The influence of patch-delineation mismatches on multi-temporal landscape pattern analysis

Investigations of land-cover change often employ metrics designed to quantify changes in landscape structure through time, using analyses of land cover maps derived from the classification of remote sensing images from two or more time periods. Unfortunately, the validity of these landscape pattern analyses (LPA) can be compromised by the presence of spurious change, i.e., differences between map products caused by classification error rather than real changes on the ground. To reduce this problem, multi-temporal time series of land-cover maps can be constructed by updating (projecting forward in time) and backdating (projecting backward in time) an existing reference map, wherein regions of change are delineated through bi-temporal change analysis and overlaid onto the reference map. However, this procedure itself creates challenges, because sliver patches can occur in cases where the boundaries of the change regions do not exactly match the land-cover patches in the reference map. In this paper, we describe how sliver patches can inadvertently be created through the backdating and updating of land-cover maps, and document their impact on the magnitude and trajectory of four popular landscape metrics: number of patches (NP), edge density (ED), mean patch size (MPS), and mean shape index (MSI). In our findings, sliver patches led to significant distortions in both the value and temporal behaviour of metrics. In backdated maps, these distortions caused metric trajectories to appear more conservative, suggesting lower rates of change for ED and inverse trajectories for NP, MPS and MSI. In updated maps, slivers caused metric trajectories to appear more extreme and exaggerated, suggesting higher rates of change for all four metrics. Our research underscores the need to eliminate sliver patches from any study dealing with multi-temporal LPA.     

Neutral model analysis of landscape patterns from mathematical morphology

Mathematical morphology encompasses methods for characterizing land-cover patterns in ecological research and biodiversity assessments. This paper reports a neutral model analysis of patterns in the absence of a structuring ecological process, to help set standards for comparing and interpreting patterns identified by mathematical morphology on real land-cover maps. We considered six structural classes (core, perforated, edge, connector, branch, and patch) on randomly generated binary (forest, non-forest) maps in which the percent occupancy (P) of forest varied from 1% to 99%. The maps were dominated by the patch class for low P, by the branch and connector classes for intermediate P, and by the edge, perforated, and core classes for high P. Two types of pattern phase changes were signaled by abrupt transitions among the six structural classes, at critical P thresholds that were indicated by increased variance among maps for the same P. A phase change from maps dominated by the patch class to maps dominated by the branch and connector classes was related to the existence of a percolating cluster of forest, and the P threshold varied depending on the co-existence of the core class. A second phase change from the edge class to the perforated class was related to the existence of a percolating cluster of non-core (including non-forest) and represents a change of context from exterior to interior. Our results appear to be the first demonstration of multiple phase changes controlling different aspects of landscape pattern on random neutral maps. Potential applications of the results are illustrated by an analysis of ten real forest maps. The U.S. Government's right to retain a non-exclusive, royalty-free license in and to any copyright is acknowledged.     

Estimating landscape pattern metrics from a sample of land cover

Although landscape pattern metrics can be computed directly from wall-to-wall land-cover maps, statistical sampling offers a practical alternative when complete coverage land-cover information is unavailable. Partitioning a region into spatial units and then selecting a subset (sample) of these units introduces artificial patch edge and patch truncation effects that may lead to biased sample-based estimators of landscape pattern metrics. The bias and variance of sample-based estimators of status and change in landscape pattern metrics were evaluated for four 120-km × 120-km test regions with land cover provided by the 1992 and 2001 National Land-Cover Data of the United States. Bias was generally small for both the estimators of status and estimators of change in landscape pattern, but exceptions to this favorable result exist and it is advisable to assess bias for the specific metrics and region of interest in any given application. A 10-km × 10-km sample block generally yielded larger biases but smaller variances for the estimators relative to a 20-km × 20-km sample block. Stratified random sampling improved precision of the estimators relative to simple random sampling. The methodology developed to determine properties of sample-based estimators can be readily extended to evaluate other landscape pattern metrics, regions, and sample block sizes.     

