New measures of Urba Sprawl - A Short Summary

Transcription

1 Ecological Idicators 0 (200) Cotets lists available at ScieceDirect Ecological Idicators joural homepage: ww w. elsevier. com/ locate/ eco lid Urba permeatio of ladscapes ad sprawl per capita: New measures of urba sprawl Joche A.G. Jaeger a, *, Reé Bertiller b, Christia Schwick c, Duca Caves d, Felix Kieast e a Cocordia Uiversity Motréal, Departmet of Geography, Plaig ad Eviromet, 455 de Maisoeuve Blvd. W., Suite H255, Motréal, Québec H3G M8, Caada b Dipl. Forstigeieur ETH, Wald Natur Ladschaft, Brugasse 53, CH-8400 Witerthur, Switzerlad c Dipl. Geogr., Die Geographe schwick + spichtig, Glasmalergasse 5, CH-8004 urich, Switzerlad d School of Architecture ad Ladscape Architecture, Uiversity of British Columbia, Mai Mall, Vacouver, British Columbia V6T 4, Caada e Swiss Federal Istitute of Forest, Sow ad Ladscape Research WSL, ürcherstrasse, CH-8903 Birmesdorf, Switzerlad A R T I C L E I N F O A B S T R A C T Article history: Received 2 Jauary 2009 Received i revised form 7 July 2009 Accepted 8 July 2009 Keywords: Clumpig Dispersio Ladscape idices Ladscape structure Scale Spatial heterogeeity Suitability criteria Switzerlad Urba sprawl (dispersed urba developmet) has icreased at alarmig rates i Europe ad North America over the last 50 years. Quatitative data are urgetly eeded i moitorig systems for sustaiable developmet. However, there is a lack of reliable measures of urba sprawl that take ito accout the spatial cofiguratio of the urba areas (ot just amout). This paper itroduces four ew measures of urba sprawl: degree of urba dispersio (DIS), sprawl (TS), degree of urba permeatio of the ladscape (UP), ad sprawl per capita (SPC). They characterize urba sprawl from a geometric poit of view. The measures are related through TS = DIS urba area, UP = TS/size of the ladscape studied, ad SPC = TS/umber of ihabitats. The paper ivestigates the properties of the ew measures systematically usig 3 suitability criteria which were derived from a clear defiitio of urba sprawl as discussed i a previous paper. The scale of aalysis is specified by the so-called horizo of perceptio. Secod, the ew measures are applied to three examples from Switzerlad. Subsequetly, the measures are briefly compared to other measures of urba sprawl from the literature. We demostrate that UP is a itesive ad area-proportioately additive measure ad is suitable for comparig urba sprawl amog regios of differig size, while SPC is most appropriate whe comparig sprawl i relatio to huma populatio desity. The paper also provides practical advice for calculatig the ew measures. We coclude that the ew method is more suitable tha previous methods to quatify the idicator urba sprawl i moitorig systems as this method distiguishes the pheomeo of urba sprawl from its various causes ad cosequeces. This article is part II of a set of two papers. ß 2009 Elsevier Ltd. All rights reserved.. Itroductio The quatificatio of urba sprawl is a prerequisite for establishig verifiable objectives for evirometal quality (e.g., limits to curtail urba sprawl), idetificatio of treds ad chages i treds (i time ad space), for detectig statistical relatioships betwee urba sprawl ad ecological effects, ad for a uambiguous determiatio of thresholds i the effects of urba sprawl. Therefore, quatitative iformatio about the degree of urba sprawl is urgetly eeded to prepare a suitable idicator for moitorig systems o regioal ad atioal scales. I additio, it DOI of origial article: 0.06/j.ecolid * Correspodig author. Tel.: x548; fax: addresses: jjaeger@alcor.cocordia.ca (Joche A.G. Jaeger), ree@bertiller.ch (R. Bertiller), schwick@hispeed.ch (C. Schwick), duca@caves.org (D. Caves), felix.kieast@wsl.ch (F. Kieast). is useful for comparig ladscape-maagemet scearios ad to more effectively commuicate scietific evidece to politicias ad other decisio-makers. However, measures of urba sprawl that have bee proposed i the literature (Ewig et al., 2003; Razi ad Rosetraub, 2000; Wilso et al., 2003; Davis ad Schaub, 2005; Tsai, 2005; Kasako et al., 2006; Frekel ad Ashkeazi, 2008; Scheider ad Woodcock, 2008; Torres, 2008) report may differig dimesios. For example, Torres (2008) distiguished eleve characteristics of urba sprawl ad applied 42 differet metrics to capture seve of these characteristics. While this approach provides researchers ad plaers with a wealth of iformatio, ot all of it eeds to be reported i moitorig systems. Therefore, we advocate a more systematic approach based o suitability criteria (origially described i Jaeger et al., 2009) to focus o the core pheomeo by disetaglig it from its causes ad cosequeces. The 3 suitability criteria described by Jaeger et al. are: () ituitive iterpretatio, (2) mathematical simplicity, (3) modest data X/$ see frot matter ß 2009 Elsevier Ltd. All rights reserved. doi:0.06/j.ecolid

2 428 J.A.G. Jaeger et al. / Ecological Idicators 0 (200) Fig.. The relatioships betwee the four measures of urba sprawl DIS, TS, UP, ad SPC. DIS, UP, ad SPC are itesive measures (shaded), ad TS is a extesive measure (white box). N ihab = umber of ihabitats of the reportig uit; A urba = amout of urba area i the reportig uit, A reportig uit = size of the reportig uit. requiremets, (4) low sesitivity to very small patches of urba area, (5) mootoous respose to icreases i urba area, (6) mootoous respose to icreasig distace betwee two urba patches whe withi the scale of aalysis, (7) mootoous respose to icreased spreadig of three urba patches, (8) same directio of the metric s resposes to the processes i criteria 5, 6 ad 7, (9) cotiuous respose to the mergig of two urba patches, (0) idepedece of the metric from the locatio of the patter of urba patches withi the reportig uit, () cotiuous respose to icreasig distace betwee two urba patches whe they move beyod the scale of aalysis, (2) mathematical homogeeity (i.e., itesive or extesive measure), ad (3) additivity (i.e., additive or area-proportioately additive measure) (see below Sectio 3). They are based o the followig defiitio: Urba sprawl is visually perceptible. A ladscape suffers from urba sprawl if it is permeated by urba developmet or solitary buildigs. The more urba area preset i a ladscape ad the more dispersed the urba patches, the higher the degree of urba sprawl (Jaeger et al.). The applicatio of the criteria to three existig metrics (amout of urba area, proximity ad cotagio) i Jaeger et al. demostrated that all three are severely limited i their suitability as a measure of urba sprawl. I fact, we are ot aware of ay existig measures of urba sprawl that meet all 3 criteria (see below Sectio 5). Throughout the remaider of this paper, we use the followig termiology: urba patches are patches of urba area, urba poits (or urba locatios ) are poits located withi urba area... Objectives higher their cotributio. As a perso placed at a particular locatio perceives the surroudig ladscape (e.g., whe hikig or seekig recreatio) up to a certai maximum distace, there is a upper limit to the distaces that are take ito accout. Therefore, whe the distaces betwee two locatios are larger tha this maximum distace, urba developmet at the two locatios is cosidered idepedetly. We call this maximum distace (cutoff radius) the horizo of perceptio (HP). It represets the scale at which urba sprawl is ivestigated, e.g., at a more local scale (HP is low), or at a more regioal scale (HP is high). This correspods well with geeral isights from spatial aalysis i that the degree of clumpig or dispersio of some type of lad cover ca be studied o differet scales. For example, a poit patter ca exhibit clumpig o a small scale while showig a evely spaced distributio o a larger scale (e.g., Forti ad Dale, 2005). Such a upper limit of distaces take ito accout excludes situatios where the cotributio of oe place (e.g., urich) to urba sprawl were iflueced by urba developmet processes i places far away (e.g., Copehage), ad prevets the sprawl measures from beig domiated by cotributios of large distaces. The degree of urba dispersio (DIS) is the average weighted distace betwee ay two poits chose radomly withi the urba areas i the ladscape ivestigated, where the secod poit is chose withi a distace less tha the horizo of perceptio (Fig. 2). A weightig of the distaces is ecessary to meet the suitability criteria (i particular criterio 7, see below Fig. 3). The weightig ca be ituitively uderstood as describig the effort for deliverig some service from oe of the two poits to the other, or for providig some kid of ifrastructure betwee the two poits The mai objective of this paper is to itroduce a reliable method to quatify the degree of urba sprawl. The method comprises a set of four related metrics called urba permeatio, urba dispersio, sprawl, ad sprawl per capita (Fig. ). We developed the ew method i order to meet all 3 criteria described i Jaeger et al. To illustrate the ew method, we used a set of examples from Switzerlad, ad briefly compared the ew method with several existig methods. 2. Defiitio of the ew measures of urba sprawl 2.. Motivatio ad verbal defiitio The ew measures make use of the uderstadig that the degree of urba sprawl icreases with both icreasig amout of urba area ad icreasig dispersio. Accordigly, the ew metrics characterize the patter of urba areas i a geometric perspective ad their calculatio is based o all distaces betwee ay two poits located withi the urba areas, i.e., the cotributio of each pair of urba poits to the measure is based o the distace betwee the two poits. The farther apart the two poits, the Fig. 2. Illustratio of the ew metrics ad the horizo of perceptio (HP). The metrics are based o the distaces betwee pairs of poits withi the urba area. The arrows idicate examples of distaces betwee oe locatio (i the ceter of the circle) ad all other locatios with a distace less tha HP. The degree of urba dispersio (DIS) is the average effort required to coect from oe radomly chose poit to aother poit withi the horizo of perceptio of that first poit (e.g., effort per m 2 of urba area). Total sprawl (TS) is the average effort required to coect from every poit to aother poit withi the horizo of perceptio of the first poit (see text). The weightig fuctio that specifies the effort for coectig the poits is illustrated i Fig. 3d.

