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Landscape Ecology 13: 111–131, 1998.
c 1998 Kluwer Academic Publishers. Printed in the Netherlands.
Analysis of fine-scale spatial pattern of a grassland from remotely-sensed
imagery and field collected data
Agustín Lobo1, Kirk Moloney2, Oscar Chic3 and Nona Chiariello4
1
Institut de Ciències de la Terra (CSIC), Martí Franqués s/n, 08028 Barcelona, Spain; 2 Department of Botany,
Iowa State University, Ames, Iowa 50011-10220, USA; 3 Institut de Ciències del Mar, P. Juan de Borbón s/n, 08039
Barcelona, Spain; 4 Department of Biological Sciences, Stanford University, Stanford, California 94305, USA
(Received: 3 July 1997; accepted 2 August 1997)
Keywords: vegetation pattern, serpentine grassland, disturbance, Thomomys bottae, geostatistics, remote sensing,
fractal, spatial simulation, NDVI, Fast Fourier Transform
Abstract
An important practical problem in the analysis of spatial pattern in ecological systems is that requires spatiallyintensive data, with both fine resolution and large extent. Such information is often difficult to obtain from
field-measured variables. Digital imagery can offer a valuable, alternative source of information in the analysis
of ecological pattern. In the present paper, we use remotely-sensed imagery to provide a link between field-based
information and spatially-explicit modeling of ecological processes. We analyzed one digitized color infrared
aerial photograph of a serpentine grassland to develop a detailed digital map of land cover categories (31.24 m 50.04 m of extent and 135 mm of resolution), and an image of vegetation index (proportional to the amount of
green biomass cover in the field). We conducted a variogram analysis of the spatial pattern of both field-measured
(microtopography, soil depth) and image-derived (land cover map, vegetation index, gopher disturbance) landscape
variables, and used a statistical simulation method to produce random realizations of the image of vegetation index
based upon our characterization of its spatial structure. The analysis revealed strong relationships in the spatial
distribution of the ecological variables (e.g., gopher mounds and perennial grasses are found primarily on deeper
soils) and a non-fractal nested spatial pattern in the distribution of green biomass as measured by the vegetation
index. The spatial pattern of the vegetation index was composed of three basic components: an exponential trend
from 0 m to 4 m, which is related to local ecological processes, a linear trend at broader scales, which is related to
a general change in topography across the study site, and a superimposed periodic structure, which is related to the
regular spacing of deeper soils within the study site. Simulations of the image of vegetation index confirmed our
interpretation of the variograms. The simulations also illustrated the limits of statistical analysis and interpolations
based solely on the semivariogram, because they cannot adequately characterize spatial discontinuities.
Introduction
Spatial patterns in ecological systems are the result of
an interaction among dynamical processes operating
across a range of spatial and temporal scales (Wiens
1989; Urban et al. 1987). Spatial issues have interested ecologists for a long time (i.e., the concept of the
Address for correspondence: Agustín Lobo, Institut de Ciències
de la Terra (CSIC), Martí Franqués s/n, 08028 Barcelona, Spain;
Tel. 34 3 330 2716; Fax. 34 3 411 0012; E-mail: alobo@ija.csic.es,
alobo@eno.princeton.edu
landscape as a shifting mosaic of patches can be traced
at least as far back as Watt’s (1947) seminal work on
pattern and process), and have been receiving increasing attention by ecologists over the last several years.
For instance, the idea of the landscape as a shifting
mosaic began to receive serious theoretical and experimental attention in the mid-1970’s (Levin and Paine
1974; Whittaker and Levin 1977; Steele 1978). More
recently, landscape ecology has developed this concept further to view the landscape as a complex, hierarchically organized, spatio-temporal mosaic, where
112
there are strong relationships coupling spatial pattern
to process (Forman and Godron 1986; Turner 1989;
Turner et al. 1991). This work has shown that, not
only does the spatial pattern of a landscape result from
dynamic ecological processes, but landscape pattern
can also directly influence the dynamics of the system
(Turner 1989; Levin 1992): biogeochemical cycling is
modified by landscape structure (Pastor and Post 1988;
Turner 1987); many life history traits can be interpreted as being an evolutionary response to spatial heterogeneity (Levin 1992); and the ecology of mobile
organisms is dependent on the spatial pattern of the
landscape (i.e., O’Niell et al. 1988; Milne et al. 1989;
and Palmer 1992), a fact with profound consequences
for conservation ecology and the design of natural preserves. As Milne (1992) notes, “the functional roles
of landscape patches may vary markedly as a consequence of patch mosaic structure”. The spatial analysis
of environmental variables leads to the conclusion that
they are scale dependent, a point that is well illustrated
by Richardson’s simple examples of the dependence of
estimates of the length of coastlines and frontiers on the
unit length used (in Mandelbrot 1983). Because of this
apparent scale dependence, the analysis of the spatial
pattern of ecological variables is best conducted over
a range of scales, as has been effectively demonstrated
in a number of recent studies. For example, Johnson
et al. (1992) examined the constraining effects of the
spatial structure of vegetation in a semi-arid grassland
on beetle movement and found that beetles exhibit a
complex behavioral response that varies across a range
of spatial and temporal scales, depending in part on the
spatial structure of the local plant community. Reed et
al. (1993) have effectively demonstrated that the correlation between the state of an environmental variable
and plant community composition may appear to be
significant when sampled at some spatial scales and not
at others. They further caution that our interpretation
of the importance of a particular environmental factor
in determining community composition may depend in
a critical way upon the sample scale we choose to use
in any given study. Also, Biondini and Grygiel (1994)
have shown in a modeling study that the outcome of
interspecific competition among plant species in a natural landscape may depend upon an interplay between
the scaling relationships of nutrient acquisition and the
spatial distribution of critical soil nutrients, such as
nitrogen.
