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