3 J.A.G. Jaeger et al. / Ecological Idicators 0 (200) (see below). The value of DIS does ot deped o the amout of urba area because the average effort of all pairs of poits is cosidered (i.e., the expected value). Therefore, the farther away the ewly built buildigs from the existig oes, the higher the expected effort; ad the more clumped the buildigs (i.e., closer to each other), the lower the expected effort (Fig. 2). DIS is a itesive measure (criterio 2, see Jaeger et al. ad Appedix C). The secod measure is called sprawl (TS), ad is defied as the average sum of the weighted distaces betwee all poits i the urba area ad radomly chose secod poits where each secod poit is ot farther away from the first poit tha the horizo of perceptio. This measure ca be ituitively uderstood as the expected effort for deliverig some service from all urba area (e.g., from every buildig) to radomly chose delivery poits withi the startig poit s rage of delivery (=horizo of perceptio). The value of TS always icreases whe ew areas are developed somewhere i the ladscape ivestigated (see below for more details). It follows from this defiitio that TS is the product of DIS ad the amout of urba area (Fig. ). It is a extesive measure (criterio 2). The third measure is defied as TS divided by the size of the reportig uit ad is called degree of urba permeatio of the ladscape (UP; Fig. ). It is a itesive measure. This measure ca be ituitively uderstood as the average effort for deliverig some service from all urba area (e.g., from all buildigs) preset withi km 2 of ladscape o average, to radomly chose delivery poits withi the horizo of perceptio of each startig poit. The fourth measure is called sprawl per capita (SPC) ad is defied as TS divided by the umber of ihabitats i the reportig uit. It is a itesive measure i relatio to the umber of ihabitats rather tha the size of the reportig uit, ad therefore ca be compared amog regios of differig size o a per capita basis. This metric establishes the relatioship betwee the other three (purely geometric) sprawl metrics ad populatio desity (Fig. ) Derivatio of the formulas The metrics are applied to a biary map of the ladscape that distiguishes built-up patches (urba patches) from ope ladscape. The the formulas correspodig to the four measures defied above are as follows: permeatio uits/m 2 ) = (UPU/m 2 ). The itegrals for ~x ad ~y are computed over the urba areas (with particular attetio give to the boudaries, see below). The fuctio f ðj~x ~yjþ ca assume differet forms ad is determied below. The calculatio of the three metrics ca be easily based o a grid of cells (e.g., a ASCII-grid), where the itegrals are approximated by sums over small cells of urba area, e.g., squares of width b: 0 DISðbÞ ¼ X i f ðd i j Þ þ WCCðbÞA; (4) i¼ i j¼ TSðbÞ ¼ b 2 DISðbÞ ¼ b 2 X b 2 UPðbÞ ¼ DISðbÞ A reportig uit 0 b 2 X A reportig uit i¼ i i¼ i X i j¼ X i j¼ f ðd i j Þ þ WCCðbÞA; (5) f ðd i j Þ þ WCCðbÞA; (6) where i deotes the umber of urba cells that are closer to cell i tha the horizo of perceptio, d ij is the distace betwee (the ceters of) cell i ad cell j, ad WCC(b) is the withi-cell cotributio. The withi-cell cotributio WCC(b) of each urba cell eeds to be icluded, i.e., i the case j = i ot just 0 is added (as suggested by the term i brackets as d ii = 0) but the value of the cell itself. The larger the cell size, the more relevat the value of the withi-cell cotributio. Each withi-cell cotributio is rather small i compariso to the sum over j (i Eqs. (4) (6)) but it will ifluece the result, i particular whe there are oly a few urba cells i the ladscape. I practical terms, these values eed to be calculated oly oce ad the ca be looked up i a table (Table ). Depedig o how the boudaries of the reportig uit are treated (see below), i may iclude urba cells that are outside of the reportig uit but withi the distace HP of cell i. The smaller the size of the cells, the better the approximatio of the true values of DIS, TS, ad UP: lim DISðbÞ ¼ DIS; b 7! 0 lim TSðbÞ ¼ TS; b 7! 0 lim UPðbÞ ¼ UP: (7) b 7! 0 Degree of urba dispersio ¼ DIS ¼ f ðj~x ~yjþd~yd~x; () A urba ~x 2 urba areas ~y 2 urba areas ad j~x ~yj < HP d~y! y 2 urba areas ad j~x ~yj < HP Total sprawl ¼ TS ¼ f ðj~x ~yjþd~yd~x; (2) ~x 2 urba areas ~y 2 urba areas ad j~x ~yj < HP d~y ~y 2 urba areas ad j~x ~yj < HP Urba permeatio of the ladscape ¼ UP ¼ f ðj~x ~yjþd~yd~x; (3) A reportig uit ~x 2 urba areas ~y 2 urba areas ad j~x ~yj < HP d~y ~y 2 urba areas ad j~x ~yj < HP where A urba is the amout of urba area withi the reportig uit, A reportig uit is the size of the reportig uit, HP is the horizo of perceptio, ad f ðj~x ~yjþ is the weightig fuctio for the distaces betwee ay two poits ~x ad ~y icludig the uit (urba 2.3. Choice of a weightig fuctio The choice of the weightig fuctio is based o the suitability criteria. A liear fuctio, e.g., f ðj~x ~yjþ ¼ j~x ~yj, does ot meet