An important practical problem for the analysis
of spatial scale dependence is that requires spatiallyintensive data. Spatial statistics, which have experi-
enced a notable advance in the last decade, are being
assimilated into the ecological literature (Matheron
1965; Journel and Huijbregts 1978; Clif and Ord
1981; Ripley 1981; Diggle 1983; Upton and Fingleton 1985; Cressie 1993; see Turner et al. 1991 for
an ecologically-oriented review). However, no matter how sophisticated the methods and tools may be,
they cannot counteract the paucity of spatial data that
often exists in ecological studies. Appropriate data for
the analysis of spatial-dependence needs to have both
a fine resolution and a broad extent. Such information is difficult to obtain for field-measured variables.
Therefore, related variables that can be derived from
digital imagery become a valuable source of data in
characterizing spatial dependence in ecological systems. We use this approach in the present paper to
study the relationship between pattern and processs in
a serpentine, annual grassland located at Jasper Ridge,
San Mateo County, California, USA. One reason for
using this as a study site is that grasslands, particularly annual grasslands, are good objects for studying
scaling issues because they are more accessible and
have simpler vertical dimension than other terrestrial
ecosystems.
In an earlier paper (Lobo et al., in press), we focused
on the development of a vegetation map from high
resolution remotely-sensed imagery, using a method
based on image segmentation and discriminant analysis. We showed that this method was able to discriminate among complex, ecologically meaningful land
cover categories that were not identifiable with more
conventional image classification methods. Here, we
develop the use of remotely-sensed imagery to provide
a link between field-based information and spatiallyexplicit modeling, which is done through a comparative study of the fine-scale spatial structure of several
ecological variables measured in a small portion of
the serpentine grasssland at Jasper Ridge. We compare
spatial relationships across a range of scales of both
field-measured (microtopography, soil depth, disturbance) and image-derived (land cover map, vegetation
index) landscape variables and use a statistical simulation method to produce random realizations of the
spatial pattern of the vegetation index based upon our
characterization of spatial structure. Results of these
analyses provide deeper insight into the spatial components of the ecology of this site than could be gained
by studying field-measured or image-derived variables
alone.
The serpentine grassland represents an ideal model
system for studying changes in vegetation at the land-
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scape scale, as community composition shifts over relatively short distances. As many as 25,000 plants can
be found per m2 (Huenneke et al 1990) and, as shown
here, environmental conditions affecting species distributions (primarily soil depth, disturbance rates, and
moisture availabilty) can change significantly over distances of less than a meter. As a consequence, although
we are only studying a small area of the grassland, this
represents a broad range of conditions across the serpentine landscape and our approach could be easily
used at a much broader scale for ecological systems
that are coarser grained than the Jasper Ridge serpentine grassland. The primary advantage of our approach
is that it provides valuable insights into the spatial relationships structuring ecological systems by coupling
very detailed, spatially-complete, and easily obtained
data (remotely sensed imagery) to sparse, and often
hard to collect, field data.
Study site
Our study site was located in the Jasper Ridge Biological Preserve of Stanford University (for a detailed
description of the ecology of this site, see Hobbs 1985;
Hobbs and Hobbs 1987; Moloney et al. 1991; Hobbs
and Mooney 1991; Moloney 1993; Wu and Levin
1994; Moloney and Levin 1996). Jasper Ridge is a low
lying ridge (maximum elevation of 189 m) situated
in the foothills of the Santa Cruz Mountains of California on the San Francisco Peninsula. The climate
is Mediterranean with an average annual rainfall of
500 mm and very little or no precipitation from May to
September. The crest of the ridge is bisected by serpentine soils, which are shallow, nutrient-poor soils high in
some heavy metals (Walker 1954; McNaughton 1968;
Hobbs and Mooney 1991). The serpentine soils support
a grassland community that is dominated by a diverse
array of native annual forbs, but also includes perennial forbs, bunch grasses, and annual grasses most
of which are also native. Dominance by annuals that
have little seed carry-over from year to year results in
rapidly changing population dynamics. Similar grassland communities occur on serpentine soils throughout
west-central California and are considered remnants of
native grassland that once occurred on more fertile soils
(Murphy and Ehrlich 1989).
The role of pocket gopher (Thomomys bottae) disturbance in the serpentine grassland of Jasper Ridge
has been studied in the field by Hobbs and Mooney
(1985 and 1991), who found that gophers turn over a
significant proportion of the grassland soil each year,
that distribution of gopher mounds is clumped and that
this type of disturbance has a large impact on the spatial
structure of the plant community.
Methods
For our analyses, we derived three spatial variables
from aerial photographs of the Jasper Ridge serpentine
grassland: a land cover classification of 4 discrete vegetation types for 1994, a vegetation index for 1994,
and location of gopher disturbances for the period
1988–1991. In addition, topographic height and soil
depth were measured in the field at discrete points
arranged in a regular (hexagonal) grid. Contingency table analysis and spatial statistics were used to
determine the spatial relationships among these variables and an inverse Fourier transform method was
used to simulate the spatial structure of the vegetation index. Image processing and Geographic Information System operations were performed with Grass
4.0 (U.S. Army Corps of Engineers 1991). Statistical
analyses, with the exception of calculation of crosscorrelograms, were performed with S-PLUS (Statistical Sciences 1993). Cross-correlograms were computed using GSLIB (Deutsch and Journel 1992).
Land cover
Digitized color infrared (CIR) aerial photographs of
the Jasper Ridge serpentine grassland were used to
develop a detailed digital map of land cover categories
in a small area of the grassland, as well as an image
of a vegetation index and a map of gopher disturbance
(see Sections 4.2 and 4.3). A detailed explanation of
the image processing methods is presented elsewhere
(Lobo et al., in press). We used a series of aerial photographs taken from 1988 to 1992 during the Spring.
Dates of the flights used in acquiring the photographs
(29 March 1988, 10 April 1989, 9 April 1990 and 24
April 1992) were set to capture vegetation at a similar
phenological state.
After digitizing a section of the scene for each year,
images were georectified. The region intersected by all
4 georectified digital CIR images became the area of
study, covering a 31.24 m 50.04 m region with a
spatial resolution of 135 mm.
