Plant Ecology 134: 97–111, 1998. 97c 1998 Kluwer Academic Publishers. Printed in Belgium.

Mapping historical forest types in Baraga County Michigan, USA as fuzzysets

Daniel G. BrownDepartment of Geography, Michigan State University, East Lansing, MI 48824-1115, USA

Received 19 November 1996; accepted in revised form 20 July 1997

Key words: Fuzzy sets, General Land Office survey, Geostatistics, Historical vegetation mapping

Abstract

Data on tree location and species in a portion of Northern Michigan were gathered from General Land Office (GLO)survey notes (ca. 1850), digitized, and generalized to represent forest types. Fuzzy membership values describingthe degree of membership of each species in each forest type were derived from (a) semantic information in theforestry literature and (b) a fuzzy clustering routine applied to data from randomly placed circular plots. The fuzzymembership values assigned to each tree point for each forest type were interpolated to form continuous surfacesusing kriging and co-kriging. Advantages of this method over traditional discrete mapping methods include: (a)multiple options are available for the display and analysis; (b) classification uncertainty and the continuity of naturalvegetation can be represented; and (c) the classification scheme is applied systematically across the entire map areaand can be altered to produce alternative maps. The subset of available display and analytical products presentedinclude: discrete forest type maps; a surface representing the confusion between forest types; fuzzy logical overlaysof forest types; and discrete class maps with color value altered within each class to indicate degree of confusionat each location.

Introduction

Maps of historical vegetation patterns provide inform-ation against which current patterns of vegetation canbe compared. The overwhelmingmajority of the veget-ation cover in the developed world (e.g., Europe andthe United States) has been disturbed by human occu-pation and use of the land. Relatively recent calls forpreserving the ‘natural state’ of ecosystems and vegeta-tion communities has required that land managers andscientists achieve a clearer understanding of what isthe ‘natural state’ of ecosystems. Although it may beargued that attempts at gaining such knowledge are notlikely to be successful because of the dynamic natureof ecosystems, the search for an understanding of thefunctioning of ecosystems and the impact that humanactivity has had on them will probably yield valuableinformation. It is this search for understanding aboutnatural ecosystems and ways to manage them that pre-serve their health that has lead to regional ecosystem

mapping efforts (e.g., Albert 1995) and to efforts athistorical vegetation mapping.

Mappable data on vegetation prior to widespreadhuman settlement are rare because data collectionprior to vegetation disturbance by humans requiredextraordinary forethought. In the United States, theRectangular (i.e., ‘township and range’) Survey hasresulted in the collection and archival of an extensiveand fairly consistent data set on vegetation prior towidespread settlement by people of European ancestry(White 1984). Surveyors were instructed to record treespecies and locations in the process of surveying theland for sale and/or settlement. Although the data setwas not originally collected for the purpose of produ-cing vegetation maps and was collected at a time whenmany of the lands were occupied, and the vegetationaltered, to some degree by Native Americans (ca. 1850in the Northern Michigan study area), these data rep-resent the best source from which vegetation maps ofthis time period have been generated (Veatch 1959;

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Marschner 1975; Finley 1976; Brewer et al. 1984;Williams 1989; Price 1994).

A primary limitation of any effort to map foresttypes from the tree data in the surveyor’s notes isthe inherent uncertainty and subjectivity in the place-ment of boundary lines. Subjectivity also characterizesthe development of classification schemes by whichgroupings of trees are assigned to categories. Becauseof their subjectivity and the fact that digitization ofmany of these efforts is carried out only after the finalmap has been created, the manual methods also sufferbecause alternative interpretations (i.e., using alternateclassification schemes) can be equally as time con-suming to implement as the original. Automating theprocess of converting the trees as points to forest typesas polygons in a geographic data base allows for altern-ate interpretations of the same data without repetitivemanual data collection and interpretation efforts. Anautomated method was developed for identifying andmapping forest types from the point data stored in theGLO survey notes, so that alternative mappings of suchforest types might be explored. Therefore, the goals ofthis paper are to (1) present an automated method forthe classification and interpolation of historical foresttypes, and (2) illustrate the analysis and visualizationuses to which a multilayer representation of historicalforest types can be put.

The automation effort presented here is an improve-ment because it (a) allows for the characterization ofthe spatial patterns of ambiguity in the resultant dataproducts, (b) permits the representation of gradualtransitions, or ecotones, between forest type polygons,and (c) can incorporate information on environmentalvariables (e.g., soil wetness). The end product of themapping process is not a paper map but rather a digitalrepresentation of multiple forest types. Separate layersare maintained to represent fuzzy memberships at eachlocation in each forest type. The resultant multi-layermodel involves representation of forest types as fuzzysets (Zadeh 1965; Burrough 1989), and allows for rep-resentation of the fact that any given location in an area(a) could belong to more than one category (i.e., thecategories themselves are fuzzy and the assignmentof a location to a class cannot be made with 100%certainty; Dale 1988) and (b) is less certain in placeswhere the neighboring trees are not clearly indicativeof one forest type and more certain where the treesmore clearly suggest one forest type (i.e., the spatialobjects, or polygons, are fuzzy). Figure 1 illustrateshow the classification and interpolation processes canbe fuzzy.

