Scale effects in uncertainty modeling of presettlement vegetation distribution

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Scale effects in uncertainty modeling ofpresettlement vegetation distributionE.-H. Yoo a & A.B. Trgovac aa Department of Geography , University at Buffalo , Buffalo, NY,USAPublished online: 23 May 2011.

To cite this article: E.-H. Yoo & A.B. Trgovac (2011) Scale effects in uncertainty modeling ofpresettlement vegetation distribution, International Journal of Geographical Information Science,25:3, 405-421, DOI: 10.1080/13658816.2010.518390

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International Journal of Geographical Information ScienceVol. 25, No. 3, March 2011, 405421

Scale effects in uncertainty modeling of presettlement vegetationdistribution

E.-H. Yoo and A.B. Trgovac

Department of Geography, University at Buffalo, Buffalo, NY, USA

(Received 2 August 2009; final version received 22 August 2010)

This article proposes a geostatistical interpolation method, namely, block indicatorkriging, as a means to model uncertainty-quantified presettlement vegetation surfaces.We demonstrate its potential in landscape ecology for improving the quality of theresulting surfaces using indicator-coded presettlement land survey record data and theareal proportion of each species. The geostatistical interpolation method presented inthis study explicitly models support differences between the source data and the pre-diction surface, while taking into account spatial dependence present in multiscalepresettlement land survey record data. In this case study, we demonstrate that block indi-cator kriging with areal proportion data substantially increases the prediction accuracyand coherence using witness tree species data recorded in the northeast of Minnesota.The relative merit of the proposed geostatistical interpolation method to other con-ventional geostatistical approaches is illustrated through a comparative analysis, wherecontinuous vegetation surfaces obtained from other approaches are compared withthose obtained from the block indicator kriging with areal proportion data in termsof the prediction accuracy and consistency.

Keywords: block indicator kriging; presettlement land survey records; spatial depen-dence; support differences

1. Introduction

Survey notes from the US General Land Office as well as private land companies pro-vide a record of the vegetation found across the United States before European settlement.Historically, these presettlement land survey records (PLSR) have been used to reconstructthe spatial distribution of individual tree species as well as vegetation communities (Howelland Kucera 1956, Delcourt and Delcourt 1974, Grimm 1984). With many applications ofsuch distributions, from providing a baseline for the ecological restoration of disturbedareas to examining forest changes over time, the task of creating a continuous distributionof presettlement vegetation often employs spatial interpolation as evidenced by the existingliterature (Brown 1998, Batek et al. 1999, He et al. 2000, Cogbill et al. 2002, Rathbun andBlack 2006, Wang 2007).

Generally speaking, spatial interpolation falls into two categories depending on thenature of the resulting surface: deterministic and probabilistic. Deterministic interpolationmethods are preferred in a situation where sufficient knowledge about the phenomenon

*Corresponding author. Email: [email protected]

ISSN 1365-8816 print/ISSN 1362-3087 online 2011 Taylor & FrancisDOI: 10.1080/13658816.2010.518390http://www.informaworld.com

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406 E.-H. Yoo and A.B. Trgovac

of interest is available. Under such circumstances, a thorough understanding of the phe-nomenon allows for a complete description of the sample data as well as a robust predictionof unknown values. When such spatial interpolation methods are applied in ecologicalstudies (Batek et al. 1999, He et al. 2000, Cogbill et al. 2002), potential errors associatedwith the reconstructed vegetation surfaces are implicitly assumed to be negligible. Unlikedeterministic methods, probabilistic approaches, which include geostatistical models andBayesian hierarchical spatial models, typically acknowledge a lack of understanding andprovide mappable uncertainty-quantified probabilities associated with the predicted pre-settlement vegetation surface (Hershey 2000, Rathbun and Black 2006, He et al. 2007,Wang 2007). Recognizing that the available information regarding presettlement vegeta-tion is incomplete, it is necessary to quantify the spatial uncertainty associated with thereconstructed vegetation distribution obtained using PLSR. Despite its significance, uncer-tainty assessment of the reconstructed presettlement surfaces has often been overlooked inexisting studies.

In addition to reporting the uncertainty associated with the prediction surface, anotherimportant issue in mapping the spatial distribution of presettlement tree species is the selec-tion and change of support associated with the sampled point data and the reconstructedsurface. Here, support pertains to the areal extent of a datum and a sought-after prediction(Journal and Huijbregts 1978, Olea 1991), and it has been referred to as spatial resolution,spatial scale, or grain in the relevant literature (Turner 1989, Quattrochi and Goodchild1997, Atkinson and Tate 2000, Dungan 2001). From the vast research conducted usingPLSR, it is clear that issues relating to spatial scale have drawn researchers attentions.Some investigators have focused on the spatial scale of the source data and have exploredhow a selectively sampled subset of the point data affects the quality of the reconstructedsurface or the landscape pattern (He et al. 2000, Manies and Mladenoff 2000, Wang andLarsen 2006). Others have focused on the target surfaces and have examined how thechoice of the spatial scale of the prediction surface affects the landscape pattern analy-sis (Delcourt and Delcourt 1996, Wu 2004). Both questions are valid as the quality of theprediction surface is affected by both the spatial scale of the observation (source data) andthe spatial scale of the prediction (target surfaces). In fact, the consideration of supporteffects in the context of spatial interpolation is a part of a larger framework of scalingissues previously identified in various disciplines including geography (Atkinson and Tate2000, Dungan 2001, Gotway and Young 2002).

