## Abstract

Patch-based landscape metrics can be biased by the boundaries and the extent of a reporting unit if the boundaries fragment patches. We call this the “boundary problem”. The effective mesh size *m*
_{eff} is a convenient method to quantify landscape fragmentation, that is based on the probability that two points chosen randomly in a region will be connected, e.g., not be separated by roads, railroads, or urban development. The cutting-out (CUT) procedure, used in the original computation of *m*
_{eff}, suffers from the boundary problem because the boundaries of the reporting units are considered to be additional barriers. Therefore, *m*
_{eff} will be underestimated, particularly if reporting units are embedded within the broader landscape. In this paper, we present a solution to overcome this limitation by a new method called “cross-boundary connections” (CBC) procedure. It attributes the connections between two points that are located in different reporting units to both reporting units. We systematically compare the CBC procedure to the CUT procedure and show that the boundary problem is intrinsic to the CUT procedure, while the CBC procedure is independent of the size and administrative boundaries of reporting units. In addition, we elucidate the superior performance of the new procedure in the case study of South Tyrol where *m*
_{eff} is being used for sustainability reporting on the level of municipalities. The new CBC procedure eliminates the bias due to the boundaries and the size of reporting units in measuring landscape fragmentation through *m*
_{eff}.

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## References

Autonomous Province of South Tyrol (1991a) Map of areas of development. Scale 1: 5000. Provincial Statistical Institute

Autonomous Province of South Tyrol (1991b) Map of municipalities. Scale 1:5000. Department for Spatial Statistical Information

Autonomous Province of South Tyrol (2001) Map of the road and railway network. Scale 1:5000. Department for Street Services

Chandler D (1987) Introduction to modern statistical mechanics. Oxford University Press Inc., New York, NY, 274 pp

Collinge SK (1996) Ecological consequences of habitat fragmentation: implications for landscape architecture and planning. Landscape Urban Plann 36:59–77

Collinge SK (1998) Spatial arrangement of habitat patches and corridors: clues from ecological field experiments. Landscape Urban Plann 42:157–168

Esswein H, Jaeger J, Schwarz-v Raumer H-G, Müller M (2002) Landschaftszerschneidung in Baden-Württemberg. Zerschneidungsanalyse zur aktuellen Situation und zur Entwicklung der letzten 70 Jahre mit der effektiven Maschenweite. Technical Report of the Center for Technology Assessment in Baden-Württemberg, no. 214, Stuttgart, Germany, 124 pp

Esswein H, Schwarz-v Raumer H-G, Kaule G (2003) Analyse der Landschaftszerschneidung in Baden-Württemberg hinsichtlich belastungsempfindlicher Räume. Programm Lebensgrundlage Umwelt und ihre Sicherung (BWPLUS). Technical Report, Stuttgart, Germany, 40 pp

Forman RTT (1995) Land Mosaics. The ecology of landscapes and regions. Cambridge University Press, Cambridge, 632 pp

Forman RTT, Alexander LE (1998) Roads and their major ecological effects. Ann Rev Ecol Syst 29:207–231

Forman RTT, Sperling D, Bissonette JA, Clevenger AP, Cutshall CD, Dale VH, Fahrig L, France R, Goldman␣CR, Heanue K, Jones JA, Swanson FJ, Turrentine T, Winter TC (2003) Road Ecology: science and solutions. Island Press, Washington DC USA, 481 pp

Gerlach G, Musolf K (2000) Fragmentation of landscapes as a cause for genetic subdivision in bank voles. Conserv Biol 14(4):1066–1074

Gulinck H, Wagendorp T (2002) References for fragmentation analysis of the rural matrix in cultural landscapes. Landscape Urban Plann 58:137–146

Haines-Young R, Chopping M (1996) Quantifying landscape structure: a review of landscape indices and their application to forested landscapes. Progr Phys Geogr 20(4):418–445

Hanski I (1999) Metapopulation ecology. Oxford University Press, Oxford, UK, 313 pp

Hargis CD, Bissonette JA, David JL (1998) The behaviour of landscape metrics commonly used in the study of habitat fragmentation. Landscape Ecol 13:167–186

Harris LD (1984) The fragmented forest: island biogeography theory and the preservation of biotic diversity.␣University of Chicago Press, Chicago IL, USA, 211 pp

