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Uncertainty in the difference between maps of future land change scenarios

It is essential to measure whether maps of various scenarios of future land change are meaningfully different, because differences among such maps serve to inform land management. This paper compares the output maps of different scenarios of future
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  SPECIAL FEATURE: ORIGINAL ARTICLE  Land use and ecosystems Uncertainty in the difference between maps of futureland change scenarios Robert Gilmore Pontius Jr.  • Neeti Neeti Received: 14 June 2008/Accepted: 16 October 2009/Published online: 8 December 2009   Integrated Research System for Sustainability Science, United Nations University, and Springer 2009 Abstract  It is essential to measure whether maps of various scenarios of future land change are meaningfullydifferent, because differences among such maps serve toinform land management. This paper compares the outputmaps of different scenarios of future land change in amanner that contrasts two different approaches to accountfor the uncertainty of the simulated projections. The sim-pler approach interprets the scenario storyline concerningthe quantity of each land change transition as assumption,and then considers the range of possibilities concerning thevalue added by a simulation model that specifies the spatialallocation of land change. The more complex approachestimates the uncertainty of future land maps based on avalidation measurement with historic data. The techniqueis illustrated by a case study that compares two scenarios of future land change in the Plum Island Ecosystems of northeastern Massachusetts, in the United States. Resultsshow that if the model simulates only the spatial allocationof the land changes given the assumed quantity of eachtransition, then there is a clearly bounded range for thedifference between the raw scenario maps; but if theuncertainties are estimated by validation, then the uncer-tainties can be so great that the output maps do not showmeaningful differences. We discuss the implications of these results for a future research agenda of land changemodeling. We conclude that a productive approach is touse the simpler method to distinguish clearly betweenvariations in the scenario maps that are due to scenarioassumptions versus variations due to the simulation model. Keywords  Accuracy    Calibration    Geomod   Model    Simulation    Validation Introduction Research objectiveScientists use scenario modeling in conjunction with landchange simulation to envision the implications of choicesthat are likely to influence the quantity and spatial alloca-tion of future land types (Kok et al. 2007). The solid lines and boxes of Fig. 1 illustrate a common approach to landchange scenario analysis. The exercise begins with thecreation of storylines, since a land change scenario is astory told in words, numbers, and maps concerning how thefuture could unfold. Typically, investigators construct abusiness-as-usual storyline that assumes continuation of past trends, and then create alternative scenario storylinesthat assume changes from past trends (Meadows et al.2004). Each qualitative storyline then inspires inputs for aland change simulation model, since these models aredesigned to portray various future landscapes, based on theassumptions that they are given. The land change modelultimately produces maps of categories of future landtransitions. Investigators presume that they can learnsomething by comparing the differences among the mapsthat the model produces for the various scenarios. Thiscomparison can be helpful if the model portrays accuratelythe landscape that would occur if decision makers were toact in accordance with the assumptions of the qualitativestoryline. This comparison can be misleading if the maps Edited by Mitsuru Osaki and Ademola Braimoh, HokkaidoUniversity, Japan.R. G. Pontius Jr. ( & )    N. NeetiSchool of Geography, Clark University, 950 Main Street,Worcester, MA 01610-1477, USAe-mail:  1 3 Sustain Sci (2010) 5:39–50DOI 10.1007/s11625-009-0095-z  contain substantial uncertainty that is not communicatedclearly. The purpose of this paper is to compare twoapproaches to address the uncertainty in the maps that landchange scenario models can produce and to examine theimplications of the uncertainty for the interpretation of thedifferences between scenarios.The dashed lines and boxes of Fig. 1 show the additionalsteps in methodology that this paper proposes. This paperpresents two approaches to perform this part of scenarioanalysis. The proposed methods consider potentialadjustments to each raw scenario map in order to generatean additional adjusted map that reveals the map’s possibleuncertainty. Then we measure the differences between theadjusted maps from the two different scenarios. Throughthis process, the methods of this paper allow one to addressthe question, ‘‘Are there meaningful differences betweenthe raw scenario maps, given the model’s uncertainty?’’ Inother fields of research, scientists use statistical methods todetermine whether observed differences are meaningful,given the level of certainty of the evidence. This paperproposes similar types of methods to address this questionfor scenario maps. We illustrate the methods for a studyarea in the Plum Island Ecosystems of northeastern Mas-sachusetts, in the United States, which is a ‘‘Long TermEcological Research’’ site of the United States’ NationalScience Foundation (Fig. 2).Literature reviewThis paper is the next in a sequence that examines theaccuracy of models and the implications of those accura-cies for projections of future land change. This researchpath started several years ago when Pontius (2000) showed how to separate quantity error from allocation error in thecontext of land change model validation. Pontius et al.(2004) built on those concepts to measure the accuracy of a land change model with respect to a null model at multipleresolutions using historic data, but that work did notexamine the implications of the validation measurement forfuture projections. Pontius and Spencer (2005) showed how to extrapolate the overall accuracy of a model into thefuture at multiple resolutions, but their method neither Business-as-Usual Story Alternative Story Land Change Model Business-as-Usual MapAlternative Map Raw Scenario Comparison Uncertainty AssumptionsBusiness-as-Usual Map adjusted for uncertainty Alternative Map adjusted for uncertainty Scenario Comparisonadjusted for uncertainty Fig. 1  Solid lines  show a general approach to scenario modeling, dashed lines  show this paper’s additional methodological contributionto account for uncertainty Fig. 2  Twenty-six towns of thePlum Island Ecosystems innortheastern Massachusetts,United States40 Sustain Sci (2010) 5:39–50  1 3  produced a map to allow visualization nor quantified thevarious accuracies among multiple transitions. Pontiuset al. (2006) addressed those two aspects, but they analyzedthe uncertainty of only one business-as-usual projectioninto the future. This paper takes the next logical step in thispath of inquiry, i.e., to compare scenarios that areremarkably different on the raw surface, in order to seewhether there are meaningful differences between thescenario maps when the uncertainties are considered. Thistopic is important since Pontius et al. (2008a) applied avalidation procedure to 13 applications of peer-reviewedmodels and found that in only one case did the modelproduce a larger amount of correctly simulated change thanerror at the resolution of the raw data. Validation mea-surements show that model outputs contain substantialuncertainties when they simulate past land change, so it iscrucial that we have tools to understand the uncertainties asthese models simulate future land change.The world of land change modeling is large and grow-ing. There are numerous approaches, and several differentmodels can be found that illustrate each type of approach,so it can be difficult to know the most enlightening way toproceed. Popular approaches include neural nets, which aremachine learning algorithms that mimic the human brain(Pijanowski et al. 2005). In addition, there are other models that possess various characteristics of intensive calibrationalgorithms (Goldstein 2004; Silva and Clarke 2002). Some of these are combined with cellular automata (Almeidaet al. 2003; Clarke and Gaydos 1998; Engelen et al. 2003). Statistical models are yet another category that reliesheavily on calibration using historic data (Hilferink andRietveld 1999; McConnell et al. 2004). For example, the statistically based CLUE model is used frequently in con- junction with scenario modeling (Verburg et al. 2002). Agent-based models are a class that simulates the decision-making of human actors who influence land change, andthus can contain a very large number of control parameters(Brown et al. 2005; Castella and Verburg 2007; Manson and Evans 2007). Yet other models use an integration of  many techniques (Pontius et al. 2007). These models contain parameters that need to be set foreach model run. There can be substantial uncertaintyconcerning which parameters should be modified and howthey should be set to portray a particular scenario. Somemodels allow the user to set the parameters that determinethe quantity of each transition separately from the para-meters that determine the spatial allocation of the transi-tions. This feature is helpful when the scenario storylinesdictate the quantity of each transition and then model’s jobis to simulate the spatial allocation of the land transitions.This paper’s methods rely heavily on the importantconceptual distinction between (1) the quantity of landtransitions, and (2) the spatial allocation of land transitions.We hope we have designed the methods of this paper sothat they are relevant to a variety of research programs thatuse various modeling approaches.Sources of uncertaintyThere are numerous potential sources of uncertainty forsimulation models (Berk et al. 2002; Fang et al. 2006; Messina et al. 2008; Santner et al. 2003). We group these sources under three headings: the data, the model, andfuture land change processes. First, data are likely