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123 


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 



124 


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 



134 


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).