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

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11.3 Inversion of the temperature proﬁle 425

Since the average of the f

is deﬁned as

f =



k=1

(11.68)

we ﬁnd

−

f =−N

−1

− N

−1

−···+(1 − N

−1

) f

−···−N

−1

(11.69)

Inserting this equation into (11.67) leads in matrix notation to

Af − A

g + γ H f = 0 (11.70)

where the N × N matrix H is given by

H =







1 − N

−1

−N

−1

... −N

−1

−N

−1

1 − N

−1

... −N

−1

. ...

−N

−1

−N

−1

... 1 − N

−1







(11.71)

Inverting (11.70) leads to the ﬁnal solution

f = (A

A + γ H )

−1

g (11.72)

This solution is due to Phillips (1962) and Twomey (1963).

11.3.3 Chahine’s relaxation method

Let us consider the solution to the radiative transfer equation for nadir observation

in the M measurement channels at the top of the atmosphere as discussed above,

see (11.44)

= B

) +



(p)

∂T

(p)

∂ ln p

d ln p, i = 1,...,M (11.73)

The Planck function is given by

(p) = B

[

T ( p)

]

aν

exp

[

bν

/T ( p)

]

− 1

(11.74)

We will assume that the weighting functions for the individual channels can

be chosen in such a way that each weighting function attains its maximum at a

different altitude level or, equivalently, within a pressure layer of thickness 

ln p.

Employing the mean value theorem for integrals, we can approximate the observed

426 Remote sensing applications of radiative transfer

radiance

by means of

≈ B

) + B

)

∂T

∂ ln p





ln p (11.75)

Note that p

is the pressure level for which ∂T

/∂ ln p attains its maximum value,

and that 

ln p can be understood as the effective width of the i-th weighting

function in logarithmic pressure coordinates.

Let T



stand for an estimate of the temperature at level p

. Inserting this estimate

into (11.75), we obtain an estimate for the observed radiance



≈ B



) + B



)

∂T



∂ ln p





ln p (11.76)

Division of (11.75) by (11.76) leads to

− B

)



− B



)

≈

)



)

∂T

∂ ln p



∂T



∂ ln p



(11.77)

Experience indicates that for a temperature change from T to T + T the Planck

function varies much more strongly than the weighting function associated with

this particular temperature change. In other words, the change of the transmission

versus pressure remains virtually constant with respect to moderate temperature

changes. Thus the second factor on the right-hand side involving the ratio of the

weighting functions is approximately equal to 1.

Assuming that the surface contribution in (11.77) is negligible we ﬁnd the simple

relaxation equation



≈

)



)

(11.78)

which can be used in the following way: insert the measured radiance

I in the

left-hand side numerator of (11.78). By employing an estimated temperature T



the

Planck function B



) can be evaluated as well as I



from (11.76) so that for a

ﬁxed channel i the temperature T

can be found by inverting B

If necessary, the same form (11.76) of the relaxation formula can be used in case

that T ( p

) and T

) ≈ T



) are known to include the surface contribution. In

this case B

) must be subtracted from

in (11.78) and from I



in (11.76).

The relaxation equation (11.78) has been originally derived by Chahine (1970).

We will now brieﬂy discuss the application of this equation. In the strong absorp-

tion regions of the 15

µm band of CO

the upwelling radiance measured by the

satellite instrument at the top of the atmosphere results from emissions of the

11.3 Inversion of the temperature proﬁle 427

upper atmospheric regions. For the weaker absorption bands this radiation origi-

nates from progressively lower atmospheric regions. One selects a particular set

of M frequency bands which is characterized by overlapping weighting functions

covering the altitude range of interest.

The algorithm for Chahine’s relaxation method is given by the following steps.

(1) Make use of the radiance observations

for all channels i = 1,...,M.

(2) Specify the constant mixing ratio of CO

, the response functions φ

of the instrument

for all channels, and select the pressure levels p

for which the weighting functions

attain their maximum.

(3) Prescribe an initial estimate T

(0)

(i = 1,...,M) for the temperature proﬁle, for exam-

ple, by using a reasonable proﬁle from a climatology.

