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recorded by the sensor for each pixel (Fig. 8.2b) is the weighted
sum of the pure spectra of each material in the pixel's IFOV
(Fig. 8.2c). The goal is SMA is to ''unmix'' the relative propor-
tion of endmembers that best model the measured spectrum; in
general, these are assumed to be correlated with the area covered
by each material present (Fig. 8.2d). The output of SMA is a
set of images representing the fractional cover of each endmem-
ber, with digital numbers (
DN
) values typically scaled between
zero and 1 (i.e., zero representing ''not present'' and 1 repre-
senting 100% cover), in addition to an image that summarizes
the difference between the modeled and measured spectrum
(Adams,
et al
., 1986), usually in terms of the root mean square
(RMS) error.
For a given pixel, SMA can be described mathematically
as follows (Adams
et al
., 1986; Smith
et al
., 1990; Roberts
et al
., 1998a):
Despite the strengths of SMA to estimate sub-pixel com-
ponents, the implementation of SMA is based on several sim-
plifications that limit its applicability in urban environments.
First, all pixels in the scene are modeled with an invariant set of
endmembers, usually between three and five (Powell
et al
., 2007;
Franke
et al
. 2009; Myint andOkin, 2009). As a result, the selected
endmembers may not effectively model all elements in the scene
or a pixel may be modeled with endmembers that are not present
in its IFOV. Either situation results in decreased accuracy of
estimated fractions (Sabol
et al
., 1992; Roberts
et al
., 1998b).
Second, simple SMA assumes that the spectral properties of each
endmember are constant across the scene, while only the frac-
tions themselves vary; therefore, the procedure cannot effectively
account for endmember variability (Song, 2005). This can be
especially problematic in urban landscapes, which are generally
characterized by a high degree of within-class variability, espe-
cially for impervious materials (Kressler and Steinnocher, 2001;
Herold
et al
., 2004; Franke
et al
. 2009; Myint and Okin, 2009;
Weng and Lu, 2009).
An extension of simple SMA that addresses these challenges is
multiple endmember spectral mixture analysis (MESMA), which
allows both the number and type of endmembers to vary on a per-
pixel basis (Roberts
et al
., 1998b). Implementation of MESMA
involves building a large, comprehensive spectral library that
accounts for the spectral variability of the scene and applying a
series of simple SMA models, based on different combinations
of library endmembers, to every pixel in the image. The best
model is selected for each pixel based on several measures
of fit, which can include the allowed range of endmember
fractions, maximum residuals allowed for each wavelength, and
the maximum allowed RMS error (Roberts
et al
., 1998b). Once
the best model for each pixel is selected, the per-pixel fractions are
often generalized into land-cover components of interest (e.g.,
vegetation, impervious surface, soil), and an image of fractional
cover is generated for each component. Fractions generated
from MESMA are usually compared to fractions derived from
high spatial resolution imagery (e.g., aerial photographs), and
agreement between modeled and reference fraction cover is used
to refine endmember selection and/or model constraints until
a satisfactory level of accuracy is achieved (Rashed
et al
., 2003;
Powell
et al
., 2007; Powell and Roberts, 2008; Rashed, 2008).
MESMAhas been successfully applied tomoderate-resolution
imagery to map subpixel fractional V-I-S components in sev-
eral environments, including Los Angeles (Rashed
et al
., 2003;
Rashed, 2008), Phoenix (Myint and Okin, 2009), and several
cities in the Amazon region of Brazil (Powell
et al
., 2007; Powell
and Roberts, 2008). Additionally, Franke
et al
. (2009) applied
MESMA to hyperspectral imagery to generate a hierarchical
map of urban land cover, ranging from pervious/impervious
layers at the coarsest level to maps of specific urban materials
and vegetation species at the finest level. Steps to implement
MESMA for mapping V-I-S components in urban environments
are summarized in Fig. 8.3 and discussed in detail below.
K
DN
i
=
F
j
ยท
DN
i
,
j
+
e
i
(8.1)
j
=
1
where
DN
i
is the measured value of a pixel in band
i
in DNs
recorded by the sensor or in units of radiance or reflectance,
F
j
is the fraction of endmember
j
present in the pixel's IFOV,
DN
ij
is the value of the endmember
j
in band
i
,and
e
i
is the residual,
or the difference between observed and modeled DNs for band
i
.Thereare
N
bands in the dataset, and
K
endmembers in the
mixture model. In addition, the mixing equation is subject to the
following constraint:
K
F
j
=
1,
(8.2)
j
=
1
i.e., the fraction of endmembers for each pixel must sum to 1 (or
100% cover). Per-pixel RMS error is effectively the mean residual
across all bands, given by:
N
e
i
i
=
1
RMSE
=
.
(8.3)
N
SMA models consist of a set of endmembers that represent
the basic spectral components of the landscape, in addition to
an endmember that represents shade to account for variations
in brightness due to illumination, topography, or other sources
of surface variability (Adams
et al
., 1986; Smith
et al
., 1990;
Dennison
et al
., 2004). If pixel values in an image are recorded
in units of reflectance, the shade endmember will have values of
zero in all bands; for an image recorded in units of radiance or
raw DNs, the shade endmember will consist of non-zero values
related to atmospheric scattering.
Givenanimagethatconsistsof
N
bands, the goal is to
simultaneously solve a total of
N
+
1 linear equations (Equations
8.1 and 8.2 above) for each pixel, thereby estimating the fractions
of each endmember present (
F
j
), while minimizing the residuals
(
e
i
) on a per-pixel basis. For a given application, the validity of
the SMA model is assessed based on (a) whether the estimated
fractions are physically realistic, i.e., fraction values are between
0 and 1, (b) the overall goodness of fit, usually measured by
RMS error, and (c) the spatial distribution of model error
throughout the scene. Based on such assessments of model
fit, model parameters may be adjusted until the results are
satisfactory (Roberts
et al
., 1998a).
8.2.2
Endmember selection
Many researchers have noted that the most important step in
applying SMA is the selection of appropriate endmembers (e.g.,
Tompkins,
et al
., 1997) (Fig. 8.3, Step 1). Endmembers used
in SMA may be collected in a laboratory or field setting using
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