NASA Technical Reports Server (NTRS) 19920008737: A Variational Formalism for the Radiative Transfer Equation: Prelude to Model 3

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Chapter IV

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A Variational Formalism for the Radiative
Transfer Equation: Prelude to MODEL III

Gary L. Achtemeier
Office of Climate and Meteorology
Division of Atmospheric Sciences

Illinois State Water Survey
Champaign, IL 61820

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1. Introduction

The MODEL III variational data assimilation model is the third

of four general assimilation models designed to blend weather data

measured from space-based platforms into the meteorological data

mainstream in a way that maximizes the information content of the

satellite data. Because there are many different observation

locations and there are many instruments with different measurement

error characteristics, it is also necessary to require that the

blending be done to maximize the information content of the data

and simultaneously to retain a dynamically consistent and

reasonably accurate description of the state of the atmosphere.

This is ideally a variational problem for which the data receive

relative weights that are inversely proportional to measurement

error and are adjusted to satisfy a set of dynamical equations that

govern atmospheric processes.?! Because of the complexity of this

i . .

type of variational problem, we have divided the problem into four
variational models of increasing complexity. The first, MODEL I,
includes as dynamical constraints the two horizontal momentum
equations, the hydrostatic equation, and an integrated continuity
equation. The second, MODEL II includes as dynamical constraints,
the equations of MODEL I plus the thermodynamic equation for a dry
atmosphere. MODEL III includes the equations of MODEL II plus the
radiative transfer equation.

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The advantage of MODEL III over the previous two models is
that radiance, the atmospheric variable measured by satellite,
becomes a dependent variable. In the previous versions, mean layer
temperatures that had been retrieved from the radiances by some
method, were included in the assimilation by substituting them in
place of the rawinsonde temperatures. Now both rawinsonde
temperatures and satellite radiances are included independently in
the assimilation.

Our approach to the development of MODEL III has been to
divide the problem into three steps of increasing complexity.
Chapter IV deals with the first step, a variational version of the
classical temperature retrieval problem that includes just the
radiative transfer equation as a constraint. The radiances for each
of the four TOVS MSU microwave channels are dependent variables.
These plus temperature constitute a set a five adjustable
variables. Each radiance is related to the temperature through its
radiative transfer equation. There are therefore four dynamic
constraints in this first variational problem.

Chapter V summarizes the second step which combines the four
radiative transfer equations of the first step with the equations
for a geostrophic and hydrostatic atmosphere. This step is intended
to bring radiance into a three-dimensional balance with wind,
height, and temperature. The use of the geostrophic approximation
in place of the full set of primitive equations allows for an
easier evaluation of how the inclusion of the radiative transfer
equation increases the complexity of the variational equations.

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The third and final step includes the four radiative transfer
equations with the fully nonlinear set of primitive equations, ie.,
MODEL III.

2 . A Variational Retrieval Algorithm

The radiative transfer equation is the only variational
constraint. It takes the form

OQ

B-B 0 w 0 -fw'Tdz - 0 (1)

o

where B is the brightness temperature as computed from radiance
measured at the satellite and T is the mean layer temperature of an
incremental depth of the atmosphere, dz. The weight, w 0 , is the
transmittance of the total atmosphere from the surface (where the
surface brightness temperature, B 0 , is measured) to the space-based
observation platform. The weights, w', are proportional to the
transmittance from some level within the atmosphere to the
satellite. In order to make the variational derivations from (1)
compatible with the larger set of variational equations in MODEL I
and MODEL II, we will make the following modifications in (l).
First, the brightness temperature is replaced by the skin
temperature, T 0 , and the weight, w 0 , will become a skin level
surface weight. Second, (1) is converted from the z to the sigma
vertical coordinate. In this conversion,

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J w'Tdz- J t ^T[f(T) ] da- J wTda

( 2 )

Now f (T) , a small conversion term that results from the changeover
to sigma coordinates, will be combined with the weights and not
subjected to variation. This approach avoids complicated nonlinear
equations that will otherwise arise through the variational
formations. The f(T) and the weak temperature dependence in the
weights will not be held constant however. At each step of a
converging iterative process, the small temperature dependencies
will be updated with adjusted temperatures. With these
modifications, (1) becomes,

oo

B-fwTdo- 0 (3)

o

The next step is to bring (3) into dimensional compatibility with
the more general variational models. Let,

T-Qt'-SHt' (4)

R

and

T'-T r +

_F_

rj*H

(5)

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so that.

