1

N0

∫

∞

0rn(r) dr =

1

N0

m1 =1

N0

N0 exp(

µ+1

2σ2)

, (53)

8


= rm exp(

1

2σ2)

≡ rm exp(

1

2ln2 S

)

, (54)

area =∫

∞

04πr2n(r) dr = 4πm2 = 4πN0 exp

(

2µ+ 2σ2)

, (55)

= 4πN0r2m exp

(

2σ2)

≡ 4πN0r2m exp

(

2 ln2 S)

, (56)

and

volume =∫

∞

0

4

3πr3n(r)dr =

4

3πm3 =

4

3πN0 exp

(

3µ+9

2σ2)

, (57)

=4

3πN0r

3m exp

(

9

2σ2)

≡ 4

3πN0r

3m exp

(

9

2ln2 S

)

. (58)

For a lognormal distribution the effective radius is

re =m3

m2

=exp

(

3µ+ 92σ2)

exp(

2µ+ 42σ2) = exp

(

µ+5

2σ2)

, (59)

= rm exp(

5

2σ2)

≡ rm exp(

5

2ln2 S

)

. (60)

2.3 Log-Normal Distributions of Area and Volume

The log-normal area density distribution is

a(r) =A0√2π

1

σa

1

rexp

[

−(ln(r)− µa)2

2σ2a

]

(61)

where A0 is the total aerosol surface area per unit volume, µa is the radiusof the median area and σa the geometric standard deviation. These con-stants can all be related to the description of a log-normal number densitydistribution starting from

a(r) = 4πr2n(r) = 4πr2N0√2π

1

σ

1

rexp

[

−(ln r − µ)2

2σ2

]

(62)

Making the substitution r2 = exp(2 ln r) and completing the square gives

a(r) = 4πN0√2π

1

σ

1

rexp

[

2µ+ 2σ2]

exp

[

−(ln r − (µ+ 2σ2))2

2σ2

]

(63)

Equating with Equations 61 and 62 gives

A0√2π

1

σa

1

rexp

[

−(ln(r)− µa)2

2σ2a

]

= 4πN0√2π

1

σ

1

rexp

[

2µ+ 2σ2]

exp

[

−(ln r − (µ+ 2σ2))2

2σ2

]

(64)

9


which is true if

A0

σa

=4πN0

σexp

[

2µ+ 2σ2]

(65)

and

(ln(r)− µa)2

2σ2a

=(ln r − (µ+ 2σ2))

2

2σ2(66)

From Equation 55

A0 = 4πN0 exp(

2µ+ 2σ2)

(67)

which gives σa = σ when inserted into Equation 65. This shows that is thenumber density distribution is log-normal then the surface area density distri-bution is log-normal with the same geometric standard deviation. Applyingthis result to Equation 66 gives

µa = µ+ 2σ2. (68)

This states that the area median radius is greater than the median radius.Equivalent expressions can be calculated for a volume density distribution,v(r) defined in Section 1.5. The log-normal volume density distribution is

v(r) =V0√2π

1

σv

1

rexp

[

−(ln(r)− µv)2

2σ2v

]

(69)

where V0 is the total aerosol volume per unit volume, µv is the radius ofthe median volume and σv the geometric standard deviation. These con-stants can all be related to the description of a log-normal number densitydistribution starting from

v(r) =4

3πr3n(r) =

4

3πr3

N0√2π

1

σ

1

rexp

[

−(ln r − µ)2

2σ2

]

(70)

Making the substitution r3 = exp(3 ln r) and completing the square gives

v(r) =4

3π

N0√2π

1

σ

1

rexp

[

−(ln r − µ)2 − 6σ2 ln r

2σ2

]

(71)

Complete the square

v(r) =4

3π

N0√2π

1

σ

1

rexp

[

3µ+ 4.5σ2]

exp

[

−(ln r − (µ+ 3σ2))2

2σ2

]

(72)

10


Equating with Equations 69 and 72 gives

V0√2π

1

σv

1

rexp

[

−(ln(r)− µv)2

2σ2v

]

=4

3π

N0√2π

1

σ

1

rexp

[

3µ+ 4.5σ2]

exp

[

−(ln r − (µ+ 3σ2))2

2σ2

]