4 430 J.A.G. Jaeger et al. / Ecological Idicators 0 (200) criterio 7 (mootoous reactio to icreased spreadig of three urba patches) because it does ot distiguish betwee a cofiguratio of three urba patches i a row where two are clumped together ad a cofiguratio where they are distributed evely, while the distace of the two patches at the eds is kept costat (Fig. 3a). I order to icrease whe the three patches are distributed more evely, the weightig fuctio has to icrease more slowly tha liearly. A coveiet way to fid a suitable fuctio is to propose a sesible behaviour of such a fuctio ad solve the resultig differetial equatio. Firstly, the fuctio should start at f(0) = 0; secodly, it should start with a slope of at x = 0; ad thirdly, whe the distace betwee the two poits is icreased by a certai percetage, the value of the weightig fuctio should icrease by a certai proportio (costat factor) of that percetage, i.e., the icrease Df over f should be a certai proportio g of the icrease i the distace Dx over the distace x: D f ðxþ f ðxþ ¼ g Dx x : (8) This leads to the differetial equatio: d f ðxþ ¼ g f ðxþ dx x ; (9) which has the solutio f(x) = Cx g where C is a costat. The possible rage of g follows from the coditio that a icrease i x shall result i a higher value of f i the situatio i Fig. 3a, i.e.: D f ðxþ þ D f ðs xþ ¼ g Dx x Cxg g Dx s x Cðs xþg > 0: (0) It follows that x g > (s x) g whe s x > x, which implies 0 < g <. The simplest of these fuctios is the square root fuctio, p i.e., g = /2. However, as the slope of f ðxþ ¼ C ffiffi x is ifiite at x = 0, we slightly modify this fuctio to start at a poit where the slope is. This ca be achieved by shiftig p it dow ad to the left by a small amout ad choosig C ¼ ffiffiffi 2. The resultig fuctio is: f ðj~x ~y jþ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 j~x ~y j m þ UPU m ; () 2 where UPU deotes urba permeatio uits. I priciple, other choices of 0 < g < are possible, but g = /2 is the simplest ad most coveiet value for the calculatios. Usig this weightig fuctio, the formulas of the three measures assume the form: Fig. 3. Illustratio of three patches of urba area i a liear cofiguratio for derivig the weightig fuctio. I cofiguratio (a), the urba patches are more clumped tha i (b). The sum of the distaces is the same i both cofiguratios (d + d 2 + d 3 = d 4 + d 5 + d 3 ). Therefore, the weightig fuctio has to icrease less tha liear to the distace. (c) Whe the urba patch i the cetre is moved by Dx, the sum of the weightig fuctios for x ad s x should icrease, i.e., Df(x) + Df(s x) > 0 (see text for details). (d) The weightig fuctio f(d ij ) accordig to Eq. (). (e) The resultig values of DIS (ad UP accordigly) for the cofiguratios show i (a) (c) (for d = 50 m ad d 3 = 000 m). The value of DIS is lowest i (a) ad highest i (b). Eve for relatively simple cofiguratios of urba area, it is ot possible to solve the itegrals aalytically. Therefore, some way of approximatio or umerical calculatio is eeded (see Appedix A for a practical suggestio). I the approximatio by a grid of cells as DIS ¼ A urba rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 j~x ~y j ~x 2 urba areas ~y 2 urba areas ad j~x ~yj < HP d~y m þ d~yd~x UPU m ; (2) 2 ~y 2 urba areas ad j~x ~yj < HP rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 j~x ~y j TS ¼ A urba DIS ¼ ~x 2 urba areas ~y 2 urba areas ad j~x ~yj < HP d~y m þ d~yd~x UPU m ; (3) 2 ~y 2 urba areas ad j~x ~yj < HP A urba UP ¼ DIS A reportig uit rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 j~x ~y j ¼ A reportig uit ~x 2 urba areas ~y 2 urba areas ad j~x ~yj < HP d~y m þ d~yd~x UPU m : (4) 2 ~y 2 urba areas ad j~x ~yj < HP

5 J.A.G. Jaeger et al. / Ecological Idicators 0 (200) suggested above, the formulas are: 0 DISðbÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! X i 2d i j m þ TSðbÞ ¼ b 2X i¼ i i¼ i b 2 j¼ j¼ X i j¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2d i j m þ! UPU m 2 UPU m 2 þ WCCðbÞ A; (5) þ WCCðbÞ A; (6) X UPðbÞ ¼ A reportig uit i¼ i 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! X 2d i j m þ UPU m þ WCCðbÞ A: (7) 2 WCC(b) is a fuctio of cell size b ad its value ca be approximated for 0 < b < 000 m by the formula: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi UPU WCCðbÞ ¼ 0:97428b= m þ :046 0: m : (8) 2 Its values are give i Table. For example, if a ladscape icludes oly oe urba cell of width b, the value of DIS equals WCC(b), e.g., 2.96 UPU/m 2 for b = 5 m ad TS(b) = b 2 WCC(b) = UPU. Agai, the smaller the size of the cells, the better the approximatio of the true values of DIS, TS, ad UP (Eq. (7)). The miimum dispersio for a give amout of urba area is assumed if they are clumped together i the shape of a circle (see Fig. i Jaeger et al.). However, if the reportig uit is much larger tha HP, the lowest value is assumed if each buildig is outside of the horizo of perceptio of all other buildigs (which is oly possible if the amout of urba area is rather low) Treatmet of the boudaries of reportig uits There are two optios of how to treat the boudaries of reportig uits:. Cuttig-out procedure: Oly the distaces amog urba poits located withi the reportig uit are take ito accout, i.e., everythig outside the boudary is eglected. 2. Cross-boudary coectios (CBC) procedure (Moser et al., 2007): All distaces betwee the urba poits withi the reportig uit ad ay other urba poits that are smaller tha the horizo of perceptio are take ito accout regardless which reportig uit the surroudig urba poits are located i, i.e., the secod poits iclude urba areas withi a buffer zoe aroud the reportig uit width of the horizo of perceptio (Fig. 4). The cuttig-out procedure has the advatage that o data are eeded from areas outside of the reportig uit ad that, as a cosequece, the results are ot iflueced by urba developmet outside of the reportig uit. This correspods to cuttig the reportig uit out from its cotext. However, it has the disadvatage that the true cotext of the urba areas located close to the boudary is oly partly cosidered eve though these parts of the reportig uit will actually be iflueced by all developmet processes surroudig them, icludig those o the other side of the boudary (Fig. 4). For example, a huma beig seekig recreatio will perceive this locatio as sufferig from urba sprawl if there are may developed areas visible, regardless of whether the buildigs are located iside or outside of the reportig uit. I additio, the calculatios for adjacet reportig uits usig the cuttig-out procedure are ot well Table Values of the withi-cell cotributio WCC(b) to the value of DIS (ad the other metrics) used i Eqs. (4) (6) as a fuctio of cell width (b). Cell width b (i m) Withi-cell cotributio WCC(b) (i UPU/m 2 ) related to the results for the combiatio of several adjacet reportig uits because all the distaces betwee urba areas located i reportig uit A ad those i reportig uit B are eglected whe calculated separately (but icluded whe their combiatio is aalyzed). The smaller the reportig uits, the larger this bias. The CBC procedure has the importat advatage that all poits withi urba areas are treated equally regardless of how close they are to the boudary of some reportig uit. No distaces betwee ay two poits of urba area that are smaller tha HP are eglected. If they are across the boudary betwee two reportig uits they are take ito accout i the sprawl calculatios of both reportig uits (Fig. 4). This procedure solves the so-called Fig. 4. Illustratio of applyig the cross-boudary coectios procedure to determie urba permeatio (UP). Show are two very small urba patches i the reportig uit A ad oe very small urba patch i the reportig uit B. All distaces betwee poits withi urba areas ad other urba poits located withi the horizo of perceptio (HP) of the first poit are take ito accout, eve whe the other urba poits are located i other reportig uits. The buffers are of width HP to idicate the area aroud a reportig uit withi which urba poits may be icluded i the calculatio of the value of UP.

