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

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11.2 Remote sensing based on short- and long-wave radiation 405

110

105

100

95 90



−1

Fig. 11.3 Comparison of the optical air mass of a plane–parallel atmosphere |µ

−1

(dashed curve) with the optical air mass m as obtained from (11.7) (solid curve).

In (11.6) we have introduced the optical air mass, m. For a perfectly plane–

parallel atmosphere we would have m =|µ

−1

. Kasten and Young (1989) sug-

gested a simple approximation for m as a function of ϑ

which is accurate enough

for most practical purposes

m ≈ [|µ

|+0.50572(ϑ

− 83.92)

−1.6364

]

−1

(11.7)

This equation applies to a clear and dry atmosphere containing no aerosols, clouds

or absorbing gases. In principle the optical air mass is wavelength dependent.

Equation (11.7) is to be interpreted in a manner that m multiplied by the total

scattering optical thickness of the Rayleigh atmosphere can be used to determine the

attenuation of the total solar radiation reaching the surface of the Earth. Figure 11.3

compares the idealized values |µ

−1

of a plane–parallel atmosphere with m of

the real atmosphere which was calculated by means of (11.7). From the ﬁgure it

is concluded that the plane–parallel approximation is acceptable for solar zenith

angles ϑ

≥ 100

◦

. For smaller ϑ

values the error becomes increasingly larger. If

the Sun is close to the horizon, the neglecting of spherical effects gives rise to big

errors. For ϑ = 90

◦

we have |µ

−1

=∞in contrast to m = 37.92.

In an atmosphere consisting of air molecules, aerosol particles and trace gases

the optical depth is given by the sum of the three individual optical depths, i.e.

τ = τ

air

+ τ

gas

+ τ

aer

(11.8)

Since τ

air

can be computed from standard atmospheric data, the optical depth of

the atmospheric aerosol can be directly retrieved if no gas contributions to the

406 Remote sensing applications of radiative transfer

ln E

Fig. 11.4 The Bouguer–Langley method to determine S

0,ν

.Form = 1 the Sun is

in the local zenith. The solid line is the linear regression.

total optical depth exist. This type of measurement is a common procedure for

ground-based observations.

In order to guarantee a high accuracy to determine τ

aer

, the instrument must be

well calibrated with respect to the extraterrestrial solar ﬂux density S

0,ν

. If this

requirement cannot be met the so-called Bouguer–Langley method may be applied.

This procedure requires a plot of ln E

versus m by taking measurements for various

solar zenith angles. Figure 11.4 depicts an example for a set of measurements. Based

on the evaluation of (11.4), from a linear regression of the resulting data points two

important quantities can be extracted: (i) the slope of the regression line is equal to

τ , and (ii) the intersection of the straight line with the ordinate gives an estimate of

ln E

. The Bouguer–Langley method has been employed to determine the spectral

solar constant before satellite measurements were available.

Optical thickness observations for trace gases make use of a similar principle. Let

us assume two neighboring wave number points (ν, ν



) with maximum absorption

of a particular gas at ν and very small absorption at ν



. If we then measure E

and



, according to (11.4) and (11.6) we can compute

gas

− τ



gas





0,ν



− ln



0,ν









0,ν



− ln







(11.9)

If at ν



the absorption of the gas is small enough, τ



gas

may be neglected yielding the

optical thickness τ

gas

. Clearly, the observation is not sensitive to uncertainties in the

calibration constant. This relation is strictly valid only if the calibration constants

for the wave numbers ν and ν



do not differ from each other.

11.2 Remote sensing based on short- and long-wave radiation 407

Example: determination of the aerosol and ozone optical depth

A Sun photometer can be used to determine the total optical depth between the

location of the instrument and the top of the atmosphere. From (11.4) and (11.8)

the total optical depth is given by

τ = τ

air

+ τ

gas

+ τ

aer

=−



0,ν



(11.10)

This equation can be used to obtain the optical depth of the aerosol particles τ

aer

First we determine the optical depth τ

air

describing the Rayleigh scattering of the air

molecules. For this we need the scattering cross-section of an air molecule. Due to

Rayleigh (1918) and Cabannes (1929) the scattering cross-section for anisotropic

gaseous molecules in random orientation is given by

sca,air

8π

− 1)

6 + 3δ

6 − 7δ

(11.11)

where λ = 1/ν is the wavelength, N is the number of air molecules per unit volume,

n is the refractive index of the air, and δ the anisotropy factor of the gas molecules.

