Cost distance, least-cost paths and corridors

How far is one forest from another, for an animal that has to cross the fields, villages and roads between them? Not the 29 km of a ruler: it depends on what lies in the way. In this part you measure distance as a cost, the sum of the resistances crossed, the way the R package gdistance does and as Terraflux's cost operators reproduce it to the last digit. You compute the accumulated cost of reaching every cell from the large southern forest around Hannover, read it as isochrones, trace the least-cost paths to four other forests, compare their costs with straight distances, and widen the best path into a least-cost corridor. At the end you will have the tools behind most corridor designs, and know their assumptions.

The idea#

A landscape as a grid of connections#

gdistance (van Etten, 2017) describes a raster as a graph: every cell is a node connected to its 4, 8 or 16 neighbours, and each connection has a value, the transition. For movement the transition is a conductance: how easily a walker passes from one cell to the next. It is made from the cells' values by a function of the two (the mean, the minimum...), usually of the conductance of each cell, 1 / its resistance.

Two corrections follow. A diagonal step is longer than a side step (√2 times), so the conductance between two cells is divided by the distance between their centres (geographic correction, type c for costs); on a grid in degrees, those distances shrink towards the poles, which the correction takes into account. The cost of a step is then its length divided by its conductance: the length times the mean resistance.

Isochrones and corridors#

An accumulated cost map can be read like a map of travel time: lines of equal cost, isochrones, enclose the land a disperser can reach for a given effort. And instead of a single path, which a real animal is unlikely to follow exactly, planners draw a least-cost corridor: every cell through which the best route between two areas is at most a few percent dearer than the least cost (Beier et al., 2008). The corridor shows the room there is on the ground and the alternatives a path hides.

When to use it#

• Distances between populations or reserves that account for the matrix: for habitat graphs (least-cost links), for isolation in genetic studies, for dispersal models.
• Designing corridors between two areas (Beier et al., 2008), and comparing them.
• Accessibility: how far, in effort, every place is from a source (a reserve, a road, a town).

What it assumes: the animal knows the landscape and takes the best route. For animals exploring unknown country, random walks and circuit theory are better models (part 2, circuit connectivity). Least-cost paths are also sensitive to the resistance values (Etherington, 2016): treat the path as one plausible route among many.

What you need#

• A resistance: the class resistance of resistance surfaces, from clc_100m.tif (CORINE Land Cover around Hannover at 100 m): forest 1, arable land 20, towns and water 100, and so on.
• Origins and goals: areas (a raster mask) or points (vector features). Here the southern forest (patch 55, 9,151 ha) as an area, and five forests as points, forests.geojson: one point in each of the southern forest (55), the north-eastern one (96), the south-western (8), the north-western (30) and the far north-eastern (110), at the cell of each forest nearest its centre.

The application's template Least costs and random walks (Templates > Connectivity) has the accumulated cost and the passages of part 2 on a landscape drawn by chance.

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Build the model#

1. The transition#

cost.transition (library subject Grid distances) takes the resistance as map. With reciprocal = true it makes, for every pair of neighbours, 1 / the function of their values: with function = "mean", 1 / the mean resistance, the conductance of the connection (gdistance's 1 / transition(r, mean)). directions = 8 connects each cell to its eight neighbours. Its output, a transition, is what every other cost operator reads.

cost.geo_correction then divides each conductance by the distance between the cells' centres (type = "c"): after it, costs are in the map's units (metres) times resistance.

2. The accumulated cost from the southern forest#

The southern forest as a mask: the patches of the forest numbered as in the habitat networks series (conn.graph_from_raster, conn.nodes, conn.to_map), then raster.calc coalesce(p == 55, 0), 1 in the forest and 0 elsewhere. cost.acc_cost takes the corrected transition and the mask as origins (cells not 0), and gives the accumulated cost from the nearest origin cell to every cell (infinite where no route exists).

3. Least-cost paths and costs between points#

vector.read reads the five points. cost.shortest_path takes the transition, an origin (the first point of a layer) and goals, and gives the lines of the least-cost paths (one per goal) and a map of the share of paths through each cell. cost.distance gives the least cost from every point to every other (a table).

format = "terraflux/1"

[[nodes]]
id = "land"
op = "raster.read"
params = { path = "clc_100m.tif" }

[[nodes]]
id = "resistance"
op = "raster.reclassify"
inputs = { map = "land.raster" }
params = { rules = "23 24 25 -> 1\n29 -> 2\n21 26 27 32 35 36 -> 5\n18 20 -> 10\n12 -> 20\n10 11 -> 30\n7 8 9 -> 50\n1 2 3 4 5 6 40 41 -> 100\n* -> 20" }
note = "A resistance per class."

[[nodes]]
id = "transition"
op = "cost.transition"
inputs = { map = "resistance.map" }
params = { function = "mean", directions = 8, reciprocal = true }
note = "1 / the mean resistance of each pair of neighbours."

[[nodes]]
id = "corrected"
op = "cost.geo_correction"
inputs = { transition = "transition.transition" }
params = { type = "c" }
note = "Divided by the distance between the cells' centres."

[[nodes]]
id = "graph"
op = "conn.graph_from_raster"
inputs = { map = "land.raster" }
params = { habitat = [23, 24, 25] }

[[nodes]]
id = "patches"
op = "conn.nodes"
inputs = { map = "land.raster" }
params = { habitat = [23, 24, 25] }

[[nodes]]
id = "patch"
op = "conn.to_map"
inputs = { graph = "graph.graph", table = "patches.table" }
params = { column = "id" }

[[nodes]]
id = "south"
op = "raster.calc"
inputs = { p = "patch.map" }
params = { expression = "coalesce(p == 55, 0)" }
note = "The southern forest: 1 inside, 0 elsewhere."