(4) For the n-th step, insert T

(n)

into (11.73) and employ an accurate quadrature formula

to determine the integral over d ln p.

(5) Make a comparison of the n-th iterate I

(n)

with the measurements

for i = 1,...,M.

If all the residuals R

(n)

− I

(n)

are smaller than a given bound ε, then T

(n)

is the solution for the temperature proﬁle. If |

− I

(n)

>εapply the relaxation

equation (11.78) in the form

(n)

≈



(n+1)





(n)



(11.79)

Substituting (11.74) into (11.79) gives the estimate

(n+1)

) = bν

/ ln

1 −



1 − exp



bν

(n)



(n)

(11.80)

for the iteration step n + 1.

(6) Return to step 4 and re-evaluate (11.73) by appropriately interpolating the temperature

proﬁle from the set ( p

, T

(n+1)

) yielding I

(n+1)

. Repeat the above steps until the residuals

are smaller than the bound ε.

(7) Use (linear) interpolation to evaluate the temperature proﬁle at pressure levels other

than p

, i = 1,...,M.

11.3.4 Smith’s iterative inversion method

As an alternative to Chahine’s relaxation algorithm we will now discuss an iterative

method which was originally introduced by Smith (1970). In this approach (11.73)

will be iteratively solved. We start by writing the n-th iteration step of (11.73) as

(n)

= B

(n)

) +



(n)

(p)

∂T

∂ ln p

d ln p (11.81)

428 Remote sensing applications of radiative transfer

For step n + 1 we write analogously

= I

(n+1)

= B

(n+1)

) +



(n+1)

(p)

∂T

∂ ln p

d ln p (11.82)

i.e. we use the observations

as a ﬁrst guess in the (n + 1)-th iteration step.

The difference between (11.82) and (11.81) is

(n+1)

− I

(n)



(n+1)

) − B

(n)

)



)





(n+1)

(p) − B

(n)

(p)



∂T

∂ ln p

d ln p

(11.83)

The main simpliﬁcation of this method results from the assumption that for each

channel i the difference of the Planck functions under the integral sign does not

depend on the entire atmospheric pressure range but only on the temperature at level

p. Assuming further that the difference B

n+1

(p) − B

(p) depends only weakly on

the pressure level, the integral term in (11.83) can be approximated as





(n+1)

(p) − B

(n)

(p)



(p) ≈



(n+1)

(p) − B

(n)

(p)



(0)

−



(n+1)

) − B

(n)

)



)

(11.84)

Since T

(0) = 1, we obtain for the differences of the nadir radiance between the

(n + 1)-th and n-th step

− I

(n)

= B

(n+1)

(p) − B

(n)

(p) (11.85)

The last expression can be solved for the Planck function in layer i for the (n + 1)-th

iteration step yielding

(n+1)

(p) = B

(n)

(p) +

− I

(n)

(11.86)

or, equivalently for the temperature, we ﬁnd

(n+1)

(p,ν

) =

bν



1 + aν

(n+1)

(p)



(11.87)

For arbitrary p the best approximation for the temperature can be found by weight-

ing all temperature proﬁles for the individual channels i with their own kernel

(p)

(n+1)

(p) =



i=1

(n+1)

(p,ν

(p)



i=1

(p)

(11.88)

11.3 Inversion of the temperature proﬁle 429

where we have deﬁned

(p) =



(p), p < p

(p), p = p

(11.89)

Using the above formulas the following steps constitute Smith’s iterative retrieval

method.

(1) Provide an initial guess for the temperature proﬁle T

(n)

(p) for n = 0.

(2) Compute B

(n)

(p) employing the expression (11.74) for the Planck function. With known

Planck function we can proceed to calculate I

(n)

using (11.81).

(3) Compute B

(n+1)

(p) and T

(n+1)

using (11.86) and (11.87). This can be done for arbitrary

pressure p.

(4) Using (11.88) a new improved approximation for the temperature proﬁle T

(n+1)

(p) can

be found in the (n + 1)-th iteration step.