To the authors knowledge, however, none of the existing literature on the process ofmapping presettlement tree species explicitly accounts for the change of support from pointsurvey data to the predicted vegetation surface. Spatial interpolation changes the supportof a variable, creating a new variable that is related to the original data but has differ-ent statistical and spatial properties (Chils and Delfiner 1999, Gotway and Young 2004).Therefore, it is necessary to consider the effects that may arise due to changes of supportwhen creating interpolated surfaces of presettlement vegetation.

Given the significance of spatial scale on the model prediction, another intriguing lineof research is to describe the complex spatial structure of the forest using the PLSR dataat multiple spatial scales. Geostatistical applications in the earth sciences typically involvearea (or volume) averages of the primary variable being estimated, mainly because dataare defined over areal support rather than point support (Journel 1999). Although pointdata provide high-resolution information at a single location, areal average data providea general description of the data at a coarser resolution, for example, as a nonstationarylocally varying mean. In addition, the kriging prediction at an unsampled location retainsonly the nearest conditioning data when large sample data are used, which may result in the

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International Journal of Geographical Information Science 407

loss of valuable large scale information (Goovaerts 1997). Therefore, conditioning the pre-diction of unsampled value to both local information and areal data will provide valuableinformation to build a numerical model.

The challenge of incorporating such areal average data in the task of mapping the pre-settlement tree species is to design a set of meaningful areal units over which the arealproportion of each species is determined. This problem has long been identified as a fun-damental aspect of data analysis in geography, known as the modifiable areal unit problem(Openshaw and Taylor 1979). One approach to address this question is to design areal unitsspecifically to meet some goals (Flowerdew and Green 2001). In the context of mappingof the spatial distribution of presettlement vegetation, such a goal would be to capturethe regional effects in tree species distribution that may not be captured using PLSR data.Recognizing the influence that environmental factors have on the spatial distribution of treespecies and the overall forest mosaic, the utilization of PLSR data at areal units congruentwith relevant environmental characteristics may prove to be a meaningful covariate.

In summary, we aim to reconstruct uncertainty-quantified presettlement vegetationsurfaces using multiscale PLSR data. The objectives are threefold: (i) mapping theuncertainty-quantified spatial distributions of common tree species using both indicator-coded PLSR data and their spatial average values, (ii) accounting for the supportdifferences between the source data and the prediction surface through block indicatorkriging, and (iii) assessing the effects of the areal data on the predictive capabilities of thegeostatistical model across various spatial scales of prediction.

2. Background

The study area for this investigation is Lake County, MN, USA. Situated in the north-eastern portion of the state, Lake County is bounded by Ontario, Canada, to the northand Lake Superior to the south (Figure 1a). The county lies within the Laurentian MixedForest ecological province, which is characterized by broad expanses of coniferous andmixed hardwood forests as well as localized areas of bogs and swamps. Before Europeansettlement, the uplands were dominated by conifers, especially white (Pinus strobes),red (Pinus resinosa), and jack (Pinus barksiana) pines, whereas quaking aspen (Populustremuloides)paper birch (Betula papyrifera) associations were found in disturbed areas(Friedman et al. 2001). The lowland forests were populated with tamarack (Larix laricina)and black spruce (Picea mariana).

The original bearing tree records for Minnesota have been published online (MinnesotaDNRGIS Data Deli 2009) and contain a wealth of ecological information including speciestype, location, distance, and direction from the survey corners and tree diameter. In thisstudy, we focus on the species attribute of the 16,192 witness trees located within the studyregion. The data were transformed into an indicator (or binary) variable depicting the pres-ence or absence of individual tree species at each survey location. For illustrative purposes,only three common tree species aspen (Populus sp.), jack pine, and paper birch wereselected for this study considering that they have sufficient stature as witness trees duringa land survey and distinct spatial distributions widespread with small clustered patches,highly concentrated at the center, and evenly spread all over the study region, respectively(see Figure 1df).

The spatial patterns of the presettlement forest are likely to be related to the patchesof environmental conditions, that the individual tree species located within the same areaare commonly exposed to (Friedman et al. 2001). The geologic features of Lake Countyshow the strong influence of the regions glacial history. The southern portion of the

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Figure 1. (a) The study area of Lake County encompasses the northeastern portion of Minnesota.Within the study area, a total of 1846 four tree-plots, that is, survey corner locations where fourindividual species were present, were identified; (b) contour map for elevation; (c) soil texture ofthe study area; (d), (e), and (f) show survey locations revealing the presence of aspen (a total of1281), jack pine (a total of 1831), and paper birch (a total of 2104) with point symbols; (g), (h), and(i) are proportions of each tree species, aspen, jack pine, and paper birch, over 35 environmentallyhomogeneous zones.

