Zdunkowski W., Trautmann T., Bott A. Radiation in the atmosphere: A course in Theoretical Meteorology

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7.1 The shape of single spectral lines 215

mechanisms need to be considered since they are in the same order of magnitude. As

already mentioned, in the meteorologically relevant part of the Earth’s atmosphere

natural line broadening remains negligibly small in comparison to collisional and

thermal broadening.

In order to combine the pressure and Doppler broadening effect we apply Doppler

broadening to a line which has already been broadened by the pressure effect.

This is achieved by replacing in (7.32)–(7.34) the Dirac δ-function, expressing the

monochromatic emission, by the Lorentz line-shape factor, expressing the pressure

broadening. Instead of (7.34) we then obtain

= S



∞

−∞









ν − ν





dν



(7.41)

By comparing (7.36) with (7.40) one may easily verify that







= f





− ν



(7.42)

Substituting this expression into (7.41) yields the absorption coefﬁcient of the Voigt

line

ν,V

= S



∞

−∞





− ν





ν − ν





dν



= Sf

(ν − ν

)

(7.43)

Here, f

(ν − ν

) is the line-shape factor of the Voigt line which is given by the

convolution of the Lorentz and the Doppler line-shape factors, i.e.

(ν − ν

) = f

(y) =



∞

−∞

(x) f

(y − x)dx (7.44)

with x = ν



− ν

and y = ν − ν

. This equation reﬂects the fact that the signal of

the product of two spectra is equivalent to the convolution of the individual signals.

Substituting (7.18) and (7.40) into (7.44) yields for the Voigt line-shape factor

(ν − ν

) =

√

ln 2

√



∞

−∞

exp





−



√

ln 2







+ (y − x)

dx (7.45)

Integration of this equation over all frequencies shows that the Voigt line-shape

factor is also normalized. Furthermore, it may be easily veriﬁed that in the lim-

its α

→ 0 and α

→ 0 the Voigt line approaches the Lorentz and Doppler line,

respectively.

Unfortunately, it is not possible to give an analytical solution to the integral in

(7.45). However, it causes no problems to ﬁnd a numerical solution. Nevertheless,

in the literature there exist several analytical approximations for the Voigt line

216 Transmission in spectral lines and bands of lines

(e.g. Penner, 1959; Fels, 1979). Fomichev and Shved (1985) suggest the following

formula

(ν − ν

) =

ln 2

1 − ξ

exp(−ln 2η

) +

πα

(1 + η

)

−

ξ(1 − ξ )

πα



2ln2

+ 1 + ξ





0.066 exp(−0.4η

) −

40 − 5.5η

+ η



(7.46)

with ξ = α

/α

and η = (ν − ν

)/α

. The term α

is the half-width of the Voigt

line and is given by

= 0.5



+ 4α



+ 0.05α





1 −

2α

+ 4α





(7.47)

Generally, this approximation yields results with an accuracy of less than 3%.

Numerical evaluation of (7.46) reveals that the approximate form of the Voigt line-

shape factor is also normalized and it approaches the Doppler and Lorentz line for

→ 0 and α

→ 0.

As an example, Figure 7.5 shows the three line types for the half-widths α

= α

The curves are plotted as function of x = (ν − ν

)/α

and they are normalized

with respect to f

(0). From this ﬁgure we conclude that collisional line broadening

dominates in the line wings, whereas near the line center f

> f

. The effect of the

Voigt line is to increase the absorption in the wings and to decrease it in the center

of the line as compared to the Lorentz and the Doppler line. The same behavior of

the Voigt line is also observed for other choices of the half-widths with α

 α

However, if there is practically total absorption near the center of the spectral

line, the simpler Lorentz shape can still be used instead of the more complicated

Voigt shape. This is due to the fact that, for very strong absorption, the particular

shape of the line near its center is of secondary importance. A detailed discussion

of the Voigt proﬁle can be found, for example, in Uns¨old (1968).

7.2 Band models

Inspection of the absorption spectrum in the short-wave and long-wave spec-

tral region reveals that there exist many absorption lines for different molecular

species. With increasing spectral resolution the structure of the absorption spectrum

7.2 Band models 217

0.8

0.6

0.4

0.2

−10 −5 0 5 10

(x)

f(x)

Fig. 7.5 Comparison of the Lorentz, the Doppler and the Voigt line-shape factors

with α

= α

and x = (ν − ν

)/α

increases in complexity. For a particular gas the positions and the strengths of the

spectral lines are calculable with the help of the quantum theory of radiation. It is

found that there exist approximately 50 000 relevant lines for water vapor which

have to be considered in detailed spectral calculations for atmospheric problems.