Heinz Center (The H. John Heinz III Center for Science, Economics and the Environment) (2002) The State of the Nation’s Ecosystems: measuring the lands, waters, and living resources of the United States. Cambridge University Press, New York, USA. Available online at http://www.heinzctr.org/ecosystems/report.html

Jaeger JAG (2000) Landscape division, splitting index, and effective mesh size: new measures of landscape fragmentation. Landscape Ecol 15(2):115–130

Jaeger JAG (2002) Landschaftszerschneidung. Eine transdisziplinäre Studie gemäß dem Konzept der Umweltgefährdung. Ulmer-Verlag, Stuttgart, Germany, 447 pp

Jaeger J, Esswein H, Schwarz-v Raumer H-G, Müller M (2001) Landschaftszerschneidung in Baden-Württemberg:␣Ergebnisse einer landesweiten räumlich differenzierten quantitativen Zustandsanalyse. Naturschutz und Landschaftsplanung 33(10):305–317

Keller I, Largiadèr CR (2003) Recent habitat fragmentation caused by major roads leads to reduction of gene` flow and loss of genetic variability in ground beetles. Proc Royal Soc London B 270:417–423

Keyghobadi N, Roland J, Strobeck C (2005) Genetic differentiation and gene flow among populations of the alpine butterfly,

*Parnassius smintheus*, vary with landscape connectivity. Mol Ecol 14(7):1897–1909Legendre P, Legendre L (1998) Numerical ecology. 2nd English edn. Elsevier Science BV, Amsterdam, 853 pp

Li H, Wu J (2004) Use and misuse of landscape indices. Landscape Ecol 19:389–399

McGarigal K, Cushman SA, Neel MC, Ene E (2002) FRAGSTATS: Spatial Pattern Analysis Program for Categorical Maps. Computer software program produced by the authors at the University of Massachusetts, Amherst. Available online at www.umass.edu/landeco/research/fragstats/fragstats.html

McGarigal K, Marks BJ (1995) FRAGSTATS: spatial pattern analysis program for quantifying landscape structure. General Technical Report PNW-GTR-351, USDA Forest Service, Pacific Northwest Research Station, Portland, Oregon, USA, 122 pp

Mladenoff FD, Verheyden C, Jouventin P (1999) Predicting grey wolf landscape recolonization: logistic regression models vs. new field data. Ecological Appl 9:37–44

O’Malley R, Cavender-Bares K, Clark WC (2003) Providing “better” data – not as simple as it might seem. Environment 45:8–18

O’Neill RV, Hunsaker CT, Timmins SP, Jackson BL, Jones KB, Riitters KH, Wickham JD (1996) Scale problems in reporting landscape pattern at the regional scale. Landscape Ecol 11:169–180

Pan-European Biological and Landscape Diversity Strategy (PEBLDS) [online] URL: http://www.strategyguide.org/fulltext.html.

Peter U, Meier S (2003) Zerschnittene Landschaft – ein Problem im Kanton Aargau? Umwelt Aargau 22:29–32

Reck H, Kaule G (1993) Straßen und Lebensräume. Ermittlung und Beurteilung straßenbedingter Auswirkungen auf Pflanzen, Tiere und ihre Lebensräume. Forschung Straßenbau und Straßenverkehrstechnik, Heft 654. Bonn, Bad Godesberg, 230 pp

Riitters KH, O’Neill RV, Hunsaker CT, Wickham JD, Yankee DH, Timmins SP, Jones KB, Jackson BL (1995) A vector analysis of landscape pattern and structure metrics. Landscape Ecol 10:23–39

Riitters KH, Wickham JD, O’Neill RV, Jones KB, Smith ER, Coulston JW, Wade TG, Smith JH (2002) Fragmentation of Continental United States Forests. Ecosystems 5:815–822

Roedenbeck IA, Esswein H, Köhler W (2005) Land-schaftszerschneidung in Hessen. Entwicklung, Verg-leich zu Baden-Württemberg und Trendanalyse als Grundlage für ein landesweites Monitoring. Natur-schutz und Landschaftsplanung 37(10):293–300

Saunders D, Hobbs R, Margules C (1991) Biological consequences of ecosystem fragmentation: a review. Conserv Biol 5:18–32

Saura S, Martinez-Millan J (2001) Sensitivity of landscape pattern metrics to map spatial extent. Photogramm Eng Remote Sensing 67:1027–1036

Spellerberg IF (2002) Ecological effects of roads. Land reconstruction and management vol 2. Science Publishers, Enfield NH, 251 pp