(73)

which is true if

V0

σv

=4

3πN0

σexp

[

3µ+ 4.5σ2]

(74)

and

(ln(r)− µv)2

2σ2v

=(ln r − (µ+ 3σ2))

2

2σ2(75)

From Equation 57

V0 =4

3πN0 exp

(

3µ+ 4.5σ2)

(76)

which gives σv = σ when inserted into Equation 74. This shows that is thenumber density distribution is log-normal then the volume density distribu-tion is log-normal with the same geometric standard deviation. Applyingthis result to Equation 75 gives

µv = µ+ 3σ2. (77)

They relationships area and volume density log-normal parameters and thenumber size distribution parameters are summarised in Table 1.

Lastly Table 2 shows derived values for a log-normal distributions over arange of rm and with S = 1.5.

2.4 A Different Basis State

While the shape of the lognormal distribution is controlled its median radius,rm, and its geometric standard deviation, S. An alternative is the effectiveradius, re, and volume per particle, v = V/N0. If these parameters areadopted rm and S can be recovered as follows:

S = exp

√

√

√

√−ln(

3v4πr3e

)

3

(78)

rm =(

3v

4π

)(5/6) 1

r(3/2)e

(79)

11


Table 1: Relationships area and volume density log-normal parameters andthe number size distribution parameters, N0, µ and σ.

Distribution Parameters Relation to NumberDensity Parameters

Area densityA0 = 4πN0 exp (2µ+ 2σ2)µa = µ+ 2σ2

σa = σ

Volume densityV0 =4

3πN0 exp (3µ+ 4.5σ2)

µv = µ+ 3σ2

σv = σ

Table 2: Values derived for a log-normal number density size distributionwith S = 1.5.

MedianRadius

MeanRadius

EffectiveRadius

AreaMedianRadius

VolumeMedianRadius

µm µm µm µm µm0.1 0.1 0.3 0.3 0.40.2 0.3 0.7 0.5 0.80.5 0.6 1.7 1.3 2.11 1.3 3.3 2.6 4.22 2.5 6.6 5.2 8.55 6.4 17 13. 2110 13 33 26 4220 25 67 52 8550 64 166 131 211100 127 332 261 423

12


3 Other Distributions

3.1 Gamma Distribution

Figure 2: Normalised gamma distribution.

The gamma distribution is given by Twomey (1977) as

n(r) = ar exp (−br) , (80)

where a and b are positive constants. The mode of the distribution occurswhere r = b−1 and it falls off slowly on the small radius side and exponentiallyon the large radius side. The i-th moment of the gamma distribution is givenby

mi = ab−2−iΓ (2 + i) , (81)

= ab−2−i (1 + i)!. (82)

If the constant a is used to denote the total number density then the nor-malised distribution (see Figure 2) can be expressed

n(r) = ab2r exp (−br) , (83)

13


which has moments defined by

mi = ab−i (1 + i)!. (84)

3.2 Modified Gamma Distribution

Figure 3: Normalised modified gamma distribution (α = 2, b = 100, γ = 2)and gamma distribution (b = 10).

The modified gamma distribution is given by Deirmendjian (1969) as

n(r) = arα exp (−brγ) . (85)

The four constants a, α, b, γ are positive and real and α is an integer. The

mode of the distribution occurs where r =(

αbγ

)1/γand the moments of this

distribution are2

mi =a

γb−

α+1+iγ Γ

(

α + 1 + i

γ

)

. (86)

2See 3 · 478/1 of Gradshteyn and Ryzhik (1994).

14


Hence the integral of the modified gamma distribution is

∫

∞

0n(r) dr =

a

γb−

α+1

γ Γ

(

α + 1

γ

)

. (87)

In order that the first parameter, a, is the total aerosol concentration it isconvenient to define the normalized modified gamma distribution (see Fig-ure 3) as

n(r) = arα exp (−brγ)1γb−

α+1

γ Γ(

α+1γ

) , (88)

where a, α, b, γ are positive and real but α is no longer constrained to be aninteger. The moments of this distribution are given by

mi = ab−iγ

Γ(

α+i+1γ

)

Γ(

α+1γ

) . (89)

3.3 Inverse Modified Gamma Distribution

The inverse modified gamma distribution is defined by Deepak (1982) as

n(r) = ar−α exp(

−br−γ)