Land-cover and structural changes in a western Norwegian cultural landscape since 1865, based on an old cadastral map and a field survey

Many studies of land-cover and structural changes in cultural landscapes have used historical maps as a source for information about past land-cover. All transformations of historical maps onto modern coordinate systems are however burdened with difficulties when it comes to accuracy. We show that a detailed land survey of the present landscape may enable transformation of an old cadastral map directly onto the present terrain with very high accuracy. The detailed resulting map enabled us to locate remnants of semi-natural grasslands and man-made structures with continuity from 1865 and to test hypotheses about relationships between landscape changes and landscape characteristics. The main land-cover change 1865–2002 was decrease of arable fields, and addition of three new land-cover classes: horticultural, orchard and abandoned areas. Of the 330 man-made structures present in 1865, only 58 remained in 2002, while 63 new structures had been built after 1865. We found that semi-natural grasslands with continuity since 1865 were situated on ground with significantly lower production capacity than mean 1865 production capacity. The man-made structures with continuity since 1865 were also associated with areas with significantly lower production capacity than the 1865 mean, situated in significantly steeper terrain but not further from the hamlet. Our study illustrates the potential of digitised and accurately transformed historical cadastral maps combined with detailed field surveys for analysis of land-cover and structural changes in the cultural landscape.     

The influence of scale and the spatial characteristics of landscapes on land-cover mapping using remote sensing

Statistical analyses provide a means for assessing relationships between landscape spatial pattern and errors in the estimates of cover-type proportions as land-cover data are aggregated to coarser scales. Results from a multiple-linear regression model suggest that as patch sizes, variance/mean ratio, and initial proportions of cover types increase, the proportion error moves in a positive direction and is governed by the interaction of the spatial characteristics and the scale of aggregation. However, the standard linear model does not account for the different directions of scale-dependent proportion error since some classes become larger and others become smaller as the scene is aggregated. Addition of indicator variables representing class-type significantly improves the performance by allowing the model to respond differently to different classes. A regression tree model provides a much simpler fit to the complex scaling behavior through an interaction between patch size and aggregation scale. An understanding of the relationships between landscape pattern, scale, and proportion error may advance methods for correcting land-cover area estimates. Such methods could also facilitate high-resolution calibration and validation of coarse-scale remote-sensing-based land-cover mapping algorithms. Ongoing initiatives to produce global land-cover datasets from remote sensing, such as efforts within the IGBP and the EOS MODIS Land-Team, include significant emphasis on high level calibration and validation activities of this nature.     

Neutral models for testing landscape hypotheses

Neutral landscape models were originally developed to test the hypothesis that human-induced fragmentation produces patterns distinctly different from those associated with random processes. Other uses for neutral models have become apparent, including the development and testing of landscape metrics to characterize landscape pattern. Although metric development proved to be significant, the focus on metrics obscured the need for iterative hypothesis testing fundamental to the advancement of the discipline. We present here an example of an alternative neutral model and hypothesis designed to relate the process of landscape change to observed landscape patterns. The methods and program, QRULE, are described and options for statistical testing outlined. The results show that human fragmentation of landscapes results in a non-random association of land-cover types that can be describe by simple statistical methods. Options for additional landscape studies are discussed and access to QRULE described in the hope that these methods will be employed to advance our understanding of the processes that affect the structure and function in human dominated landscapes.     

Analysis of land-cover transitions based on 17th and 18th century cadastral maps and aerial photographs

This paper explores the possibility of using non-geometric cadastral maps from the 17th and 18th century together with aerial photographs from 1945 and 1981 to analyse land-cover change in south-east Sweden. Habitats rich in plant species in the European rural landscape seem to be correlated with a long continuity of management. Accurate spatial data from historical data sources are fundamental to understand patterns of vegetation and biodiversity in the present-day landscape. However, traditional methods for rectification of non-geometric maps using corresponding points from orthophotos or modern maps are not satisfying, as internal inaccuracies will remain in the maps. This study presents a method to rectify the maps by local warping, thereby eliminating geometrical irregularities. Further, the land-cover changes were calculated and presented as transition matrices. The extent of arable fields and grasslands were analysed in relation to soil characteristics and continuity of management. The results show a dynamic relation between grassland and arable field, albeit the overall proportions remained almost the same between 17th and 18th centuries: 60% grassland to 32% arable field. The most substantial changes in land-cover were prior to 1945. Today there is 18% grasslands left in the study area, while 56% of the land-cover is arable field. Approximately 8% of present-day land-cover is semi-natural grassland 300 years of age or more. Compared to 300 years ago there is only 1% grassland left on peat and 2% on clay. In contrast, grassland covers associated with bare bedrock have been fairly stable in size. All semi-natural grasslands with a long continuity of management were situated on shallow soils, less than 50 cm depth. The major conclusions from this study are that (i) correctly rectified, old maps are very useful to address questions of land-cover changes in historical time, (ii) general trends in land use over 300 years in this hemi-boreal landscape seem to underestimate the full dynamics of land use change, and (iii) only a small proportion of the semi-natural grassland area had a 300 year continuity of management.     