Trombulak SC, Frissell CA (2000) Review of ecological effects of roads on terrestrial and aquatic communities. Conserv Biol 14(1):18–30

Turner MG, O’Neill RV, Gardener RH, Milne BT (1989) Effects of changing spatial scale on the analysis of landscape pattern. Landscape Ecol 3:153–162

Turner MG, Gardner RH, O’Neill RV (2001) Landscape ecology in theory and practice: pattern and process. Springer, New York

UN Convention on Biological Diversity. [online] URL: http://www.biodiv.org/programmes/areas/forest/default.asp

Verboom J, Fopppen R, Chardon P, Opdam P, Luttikhuizen P (2001) Introducing the key patch approach for habitat networks with persistent populations: an example for marshland birds. Biol Conserv 100(1):89–101

Wade TG, Riitters KH, Wickham JD, Jones KB (2003) Distribution and causes of global forest fragmentation. Conservation Ecology 7(2):7, [online] URL: http://www.consecol.org/vol7/iss2/art7/

Wu J (2004) Effects of changing scale on landscape pattern analysis: scaling relations. Landscape Ecol 19:125–138

Zebisch M, Wechsung F, Kenneweg H (2004) Landscape response functions for biodiversity – assessing the impact of land-use changes at the county level. Landscape Urban Plann 67:157–172

## Acknowledgements

We thank Hans-Georg Schwarz-von Raumer for the AVENUE scripts that we initially used for the calculations of *m*
_{eff} according to the cutting-out procedure, as well as Oluwayemisi Dare, Kevin McGarigal, Kerri Widenmaier, Marc Zebisch, and two anonymous reviewers for their helpful comments on the manuscript. Verena Grüner provided help and advice for data processing. The work by JAGJ was supported through a postdoctoral research scholarship from the German Research Foundation (grant number JA-1105/1-1).

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## Appendix A

### Appendix A

### Some useful characteristics of the CBC procedure

#### Definitions

A landscape metric, say *F*, is called “** intensive**”, if \(F\left({\lambda \cdot \Phi} \right)=F\left(\Phi \right)\) for all area configurations Φ and all λ ϵ

*N*with λ · Φ defined as the multiplication of the region represented by Φ in the same spatial arrangement of patches (cf. Chandler 1987, pp. 22–25; Legendre and Legendre 1998, p. 31). For example, for \(\Phi =\left\{ {1\ \hbox{ha},\ 4\ \hbox{ha},\ 5\ \hbox{ha}} \right\}\) a multiplication by λ = 2 results in 2Φ = {1 ha, 1 ha, 4 ha, 4 ha, 5 ha, 5 ha}, etc.

A landscape metric, say *F*, is called “** area-proportionately additive**” if the value of

*F*for the combination of two area configurations Φ

_{1}and Φ

_{2}(with total areas

*A*

^{(1)}

_{total}and

*A*

^{(2)}

_{total}) is given by

This is analogous to the way the temperature or concentration of a liquid is determined: when two liquids are mixed, the concentration of the mixture becomes

with *V*
_{
j
} and *c*
_{
j
} denoting the volumes and concentrations. This means that each part (e.g., Φ_{1} and Φ_{2}) contributes proportionally to its size, even if␣each part has a different spatial structure.

The characteristics of being intensive or area-proportionately additive are interrelated. “Area-proportionately additive” means more than “intensive”. In fact, every area-proportionately additive quantity is intensive. The reverse generally does not hold. Average patch size is an example of an intensive measure which is not area-proportionately additive.

#### On the case that two or more parts of a patch are located within a reporting unit

Whether the parts of a patch that are located within a reporting unit are connected inside or only outside the reporting unit does not influence the value of *m*
_{eff}.

### Proof

Let *A*
_{1} and *A*
_{2} be two parts of a single patch that are located within a reporting unit, as shown in Fig. 6

.

The general formula of *m*
_{eff} according to the CBC procedure (see Eq. 3) is \(m_{\rm eff}^{\rm CBC} =\frac{1}{A_{\rm total}}\sum\limits_{i=1}^n {A_i \cdot A_i^{\rm cmpl}}\). In the case shown in Fig. 6, it holds \(A_1^{\rm cmpl} =A_2^{\rm cmpl}\), and thus,

Consequently, the value of *m*
_{eff} according to the cross-boundary connections procedure is the same in both cases if *A*
_{1} and *A*
_{2} are disconnected within the reporting unit, or if they are connected, i.e., one patch size of (*A*
_{1} + *A*
_{2}). The same is true if the number of parts within the reporting unit is larger than two. The value of *m*
_{eff} does not depend on the number of fractions that are cut away by boundaries of a reporting unit, because the probability that a randomly chosen point is found within a group of several fractions of a patch within a reporting unit equals the sum of these fractions. The connections between two points, located one in *A*
_{1} and the other in *A*
_{2}, are not affected by whether they are running within or outside of the reporting unit.