. (90)

The mode of the distribution occurs where r =(

αbγ

)

−1/γand it falls off

slowly on the large radius side and exponentially on the small radius side.The moments are defined by

mi =a

γb−

α−1−iγ Γ

(

α− 1− i

γ

)

. (91)

The normalized inverse modified gamma distribution can be defined

n(r) = ar−α exp (−br−γ)1γb−

α−1

γ Γ(

α−1γ

) . (92)

The moments of this distribution are given by

mi = abiγ

Γ(

α−1−iγ

)

Γ(

α−1γ

) . (93)

15


Figure 4: Normalised inverse modified gamma distribution (α = 2, b =0.01, γ = 2) and gamma distribution(b− 10).

3.4 Regularized Power Law

The regularized power law is defined by Deepak (1982) as

n(r) = abα−2 rα−1

[

1 +(

rb

)α]γ , (94)

where the positive constants a, b, α, γ mainly effect the number density, themode radius, the positive gradient and the negative gradient respectively.The mode radius is given by

r = b

(

α− 1

1 + α(γ − 1)

)1/α

, (95)

and the moments by

mi = abi

α

Γ(1 + i/α)Γ(γ − 1− i/α)

Γ(γ). (96)

16


Hence the normalised distribution is

n(r) = aαγbα−2 rα−1

[

1 +(

rb

)α]γ . (97)

17


4 Modelling the Evolution of an Aerosol Size

Distribution

For retrieval purposes it is necessary to describe the evolution of an aerosolsize distribution. Consider the case where an aerosol size distribution isdescribed by three modes which are parametrized by a mode radius, rm,i anda spread, σi. We wish to alter the mixing ratios, µi, of each of the modes toachieve a given effective radius re. How do we do this?

Firstly calculate the effective radius of each of the modes according to

re,i = rm,i exp(

5

2ln2 Si

)

. (98)

If re ≤ re,1 then µ =

100

and rm =

re,i/ exp(

52ln2 S1

)

rm,2

rm,3

.

Similarly if re ≥ re,3 then µ =

001

and rm =

rm,1

rm,2

re,i/ exp(

52ln2 S3

)

.

If re,1 < re < re,3 then µ1 and is estimated by linearly interpolatingbetween [0,1] as a function of re i.e.

µ1 =re − re,1re,3 − re,1

(99)

We now have two equations

µ1 + µ2 + µ3 = 1(100)

µ1r3m,1 exp

(

92ln2 S1

)

+ µ2r3m,2 exp

(

92ln2 S2

)

+ µ3r3m,3 exp

(

92ln2 S3

)

µ1r2m,1 exp(

2 ln2 S1

)

+ µ2r2m,2 exp(

2 ln2 S2

)

+ µ3r2m,3 exp(

2 ln2 S3

) = re(101)

and two unknowns i.e. µ2 and µ3. The second equation is simplified bysubstitution i.e.

Aµ1 +Bµ2 + Cµ3

Dµ1 + Eµ2 + Fµ3

= re (102)

and the two equations solved to give

µ2 =reE −B − µ1(A− B + re(E −D))

C −B + re(E − F )(103)

µ3 =reF − C − µ1(A− C + re(F −D))

B − C + re(F − E)(104)

18


5 Optical Properties

5.1 Volume Absorption, Scattering and Extinction Co-efficients

The volume absorption coefficient, βabs(λ, r), the volume scattering coeffi-cient, βsca(λ, r), and the volume extinction coefficient, βext(λ, r), representthe energy removed from a beam per unit distance by absorption, scattering,and by both absorption and scattering. For a monodisperse aerosol they arecalculated from

monodisperse only

βabs(λ, r) = σabs(λ, r)N(r) = πr2Qabs(λ, r)N(r),βsca(λ, r) = σsca(λ, r)N(r) = πr2Qsca(λ, r)N(r),βext(λ, r) = σext(λ, r)N(r) = πr2Qext(λ, r)N(r),

(105)

where N(r) is the number of particles per unit volume at some radius, r. Theabsorption cross section, σext(λ, r), the scattering cross section, σext(λ, r),and the extinction cross section, σext(λ, r), are determined from the extinc-tion efficiency factor, Qext(λ, r), extinction efficiency factor, Qext(λ, r), ex-tinction efficiency factor, Qext(λ, r), respectively.

