Spatial Modelling and Scaling Issues - Environmental Modelling: Finding Simplicity in Complexity - page 71

coarser data since the low frequency values are excluded
during aggregation. The dominant procedure retains the
values of the original data but may alter the spatial pattern
(including spatial autocorrelation) at coarse resolutions
(Bian, 1997). Categorical data are usually aggregatedusing
thismethod. It should be noted that this procedure creates
bias because types with low proportions will diminish or
disappear while others will increase with scales depending
on the spatial patterns and probability distributions of
the cover types.
The linear convolution is a useful technique for degrad-
ing remotely sensed data. The degraded radiance value
Y ( t ) at point t is the convolutionof the scene radiance Z ( t )
with the system point spread function p ( u ) (Rosenfeld
and Kak, 1982; Collins, 1998):

power law (Braun et al ., 1997; Zhang et al ., 1999; Pradhan
et al ., 2006). Note that the spatial distribution of the frac-
tal dimension has to be determined if we need to scale a
parameter spatially (Zhang et al ., 1999). The limitation of
the fractal method is thatmultifractal properties of objects
may exist over a broad enough range of scales (e.g. Evans
and McClean, 1995), and the unifractal relationship may
break down when the measurement scales are very small
or very large.
The data-fusionmethod here is referred as to techniques
that combine data from multiple scales and sources to
achieve information that is potentially more effective
and accurate at the interest objects. Data fusion in a
geospatial domain often combines diverse sources with
various resolutions and projections into a unified dataset
which includes all of the data points and time steps from
the input datasets (Pohl and Van Genderen, 1998). The
fused data allow for exploitation of the different spatial
and temporal characteristics in environmental variables.
The simple approach to data fusion is to navigate
and register multiple datasets (such as multisensors and
multispectral data). This processing is related to data
reprojection and resampling to create a same format and
scale dataset suitable for both exploratory and inferential
environmental analysis (Jones et al ., 1995).
Sophisticated algorithms of data fusion are used to
generate data with high temporal and spatial resolu-
tions. These data usually contain detailed heterogeneous
biophysical properties suitable for local and regional envi-
ronmental modelling. In remote sensing perspective, data
with high temporal resolutions generally have low spa-
tial resolution, such as daily data in the AVHRR data
(1000m) and MODIS data (500-1000m). In contrast,
data with high spatial resolution are often accompanied
with low temporal resolutions, such as the 30m Landsat
ThematicMapper (TM), which repeats every 16 days. The
temporal resolution in good quality TM data is greatly
degraded because of high frequency of cloud contamina-
tion. To overcome this problem, several fusion methods
are developed to generate datasets with both high spa-
tial and temporal resolutions by fusing the high spatial
resolution data (such as Landsat) with the high tem-
poral frequency of coarse-resolution sensors (such as
MODIS and AVHRR) (e.g. Gao et al . 2006; Scott et al .,
2007). For example, the empirical fusion technique of the
spatial and temporal adaptive reflectance fusion model
(STARFM, Gao et al ., 2006) is developed using one or
several pairs of Landsat TM and MODIS images acquired
on the same day. The developed model is then applied to
simulate daily data at Lansat sale using daily MODIS data

Y ( t ) = p ( u ) Z ( t − u ) du (5.2)

u

Generally, all parameters are vectors representing position
in a two-dimension space. The system point-spread func-
tion can be approximated by a probability distribution,
such as a Gaussian function.
The fractal method provides the possibility of interpo-
lating an object with self-similarity at various scales (both
upscaling and downscaling). In general, a scale-dependent
parameter can be defined as:

F ( d ) = f ( r ) g ( d ) (5.3)

where d is a scaling factor, g ( d ) is a scaling function, f ( r )
represents the reality of an object at scale r and F ( d )isa
parameter value at measurement scale d .
If an object has fractal (unifractal) properties, the
scaling of the object F can be described as a power law
(Mandelbrot, 1982):

Ad α F ( d ) = (5.4)

α = D − L + 1 (5.5)

where D is the fractal dimension, L is a Euclidean dimen-
sion and A is the amplitude or prefactor, which is related
to lacunarity of an object.
This universal scaling law has been widely applied
in biological and environmental research. For example,
metabolic rate for a series of organisms ranging from
the smallest microbes to the largest mammals is a power
function of mass (West et al ., 2000). The number of
species found in a censured patch of habitat on the area
of that patch can be described by a power function of area
(Harte, 2000). Hydrological parameters and morpholog-
ical attributes are clearly special scaling features with this