is defined called ”image endmembers”, representing the 
spectral reflectance of the different cover types. Different 
approaches have been used to define these endmembers. 
These include pixel vectors, training areas, laboratory data, 
or a combination or these methods. When mixed using the 
appropriate rule, these endmembers reproduce all of the 
pixel spectra. The maximum number of endmembers is 
limited by the number of spectral bands of the satellite 
image. Due to the fact that some bands are highly 
correlated, the number of endmembers necessary to explain 
an image adequately is in general smaller than the number 
of bands. To identify the intrinsic dimensionality of the 
data, the principal component analysis may be used. The 
number of components showing meaningful information 
equals the smallest number of endmembers needed to 
construct a linear mixture model (Settle and Drake, 1993). 
The endmembers are selected from areas which show only 
or almost only the surface material in question, and which 
receive maximum illumination. In addition an endmember 
called ”shade” is introduced, which accounts for variations 
in lighting at all scales (e.g. changes in incidence angles, 
shadows cast by topographic features, subpixel shadows 
cast by trees, and so forth). Once the endmembers are 
defined the fractions of each endmember in each pixel may 
be calculated by applying the appropriate mixing rule. A 
general equation for mixing is (Adams et al., 1989): 
N 
DN.= YF -DN, +E, 
(1) 
n=1 
where 
N 
YES Q) 
n=l 
with 
DN, radiance in channel c, 
N number of endmembers , 
2 fraction of endmember n, 
DN,. radiance of endmember n in channel c, 
E. error for channel c of the fit of N spectral 
endmembers. 
Equation (1) converts the DN value of each pixel in each 
channel to the equivalent fraction (F,) of each endmember 
as defined by the endmembers (DN,.). The error (E,) 
accounts for that part of the DN-value which is not 
described by the mixing rule. Equation (2) introduces the 
constraint that all fractions of a pixel must sum to one. 
Three ways exist to evaluate the results of the spectral 
mixture analysis. These are the visual analysis, the 
calculation of the root-mean-squared (rms) error, and the 
calculation of the fraction overflow (Adams et al., 1989). 
With the visual analysis of the fraction images, the analyst 
determines whether they results consistent with other 
information existing about the area in question. If the 
patterns do not correspond with the additional information 
obtained by ground truthing or other sources then the 
model constructed may not be correct. 
380 
The second test is the calculation of the rms error. It is 
based on the E, term of equation 1, squared and summed 
over all M image channels (see (3)) (Adams et al., 1989), 
ki 1/2 
o CE, 
c=l 
with 
€ root-mean-squared (rms) error 
k number of Channels 
The rms error is calculated for every pixel individually and 
can also be visualized as an image. It may also be 
calculated for the whole image, showing the overall rms 
error. A small rms error is an indication of a 
mathematically good model. A high rms error indicates that 
the model has not been constructed correctly. 
The third test is the computation of the fraction overflow. 
Reason dictates that the fractions of the land cover 
components should lie between zero and one, but if the 
model is not constructed correctly fractions may fall 
outside this range. As the endmembers are supposed to 
represent 100 96 of the land cover in question, any pixel 
having a higher portion of the land cover as compared to 
the endmember, will have a fraction higher than one. To 
satisfy the constraint that all the fractions of a pixel must 
sum to one, the fraction of another endmember of this pixel 
will be below zero. 
If the model is not satisfactory according to the tests 
described above, the endmembers must either be changed, 
deleted, or additional endmembers must be defined. The 
following rules aid in the selection of new endmembers. An 
overflow in a fraction image is an indication for a pixel, 
which represents the land cover better then the pixel used 
for the definition of this endmember up to now. An 
overflow and a high rms error in a pixel may be due to an 
unmodelled endmember which is represented by that pixel 
(Adams et al., 1989). 
2.2. Results 
The endmembers selected for the analysis represent 
vegetation, water, built-up areas, and shadow. The 
limitation to this number of endmembers was confirmed by 
the principal component analysis. To define the 
endmembers, pixel vectors were examined which only or 
nearly only represent the land cover in question. The pixel 
vector chosen for vegetation has a very high value in the 
near infrared band 4, as this band is best for picking up 
vegetation. The endmember for water is defined by a pixel 
vector located in a faster flowing part of the Danube in the 
North of Vienna, and the endmember for built-up areas is 
defined by a pixel vector located in an administration 
building. As shadow represents areas not or badly 
illuminated, this endmember was defined as zero in all six 
bands although it is possible that the shade endmember is 
greater than zero, owing to instrumentation offsets and/or 
gain, skylight scattering, and so forth (Adams and Smith, 
1986). Endmembers were selected for both images, using 
the guidelines described above. Table 1 shows the values of 
these endmembers for both images. As can be expected the 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B7. Vienna 1996