6 432 J.A.G. Jaeger et al. / Ecological Idicators 0 (200) boudary problem (Moser et al., 2007). It has bee applied to other ladscape metrics before, e.g., to the effective mesh size metric for quatifyig the degree of ladscape fragmetatio (Moser et al., 2007). The oly potetial disadvatage of this treatmet is that data outside of the reportig uit withi a buffer width of HP eed to be available. As a cosequece, the calculatio of the measures accordig to the CBC procedure ca be performed i a two-step procedure whe a approximatio based o raster cells is used. First, the values of UP for every cell of urba area ca be calculated takig ito accout the distaces to all other urba cells closer tha HP. Secod, the cells that are actually part of the reportig uit of iterest are selected ad their cotributios to UP are added up. Their sum is divided by the size of the reportig uit, resultig i the value of UP. Because of the advatages of the CBC procedure, this is the most appropriate method. I additio, it has the advatage that the metrics UP ad DIS the are rigorously area-proportioately additive ad TS is additive (criterio 3, see Appedix B). However, i some cases, the cuttig-out procedure may also be useful, e.g., whe the data about the areas outside of the reportig uit are ot available. 3. Examiig the suitability criteria The ew measures meet all 3 suitability criteria well or very well. Urba permeatio (UP) meets the criteria directly (Table 2). UP is the mai measure of urba sprawl proposed i this paper accordig to the defiitio preseted above. The other three metrics are related to UP (Fig. ) ad the criteria apply accordigly (i.e., with a few modificatios). Total sprawl (TS) beig the product of UP ad the size of the reportig uit, is a extesive measure (criterio 2). It is also the product of the degree of dispersio ad the amout of urba area (rather tha...ad the amout of urba area per uit area of the ladscape ; criterio i Table 2, item (3)). Cosequetly, TS is a additive measure (whe the cross-boudary coectios procedure is used; Fig. 4; criterio 3). Criteria 2 apply directly to TS without modificatio i the same way that they apply to UP (Table 2). Sprawl per capita (SPC) is itesive i relatio to populatio size, i.e., its value ca be compared amog regios with differig umbers of ihabitats (criterio 2). It is also the product of the degree of dispersio ad the average urba area take up by each ihabitat i the regio ivestigated (rather tha...ad the amout of urba area per uit area of the ladscape ; criterio i Table 2). As a cosequece, SPC is populatio-proportioately additive, i.e., the value of SPC for the combiatio of two (or more) reportig uits is the populatio-proportioate average of the values of the reportig uits (whe the cross-boudary coectios procedure is used; criterio 3). Criteria 2 apply directly to SPC i the same way that they apply to UP without modificatio (Table 2). The degree of dispersio (DIS) is the ratio of UP ad the amout of urba area per km 2 ad therefore is a itesive measure with regard to both ladscape size ad amout of urba area (criterio 2). Its value ca be compared amog regios of differig proportios of urba area. Criteria 2 3, 6 7, ad 9 0 apply directly to DIS without modificatio i the same way as they apply to UP (Table 2). Criteria, 4 ad apply with slight (obvious) adjustmets. As DIS ca icrease or decrease whe ew urba area is added to a ladscape, criterio 5 does ot apply to DIS. Cosequetly, criterio 8 does ot apply to DIS, either. Criterio 3 applies to DIS i a modified form, e.g., the value of DIS for the combied reportig uit ca be calculated via the value of UP. 4. Three examples from Switzerlad We applied the ew metrics to three examples from Switzerlad (Sursee, Chur, ad Lugao; Fig. 5) as a illustratio ad to ehace the ituitive uderstadig of the metrics. Each example regio is a circle of size 3.95 km 2, i.e., it has a diameter of 2,045 m (Fig. 5). The examples are based o the VECTOR25 data by Swisstopo, Bere, for Historic maps were digitized for 960 ad 935. We compared the results for two horizos of perceptio (2 ad 5 km). The settlemet patter outside the circles withi the horizo of perceptio also iflueced the values of the metrics through the cross-boudary coectios procedure. Therefore, each characterizatio of the three regios icludes a brief descriptio of the surroudigs of the circles. The Sursee regio is located i the Swiss Lowlads ad is domiated by agriculture. The area icludes may small villages ad hamlets, ad cotais o larger tows. The settlemets are embedded i the valleys of soft chais of hills ruig from the southeast to the orthwest. The settlemets are evely distributed across the ladscape, ad this patter is cotiued withi 5 km aroud the circle. The secod example is Chur which is located o a alluvial coe i a valley i the Alps with steep slopes. From there it grows ito the valley bottom of the river Rhie which flows from the southwest to the ortheast. A chai of a small umber of villages follows the river, ad this chai is cotiued outside the circle, but there the umber of villages is rather low. The third example, Lugao, is located o a lake (to the southeast of the city). It is bordered by moutai rages to the west ad to the east. The developmet of settlemets proceeded alog the valley bottoms from the south to the orth. To the orth of the circle show, the umber of settlemets is greatly reduced, ad oly a thi chai of villages cotiues. To the south, the settlemet area is bordered by aother lake, so there are almost o settlemets outside the circle i this directio. Urba areas used i these examples iclude residetial ad idustrial areas. Oly those traffic areas are icluded that are located withi the settlemets. Roads i the ope ladscape are ot icluded because they do ot cotribute to urba sprawl accordig to our defiitio (see above) but costitute a differet topic (i.e., ladscape fragmetatio, e.g., Jaeger et al., 2008; Girvetz et al., 2008). Some areas that are itesively used by humas, e.g., golf courses or outdoor sports facilities, are ot icluded, either. However, the buildigs located i such areas are take ito accout. With icreasig horizo of perceptio, the values of the urba sprawl metrics also icrease. Therefore, the values for the 5 km horizo of perceptio are always higher tha those for the 2 km horizo of perceptio. Both the amouts of urba area ad their icreases betwee 935 ad 2002 are very similar i Sursee ad Chur (+ 3%), whereas Lugao has more urba area ad a relative icrease more tha twice as high (+230%) (Table 3). At all three times (935, 960, 2002), urba permeatio was highest i the Lugao regio ad lowest i the Chur regio (Fig. 6a). Betwee 960 ad 2002, UP has icreased by more tha three times as much as betwee 935 ad 960 i all three regios. I geeral, UP icreases more tha urba area does, if DIS icreases; if UP icreases less tha urba area, the DIS decreases. For the 2 km horizo of perceptio, DIS is highest i Lugao. DIS has icreased rather uiformly with icreasig urba area i Lugao for both horizos of perceptio (Fig. 6b ad c). There were already may small villages aroud the tow of Lugao i 935 which were at distaces closer tha 2 km to each other ad therefore relevat for both horizos of perceptio (Fig. 5), ad dispersio was already high. By 960, ew urba areas had bee