[[nodes]]
id = "cost"
op = "cost.acc_cost"
inputs = { transition = "corrected.transition", origins = "south.map" }
note = "The cost of the cheapest route from the forest to every cell."

[[nodes]]
id = "forests"
op = "vector.read"
params = { path = "forests.geojson" }
note = "A point in each of five forests; the first is the southern one."

[[nodes]]
id = "paths"
op = "cost.shortest_path"
inputs = { transition = "corrected.transition", origin = "forests.vector", goals = "forests.vector" }

[[nodes]]
id = "least_costs"
op = "cost.distance"
inputs = { transition = "corrected.transition", from = "forests.vector" }

To keep the paths, connect paths.lines to a vector.write (a GeoPackage or GeoJSON); the application also draws them over a map when you open them.

Read the results#

The accumulated cost and its isochrones#

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The isochrones are far from circles. They stretch north along the belt of woods and pastures in the centre-west, and pinch against Hannover in the east, where the cost rises fastest: the land east of the city lies beyond the 300 and 400 lines though it is no farther than the north-west. Lake Steinhude makes a hole of high cost north of the 100 line. The south-western forest lies just beyond the isochrone of 100 (as much effort as crossing 100 km of forest, or 5 km of arable land); the north-eastern forest near the 200 line.

Least-cost paths and costs#

The paths follow the cheap land: from the southern forest they run north through the centre-west, then branch west to the north-western forest and east to the north-eastern ones; the path to the south-western forest goes north first, to cross the farmland where it is narrowest. They are far longer than straight lines: 45 km to the north-eastern forest (29 km straight), 71 km to the far north-eastern one (43 km straight).

The ratio of cost to distance is the landscape's "roughness" between two forests. The northern forests are cheap to join (3 to 4 thousand per km: much of the way through woods and pasture); the southern forest is costly to leave towards the north-east (7.5: open farmland, then Hannover's suburbs). A habitat graph with straight distances treats these pairs as alike; with least costs it does not (least-cost links).

A least-cost corridor#

The same model gives a corridor between two areas: the accumulated cost from the southern forest plus the accumulated cost from the north-eastern forest (a second cost.acc_cost from a mask of patch 96, and a raster.calc a + b) is, in each cell, the cost of the best route between the two forests through that cell. Its least value (202,287 here) is the cost of the least-cost path; cells a little above it are on routes almost as good.

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Within 2% of the least cost lie 4,003 cells (40 km²); within 5%, 7,133; within 10%, 14,771. The corridor is narrow where every route must pass the same place, as between Lake Steinhude and the villages to its east (a pinch point for this pair), and wide in the woods of the centre-west, where routes of almost equal cost run side by side. The threshold is a choice: Beier et al. (2008) discuss choosing it so that the corridor has a width the species needs along its whole length.

Settings that change the answer#

• The resistance, above all its contrasts (see the sensitivity analysis).
• Neighbours: 4 (no diagonals: paths look like staircases and costs are higher), 8 (the usual), 16 (knight's moves too: smoother paths, a little cheaper).
• The transition function: the mean of the two cells' conductances, or the minimum (a barrier cell blocks the connection), or a harmonic or geometric mean.
• The corridor's threshold: 2%, 5% or 10% give corridors of very different widths.
• Points or areas: from a point, the path starts at that cell; from an area, from its cheapest edge. Use areas for reserves.

Common pitfalls#

• Resistance of 0 or no value. A cell without a value is not in the graph: a barrier. A resistance of 0 makes an infinite conductance; avoid it.
• Forgetting the geographic correction. Without it, diagonal steps cost the same as side steps, and costs are not in map units.
• Comparing costs between resistance surfaces. Costs scale with the resistance; compare paths and rankings, not raw costs.
• Paths at the map's edge. A path that hugs the edge may want to go around something outside the map. Use a study area larger than the region of interest.
• One path as "the" corridor. A path is one route; draw the corridor and say how wide it is.

Going further#

• Part 2: random walks and randomised shortest paths: from the best route to walkers that do not know the way.
• Costs and corridors: what restoring a corridor gains.
• Least-cost links: least costs in a habitat graph.
• Circuitscape: currents instead of a path.
• The operators: cost.transition, cost.geo_correction, cost.acc_cost, cost.shortest_path, cost.distance, cost.to_map, cost.math, cost.combine.

References#

• Adriaensen, F., Chardon, J. P., De Blust, G., Swinnen, E., Villalba, S., Gulinck, H., & Matthysen, E. (2003). The application of 'least-cost' modelling as a functional landscape model. Landscape and Urban Planning, 64(4), 233–247. https://doi.org/10.1016/S0169-2046(02)00242-6
• Beier, P., Majka, D. R., & Spencer, W. D. (2008). Forks in the road: choices in procedures for designing wildland linkages. Conservation Biology, 22(4), 836–851. https://doi.org/10.1111/j.1523-1739.2008.00942.x
• Dijkstra, E. W. (1959). A note on two problems in connexion with graphs. Numerische Mathematik, 1(1), 269–271. https://doi.org/10.1007/BF01386390
• Etherington, T. R. (2016). Least-cost modelling and landscape ecology: concepts, applications, and opportunities. Current Landscape Ecology Reports, 1(1), 40–53. https://doi.org/10.1007/s40823-016-0006-9
• van Etten, J. (2017). R package gdistance: distances and routes on geographical grids. Journal of Statistical Software, 76(13). https://doi.org/10.18637/jss.v076.i13