Impact of scale on morphological spatial pattern of forest

Assessing and monitoring landscape pattern structure from multi-scale land-cover maps can utilize morphological spatial pattern analysis (MSPA), only if various influences of scale are known and taken into account. This paper lays part of the foundation for applying MSPA analysis in landscape monitoring by quantifying scale effects on six classes of spatial patterns called: core, edge, perforation, branch, connector and islet. Four forest maps were selected with different forest composition and configuration. The sensitivity of MSPA to scale was studied by comparing frequencies of pattern classes in total forest area for various combinations of pixel size (P) and size parameter (S). It was found that the quantification of forest pattern with MSPA is sensitive to scale. Differences in initial composition and configuration influence the amount but not the general tendencies of the variations of morphological spatial pattern (MSP) class proportions with scale. Increase of P led to data generalization resulting in either a removal of the small size features or their potential transformation into other non-core MSP classes, while an increase of S decreases the MSP core area and this process may transform small core areas into the MSP class islet. We established that the behavior of the MSPA classes with changing scale can be categorized as consistent and robust scaling relations in the forms of linear, power, or logarithmic functions over a range of scales.     

Watershed analysis of forest fragmentation by clearcuts and roads in a Wyoming forest

Remotely sensed data and a Geographic Information System were used to compare the effects of clearcutting and road-building on the landscape pattern of the Bighorn National Forest, in north-central Wyoming. Landscape patterns were quantified for each of 12 watersheds on a series of four maps that differed only in the degree of clearcutting and road density. We analyzed several landscape pattern metrics for the landscape as a whole and for the lodgepole pine and spruce/fir cover classes across these maps, and determined the relative effects of clearcutting and road building on the pattern of each watershed. At both the landscape- and cover class-scales, clearcutting and road building resulted in increased fragmentation as represented by a distinct suite of landscape structural changes. Patch core area and mean patch size decreased, and edge density and patch density increased as a result of clearcuts and roads. Clearcuts and roads simplified patch shapes at the landscape scale, but increased the complexity of lodgepole pine patches. Roads appeared to be a more significant agent of change than clearcuts, and roads which were more evenly distributed across a watershed had a greater effect on landscape pattern than did those which were densely clustered. Examining individual watersheds allows for the comparison of fragmentation among watersheds, as well as across the landscape as a whole. Similar studies of landscape structure in other National Forests and on other public lands may help to identify and prevent further fragmentation of these areas.     

A factor analysis of landscape pattern and structure metrics

Fifty-five metrics of landscape pattern and structure were calculated for 85 maps of land use and land cover. A multivariate factor analysis was used to identify the common axes (or dimensions) of pattern and structure which were measured by a reduced set of 26 metrics. The first six factors explained about 87% of the variation in the 26 landscape metrics. These factors were interpreted as composite measures of average patch compaction, overall image texture, average patch shape, patch perimeter-area scaling, number of attribute classes, and large-patch density-area scaling. We suggest that these factors can be represented in a simpler way by six univariate metrics - average perimeter-area ratio, contagion, standardized patch shape, patch perimeter-area scaling, number of attribute classes, and large-patch density-area scaling.     

Surface metrics for landscape ecology: a comparison of landscape models across ecoregions and scales

Context

The patch-mosaic model is lauded for its conceptual simplicity and ease with which conventional landscape metrics can be computed from categorical maps, yet many argue it is inconsistent with ecological theory. Gradient surface models (GSMs) are an alternative for representing landscapes, but adoption of surface metrics for analyzing spatial patterns in GSMs is hindered by several factors including a lack of meaningful interpretations.

Objectives

We investigate the performance and applicability of surface metrics across a range of ecoregions and scales to strengthen theoretical foundations for their adoption in landscape ecology.

Methods

We examine metric clustering across scales and ecoregions, test correlations with patch-based metrics, and provide ecological interpretations for a variety of surface metrics with respect to forest cover to support the basis for selecting surface metrics for ecological analyses.

