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Explicit Consideration of Multiple
Landscape Scales While Selecting
Spatial Resolutions
-
John B. Collins and Curtis E. Woodcock
Abstract. - The information content of a digital image - defined as
the spatial variation of the image data - depends on data resolution.
Spatial data may vary at a number of different scales simultaneously,
so the choice of an appropriate resolution depends on the information
content at the landscape scale relevant to a particular application. A
useful framework for studying variation at different scales in digital
images is provided by a nested hierarchical model of the landscape.
The landscape is modeled as being composed of spatial units of a typical size, which are themselves composed of smaller units, and so on.
The overall variation of the image data can be decomposed into variation due to effects of different levels of such a hierarchy. The hierarchical decomposition of the variation in a digital image is reflected by a
similar hierarchical decomposition of the scene semivariance. Decomposed semivariograms are used to assess the relationships between data
resolution and information content at particular landscape scales.
INTRODUCTION
The issue of scale is of fundamental importance to disciplines concerned
with phenomena which vary spatially. Yet the relationship between the scale at
which a phenomenon is observed and that phenomenon's characteristic scale of
variation is not completely understood. In this discussion, scale of observation
is defined as the spatial interval over which observations are made. When data
are collected by remote sensing devices, this term corresponds to sensor resolution, or size of the Instantaneous Field of View (IFOV). The characteristic
scale of a phenomenon is the range of resolutions over which that phenomenon
may be characterized. These definitions correspond to those proposed by Quattrochi ( 1993).
Research Assistant, Boston Liniversiry Department of Geography. Boston. MA,
Associm Professor of Geography. Boston Universily, Boston, MA.
In remote sensing investigations, resolution is an important consideration.
The information contained in an image, which can be identified as the overall
variance of the image data, declines as resolution becomes more coarse. The
simplest view of the issue is one which regards the selection of an appropriate
resolution as a compromise between greater information content (fine resolu-.
tions) and more manageable data volume (coarse resolutions). But for some
applications, greater spatial variation is not always desireable. For example,
classification of remotely sensed imagery is most successful when pixel size is
large enough that some amount of averaging of within-class variation occurs
(Markham and Townshend, 1981).
The fact that variation of spatial data occurs at multiple scales has an
explanation in the field of landscape ecology. It is becoming an accepted tenet
of that field that landscapes are often arranged hierarchically (O'Neill et al.,
1986). That is, landscapes are often composed of objects which are internally
homogeneous and externally variable with respect to spatially varying
processes. These objects may be composed of smaller objects, and so on. For
example, a landscape may be composed of a number of watersheds, each containing a collection of vegetation stands, which each contain individual plants.
Associated with each level of such a hierarchy is a set of processes acting at
that level. The range of sizes of objects on a given level approximately defines
the spatial frequencies at which those processes vary - i.e. their characteristic
scales. So multiple scales of variation of spatial data arise from a scene's
hierarchical structure.
The choice of an appropriate resolution for a particular investigation
depends on prior knowledge of the characteristic scale of variation of the
phenomenon of interest. This paper presents methods to help make such a
choice under explicit consideration of the hierarchical nature of landscapes. A
review of the use of hierarchical partitionings of spatial data is presented, followed by an explanation of geostatistical methods for investigating resolutiondependent characteristics of digital images.
SPATIAL HIERARCHIES
A number of studies use a hierarchical model to identify multiple
landscape scales. A common paradigm involves partitioning a scene into
objects of regular size and shape - usually squares or rectangles - each of
which are characterized by some aggregation of the underlying spatially distributed variable (Grieg-Smith, 1963; Townshend and Justice, 1995). Then there
is a clear and explicit relationship between any level of the hierarchy and a
scale of observation. By examining some attribute of each level of the hierarchy - especially the data variance - multiple scales of variation may be
revealed.
An alternate approach is the use of hierarchies of irregular-shaped objects,
such as political boundaries (Moellering and Tobler, 1972). The characteristic
scales associated with any level are identified with the range of sizes of objects
on that level. However, the hierarchies so constructed can be rather arbitrary.
Woodcock et al. (1994) describe a forest mapping project which makes use of a
hierarchical model dividing the landscape into ecologically meaningful units.
(see also Woodcock and Harward, 1992; Collins et al., 1995). Under this
model, the entire image is considered to be composed of a number of vegetation classes such as conifer, hardwood, and brush. Each of these can be broken down into smaller homogeneous patches, referred to as regions. Within
the conifer class, for example, a region corresponds to a forest stand. Regions,
in turn, are collections of individual plants. Such a hierarchy can be associated
with ecosystem processes, which tend to be more homogeneous within units
and heterogeneous between units at any level.