Figure 1. Forest types are fuzzy in terms of both (A) the classificationschema and (B) the polygon features. In (A) heavier solid linesrepresent stronger affinities of species to forest types and the dashedlines represent weakest affinities. All species are members of allforest types to some degree. In (B), Point 1 is clearly within onepolygon, Point 3 is ambiguously within a polygon, and Point 2 ismost ambiguous.

The study area and data

Baraga County is in Michigan’s upper peninsula andborders the southern shore of Lake Superior, the north-ernmost of the Great Lakes. The climate of the countyis cool, humid continental. Many of the landforms inBaraga County have glacial origins. The county wasessentially deglaciated by 10 500 BP, but the northerntownships were covered by deposits associated with theMarquette readvance at 9,900 BP, which left behind theKeewenaw moraine (Saarnisto 1974; Drexler 1977).The most prominent physiographic region is the Peshe-kee Bedrock Highlands in east-central Baraga County(Figure 2). This hilly, remote area has thin glacialdrift covering Precambrian rocks,primarily granite andgneiss. In eastern Baraga County the southern edge ofthe Keewenaw moraine adjoins the Peshekee High-lands. The flat, largely undissected Baraga OutwashPlains contrast with the more rolling and dissectedMenge Creek Outwash Plains. At the southern edge ofthe Baraga Plains the soils become wetter and swampsare widespread, as the plains merge into the SturgeonRiver floodplain. This river runs east-west across thecenter of the county, turns north, and eventually exitsthe county in the extreme NW corner (Figure 2). TheSouth Baraga Ground Moraine covers most of south-ern Baraga County, except for a small outlier of thePeshekee Highlands, called the Covington BedrockHighlands. The Peshekee or Covington Highlands, as

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Figure 2. Physiographic regions of Baraga County, after Berndt(1988) and Barrett et al. (1995), and location of study area.

well as the South Baraga Ground Moraine, are inter-mittently covered with a layer of silty, probably eolian,sediments ranging up to 60 cm in thickness.

Baraga County was surveyed between 1846 and1853. The data stored in the General Land Office(GLO) survey notes consist of individual tree spe-cies and locations, in distance and azimuth coordin-ates from survey section corners; summaries of relat-ive forest composition, recent disturbances, and majorlandscape features (e.g., wetlands) for section lines orsegments of section lines; and sketch maps of the land-scape features in an entire section or township. Becausethey represent the least subjective, most spatially andthematically precise, and most consistent of the datastored in the survey notes, the point data are the mostreliable and the others are not used in this mappingeffort.

Figure 3. Point locations of trees recorded in the General LandOffice survey notes (1846–1853) for Baraga County in Michigan’supper peninsula.

Surveyors were instructed to record the locations oftwo or four (depending on the year of the survey) treesat each section corner (each corner is about 1.6 km fromits nearest neighbors), one tree at the quarter-sectioncorner (i.e., the midpoint between section corners ona section line), and any trees that fell on the sec-tion line between section and quarter-section corners(White 1984). Section and quarter-section corner treesare referred to as witness trees and other trees fallingon section lines are referred to as bearing trees. Thenumber of section corner and quarter-section cornertrees was fairly consistent throughout the county (twoand one, respectively), but the number of line trees ona given section line varied in number (from zero to adozen or more) and spacing.

Copies of the transcribed GLO survey field noteswere obtained at the State of Michigan Library, Lans-ing. The distance and direction of all bearing and wit-ness trees from section corners, codes describing thesection corner by township, range and section, a codeindicating species (based on common names recor-ded by surveyors), and diameter at breast height wereentered into a spreadsheet and processed with a FOR-TRAN program to calculate the state plane coordinatesof the trees relative to a section corner database, alsoobtained from the State of Michigan. In total 12 760trees were recorded and used in this project (Figure 3).

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The primary limitation of the tree data stored in theGLO survey notes is probably their sparseness (Fig-ure 3). The average density of trees over the entirecounty is about 5 trees per km2 (or about 13 trees persurvey section). The locations of trees form a grid pat-tern with 2.59 km2 (or 1 mi2) areas within which notrees were sampled.

The potential for erroneous vegetation reconstruc-tions based on the GLO records has been addressed byCottam (1949) and Bourdo (1956). Surveyors tendedto ignore small-diameter trees. Because the survey-ors used common names for the trees, assignment ofeach tree to its proper genus and species was imper-fect. Some species may have been selectively avoidedor used depending on their average size, bark type,and branching configurations. Although such prob-lems make precise vegetation reconstruction difficult,the GLO data are the best data available for describinglandscape conditions prior to European settlement.

In addition to the tree data, digital soil survey datawere obtained from the Soil Conservation Service (nowthe Natural Resources Conservation Service) and usedin the construction of vegetation type maps (Berndt1988). The maps were interpreted from 1:20 000 scaleaerial photographs. Although many soil attributes wererecorded for each soil type, soil wetness was the onlyattribute used. A real number value representing soilwetness was assigned to each soil polygon based on itsdrainage class and, to some extent, the texture of thesoil. The resulting values ranged from 2.0 to 8.0 withmost being whole numbers assigned on the basis ofthe drainage class. Non-integer values were assignedin some cases where texture was believed to mediatesoil wetness. The approach is very similar to that takenby Schaetzl (1986). The resulting wetness index wasused to improve the mapping of the forest types.