(5) In the ﬁnal step we compare the computed nadir radiances I

(n)

and

.IfR

(n)

− I

(n)

<ε for all i, then T

(n+1)

(p) is the solution for the temperature pro-

ﬁle. If R

(n)

>ε, then repeat steps 1–5 until the residual term meets the speciﬁed

bound ε.

Let us now discuss some of the advantages of Smith’s iterative method.

(1) For the derivation of the temperature proﬁle there is no prescription of the analytic

form of the proﬁle. This is in contrast to Chahine’s method where the total number of

discrete points of the retrieved temperature proﬁle directly depends on the total number

M of radiance observations.

(2) With Smith’s method the temperature proﬁle can be evaluated for arbitrary p. Chahine’s

relaxation method, on the other hand, employs a linear interpolation of the retrieved

discrete temperature values T

(n)

)(i = 1,...,M).

Figure 11.11 shows a comparison of retrieved temperature proﬁles using

Chahine’s relaxation method and Smith’s iteration procedure. This ﬁgure is due

to Liou (1980) who computed the synthetic radiances for the six VTPR (Vertical

Temperature Proﬁle Radiometer) channels centered at the wave numbers 669.0,

676.7, 694.7, 708.7, 723.6, and 746.7 cm

−1

of the 15 µmCO

band (solid curve

without dots). The VTPR was ﬂown on the NIMBUS 4 satellite in the early 1970s.

As initial guess an isothermal temperature proﬁle of 300 K was employed, and the

surface temperature was held ﬁxed at 279.5 K. For Chahine’s relaxation method

the residual was set to ε = 10

−2

. The ﬁnal result (solid curve with black dots) was

obtained after only four iteration steps. The dots represent the discrete pressure lev-

els p

. It is seen that the difference between the actual and Chahine’s temperature

proﬁle amounts to typically 2–3 K with similar over- and underestimations. The

result obtained with Smith’s iterative method is depicted as the dashed line in the

ﬁgure. Smith’s method converged after about 20 iteration steps whereby the bound

430 Remote sensing applications of radiative transfer

100

1000

200 220 240 260 280

Temperature (K)

Pressure (hPa)

300 320

Fig. 11.11 Comparison between the temperature proﬁle retrieval employing

Chahine’s relaxation and Smith’s iteration method for the six VTPR channels.

The thick solid line represents the actual temperature proﬁle. (Redrawn from Liou

(1980), with permission from Elsevier.)

ε was set to 0.02. It is noteworthy that a reduction of the size of ε did not improve the

temperature proﬁle. One can see that the retrieved proﬁles exhibit less variability

than the true proﬁle. Moreover, both methods are not capable of recovering the tem-

perature proﬁle in the upper atmosphere, because the VTPR channels were selected

in a manner that the peak for the highest weighting function lies near 30 hPa.

We do not wish to conclude this section without giving special credit to two

pioneers in the ﬁeld of remote sensing of temperature proﬁles. King (1956) was the

ﬁrst to show in which way the vertical temperature distribution could be inferred

from satellite radiance scan measurements. Kaplan (1959) pointed out that the

vertical temperature proﬁle of the atmosphere can be inferred from the spectral

distribution of the emission spectrum as observed by a satellite. We have already

made use of his observation that the upper part of the atmosphere can be seen by

probing the radiance of the emitting gas at the band center. By properly selecting

spectral regions in the wings of the band we may view the entire atmosphere.

Many papers, too numerous to be listed here, have been written on remote sensing

techniques. Moreover, several textbooks on remote sensing are now available for

detailed studies, for example Houghton et al. (1984) and Stephens (1994).

11.4 Perturbation theory and ozone proﬁle retrieval 431

11.4 Radiative perturbation theory and ozone proﬁle retrieval

We have pointed out already that ozone is one of the most important trace gases

in the Earth’s atmosphere. It is well known that the concentration of this trace

gas strongly depends on altitude. Typically, in the free troposphere the volume

concentration is approximately 0.05 ppmv, whereas in the altitude region between

about 20 and 40 km ozone has a pronounced maximum of about 10 ppmv.