T~-^ (T r + — T")

R

R

( 6 )

Here g is gravity, H=10 km is a reference height, R is the
universal gas constant, F is the Froude number, and R 0 is the
Rossby number. The subscript R refers to a reference atmosphere
and the notation " refers to departures from the reference
atmosphere. Substitution of (6) in place of T in (3) gives,

00 oo

B--&H [ fwT R do+— fwr"do] =0 (7)

R J o R °o

Further, we partition B = B R + B m and define

( 8 )

It follows then, that

gH_F.

R R

( 9 )

Finally, upon suppression of the double primes, the radiative
transfer equation becomes,

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K

B- E

Jc-1

( 10 )

Now there are four TOVS microwave channels each with an
independent measurement of the brightness temperature. Let Bj be
the brightness temperature perturbation for the jth channel. The J
constraining equations are,

K

m r B rE w * v T *’° (11)

Jc-l

The functional to be minimized is

f Ida

( 12 )

where

K

j

f-E * * ( T k - T k°) it ,

k-1 j - 1

*E if

t m.

J-l

(13)

Performing the variations upon T and B as shown by Achtemeier, et
al. (1986) yields the following Euler-Lagrange equations,

6T: n k (T k -T k °)-J2w kj X r 0

(14)

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for each k and,

6 Bi Kj (Bj-Bf) +Xj-0

( 15 )

for each j . Variation upon the J Lagrange multipliers restore the
original constraints (11) . These equations are linear and may be
easily reduced to one diagnostic equation in temperature. First,
eliminate reference to the Lagrangian multipliers by substituting
(15) into (14) . Then substituting for Bj gives the adjusted
temperature as a function of weight functions and observed
variables,

\

n * T * + £ £ n 3 W U W ii T r F k- 0 <16)

for each k. Here

**- w * r * + £ * jBf

j-i

( 17 )

Equation (16) can be easily solved with a standard matrix
inversion package to retrieve the variationally adjusted
temperature profile. At most two cycles with the weight functions
updated with adjusted temperatures are required for convergence to
a final adjusted temperature.

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4 . Results

In order to properly interpret the results of the example of
variational adjustment with the radiative transfer equation, one
must be aware that three sets of weights appear in (16) . The
weights, w^. , are the transmittance weights for the ith level and
the jth microwave channel. They are not subject to the variational
adjustment and remain unchanged with the exception of minor
adjustments for temperature sensitivity. The variational weights,
TTj and 7T k , carry the relative importance of the jth microwave
channel and the temperature at the kth level. It is the choice of
the variational weights that are important in interpreting the
results.

Consider a temperature profile that is to be retrieved from
MSU brightness temperatures. It is to be made halfway between two
rawinsonde sites. The rawinsonde soundings are given by A and B in
Fig. 1. Sounding A is cold up to the tropopause (about 220 mb) and
then it becomes isothermal up to 60 mb. Sounding B is warm from
the surface to 170 mb and then becomes colder than A in the layer
from 170 mb to 60 mb. Its tropopause is located at 100 mb.

The first guess or "observed" sounding that will enter into
the temperature part of the variational analysis is the mean of A
and B. It is given by M in Fig. 1. Now suppose that the true
sounding is given by T. Note that M=T from the surface to 230 mb
and from 50 mb to the top.

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Next, the brightness temperatures, Bj, were calculated from
(10) using the true temperature sounding. Thus the Bj° that enter
(17) are true and the T k ° are approximate. However, only the
temperatures between 100-230 mb need adjustment. The observational
error for the temperature was 0.7 K and the weight accorded to the
temperature was,

n k ±-- 1.0 (18)

Fig. 2 shows the results of three retrievals between 500 mb
and the top. The dashed line is the difference M-T between the
true and first guess temperature soundings. The other curves are
the differences between the adjusted and the true temperatures for
7Tj that ranged in values from 10 to 100 to 1000. Note that the
weights for the four MSU channels and hence the brightness
temperatures were always equal.

Fig. 2 shows that increasing the brightness temperature
weights progressively reduced the differences between the adjusted
and true temperature soundings but by only 2.5 K. However, the
retrievals also spread the adjustments throughout the depth of the
sounding. Therefore, improvements where the M-T residuals were
nonzero were offset by degraded temperatures throughout the
reminder of the sounding - the errors being almost 2 K at 250 mb
with lesser error elsewhere.