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county is covered by the remains of the terminal and ground moraine of the Superiorlobe (Hobbs and Goebel 1982). Till plains, drumlin fields, and peat bogs dominate themidsection of the county whereas scoured bedrock and glacial lakes are prevalent to thenorth. The soils of the county also reflect the glaciation of the region with coarse loamysoils dominating the northern portion, sandy and hemic soils found in the center, anda thin covering of soils of various textures that comprise the moraine landforms in thesouth.

Recognizing that such diverse landscape and edaphic conditions of the study area mayafford an opportunity to study the spatial distribution of presettlement tree species acrossvarious environmental gradients, we identified 35 zones sharing similar environmentalcharacteristics. More specifically, two environmental covariates, elevation and soil textureshown in Figure 1b and c, were considered as significant drivers to influence the distri-bution of the selected tree species (Friedman et al. 2001, Lichstein et al. 2002, Bolligerand Mladenoff 2005, He et al. 2007), and a set of areal units sharing a relatively homo-geneous soil texture and elevation is identified from them. Both the elevation and the soiltexture data are from Minnesota DNR GIS Data Deli (2009). The areal units are usedas a spatial basis for the areal proportion data, whose attribute values are calculated asthe ratio between the number of individuals of the species (aspen, jack pine, and paperbirch) and the number of individuals of all species occurring within each areal unit. Theseareal proportion values amount to the relative abundance determined for each tree speciesat a regional scale, which complements the fine-scale composition and local variation ofthe individual tree records by accounting for environmental phenomena that take place orare present at a larger spatial scale. Figure 1gi, respectively, illustrates the spatial dis-tribution of areal proportions occupied by the three tree species. These areal proportionsfor each tree species are used as auxiliary data in the uncertainty-quantified vegetationsurface mapping as well as informed areal data in the model validation. Such areal datahave rarely been used in previous studies as a source of additional information, althoughthey can be substantially effective to increase the accuracy of predicted vegetation surface(see Section 4.2). We chose the coherence property of the predicted probabilities as a cri-terion to evaluate the model performances, that is, if the informed areal proportion value ateach zone is reproduced by the spatial average of predicted probabilities within that zone(see Section 4.3).

Finally, it is known that plot-scale species aggregation often provides useful informa-tion in addition to the absolute value obtained from the indicator transformed PLSR atsurvey locations (Cottam and Curtis 1956). For example, various measures of aggregation,such as relative frequency and numerical dominance, observed at the plot level may sug-gest patterns of spatial dependence associated with the reproductive or ecological nature ofthe species (Friedman et al. 2001). Although spatial distributions of individual tree speciesoccurrence at a single location may be of value in economic estimates, they are of lim-ited utility in landscape ecology and more likely to be subject to confounding factors andmeasurement errors. In this study, the degree of aggregation is quantified by the numberof individuals at each plot site, and this information is used as a source in the model vali-dation (see Section 4.2). To increase the reliability of plot-level data, such validation dataare taken only at survey corner locations in the PLSR data where four individuals werepresent, referred to as four tree-plot hereafter (Friedman et al. 2001). The imposition ofthis additional constraint identified only a total of 1846 four tree-plots (Figure 1a) from theoriginal 6769 survey corner sites.

In what follows, we will illustrate a geostatistical framework that integrates the multi-scale PLSR data for model predictions, while taking into account their spatial correlations

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across various spatial scales and the support differences among multiscale PLSR data andprediction surfaces.

3. Methods

Consider the problem of modeling the uncertainty about the tree species attribute at anunsampled location. Let S(u) denote a discrete random variable (RV) associated with apresettlement tree type {sk , k = 1, . . .,K} present at any location uwithin a study domain A.Here u denotes a vector of spatial coordinates, that is, u = (x, y). The uncertainty intrinsicto the spatial classification of a tree species at location u can be modeled by the conditionalprobability distribution function of the discrete RV S(u) derived from survey data at nlocations {u , = 1, . . ., n} (Journel 1983, Goovaerts 1997). At each survey location, thespecies attribute is transformed into a vector of K local prior probabilities correspondingto the K tree species as P{S(u) = sk} = 1, if the kth tree species was witnessed at uand 0 otherwise. For the kth tree species, indicator-coded data can be arranged in a (n 1) vector ik = [ik(u), = 1, . . . , n]T , where ik(u) denotes a realization of the indicatorRV Ik(u) corresponding to the kth tree species at the th survey location u with Ik(u)= 1, if S(u) = sk and 0, otherwise. Here, the superscript T denotes the transposition of avector or matrix.