The 1992 image was classified into four land cover
cateogories, using processing methods based on image
segmentation and discriminant analysis (Lobo 1997).
114
These methods discriminate among natural cover classes characterized by a highly textured surface and produce results superior to conventional “per-pixel” classification methods. As a result of this analysis, the
following categories were identified in the digitized
CIR image (Fig. 1):
1. “Bunchgrasses”: sites dominated by the annual
plant species Linanthus, and the tarweeds (genera Calycadenia and Hemizonia); tall, perennial
bunchgrasses present; and less than 1% cover by
exposed rocks.
2. “Dense annuals”: sites dominated by a dense carpet of short-lived, early flowering annuals, few tarweeds and no perennial grasses present; 1% to 10%
cover by small exposed rocks.
3. “Sparse annuals”: sites with only sparsely distributed annual plant species, no perennial grasses
present; gravely terrain.
4. “Bare soil”: no flowering plants present; terrain
composed of gravel, small stones and rocks.
Information from this map was then used to study
the distribution of other ecological variables with
respect to the four general cover categories.
image-recognizable gopher disturbances in the study
area for 1992, although some were visible in the image
outside the study area. The disturbance map was produced by combining the locations of disturbances for
1988–1990 into a single digital map. We analyzed the
distribution of gopher mounds with respect to the four
cover categories through a contingency table analysis.
Microtopography and soil depth
Soil depth and topographic height were determined for
481 sample points arranged as a hexagonal array in
the aforementioned 30 m 30 m experimental plot.
Soil depth was determined at each point by pushing a
soil probe into the soil until it hit bedrock. Topographic height was determined by conventional surveying
techniques using a theodolite and a metric pole.
The microtopography within the experimental plot
consisted of two elements: a slope and a number of
small ridges. The slope was treated as a trend and filtered out prior to analyzing the relationship between
the spatial structure of local topography and other ecological variables, such as soil depth and vegetation
structure (Fig. 2).
Normalized Difference Vegetation Index (NDVI)
Spatial analysis
A Normalized Difference Vegetation Index (NDVI)
map, coinciding with the land cover map, was constructed by computing the normalized difference of
the infrared and red components of the digitized CIR
image from 1992. NDVI strongly contrasts green vegetation against non-vegetated areas because of the
large difference in reflectivity shown by green biomass between the red and near-infrared wave lengths.
Since the images from Jasper Ridge have not been
radiometrically calibrated, the NDVI values presented here are not directly comparable to NDVI values
found elsewhere. Nevertheless, since we have calculated NDVI using linear transforms of near infrared
and red reflectance, the essential physical meaning of
NDVI as a proxy to projected green area cover remains
unchanged (Price and Bausch 1995).
Gopher disturbance
A digital map of gopher disturbances, coincident with
the land cover and NDVI maps, was produced by
interactive classification of disturbances in the color
infrared photographs of the Jasper Ridge grassland.
Recently produced gopher disturbances were easily
recognized in the 1988–1990 images. There were no
We studied the spatial structure of NDVI, soil depth
and topographic height by calculating their semivariograms. Semivariograms are plots of the average
square difference between the values of a spatial variable at pairs of points separated by a lag distance,
against the lag. Semivariograms are now being commonly employed in the analysis of remotely sensed
imagery (Curran 1988; Jupp et al. 1988a and b; Woodcock et al. 1988a and b; Curran and Dungan 1989;
Ramstein and Rafy 1989; Webster et al. 1989; Cohen
et al. 1990) and in ecological studies (Robertson 1987;
Robertson et al. 1988; Rossi et al. 1992; Biondini
and Grygiel 1994; Palmer 1990). We refer the reader
to Rossi et al. 1992 for an introduction to the use of
geostatistics in an ecological context and for the basic
terminology.
In this paper semivariograms were computed with
two methods. The first method was used to compute
semivariograms for data that were available for all map
locations (NDVI). In this case, 200 pairs of points were
randomly selected over the image for each lag distance,
and the average square difference of the corresponding
NDVI values was computed. Distances ranged from
1 to 220 pixels (0.135 m to 29.7 m) by steps of 1
115
Figure 1. Image-derived map of 1992 with the perimeters of gopher disturbances for the period 1988–1991 overlaid. Dark grey: tall bunchgrasses;
grey: dense annuals; light grey: sparse annuals; white: gravel, small stones and rocks.
pixel (0.135 m). Semivariograms were independently
computed 30 times, and the average of the 30 means
and their 95% confidence intervals were plotted. In
total, each semivariogram consisted of 6000 pairs of
points per lag (Fig 3). Separate NDVI semivariograms
were also calculated for each land cover category.
Usual methods were used to calculate semivariograms for data available only from the 481 field sample points (soil depth and topographic height), which
were available as (x,y,z) coordinates. For each pair of
points, the distance lag between them in the (x,y) plane
and the square differences for soil depth and topographic height were computed. Pairs of points were grouped
according to lag distance intervals of 1.467 m. The
resulting lags ranged from 1.467 m to 20.533 m. The
same method for calculating semivariograms was also
applied to NDVI data corresponding to the 481 data
locations used in calculating soil depth and topographic height in the field.
We examined the importance of directional
anisotropy in the spatial distributions of soil depth,
topographic height, and NDVI by calculating 4 separate directional semivariograms (0–180, 90–270,
45–225 and 135–325, with the 0–180 direction
being the horizontal axis of the map). The directional semivariograms were calculated with a tolerance of
22 300 .
Spatial relationships between different pairs of
variables have been studied by means of crosscorrelograms to facilitate the interpretation of spatial
statistics calculated with variables measured in different units.