Results

We identify several factors complicating the interpretation of surface metrics from a landscape perspective. First, not all surface metrics are appropriate for landscape analyses. Second, true analogs between surface metrics and patch-based, landscape metrics are rare. Researchers should focus instead on how surface measures can uniquely measure spatial patterns. Lastly, scale dependencies exist for surface metrics, but relationships between metrics do not appear to change considerably with scale.

Conclusions

Incorporating gradient surfaces into landscape ecological analyses is challenging, and many surface metrics may not have patch analogs or be ecologically relevant. For this reason, surface metrics should be considered in terms of the set of pattern elements they represent that can then be linked to landscape characteristics.

     

The transiogram as a graphic metric for characterizing the spatial patterns of landscapes

Context

Landscape metrics play an important role in measurement, analysis, and interpretation of spatial patterns of landscapes. There are a variety of different landscape metrics widely used in landscape ecology. However, existing landscape metrics are mostly non-graphic and single-value indices, which may not be sufficient to describe the complex spatial correlation and interclass relationships of various landscapes. As a transition probability diagram over the lag distance, the transiogram, which emerged in recent years, essentially provides a new graphic metric for measuring and visualizing the auto and cross correlations of landscape categories.

Objectives

To explore the capability of the transiogram for measuring spatial patterns of categorical landscape maps and compare it with existing landscape metrics.

Methods

Sixteen commonly-used landscape metrics and transiograms (including auto- and cross-transiograms) were estimated and compared for land cover/use classes in four areas with different landscapes.

Results

Results show that (1) these transiograms can provide visual information about the proportions, aggregation levels, interclass adjacencies, and intra-class/interclass correlation ranges of landscape classes; (2) sills and auto-correlation ranges of transiograms are correlated with the values of some landscape metrics; and (3) the peak height ratios of idealized transiograms can effectively represent the juxtaposition strength of neighboring class pairs.

Conclusions

The transiogram can be an effective graphic metric for characterizing the auto-correlation of single classes (through auto-transiograms) and the complex interclass relationships, such as interdependency and juxtaposition, between different landscape classes (through cross-transiograms).

     

Landscape patterns from mathematical morphology on maps with contagion

The perceived realism of simulated maps with contagion (spatial autocorrelation) has led to their use for comparing landscape pattern metrics and as habitat maps for modeling organism movement across landscapes. The objective of this study was to conduct a neutral model analysis of pattern metrics defined by morphological spatial pattern analysis (MSPA) on maps with contagion, with comparisons to phase transitions (abrupt changes) of patterns on simple random maps. Using MSPA, each focal class pixel on a neutral map was assigned to one of six pattern classes—core, edge, perforated, connector, branch, or islet—depending on MSPA rules for connectivity and edge width. As the density of the focal class (P) was increased on simple random maps, the proportions of pixels in different pattern classes exhibited two types of phase transitions at threshold densities (0.41 ≤ P ≤ 0.99) that were predicted by percolation theory after taking into account the MSPA rules for connectivity and edge width. While there was no evidence of phase transitions on maps with contagion, the general trends of pattern metrics in relation to P were similar to simple random maps. Using an index P for comparisons, the effect of increasing contagion was opposite that of increasing edge width.     

Impact of data integration technique on historical land-use/land-cover change: Comparing historical maps with remote sensing data in the Belgian Ardennes

Historical reconstructions of land-use/cover change often require comparing maps derived from different sources. The objective of this study was to measure land-use/cover changes over the last 225 years at the scale of a Belgian landscape, Lierneux in Ardennes, on the basis of a heterogeneous time series of land cover data. The comparability between the land-cover maps was increased following a method of data integration by map generalisation. Two types of time series were built by integrating the maps either by reference to the initial map of the time series or by pair of successive maps. Land-cover change detection was performed on the initial time series without data integration and on the two types of integrated time series. Results reveal that land cover and landscape structure have been subject to profound changes in Lierneux since 1775, with an annual rate of change at the landscape level of up to 1.40%. The major land-cover change processes observed are expansion of grasslands-croplands and reforestation with coniferous species, leading to amore fragmented landscape structure. The annual rates of land-cover change estimated from integrated data are significantly different from the annual rates of change estimated without a prior integration of the data. There is a trade-off between going as far back in time as possibleversus performing change detection as accurately as possible. This revised version was published online in July 2006 with corrections to the Cover Date.     

