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)
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