(Z1) Conductance index

Background.—The conductance index is a measure of landscape connectivity. Connectivity, in turn, is a complex concept, but essentially refers to the facilitation or impedance of ecological flows (e.g., organisms, materials, energy) across the landscape in space and/or time, and it is process-dependent. Process-dependency means that how the landscape pattern affects connectivity depends on the particular ecological flow or process under consideration. Connectivity is also a synoptic concept as it integrates multiple aspects of landscape pattern, including the aggregation, subdivision and isolation attributes of the landscape mosaic and potentially other aspects of pattern such as patch size, edge density and contrast. As such, many of the configuration metrics included in FRAGSTATS can affect landscape connectivity and could in their own right be considered connectivity metrics. However, each of these other distinct metric groups can be used to address pattern and process unrelated to landscape connectivity per se, and thus they have been separated out as distinct groups. The conductance index, on the other hand, is devoted exclusively to landscape connectivity.

Algorithm.—There many ways to conceptualize and measure connectivity. Currently, FRAGSTATS includes only a single connectivity metric that we call Conductance. The conductance metric is based on the resistant-kernel estimator described previously (see documentation on cell-based metrics). Comparative evaluations have demonstrated that resistant kernels are superior to other common methods such as factorial least cost paths, circuitscape and graph theoretic approaches in representing connectivity (e.g., Kumar et al. 2022, Lumia et al 2024). Briefly, resistant kernels are a hybrid between two standard approaches: standard kernels and least-cost paths based on resistant surfaces. The conductance metric is derived as described previously for resistant kernels, but with the following modifications:

1. First, for each focal cell, built a resistant kernel as described previously. Recall that the resistant kernel has a value of 1 at the focal cell and decreases monotonically to zero (or it is truncated) as it spreads outward from the focal cell.
   
   
2. Next, assign source probabilities to the input raster. The source probability represents the probability of a cell serving as a “source” for the ecological flow of interest (e.g., the probability of being occupied by a focal species). Source probability of < 1 results in a proportional decrease in the kernel density; i.e., the height of the kernel is decreased in proportion to the source probability of the focal cell. Thus, the kernel built from a focal cell with a source probability < 1 contributes proportionally less to the computed conductance of each cell (see below). A source probability of 1 results in no discounting of the resistant kernel. A source probability of 0 results in no resistant kernel and thus no contribution to the final conductance. There are three options for specifying source probabilities:
   
   
3. 1. Use Probability Table.--assign a source probability to each patch type via a table. FRAGSTATS uses the file extension .fln for the source probability table and will look for files with this extension by default when navigating. The .fln extension is not mandatory, but using it can help keep files organized. The syntax for this comma-delimited ASCII file is as follows:

FLN_TABLE

CLASS_LIST_LITERAL(1stClassName, 2ndClassName, etc.)

CLASS_LIST_NUMERIC(1stClassID, 2ndClassID, etc.)

Class1probability, Class2probality, etc.

• • Comment lines start with # and are allowed anywhere in the table.
  • FLN_TABLE must be specified in the first line.
  • Two types of class lists are allowed CLASS_LIST_LITERAL() and CLASS_LIST_NUMERIC(), but only the first one encountered is considered, so you only need one of these lines.
  • Literal class names (1stClassName, 2ndClassName, etc.,) are character strings and cannot contain spaces.
  • Class Ids (1stClassID, 2ndClassID, etc.) are integer values corresponding to class values in the grid.
  • With regards to the source probabilities specified, the order must match the order of the classes given in the list.
  • Source probabilities must be given on the probability scale 0-1.
  • If the input landscape has background (not the same as nodata), you need to include the background class value specified in the grid properties during data import, otherwise all background cells will be given a zero source probability.
  • This table can be created and managed using any text editor and then simply saved as a comma delimited file (.csv).

For example, the following file would be suitable:

FLN_TABLE

CLASS_LIST_NUMERIC(2, 3, 4, 5, 6)

0, 0.1, 0.5, 1, 0

1. 1. Use Probability Raster.--Supply a source probability raster, in which cells are assigned integer values representing the source probability (0-1) times 100. Note, in contrast to the source probability table option above, the raster must have values in the range 0-100, representing 100 times the actual probability. This allows the input raster to be in an integer format and thus less demanding on the computer memory requirements.
      
      
   2. No Source Probabilities.--specify no source probabilities (equivalent to setting the source probability for all cells to be 1). Note, this option is used when you want all cells to be treated equally as sources for the ecological flows.

3. Next, multiply the resistant kernel by the source probability of the focal cell. In this manner, the resistant kernel values for all cells in the kernel are decreased proportional to the source cell probability, as follows:

rs_ij = r_ij × s_i

where:   rs_ij = source-adjusted resistant kernel value of window cell j from focal cell i

d_ij = kernel density of window cell j from focal cell i

s_i = source probability of focal cell i

4. Next, add the source-adjusted resistant kernel above (i.e., all the cells within the kernel derived for the focal cell) to the raw conductance raster so that it accumulates kernels as they get built for the focal cells.

z_ij = ∑rs_ij

where:   z_ij = raw conductance value of cell ij

rs_ij = source-adjusted resistant kernel value of window cell j from focal cell i summed across i kernels

5. Lastly, divide the raw conductance value above by a standardizing constant, which is equal to the number of standard kernel windows derived from focal cells with a source probability > 0 overlapping each cell. Note, the count of overlapping standard kernel windows is based on the FULL standard kernel derived from a non-resistant landscape (i.e., cost = 1 for all cells), not the realized resistant kernel above, so as to represent the “potential” number of overlapping kernels. In addition, only kernels built from source cells with non-zero probability are tallied. This standardization is devised to account for the conductance bias near the landscape boundary, as these cells will have fewer overlapping kernels as a result of being near the landscape boundary, as follows:

z*_ij = z_ij/ c_ij

where:   z*_ij = final conductance of cell ij

z_ij = raw conductance of cell ij

c_ij = count of number of standard kernels (i.e., nonresistant kernels) overlapping cell ij