The partitioning of a scene into ecologically meaningful units prior to
analysis constitutes explicit identification of the characteristic scales under
study. Knowledge of these characteristic scales is important for assessing the
relationship between resolution and information content for a given application.
Statistical Analysis of Hierarchal Data
A general and mathematically precise definition of a hierarchical scene
model is needed. Consider a regionalized variable Z which depends on vector
location s within some domain set D!o). When discussing analysis of digital
images, D(o) is a subset of two-dimensional space R ~ and
, Z could be radiance
in some wavelength band or any other spatially-varying quantity. The hierarchical model consists of partitioning the domain set D(o) into a family of objects
(sets) {D(*)},which form level 1 of a hierarchy. Each of the level-1 objects
may be divided into a number of level-2 objects, which can collectively be
referred to as
Further partitions proceed in an obvious manner. If level
n is the highest level of the hierarchy, then the family of domain sets {D(,)}
is
the collection of infitesimal points on which Z is defined. Any point s E D(o)
will be contained in exactly one set associated with each level of the hierarchy.
Such a set of level k is D ( k ) ( ~ )This
.
terminology is illustrated by Figure 1.
Figure 1. A hierarchical partitioning of a scene. A. The set of level-1 objects
forming a partition of D(0). B. The level-2 objects {D(2)}further partitioning the level-1
objects.
Determination of the relative contributions of different levels of a hierarchy to overall variation in a data set is the goal of Hierarchical Analysis of
Variance (Moellering and Tobler, 1972). Under the Hierarchical ANOVA
model, the observed value at a point in the scene is considered equal to the
overall mean for the area, plus a deviation, or effect, characteristic of each level
o_f the big-archy. The effect of level k of a hierarchy at point s is defined as
D(k)(s)-D(k-l)(~),
where D(k)(s)is the mean value of Z over the level-k object
containing s . Note that each effect constitutes a regionalized variable in its
own right. Further elaboration of the Hierarchical ANOVA model can be
found in standard texts on the subject (e.g. Dunn and Clark, 1974). The model
leads to the following important result.
where o * ~represents the overall variance of Z , and 02(i)represents the variance of the effect of level i . The overall data variance decomposes into the
sum of variances of each effect. Unfortunately, the ability to make statistical
inferences from a hierarchical partitioning of variance relies on the assumption
of independence of observations. Since this assumption is badly violated for
most spatial data sets, the technique is suited only for descriptive purposes.
When the regionalized variable Z exhibits appropriate stationarity characteristics, its variation is described by the scene semivariogram (Isaaks and
Srivastava, 1989; Cressie, 1993).
where h is a vector lag. Similarly, relationships between two regionalized variables Y and Z are described by the cross-semivariogram.
The parameters describing the scene semivariogram - its range, sill, and
derivative at the origin - can be shown to be related to quantifiable attributes
of a scene (Woodcock et al. 1988a+b).
Since the semivariogram provides a better characterization of the variation
of a regionalized variable than does the variance, it is useful to ask whether
there is a geostatistical analogy to Eq. 1. Collins et al. (1995) present an
example to show that under the Hierarchical ANOVA model, the scene
semivariogram decomposes as
where yZ represents the overall scene semivariogram, y(, represents the
semivariogram associated with the effect of level i of the hierarchy, and yG)(k)
is the cross-semivariogram between the effects of levels j and k. It is interesting to note that when overall scene variance is considered (Eq. 1) all crossproduct terms vanish, but that this is not the case for spatial variance. The
cross-semivariograms in (4) arise as the result of fine-scale continuity across
boundaries of larger-scale objects (Collins and Woodcock, manuscript in
preparation). They generally have small magnitudes and relatively short
ranges, and approach zero at large lags. The relevance of Eq. 4 to the present
study is its indication that semivariograms associated with particular levels of a
hierarchy - i.e. with different characteristic scales in the landscape - may be
discussed individually.
EFFECT OF RESOLUTION ON SPATIAL STRUCTURE
The scale-specific semivariograms included in Eq. 4 may be used to
retrieve quantitative information about scene variation at arbitrary resolutions.
