Characterizing urban subpixel composition using spectral mixture analysis - Urban Remote Sensing: Monitoring, Synthesis and Modeling in the Urban Environment - page 109

a measure of fit is desired (i.e., residuals for each band) (Adams
et al ., 1986). However, even given data with very fine spectral
resolution (i.e., a large number of bands), there is a practical limit
to the number of endmembers that can be modeled, because a
large enough number of endmembers can model virtually any
measured spectrum, though the resulting combination may be
physicallymeaningless. Thus, accuracy of fraction estimates tends
to decrease as the number of endmembers increases, and as a
result, the maximum number of allowed endmembers is usually
no more than four or five, regardless of the spectral resolution
of the data (Sabol et al ., 1992; Song 2005). The most rigorous
and exhaustive implementation of MESMA is to test all possible
combinations of two-, three-, and four- (e.g., Rashed et al ., 2003;
Powell et al ., 2007), and in some environments, five-endmember
models (e.g., Myint and Okin, 2009). However, limiting allowed
combinations of endmembers reduces computational time and
can improve fraction results. Specification of allowed model
combinations, therefore, should be supported by close inspection
of high resolution reference data to identify the most common
combinations of materials in a specific urban area.
An endmember representing shade is commonly included
in every model in order to account for variations in surface
brightness (Adams et al ., 1986; Dennison and Roberts, 2003;
Rashed et al ., 2003; Powell et al ., 2007). To increase computational
efficiency and spectral separability, models are often limited
to permutations of different classes of materials (e.g., Rashed
et al ., 2003; Powell and Roberts, 2008; Rashed 2008; Myint and
Okin, 2009). However, because impervious materials exhibit high
sub-pixel spatial and spectral heterogeneity, it may be valuable
to test models that include more than one impervious surface
endmember (e.g., Powell et al ., 2007).
Once allowed endmember combinations are identified, every
model is tested for every pixel in the image. Constraints are
defined to select candidate models, most commonly based on the
following: (a) Bright fractions are physically realistic, i.e., ideally
bright fractions are constrained between zero and 1, though
some applications relax the constraint to account for sensor
noise and the imprecision of per-pixel analysis introduced by
the modular transfer function (MTF) of the sensor (Roberts
et al ., 1998b; Townshend et al ., 2000). (b) The maximum shade
fraction is limited (most commonly between 0.50 and 0.80),
because dark pixels can often be modeled by a high shade
fraction and a small fraction of virtually any bright endmember.
(c) A RMS error constraint is also applied, most commonly 2.5%
reflectance, roughly equivalent to 6.4 DN for Landsat data, so that
all candidate models have a minimal goodness-of-fit (Roberts
et al ., 1998b; Okin et al ., 1999; Dennison and Roberts, 2003;
Powell et al ., 2007).
For each pixel, the best model at each level of complexity (i.e.,
two-, three-, four-, or five-endmember) is identified as the model
which meets all constraints and has the minimum RMS error,
as this is assumed to be the best fit (e.g., Painter et al ., 1998).
If no model meets all of the constraints, the pixel remains
unmodeled. At this stage, each pixel could be associated with
several candidate models. Selection of the overall best-fit model
depends on the goals of the analysis, but several factors concern-
ing model complexity should be considered (Fig. 8.3, Step 3).
First, there is a negative correlation between increasing model
complexity and estimated endmember fraction accuracy (Sabol
et al ., 1992). There is also a negative correlation between model
complexity and RMS error; i.e., RMS error tends to decrease
as model complexity increases because additional degrees of

freedom reduce overall error (Okin et al ., 1999; Dennison and
Roberts, 2003; Powell and Roberts, 2008). Finally, there is a pos-
itive correlation between model complexity and computational
expense (Roberts, et al ., 1998b). Several methods of compar-
ing models of different degrees have been proposed, including
ranking models by RMS error, which almost always results in
selection of the most complex model (Painter et al ., 1998); rank-
ing by degree of complexity, where the simplest model is favored
(Roberts et al ., 1998b; Powell et al ., 2007); comparing the relative
magnitude of fractions and eliminating models with very small
fractions (Okin et al ., 1999); and considering more complex
models only when they result in a significant reduction of RMS
error (Roberts et al ., 2003; Powell and Roberts, 2008).
Endmember combinations, model constraints, and model
selection rules are specific to the application. Each of these fac-
tors can be adjusted to better fit the conditions of the scene
and goals of the analyst. For example, a commonly recognized
source of confusion in urban environments is the spectral sim-
ilarity between bright impervious spectra and dry soil spectra
(e.g., Ridd, 1995; Small, 2001; Kressler and Steinnocher, 2001).
One strategy to reduce confusion between impervious and soil
spectra is to segment the dataset into ''urban'' and ''non-urban''
land cover and apply different models to each portion of the
landscape (e.g., Myint and Okin, 2009). Another strategy is to
include impervious endmembers only in the most complex mod-
els (e.g., Powell and Roberts, 2008), as urban environments are
characterized by high spectral complexity relative to other land-
cover types. In a given study, therefore, the specific models tested
and constraints employedmay be adjusted following preliminary
accuracy assessment of fractions.

8.2.4 Mapping fraction images

The output of MESMA is a set of images: (a) an image indicating
the ''winning'' model for each pixel; (b) fractional abundance
images, indicating the estimated fraction of each endmember;
and (c) an image of RMS error. Often, the fraction images are
combined to create abundance images of each generalized class of
materials (i.e., % vegetation, % impervious surfaces, % soil). For
each generalizedmaterial, the pixels values are scaled to represent
the fractional abundance of that component (Fig. 8.3, Step 4).
However, information on the winning model can be retained,
providing multiple ''scales'' of class information, particularly
when hyperspectral data are analyzed (Franke et al ., 2009).
While shade is an integral component of an urban landscape,
it is often a function of topographic effects, shadowing due to
surface roughness, and solar zenith angle. Shade is therefore
usually considered a variant on endmember brightness rather
than a land-cover component itself (Adams et al ., 1986; Smith
et al ., 1990; Rashed et al ., 2003). To more accurately represent
the physical abundance of materials in the scene, the fractions
of each pixel may be shade-normalized; i.e., each non-shade
fraction is divided by the sum of all non-shade fractions used
to model the pixel (Adams et al ., 1986). For example, if a
pixel is modeled by a three-endmember combination of shade,
vegetation, and impervious, the shade-normalized impervious
fraction ( Imp sh_norm ) would be calculated as follows:

Imp
Imp + Veg .

Imp sh_norm = (8.4)