Modification of the effective mesh size for measuring landscape fragmentation to solve the boundary problem

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.

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 Φ₁ and Φ₂ (with total areas A ⁽¹⁾_total and A ⁽²⁾_total ) is given by

$$\eqalign{F\left({\Phi_1 \cup \Phi_2 } \right)= \frac{A_{\rm total}^{(1)}}{A_{\rm total}^{(1)} +A_{\rm total}^{(2)}}\cdot F\left({\Phi_1 } \right) \cr +\frac{A_{\rm total}^{(2)}}{A_{\rm total}^{(1)} +A_{\rm total}^{(2)}}\cdot F\left({\Phi_2 } \right).} $$

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

$$ c=\frac{V_1 }{V_1 +V_2 }c_1 +\frac{V_2 }{V_1 +V_2 }c_2 $$

with V _j and c _j denoting the volumes and concentrations. This means that each part (e.g., Φ₁ and Φ₂) 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,

$$\begin{array}{*{20}c} m_{\rm eff}^{\rm CBC} = \frac{1}{A_{\rm total} }\left(A_1 \cdot A_1^{\rm cmpl} +A_2 \cdot A_1^{\rm cmpl} \right.\cr \left.\qquad\qquad +A_3 \cdot A_3^{\rm cmpl} + \ldots + A_n \cdot A_n^{\rm cmpl} \right) \cr = \frac{1}{A_{\rm total}} \left({\left({A_1 +A_2 } \right)\cdot A_1^{\rm cmpl}} \right.\cr \left.\qquad\qquad + A_3 \cdot A_3^{\rm cmpl} +\ldots + A_n \cdot A_n^{\rm cmpl} \right). \end{array} $$

 

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 Φ₁ and Φ₂ 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 Φ₁ ∪ Φ₂ 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 Φ₁ and Φ₂ 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 Φ₁ ∪ Φ₂ results in

$$ \eqalign{ m_{\rm eff} \left({\Phi _1 \cup \Phi _2 } \right)=\frac{1}{A_{\rm total}^{(1)} +A_{\rm total}^{(2)} }\left({\sum\limits_{i=1}^{n_1 } {\left( {A_i^{(1)} \cdot A_i^{(1),\;{\rm cmpl}}} \right)} +\sum\limits_{j=1}^{n_2 } {\left({A_j^{(2)} \cdot A_j^{(2),\ {\rm cmpl}}} \right)} } \right) \cr =\frac{A_{\rm total}^{(1)} }{A_{\rm total}^{(1)} +A_{\rm total}^{(2)} }\frac{1}{A_{\rm total}^{(1)} }\sum\limits_{i=1}^{n_1 } {\left( {A_i^{(1)} \cdot A_i^{(1),\; {\rm cmpl}} } \right)} +\frac{A_{\rm total}^{(2)} }{A_{\rm total}^{(1)} +A_{\rm total}^{(2)} }\frac{1}{A_{\rm total}^{(2)} }\sum\limits_{j=1}^{n_2 } {\left({A_j^{(2)} \cdot A_j^{(2),\ {\rm cmpl}} } \right)} \cr =\frac{A_{\rm total}^{(1)} }{A_{\rm total}^{(1)} +A_{\rm total}^{(2)} }\cdot m_{\rm eff} \left({\Phi _1 } \right)+\frac{A_{\rm total}^{(2)} }{A_{\rm total}^{(1)} +A_{\rm total}^{(2)} }\cdot m_{\rm eff} \left({\Phi _2 } \right). \cr } $$

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.