Simulation of landscape variables
We have used a form of stochastic simulation to study
more fully the components making up spatial pattern
in the Jasper Ridge grassland image, focusing on an
116
Figure 2. 3-D representations of topographic height and soil depth data. The trend was obtained by 2-D smoothing and then substracted from
the raw topography to produce the detrended topography. Note the visual correspondence between the detrended topography and the soil depth.
analysis of NDVI. Stochastic simulation approaches
are being used increasingly in studies employing geostatistics, both as a complement to interpolation techniques (such as kriging) and as an alternative method
for studying 2-dimensional (2-D) spatial patterns (see
Deutsch and Journel 1992 for a practical introduction
to the topic). Here we use the Fourier integral method
because of its simplicity and its ability to generate stochastic 2-D fields of a spatial variable from the semivariogram. The Fourier integral method is described
by Pardo-Igúzquiza and Chica-Olmo (1993), to whom
we refer the reader for an extended explanation. A brief
description of the method is given in Appendix I. The
method is based on the facts that (i) the Fourier transform (FT) of a real function (an image is a 2-D real
function) produces a complex spectrum, which can
be decomposed into an amplitude and a phase spec-
trum (which are 2-D as well); (ii) the FT of the autocorrelation function (a.c.f.) of the same real function
produces the square of the amplitude spectrum, also
known as the spectral density function; and (iii) the
FT is reversible, hence the inverse FT (FT,1 ) applied
to an amplitude and phase spectra pair of an image
would reproduce the original image. We take advantage of these three facts to combine different pairs of
amplitude and phase spectra and backtransform them
to create different simulated images. Modeled a.c.f.
can be produced from particular components of the
modeled semivariogram and thus be used to clarify the
interpretation of these components by visualizing their
corresponding patterns (Fig. 4). As it is common with
digital imagery, the Fast Fourier Transform (FFT) algorithm is used here (Gonzalez and Wintz 1977, Richards
1993).
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Figure 3. Omnidirectional semivariogram of NDVI. white dots: observed values with the dipole method; thin solid lines: 95% confidence
interval; black dots (appear as a thick solid line): observed values using all possible pairs for each lag. See Section 4.5.
Results
Associations among landscape variables
Several interesting spatial relationships among cover
categories can be observed in the vegetation map of
the study site (Fig. 1c). Areas that were classified as
“bunchgrass” form long narrow bands running perpendicular to the overall slope of the site, with “dense
annuals” appearing to be mostly distributed around the
“bunchgrass” category. “Sparse annuals”, on the other
hand, appear to occupy other broad areas of the site.
Our analysis shows that gopher disturbances
departed significantly from the general null hypothesis of random association among the four 1992 cover categories (Table 1; X2 = 257.515, d.f = 3, p <
0.001). Areas identified as bunchgrasses in the 1992
image were more highly disturbed, based on proportional representation, than the other areas in the 1988–
1990 period: 1.79 times the rate expected at random.
Areas covered by the remaining categories had fewer
disturbances than expected.
Bunchgrasses also significantly differed from the
rest of land cover categories in terms of NDVI, soil
depth and detrended topography (Table 2). The same
results are observed for disturbed areas, which is to be
expected given the high degree of coincidence with
Table 1. Contingency table between gopher disturbance and terrain
categories. Values in brackets are the expectations under the null
hypothesis of non interaction. This hypothesis is rejected by the X2
test (X2 = 257.515, d.f. = 3, n = 2000, p < 0.001).
Terrain category
Gopher disturbance
No gopher disturbance
Bunch grasses
Annuals 1
Annuals 2
Bare soil
482 (269.328)
288 (378.289)
219 (314.562)
29 (55.786)
2164 (2376.637)
3428 (3337.711)
2871 (2775.438)
519 (492.214)
the bunchgrass areas (Table 3). Bunchgrasses (and
disturbances) occurred on deeper soils with higher
topographic relief than the other vegetation categories
(Tables 2 and 3). NDVI values were also higher on
areas classified as bunchgrass or disturbed (Tables 2
and 3). The latter result indicates that there is greater
aboveground biomass in bunchgrass areas, as would
be expected for this ecosystem, and that the vegetation
recovers rapidly from localized disturbances, which is
not surprising for a community predominantly comprised of annual plant species.
NDVI and soil depth were linearly correlated
(r2 = 0.17, p < 0.00001; Fig. 5a) although with a large
degree of scatter. The correlation increases if NDVI
and soil depth values are averaged by patches of the
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Figure 4. Flow chart of the simulations. Images (a), (b), (c) and (d) correspond to the ones represented in Fig. 11.
Table 2. Significance test comparing average values of NDVI, soil depth
and detrended topography for a subset of cells from the areas assigned
to the terrain category “Bunchgrass” versus the remaining terrain categories (“Other”). (Standard deviations are indicated in parentheses).
Table 3. Significance test comparing average values of NDVI, soil
depth and detrended topography for a subset of cells from areas identified as “disturbed” versus “undisturbed”. (Standard deviations are
indicated in parentheses.)
Variable
df
Bunchgrass
Other
t-value
p
Variable
d.f.
NDVI
390
< 0.001
NDVI
479
390
–188.526
(157.828)
209.687
(131.459)
–0.771
(3.821)
13.726
Soil depth
(mm)
Detrended
topography
(mm)
73.634
(137.347)
365.805
(202.871)
2.563
(3.006)
8.430
< 0.001
7.323
< 0.001
Soil depth
(mm)
Detrended
topography
(mm)
390
Disturbed
–49.750
(182.474)
479 339.278
(215.194)
479
2.643
(4.305)
Undisturbed
t-value
p
–136.593
(179.676)
239.187
(162.890)
–0.181
(3.702)
2.786
< 0.0055
3.453
< 0.0006
4.348
< 0.001
119
same land cover type according to the map of Fig. 1
(r2 = 0.61, p < 0.00001; Fig. 5b), which is a consequence of both variables being spatially autocorrelated
at a local scale.