A more extensive analysis of the behavior of (16) found that
the retrievals were sensitive to the vertical distribution of the

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weights for the temperature hence the errors of observation for the
temperature. If there existed some independent observations that
could be used to estimate the accuracy of the first guess
temperature as a function of height, then the retrievals could be
focused into those locations where the M-T residuals were greatest.
Consider possible accuracy functions given in Fig. 3. The
effective temperature error at 150 mb is doubled by f(l) and is
tripled by f ( 2 ) . Therefore, the weights accorded to the
temperature there are decreased by a factor of four for f ( 1 ) and a
factor of nine for f ( 2 ) .

Fig. 4 shows the residuals between the adjusted and true
temperature profiles for the three retrievals when the accuracy
function f(l) was applied to the temperature weights. The initial
residual has been reduced by approximately 6 K. Fig. 5 shows the
results for f ( 2 ) . Additional reductions in the residuals over f(l)
results were found between 150 and 100 mb. Fig. 6 summarizes the
resulting temperature soundings for f(0), f(l), and f(2) if the
weights for the brightness temperatures were n. = 1000. The
improvement of f ( 2 ) over f(l) is apparent between 150 and 100 mb
but elsewhere the differences between the two retrievals are only
a few tenths of a degree. This suggests that it is the shape of
the accuracy function, not the magnitude, that determines where the
variational adjustment will be focused.

Fig. 7 shows part of the temperature soundings T and M between
250 mb and 50 mb. The curve identified by VI is the sounding that
was obtained with the conditions that the weights for the first

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guess temperature were constant with height. The sounding V2
results from the application of f ( 2 ) to the temperature weights.

The first step in the variational analysis of the radiative
transfer equation succeeded in producing a variational algorithm
that could be used to retrieve temperature from the four MSU
channel brightness temperatures given a first guess temperature
sounding. The results showed that the variational retrievals were
subject to the same limitations as are retrievals by other methods,
inability to accurately resolve temperatures near the tropopause
spreads error though the whole retrieved sounding, unless some
temperature accuracy function is employed to focus the retrieval.
The identification of a data set that could be used for a
temperature accuracy function and the derivation of the same is
beyond the scope of this study.

REFERENCE

Achtemeier, G. L. , H. T. Ochs, III, S. Q. Kidder, R. W. Scott, J.
Chen, D. Isard, and B. Chance, 1986: A variational
assimilation method for satellite and conventional data:
Development of basic model for diagnosis of cyclone systems.
NASA Con. Rept. 3981, 223 pp.

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FIGURE CAPTIONS

Figure l. Two typical temperature soundings A and B; the mean of
A and B, sounding M; and true temperature sounding T used for
sensitivity studies of variational temperature retrievals.

Figure 2. Dashed line: differences between the mean or first guess
temperature sounding and true sounding. Solid lines:
differences between variational temperature retrievals and
true temperature sounding for the following choices of
brightness temperature weights; sounding 1 (10) , sounding 2
(100), sounding 3 (1000).

Figure 3 .
Figure 4.
Figure 5.

Curves for hypothesized temperature accuracy functions.
Same as Fig. 2 but for f(l).

Same as Fig. 4 but for f(2).

Figure 6. Differences between first guess and true temperature

(dashed line) and variational temperature retrievals and true
temperature for brightness temperature weights equal to 1000
for f (0) , f (1) , and f (2) .

Figure 7. Parts of temperature soundings T and M between 250 mb
and 50 mb. Sounding VI is temperature retrieval with f (0) and
sounding V2 is temperature retrieval with f(2).

Figure l. Two typical temperature soundings A and B; the mean of
A and B, sounding M; and true temperature sounding T used for
sensitivity studies of variational temperature retrievals.

PRESSURE (UB)

Figure 2. Dashed line: differences between the mean or first guess
temperature sounding and true sounding. Solid lines:
differences between variational temperature retrievals and
true temperature sounding for the following choices of
brightness temperature weights; sounding 1 (10), sounding 2
(100), sounding 3 (1000).

PRESSURE (MB)

PRESSURE (MB)

Figure 5. Same as Fig. 4 but for f(2).

Figure 6. Differences between first guess and true temperature

(dashed line) and variational temperature retrievals and true
temperature for brightness temperature weights equal to 1000
for f (0) , f ( 1) , and f (2) .

PRESSURE (MB)

Figure 7 . Parts of temperature soundings T and M between 250 mb
and 50 mb. Sounding VI is temperature retrieval with f (0) and
sounding V2 is temperature retrieval with f(2).