International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004
cover (i.e. higher increase, lower decrease, etc) according to a
threshold that indicates a specific degree of certainty. In the
third phase, the boundaries of local zones within a given urban
region (e.g. urban blocks, census tracts, zip codes) are used to
average the magnitudes of change in each land cover type from
the pixels that fall within individual zones. Next, landscape
metrics are applied to these zonal averages to quantify the
ecological patterns of a given magnitude of change in a given
land cover within a given local zone. The calculated metrics can
then be used to compare urban local areas in terms of the
ecological patterns of change associated with urban
densification and other drivers of internal morphological change
in urban areas. The following sections outline in more detail the
nethods used in the three phases described above.
2.2 MESMA
Spectral mixture analysis (SMA) (Adams et al., 1993) is one of
the techniques proposed to provide a solution to the problem of
mixed pixels in urban satellite imagery with medium spatial
resolution (i.e. 20m or lower). Findings from recent studies
indicate that although SMA provides superior results to
traditional per-pixel classification techniques when applied to
urban imagery, a considerable degree of error may be
associated with SMA models (Ward et al., 2000; Rashed et al.,
2001; Small, 2001). This is because the standard SMA model
implements an invariable set of endmembers to model the
spectra in all the pixels within an image. This assumption fails
to account for the fact that, due to the diversity of urban
materials, the number and type of components within the
satellite sensor's field of view are variable. Roberts et al. (1998)
proposed a solution to this problem by developing a modified
SMA algorithm that allows the number and type of
endmembers to vary for each pixel in an image. This technique
is referred to as multiple endmember spectral mixture analysis
or MEMSA. Thus, MESMA can be described as a modified
linear SMA approach in which many simple SMA models are
first calculated for each pixel in the image. The objective is then
to choose, for every pixel in the image, which model amongst
the candidate models provides the best fit to the pixel spectrum
while producing physically reasonable fractions (Roberts et al.,
1998; Rashed et al., 2003).
The procedure of applying MESMA to urban satellite imagery
is described in detail in Rashed et al (2003). However, it is
worthwhile to highlight here some important aspects of this
procedure to provide a context for the case study described in
this paper. Applying MESMA to a single-date image starts by
selecting a set of candidate endmembers believed to represent a
relatively pure spectral response of the target materials in the
scene. A conceptual framework for urban landscape ecology
devised by Ridd (1995) becomes very useful in this regard.
Ridd suggested that urban areas may be described in terms of
proportions of Vegetation (V), Impervious surfaces (I), and Soil
(S) The process of endmember selection is commenced by
applying the Pixel Purity Index (PPI) method, developed by
Boardman et al. (1995), to screen all the pixels in the image in
terms of their relative purity. In the next step, a series of
standard SMA models are applied based on a variety of possible
combinations of the selected endmembers. The performance of
all models is evaluated so that a smallest subset of candidate
models can be selected for every pixel in the image. A reliable
candidate model is one that produces physically realistic
fractions (i.e. 0-100965 range) and does not exceed a certain
threshold of error. From the selected candidate models, an
optimal model is then identified for each pixel based on the
classical maximal covering problem, originally introduced by
Church and ReVelle (1974). Finally, the fraction values
produced by these optimal models are utilized to map the
abundance of general land cover components in the urban scene
at a given point of time.
The final results from applying MESMA to each single-date
image are validated using aerial photos, following a process
described in Peddle et al. (1999). In this process, a stratified
adaptive cluster sampling (SACS) method is used to identify a
number of test sites on the aerial photos. The accumulation of
corresponding endmember fractions is calculated to estimate the
area of a test site based on the land cover fractions modeled by
MESMA. The accuracy of each endmember fraction is
estimated as the mean of the absolute difference (%) between
actual and modeled cover estimates derived from aerial photos
and MESMA, respectively.
2.3 Change Analysis through Fuzzy Logic
Fuzzy methods in remote sensing have received growing
interest in recent years (Foody, 1999; Foody, 2001). This paper
presents a methodology in which fuzzy models operate in a
subservient role to MESMA models. The application of fuzzy
logic to quantifying magnitudes of urban land cover change is
highly appealing because. of its capability to deal with
uncertainties such as in the case when one cannot accurately
identify a threshold value to separate areas of change from areas
of no-change. Fuzzy logic is a superset of Boolean logic that
has been extended to handle the concept of partial truth values
between “certainly true” and “certainly false.” Fuzzy sets are
sets without sharp boundaries in which the transition between
membership and non-membership is gradual (Zadeh, 1975).
This gradient corresponds to the degree to which an element
(e.g. a pixel) is compatible with the concept represented by a
fuzzy set. Thus, elements may belong to the fuzzy set to a
greater or lesser degree as indicated by a larger or smaller
membership grade.
Once MESMA fractions are calculated for each individual date,
change can be identified in a straightforward way by
subtracting each class of land cover fractions at Date 1 from
their corresponding fractions at Date 2. The resultant fractional
differences represent the temporal change in land cover
abundance in each pixel in absolute terms. To assess the
magnitude of this temporal change, the proposed methodology
applies fuzzy sets representing varying degrees of change (high
increase, lower decrease, etc). This “fuzzification” of change
involves two steps. The first is to translate the concept of
magnitude of change into fuzzy sets using sigmoidal (or S-
curve) membership functions, which are very effective in
modeling continuous, nonlinear phenomena (Cox, 1999). The
case study described in this paper utilizes five fuzzy sets
representing the following levels of change: higher increase,
lower increase, no change, lower decrease, and higher decrease.
The equations used to generate these functions are listed and
discussed in Rashed et al. (2004). The second step of the
fuzzification process is to apply these fuzzy sets to the ‘change-
in-fraction images,” which represent change in endmember
fractions. This step results in new images representing the five
magnitudes of change in each land cover class on a pixel-by-
pixel basis.
2.4 Landscape Metrics
Landscape metrics are indices developed for categorical map
patterns. Their development has been based on both information
theory and fractal geometry (Herold et al., 2002). Categorical
map patterns represent data in which the ecosystem property of
interest is represented as a mosaic of patches. Patches represent
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