Spatial analysis
Visual inspection of the omnidirectional semivariogram of NDVI reveals that it is composed of 3 basic
components: (1) an exponential trend dominating at
small scales, with a range of approximately 4 m, (2) a
linear trend dominating at broader scales, and (3) a
superimposed periodic structure, indicated in part by
the local minimum seen at 7.5 m (Fig. 6). The exponential and linear components can be modeled (Fig. 6a)
as:
^(h) = 0
for h = 0
^(h) = a + c1 (1 , exp(,h=r)) + c2 h for h > 0
(1)
where ^ stands for the semivariance, h for the lag distance and the parameters are estimated by non-linear
weighted least-squares (Table 4). The residuals, after
removal of the exponential and linear components,
clearly exhibit a periodic structure (Fig. 6b). The latter can be modeled approximately using trigonometric
functions (Table 4):
^ (h) = c3 (2h=b) + c4 sin(2h=b)
(2)
Figure 6 also includes a power (fractal) model,
which provided a poorer fit than model (1) (Table 4):
^ (h) = a2 hb2
(3)
The exponential trend in the NDVI semivariogram,
which dominates at short distances, is indicative of
strong local spatial correlation in the distribution of
biomass that extends over distances up to about 4 m.
Similar patterns at that distance can be observed in the
correlograms (approximately equal to 1, ^(h)) for soil
depth and detrended topography (Fig. 7). Although
the spatial patterns of these three variables – NDVI,
soil depth and detrended topography – are similar in
structure between lags of 0 and 4 m, they could be
totally independent of one another. However, crosscorrelation analysis among these three variables shows
that there is a strong correspondence in their spatial
structure, at least at the local scale (Fig. 7). The lat-
ter result extends the results of the contingency table
analyses of the previous section into a spatial context,
demonstrating again the strong correlation between
soil depth, detrended topography and NDVI.
The linear trend dominating the semivariogram of
NDVI at distances greater than 4 m is closely associated with the topographic relief of the study site, as can
be seen in an analysis of directional semivariograms
of NDVI (Fig. 8). (There is a distinct anisotropy in the
spatial distribution of NDVI values.) The linear trend
observed in the global semivariogram was not present
in the 90–270 and 45–225 directions, but is a strong
element in the 0–180 and the 135–315 directions.
Comparing these results to a plot of raw topography
shows that the linear trend in the semivariogram dominates in a direction parallel to the slope, indicating
a systematic change in NDVI values in this direction
(Fig. 2).
The periodic component of the NDVI semivariogram derives from the fact that NDVI values are higher on bunchgrass-dominated ridges, which are more or
less evenly distributed as strips along the vertical axis
of the image (oriented primarily in the 90–270 direction as seen in Fig. 1). This interpretation is strengthened by the fact that the period estimated for (2) is 10 m,
which corresponds closely to the distances between
topographic ridges at the study site (cf., Fig. 1 and
Table 4). Also, inspection of semivariograms for NDVI
calculated for individual land cover types (Fig. 9) indicates that semivariograms of NDVI computed for the
“dense annuals” and “sparse annuals” do not exhibit the periodic structure found in the omnidirectional
semivariogram computed across all cover categories
(Fig. 9). However, the semivariogram for the “bunchgrass” cover category, which is closely associated with
the topographic ridges, does contain a within-category
periodicity. More evidence for the interpretation of the
periodic structure of the NDVI semivariogram comes
from the simulation results presented in the next section, and from the fact that the periodic structure is
more evident in directional semivariograms of NDVI
computed parallel to the slope (0–180) rather than
perpendicular to the slope (90–270) (Fig. 8).
The periodic element was also present in the omnidirectional semivariograms for soil depth and detrended topographic height (Fig. 10). This is consistent with
the observed co-occurrence of bunchgrasses, deeper
soils and higher NDVI values on ridges, as described
earlier. The semivariogram of the raw topographic
height was totally dominated by the slope, which
appears in the semivariogram as a parabola.
120
Figure 5. Scatter plot of NDVI vs. soil depth values. (a), plot of observed soil depth values vs. the corresponding individual pixel NDVI value;
(b), plot of per-patch means, where patches are defined by the classified image (Fig. 1). 1, bunchgrass patches; 2, patches of dense annuals;
patches of sparse annuals. Only patches with more than 5 soil depth observations are used to compute the means. Patches of bare soil were too
small to contain more than 5 observations of soil depth and are not represented.
Table 4. Parameters and statistics of the models fitted by weighted non-linear least squares to the semivariograms.
NDVI
Soil depth
Parameter
Model [1]
(image)
(grid)
Model [2]
parameter
(image)
Model [3]
parameter
(grid)
Model [4]
a1
c1
c2
r1
R2
DF
0.0066
0.0276
0.0001
1.1469
0.999
219
0.0089
0.2487
0.0024
1.6654
0.925
14
c3
c4
b
–
R2
DF
0.0005
–0.0001
10.0840
–
0.202
219
a2
b2
–
–
R2
DF
0.024
0.144
–
–
0.689
219
–
c5
–
r2
R2
DF
–
29258.0
–
0.832
0.844
14
121
,
,
Figure 6. (a) Omnidirectional semivariogram of NDVI. dots: observed values; solid line: model [1] (
^ (h) = a + c1 (1
exp( h=r )) +
c2 h for h > 0) fit to the data; dashed line, power (fractal) model [3] (^ (h) = a2 hb2 ) fit to the data. (b) Residuals of the previous model [1]
^ (h) = c3 cos(2h=b) + c4 sin(2h=b)) fit to the residuals (solid line). See Table 4 for values and statistics.
(dots), and periodic model [2] (
The semivariogram of soil depth does not exhibit
a linear trend and its non-periodic component can be
modeled by (Table 4):
^(h) = 0
for h = 0
^(h) = a2 + c4 (1 , exp(,h=r2 )) for h > 0
(4)
NDVI was the only variable for which a resolution
of 0.135 m per pixel was available. For soil depth and
topography only sample points arranged in the sample grid were available. In order to assess the impact
of the loss of spatial precision in computing semivariograms for these variables, NDVI values were sampled
from the image at the same grid positions as soil depth
and topographic height and the semivariogram of this
“sampled NDVI” was computed (Fig. 10). The global
omnidirectional semivariogram for “sampled NDVI”
was a good approximation to the “full resolution”
semivariogram. The same was true for semivariograms
of “sampled NDVI” computed over categories corresponding to “dense annuals” and “sparse annuals”, but
not for the ones computed with sampled NDVI values
over the “bunch grasses” category. As a consequence,
“per category” semivariograms of soil depth and topography are not presented.