Although the indicator data associated with the individual tree species occurrence at thesurvey locations provide local details of individual species distribution, it would be usefulto consider spatial variations at a larger spatial scale to understand the complex spatiallystructured properties of the forest. The areal proportions of individual tree species overselectively chosen spatial units may provide a landscape spatial pattern that is reflective ofthe environmental factors which take place at that scale. More specifically, consider a setof areal units that yield a (M 1) vector fk = [fk (w), = 1, . . .,M]T of areal proportionsof the kth tree species occurring over all survey sites within that areal unit. The th arealunit is denoted as w = w(u) centered at u and its attribute value is calculated as alinear average of the indicator data as fk(w) = 1nw

nw=1 ik(u), u w for k = 1, . . ., K,

where nw denotes the total survey sites within the areal unit. This transformation of pointdata to areal proportion values amounts to the rescaling of PLSR data to a process-basedcomponent observed at a coarser spatial scale.

Given a set of point indicator data and areal proportions {ik , fk , k = 1, . . . , K} for theindividual tree species, K conditional probabilities at an unsampled location u0 equal a setof K conditional expectation values of the corresponding indicator RVs, that is, pk(u0) =P{S(u0) = sk|ik , fk} = E{ik(u0)|ik , fk} for k = 1, . . ., K. The conditional probability for thek-th tree species at unvisited location u0 can be predicted as a linear combination of n pointindicator data ik and M areal proportion values fk as

pk(u0) =n

=1k(u0)ik(u) +

M=1

k(u0) fk(w)

= iTk k(u0) + fTk k(u0)(1)

where k(u0) = [k(u0), = 1, . . . , n]T denotes a (n 1) vector of weights assigned topoint indicator data ik and k(u0) = [k(u0), = 1, . . . ,M]T denotes a (M 1) vector ofweights assigned to areal proportion values fk . These two sets of weights k(u0), k(u0)for the kth tree species are the solution of (n + M + 1) linear equations, often referred toas an indicator ordinary kriging (IOK) system (see Appendix 2). Note that the above IOK

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system with areal proportion data does not necessarily call for regular supports for arealdata. The extended IOK system is valid as long as its attribute values are a linear averageof point data. When the unknown value is related to different attributes, for example, pointindicator or areal proportions of co-occurrent species, the IOK system is easily extendedto cokriging. Refer to Journel (1983), Goovaerts (1997), and Liu (2007) for further detailson IOK without/with area average values or indicator ordinary cokriging.

Here we assume that the necessary conditions for IOK have been met, that is, pointindicator data ik and areal proportions values fk are linked to a realization of an intrinsicstationary point random function {Ik(u), u A} with a constant but unknown mean anda stationary variogram model k(h) as a function of the separation vector h =||u u||between any two pairs of locations u, u A within the study region. The consistent IOKpredictions with the areal proportion values used are guaranteed by the adequate modelingof area-to-area variogram between any pair of areal proportion values and area-to-pointvariogram between any pair of areal proportion values and point indicator data as well aspoint predictions. The coherence, referred to as mass-preserving or pycnophylactic prop-erty (Tobler 1979, Lam 1983, Wahba 1990), of the prediction probabilities, that is, the arealproportion value is reproduced when point conditional probability predictions are reaggre-gated within the support of the areal datum considered, is guaranteed through a consistentmodeling of the area-to-area and area-to-point variograms (see Appendix 1).

In the application of indicator kriging to presettlement forest studies, the support ofthe reconstructed vegetation surfaces is typically larger than the point support of the wit-ness tree data. The continuous surface of each tree species distribution is approximatedby a finite collection of block IOK prediction values. The task is to model the uncertaintyabout the proportion of a block v occupied by the kth tree species by a block conditionalprobability P{S(v) = sk|ik , fk}. Although various techniques are available to derive a blockconditional probability from point predictions (Journel and Huijbregts 1978, Isaaks andSrivastava 1989, Saito and Goovaerts 2002), here we use a numerical approximation rep-resented by the arithmetic average of point or pseudo-point prediction values within theblock. Then, the pth block IOK prediction, denoted as pk(vp), is calculated from a weightedaverage of the discrete point probabilities within the support vp as

pk(vp) 1nv

nvj=1

pk(uj), uj vp (2)

where nv denotes the number of pseudo-points within the pth target block vp.

4. Results and discussion

In the following subsections, the indicator approach is used to predict the conditional prob-abilities of each tree species occurring at various spatial scales both with and without theareal proportion data. The influence of spatially aggregated data on the models predictivecapabilities is assessed in terms of the prediction accuracy to the degree of species aggre-gation at a tree-plot scale and the consistency of the predicted probabilities at multiplescales of prediction.

4.1. Spatial continuity modeling

The spatial continuity (dependence) of the individual tree species is modeled through a var-iogram and the corresponding model parameters are summarized in Table 1. Experimental

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Table 1. Parameters of point indicator variogram models.

Structure (I) Structure (II)

Taxa Type Nugget Sill Rangea Sill Range

Aspen Exponential 0.62 0.22 1.60 0.16 12.70Jack pine Exponential 0.28 0.47 1.40 0.25 24.00Paper birch Spherical 0.53 0.36 1.20 0.11 15.00

Note: aThe distance is measured on the kilometer scale (km).

variograms computed from the 16,192 indicator-coded data are fitted using either an expo-nential or a spherical model with two nested structures. A lag distance of approximately0.6 km was used and variograms were computed to a maximum distance of 30 km, giventhat the dimensions of the study area are approximately 58 km (EW) and 138 km (NS).