Methods

Maps of ‘presettlement’ vegetation (e.g., Veatch 1959;Marschner 1975; Finley 1971; Brewer et al. 1984; andPrice 1994) have usually been constructed through amanual process involving plotting the tree locationson a base map, with some indication of species, andinterpreting some set of forest types from the spatialorganization of species. Occasionally, the line sum-maries and sketch maps in the survey were used tosupplement the tree data (Brewer et al. 1984), as weredata on soils (Veatch 1959). The end goal of the manu-al mapping efforts has been the production of paper

maps. Others have reconstructed historical vegetationpatterns from GLO notes for input to a geographicinformation system (GIS), but have limited their datasource to the less reliable surveyor’s sketch maps ofvegetation types (Bandi & Huggins 1994) or have takena probabilistic approach to representing discrete foresttypes from the tree data (Batek 1995). These effortshave resulted in the production of digital replicas ofpaper maps of historical vegetation.

The procedure employed here to produce a fuzzyrepresentation of forest types contained three primarysteps: (1) the identification of how species cluster toform forest types (i.e., classification) and the expres-sion of those clusters as fuzzy membership values, (2)the interpolation of membership values from tree pointsto grid cells for each forest type, and (3) the assignmentof sites to discrete and fuzzy forest types.

Classification methods

Although the classification process used differs fromthe manual methods of mapping forest types in thatthe classification scheme must be defined a priori, itformalizes the classification and ensures that it can beapplied consistently throughout the map area. Manu-al methods of mapping typically involve identifyingclusters or groups of species on a plot of trees andbounding those with a line. The clusters and groupscan change in composition throughout the map. Bypre-defining the classification scheme, the subjectiv-ity and inconsistency of the classification scheme isreduced.

The classification is fuzzy in that there are many-to-many relationships between species and forest types(Figure 1A). Each species belongs to each forest typeto some degree. In some cases, for example Jack Pinein a Northern Hardwood Forest, the memberships willbe very low. In others, for example Jack Pine in aJack Pine Forest, the memberships will be very high.The membership values represent the degree to whicha given forest type exists given a certain species. Thedifference between this and a traditional probabilisticapproach is that probability represents the likelihoodthat a given class exists, assuming that one and only oneclass can exist at one location, whereas the fuzzy mem-bership values indicate that each forest type exists ateach location (i.e., tree) to some degree. Because mul-tiple trees in a spatial neighborhood must be consideredin the interpretation of forest types, these membershipvalues should only be interpreted after they are inter-polated, using a spatial averaging process.

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Definition of fuzzy membership values (FMVs) iscentral to the application of fuzzy sets to any problem.In this project, FMVs for each species in each foresttype were defined using two alternative approaches.The first relied on expert knowledge presented in the lit-erature regarding the expected forest types in the studyarea. The second approach used observable spatial-statistical clusters of species occurring in the data. Ineach of these methods the end result is a set of FMVsthat numerically described the relationships betweentree species, for which data are available as point obser-vations, and forest types, which were to be generatedthrough the generalization process. Once defined, theforest type membership values were assigned to eachtree point based on its species for interpolation.

Semantic descriptorsSeven forest type classes were defined by combin-ing many of the classes described by Eyre (1980).The combinations were similar to the general classespresented by Barnes and Wagner (1981). The rela-tionships between the forest types in Eyre (1980) andtree species are described qualitatively in Burns &Honkala (1990). Those descriptions were used to con-struct FMVs.

Quantitative values indicating the membership of agiven species within each forest type were estimatedfollowing three steps: (1) the forest type compositionswere constructed by assigning each species to one offour categories of abundance within each forest type(A, B, C, or D); (2) assumptions about the relativestrengths of each of the categories of membership weremade; and (3) each species was assigned a numericestimate of its relative strength of membership in eachforest type such that all estimates for a given that spe-cies sum to one and the constraints defined in (1) and(2) were maintained.

Table 1 lists four categories, representing the rel-ative degree to which each species was a constituentof a given forest type, and the words used to inferthe relationships. The following rules were assumed inthis particular application: species in category A wereapproximately three times more abundant than those incategory C; species in category B were approximatelytwo times more abundant than those in category C; andcategory D always had a FMV of 0.03.

The fourth category represents all tree species thatwere never indicated as components of a given foresttype and a very low level of membership was assumed.Table 2 lists the relative abundance assumed for each

species in each forest type along with the resultingFMVs. The seven forest type classes selected on thebasis of Eyre (1980) and Barnes & Wagner (1981)were labeled Northern Hardwoods (Hwd), MixedPines (Pine), Conifer Upland (CUp), Conifer Bog(CBg), Deciduous Swamp (DSp), Jack Pine (Jack),and Aspen-Birch (AsB). Each has been described witha unique species composition and usually represents adistinct set of environmental and/or historical condi-tions.