The Earth’s atmosphere can be treated as a single gas having δ = 0.031. Equation

(11.11) can be used to derive a simple expression for the total Rayleigh optical

depth of the atmosphere, see Allen (1963)

air,tot

= 0.008569λ

−4

(1 + 0.0113λ

−2

+ 0.00013λ

−4

)

(11.12)

where the wavelength λ has to be inserted in units of

µm. Note that due to the

wavelength dependence of the refractive index τ

air,tot

slightly differs from the ideal

−4

law of Rayleigh scattering, see Section 9. For a wavelength of 0.55 µmwe

obtain τ

air,tot

= 0.0973. At an arbitrary pressure level p the Rayleigh optical depth

as measured from the top of the atmosphere is obtained from

air

(p) =

air,tot

with p

= 1013.25 hPa (11.13)

An important absorbing gas in the UV/VIS spectral region is ozone. By assuming

that O

is the only trace gas absorber within the atmosphere, for the determination

of τ

aer

in (11.10) we need to know τ

gas

= τ

. We select the two wavelengths λ

and λ



where the ozone absorption is relatively high (λ) and rather low (λ



). The

differential absorption technique as described before allows us then to write

− τ







0,ν



− ln







(11.14)

where the spectral solar constants S

0,ν

, S

0,ν



are known.

408 Remote sensing applications of radiative transfer

Table 11.1 Examples for Dobson wavelength pairs (λ, λ



)

and the corresponding differences for the absorption

coefﬁcient of ozone. See also Lenoble (1993).

(λ, λ



) k

abs,ν,O

− k

abs,ν



Pair (nm) (cm atm)

−1

A (305.5, 325.4) 4.025

B (308.8, 325.1) 2.625

C (311.4, 332.4) 1.842

D (317.6, 339.8) 0.829



(332.4, 453.6) 0.108

Equation (11.14) follows from (11.10) by assuming that there is no wavelength

dependence of τ

air

and τ

aer

. To fulﬁll this requirement as closely as possible, we

must choose λ and λ



in close proximity within the spectrum. The aerosol optical

depth varies only very weakly with wavelength, and since the optical depth of

the air molecules varies approximately with λ

−4

, an expression such as (11.14) is

acceptable. To improve the treatment and make it even more correct, one may add to

the right-hand side of (11.14) a small correction term to account for the difference

in τ

air

− τ



air

, and an even smaller correction τ

aer

− τ



aer

. While the correction for

Rayleigh scattering can be carried out with high precision, physically reasonable

information from a suitable standard aerosol model is required to obtain the aerosol

correction, e.g. Shettle and Fenn (1979), d’Almeida et al. (1991), Hess et al. (1998).

As a particular example, Table 11.1 presents the generally adopted differences

of the ozone absorption coefﬁcient for ﬁve different wavelength pairs A, B, C, D,



which have been selected by the Dobson Network for total ozone measurements.

Utilizing these data the total ozone column u

can be evaluated from

− τ



abs,ν,O

− k

abs,ν



(11.15)

In contrast to the total vertical optical depth, it is much more difﬁcult to obtain

an optical depth proﬁle as a function of altitude. An important technique to achieve

this goal are spaceborne solar occultation measurements as shown in Figure 11.5.

Here, the instrument views the Sun along a path tangential to an interior atmospheric

surface S. This surface is deﬁned by the minimum distance z

which is known as

the tangent height. Such observations cannot be performed continually. Only a

rather narrow time interval just after sunrise and before sunset, as viewed from the

satellite, can be used for the observations. As the satellite moves along its track,

the measurements can be performed for different tangent heights. In this manner

11.2 Remote sensing based on short- and long-wave radiation 409

satellite

Sun

day

Earth

night

top of the atmosphere

Fig. 11.5 Solar occultation measurements and tangent height z

an optical depth retrieval versus altitude may be achieved. According to (11.6) this

can be expressed as

τ (z

) = 2



∞

ext,ν

(z)

dz (11.16)

where ds is the path element along the viewing direction of the satellite. The above

formula further assumes an atmosphere consisting of concentric spherical shells.

The factor 2 implies a symmetry in the atmospheric extinction properties with

respect to z

. Utilizing the occultation extinction observations it is possible to ﬁnd

τ (z

) so that, in principle, an inversion of (11.16) can be carried out to ﬁnd the

extinction proﬁle k

ext,ν

(z).

Let us assume that at a given wavelength only absorption due to an individual

gas occurs. In this case the extinction coefﬁcient reduces to

abs,ν

(z) = σ

abs,ν

N (z) (11.17)

where σ

abs,ν

is the molecular absorption cross-section in units of m

, and N (z)is

the volume number concentration of the particular gas in question, i.e. the number

of gas molecules per unit volume of air located at altitude z. Assuming that the

absorption coefﬁcient is known, with the help of the occultation measurements the

vertical concentration proﬁle of the gas may be retrieved. In general, a correction

for molecular scattering will be necessary.