122
Figure 7a. (a) Cross-correlograms. Open circles, NDVI and soil depth; filled circles, detrended topographic relief and soil depth; open diamonds,
detrended topographic relief and NDVI.
Figure 7b. (b) Autocorrelograms of soil depth (open squares) and NDVI (filled squares). In both cases the dashed lines indicate 95% confidence
interval.
Simulations
Figure 11 shows results of the FFT method used to simulate patterns from the central part of the NDVI image
(Fig. 11a). FFT of this image produced the observed
phase and amplitude spectra. An FFT,1 applied to this
pair of observed spectra renders the original image.
Fig. 11b has been produced by applying FFT,1 to a
pair composed of a randomized phase spectrum and
the original amplitude spectrum.
A modeled amplitude spectrum, along with the
observed phase spectrum, has been used to produce
Fig. 11c by FFT,1 . This modeled amplitude spectrum
was computed from the exponential component of the
semivariogram model fit to the observed omnidirectional semivariogram of NDVI (equation (1); Fig. 6):
the exponential element was converted to an a.c.f. and
then the a.c.f. was transformed by FFT to produce a
power spectrum. The square root of the power spectrum
was the amplitude spectrum used to produce Fig. 11c.
123
Figure 8a. (a) Directional semivariograms of NDVI. Dots, 0–180 direction; crosses, 90–270 direction.
Figure 8b. (b) Fitted linear components of the model [1] for each global directional semivariogram.
Note that the main spatial discontinuities of Fig. 11a
can be recognized in Fig. 11c and not in Fig. 11b,
which indicates that this important part of the spatial
information is present in the phase spectrum and not
in the power spectrum. Instead, the linear ridges are
present in Fig. 11b and not in Fig. 11c, which is a consequence of not including the periodic component of
the semivariogram in the calculation of Fig. 11c.
The same modeled amplitude as in Fig. 11c was
used in Fig. 11d, but in this case the second element of
the pair, the phase spectrum, was a randomization of
the observed phase spectrum. Fig. 11d is thus, a realization of a “generalized landscape” that corresponds
to the exponential component of (1).
Discussion
The spatial structure of the grassland as described from
the results presented above, can be summarized by
124
Figure 9. Omnidirectional semivariograms of NDVI within each cover category.
(i) the co-occurrence of gopher disturbance, deeper
soils, higher NDVI and bunchgrass-dominated vegetation; and by (ii) the nested structure of the semivariograms for NDVI, which are dominated by an exponential increase at short lags and a periodic structure
at larger lags. Both features have to be considered in
developing suitable models of the ecological dynamics of this system, particularly if spatial relationships
are to be taken into account. Currently, a spatiallyexplicit model of the ecological dynamics of the Jasper
Ridge serpentine grassland exists, which considers the
demographic dynamics of a subset of the plant species
125
Figure 10. Omnidirectional semivariograms of NDVI and of the field – sampled variables soil depth (diamonds), topographic relief (triangles)
and detrended topographic relief (squares). Values have been standardized for comparison. NDVI values (circles) have been sampled from the
image at the same sampling locations.
occurring at the site (all annuals) and has been used
to study the impact of gopher mound production on
the distribution of those species (Moloney and Levin
1996). Unlike the relationships reported here, disturbances in the model are generally distributed with equal
probability throughout the landscape, although spatial
and temporal autocorrelation within the disturbance
regime have been accounted for. However, a subset
of model runs has been conducted within which there
were two distinct types of habitat, highly disturbed
sites versus completely undisturbed sites (Moloney and
Levin 1996). The results of the latter model runs were
fundamentally different from the results of the standard model consisting of a homogeneous landscape,
emphasizing the importance of considering the spatial
distribution of ecological factors, such as those studied
here.
Our finding that gopher disturbances were concentrated on topographic ridges, suggests that there may
be some very important constraints operating to restrict
the activity of gophers, such as shallow soils and local
availability of suitable food plants. A more realistic
simulation model than is currently being employed
would incorporate some of these relationships into
its dynamics when predicting the distribution of plant
species and disturbances. Although our estimates of
gopher disturbance might be biased by easier detection on a bunchgrass background, our results indicate
that the high diversity of annuals at Jasper Ridge that
has been reported elsewhere (Huenneke et al. 1990) is
not only the result of disturbances producing patches
of different ages, but must also be the result of spatial
variation in soil structure.
Observations at a broader scale indicate that the
occurrence of deeper soils on the ridges in the study site
are most likely related to intrusions of non-serpentine
lithologies from the surrounding landscape. The quality of soil in ridges is perhaps enhanced by intense
gopher activity resulting in higher NDVI values. One
effect of gopher activity is the removal of the soil crust,
which enhances water penetration into the soil. Competition among plant species is also lower on mounds
(Hobbs and Mooney 1985). However, mound soils on
pure serpentine can actually reduce plant growth due
126
Figure 11. a. Subscene of the NDVI image obtained from the digitized CIR image. Light tones represent high NDVI values, dark tones represent
low NDVI values. A Fourier transform of this scene renders the original phase and the original amplitude. The other 3 images were obtained by
inverse Fourier transform as follows (see Sections 3.6 and 4.3).
b. Using the original amplitude and a randomization of the original phase.
c. Using the original phase and a modeled amplitude (from an exponential autocovariance function).
d. Using a randomization of the original phase and the above modeled amplitude.
to changes in soil chemistry, such as lowering N and
P availability (Koide et al. 1987). All of these factors
taken together indicate that there can be very complex relationships among patterns of disturbance, soil
structure, and the distribution of plant species. However, what is clear from this study is that there is a close
association between the distribution of bunchgrasses
and deep soils and between annuals and shallow soils.