The fitted point variogram models for all the tree species have relatively high sillvalues with various ranges reflecting the spatial continuity of each tree species. Amongthree species considered, aspen has the smallest population size (1281 occurrences among16,192 witness trees) and occurred at a lower level of aggregation, which is also evidencedin the highest nugget effect (62%) of the fitted variogram model. Ecologically, paper birchis capable of colonizing disturbed areas rapidly on a wide variety of soil types but is rel-atively short-lived and susceptible to fire. Such traits may yield the highest abundanceamong the three species considered but may not necessarily yield in aggregation in its spa-tial distribution, as evidenced by the high nugget effects (53%) and medium level (36%)of spatial dependence at a short distance (1.2 km). Jack pine, the most clustered amongthe three species, is capable of forming selective stands, particularly on sandy soils. Oncepast the sapling stage, the bark of the jack pine affords it some protection from fire allow-ing mature individuals to survive through this particular disturbance type. Additionally,jack pine exhibits fire-mediated serotiny allowing it to be among the first to establish itselfon the recently disturbed landscape, and potentially form monospecific stands. Althoughthis study did not include an investigation into edaphic conditions or disturbance regimes,it might be assumed that jack pines clustered distribution with a strong spatial conti-nuity (approximately 50% of the spatial variability up to 1.4 km) may be due to thesecharacteristics.

The point-to-point, point-to-area, and area-to-area variogram associated with the arealproportion values for the IOK prediction are computed from the fitted point variogrammodels of each tree species (Table 1).

4.2. Spatial prediction with areal proportion data

Given the model of spatial continuity, uncertainty associated with the individualtree species occurrence at any location is assessed through indicator kriging modelswithout/with areal proportion data. For both cases, the unknown probability of each treespecies occurring is predicted using point indicator data, but the areal proportion valuesare considered simultaneously only in one of the kriging systems.

The two IOKmodels are evaluated in terms of their prediction capabilities to the degreeof species aggregation at the plot scale by comparing the number of individuals per speciesat each plot with the local average of predicted probabilities at the collocated locations.

Given that the plot data are collected using a certain distance measure from a cornerlocation to a bearing tree, generally refered to as point to plant methods (Cottam and Curtis

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1956), the comparative analysis calls for the identification of the collocated locations (pix-els) of the two IOK prediction surfaces with the four tree-plot sites along with the distanceused to measure from sampling point to the tree closest to it. More specifically, the spatialsetting used to derive the metrics at the four tree-plots is approximated from the calcula-tion of the average distance between sampling points and their nearest four trees and theobservation that the four tree-plots are situated approximately at 1.6 km intervals, whereasthe IOK prediction surfaces are available over a regular grid with size 0.4 0.4 km2. Insummary, at each pixel collocated with the four tree-plot sites the local average of the pre-dicted probabilities of any pixel located within 0.8 km is computed and compared with thenumber of individuals of species derived at the corresponding plot.

Table 2 shows the summary statistics, mean, and variance (within parenthesis), of thelocal average conditional probabilities of tree occurrences obtained through IOK withoutand with areal proportion data, respectively. The number of conspecific species present ata four tree-plot varies between 0 and 4, which indicates the level of aggregation occupiedby each of the three tree species. Measures of the degree of species aggregation at this plotscale often provide useful information to characterize the spatial distribution of each treespecies over the study region, although the primary interest of this study is in assessingthe reproducibility of such measures between the two prediction models. To facilitate thecomparison between the number of individuals per plot and the local average of predictedconditional probabilities, the range of the expected proportion corresponding to each num-ber of individuals per plot is adjusted by smoothing (or averaging) the local average ofconditional probabilities (0.00.9).

In general, both IOK model predictions locate within the expected range of propor-tions. However, through all categories (04) and for all three species, the local averageof IOK predictions with areal data is closer to the center of the range with smaller vari-ances. The instances where the local average of IOK predictions without areal proportionsunderestimates the degree of aggregation at tree-plot scale outnumbered those obtainedusing areal proportions for both aspen and paper birch. For aspen, the smaller populationsize and less clustered spatial distribution may result in the underestimation of IOK pre-dictions both without and with areal data, although the incorporation of areal proportiondata improves the quality of prediction by reducing the number of understimated instancesand decreasing the variances of the predictions. The aggregation level of paper birch isrelatively well described in both IOK models except at the highest aggregation (four indi-viduals per plot). This is likely due to its widespread distribution and large population size.The species aggregation pattern of jack pine is well reproduced regardless of the use ofareal data, although the incorporation of areal proportion improves the precision, as themean of the local average probabilities is closer to the center of the range and the varianceis smaller for each category.

