Glen D. Johnson

Characterizing watershed-delineated landscapes in Pennsylvania using conditional entropy profiles

When the objective is to characterize landscapes with respect to relative degree and type of forest (or other critical habitat) fragmentation, it is difficult to decide which variables to measure and what type of discriminatory analysis to apply. It is also desirable to incorporate multiple measurement scales. In response, a new method has been developed that responds to changes in both the marginal and spatial distributions of land cover in a raster map. Multiscale features of the map are captured in a sequence of successively coarsened resolutions based on the random filter for degrading raster map resolutions. Basically, the entropy of spatial pattern associated with a particular pixel resolution is calculated, conditional on the pattern of the next coarser ‘parent’ resolution. When the entropy is plotted as a function of changing resolution, we obtain a simple two-dimensional graph called a ‘conditional entropy profile’, thus providing a graphical visualization of multi-scale fragmentation patterns.Using eight-category raster maps derived from 30-meter resolution LANDSAT Thematic Mapper images, the conditional entropy profile was obtained for each of 102 watersheds covering the state of Pennsylvania (USA). A suite of more conventional single-resolution landscape measurements was also obtained for each watershed using the FRAGSTATS program. After dividing the watersheds into three major physiographic provinces, cluster analysis was performed within each province using various combinations of the FRAGSTATS variables, land cover proportions and variables describing the conditional entropy profiles. Measurements of both spatial pattern and marginal land cover proportions were necessary to clearly discriminate the watersheds into distinct clusters for most of the state; however, the Piedmont province essentially only required the land cover proportions. In addition to land cover proportions, only the variables describing a conditional entropy profile appeared to be necessary for the Ridge and Valley province, whereas only the FRAGSTATS variables appeared to be necessary for the Appalachian Plateaus province. Meanwhile, the graphical representation of conditional entropy profiles provided a visualization of multi-scale fragmentation that was quite sensitive to changing pattern.

Multiresolution fragmentation profiles for assessing hierarchically structured landscape patterns

For landscapes that are cast as categorical raster maps, we present an entropy based method for obtaining a multiresolution characterization of spatial pattern. The result is a conditional entropy profile which reflects the rate of information loss as map resolution is degraded by increasing the pixel size through a resampling filter. We choose a random filter because of desirable properties that simplify calculations. Neutral landscapes that are simulated by stochastic generating models provide a way to evaluate the behavior of conditional entropy profiles under known hierarchically scaled generating mechanisms. When the random filter is used, we provide a method to directly compute the conditional entropy profile for specified generating models. Such profiles can provide benchmarks for comparing results obtained from raster maps of actual landscapes that are classified from satellite images. These profiles appear to capture much of the information about a landscape pattern that is otherwise obtained by a suite of landscape measurements which characterize different aspects of spatial pattern.

Stochastic generating models for simulating hierarchically structured multi-cover landscapes

For simulating hierarchically structured raster maps of landscapes that consist of multiple land cover types, we extend the concept of neutral landscape models to provide a general Markovian model. A stochastic transition matrix provides the probability rules that govern landscape fragmentation processes by assigning finer resolution land cover categories, given coarser resolution categories. This matrix can either be changed or remain the same at different resolutions. The probability rules may be defined for simulating properties of an actual landscape or they may be specified in a truly neutral manner to evaluate the effects of particular transition probability rules.For illustration, model parameters are defined heuristically to simulate properites of actual watershed-delineated landscapes in Pennsylvania. Three landscapes were chosen; one is mostly forested, one is in a transitional state between mostly forested and a mixture of agriculture, urban and suburban land, while the third is fully developed with only remnant forest patches that are small and disconnected. For each landscape type, a small sample of raster maps are simulated in a Monte Carlo fashion to illustrate how an empirical distribution of landscape measurements can be obtained.