7 J.A.G. Jaeger et al. / Ecological Idicators 0 (200) Table 2 Examiatio of the ew measure Degree of urba permeatio (UP) with regard to the 3 suitability criteria for metrics of urba sprawl (+++ = very good, ++ = satisfyig or good, + = slightly fulfilled, = ot fulfilled). The other three measures DIS, TS ad SPC are closely related to UP (Fig. ) ad therefore, the suitability criteria apply accordigly, see text. For a assessmet of the three measures amout of urba area, proximity, ad cotagio usig the same criteria, see Table 2 i Jaeger et al. Suitability criteria Assessmet of the measure Degree of urba dispersio (UP) Suitability Explaatio. Ituitive iterpretatio +++ The ew metrics are based o the uderstadig of a ladscape beig the more sprawled, the more area is built-up ad the more dispersed the buildigs. Accordigly, the metric of urba permeatio uses three ituitive ideas: () it describes the average effort of deliverig some service from all urba poits (e.g., every buildig) to radomly chose delivery poits (withi a specified rage of delivery represetig the scale of aalysis); (2) its value always icreases whe ew urba areas are added; (3) it is the product of the degree of dispersio ad the amout of urba area per uit area of the ladscape. Therefore, UP is a direct expressio of the defiitio of urba sprawl used i this paper. 2. Mathematical simplicity ++ UP is a secod-order metric ad its value is calculated as a itegral over all pairs of poits withi the urba area of the ladscape ivestigated. It does ot deped o a particular cell size. Its value ca be well approximated by a sum over all pairs of cells of urba area, ad this approximatio quickly coverges towards the value of the itegral (by reducig the size of the cells). Each pair of poits (or cells) cotributes to UP accordig to their distace ad the respective value of the weightig fuctio. The formula is coceptually straightforward ad ca be calculated umerically for ay ladscape ad its patter of urba area (see Appedix A) (it does ot receive +++ because a computer is required to calculate its value). 3. Modest data requiremets +++ The eed for data is low. Maps of the areas classified as settlemets (or urba area ) are sufficiet. Usually, such maps are available i digital format (e.g., VECTOR25 :25,000 by Swisstopo Bere). This is a ideal basis for calculatig the ew sprawl metrics usig a Geographic Iformatio System (GIS). To documet historical states of urba developmet, the correspodig older maps eed to be scaed ad digitized. 4. Low sesitivity to very small patches of urba area 5. Mootoous reactio to icreases i urba area 6. Mootoous reactio to icreasig distace betwee two urba patches whe withi the scale of aalysis 7. Mootoous reactio to icreased spreadig of three urba patches 8. Same directio of the metric s resposes to the processes i criteria 5, 6 ad 7 9. Cotiuous reactio to the mergig of two urba patches 0. Idepedece of the metric from the locatio of the patter of urba patches withi the reportig uit. Cotiuous reactio to icreasig distace betwee two urba patches whe they move beyod the scale of aalysis 2. Mathematical homogeeity (i.e., itesive or extesive measure) 3. Additivity (i.e., additive or area-proportioately additive measure) +++ The cotributio of each patch of settlemet area to UP is proportioal to its size. Therefore, smaller ad smaller patches have less ad less ifluece o the metric s value (the calculatio of the average effort for coectig two poits is take i relatio to the size of patches; at o poit is the umber of patches used which would create the problem of disregardig patch size; Jaeger, 2000). ++ Whe ew urba areas are added to a ladscape, the value of UP always icreases, except for a few rare exceptioal cases of high dispersio where UP ca be slightly reduced by buildig desely (see Appedix B). The amout by which UP icreases will deped o the amout ad the relative locatio of the ew urba patches i relatio to the existig patter of urba areas. It is geerally impossible to reduce the value of UP of a ladscape by addig more urba area (except for some rare cases of buildig desely i a very dispersed situatio). +++ Whe the distace betwee two urba patches icreases (while they are still withi the horizo of perceptio of each other), the value of UP always icreases. The icrease of UP exhibits a shape that is similar to the shape of the effort fuctio (Fig. 3). +++ This criterio is met through the choice of the weightig fuctio icreasig less tha proportioally to the distace betwee poits (Fig. 3). Therefore, the value of UP icreases faster at shorter distaces, i.e., the gai i UP due to icreases i the distace to close urba patches is larger tha the loss i UP due to decreases i the distace to distat urba patches (Fig. 3). +++ The resposes of UP to the three processes referred to i criteria 5, 6 ad 7 are all icreasig, i.e., i the same directio. +++ Whe two urba patches merge, the cotributio of iter-patch distaces to UP decreases cotiuously (i.e., from pairs of poits each of which is located o a differet patch). This is a cosequece of the weightig fuctio beig a cotiuous fuctio (icludig the poit 0). The two itra-patch cotributios do ot chage. +++ The value of UP depeds oly o the spatial patter of the urba area ad o the size of the ladscape ivestigated. Therefore, the value of UP is ot chaged whe the etire patter of urba area is rotated or moved to a differet locatio withi the ladscape. +++ Whe the distace betwee two urba patches icreases beyod the horizo of perceptio (that defies the scale of aalysis) the value of UP chages cotiuously. For those parts of the iter-patch cotributio that are based o pairs of poits closer tha the horizo of perceptio, their cotributio icreases, while the other parts do ot cotribute ay more. UP chages cotiuously, because the decrease i the value of UP is proportioal to the amout of urba area ad the movemet across the HP distace is a cotiuous process. +++ UP is a itesive measure i relatio to the size of the reportig uit (i.e., UP does ot deped o the size of the reportig uit), ad its value ca be compared amog reportig uits of differig sizes. +++ UP is a area-proportioately additive metric, i.e., the value of UP for the combiatio of two (or more) reportig uits is the area-proportioate average of the values for the reportig uits (whe the cross-boudary coectios procedure is used for the calculatio of UP; Fig. 4), see proof i Appedix C. added i the form of strads at the frige of the mai tow ad rather dispersed additios to the older villages. The ew developmet by 2002 has exteded the strads ad has coected may of the surroudig villages formig elogated stripes. Therefore, dispersio has icreased eve further. DIS icreased eve more steeply i Sursee ad Chur betwee 935 ad 2002 tha i Lugao for the 2 km horizo of perceptio. However, the value of DIS first decreased i Sursee betwee 935 ad 960 (Fig. 6b). I 935, the may villages i Sursee were mostly separated by distaces larger tha 2 km ad therefore

8 434 J.A.G. Jaeger et al. / Ecological Idicators 0 (200) Fig. 5. Urba developmet i three regios from Switzerlad used to illustrate the ew urba sprawl metrics (Sursee, Chur, ad Lugao). The diameter of each ladscape is 2 km. The maps show the developmet of urba areas for three poits i time: 935 (i light grey), 960 (i dark grey), ad 2002 (i black). cotributed idepedetly to the sprawl metrics for the 2 km horizo of perceptio. The urba areas that had bee added by 960 were located close to the existig villages ad therefore were still ot perceived from eighbourig villages (thus DIS decreased). Oly after 960 did the urba areas exted farther away from the villages ad reduced the average distaces betwee the boudaries of the villages to less tha 2 km, which meas that sigificat parts of eighbourig villages were ow ofte withi the horizo of perceptio of each village. Thus, DIS icreased steeply betwee 960 ad I Chur, the urba area was ot broke up ito as may idepedet small villages i 935 at the 2 km scale as i Sursee; oly about four small villages surroud the mai tow ad are far eough to be idepedet of it, i.e., >2 km (Fig. 5). Therefore, DIS is higher i Chur tha i Sursee for the 2 km horizo of perceptio, whereas it is higher i Sursee tha i Chur for the 5 km horizo of perceptio. This is clearly visible i the map of Sursee (Fig. 5) as each village icludes i its 5 km horizo of perceptio three to five of its surroudig villages. This implies a much more scattered distributio of the urba areas at this scale tha the distributio i