For a collection of particles, the volume coefficients are given by

βabs(λ) =∫

∞

0σabs(λ, r)n(r) dr =

∫

∞

0πr2Qabs(λ, r)n(r) dr, (106)

βsca(λ) =∫

∞

0σsca(λ, r)n(r) dr =

∫

∞

0πr2Qsca(λ, r)n(r) dr, (107)

βext(λ) =∫

∞

0σext(λ, r)n(r) dr =

∫

∞

0πr2Qext(λ, r)n(r) dr, (108)

where n(r) represents the number of particles with radii between r and r+drper unit volume. It is also useful to define the quantities per particle i.e.

σ̄abs(λ) =∫

∞

0σabsn(r) dr

/

∫

∞

0n(r) dr =

βabs(λ)

N0

, (109)

σ̄sca(λ) =∫

∞

0σscan(r) dr

/

∫

∞

0n(r) dr =

βsca(λ)

N0

, (110)

σ̄ext(λ) =∫

∞

0σextn(r) dr

/

∫

∞

0n(r) dr =

βext(λ)

N0

, (111)

where σ̄abs(λ), σ̄sca(λ) and σ̄ext(λ) are the mean absorption cross section, themean scattering cross section and the mean extinction cross section respec-tively.

19


5.2 Back Scatter

to be done

5.3 Phase Function

The phase function represents the redistribution of the scattered energy.For a collection of particles, the phase function is given by

P (λ, θ) =1

βsca

∫

∞

0πr2Qsca(λ, r)P (λ, r, θ)n(r) dr. (112)

5.4 Single Scatter Albedo

The single scatter albedo is the ratio of the energy scattered from a particleto that intercepted by the particle. Hence

ω(λ) =βsca(λ)

βext(λ). (113)

5.5 Asymmetry Parameter

The asymmetry parameter is the average cosine of the scattering angle,weighted by the intensity of the scattered light as a function of angle. Ithas the value 1 for perfect forward scattering, 0 for isotropic scattering and-1 for perfect backscatter.

g =1

βsca

∫

∞

0πr2Qsca(λ, r)g(λ, r)n(r) dr (114)

5.6 Integration

The integration of an optical properties over size is usually reduced fromthe interval r = [0,∞] to r = [rl, ru] as n(r) → 0 as r → 0 and r → ∞.Numerically an integral over particle size becomes

∫ ru

rl

f(r)n(r) dr =n∑

i=1

wif(ri) (115)

where wi are the weights at discrete values of radius, ri.For a log normal size distribution the integrals are

βext(λ) =N0

σ

√

π

2

∫ ru

rl

rQext(λ, r) exp

−1

2

(

ln r − ln rmσ

)2

dr (116)

20


=N0

σ

√

π

2

n∑

i=1

wiriQext(λ, ri) exp

−1

2

(

ln ri − ln rmσ

)2

=n∑

i=1

w′

iQext(λ, ri)

βabs(λ) =n∑

i=1

w′

iQabs(λ, ri)

βsca(λ) =n∑

i=1

w′

iQsca(λ, ri)

P (λ, θ) =N0

σ

√

π

2

1

βsca

∫ ru

rl

rQsca(λ, r)P (λ, r, θ) exp

−1

2

(

ln r − ln rmσ

)2

dr

=N0

σ

√

π

2

1

βsca

n∑

i=1

wiriQsca(λ, ri)P (λ, ri, θ) exp

−1

2

(

ln ri − ln rmσ

)2

=1

βsca

n∑

i=1

w′

iQsca(λ, ri)P (λ, ri, θ)

g =N0

σ

√

π

2

1

βsca

∫

∞

0rQsca(λ, r)g(λ, r) exp

−1

2

(

ln r − ln rmσ

)2

dr

=N0

σ

√

π

2

1

βsca

n∑

i=1

wiriQsca(λ, ri)g(λ, ri) exp

−1

2

(

ln ri − ln rmσ

)2

=1

βsca

n∑

i=1

w′

iQsca(λ, ri)g(λ, ri)

where

w′

i =N0

σ

√

π

2ri exp

−1

2

(

ln ri − ln rmσ

)2

wi (117)

5.7 Formulae for Practical Use

As part of the retrieval process it is helpful to have analytic expression for thepartial derivatives of βext (Equation 116) with respect to the size distributionparameters (N0, rm, σ).