tocontain many types of uncertainty, so the model is likelyto read some erroneous information. For this paper, weuse the best available digital maps and assume the dataare correct, as is frequently done, in spite of the fact thatwe suspect that all data have errors. Secondly, modelscontain various types of uncertainty associated with howaccurately their algorithms express important processesand use those relationships to simulate land transitions.This paper considers one way to quantify this type of uncertainty by performing a validation exercise, but wedo not examine the qualitative internal workings of thecomponents of the computer algorithms as would benecessary for a more comprehensive analysis. Thirdly,future land change processes can be uncertain becausedecision making involves human free will, which can benon-stationary at a variety of levels. This paper assumesthat the scenario’s qualitative storyline expresses the non-stationary aspects of future decision making, while thesimulation model attempts to extrapolate stationary pro-cesses. Our approach concerning these three types of uncertainties is likely to make our results underestimatethe total amount of uncertainty. Materials and methods DataWe illustrate the approach using maps of two land covercategories, i.e., forest versus non-forest, for 1971, 1985,and 1999, which we call reference maps. These data wereoriginally produced using aerial photography, and thentranslated into vector maps that are freely available fromthe State of Massachusetts ( State has never assessed the accuracy of these maps.The Human-Environment Regional Observatory (http:// converted the maps toraster format with a pixel resolution of 30 m. The inde-pendent variables derive from the same source and are usedto drive the simulation of the spatial allocation of forestchange. These variables are slope, surficial geology, and 21categories of land use in 1971. Sustain Sci (2010) 5:39–50 41  1 3  Raw maps for two scenariosThe two future scenarios differ, starting with their story-lines concerning the quantity of each land transition. Thescenarios are purposely designed to be substantially dif-ferent in this respect in order to illustrate this paper’sconcepts.The business-as-usual scenario assumes a continuationof historic trends. The recent past shows a gross forest losson 4% of the landscape and a gross forest gain on 1% of thelandscape between 1985 and 1999. The business-as-usualscenario assumes these constant rates of annual incrementin area until all the forest of 1999 has been lost, which isprojected to occur at approximately 2130. The business-as-usual scenario also assumes a constant annual area of grossforest gain over the next 200 years as Fig. 3 portrays.Figure 4 shows analogous information for an alternativescenario that assumes substantial changes from the historicprocesses of land transformation. It assumes an extremecase that has no gross forest loss, while forest experiences agross gain until 2130, at which point the study area is 75%forest. Beyond 2130, the alternative scenario portrays asteady state landscape.For each of the two scenarios, the land change modelGeomod takes the quantities of projected land transitionsand allocates them in space to produce maps of the future(Pontius et al. 2001). Geomod calibrates a relationship between the land cover of 1999 and the three independentvariables mentioned in this paper’s Data section in order togenerate suitability maps that it uses to allocate futuretransitions spatially. For this application of Geomod, landchange is not stratified by sub-region and is not constrainedto occur on the border between forest and non-forest.Range for difference between maps, given storylinesOur first approach to uncertainty analysis computes therange for possible differences in the scenario maps, giventhe quantities of the transitions that the scenario storylinesassume at each point in time. These quantities constrain thedifference between the business-as-usual scenario map andthe alternative scenario map. The raw scenario maps areguaranteed to agree concerning persistence of both forestand non-forest during the first several decades of the sim-ulation regardless of the simulated spatial allocationbecause both scenario storylines call for persistence onmuch of the landscape. The scenario maps will never agreeconcerning forest loss because the alternative scenarionever simulates forest loss. The least possible differencebetween the scenario maps would occur if the simulationmodel were to allocate forest gain as similarly as possible inthe two scenarios; the greatest possible difference betweenthe scenario maps would occur if the simulation model wereto allocate forest gain at entirely different places for eachscenario. We use the following notation to compute the leastpossible difference and the greatest possible differencebetween the scenario maps due to the spatial allocationalgorithm, given the quantities dictated by the storylines. 