A further modiﬁcation of the experiment is possible as will be discussed next.

Let us assume that the molecular scattering effects either are exactly known or

can be totally neglected, and that the same is true for gaseous absorption. The

only extinction that remains is due to a population of Mie scatterers. Assuming

410 Remote sensing applications of radiative transfer

spherical particles of radius r and a particle number density concentration N (z, r ),

the extinction coefﬁcient for this population is given by

ext,ν

(z) =

∞



πr

ext,ν

N (z, r) dr (11.18)

Here Q

ext,ν

is the Mie extinction efﬁciency for spherical particles, see Chapter 9. In

principle it is possible to obtain the vertical proﬁle of the particle number density

concentration N (z, r ) by inversion. However, such an inversion is rather difﬁcult

because further assumptions have to be made to constrain the ill-posed problem

in a favorable way. For example, for the determination of Q

ext,ν

the chemical

composition of the particles has to be speciﬁed in advance.

If a set of wavelengths is used simultaneously, an inversion of N (z, r ) is feasible

under certain circumstances. We will not give details and refer to the primary

literature, cf. King et al. (1978). In connection with the Stratospheric Aerosol and

Gas Experiment II (SAGE II) see Wang et al. (1989) or Livingston and Russell

(1989).

In contrast to the above very complicated situation, an inversion of atmospheric

column contents of trace gases can be carried out as follows. Let us assume that

some remote sensing experiment provides us with τ . In case of gaseous absorption

it is then possible to retrieve the entire column content,



∞

N (z)dz, i.e. the total

number of gas molecules per m

in an atmospheric column having a 1 m

cross-

section. A similar approach can be used in case of aerosol particles. This procedure

makes it possible to invert a mean particle number density for the entire atmospheric

column.

Figures 11.6a–d depict four typical situations of nadir viewing short-wave remote

sensing. Figure 11.6a shows the general situation where three contributions must be

accounted for describing the instrument’s signal: (i) direct solar radiation reﬂected

at the Earth’s surface and directly transmitted to the instrument, (ii) solar radiation

scattered by air molecules and aerosol particles and then transmitted to the instru-

ment, and (iii) solar radiation being reﬂected by the cloud tops and transmitted to

the satellite. This ﬁgure applies to a relatively transparent atmosphere with some

embedded clouds lying over a strongly reﬂecting surface.

Reﬂection of short-wave radiation at the surface and at the top of the clouds

plays an important role for remote sensing performed in an absorption band of an

atmospheric trace gas. This situation, shown in Figure 11.6b, is important because

the application of Beer–Lambert’s extinction law allows the establishing of a direct

relation between the transmission of the directly transmitted solar radiation and

altitude. If multiple scattering is of relevance, the relationship would turn out to be

much more involved and complicated to be exploited.

11.2 Remote sensing based on short- and long-wave radiation 411

(a) (b)

Fig. 11.6 Nadir viewing geometry for different short-wave remote sensing obser-

vations. Dashed lines denote contributions of minor importance. See also text.

The case of a turbid atmosphere lying over a dark surface is illustrated in

Figure 11.6c. In order to detect a noticeable signal at the instrument, the atmosphere

must scatter the incoming solar radiation in the upward direction. Furthermore,

the atmosphere itself should not be opaque. Contributions due to ground reﬂection

are of secondary importance, since the ground is assumed to be dark.

Finally, Figure 11.6d depicts the situation that the backscattered solar radiation

is observed in the very strong absorption band in the UV-A and UV-B spectral

region (λ<330 nm). For very short wavelengths (λ<300 nm) the ozone absorp-

tion optical depth is so large that solar photons entering the top of the atmosphere

are absorbed before they reach the surface. Taking advantage of the smoothly

varying absorption cross-sections of ozone molecules, the location of the atmo-

spheric backscatter contributions can be varied through the entire altitude range

of the Earth’s atmosphere. In the ﬁgure we have λ

<λ

. If clouds

occur in the pixel observed by the satellite instrument, then it is of key impor-

tance to know both the location of the cloud top as well as the cloud cover for the

pixel.

The limb viewing geometry is depicted in Figure 11.7. The observing instrument

and the Sun are located to the left and to the right of the local vertical with respect

412 Remote sensing applications of radiative transfer

Fig. 11.7 Limb viewing geometry for short-wave remote sensing.