Assignment of relative abundance categories, therelative weights of the categories, and the classes them-selves all involve subjective interpretation. However,because the mapping process is automated, anychanges, corrections, or alternative interpretations ofthese relationships can easily be translated into altern-ative forest type representations.This a priori approachto forest type definition has the advantage of usingexisting ecological knowledge in the identification offorest types, but is limited when the local forest pat-terns are different from the general patterns describedin the literature. Species associations vary from regionto region and site to site and defined classes may notfit the forest patterns in particular areas. The methodalso has a high degree of subjectivity, but it has theadvantage over many manually defined classificationsin that it can also be applied consistently across themap.

Fuzzy clusteringIn the second classification approach, naturally occur-ring spatial-statistical clusters of species in the datawere identified to form the basis of alternative FMVs.Two-hundred circular plots were randomly located andthe species composition of each plot was enumerated(Figure 4). The plots each had a diameter of approx-imately 2 300 meters, large enough to encircle all fourcorners of a survey section, to ensure that each plot hadan adequate sample of trees. Plots were removed fromconsideration if they had fewer than 10 trees, to reduceedge effects, or if the distance between the circle centerand that of any other circle was less than or equal tothe radius.

The fuzzy c-means (FCM) clustering algorithm byBezdek (1981) and Bezdek et al. (1984) was used toassign each circular plot to a set of clusters defined bythe species composition of each plot. FCM uses an iter-ative algorithm such that an initial, randomly definedset of clusters converge to a naturally occurring set.The number and composition of the clusters depended

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Table 1. Examples of words used within the ecological literature (especiallyBurns & Honkala 1981; Eyre 1980) to identify the degree to which a species is acomponent of a forest type. The four groupings are the author’s.

Decreasing strength of relationship

A B C D

‘major component’ ‘minor component’ ‘occurs’ no mention

‘principal component’ ‘common’ ‘infrequent’

‘prime constituent’ ‘associated’ ‘rare’

‘dominant’ ‘occasional’

‘pure stands’

Table 2. Relative abundance categories using semantic descriptors for each species in each of seven forest types and the resultingfuzzy membership values.

Species Relative abundance categories Fuzzy membership values

Hwd Pine CUp CBg DSp Jack AsB Hwd Pine CUp CBg DSp Jack AsB

Sugar Maple A C B D D D C 0.39 0.13 0.25 0.03 0.03 0.03 0.13

White Pine C A A B C B B 0.08 0.22 0.22 0.13 0.08 0.13 0.13

Ylw. Birch A C A C B C B 0.23 0.08 0.23 0.08 0.14 0.08 0.14

Red Maple A B A B B C B 0.21 0.13 0.21 0.13 0.13 0.07 0.13

Cedar C C C A B D D 0.12 0.12 0.12 0.35 0.22 0.03 0.03

Balsam Fir B B B A B C C 0.15 0.15 0.15 0.24 0.15 0.08 0.08

Tamarack D D D A C D D 0.03 0.03 0.03 0.64 0.22 0.03 0.03

Blk. Spruce B B B A B D B 0.15 0.15 0.15 0.24 0.15 0.03 0.15

Hemlock B C A C C D C 0.20 0.11 0.32 0.11 0.11 0.03 0.11

Black Ash C D C B A D D 0.13 0.03 0.13 0.25 0.39 0.03 0.03

Ironwood B D C D D D D 0.56 0.03 0.29 0.03 0.03 0.03 0.03

White Ash B C C D B D D 0.30 0.16 0.16 0.03 0.30 0.03 0.03

Aspen B B D B C B A 0.16 0.16 0.03 0.16 0.09 0.16 0.26

Alder B D D D C D C 0.43 0.03 0.03 0.03 0.23 0.03 0.23

Jack Pine D B B D D A C 0.03 0.22 0.22 0.03 0.03 0.35 0.12

Red Pine D A A D D B D 0.03 0.34 0.34 0.03 0.03 0.21 0.03

White Birch C B B D D B A 0.10 0.18 0.18 0.03 0.03 0.18 0.29

Basswood A C C D C D D 0.45 0.15 0.15 0.03 0.15 0.03 0.03

Blk. Cherry A C C D D D C 0.45 0.15 0.15 0.03 0.03 0.03 0.15

Red Oak A D B D D D D 0.52 0.03 0.33 0.03 0.03 0.03 0.03

Elm C D C C A D D 0.15 0.03 0.15 0.15 0.45 0.03 0.03

on the natural groupings of species that occurred in thelandscape at the scale of the plot size. The clusteringalgorithm was fuzzy in that each plot was assigned toeach of the identified clusters to some degree depend-ing on the species composition (expressed as propor-tions), the degree being indicated by a fuzzy member-ship value ranging from zero to one.