∂βext

∂N0=

1

σ

√

π

2

∫

∞

0rQext(λ, r) exp

[


2σ2

]

dr, (118)

∂βext

∂rm=

N0

rmσ3

√

π

2

∫

∞

0(ln r − ln rm)rQ

ext(λ, r) exp

[


2σ2

]

dr, (119)

21


∂βext

∂σ= −N0

σ2

√

π

2

∫

∞

0rQext(λ, r) exp

[


2σ2

]

dr

+N0

σ

√

π

2

∫

∞

0

(ln r − ln rm)2

σ3rQext(λ, r) exp

[


2σ2

]

dr,

=N0

σ2

√

π

2

∫

∞

0

[

(ln r − ln rm)2

σ2− 1

]

rQext(λ, r) exp

[


2σ2

]

dr.

(120)

To linearise the retrieval it is better to retrieve T (= lnN0) rather than N0.In addition to limit the values of rm and σ to positive quantities it is better toretrieve lm (= ln rm) and G (= ln σ). In terms of these new variables volumeextinction coefficient for a log normal size distribution is

βext(λ) =expT

expG

√

π

2

∫

∞

−∞

e2lQext(l, λ) exp

[

− (l − lm)2

2 exp(2G)

]

dl. (121)

The partial derivatives of the transformed parameters (Equation 121) are

∂βext

∂T=

expT

expG

√

π

2

∫

∞

−∞

e2lQext(l, λ) exp

[

− (l − lm)2

2 exp(2G)

]

dl, (122)

∂βext

∂lm=

expT

exp(3G)

√

π

2

∫

∞

−∞

(l − lm)e2lQext(l, λ) exp

[

− (l − lm)2

2 exp(2G)

]

dl, (123)

∂βext

∂G= − expT

exp(G)

√

π

2

∫

∞

−∞

e2lQext(l, λ) exp

[

− (l − lm)2

2 exp(2G)

]

dl

+expT

expG

√

π

2

∫

∞

−∞

(l − lm)2

exp(2G)e2lQext(l, λ) exp

[

− (l − lm)2

2 exp(2G)

]

dl,

=expT

exp(G)

√

π

2

∫

∞

−∞

[

(l − lm)2

exp(2G)− 1

]

e2lQext(l, λ) exp

[

− (l − lm)2

2 exp(2G)

]

dl.

(124)

22


5.8 Cloud Liquid Water Path

The mass l of liquid per unit area in a cloud with a homogeneous size distri-bution is given by

l = ρ∫

∞

0

4

3πr3n(r) dr × z (125)

where ρ is the density of the cloud material (water or ice) and z is the verticaldistance through the cloud. The liquid water path is usually expressed asgm−2. Note that

τ = βext × z (126)

so

l = ρτ

∫

∞

043πr3n(r) dr

βext(127)

=4

3πρτ

∫

∞

0 r3n(r) dr∫

∞

0 πr2Qext(λ, r)n(r) dr. (128)

For drops large with respect to wavelength we assume Qext(λ, r) = 2. Hence

l =4

6ρτ

∫

∞

0 r3n(r) dr∫

∞

0 r2n(r) dr=

2

3ρτre (129)

So for a water cloud (ρ = 1 g cm−3) of τ = 10, re = 15µm we get

l =2

3× 1× 10× 15 g cm−3µm = 100 gm−2 (130)

While for an ice cloud (-45 ◦C, ρ = 0.920 g cm−3) of τ = 1, re = 25µm weget

l =2

3× 0.92× 1× 25 g cm−3µm = 15 gm−2 (131)

.