#### On the mathematical property of *m*
^{CBC}_{eff}
to be
area-proportionately additive

The effective mesh size, when calculated according to the CBC procedure, is an area-proportionately additive quantity without any restrictions.

### Proof

Let Φ_{1} and Φ_{2} be two area distributions \(\Phi_1 =\left\{ {A_i^{(1)} \left| {i=1,\ldots,\ n_1 } \right.} \right\},\ \Phi _2 =\left\{ {A_i^{(2)} \left| {i=1,\ldots,n_2} \right.} \right\}\) with total areas *A*
^{(1)}_{total}
and *A*
^{(2)}_{total}
. The joint configuration Φ_{1} ∪ Φ_{2} has *n*
_{3} patches where *n*
_{3}≤ *n*
_{1} + *n*
_{2} because either none of the patches has parts located in Φ _{1} and Φ _{2} at the same time (and then *n*
_{3} = *n*
_{1} + *n*
_{2}), or one or more of the patches have parts located in Φ_{1} and Φ_{2} at the same time (and then *n*
_{3} < *n*
_{1} + *n*
_{2}).

In the first case, all \(A_i^{(1),\;{\rm cmpl}} \) are different from all \(A_j^{(2),\ {\rm cmpl}} \), and *m*
_{eff} of the joint configuration Φ_{1} ∪ Φ_{2} results in

In the second case, there are patches with \(A_i^{(1),\;{\rm cmpl}} =A_j^{(2),\;{\rm cmpl}}\), and either *A*
^{(1)}_{
i
}
and *A*
^{(2)}_{
j
}
are connected or not connected (as shown in Fig. 6). In either case, their contribution to *m*
_{eff} is the same as \(A_i^{(1)} \cdot A_i^{(1),\; {\rm cmpl}} +A_j^{(2)} \cdot A_j^{(2), {\rm cmpl}} = \left({A_i^{(1)} +A_j^{(2)} } \right)\cdot A_i^{(1),\;^{\rm cmpl}} = A_k^{(1+2)} \cdot A_k^{(1+2), {\rm cmpl}}\) as discussed above (in section “On the case that two or more parts of a patch are located within a reporting unit”). Therefore, the sum \(\mathop {\sum\limits_{k=1}^{n_3}} {\left( {A_k^{(1+2)} \cdot A_k^{(1+2),{\rm compl}}} \right)} \) can be written as the two sums \(\sum\limits_{i=1}^{n_1 } {\left({A_i^{(1)} \cdot A_i^{(1),\; {\rm cmpl}} } \right)}+\sum\limits_{j=1}^{n_2 } {\left({A_j^{(2)} \cdot A_j^{(2), {\rm cmpl}}} \right)} \), and the relationship above is also valid, i.e., \(m_{\rm eff} \left({\Phi _1 \cup \Phi _2 } \right)<$> <$>=\frac{A_{\rm total}^{(1)} }{A_{\rm total}^{(1)} +A_{\rm total}^{(2)} }\cdot m_{\rm eff} \left({\Phi _1 } \right)+\frac{A_{\rm total}^{(2)} }{A_{\rm total}^{(1)} +A_{\rm total}^{(2)} }\cdot <$> <$>m_{\rm eff} \left({\Phi _2 } \right).\) This means that *m*
^{CBC}_{eff}
is an area-proportionately additive quantity.

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Moser, B., Jaeger, J.A.G., Tappeiner, U. *et al.* Modification of the effective mesh size for measuring landscape fragmentation to solve the boundary problem.
*Landscape Ecol* **22**, 447–459 (2007). https://doi.org/10.1007/s10980-006-9023-0

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DOI: https://doi.org/10.1007/s10980-006-9023-0

### Keywords

- Cross-boundary connections procedure
- Cutting-out procedure
- Scale
- Spatial extent
- Landscape metrics
- Landscape indices
- Spatial heterogeneity
- Environmental indicators
- Environmental monitoring
- South Tyrol