The incorporation of areal data in IOK prediction clearly improves the model capabili-ties to reproduce the species aggregation pattern at a tree-plot scale, although the degree ofcontribution of areal data depends on the spatial pattern and the abundance of each species.The more dispersed and scarce is a species, the more informative the areal data are in theIOK predictions.

4.3. Support differences in spatial prediction

The prediction of a conditional probability of individual tree species occurrence at a singlelocation is rarely a goal of ecological studies per se, but it is typically a preliminary step

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Table2.

The

summaryof

IOKpredictionsatthetree-plotlevelwithout

(wo)

/withareald

ata.

The

numberof

individualsperplot

a

Species

Arealdata

0(00.1)b

1(0.10.3)

2(0.30.5)

3(0.50.7)

4(0.70.9)

Aspen

woc

0.0507

(0.0047)

0.1362

(0.0091)

0.1933

d(0.0075)

0.2840

(0.0175)

0.3741

(0.0156)

with

0.0283

(0.0017)

0.1804

(0.0028)

0.3294

(0.0037)

0.5001

(0.0070)

0.6530

(0.0082)

Jack

pine

wo

0.0216

(0.0029)

0.2714

(0.0099)

0.4293

(0.0105)

0.6105

(0.0109)

0.7834

(0.0107)

with

0.0179

(0.0018)

0.2663

(0.0069)

0.4402

(0.0074)

0.6344

(0.0069)

0.8167

(0.0072)

Paper

birch

wo

0.0542

(0.0034)

0.2086

(0.0051)

0.3467

(0.0072)

0.4694

(0.0091)

0.6044

(0.0054)

with

0.0449

(0.0019)

0.2113

(0.0033)

0.3753

(0.0042)

0.5167

(0.0057)

0.6763

(0.0026)

Notes:a

takenonlyatthefour

tree-plots;b

theadjusted

rangeof

proportions(00.9)correspondingtothenumberof

individualsperplot.;cwithoutarealproportiondata;d

bold

type

indicatesinstanceswhereagivendegree

ofaggregationisunderestim

ated

bythecorrespondinglocalaverage

ofIO

Kpredictions.

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in creating a map of vegetation distributions or in a subsequent landscape pattern analy-sis (Delcourt and Delcourt 1996, Manies and Mladenoff 2000, Friedman et al. 2001, Wu2004, Wang and Larsen 2006). In various applications of spatial interpolation to producepresettlement vegetation surfaces, however, support differences between the source dataand the sought-after prediction surfaces are often overlooked.

A comparative analysis is conducted to demonstrate the consequences of ignoringsupport differences on indicator kriging predictions, where three kriging models centroid-based IOK, block IOK without areal data, block IOK with areal data are used toreconstruct the surface over a regular grid of four different cell sizes (0.8, 1.6, 2.4, 3.6 km).At each scale of prediction, the coherence property of the three model predictions isassessed to determine whether the informed areal proportion value of each areal unit isreproduced when the predicted conditional probabilities within the corresponding arealunit are reaggregated. Here, it is assumed that the PLSR data-driven areal proportion val-ues are representative of the true proportions of each tree species within that areal unit. Theprediction errors are summarized through mean absolute prediction error (MAPR), definedas MAPR = 100M

Mp= 1

fk(vp) pk(vp), where vp denotes the pth block constituting theprediction surface and M denotes the number of areal proportion data.

Note that the objectives of this comparative analysis are not only to evaluate the conse-quences of ignoring support differences between source data and target surface but also toquantify the influence of areal data. The contribution of the areal proportion data is assessedthrough a sensitivity analysis where kriging prediction is obtained using a random sampleof various sizes (30100% by 10% increments) of indicator point data accompanied byareal proportion data, which are derived from the exhaustive indicator point data.

Centroid-based IOK mimics the conventional approach where the prediction supportis collapsed into the centroid (or grid node) of the corresponding block, whereas the othertwo block indicator kriging models account for support differences between the sourcedata and the prediction surface. The block prediction probability at each scale of analy-sis is numerically approximated by the spatial average of pseudo-point IOK or block IOKpredictions without/with areal proportion data on grid cells at spatial resolution 0.4 0.4km2. The only difference between the two block IOK models is that the latter incorporatesareal proportion values as auxiliary data. Figure 2 shows the maps of the paper birch dis-tribution obtained through three different kriging models at the scale of prediction 1.6 km.The centroid-based IOK yields a smooth surface mixed with lots of salt and pepper asshown in Figure 2a, whereas block IOK prediction without areal proportion data in Figure2b yields the smoothest surface among the three results. The abrupt changes shown in thecentroid-based IOK prediction surface are attributed to the consequence of ignoring thesupport of prediction, where the unknown at the centroid of each pixel constituting theprediction surface is directly influenced by the nearest local point indicator data. In con-trast, the prediction surfaces obtained from block IOK tend to smooth out those artificialvariations of predicted conditional probabilities, where the influence of areal data in theblock IOK (Figure 2c) enhances local details while describing a spatially varying patternof species occurrences.