5.9 Aerosol Mass

Consider the measurement of optical depth and effective radius made by aimaging instrument. How can this be related to the mass of aerosol presentin the atmosphere? Consider a volume observed by the instrument whose

23


area is A. If ρ is the density of the aerosol and Z is the height of this volumethen the total mass of aerosol, M , in the box is given by

M = ρ× v ×N × A× Z (132)

where N is the number of particles per unit volume and v is the averagevolume of each particle. If we divide both sides by A we obtain the mass perunit area m, i.e.

m = ρ× v ×N × Z (133)

This formula can re-expressed in terms of more familiar optical measurementsof the volume. First note that the optical depth is related to the βext by

τ = βext × Z (134)

so that

m =ρ× v ×N × τ

βext=

ρ× v × τ

σ̄ext(135)

For a given size distribution n(r) the average volume of each particle is

v =

∫

∞

043πr3n(r) dr

N(136)

so that the mass per unit area is given by

m =ρτ

βext

∫

∞

0

4

3πr3n(r) dr (137)

The important thing to note here is that N disappears explicitly from theequation.

In terms of re the mass per unit area is given by

m =4πρτ

3βext

∫

∞

0r3n(r) dr ×

∫

∞

0 r2n(r) dr∫

∞

0 r2n(r) dr=

4ρτ

3Q̃extre (138)

24


Substance Density (g cm−3) Reference

Ice 0.92Volcanic ash 2.42±0.79 Bayhurst, Wohletz and Mason (1994)Water 1

Table 3: Density of some materials that form aerosols.

which uses an area weighted extinction efficiency

Q̃ext =βext

π∫

∞

0 r2n(r) dr=

∫

∞

0 σextn(r) dr∫

∞

0 πr2n(r) dr=

∫

∞

0 πr2Qextn(r) dr∫

∞

0 πr2n(r) dr(139)

If the size distribution is much larger than the wavelength then Qext → 2and Q̃ext ≈ 2 and Equation 129 becomes identical to Equation 138.

If the aerosol size distribution is log-normal with number density N0,mode radius rm and spread σ then the integral in Equation 137 can becompleted analytically i.e.

m =ρτ

βext

4

3πN0r

3m exp

(

9

2σ2)

(140)

Typically ρ is in g cm−3, N0 is in cm−3, rm is in µm, and βext is in km−1 sothat the units of m are

gcm3

1cm3µm3

1

km=

g

10−6 m3

1

10−6 m310−18 m3103 m = 10−3 g m−2 (141)

Table 3 list the bulk density of some aerosol components.If the effective radius, re, is known rather than rm then we can use the

relationship between re and rm

re = rm exp(

5

2σ2)

, (142)

to get

m =4ρτπN0

3βextr3e exp

(

−15

2σ2)

exp(

9

2σ2)

=4ρτπN0

3βextr3e exp

(

−3σ2)

. (143)

Equating this expression to Equation 129 gives

Q̃ext =βext

πN0r2e exp (−3σ2). (144)

25


which is true for a log-normal distribution.For a multi-mode lognormal distribution where the ith model is parame-

terised by Ni, ri, σi and density ρi we have

m =τ × ρ×N × v

βext=

τ∑n

i=1 ρiNivi∑n

i=1 Niσ̄exti

(145)

where σ̄exti is the extinction cross section per particle for the ith mode. Re-

member the volume per particle for the ithmode is

vi =4

3πr3i exp

(

9

2σ2i

)

(146)

Hence

m = τ

∑ni=1 ρiNi

43πr3i exp

(

92σ2i

)

∑ni=1 Niσ̄ext

i

(147)

=4

3πτ

∑ni=1 ρiNir

3i exp

(

92σ2i

)

∑ni=1 Niσ̄ext

i

(148)

26


References

Bayhurst, G. K., Wohletz, K. H. and Mason, A. S. (1994). A method for char-acterizing volcanic ash from the December 15, 1989 eruption of Redoubtvolcano, Alaska, in T. J. Casadevall (ed.), Volcanic Ash and Aviation

Safety, Proc. 1st Int. Symp. on Volcanic Ash and Aviation Safety, USGSBulletin 2047, Washington, pp. 13–17.

Deepak, A. (ed.) (1982). Atmospheric Aerosols, Spectrum Press, Hampton,Virginia.

Deirmendjian, D. (1969). Electromagnetic scattering on spherical polydisper-

sions, Elsevier, New York.

Gradshteyn, I. S. and Ryzhik, I. M. (1994). Table of integrals, series, and

products, Academic Press, London.

Twomey, S. (1977). Atmospheric aerosols, Elsevier, Amsterdam, The Nether-lands.

27

Index

mean, 1median, 1mode, 1moment, 1

size distribution, 1standard deviation, 1

variance, 1

28

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