Truly exact calculations of the absorption spectrum are carried out by the so-called

line-by-line models, meaning that the absorption coefﬁcient must be determined

along the shape of each individual spectral line. In order to determine the radiation

ﬁeld these calculations have to be performed for many altitudes within the atmo-

sphere because gaseous concentrations, half-widths and line intensities depend on

pressure and temperature.

One can easily imagine that radiative transfer by the exact line-by-line method

requires an extraordinary computational effort. Such benchmark calculations are

normally carried out for a few representative model atmospheres only. For routine

purposes, however, one needs to design simpler approaches which do not treat the

spectroscopic structure in detail, but still take the main features of the absorption

spectrum into account. These are the so-called band models.

A view of the absorption spectrum gives the impression that the positions of

the lines are randomly placed and that the line intensities are distributed accord-

ing to some statistical law. In reality, the spectral arrangement of the lines is not

completely random since line positions and intensities can be determined via quan-

tum mechanical laws. Nevertheless, random line position and statistical distribution

218 Transmission in spectral lines and bands of lines

laws for line intensities may be used to determine the absorption of simple band

models.

7.2.1 Mean absorption in a single Lorentz line

In the following we will ﬁrst discuss the simple but important case of absorption by

a single Lorentz line in a homogeneous atmosphere. The resulting mean absorption

of such a spectral line will be used to ﬁnd the limiting forms of very weak and very

strong absorption. These results in turn will be needed as closure assumptions of

various band models to be discussed later.

The monochromatic absorption A

(u) in a single line depends on the spec-

tral absorption coefﬁcient and the optical mass u of the gas in the following

way

(u) = 1 − exp

(

−k

)

= 1 − T

(u) with T

(u) = exp

(

−k

)

(7.48)

where u was deﬁned in (2.118). T

(u) is the monochromatic transmission of the

single line. This fundamental quantity represents that part of the incident radiation

which remains after passage of the optical mass.

The single line model can be extended to more general situations involving the

absorption in several well-separated lines so that no appreciable line overlap ocurs.

If there exist N different absorbers with absorption coefﬁcients k

ν,i

and optical

masses u

the total monochromatic transmission is given by

= exp



−



i=1

ν,i



i=1

ν,i

) (7.49)

indicating that the individual monochromatic transmissions can be multiplied to

yield the total monochromatic effect. It should be noted, however, that this multi-

plication property of the transmissions is not strictly true for the transmission of

band intervals.

A convenient way to deﬁne the spectrally integrated absorption is by means of

the equivalent width W (u)

W (u) =



∞

−∞

(u)dν =



∞

−∞

[1 − exp

(

−k

)

]dν (7.50)

The relationship between W and u is also called the curve of growth. The notion

of the equivalent width stems from the fact that a rectangular line with width W ,

whose line center is completely absorbed (A

= 1), gives rise to the same area

under A

7.2 Band models 219

Instead of W (u), which has the dimension of a frequency, one deﬁnes the average

absorption A(u) of a single line over a frequency interval ν taken sufﬁciently wide

A(u) =

W (u)

ν



∞

−∞

(u)dν =

ν



∞

−∞

[1 − exp

(

−k

)

]dν (7.51)

Introducing in this equation the absorption coefﬁcient of the Lorentz line (7.15)

and using the substitutions

u =

2πα

, x =

ν − ν

(7.52)

we obtain the expression

u) =

ν



∞

−∞



1 − exp



−

(1 + x

)



dx (7.53)

Integration by parts results in

u) =

uα

ν



∞

−∞

exp(−2

u/(1 + x

))

(1 + x

)

uα

ν



∞

−∞

exp(−2

u/(1 + x

))

(1 + x

)

dx −

uα

ν



∞

−∞

exp(−2

u/(1 + x

))

(1 + x

)

(7.54)

In the next step we may treat the variable

u as a parameter in the integral of

(7.53). Thus, the ﬁrst and second derivative of A with respect to

u are

2α

ν



∞

−∞

exp(−2

u/(1 + x

))

(1 + x

)

dx,

(7.55)

=−

4α

ν



∞

−∞

exp(−2

u/(1 + x

))

(1 + x

)

Substituting these expressions into (7.54) gives an ordinary linear differential

equation of second order for A(

+ 2

− A(

u) = 0 (7.56)

This differential equation can be easily solved by means of Laplace transforms. Let

us deﬁne the Laplace transform of A(

u)as

[

]



∞

exp

(

−s

)

u)d

u = y(s) (7.57)

Many authors denote the average absorption by

A(u). This notation is unnecessarily complicated since averaging

implies an integration over ν which is just as well indicated by A(u).