In geostatistical parlance, sensor resolution - the size and shape of the IFOV is called the support of the measurements. There exists a well-developed
theory of the effect of changes of support on regionalized variables (Rendu,
1978; Zhang et al., 1990), and its applications in the context of remote sensing
(Jupp et al., 1988+1989; Atkinson and Curran, 1995). This section discusses
some of the ways in which information relating to scale of observation can be
derived from semivariograms. Notationally, subscripts indicative of levels in a
hierarchy are dropped, as the ideas apply to any regionalized variable with
finite variance. The relevance to multiple scales of variation comes from the
indication of Eq. 4 that these methods can be applied to scale-specific
semivariograms.
Regularization and Deregularization
As was noted above, the effect of a level in a hierarchy is a regionalized
variable, and is defined at all points in a scene. However, point-scale observations are never available from a remote sensing device, since there is always
some averaging of scene radiance over the sensor IFOV. That is, the data and
the semivariograms derived from it are regularized. It is desireable to derive
the point-scale, or punctual semivariograms from the regularized ones.
In the theoretical development which follows, the support of observations
is modeled as a set in the domain space D(o) over which the continuouslyvarying values of some effect are averaged. A useful quantity is the average
punctual semivariance between two sets in the scene, denoted A and B .
"XA ,B ) =
1
J J
Mes(A )Mes(B ) A
y(z-z') dz' dz
Mes(*) denotes the Lebesque measure of a set (i.e. its area in this context).
This definition is particularly useful when the sets A and B are shifts of a sensor IFOV. Specifically, let w denote some support, and let w h denote its translation by distance h . The regularized semivariogram at a lag of h can be
found from the punctual semivariogram by (Rendu, 1978)
yw(h) =y(w;wh) -y(w,w)
(6)
This relationship cannot be inverted to find the punctual semivariogram from
its regularized counterpart. But Atkinson and Curran (1995) present a numerical technique based on this relationship which may be used to estimate it.
The effects of regularization are most pronounced on higher levels of a
scene hierarchy, where the characteristic scales of variation are the smallest.
When scales of variability are much smaller than the support, 'microstructures'
in the data are undetected and spatial variation is lost. This phenomenon plus
the effect of measurement error can possibly lead to an observed 'nugget
effect' or a non-zero intercept of a regularized semivariogram. At scales larger
than that at which data are collected, the nugget effect shows no spatial structure. This leads to a simple relationship between the nugget effect observed
via support w (Nw) and that observed at a larger support W (Nw):
The nugget effect behaves differently under changes of support than does
the spatially correlated component of variation. Thus it is useful to express the
regularized semivariogram as (Zhang et al., 1990):
where j, is a regularized zero-nugget semivariogram, from which a punctual
zero-nugget semivariogram y* may be estimated. Then using 6, 7, and 8 the
semivariogram for any support W can be expressed as
Since parameters describing semivariograms relate to properties of the scene
under study, the ability to determine semivariograms for arbitrary supports is
useful. When used with a hierarchical decomposition of a scene
semivariogram, it can yield useful information concerning the relationship
between image resolution and spatial variation at different characteristic scales.
Determination of Variance at Arbitrary Resolutions
As was mentioned above, the overall scene variance is perhaps the best
measure of its information content. As a rule, the variance decreases as the
size of support increases. But the precise rate of decrease depends on the autocorrelation structure of the image, as revealed by the semivariogram. If autocorrelation between points is strong, then regularization will have little effect
since averaging over the IFOV will tend to occur between points with similar
values.
This assertion is justified mathematically. Let D(o) represent an area being
studied. If this area has been sampled on supports w , then the variance of supports W within L) (0) is (Zhang et al., 1990)
r
VW,D(,)) =
-I
The first term accounts for the nugget effect among the supports W , and the
last two terms deal with the spatial1y-correlated component of variation. Eq.
10 gives the variance for arbitrary resolutions, and so is useful for a number of
purposes.
CONCLUSION
Methods to determine the effect of sensor resolution on different
landscape scales have been presented. Characteristic scales of variation can be
identified by partitioning a scene hierarchically into ecologically meaningful
units. In order not to restrict the discussion to a particular classification of
scene objects, a general hierarchical model has been used. Use of different
landscape hierarchies may be called for in different situations, but the ideas discussed here continue to apply.
Semivariograms may be calculated characterizing the spatial variation at
each landscape scale, and may be used to derive information about scene structure at arbitrary resolutions. Throughout the discussion above, the assumption
has been made that the data are second-order stationary at all scales, a requirement for the calculation of semivariograms. This assumption needs to be
justified in order for the analysis of semivariance to be appropriate. If this condition is met, the relationships between resolution and semivariance presented
here are general with respect to directional qualities of variation. All results
apply equally for isotropic and non-isotropic semivariograms.
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