The results of coherence assessments are summarized in Figure 3. The coherence prop-erty quantified by MAPR is summarized per species, where the three kriging models areevaluated at various spatial scales of prediction using a subset of point indicator datawithout/with areal proportion data. Paper birch (Figure 3c) has the highest MAPR amongthree species, which is related with its widely distributed and moderately concentrated spa-tial pattern. In contrast, MAPR of jack pine is consistently low regardless of the spatialscales of prediction or the kriging model considered. The incorporation of areal proportion

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Figure 2. Conditional probability prediction surface of paper birch at the scale of prediction 1.6km through (a) centroid-based IOK, (b) block IOK without areal proportion data, and (c) block IOKwith areal proportion data.

data reduces the MAPR for all species, but the effects of areal data are different amongthem due to their distinct spatial patterns. As expected, the smaller the sample of the pointindicator data used, the more influential the areal proportion data are when incorporatedinto the kriging prediction. This can be seen particularly at finer spatial scales (0.8 km),evidenced by the larger gaps among three lines in Figure 3ac.

Another interesting result is that a critical spatial scale of prediction is found andabove this threshold the consideration of support differences between source data and tar-get surface affects the coherence of prediction surfaces. For example, when the surface isconstructed at a relatively coarse scale, a grid of cell size 3.6 km, the consideration of thedifferences between point and block support of the prediction surface affects the coherenceproperty of the prediction surface, which is summarized using MAPR. This is shown as asignificant difference between the two lines with the symbol circles and that with asteriskat the target scale of 3.6 km for all species. Caution should be taken in this generalization,however, as the results may vary depending on the spatial extent of the study and the spatialcontinuity of species.

A factor closely related to support differences in the assessment of the conditional prob-ability surface is the effects of the areal proportion data, shown in the differences betweenthe line with the square symbols and other two lines. Across all conditions considered, theincorporation of the areal proportion values in the kriging prediction improves the coher-ence quality of the prediction surface. The degree of influence of the areal proportion data,however, is different depending on the spatial patterns of species, the spatial scale of targetsurface, and the percent portion of the point sample data used. When the full set of pointindicator data is used (100%) to predict the conditional probabilities at a finest spatial scale(0.8 km), the influence of areal data is minimal and the spatial structure of the individualspecies can account for the differences [see the differences among the three symbols onthe right axis (100%) of the three graphs in the first column of Figure 3]. The incorpora-tion of areal data is the most effective at reducing the prediction errors (MAPR) for aspen,the tree species with the lowest level of spatial dependence. The opposite is also true asthe prediction errors associated with jack pine, the species with the highest level of spatialconcentration, are reduced minimally across all spatial scales of prediction. At a coarser

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30 50 70 100Percentage Percentage Percentage Percentage

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Centroid IOK Block IOK without areal data Block IOK with areal data

Figure 3. Coherence assessments of three kriging models (denoted by different symbols) perspecies [(a) aspen, (b) jack pine, and (c) paper birch] at various spatial scales of prediction (0.8,1.6, 2.4, 3.6 km) using a random sample of point indicator data of different sizes (30100% of the16,192 indicator-coded witness tree records). The line with circles () denotes mean absolute predic-tion error (MAPR) associated with the centroid-based IOK predictions, and the line with asterisks() and squares (), respectively, denote MAPR of block IOK without/with areal proportion data.scale of prediction, the effect of areal data is mixed with the support differences, althoughsensitivity analyses support the preliminary conclusion that the inclusion of areal propor-tion data in IOK improves the quality of the prediction surface as shown by lower overallMAPR.

Finally, the coherence of the prediction surface deteriorates as the target supportincreases. In particular, the IOK with areal data at a coarser scale (3.6 km) yields largerMAPR for three species. The increase in the MAPR is consistent regardless of the samplesize of the indicator point data. It should be noted that the MAPR of block IOK predictionswith areal proportion data reported in this study are not exactly 0 due to the approximationof survey locations to quasi-point support using a grid with 0.4 km resoulution. The MAPRof any block IOK with areal proportion data will be smaller if a very dense grid is imposedon the study region or if the point indicator data are moved to the nearest discretizationgrid nodes.

5. Conclusion

Geostatistical approaches have been used in several ecological studies to reconstruct thespatial distributions of individual tree species and historical vegetation communities using

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PLSR data. In these previous studies, geostatistics allows for the interpolation of the prob-ability distribution of a tree species occurrence at unvisited locations. However, the changeof support from the PLSR data to the prediction surfaces has been ignored by assumingthat support of the prediction surfaces is sufficiently represented by the set of centroids ofa grid overlaid in the study region, regardless of the size of the grid. In addition, the mul-tiscale properties of PLSR data, such as the regional distribution of a single tree speciesdescribed by the areal proportion, are rarely explored.