220 Transmission in spectral lines and bands of lines

Then the properties of the Laplace transform yield





= sy(s) −A(0), L





= s

y(s) − sA(0) −

(0)

(7.58)

The two boundary conditions A(0) and (dA/d

u)(0) to evaluate the Laplace

transforms can be obtained very easily. From physical reasoning we must have

A(0) = 0. To obtain the ﬁrst derivative of A at

u = 0 we expand (7.51) in a power

series. For small values of u we ﬁnd

u  1: A(u)ν = Su, A(

u) =

2πα

ν

(0) =

2πα

ν

(7.59)

The additional transform rule

[

uf(

]

=−

with L

[

f (

]

= f (s) (7.60)

can then be used to formulate (7.56) in transformed space as

s + 2

s(s + 2)

= 0 (7.61)

It may be easily veriﬁed that the solution to this differential equation is given by

y(s) =

(s + 2)s

(7.62)

where C is a constant of integration.

The inverse Laplace transform of y(s) may be found in transform tables and is

given by

−1

[

y(s)

]

= A(

u) = C

u exp

(

−

)

[

(

u) + I

(

]

(7.63)

where I

, I

are the modiﬁed Bessel functions of the ﬁrst kind. The constant C is

evaluated with the help of (7.59) and by observing that

(0) = 1, I

(0) = I

(0) = 0

(

= I

(

u), 2

(

= I

(

u) + I

(

(7.64)

This results in

C =

2πα

ν

(7.65)

which is required to evaluate (7.63). As ﬁnal result we obtain the average absorption

of an isolated Lorentz line

u) =

2πα

ν

u exp

(

−

)

(

u) + I

(

u)]

(7.66)

7.2 Band models 221

This important result is due to Ladenburg and Reiche (1913) who originally used

the Bessel functions of the ﬁrst kind J

and J

with pure imaginary argument to

express A(

u). In the literature the Ladenburg and Reiche function L(

u) is deﬁned as

u) =

u exp

(

−

)

(

u) + I

(

u)]

(7.67)

so that

u) =

2πα

ν

(7.68)

The Laplace transform method to ﬁnd the Ladenburg and Reiche function is due to

Zdunkowski (1974). The original method to obtain L(

u) is outlined in Appendix

7.6.2.

Let us discuss the average absorption of an isolated Lorentz line for the two

important cases

u  1 and

u  1. For small

u the I

may be expressed as inﬁnite

series of the form (Abramowitz and Stegun, 1972)

(x) =

∞



k=0

k! (n + k + 1)





2k+n

(7.69)

where (x) is the Gamma function which is deﬁned by the deﬁnite integral

(x) =



∞

exp

(

−t

)

x−1

dt (7.70)

For a positive integer n one ﬁnds (n + 1) = n! The leading terms of the power

series for I

and I

are then given by

(

u) = 1 +









+···

(

u) =









+···

(7.71)

For

u  1 it is sufﬁciently accurate if we retain only the ﬁrst two terms of

these series. Furthermore, since exp

(

−

)

≈ 1 −

u the third-order expansion of the

average absorption is found to be

u) ≈

2πα

ν



u −

−



(7.72)

The so-called weak line approximation is deﬁned as the linear part of the absorption

law (7.72)

(

u) ≈

2πα

ν

u or A

(u) ≈

ν

(7.73)

222 Transmission in spectral lines and bands of lines

This very important result shows that very weak absorption varies linearly with u.

Moreover, it is found that the absorption is independent of the half-width of the

spectral line. Since in (7.51) no special form of k

is involved, it may be easily seen

that for u  1 the general form of the weak line approximation (7.73) is valid for

arbitrary line shapes.