These results indicate that it would be useful to include
a resource-based approach in modeling the ecological
dynamics of this system, in addition to the demographically based approach that is currently being employed
(Moloney and Levin 1996).
The semivariograms of NDVI facilitate a hierarchical analysis of the landscape that can aid in improving the design of the current spatially-explicit model of Jasper Ridge and are quite useful in suggesting
the relevant spatial scales involved in determining the
spatial structure of this system. All three components
– the exponential, the linear trend and the periodic
structure – are present at all lags, but with very different impacts depending on the particular lag value
considered. The fine scale correlations represented by
the exponential component represent the approximate
range over which ecological relationships are relatively homogeneous (+4 m). The landscape element that
generates the periodic spatial component of the distribution of NDVI can be traced to the presence of
topographic ridges consisting of significantly deeper
soils that are more or less evenly distributed in narrow bands oriented perpendicular to the slope. In contrast, the mechanism producing the linear trend has
not been identified. It is related to a systematic change
in elevation within the study site, but the underlying
ecological variables affecting NDVI values at different
positions along the slope are not known at this time.
Further investigation involving a larger extent of the
grassland and more field information (in particular of
water-related variables) is clearly required if we are to
understand the factors producing this relationship.
The parameters of the semivariogram model for
NDVI (Table 5) reveal that most of the spatial variation
at short lags is contained within the exponential term,
while most of the variation at longer lags is contained
127
within the linear trend. A shift in structure such as this
is coincident with a widely held view of complex semivariograms being the result of nested spatial processes
(Burrough 1983b). Nested processes are often immediately interpreted as being fractal in structure, which
would result in a linear semivariogram on a log-log
scale (Burrough 1983a). However, plots in a log-log
scale can often hide the inadequacy of a power law
to fit an observed semivariogram. The arithmetic scale
reveals that, in the case discussed here, an exponential
model is not appropriate for characterizing the spatial
structure in our study area.
Fractals have become quite popular in landscape
ecology, probably due in part to the impressive simulations of topographic relief using relatively simple
fractal algorithms (Voss 1988; Saupe 1988). Palmer
(1992) introduced the simulation of 2-D fields using
fractal models as a method for generating landscapes
as part of a spatially-explicit ecological model, and
fractal geometry had been used by ecologists as a landscape metric (see Milne 1991 and 1992 for a review).
Fractals are also theoretically appealing because they
can be related to spatial processes through a generalization of the Bachelier-Wiener process that describes
Brownian motion (Burrough 1983a; Oliver and Webster 1986). However, fractal models are rarely selected as an appropriate authorized model for kriging.
Semivariograms computed from remotely-sensed digital imagery often do not conform to the fractal model
(Curran 1988; Cohen et al. 1990; Webster et al. 1989;
Ramstein and Raffy 1989; Woodcock et al. 1988b).
Several examples of semivariograms that depart from a
fractal model can be found in Burrough (1983b), Mark
and Aronson (1984) and Armstrong (1986) for soil
variables and Palmer (1988) and Leduc et al. (1994) for
vegetation variables. In the latter two cases, spatial patterns were in fact defined as fractals because the slope
of a straight line fitted to a log-log transform of the
semivariogram yields a fractal dimension D between
1 and 2. This definition, which ignores self-similarity
as a necessary condition, implies that a linear fit can
be considered as a first approximation. Nevertheless,
even if fractal models can be used as an approximation for some applications, it is important to stress that
semivariograms of many environmental variables are
not adequately described by fractal models. In other
cases, assuming a fractal model can be of little use. As
stated by Cressie (1993): “: : : it would be overly optimistic to expect that a complex physical process could
be summarized by just the fractal dimension D : : : For
more precise information over different spatial scales,
one could use the semivariogram : : : However, estimated variograms usually demonstrate a much richer structure than a linear one in the log-log scale”.
Multi-fractal models (in which different parameters
are used for different lag segments; Burrough 1983b;
Biondini and Grygiel 1994) can be fitted in some cases,
but unless some empirical or theoretical indication of
their appropriateness is available and/or sharp discontinuities in the variogram are observed, this procedure
could be understood to be simply an attempt to approximate a curved line (on a log-log scale) by means of a
number of straight segments.
A spatial process that is characterized by an exponential semivariogram will correspond to a landscape
that is divided into regions whose boundaries occur
as a Poisson process (Burrough 1983b). Hence the
exponential component for NDVI found in this study
indicates that at shorter lags the different mechanisms
creating pattern are not nested. Multifractal processes,
on the other hand, are characterized by semivariograms
that can be modeled by a sequence of power curves (in
which each power corresponds to a different fractal
dimension). It is important to note that, if the processes that generate the multifractal tend to overlap instead
of having defined transitions, the multifractal semivariogram becomes more smoothed towards a single
exponential. Therefore, it is normal to expect natural
spatial patterns to lie in between these two extremes.
Values of the intercept term a in (1), which account
for what is called the “nugget” variance in the geostatistics literature, also have an ecological significance. The “nugget” represents unexplained variance
at a scale below the resolution of the analysis and may
represent either measurement error or variance attributable to processes occurring below the resolution of the
analysis. Interpretation of the relative importance of
the “nugget” variance can be obtained by dividing it
by the “sill” to determine the proportion of the variance that is unexplained by the semivariogram model.