Although a detailed description of the FCMalgorithm is beyond the scope of this discussion, men-tion of one of the parameters is appropriate. Theweighting exponent (m) determines the fuzziness ofthe membership assignments. The weighting exponent

can vary between one, which indicates a hard classi-fication similar to traditional statistical clustering, and1. As m increases the distinctions between clustersbecome less and less obvious (i.e., they are blurred).For the sake of demonstration, m was fixed at a valuewhich results in fairly ‘hard’ clusters, 1.25. Accordingto Bezdek et al. (1984, p. 193), ‘the best strategy forselecting m at present seems to be experimental.’

Output from the FCM program given by Bezdeket al. (1984) includes descriptions of the clusters (i.e.,forest types) and statistics describing the overall fitof the classification (i.e., the partition coefficient and

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Figure 4. Locations of sample plots used for fuzzy clustering.

Figure 5. Smoothed experimental semi-variograms for the fuzzymembership values of each of the forest types: (A) Forest typesdefined using qualitative descriptions. (B) Forest types definedthrough fuzzy clustering of plot data.

Figure 6. Cross-variograms for the fuzzy membership values ofeach of the forest types and soil wetness: (A) Forest types definedusing qualitative descriptions. (B) Forest types defined through fuzzyclustering of plot data.

entropy). The algorithm was run at three different start-ing points (i.e., with three different random numbers)and for each number of clusters between three and sev-en. The results for five clusters were among the moststable to changes in the random starting point and hadentropy and partition coefficients which indicated thepresence of better developed clusters than at four, six,or seven clusters.

The average proportional species compositions ofplots within each cluster, presented in the output of theclustering routine as cluster centers, were normalizedso that the values summed to one for each species(Table 3). The normalized cluster centers served asthe FMVs for each species on each forest type (i.e.,the degree to which each species is a member of eachforest type). The order of clusters and their names hadno physical meaning. Note the similarities, however, inrelative FMVs between Type 2 and Conifer Bog, Type 3and Conifer Upland, Type 4 and Northern Hardwoods,and Type 5 and Jack Pine (Tables 2 and 3). Littlecorrespondence can be found between Type 1 and anyof the a priori classes, calling into question the validity

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Table 3. Mean species composition of five clusters (i.e., forest types) from the fuzzy c-means classifier and theresulting fuzzy membership values.

Species Cluster mean composition Fuzzy memberships values

Type1 Type2 Type3 Type4 Type5 Type1 Type2 Type3 Type4 Type5

Sugar Maple 19.3 12.0 17.2 43.3 0.0 0.21 0.13 0.19 0.46 0.01

White Pine 5.7 3.3 2.0 1.2 2.7 0.34 0.22 0.15 0.11 0.18

Ylw. Birch 25.1 11.1 12.3 12.1 0.1 0.40 0.18 0.20 0.20 0.02

Red Maple 3.8 3.4 3.0 3.5 0.0 0.26 0.24 0.21 0.24 0.05

Cedar 8.9 8.9 11.3 6.3 0.8 0.24 0.24 0.30 0.18 0.04

Balsam Fir 14.1 14.8 6.5 9.4 3.3 0.29 0.30 0.14 0.20 0.08

Tamarack 3.5 10.6 0.2 1.3 1.7 0.20 0.52 0.06 0.10 0.12

Blk. Spruce 9.4 28.0 2.8 4.6 4.1 0.19 0.54 0.07 0.10 0.09

Hemlock 4.8 2.5 37.1 9.9 0.0 0.10 0.06 0.64 0.18 0.02

Black Ash 0.9 0.3 0.4 1.0 0.0 0.25 0.17 0.19 0.27 0.13

Ironwood 0.3 0.1 0.7 1.2 0.0 0.17 0.15 0.23 0.30 0.14

White Ash 0.0 0.0 0.0 0.1 0.0 0.19 0.19 0.19 0.22 0.19

Aspen 1.0 0.5 0.6 1.2 0.0 0.24 0.18 0.19 0.26 0.12

Alder 0.1 0.1 0.0 0.1 0.0 0.20 0.20 0.19 0.21 0.19

Jack Pine 1.0 1.2 0.2 0.2 83.3 0.02 0.02 0.01 0.01 0.93

Red Pine 0.9 0.9 1.6 1.2 1.0 0.18 0.18 0.24 0.21 0.19

White Birch 0.8 2.0 2.6 1.4 3.0 0.12 0.20 0.24 0.16 0.27

Basswood 0.7 0.2 1.4 1.7 0.0 0.18 0.13 0.27 0.30 0.11

Blk. Cherry 0.0 0.4 0.0 0.3 0.0 0.18 0.24 0.18 0.23 0.18

Red Oak 0.0 0.0 0.1 0.0 0.0 0.20 0.20 0.21 0.20 0.20

Elm 0.0 0.0 0.0 0.0 0.0 0.20 0.20 0.20 0.20 0.20

of the three classes in the literature for which there isno observable analog in Baraga County at the currentscale of analysis (Mixed Pines, Deciduous Swamp, andAspen-Birch).

Interpolation methods

In the second step of the mapping process, the mem-berships of trees are interpolated to form a continuousmembership surface for each forest type using krigingand co-kriging on the FMVs. Detailed descriptions ofthese interpolation methods are beyond the scope ofthis paper. Readers seeking background informationon these methods are directed to Isaaks and Srivast-ava (1989), Oliver et al. (1989) or Rossi et al. (1992).Semi-variogram calculation, kriging, and co-krigingwere implemented using an Arc Macro Language(AML) interface written to take advantage of the FOR-TRAN language geostatistical subroutines provided byDeutsch & Journel (1992).