The radiative effects of ozone can be summarized as follows: it absorbs practi-

cally all solar radiation between 240 and 300 nm wavelength. Therefore, the ozone

layer in the Earth’s atmosphere protects all unicellular organisms and all skin cells

of plants, animals and human beings from the dangerous lethal ultraviolet radia-

tion. Nevertheless, solar photons in the wavelength range 300 to 330 nm (UV-B

radiation) can penetrate the entire atmosphere and reach the surface. The intensity

of the solar radiation is signiﬁcantly reduced due to the absorbing ozone molecules.

Thus UV-B radiation in this spectral region may cause skin cancer for susceptible

humans.

The temperature structure and the dynamics of the stratosphere have their origin

in the absorption of solar UV radiation. Thus in the stratosphere the tempera-

ture increases with increasing altitude while in the troposphere the temperature

decreases with increasing altitude. Due to the thermal stability, vertical mixing of

trace gases is very slow in the stratosphere. This is in contrast to the situation in

the troposphere where vertical convective and turbulent processes cause a rapid

mixing of atmospheric gases. In the atmospheric window region ozone possesses

a strong absorption band near 9.6

µm and acts here as an important greenhouse

gas.

In the stratosphere and lower mesosphere ozone molecules are produced by two

steps. First, for wavelengths shorter than 242 nm molecular oxygen is photodisso-

ciated into atomic oxygen by the absorption of a photon of energy hν (Chapman,

1930a,b)

+ hν → 2 O (11.90)

The subsequent reaction of O and the remaining oxygen molecules is

+ O + M → O

+ M (11.91)

where M denotes any third atom or molecule. This third atom or molecule is required

in order to conserve energy and momentum.

In the mesosphere the concentration of molecular oxygen is so low that the

production of atomic oxygen by photodissociation is unimportant. At lower alti-

tudes the concentration of O

is higher but the solar photons have been absorbed

432 Remote sensing applications of radiative transfer

already by ozone at higher altitudes. Thus there exists an intermediate altitude range

characterized by a maximum production of ozone molecules.

In the stratosphere ozone is destroyed due to catalytic reactions (Bates and

Nicolet, 1950)

+ X → O

+ XO

O + XO → O

+ X

(11.92)

leading to a net reaction

O + O

→ O

+ O

(11.93)

Here X is the catalyst species. Due to the fact that the catalyst is not destroyed in

this reaction, only a small amount of X is necessary to destroy a large reservoir

of O

. For this catalytic reaction several gaseous species can be important, e.g.

X = H, OH, NO and Cl.

In the troposphere ozone is photochemically produced by the reactions

+ hν → NO +O

O + O

→ O

(11.94)

so that the photodissociation of a NO

molecule provides a single oxygen atom.

The latter then reacts with molecular oxygen to produce ozone. It should be pointed

out that the presence of trace gases like CO, CH

and other hydrocarbons allows

for a recycling of NO to NO

thus making up a catalytic reaction chain. In the

lower troposphere a major source of ozone is the in situ production via the above

reactions. Transport from the stratosphere represents a second source of ozone in

the upper and middle troposphere.

In the troposphere ozone plays an important role in atmospheric chemistry. Due to

the photodissociation of ozone for wavelengths shorter than 315 nm highly reactive

OH (hydroxyl) radicals are produced. These OH radicals can react with practically

all atmospheric gases, and, therefore, OH is called an atmospheric detergent. It

can be concluded that tropospheric ozone is crucial for the removal of atmospheric

pollution. For detailed information on the chemistry of the atmosphere see, for

example, Graedel and Crutzen (1994).

A very important ﬁeld of research is to study the different roles of both tropo-

spheric as well as stratospheric ozone in atmospheric chemistry and climate. For

this type of research altitude resolved ozone measurements are an essential input.

Ozone proﬁle information on a global scale can only be obtained via satellite remote

sensing.

11.4 Perturbation theory and ozone proﬁle retrieval 433

In the following we will discuss the retrieval of ozone proﬁles from atmospheric

spectrometer instruments aboard satellites which observe backscattered solar radi-

ation. The retrieval method is based on radiative transfer modeling involving the

solution to the forward and adjoint radiative transfer equation. As will be discussed

next, the adjoint problem is used to derive the weighting functions, or Jacobians,

via the linear radiative perturbation theory. In general, the Jacobians are the key

input for the inversion of atmospheric parameters.

