irection 6. 
adiance at one 
tellite detects 
finite width, 
satellite 
from equation 
unction H(X): 
(3.3) 
ponse at wave- 
assband wave- 
from equations 
e atmospheric 
bsolute, 
on of Planck's 
perature if the 
ssume the sea 
hat the emis- 
1 .0. 
pheric trans 
rom the system 
ver, when the 
system was not 
e based upon 
Wark et.al. 
973) was deve- 
e is that the 
n as the pro- 
vidual absor- 
or, nitrogen 
treated as a 
yers, in each of 
and the water 
Important cons- 
the product of 
aa transmittan 
ce calculated 
ceb and Neuen- 
transmittance 
sorbing médium, 
ce, p, for each 
nosphere is sub 
calculated 
s et.al. (1976). 
explained from 
s (sb), and 
c atmospheric 
Roberts et.al. 
continua trans- 
ng expressions: 
) -1 ) )dp (3.4.a) 
(3.4.b) 
t wavelength Al 
nadir 
r vapor (g/kg) 
atmosphere to 
ived from an 
expression similiar to the water continuum transmit 
tance (Weinreb and Hill, 1980): 
P 
t ( A, P ) « C n U,T=296 K) sec (0 ) / p / T dp (3.5) 
0 
where the notation of equation 3.5 is similiar to the 
notation of 3.4. C N is the Nitrogen absorption coef 
ficients at a reference temperature T=296 K. 
The transmittance of the uniformly mixed gases, t , 
is calculated from a simplified method of that used 
in the LOWTRAN system (Kneizys et.al., 1983, McClat- 
chey et.al., 1972). The uniformly mixed gases are 
treated as one unit with fixed concentrations from 
McClatchey et.al. (1972) as input to the model. 
Knowing the transmittances for all the atmospheric 
radiative absorbing factors, the total atmospheric 
transmittance at wavelength A , pressure, P, and tempe 
rature T is given by: 
t ( A,P,T) = t ( X,P,T)t ( A,P,T)t (,X,P,T) (3.6) 
W N U 
3.3 Emitted atmospheric radiance 
From the equation of radiative transfer (3.2), an 
expression for the emitted atmospheric radiance can be 
derived (Dalu et.al., 1979). Equation 3.2 can be writ 
ten on the following form: 
R(A 0 ) = B(A 0 ,T s )t(x 0 ,P 0 )+{l-t(X o ,p o )} B(A 0/ ,T(P)) (3.7) 
The notation of equation 3.7 follows the notation of 
quation 3.2. B is the weighted, averaged atmospheric 
Planck radiance given from: 
1 
/ EÎX ,T(p)}dt 
- ta;,p 0 ) 
BÎVT(p)} = 1 (3 - 8) 
/ dt 
In equation 3.7 and 3.8, the radiance is calculated at 
the centre wavelengths, A^, for each of the passbands. 
In numerical implementation of the algorithm, all 
the integrals are implemented as sums of finite width 
intervals across the actual passband. The weightfactor 
H (A ) is given for each of the intervals:. h=H(i), i=l, 
—-N, where N denotes the number of subintervals of 
passband. 
3.4 Tests of method 
In testing the algorithm, the transmittances and radi 
ances were calculated for different wavelengths and 
atmospheric models. The calculated values were compar 
ed to corresponding values reported by others. The 
applied atmospheric models were taken from radiosondes 
data from Norwegian meteorological stations. In table 
2, the comparisons for two different atmospheres are 
listed. The first atmosphere is dry atmosphere (0.5 cm 
water), and the second is an atmosphere containing 
approximately 2 cm water. These two atmospheres are 
fairly representative for atmospheres observed in the 
Arctic. All results in table 2 are for nadir viewing. 
From the comparisons presented in table 2, there is 
an fairly good agreement between calculated and repor 
ted transmittances. Therefore, to conclude,the deve 
loped routines for calculating the total atmospheric 
transmittance seem to give reasonable results. 
Knowing the transmittances and the atmospheric radi 
ance given by equation 3.8, the absolute surface skin 
temperature can be derived from inversion of Planck's 
radiation law. The last section of this chapter will 
deal with calculations of transmittance versus scan 
angle for different atmospheres, and the effect upon 
the surface temperature. 
In figure 3.1 are presented the total transmittance 
Table 2. Comparison of calculated and reported trans 
mittances . 
tw,c “ calculated water transmittance 
tWjWH water transmittance from Weinreb and Hill. 
t w p - water transmittance from Prabhakara et.al. 
tN c - calculated Nitrogen transmittance 
" Nitrogen transmittance from Weinreb and Hill 
wave 
length 
(urn) 
0.5 
cm water 
^w ,c 
fc w,WH 
t w ,P 
fc N,c 
tN,WH 
10.5 
.979 
.967 
.946 
1.0 
1.0 
11.5 
.955 
.933 
.968 
.945 
.966 
approximately 2 cm water 
3.5 
.880 
.740 
— 
1.0 
1.0 
3.9 
.970 
.920 
— 
.958 
.966 
10.5 
.841 
— 
.812 
— 
— 
11.5 
.748 
— 
.738 
— 
— 
versus scan angle for three different atmospheres for 
the AVHRR channel 4. In figure 3.2 are illustrated 
the introduced errors on the surface temperature by 
ignoring the varying scan angle for the three atmos 
pheres presented in figure 3.1. 
SGan angle (in degrees) 
Figure 3.1. Transmittance versus scan angle for three 
atmospheres : 
1: 0.5 cm vater content 
2: 1.3 cm water content 
3: 1.5 cm water content. 
From the figure 3.1 it is seen that the transmittance 
variations are negligible for scan angles less than ^ 
40 degrees from nadir,for the dry polar atmospheres. 
However, the effect is increasing for angles below 40 
degrees as the water content increase from 0.5 to 1.5 
cm water. 
From the curves in figure 3.2, the observed error 
introduced by ignoring the across-scan transmittance 
variations is less than approximately 0.6 K for scan 
angles less than 60 degrees for a polar atmosphere 
with water content less than approximately 1.3 cm. 
The results agree well with results reported by 
Smith et.al. (1970). They report errors of order 0.7 
K for a polar atmosphere and across-scan angles less 
than 60 degrees. 
4 AVHRR-DATA SET APPLICATIONS 
The major limitations in applying optical satellite