(The “sill” is the maximum value for the semivariance
in models that reach an asymptote at large lags and,
in a system that is second order stationary, is equal
to the variance in the system.) Exponential semivariograms are unbounded, but for practical reasons it is
conventional in geostatistics to assume an “effective”
range (3r) at which the semivariance is about 95% of
c1 (Webster and Oliver 1990). With our parameters for
the NDVI model [1] (Table 4), corresponds to a sill of
0.0328, and thus the ratio:
nugget 0:00661
=
= 0:201
sill
0:0328
128
implies that more than 20% of the spatial variance
remains undescribed by the semivariogram of NDVI
despite the high resolution, which means that there is
a high degree of heterogeneity at scales below 0.135
m. Also, using the same 3 r criterion and the models
fit to variograms computed with the more sparsely distributed grid values (Table 4), we find that soil depth
reaches its sill at a shorter range (2.496 m) than NDVI
(4.996 m). Hence the observed spatial pattern of plants
has a coarser grain than that of soil depth.
Heterogeneity of NDVI values associated with
annual plant cover (“dense annuals” and “sparse annuals”) is also observed in the nugget values of the percategory semivariograms of NDVI, in which annuals
and bunchgrasses show similar ranges (Fig. 9). Taking
the smaller size of annual plants into account, it follows that the annuals show a high degree of patchiness
at fine scales, with areas of higher biomass (and plant
density) surrounded by areas of lower biomass. This
is probably a consequence of poorer soils, which are
commonly associated with higher biodiversity (Tilman
1982). The high biodiversity of the serpentine and its
relationship with nutrient poverty (which makes efficient dominance impossible) has been reported previously by Huenneke et al. (1990).
The simulation of landscape fields by FFT using
the exponential component of the autocorrelation functions for NDVI has provided us with a tool for verifying
our interpretation of the variograms. The simulations
also demonstrate that statistical analysis and interpolations based solely on the semivariogram (or the a.c.f.)
actually miss the information carried by the phase spectrum (Fig. 11b), which can be associated with spatial
discontinuities. Interpolation techniques such as kriging commonly ignore the phase information, which
can result in an oversmoothed view of the distribution
of spatial variables that removes important information about spatial discontinuities in a pattern (Deutsch
and Journel 1992). This in turn runs the risk of producing an oversmoothed view of spatial variation in
the minds of vegetation scientists, paralleling the case
of the presentation of smoothly changing distributions
of species along complex one-dimensional gradients
(e.g., see any classic textbook on community ecology,
such as Whittaker 1975, Pianka 1978, Barbour et al.
1987). It is therefore critical that procedures for the
interpolation of ecological variables explicitly include
information on spatial discontinuities, either directly
from remotely sensed images (assuming that the phase
pattern in the image and in the ecological variables
are equivalent) or indirectly, by sampling, with suffi-
cient intensity, spatial variables in the field that have
a known functional relationship with the variable of
interest (assuming that the variable of interest is too
hard or expensive to sample intensively in the field).
Conclusions
In this paper we have used digitized CIR imagery to
obtain information on spatial variables at a scale of 30
30 m2 of extent and 0.135 m of resolution (grain). We
have used this information in conjunction with more
sparsely distributed data obtained in the field to develop
a deeper understanding of the spatial relationships controlling the distribution of vegetation and disturbance
activity at Jasper Ridge. From the analysis of these
relationships we have gained valuable information that
can be used in improving spatially-explicit models of
the Jasper Ridge grassland. We have also evaluated
our techniques and identified sources of environmental
variation that we can explore over larger extents.
Simulations of two-dimensional fields of NDVI
using the FFT of modeled autocorrelation functions
and both observed and randomized phase spectra prove
to be useful in validating the interpretation of semivariograms and in generating spatial templates for
spatially-explicit ecological models. The simulations
also illustrate the limits of spatial analysis and interpolations of landscape variables based on semivariograms
(or autocorrelation functions) solely, stressing the need
to take spatial discontinuities into account.
This study provides a good illustration of the dual
nature of spatial pattern. Some aspects can be studied through a consideration of pattern as a continuous process coupling both deterministic and stochastic
elements. However, there are also some aspects that
need a discrete representation. Using continuous variables such as NDVI (or others related to it, such as
green biomass), we focus on the continuous nature of
vegetation. In contrast, mapping of different types of
vegetation focuses on the discrete nature of the landscape produced by relatively abrupt changes in species
and plant functional type composition. Both views are
necessary in developing a complete understanding of
landscape pattern.
Acknowledgments
This work was supported in part through grants from
the Andrew W. Mellon Foundation and the National
129
Aeronautics and Space Administration (NAGW-3124)
to Simon Levin, Princeton University. A. Lobo was
supported by the Joint Program of the Ministry of Education and Science of Spain / The Fulbright Committee (FU91-50413540) as a visiting fellow in Cornell
and Princeton Universities. The authors are grateful to
facilities provided by the Administration of the Jasper
Ridge Biological Preserve. Comments by Prof. Mario
Chica Olmo and three anonymous reviewers significantly improved the original manuscript.
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Appendix
An image can be considered to be a discrete two-dimensional
signal, i; j , and be expressed in terms of its discrete FT,
r; s (Gonzalez and Wintz 1977; Richards 1993):
( )
( )
,1
X
K
( ) = K12
i; j
where
=0
(r; s) is:
(r; s) = K12
(r; s) exp[j 2(ir + js)=K ]
(5)
i
,1 KX
,1
X
K
=0 i,0
( ) exp[,j 2(ir + js)=K ]
i; j
i
(6)
Note that the complex exponentials are a compact form
of writing a sum of sines and cosines:
131
exp
(jx) = cos(x) j sin(x)
and thus the FT expresses the original function as a weighted
sum of a series of sines and cosines.
As the discrete FT (DFT) is a sum of complex exponentials, it consists of a real and a complex part and can
be decomposed into one pair of series (spectra): the phase
spectrum ( r; s ) and the amplitude spectrum ( r; s ),
which in the case of images are two-dimensional:
( )
j( )j
(r; s) = j(r; s)j exp[j (r; s)]
2
2
j(r; s)j = (Re((
h r; s)) +iIm((r; s))
))
(r; s) = tan,1 ImRe((
(( ))
r;s
r;s
where Re( ) and Im( ) are the real and imaginary part operators. We note in the passing that the spectral density (P(r,s))
is the square of the amplitude spectrum ( r; s ).
j( )j