9 J.A.G. Jaeger et al. / Ecological Idicators 0 (200) Table 3 Values of the three metrics urba dispersio (DIS), sprawl (TS), urba permeatio (UP) for two horizos of permeatio (2 ad 5 km), ad the urba areas i the three example regios show i Fig. 5 from Switzerlad for three poits i time (935, 960, 2002) (UPU = urba permeatio uits, MUPU = mega-upu). Regio Year Urba area (ha) Values of the sprawl metrics Horizo of perceptio = 2 km Horizo of perceptio = 5 km DIS 2 (UPU/m 2 ) TS 2 (MUPU) UP 2 (UPU/km 2 ) DIS 5 (UPU/m 2 ) TS 5 (MUPU) UP 5 (UPU/km 2 ) Sursee Chur Lugao the cocetrated arragemet of the tow of Chur where the tow is surrouded by oly oe or two small villages (the third at the ortheast border of the regio is almost idepedet for the 5 km horizo). This differece also explais why DIS cotiues to decrease i Sursee betwee 960 ad 2002 for the 5 km horizo of perceptio. At this scale, the ew urba areas fill i the space betwee the villages i a rather dese form, i.e., deser tha the distributio of the villages i 935 (TS behaves the same way as UP because all regios are of same size). The broke lies idicate the value of DIS for a eve distributio of urba cells width of 5 m (i.e., maximum value of DIS) ad for a cofiguratio as a circle. The area of a circle with diameter of 2 km is 33.2 ha, ad ha for 5 km; therefore, the lower curves ed at these values. For HP = 2 km, up to four circles of 2 km diameter ca fit ito the 3.95 km 2 ladscape with distaces >2 km, ad the correspodig four lies are icluded i Fig. 6b. The three examples illustrate very clearly that it is importat to keep i mid what the horizo of perceptio is whe iterpretig the values of the metrics. 5. Discussio 5.. Utility of the ew metrics For the iterpretatio of the results for a particular regio, the values of UP, DIS ad SPC should be compared. UP describes to what degree a ladscape is permeated by settlemet areas ad solitary buildigs. SPC relates sprawl to the umber of ihabitats. As idustrial areas ofte have low umbers of ihabitats, SPC ca also be defied i relatio to the umber of jobs i a regio (or to the sum of ihabitats ad jobs). Whe ew buildigs are added withi the existig urba patches (desificatio), the the values of UP ad DIS do ot chage, whereas SPC decreases accordig to the umber of ew ihabitats ad/or jobs. This correspods well with the ituitive uderstadig that urba areas of higher desities are less sprawled. Therefore, it may be coveiet to idetify sprawl as a particular combiatio of certai rages of values of UP, DIS ad SPC. This may iclude attempts to quatitatively defie sprawl based o the combied values of UP, DIS ad SPC. For example, such a quatitative defiitio could exclude city ceters from the term sprawl whe SPC is higher tha a certai threshold eve though UP is high (ad DIS itermediate). Such rages will be suggested based o empirical data from Switzerlad i a separate paper. The values of the metrics will differ depedig o the defiitio of urba area, e.g., whether or ot solitary buildigs or areas take up by trasportatio ifrastructure i the ope ladscape are icluded. Therefore, a reliable defiitio ad delieatio of urba areas is a prerequisite for the quatificatio of UP ad DIS. Tools such as VectorGe ca be used to objectively delieate urba areas (Millward, 2002, 2004). Attempts to stadardize DIS to rage betwee 0 ad should be treated with cautio for several reasos. Stadardizatio would likely affect the relatioship UP = DIS A urba. This relatioship is oe of the mai advatages of the ew metrics itroduced i this paper (Fig. ) ad should be maitaied. Stadardizatio of DIS might also compromize the validity of criterio 5 for UP. I additio, the coveiet property of UP beig itesive ad areaproportioately additive (criteria 2 ad 3) should ot be put at risk. Istead of tryig to chage the role of DIS withi its relatioship with UP, the value of DIS ca always be reported separately ad iterpreted i compariso with UP ad SPC. Oe major advatage of the ew metrics over may other ladscape metrics is that their defiitios ad values do ot deped o ay cell sizes. The cells used i Eqs. (4) (6) ad (5) (7) serve to approximate the itegrals give i Eqs. () (3) ad (2) (4), ad all cell sizes produce the same results (whe the cells are ot too large, e.g., less tha 50 m). This is a cosequece of the beig covergece behaviour of the approximatios. We performed a test of how well the approximatios coverge towards the true values (see Appedix A) ad foud that:. the values of DIS(b) coverge with the refiemet of the approximatio (i.e., icreasig umber of cells) towards the values that were calculated with Mathematica; 2. the iclusio of the withi-cell cotributio WCC(b) leads to icreased covergece of the approximatio; 3. the iclusio of the withi-cell cotributio WCC(b) i the approximatios leads to more accurate results, i particular whe the cell size is rather large. The ifluece of the withi-cell cotributio o accuracy of the results decreases whe cell size decreases. Based o these tests ad o our experiece from applyig the ew sprawl metrics to Switzerlad, we recommed the choice of b = 5 m. The cotributios of those urba patches located closer to the boudary of the reportig uit tha the horizo of perceptio are iflueced by urba areas outside of the reportig uit if these patches are withi the horizo of perceptio (Fig. 4). This is achieved through the applicatio of the CBC procedure which removes ay bias that would otherwise be produced by igorig the cotext of a reportig uit, e.g., by the cuttig-out procedure (Moser et al., 2007).

10 436 J.A.G. Jaeger et al. / Ecological Idicators 0 (200) Fig. 6. Developmet of urba permeatio (UP), urba dispersio (DIS) ad urba area i the three example regios show i Fig. 5 betwee 935 ad 2002 for two horizos of permeatio (2 ad 5 km). For compariso, the values of DIS for a regular distributio of 5 m 5 m cells ad for a solid circle (up to four circles for HP = 2 km) of urba area are idicated by broke lies (i b ad c), see text. The data poits i the ceter are values for Suitability criteria for measures of urba sprawl This paper applies the 3 suitability criteria to esure that the ew metrics meet all requiremets for measures of urba sprawl. The applicatio of suitability criteria provides a reliable approach to uderstadig the behaviour of ladscape metrics i a systematic way (Jaeger, 2000; Jaeger et al.) ad is useful for prevetig misuderstadig ad misuse of ladscape metrics which is a commo issue i ladscape ecology (Li ad Wu, 2004) Brief compariso with other measures of urba sprawl Jaeger et al. assessed three metrics i detail: amout of urba area, proximity, ad cotagio. Noe of them meets all 3 criteria.