Quantifying Spatiotemporal Patterns of Urban Land-use Change in Four Cities of China with Time Series Landscape Metrics

This paper provides a dynamic inter- and intra-city analysis of spatial and temporal patterns of urban land-use change. It is the first comparative analysis of a system of rapidly developing cities with landscape pattern metrics. Using ten classified Landsat Thematic Mapper images acquired from 1988 to 1999, we quantify the annual rate of urban land-use change for four cities in southern China. The classified images were used to generate annual maps of urban extent, and landscape metrics were calculated and analyzed spatiotemporally across three buffer zones for each city for each year. The study shows that for comprehensive understanding of the shapes and trajectories of urban expansion, a spatiotemporal landscape metrics analysis across buffer zones is an improvement over using only urban growth rates. This type of analysis can also be used to infer underlying social, economic, and political processes that drive the observed urban forms. The results indicate that urban form can be quite malleable over relatively short periods of time. Despite different economic development and policy histories, the four cities exhibit common patterns in their shape, size, and growth rates, suggesting a convergence toward a standard urban form.     

Determining an appropriate minimum mapping unit in vegetation mapping for ecosystem restoration: a case study from the Everglades,USA

This paper documents the analyses that were conducted with regards to investigating an appropriate Minimum Mapping Unit (MMU) to be used to capture the potential changes in vegetation patterns for a 10,924 square km restoration project being conducted in south Florida, USA. Spatial landscape and class metrics that were shown to change predictably with increasing grain size were adopted from previous studies and applied to a multi-scale analysis. Specifically, this study examines the effects of changing grain size on landscape metrics, utilizing empirical data from a real landscape encompassing 234,913 ha of south Florida’s Everglades. The objective was to identify critical thresholds within landscape metrics, which can be used to provide insight in determining an appropriate MMU for vegetation mapping. Results from this study demonstrate that vegetation heterogeneity will exhibit dissimilar patterns when investigating the loss of information within landscape and class metrics, as grain size is increased. These results also support previous findings that suggest that landscape metric “scalograms” (the response curves of landscape metrics to changing grain size), are more likely to be successful for linking landscape pattern to ecological processes as both pattern and process in ecological systems often operate on multiple scales. This study also incorporates an economic cost for various grain dependant vegetation mapping scales. A final selection of the 50 × 50 m grain size for mapping vegetation was based on this study’s investigation of the “scalograms”, the costs, and a composite best professional judgment of seasoned scientists having extensive experience within these ecosystems.     

Global analysis and simulation of land-use change associated with urbanization

A combination of rapid population growth and an accelerating demographic shift from rural to urbanized habitats has resulted in urbanization becoming an increasingly global phenomenon. Two alternate hypotheses describing urban landscape trajectories suggest urbanization is either leading to more homogeneous global patterns or urbanization has dichotomous trajectories of increasing dispersal or coalescence. To better understand the global variation in urban land-cover patterns and trajectories we described the variation in urban landscape structure for 120 cities distributed throughout the world assessed at circa 1990 and 2000. We coupled these data to a low-dimensional neighborhood based model of urban growth using a data-model fusion approach. Trajectories of urban growth were assessed using both the original data and model projections to 2030. The patterns of landscape change were related to both the rate of growth and income. The historical patterns of change showed a trend of increasing landscape complexity and this trend was projected to continue. Urban rate of growth was closely related to the change in several landscape metrics. Income was associated with landscape dynamics and this effect interacted with city size. Large cities were less sensitive to the income effect than small cities. Along with changes to the magnitude of each metric, the overall variation in metrics between years generally exhibited a decrease in variability and this variability was projected to continue decreasing. These findings supported the hypothesis that urban landscapes are becoming more homogeneous and that the dispersal-coalescing dichotomy represent endpoints rather than alternate states of urban growth.     