A semi-variogram, a graph of the relationshipbetween variance and distance separation, describesthe strength and scale of spatial dependence in theFMVs for each forest type, with a positive slope

indicating spatial dependence. Prior to interpolationthe calculated semi-variograms for the FMVs result-ing from each classification method (Figure 5) werevisually fitted with positive definite functions, or semi-variogram models, a requirement of kriging and co-kriging (Isaaks & Srivastava, 1989). Because differentforest types are composed of species with different dis-persal mechanisms and because their spatial patternsreflect disturbance history to varying degrees, identicalfunctions describing spatial dependence in the differ-ent forest types were not expected. Exponential modelswere used to fit all but 3 of the semi-variograms. Spher-ical models were used for the Jack Pine and Aspen-Birch types defined using semantic descriptors (Fig-ure 5A) and for Type 5 resulting from fuzzy clustering(Figure 5B). These forest types are all post-disturbancetypes and may be expected to exhibit the more abruptspatial dependence pattern that is characteristic of thespherical model when compared with the smootherexponential model.

The definitions of the forest types (i.e., usingsemantic descriptors as in Figure 5A versus fuzzyclustering as in Figure 5B) had an effect on the spa-tial pattern of forest type memberships. Although all

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semivariograms rise at short distances, indicating spa-tial dependence and interpolability, the forest typeswith the weakest spatial dependence may be difficultto interpolate (i.e., Aspen/Birch and Mixed Pines in thesemantically defined types and Type 1 in the statistic-ally defined types). This weaker spatial dependencemay indicate that these types are either (1) not presentin Baraga County with sufficient frequency to warrantmapping them, (2) weakly defined with respect to theother types, or (3) occur at a spatial scale that is toosmall for the data resolution.

Standardized ordinary co-kriging (Deutsch & Jour-nel 1992) was used to incorporate environmentalinformation into the FMV estimation process.Soil wet-ness information was extracted by overlaying each treepoint on the soil coverage. Additionally, 926 pointswith the soil wetness attribute, one in the center ofeach section, were added to the data set to improve thedensity of soil wetness sample points. In addition tosemi-variograms for each forest type category and forsoil wetness, cross-variograms were required to repres-ent the correspondence between the primary variables(i.e., forest type FMVs) and the secondary variable(i.e., soil wetness).

Cross-variograms indicate the degree of co-variation between forest type membership and soil wet-ness, and how that co-variation changes with distance(Figure 6). When the cross-variogram was positive,then membership in a class tended to increase with soilwetness; when it was negative, membership tendedto decrease with wetness. The cross-variograms weremodeled in a manner similar to the semi-variogramsfollowing the linear model of coregionalization (Isaaksand Srivastava, 1989).

Kriging and co-kriging were each performed oncefor every forest type category. Values for each of theforest type FMVs were interpolated from the tree pointsusing ordinary block kriging and co-kriging. AverageFMVs for blocks (or grid cells) of 303 m by 303 min size (approximately one-fifth the size of a sectionline on a side) were estimated. The co-kriging res-ults were very similar to the kriging results in eachclassification scheme. In fact, the Pearson correlationcoefficient (r) between the kriged and co-kriged mapsfor each type ranged from 0.979 to 0.998. The simil-arity between the representations may be attributed tothe small number of additional sample points used inthe co-kriging example.

The forest type model

The limitations of the traditional vegetation map (i.e.,that a finite set of vegetation types can be conveyed intwo dimensions through visual symbols) need not beimposed when the creation and use of a representationis carried out in a GIS. The goal of a ‘mapping’ exer-cise no longer necessarily implies the production of apaper, or even a visual, product. Each of the griddedforest type surfaces resulting from the classificationand interpolation processes (Figure 7) represents thepossibility that each grid-cell location belongs to eachof the forest type categories. This representation con-tains more information, for example about confusionbetween classes, than would a traditional map of foresttypes, wherein each location would be assigned to theone and only one type that maximized the probabil-ity that the type existed at that location. The spatialpatterns and juxtaposition of types is not well commu-nicated through the visual presentation in Figure 7, butadditional analysis and interpretation of such repres-entations can be conducted with the aid of GIS analyt-ical software. The next section describes some of theprocedures that can be followed to extract cartograph-ic and analytical products from the semi-continuousmulti-nomial forest type model described.