Let us consider the situation where the reﬂectance r(x) of the atmosphere at the

satellite position and in the viewing direction of the instrument is the required infor-

mation for a given state vector x of the atmosphere. In the following this state vector

represents the vertical proﬁle of the ozone density ρ

plus the Lambertian ground

albedo A

, i.e. x = (ρ

, A

). Thus we may use a forward radiative transfer model

to simulate r (x). The instrument itself measures the reﬂectance

r in dependence

of the unknown atmospheric state. In addition to this, the observation involves the

instrument error ε so that

r = r (x) + ε (11.95)

Clearly, the radiative transfer model depends in a nonlinear way on the state

vector x. Any retrieval method requires a linearization of the reﬂectance about a

ﬁrst guess state vector of the atmosphere which we will call x

. Using a Taylor

series expansion, omitting nonlinear terms, we may write

r(x) ≈ r(x

) +



k=1

∂r

∂x



x

with x

= (x

− x

0,k

) (11.96)

where K is the dimension of the state vector x and x

is its k-th component. As an

example, one may identify this k-th component with the ozone density in the k-th

homogeneous atmospheric sublayer.

For the inversion of the forward radiative transfer model we have to ﬁnd the

reﬂectance r as well as its linearization with respect to the individual components of

the state vector. Usually the linearization represents the computational bottleneck of

ozone proﬁle retrievals. Therefore, for fast ozone proﬁle retrievals on an operational

basis the development of efﬁcient linearized radiative transfer models represents an

important task. For the linearization process we will now employ the linear radiative

perturbation theory as described previously in Chapter 5 in more detail.

According to (5.5) the radiative transfer equation including Lambertian ground

reﬂection can be formulated in its forward formulation as

LI(z, Ω) = Q(z, Ω) (11.97)

434 Remote sensing applications of radiative transfer

where the linear differential operator L including the surface reﬂection part may

be written as (Ustinov, 2001; Landgraf et al., 2002)

L = µ

∂

∂z

+ k

ext

(z) −

sca

(z)

4π



4π



P(z, Ω



→ Ω) −

δ(z)U(µ)|µ|U (−µ



)|µ





◦ d



(11.98)

The ﬁrst two terms on the right-hand side represent the extinction of the radiance, the

third term describes the scattering process due to air molecules and aerosol particles

while the last term denotes the reﬂection by the Earth’s surface with albedo A

Again U is the Heaviside step function as deﬁned in (5.50). For simplicity we will

not treat the effect of clouds so that only cloud free scenes will be considered in

the sequel. For solar radiation the source term Q on the right-hand side of (11.97)

is taken from (5.3), that is

Q(z, Ω) =|µ

δ(z − z

)δ(µ − µ

)δ(ϕ − ϕ

) (11.99)

As discussed in Chapter 5, the radiance I has to fulﬁll vacuum boundary condi-

tions for the incoming radiation ﬁeld

top of the atmosphere: I (z

,µ,ϕ) = 0, 0 ≤ ϕ ≤ 2π, −1 ≤ µ<0

Earth’s surface: I (0,µ,ϕ) = 0, 0 ≤ ϕ ≤ 2π,0<µ≤ 1

(11.100)

Any radiative effect E is associated with a corresponding response function R

E =R, I  (11.101)

where the angular brackets <...>denote the inner product deﬁned by the inte-

gration over the full solid angle (4π) and the entire vertical extension of the model

atmosphere, see (5.6). In the present situation the radiative effect is the reﬂectance

r(x).

In Chapter 5 we treated speciﬁc examples for the response function which yield

upward, downward or net ﬂux densities. For satellite applications we have to con-

sider another type of response function. If 

= (µ

,ϕ

) is the instrument’s viewing

direction at the position of the satellite, then the response function

δ(z − z

)δ( − 

) (11.102)

extracts from (11.101) the radiance ﬁeld as seen by an ideal instrument.