11 J.A.G. Jaeger et al. / Ecological Idicators 0 (200) The amout of urba area by itself, though a importat compoet of urba sprawl ad widely used, does ot iclude iformatio about the spatial arragemet of urba areas ad therefore is ot sufficiet to measure urba sprawl. Methods from spatial aalysis that are frequetly used to assess whether poit patters are radom, clumped, or regular iclude the K fuctio ad quadrat tests of radomess (Cressie, 993; Bailey ad Gatrell, 995; Fotherigham et al., 2000; Forti ad Dale, 2005). However, these methods do ot apply to cotiuous spatial patters. They are based o couts of poit evets, rather tha cotiuous areas (which caot be couted i a poit-wise maer). However, the ew metrics are to some degree related to the K fuctio (Bailey ad Gatrell, 995), i the sese that urba locatios at a certai distace aroud each urba poit are cosidered, but here they are weighted (ot just couted). A importat advatage of the ew metrics is that they iclude both itra-patch distaces ad iter-patch distaces. This is a major reaso why the merger of two (or more) urba patches does ot produce a jump i the values of the metrics (criterio 9). Such jumps costitute a major drawback of the proximity metric itroduced by Whitcomb et al. (98; see also Gustafso ad Parker, 992, 994). I additio, the proximity metric does ot meet the directio criterio (criterio 8 i Jaeger et al.). Various other measures that have bee suggested to quatify certai aspects of urba sprawl have bee discussed i Jaeger et al., e.g., () percetage of dwelligs i sigle-uit detached houses, (2) populatio per square kilometer, ad (3) housig uits per square kilometer (Razi ad Rosetraub, 2000). These earlier measures do ot, to our kowledge, meet the 3 suitability criteria. I additio, most of them do ot explicitly accout for the eed to aalyze urba sprawl o differig scales, e.g., cotagio (see Jaeger et al.). A detailed compariso of other existig measures ad the ew metrics to substatiate these claims will be performed i a separate paper. The ew measures are secod-order metrics. Both first-order ad secod-order metrics are meaigful for quatifyig ladscape patters. Most ladscape metrics calculate first-order statistics, e.g., patch area, road desity, patch shape metrics (McGarigal ad Marks, 995). First-order statistics describe the variatio i the itesity of some process at idividual locatios (or evets), whereas secodorder characteristics summarize poit-to-poit relatioships (Wiegad ad Moloey, 2004). I geeral, secod-order properties describe the spatial depedece betwee evets at ay two locatios, i.e., they examie the correlatios or covariaces betwee evets occurrig i two distict poits or regios (Fotherigham et al., 2000: 40). To measure the spatial cofiguratio of urba areas, the distaces to all other poits withi urba area are relevat (if they are located withi the horizo of perceptio). Several other ladscape metrics have bee proposed i the literature that have secod-order properties. These iclude the ecologically scaled ladscape idex average patch coectivity (Vos et al., 200), which is the probability that a patch is coloized based o species-specific movemet distaces ad the spatial cofiguratio of habitat patches. Other examples are the effective mesh size (Jaeger, 2000; Girvetz et al., 2008), Ripley s K fuctio ad the O-rig statistic (Wiegad ad Moloey, 2004). 6. Coclusios To measure urba sprawl, the spatial arragemet of the urba areas eeds to be take ito accout. The method for quatifyig urba sprawl itroduced i this paper meets all 3 suitability criteria for measures of urba sprawl ad has produced covicig results for Switzerlad (Wisse et al., submitted for publicatio). The four ew metrics ca be used separately to characterize urba sprawl, or i combiatio to idetify urba sprawl as a specific associatio of certai value rages of the four metrics. The ew metrics are useful to measure the speed of urba developmet, idetify treds (e.g., desificatio or icreasig dispersio), compare urba sprawl amog differet regios, ad to suggest quatitative limits to curtail urba sprawl. The properties of the ew metrics are particularly coveiet for the compariso of regios of differig size because they are itesive measures (ad eve area-proportioately additive measures). The four ew metrics have recetly bee applied to Switzerlad (o a time series sice 935) i two projects that are part of the Natioal Research Programme 54 Sustaiability of the Built Eviromet by the Swiss Natioal Sciece Foudatio (Wisse et al., submitted for publicatio). The results are plaed to be used as a idicator i the Swiss Moitorig System of Sustaiable Developmet (MONET; SFSO et al., 2004) ad i the Swiss Spatial Moitorig Program (ru by the Swiss Federal Office for Spatial Developmet ARE ad the Swiss Federal Office for the Eviromet FOEN). Urba sprawl ca be measured o differet scales. Therefore, the four ew metrics iclude a parameter called horizo of perceptio (HP) that specifies the scale of aalysis. As illustrated by the three examples from Switzerlad, the scale of aalysis is importat to cosider i the iterpretatio of the results. Recommedatios for the choice of the horizo of perceptio ca be based o the followig estimatio: due to the curvature of the earth, the distace of perceptio for a huma beig (with eye-height of.80 m) is a = 4.9 km o a surface with o obstacles (calculated by usig the Pythagorea formula a 2 + (6370 km) 2 = (6370 km +.80 m) 2, where 6370 km is the average radius of the earth). Therefore, distaces betwee ad 0 km seem most suitable. The horizo of perceptio may also be chose i accordace with the type of urba developmet ivestigated ad with the historical settlemet structures. For example, if ew urba developmet reduces the distaces betwee the boudaries of eighbourig tows or villages ad this process is cosidered relevat for assessig urba sprawl, the the horizo of perceptio should be chose larger tha this distace. Based o our experiece from applyig the ew metrics to Switzerlad, we recommed choosig a value for HP of 2 ad 5 km for regios with rather small-scale settlemet structures such as Switzerlad. To ivestigate at what scales the relevat sprawl processes are takig place, we recommed to use several HPs i parallel ad to compare the results. A computer program for automated calculatio of the metrics is available from the authors. Ackowledgemets We thak Michael Wezlaff ad Beat Trachsler for their geerous programmig support. We also thak Stefa Keller for his fruitful collaboratio i software developmet. This work is part of a project withi the Swiss Natioal Research Programme (NRP 54) Sustaiable Developmet of the Built Eviromet which was fuded by the Swiss Natioal Sciece Foudatio (NSF); we particularly thak Stefa Husi from the NSF for supportig the collaboratio amog the authors of this paper. We thak Joh Lowry ad Hugh Millward for ispirig discussios about measurig urba sprawl ad two reviewers for helpful commets o the mauscript. Appedix A. O the umerical calculatio of the metrics It is coveiet to first calculate the value of S i for each cell i i the ladscape that the reportig uits of iterest are embedded i: S i ¼ i X i k¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!! 2 d ik m þ þ WCCðbÞ ; (A)

12 438 J.A.G. Jaeger et al. / Ecological Idicators 0 (200) where i is the umber of urba cells withi the HP of cell i. For ay chose reportig uit, the three metrics ca the be calculated based o the S i values of the cells located withi the reportig uit: DISðbÞ ¼ X UPU S i m ; (A2) 2 i¼ where is the umber of urba cells i the reportig uit, TSðbÞ ¼ b 2 X i¼ S i UPU m 2 ; (A3) where b is the width of the cell (i m), b 2 X UPðbÞ ¼ A reportig uit i¼ S i UPU m 2 ¼ A reportig uit TSðbÞ: (A4) The approximatio of the metrics based o cells coverges quickly. Eve cell sizes of 50 m 50 m provide good results (Fig. 7). I order to speed up the calculatio of the metrics for large reportig uits (e.g., large coutries), the urba areas ca be represeted by cells that are oly partially filled with urba developmet. This implies that the cells ca be larger tha the smallest patch of urba developmet take accout of i the calculatios, e.g., larger tha the size of solitary buildigs i the ladscape (>5 m). The calculatio is faster because it icludes fewer cells. The price paid for this advatage is that the accuracy i the calculatio of the distaces is lower. The degree of urbaizatio of a cell ca be represeted by values betwee 0 ad 00% idicatig the percetage of area of the cell covered by developmet. The formulas for UP, DIS, ad TS will the eed to be modified accordigly to iclude these percetage values. Appedix B. Examiatio of UP with regard to suitability criterio 5 Because of the horizo of perceptio, the behavior of UP is i some cases o-trivial. Whe ew urba areas are added to a ladscape, the value of UP always icreases, except for a few rare exceptioal cases where UP ca be slightly reduced by buildig desely i a very dispersed situatio. (This effect disappears for differet choices of HP.) Fig. 7. Calculatio of the degree of dispersio, DIS, for a square patch of urba area size of km 2 through approximatio of the itegral Eq. (2) by the formula based o cells of varyig size Eq. (5). (a) overall picture for cell sizes smaller tha 500 m, ad (b) logarithmic diagram for cell sizes smaller tha 00 m. The approximatio approaches the true value of UPU/m 2 (umerical calculatio usig Mathematica, see Table ) very quickly whe the size of the cells is smaller tha 50 m. (The p horizo of perceptio does ot ifluece these results as log as it is larger tha the largest distace betwee urba cells withi the 000 m* 000 m square, i.e., HP > ffiffiffi 2 km =.442 km).