Cost analysis of composite sampling for classification

When an environmental sampling objective is to classify all the sample units as contaminated or not, composite sampling with selective retesting can substantially reduce costs by reducing the number of units that require direct analysis. The tradeoff, however, is increased complexity that has its own hidden costs. For this reason, we propose a model for assessing the relative cost, expressed as the ratio of total expected cost with compositing to total expected cost without compositing (initial exhaustive testing). Expressions are derived for the following retesting protocols: (i) exhaustive, (ii) sequential and (iii) binary split. The effects of both false positive and false negative rates are also derived and incorporated. The derived expressions of relative cost are illustrated for a range of values for various cost components that reflect typical costs incurred with hazardous waste site monitoring. Results allow those who are designing sampling plans to evaluate if any of these compositing/retesting protocols will be cost effective for particular applications.

Fragmentation profiles for real and simulated landscapes

When a natural landscape is represented by a series of categorical raster maps of varying resolution, a multiresolution characterization of spatial pattern can be obtained in which entropy is computed at each resolution conditional on the next coarser resolution. The series of entropy values is plotted as a function of resolution, resulting in a multiresolution profile of fragmentation pattern in the landscape. If a categorical raster map is available at a single resolution only, a series of degraded maps at increasingly coarser resolutions is generated and the fragmentation profile is computed for this series. An algorithm has been developed for obtaining the profile directly from the single resolution map without having to generate and store the coarser resolution maps. A hierarchical stochastic model is described for simulating categorical raster maps and the fragmentation profile of the generating process is obtained in terms of the model parameters. These “process” profiles provide benchmarks for assessing empirical profiles obtained from raster maps of actual landscapes. Methods of the paper are applied to several watersheds of Pennsylvania using landcover maps derived from satellite imagery. These examples indicate that characteristic landscape types induce characteristic features in their fragmentation profiles.

Multiscale assessment of landscapes and watersheds with synoptic multivariate spatial data in environmental and ecological statistics

The paper attempts to provide a multiscale assessment of landscapes and watersheds using synoptic multivariate spatial data. Multiscale assessment is a frontier problem in environmental and ecological statistics today. The paper briefly deals with univariate surface data, multivariate signal data, and multicover categorical data, and applies stochastic conceptualization involving dendrogram trees and conditional entropies with special reference to the landscapes and watersheds of Pennsylvania.

Multiscale Analysis of the Spatial Distribution of Breeding Bird Species Richness Using the Echelon Approach

As an important component of biodiversity monitoring, species richness is a response variable of considerable interest to conservation biologists. Monitoring over a large spatial extent (such as statewide) requires an objective method for characterizing the spatial distribution of species richness. Objectivity is needed for delineation of areas that are relatively species rich or poor, and to also provide a way to compare other spatial distributions, as with temporal monitoring for change detection. The echelon approach introduced by Myers, Patil and Taillie (1995) provides a way to objectively characterize spatial structure of response variables that are at least ordinal. We apply this method to the characterization of breeding bird species richness across the state of Pennsylvania, using US-EPA EMAP hexagons which constitute a very coarse grain (635 km2). After identifying relative species rich areas at the coarse measurement scale, the hexagons covering such areas are analyzed in much finer detail, using the same echelon protocol but with a measurement scale (grain) equal to approximately 24 km2 rectangles corresponding to one sixth of a 7.5 minute USGS topographic quadrangle.

Using Covariate-Species Community Dissimilarity to Guide Sampling for Estimating Breeding Bird Species Richness

Estimating species richness over a certain geographic region becomes a problem of sampling enough area to reach the plateau of the true species-area curve. Sampling efficiency is then achieved by minimizing the area that requires measurement. Johnson and Patil (1995) suggested that covariate-directed sampling may help achieve efficiency by choosing sample units that have the greatest chance of containing different habitat and therefore different species. By retrospectively sampling the breeding bird community in Pennsylvania, various sampling protocols are investigated in conjunction with various covariate species. Results indicate which combinations of covariate species and sampling protocols may be more efficient than random sampling and which may be less efficient.