Options for mapping and analysis

Discrete maps

The simplest approach to portraying the forest typesusing the multiple surfaces is to assign each loca-tion (i.e., grid cell) to the class with the highest fuzzymembership value at that location. The result is a dis-crete classification, similar in nature to the type pro-duced by traditional mapping methods (Figure 8). Themajor landscape features of the county (Figure 2) wereapparent in Figure 8 (Barrett et al. 1995). The Baraga(Outwash) Plains, an area dominated by excessivelydrained and sandy soils, were also dominated by JackPine (Type 5). Hemlock dominated the uplands in thenorthwestern portion of the county (Type 3). A some-what linear occurrence of the lowland class (Type 2)occurred along the Sturgeon River in the extreme north-west. A band of Northern Hardwoods forest, domin-ated by Sugar Maple (Type 4), ran from southwestto northeast through the center of the county, follow-ing the Keweenaw Moraine. The southeastern portionof the county had many lowland areas, dominated by

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Figure 7. Shaded maps of kriged fuzzy membership values for five forest types identified through fuzzy c-means clustering. All maps are scaledsuch that white saturates for values below 0.4 and black saturates for values above 0.6.

Tamarack & Black Spruce (Type 2), which were inter-spersed with uplands (Types 1 and 3).

Confusion maps

The degree of confusion between different forest typesacross the landscape, based on the membership values,

107

Type 1

Type 2

Type 3

Type 4

Type 5

Figure 8. Traditional-style discrete forest type classification basedon the maximum membership value at each location.

Type 1

Type 2

Type 3

Type 4

Type 5

Figure 11. Forest type classification with degrees of membershipindicated by the intensity of the color. Only purest colors are shownin the legend.

can provide the viewer of a map of forest types or theuser of a digital representation with information abouta multi-nomial map that is not normally available. This

Figure 9. Entropy for fuzzy membership values of all 5 classes ateach location. Darker shades indicate greater entropy.

information can be used to examine ecotones, dis-count the quality of a classification where confusionis high, or re-assess the validity of a given classific-ation scheme. Entropy (Figure 9) takes advantage ofinformation theory to portray the degree to which eachlocation has equal membership across all the classesand is calculated by

h(x) =

�����1

ln(k)

kX

i=1

Vi(x )ln[Vi(x )]

�����; (1)

where h(x) is the entropy at location x, k is the numberof classes, and Vi(x) is the membership value for classi at location x (Goodchild et al. 1994).

This measure of class confusion provides a spatiallydifferentiated indicator of map quality. High entropyvalues indicate that the FMVs across all classes arefairly similar and that no one class dominates. Depend-ing on the purpose of the mapping, the map of entropymight be used in conjunction with a classified map(e.g., Figure 8), for example, to limit an analysis toonly the least ambiguous locations (i.e., those with thelowest entropy).

The degree of entropy in each of the mappedclasses was summarized by class type to assess someof the sources of ambiguity in the mapping method.Table 4 lists the mean entropy values, along with otherdescriptive information, for the seven classes describedin Table 2 and for the five classes described in Table 3.

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Table 4. Descriptive information on results of discrete classifications, determinedby assigning each cell to the class for which its membership value was maximized.Results are for kriging runs only.

Class name Total area Number of Mean polygon Mean

(km2) polygons area (km2) entropy

Semantic Types

Northern Hardwoods 1043 146 7.1 0.94

Mixed Pines 8 17 0.5 0.95

Conifer Upland 647 198 3.3 0.95

Conifer Bog 565 125 4.5 0.95

Deciduous Swamp 1 3 0.5 0.96

Jack Pine 69 2 34.7 0.87

Aspen-Birch 3 5 0.6 0.98

Statistical Types

Type 1 (Y Birch) 279 270 1.0 0.94

Type 2 (Conifer Bog) 643 134 4.8 0.93

Type 3 (Hemlock) 764 67 11.4 0.90

Type 4 (Hardwoods) 560 190 2.9 0.92

Type 5 (Jack Pine) 92 5 18.5 0.55

Table 4 lists only results for the maps produced throughkriging, but the values for the maps produced throughco-kriging were different by only 1 to 2% in mostcases. The relatively low amounts of confusion inthe Jack Pine class among the semantically definedclasses and in Type 5 among the fuzzy c-means classesresulted from the fact that most of the Jack Pines inthe county occurred in one, nearly pure, large stand.The forest types with the greatest amounts of confu-sion (i.e., entropy), including Aspen/Birch, DeciduousSwamp, and Type 1, were those which tended to occurin fairly small stands (Table 4). Also, classes with sim-ilar definitions were more likely to have high meanentropy values. Types 1 and 4 had similar species-composition definitions (Table 3) and it was likely thatthese classes were confused with each other more oftenthan with the remaining classes.The proximity of manyof the stands of these classes (Figure 8) also indic-ated that there was likely some definitional confusion.This indicates that it may be reasonable to combinethese two classes. Indeed, both Sugar Maple and Yel-low Birch were commonly listed as dominants in theNorthern Hardwoods forest type.

Fuzzy logical operations

Another use to which the maps of membership canbe put takes advantage of fuzzy logic. Categories canbe overlaid through union and intersection to represent

transitions and mixtures of two or more classes (Robin-son 1988; Burrough 1989). Transition zones, or eco-tones, can be common and maps of their locations canbe useful to users of vegetation information (Kuchler1988). Logical intersection (i.e., Boolean AND) canbe used to map transition zones as the degree to whicheach location represents all of two or more differentclasses at once. The HARD and SOFT AND opera-tions are performed by taking the minimum member-ship value across all forest types of interest or calcu-lating the product of all membership surfaces, respect-ively. Two or more classes can be combined using thelogical union operation (i.e., Boolean OR), or mappingthe degree to which each location is a member of anyone of two or more classes. The HARD and SOFTOR operations take the maximum value or sum of allmembership surfaces, respectively.