In this article, we demonstrate the use of block indicator kriging as a means to modelthe unknown presettlement vegetation surface, while accounting for support differencesbetween prediction surfaces and point-support PLSR. We also consider area-averaged indi-cator data as an alternative source of information, which has the potential for improving thequality of the prediction in the recostruction of the presettlement vegetation surface. Theuse of these areal proportion data in the endeavor of constructing uncertainty-quantifiedprobability surfaces is not free from limitation. The design of such areal units is controver-sial in that the proportion taken by each species defined over a set of such areal units variesin how the units are constituted and may, consequently, affect the quality of the result-ing surface (Flowerdew and Green 2001). In this article, a set of areal units is designedto maximize within-area homogeneity for the purpose of capturing the broadscale spatialvariation of species distribution under the assumption that tree species that are exposedto similar environmental conditions tend to have similar abundance. This may be arguableand further investigation with alternative areal units should be considered.

In the case study, we illustrated the contribution of areal proportion values on the pre-diction and a potential loss of accuracy (higher uncertainty) by ignoring support differencesbetween source data and prediction surfaces through a comparative analysis. PLSR datacollected in northeast Minnesota are used as a primary point source data and area-averagedindicator values of individual tree species are used as auxiliary data. An analysis of thecoherence property shows that incorporating auxiliary data in the prediction depends on thequality, quantity, and spatial configuration of existing data, but the quality of predictionsis improved as long as spatial dependence is present in the data. It was also clearly shownthat the indicator kriging prediction, which ignores the support differences (centroid-basedkriging), results in larger errors than block kriging through the comparative study overvarious spatial scales of prediction.

The uncertainty assessment of presettlement tree species distribution presented in thisstudy can be further extended by investigating (i) the spatial uncertainty associated withindividual tree species occurring at several locations, (ii) the block conditional probabilitiessimulated at multiple spatial resolutions, and (iii) the uncertainty propagation in the subse-quent analysis such as spatial classification or landscape pattern analysis, using indicatorstochastic simulation. Perhaps a further extension of this work would be to incorporate rele-vant environmental covariates, such as topographical and edaphic variables, into the spatialmodel to further refine the method of determining the spatial distribution of presettlementtree species.

Appendix 1. Indicator variogram with areal data

The variogram of areal proportions occupied by the kth tree species between any two arealsupports w , w is derived from the point support variogram model k(u, u + h) = k(h)as:

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k(w ,w ) = 1|w ||w |uw

uw

k(u,u) dudu (A-1)

where the variogram between any two pairs of areal units is the average point variogram k(u, u) corresponding to all possible separation vectors formed by any two locations u w and u w . Similarly, the variogram of the kth tree species occurrence at the thsurvey location u with respect to its areal proportion value determined over the th arealunit w is derived from the point variogram model k(u, u) as

k(u,w) =uw

k(u,u) du (A-2)

These regularized area-to-area and point-to-area variograms are stationary (for an arealunit with any size, i.e., |w |, , 1, . . ., M) under the assumption that the stationarypoint support variogram k(h) = k(u, u) of the kth tree species occurrence exists and theattribute value of the areal data is defined as a linear average of the point data.

This entails that the average variogram between informed point indicator-coded dataand areal average of indicator data is a function of the separation vector h = ||u w ||between the point u and any point located within the th areal unit w , that is, k(u, w) = k(h). Refer to Journel and Huijbregts (1978), Armstrong (1998), Boucher and Kyriakidis(2006), and Remy et al. (2009) for further discussion.

Appendix 2. Indicator ordinary kriging system with areal data

Data taken at different spatial scales, both point data and its linear average over areal sup-ports, can be considered simultaneously in the kriging system. The only constraint is thatthe areal data are defined as a linear average of point support data within the areal sup-port (Remy et al. 2009). The weights of IOK with areal proportions k(u0) and k(u0) areobtained by solving the following (n + M + 1) ordinary kriging system of equations:

[ uuk

uwk 1n

wuk wwk 1M

1Tn 1TM 0

][k(u0)k(u0)k(u0)

]=[ uuk tuk1

]

where uuk = [k(u ,u ),, = 1, . . . , n] denotes a (n n) matrix of autovariogramvalues between any pair of point indicator RVs corresponding to the kth tree species,and uwk =

[k(u,w), = 1, . . . , n, = 1, . . . ,M

]denotes a (nM) matrix of point-to-

area variogram values between any pair of point indicator RVs and areal proportion RVs,and wwk =

[k(w ,w ),, = 1, . . . ,M

]denotes a (MM) matrix of area-to-area var-

iogram values between any pair in areal proportion RVs. The Lagrange parameter k(u0)accounts for the unbiased constraint on the weights. Note that here, a single constraintreplaces the two nonbias conditions, such as one for the point indicator data and another forareal proportions, that is, 1Tnk(u0) + 1TMk(u0) = 1. Such a single unbiasedness constraintis allowed in the kriging system, when two variables have the same meaning and the sameexpectation, but different spatial structures (Journel and Huijbregts 1978, p. 325). In thisstudy, the areal proportion values are derived from point indicator data, which implies thattheir expected values are the same but they may have different spatial continuity models.

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Scale effects in uncertainty modeling of presettlement vegetation distribution