size of 100 by 1080 pixels. For the remaining subarea sizes, depending on 
position, they reach about 80°. Thus it becomes evident that - eigen- 
values and eigenvectors must be considered in the evaluation of the dis- 
crimination of land cover types: the relatively high orientation stability 
of the error ellipsoid for very large test areas is practically irrelevant 
because of the concurrently appearing large eigenvalues. For the smaller 
subareas - including the areas of the approx. 100 x 100 pixel size found 
to be the optimum with respect to the eigenvalues - the discrimination of 
land cover type evidently becomes a position function. 
h.h Function of the Eigenvectors: Quotients of the Direction Cosines 
The calculation of multispectrally effective atmospheric influences and 
their utilization for the purpose of correction is possible by means of 
the Covariance Matrix Method (MM) [2] - although for the determination of 
corrected ratios only. Setting the atmospheric correction v. for channel 7 
to O0 in all area subdivisions to be investigated yields as 4 relative 
atmospheric correction for the other channels 
  
  
x. 
y, = 9; - 97 2 resp. with z; = zi 
7 7 
v. zg.-qg'Z. (474-1) 
09797 3 
wherein for the area subdivision under consideration: 
j = channels 4, 5, 6 
9; = the mean gray values, channelwise 
X; - the components of the all-positive eigenvector x 
a = the quotients of the direction cosines Xy/ X7 Xp/ X4 Xg/ X7* 
As the object of the following considerations is to observe the effect of 
parameter variation and not to study error propagation, only the total 
differential is formed of equation (4.4.1). This leads to: 
dv, = de, = Zz; : dg - 9, : dz; (4.4.2) 
for the variation of atmospheric correction. 
To determine the plausible solution interval of z,the all-positive eigen- 
vector x is utilized first, hence: 
Z; > 0 (4.4.3) 
As equation (4.4.2) is underdetermined, dz, must be pre-assigned. On the 
basis of the assumptions that the error-spetific variation of the mean gray 
values dg. and dg is equal to 0 and the maximum permissible value for 
dv; is equal to 1, it follows from equation (4.4.2) that 
dz, = ~- Ei 
J 97 
Supposing that, for the determination of the upper bound to the solution of 
z., the variation for the mean gray values and the atmospheric correction 
does not exceed 1 gray value, equation (4.4.2) yields, in conjunction with 
(4.4.3) and (4.4.4), 
0 <Z; S. d (4.4.5) 
(4.4.4) 
176