Figure 10 shows the intersection (HARD AND) andunion (HARD OR) between Type 4 (Northern Hard-woods) and Type 1 (Yellow Birch Upland). Figure 10Aportrays areas of overlap and/or transition betweenType 1 and 4. These two classes tended to overlap in abroad band trending southwest to northeast through thecenter of the county, among other locations. BecauseTypes 1 and 4 were similar in definition it may beprudent to combine them. The union (Figure 10B) rep-resents the combination of the membership surfacesfor Types 1 and 4 and might be used in lieu of the twoseparate membership surfaces.

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Figure 10. Fuzzy intersection (HARD AND) and fuzzy union(HARD OR) between Type 1 and Type 4 surfaces. Shading val-ues are scaled as in Figure 6.

Shaded class maps

Finally, in order to portray forest type and data qualityinformation simultaneously, discrete categories can beassigned on the basis of maximum fuzzy membershipwith sub-categories representing the relative member-ship value. Whereas the hue and saturation of the dis-

play color represents the class to which a location isassigned, the value (i.e., intensity) of the color used todisplay that class was altered on the basis of strengthof membership (Figure 11). The number of member-ship intervals available for each class can be altered toprovide varying degrees of fuzziness in the representa-tions, where one interval is a discrete classification andunlimited intervals allows for full fuzziness within thelimits of precision in the fuzzy membership values.

Figure 11 was prepared with three membershiplevels, determined by dividing the range of mem-bership values across all maps into three categories.The lower membership values tended to occur nearthe edges of individual stands, where the types bor-der on other types. This general relationship betweencertainty of classification and distance from a borderbetween types is consistent with the findings of others’work on fuzzy boundaries (Edwards & Lowell 1996;Wang & Hall 1996).

Conclusions

Information about the forest cover in North Americaprior to European settlement is scarce and generallylimited to narrative accounts. Although the land wasnot vacant at that time and there is evidence of substan-tial impact by Native Americans, especially by burning(Williams, 1989), the GLO survey records provide animportant baseline against which modifications to thelandscape since the time of European settlement can bemeasured. This paper has demonstrated how the datastored in those records can be automated to generaterepresentations of presettlement forest types that canbe analyzed in a GIS environment.

The identification of forest types as fuzzy, ratherthan crisp, sets permits explicit recognition of: (1) thespatially variable uncertainty in the forest type repres-entation; (2) the existence of transitions between foresttypes on the landscape; and (3) the ambiguity in foresttype definitions which can make forest type polygondelineation difficult. The classification and interpola-tion methods outlined yield more objective and consist-ent representations of presettlement forest conditionsthan do manual methods. The resultant representationsare richer portrayals of the distribution of species con-tained in the GLO records than simple discrete mapsbased on the same data because they contain informa-tion on confusion between multiple forest type classesat each location.

110

It was observed that the degree of confusion in map-ping a given forest type, measured using classificationentropy, varied with both the area covered by the foresttype polygons and the similarity between the attributedescriptions of the type and other types. Forest typestending to occur in small stands were more difficult toidentify because of the spatial averaging inherent in themethod and the fact that, near forest type boundaries,the mix of species becomes more ambiguous and dif-ficult to classify into one class or another. Also, wherethe definitions of classes were more similar to otherclasses the entropy of the classes tended to be greater.

Fuzzy sets are reasonable alternatives to crisp cat-egories whenever real world objects do not fit neatlywithin information categories designed for classifica-tion or analysis. As Roberts (1986, 1989) and Dale(1988) demonstrate, this situation can be quite com-mon in plant ecology. Indeed, much plant ecologicalwork is centered on identifying categories and crisporder in a fairly non-crisp world. Fuzzy sets mayprovide a framework and methodology with which toreconcile the traditional dichotomy of vegetation com-munities as identifiable units, called the community-unit concept (Shipley & Keddy 1987), versus vegeta-tion communities as chance assemblages of independ-ent species responding to gradients, called the con-tinuum concept (MacIntosh 1967). In other words, weshould recognize the importance of categories and clas-sification for analyzing a landscape, but should alsounderstand that at any given place or time those cat-egories may be imperfect and result in ambiguity.

Acknowledgements

This paper was presented at the NSF-NCGIA andESF-GISDATA First Summer Institute for GeographicInformation, Wolfe’s Neck, Maine, 27 July – 2 August1995. The work was funded, in part, by the Center forRemote Sensing and an All University Research Grantprovided by the College of Social Science, both atMichigan State University. The author acknowledgesthe work of the following individuals in interpreting thesurvey notes and entering the data: Linda R. Barrett,Thomas W. Cate, Robert Hunckler, Johan Liebens,David S. Nolan, and Patricia Zuwerink. Intellectualsupport from Dr Randy Schaetzl and the technical sup-port provided by Jue Hu are also appreciated.

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