13 J.A.G. Jaeger et al. / Ecological Idicators 0 (200) Fig. 8. Illustratio of the four cases used for studyig the respose of UP to icreases i urba area (criterio 5). I cofiguratio (a), patch 3 was added ad UP icreased (HP was larger tha the largest distace betwee ay two urba patches, see text). (b) Whe aother patch was added (patch 4), UP icreased agai, see text (d 24 ad d 34 are ot show to avoid clutterig). (c) Whe patch 3 was added i a situatio where the patches were outside of each other s HP, the UP always icreased. (d) Oly i a situatio whe the ew patch (patch 3) is outside of the HP of patch, the UP ca icrease or decrease, depedig o the distaces betwee patches 2 ad 3 ad betwee patches ad 2, see text. Proof: Four cases are distiguished (Fig. 8; all patches show are of width b, i.e., urba cells, with o loss of geerality): I cofiguratio (a), patch 3 was added ad UP icreased: Before addig patch 3, UP was UP 2 ¼ b2 X 2 S A i ¼ b2 ð ru A ru 2 f ðd 2Þ þ WCC b Þ þ ð 2 f ðd 2Þ þ WCC b Þ ; i¼ where A ru is the area of the reportig uit. After addig patch 3, UP is usig 2 ð f ðd 4Þ þ f ðd 24 ÞÞ > 2 f ðd 2Þ, etc. This ca be cotiued for ay umber of urba cells. (c) Whe patch 3 is added i a situatio where the patches are outside of each other s HP, the UP always icreases: UP ðcþ ¼ b2 ðwcc b þ WCC b þ WCC b Þ ¼ 3 b2 WCC A b ; ru A ru which is simply the sum of the cotributios of each patch. (d) Oly i a situatio whe the ew patch (patch 3) is outside of the HP of patch, the UP ca icrease or decrease, depedig o UP 23 ¼ b2 ð A ru 3 f ðd 2Þ þ f ðd 3 Þ þ WCC b þ f ðd 2 Þ þ f ðd 23 Þ þ WCC b þ f ðd 3 Þ þ f ðd 23 Þ þ WCC b Þ ¼ b2 WCC A b þ 2 ru 3 f ðd 2Þ þ 2 3 f ðd 23Þ þ 2 3 f ðd 3Þ > b2 WCC A b þ 4 ru 3 f ðd 2Þ > UP 2 ; usig f(d 3 ) + f(d 23 ) > f(d 2 ). This holds true wherever patch 3 is located, as log as it is withi the HP of patch ad patch 2. (b) Whe aother patch was added (patch 4), UP cotiued to icrease: After addig patch 4, UP is the distace betwee patches 2 ad 3. UP 2 ¼ b2 ð A ru 2 f ðd 2Þ þ WCC b Þ 2 UP 234 ¼ b2 WCC A b þ 2 ð ru 4 f ðd 2Þ þ f ðd 3 Þ þ f ðd 23 Þ þ f ðd 4 Þ þ f ðd 24 Þ þ f ðd 34 ÞÞ > b2 WCC A b þ 2 3 ru 4 2 f ðd 2Þ þ 3 2 f ðd 3Þ þ 3 2 f ðd 23Þ ¼ b2 WCC A b þ 3 ð ru 4 f ðd 2Þ þ f ðd 3 Þ þ f ðd 23 ÞÞ > UP 23 ;

14 440 J.A.G. Jaeger et al. / Ecological Idicators 0 (200) ad UP 23 ¼ b2 A ru 2 ð f ðd 2Þ þ WCC b Þ þ 3 ð f ðd 2Þ þ f ðd 23 Þ þ WCC b Þ þ 2 ð f ðd 23Þ þ WCC b Þ : Thus, DUP ¼ UP 23 UP 2 ¼ b2 A ru 3 WCC b þ 5 6 f ðd 23Þ 6 f ðd 2Þ : Whe d 23 is similar to d 2 the DUP clearly is positive. However, if d 23 is much smaller tha d 2, this term ca become egative. For example, whe d 2 = 9900 m, HP = 0,000 m, d 23 = 200 m, the DUP ¼ b2 A ru 5:854 UPU m 2 ¼ 450:8 UPU A ru < 0; usig WCC b from Table. þ 0:988 UPU m 2 23:29 UPU m 2 Appedix C. O the mathematical property of UP to be areaproportioately additive Defiitios A ladscape metric, say F, is called itesive, if Fðl FÞ ¼ FðFÞ for all cofiguratios of urba area F ad all l 2 N where l F is defied as the multiplicatio of the regio represeted by F i the same spatial arragemet of urba patches (cf. Chadler, 987, pp ; Legedre ad Legedre, 998, p. 3). For example, for F ¼ fha; 4ha; 5hag a multiplicatio by l ¼ 2 results i 2F ¼ fha; ha; 4ha; 4ha; 5ha; 5hag, etc. A ladscape metric, say F, is called area-proportioately additive if the value of F for the combiatio of two urba area cofiguratios F ad F 2 (with areas A ðþ ad A ð2þ ) is give by FðF [ F 2 Þ ¼ A ðþ A ðþ þ A ð2þ FðF Þ þ A ð2þ A ðþ þ A ð2þ FðF 2 Þ: This is aalogous to the way that temperature or the cocetratio of a liquid is determied: whe two liquids are mixed, the cocetratio of the mixture becomes c ¼ V V þ V 2 c þ V 2 V þ V 2 c 2 with V j ad c j deotig the volumes ad cocetratios. This meas that each part (e.g., F ad F 2 ) cotributes proportioally to its size, eve if each part has a differet spatial structure. The characteristics of beig itesive or area-proportioately additive are iterrelated. Area-proportioately additive meas more tha itesive. I fact, every area-proportioately additive quatity is itesive. The reverse geerally does ot hold. Average patch size is a example of a itesive measure that is ot areaproportioately additive. Proof that UP is area-proportioately additive Urba permeatio, whe calculated accordig to the CBC procedure, is a area-proportioately additive quatity (without ay restrictios). Proof: Let F ad F 2 be two cofiguratios of urba area o o F ¼ A ðþ i ¼ ;... ;, F 2 ¼ A ð2þ i ¼ ;... ; 2 with i areas A ðþ ad A ð2þ of the two reportig uits. Calculate the values of S i for all cells i the two reportig uits based o some cell size (b) accordig to formula (A) give above. For ay reportig uit, UP is the give by formula (A4). i Therefore, the value of UP for the joit cofiguratio ðf [ F 2 Þ results i 0 b UPðF [ F 2 X 2 Þ UPU UPU A ðþ þ A S ð2þ i m 2 þx2 S i A m 2 i¼ j¼ A ðþ X ð UPU ¼ A ðþ þ A S ð2þ A ðþ i m þ A 2 Þ X UPU 2 i¼ A ðþ þ A S ð2þ A ð2þ i ; m 2 i¼ ¼ A ðþ A ðþ ð Þ þ A 2 UPðF Þ þ A ðþ A ð2þ ð Þ þ A 2 UPðF 2 Þ where is the umber of urba cells i reportig uit ud 2 is the umber of urba cells i reportig uit 2. This meas that UP is a area-proportioately additive quatity. (Note that this equatio does ot hold true whe the cuttig-out procedure is used, because the the S i -values would be differet for cells close to the boudary that separates the two reportig uits, see Fig. 4, as the coectios betwee urba poits o either side of the boudary would be missig.) Refereces Bailey, T.C., Gatrell, A.C., 995. Iteractive Spatial Data Aalysis. Pretice Hall/ Pearso Educatio, Essex, 43 pp. Chadler, D., 987. Itroductio to Moder Statistical Mechaics. 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