(a) (b) (c)

Earth

Fig. 11.8 Ground-based observation of sky radiation: (a) sky radiance; (b) solar

aureole; (c) zenith radiation during sunset or sunrise.

to the tangent point at height z

. This point is the lowermost location along the limb

viewing path. The instrument’s viewing direction is tangential with respect to the

Earth’s atmosphere.

Note that, in principle, all atmospheric volume elements located along the limb

path may deliver a certain contribution to the observed signal. It should be observed,

however, that the more distant these volume elements are relative to z

the less

important is their contribution. This is due to the fact that the outer shells of the

atmosphere contain less scattering material than the volume elements close to z

Therefore, to a very good degree of approximation the major contributions to the

signal originate from atmospheric volume elements centered around z

Figure 11.8a–c depict several observation geometries for ground-based remote

sensing of solar radiation. Figure 11.8a illustrates the daytime observation of the

sky radiance. Due to atmospheric scattering of solar radiation by air molecules,

aerosol or cloud particles, the sky radiance exhibits speciﬁc features with respect to

the viewing zenith and azimuth angle. This information can be exploited to derive

physical properties of the scattering particles. More information can be obtained

by measuring, modeling and analyzing the polarized radiation ﬁeld. This can be

understood if one realizes that the clear atmosphere (no aerosols and no clouds

present) possesses a distinct polarization pattern which is due to the polarizing

11.2 Remote sensing based on short- and long-wave radiation 413

nature of Rayleigh scatterers. In contrast to this, scattering by aerosol particles

or cloud droplets causes a depolarizing effect. This depolarizing nature can be

exploited to deﬁne speciﬁc remote sensing algorithms. We will not treat this subject

further and refer to the scientiﬁc literature (e.g. Coulson, 1988; Herman et al. 1997;

Mishchenko and Travis, 1997).

Figure 11.8b depicts the so-called solar aureole observation. By this we mean

that one observes the immediate neighborhood of the solar disk, i.e. the instrument

does not view the solar disk itself but detects diffuse radiation coming from a

narrow solid angle interval covering the solar aureole. It is obvious that this type

of observation exploits the forward scattering properties of atmospheric aerosol

particles or thin or medium thick cirrus clouds. Clearly, such measurements cannot

be performed if the sky is overcast because then the radiation ﬁeld becomes more

and more isotropic thus losing any information signature that originates from the

single scattering processes. Further information can be found in the literature (e.g.

Deirmendjian, 1957; Box and Deepak, 1981; Nakajima et al., 1983; Santer and

Herman, 1983).

Another interesting observation geometry is shown in Figure 11.8c. Here one

observes zenith radiation during dawn or dusk, i.e. when the Sun is just below the

horizon. This observation geometry makes it necessary to take the spherical nature

of the atmosphere fully into account. While the Sun moves more and more below

the horizon it illuminates only atmospheric layers higher up in the atmosphere. This

means that the entire troposphere is located in the darkness zone.

11.2.2 Methods based on thermal emission

We will illustrate the basic principles by making the following assumptions. To a

sufﬁciently high degree of approximation scattering of radiation can be neglected

in the long-wave and microwave spectral region yielding

ext,ν

= k

abs,ν

, ω

0,ν

= 0 (11.19)

As boundary conditions for the atmospheric radiation ﬁeld we will assume that

at the top of the atmosphere no downwelling radiation exists whereas the upward

directed radiation at the Earth’s surface consists of the Planckian emission of the

ground having an emissivity ε

g,ν

, plus the reﬂected part of the downwelling ﬂux

density, i.e.

−,ν

) = 0

+,ν

(0) = ε

g,ν

) + (1 − ε

g,ν

)

−,ν

(0)

(11.20)

414 Remote sensing applications of radiative transfer

(z, z

)

abs,ν

(z)

(z, z

)

Arbitrary units

Fig. 11.9 Typical functional shape of the weighting function W

(z, z

In the following we will discuss two arrangements for remote sensing in the long-

wave spectral region.

(1) Nadir-looking instrument

In z coordinates the solution of the RTE for the upwelling radiation is given by,

(see (2.122))

+,ν

) = I

+,ν

(0)T

(0, z

) +



(z)

(z, z

)dz (11.21)

with

(z, z

) = exp



−



abs,ν



)dz



(11.22)

The term

(z, z

) =

(z, z

) = k

abs,ν

(z)T

(z, z

)

(11.23)

is called the weighting function.

Usually k

abs,ν

(z) decreases with height because for gases like O

or CO

the

number concentration of the gas molecules decreases with increasing z. For such a

case Figure 11.9 shows typical vertical proﬁles of k

abs,ν

, T (z, z

) and the resulting

weighting function. In a strong absorption band T

(z, z

) increases from 0 to 1