Landscape Ecology - Lecture Notes

Quantifying Landscape Pattern: Methods and Applications

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I. Why Quantify Landscape Pattern? 

Landscape pattern analysis (LPA) has been a major part of landscape ecological research for the last few decades.  The ultimate goal of LPA is to link spatial patterns to ecological processes at different scales.  The importance of LPA lies in the needs to (1) describe and compare landscape patterns, (2) monitor and project landscape changes, and (3) help understand ecological processes underlying observed patterns, so that landscape dynamics may be better understood and managed.  Thus, appropriate and effective use of LPA methods is vital in landscape ecological research. [Li, H. and J. Wu. 2007. Landscape pattern analysis: Key issues and challenges. Pages 39-61 in J. Wu and R. Hobbs, editors. Key Topics in Landscape Ecology. Cambridge University Press, Cambridge, UK.]

Three Key Reasons for Quantifying Landscape Pattern:

• Describing and comparing the spatial patterns of landscapes
• Monitoring and predicting changes in landscape patterns
• Relating spatial pattern to ecological processes at a particular scale or across a range of scales

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II. How to Quantify Landscape Pattern?

1. Simple Indices for Individual Patches

(1) Patch shape index D:

where D is a shape index, p is the perimeter of the patch, and A is the area of the patch. D = 1 for circles; D = 1.1283 for squares; and D = 1.1968 for a rectangle (2L by L).  1/D is called "compactness" (see Forman 1995 and references therein).

(2) Perimeter/area ratio PAR: 


PAR = P / A

(3) Patch isolation index r_i:

where n is the number of neighboring patches, and dij is the distance between patch i and any neighboring patch j. Therefore, the larger the value of the index, the more isolated the patch in consideration.

(4) Patch interaction index I_i:

where I_i is the degree of interaction of patch i with n neighboring patches, Aj is the area of any neighboring patch j, and dj is the distance between the edges of patch i and any patch j. The bigger the value of the index, the stronger the interaction. This is essentially based on the law of gravity in physics: F_ij = K(M_i M_j)/d².
 


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(5) Fractal dimension, D (Mandelbrot 1977):

What is fractal?

• A fractal is a shape made of parts similar to the whole in some way (Mandelbrot 1986)
• Fractals are shapes in which an identical motif repeats itself on an ever diminighing scale
• Mathematical sets of non-integer dimension
• The classical question by Mandelbrot: "How long is the coast of Britain?"
• Example: What is the total length of the Amazon river system?

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Patterns (real or mathematical) which look similar at different scales, for example the network of airways in the lung which shows similar branching patterns at progressively higher magnifications.  Natural fractals are self-similar across a finite range of scales while mathematical fractals are the same across an infinite range. Many natural, including biological, structures are fractal (or fractal-like).  Fractals are related to "chaos" in that chaotic processes can produce fractal structures in nature, and appropriate representations of chaotic processes usually reveal self-similarity over time.  



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Classical exampes of fractals

• Contor dust: D = 0.6309
• Koch curve: D = 1.26
• Sierpinski gasket: D = 1.585 (for triangle); D = 1.8928 (for squares)

Exact fractals vs. statistical fractals

• Exact fractals are composed of geometically regular objects (like circles, triangles, and rectangles).
• Statistical fractals are those whose regular shape is reveaed by statistical analysis rather than by direct inspection of each object.
• Fractals in nature mostly are statistical fractals



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* Fractal gallery (http://sprott.physics.wisc.edu/carlson/)

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How to estimate fractal dimension?
 
• The basic formula:
Q(L) = L^D
 
where Q(L) is a quantity measured at the length scale, L, and D is the fractal dimension.
 
P = k A^D/2

D = 2 ln(P/k)/ln A
D = 2 (ln(P) - ln(k)) / ln A

where D is the fractal dimension of a shape, P is its perimeter, A is its area, and k is a constant of proportionality.
• The value of D is between 1 for simplest shapes, and 2 for most complex shapes.
• k is the constant that relates the perimeter of a shape to the square root of the area of a shape.  For instance, for a square k = 4, for a circle k = 1.264, and for a hexagon k = 3.72.

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2. Landscape Metrics or Indices (for Patch Class or the Landscape)

- O'Neill et al. (1988): Dominance, Contagion, and Fractal Dimension
- Li and Reynolds (1993): Revised version of Contagion

Some commonly used landscape metrics at the class/patch type and landscape levels (Wu et al. 2002):



[From: Wu, J., W. J. Shen, W. Z. Sun, and P. T. Tueller. 2002. Empirical patterns of the effects of changing scale on landscape metrics. Landscape Ecology 17:761-782.]

Software packages for computing landscape metrics

• FRAGSTATS  [https://fragstats.org/index.php]
  

• Documentation
  

• FragStats - Quick Quide

• FragStats - Manual

• FragStats Tutorial#1  [View]

• FragStats Tutorial#2  [View]

• Other landscape pattern analysis tools

• R Packages
• multifit: an R function for multi-scale analysis in landscape ecology

• Motif: an open-source R tool for pattern-based spatial analysis

• landscapemetrics - R package for calculating landscape metrics for categorical landscape patterns

• MSPA (Morphological Spatial Pattern Analysis)
• Mapping Spatial Patterns with Morphological Image Processing

• GuidosToolbox
• GuidosToolbox: universal digital image object analysis

• Surface Metrics
• McGarigal, K., S. Tagil, and S. A. Cushman. 2009. Surface metrics: an alternative to patch metrics for the quantification of landscape structure. Landscape Ecology 24:433-450.

• Frazier, A. E. 2016. Surface metrics: scaling relationships and downscaling behavior. Landscape Ecology 31:351-363.

• Kedron, P. J., A. E. Frazier, G. A. Ovando-Montejo, and J. Wang. 2018. Surface metrics for landscape ecology: a comparison of landscape models across ecoregions and scales. Landscape Ecology 33:1489-1504.