Analysis and simulation of land-use change in the central Arizona – Phoenix region,USA

To understand how urbanization has transformed the desert landscape in the central Arizona – Phoenix region of the United States, we conducted a series of spatial analyses of the land-use pattern from 1912–1995. The results of the spatial analysis show that the extent of urban area has increased exponentially for the past 83 years, and this urban expansion is correlated with the increase in population size for the same period of time. The accelerating urbanization process has increased the degree of fragmentation and structural complexity of the desert landscape. To simulate land-use change we developed a Markov-cellular automata model. Model parameters and neighborhood rules were obtained both empirically and with a modified genetic algorithm. Land-use maps for 1975 and 1995 were used to implement the model at two distinct spatial scales with a time step of one year. Model performance was evaluated using Monte-Carlo confidence interval estimation for selected landscape pattern indices. The coarse-scale model simulated the statistical patterns of the landscape at a higher accuracy than the fine-scale model. The empirically derived parameter set poorly simulated land-use change as compared to the optimized parameter set. In summary, our results showed that landscape pattern metrics (patch density, edge density, fractal dimension, contagion) together were able to effectively capture the trend in land-use associated with urbanization for this region. The Markov-cellular automata parameterized by a modified genetic algorithm reasonably replicated the change in land-use pattern.     

Regional land-cover conversion in the U.S. upper Midwest: magnitude of change and limited recovery (1850–1935–1993)

Land-use legacies can persist for hundreds to thousands of years, influencing plant species composition, nutrient cycling, water flows, and climate. To understand how land use has affected regional land-cover composition in Wisconsin (USA), we assessed the magnitude and direction of change in land cover between: (1) c.1850, at the onset of Euro-American settlement; (2) c.1935, the period of maximum clearing for agriculture following widespread forest logging; and (3) 1993, which, especially in northern Wisconsin, follows farm abandonment and forest recovery. We derived land-cover maps using U.S. Public Land Survey records (c.1850), the Wisconsin Land Economic Inventory (c.1935), and Landsat TM satellite data (1993). We stratified Wisconsin (145,000 km²) into two ecological provinces and used spatial error models, multinomial logistic regression, and non-metric multi-dimensional scaling ordination to examine change. Between 1850 and 1935, forest cover in the North declined from 84% to 56%, cropland increased to 24%, and mixed/coniferous forests and savannas were replaced by deciduous forests. In the South, formerly dominant savannas (69%) and prairies (6%) were mostly converted to cropland (51%) and pasture (11%). Remnant deciduous savannas and coniferous forests and savannas were replaced by deciduous forests. Remarkably little recovery to pre-settlement land-cover classes occurred from 1935 to 1993. Less cropland was abandoned than expected, and there was little net gain in coniferous/mixed forest. Based on these general land-cover classes, current cover is significantly different from that in 1850, but not from that in 1935, and thus continues to reflect historical logging and agricultural patterns. These results provide a historical framework for measuring associated changes in ecosystem function and can be used to guide restoration where desirable and feasible.     

Statistical models of landscape pattern metrics,with applications to regional scale dynamic forest simulations

Forest managers in Canada need to model landscape pattern or spatial configurationoverlarge (100,000 km²) regions. This presents a scalingproblem, as landscape configuration is measured at a high spatial resolution,but a low spatial resolution is indicated for regional simulation. We present astatistical solution to this scaling problem by showing how a wide range oflandscape pattern metrics can be modelled from low resolution data. Our studyarea comprises about 75,000 km² of boreal mixedwoodforest in northeast Alberta, Canada. Within this area we gridded a sample of 84digital forest cover maps, each about 9500 ha in size, to aresolution of 1 ha and used FRAGSTATS to compute a suite oflandscape pattern metrics for each map. We then used multivariate dimensionreduction techniques and canonical correlation analysis to model therelationship between landscape pattern metrics and simpler stand table metricsthat are easily obtained from non-spatial forest inventories. These analyseswere performed on four habitat types common in boreal mixedwood forests: youngdeciduous, old deciduous, white spruce, and mixedwood types. Using only threelandscape variables obtained directly from stand attribute tables (totalhabitatarea, and the mean and standard deviation of habitat patch size), ourstatistical models explained more than 73% of the joint variation in fivelandscape pattern metrics (representing patch shape, forest interior habitat,and patch isolation). By PCA, these five indices captured much of the totalvariability in the rich set of landscape pattern metrics that FRAGSTATS cangenerate. The predictor variables and strengths of association were highlyconsistent across habitat classes. We illustrate the potential use of suchstatistical relationships by simulating the regional, cumulative effects ofwildfire and forest management on the spatial arrangement of forest patches,using non-spatial stand attribute tables.This revised version was published online in May 2005 with corrections to the Cover Date.