010203040506070809010019992050210021502200 Time in Years    P  e  r  c  e  n   t  o   f   S   t  u   d  y   A  r  e  a non-forest to non-forestnon-forest to forestforest to non-forestforest to forest Fig. 3  Quantity of each land transition over time for the business-as-usual scenario 010203040506070809010019992050210021502200 Time in Years    P  e  r  c  e  n   t  o   f   S   t  u   d  y   A  r  e  a non-forest to non-forestnon-forest to forestforest to forest Fig. 4  Quantity of each land transition over time for the alternativescenario42 Sustain Sci (2010) 5:39–50  1 3  t   index for future year i ,  j  indices for land categories  J   number of categories  A tij  quantity of land transition from category  i  atthe beginning time of the simulation tocategory  j  at a future time  t   in percent of thestudy area for the alternative scenario  B tij  quantity of land transition from category  i  atthe beginning time of the simulation tocategory  j  at a future time  t   in percent of thestudy area for the business-as-usual scenario  L  t  |  A tij ,  B tij  least possible difference between the scenariomaps attributable to variation in simulatedspatial allocation of the transitions at futuretime  t   in percent of the study area, given  A tij and  B tij G t  |  A tij ,  B tij  greatest possible difference between thescenario maps attributable to variation insimulated spatial allocation of the transitionsat future time  t   in percent of the study area,given  A tij  and  B tij Equation 1 computes the least possible differencebetween scenarios, given the quantities of transitions in thealternative and business-as-usual scenarios. The least pos-sible difference is 100% minus the greatest possibleagreement. The double summation of the minimum func-tion gives the greatest possible agreement based on thelogic of Pontius and Connors (2009).  L  t  j  A tij ;  B tij  ¼  100 %  X  J i ¼ 1 X  J  j ¼ 1 MIN[  A tij ;  B tij  ð 1 Þ Equation 2 expresses the greatest possible differencebetween the scenario maps as 100% minus the least possibleagreement. The double summation of the maximumfunction in Eq. 2 gives the least possible agreement(Pontius and Connors 2009). The single summation in round parentheses in Eq. 2 gives the quantity of category  i at the beginning time, which is 1999 for our case study. G t  j  A tij ;  B tij  ¼  100 %  X  J i ¼ 1 X  J  j ¼ 1 MAX 0 ;  A tij  þ  B tij   X  J  j ¼ 1  B tij !"# : ð 2 Þ Estimated difference between maps based on validationPontius et al. (2006) inspires a more complex second method to analyze the uncertainties. This second uncer-tainty assessment consists of three steps: (1) to measure theuncertainty of the model using a validation procedure withhistoric information, (2) to use the results from step (1) toextrapolate the uncertainty of the model as it projects intothe future, (3) to use the results from step (2) to createadjusted scenario maps so we can measure the differencebetween them.Step 1 consists of an exercise where the quantities of thetransitions are based on linear interpolation through the1971–1985 calibration interval and then extrapolationthrough the 1985–1999 validation interval. Geomod allo-cates the transitions spatially based on calibration using the1985 forest map. This generates a simulation map for 1999,which we compare to the reference maps for 1985 and1999. Table 1 presents the quantitative foundation of thevalidation exercise for step 1. It summarizes the three-mapcomparison as a three-dimensional cross-tabulation matrix,where the round parentheses show the bottom layer thatgives the forest category in the simulation map of 1999 andthe square brackets denote the upper layer that gives thenon-forest category in the simulation map of 1999. Eachnumber in Table 1 gives a percent of the landscape.Table 1 indicates that the simulation contains errors due toboth imperfectly simulated quantities and allocations of transitions.We converted the information of Table 1 into measuresof accuracy, given in Table 2. Each conditional probability Table 1  Three-dimensional matrix to quantify the validation of thesimulated transitions from 1985 to 1999Reference 1999Forest Non-forest TotalReference 1985Forest (34.94) b [3.89] a (3.87) [0.47] a,b (38.81) [4.36] a Non-forest (0.00) a,b [1.11] (0.01) a [55.69] b (0.01) a [56.80]Total (34.94) b [5.00] (3.88) [56.16] b (38.82) [61.16]Round parentheses give the forest layer for the simulated map of 1999and square brackets give the non-forest layer for the simulated map of 1999. All numbers are in percent of the landscape a Simulated change as opposed to persistence b Correctly simulated transitions Table 2  Three-dimensional matrix of observed user’s conditionalprobabilities, given a simulated transitionReference 1999Forest Non-forest TotalReference 1985Forest (0.90) b [0.89] a (0.10) [0.11] a,b (1.00) [1.00] a Non-forest (0.04) a,b [0.02] (0.96) a [0.98] b (1.00) a [1.00]Each row is a different simulated transition. Round parentheses givethe forest layer for the simulated map of 1999 and square bracketsgive the non-forest layer for the simulated map of 1999 a Simulated change as opposed to persistence b Correctly simulated transitionsSustain Sci (2010) 5:39–50 43  1 3
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