The second important case is known as strong absorption, i.e.

u  1. Now

we introduce the asymptotic expressions for the modiﬁed Bessel functions

(Abramowitz and Stegun, 1972)

(x) =

exp

(

)

√

2π x



1 + O





(7.74)

We immediately ﬁnd the so-called strong line approximation or square-root law

(

u) ≈

2α

ν

√

2π

u or A

(u) =

ν

Sα

(7.75)

It should be noted that, contrary to the weak line law, the strong line approximation

for certain combinations of S, α

and u might give an unphysical value A > 1.

Thus, one should correctly limit the average absorption in (7.75) by 1. In contrast,

the weak line limit is always bounded.

In the following we will seek a physical interpretation for these limiting values

of the average absorption. In a pressure-broadened spectral line the monochromatic

transmission reads

(

u) = exp



−

2α

+ α



(7.76)

Figure 7.6 depicts the absorption and transmission by a single Lorentz line for

different values of the parameter

u. It can be seen that near the line center T

gradually approaches zero for increasing

u.For

u = 5 the absorption is already

complete (A

= 1) as long as ν stays within a distance of one half-width from the

line center.

For ν  α

we observe that in the denominator of T

in (7.76) the term α

can

safely be neglected in comparison to the term ν

. In fact, already for ν>10α

this gives rather accurate approximations. Furthermore, if we assume that

u is very

large, then the absorption near the line center is complete. Neglecting α

in the

denominator of (7.76) does not appreciably change the monochromatic absorption

for

u  1 and for any

u far from the line center. In summary, for these two cases

we ﬁnd the following approximate form for the monochromatic transmission

(

u) ≈ exp



−

2α



(7.77)

7.2 Band models 223

0.8

0.6

0.4

0.2

0.4

0.6

0.8

10 5 0−5 −10

0.2

A()T ()

= 0.1

u =1

u =10

νν

Fig. 7.6 Comparison of the transmission function (7.76) (full curves) with the

approximate form (7.77) (dashed curves) for different values of the variable

with x = ν/α

The accuracy of this approximation can be inferred from the dashed curves in Figure

7.6. As expected for large

u and for ν far away from the center line the approximation

is very good. For small

u, however, extremely large errors are observed near the

center line.

Using (7.52) and (7.53) it is easy to ﬁnd an expression for the average

absorption of an isolated Lorentz line for large

u. Setting ν

= 0 and

introducing the substitution ξ = 2α

u/ν

we again obtain the strong line

approximation

u) ≈

ν

√



∞

(1 − exp

(

−ξ

)

)ξ

−3/2

dξ =

2α

ν

√

2π

u (7.78)

In case of very weak absorption the linear expansion of the exponential (7.76)

leads to

(

u) ≈ 1 −

2α

+ α

(7.79)

For ν/α

 1 and small values for

u we obtain T

(

u) ≈ 1 in accordance with

Figure 7.6.

224 Transmission in spectral lines and bands of lines

7.2.2 Band model for nonoverlapping lines

The simplest band model to be treated is a spectral interval ν = N δ containing

N nonoverlapping lines of mean line spacing δ. Thus the average equivalent width

W (u) of all lines in the band can be obtained by summation over the individual

equivalent widths W

, i.e.

W (u) =



i=1

(u) (7.80)

The average band absorption then follows from

A(u) =

N δ



i=1

(u) =



i=1

(u) (7.81)

It is useful to consider the two limiting cases of weak and strong absorption. In the

weak line limit the individual lines have the average absorption A

w,i

(u) = S

u/δ.

Thus, for N nonoverlapping lines we obtain the weak line limit

(u) =

N δ



i=1

(7.82)

In this limit A is independent of the line shape. For the strong line limit of N

nonoverlapping Lorentz lines we similarly obtain

(u) =

N δ



i=1

L,i

(7.83)

In general, the band model for nonoverlapping spectral lines does not apply

to atmospheric conditions. Nevertheless, the model results will be needed in the

construction of realistic atmospheric transmission functions.

7.2.3 Random band models

For triatomic molecules such as H

O, the band structure is very complex and gives

the impression that the spectral lines are randomly spaced. We will use several steps

to construct a general random model.

First we consider an inﬁnite array of identical lines of unspeciﬁed line shape.

Suppose that N lines are distributed randomly in the interval between −Nδ/2 and

N δ/2 where δ is the mean line spacing. If a line is centered at ν = ν

within this

interval, the contribution of this line to the absorption coefﬁcient at the center of the