Figure 1.
Example Instances of Eight Virtual Landcapes
Example virtual landscape geometries (7 km × 7 km section). (A) von Neumann and (B) Moore neighbourhoods in a raster grid; (C) hexagonal; (D) Dirichlet tessellation; CGD tessellation with a mean of (E) four, (F) nine, and (G) 16 raster cells per km2; (H) land cover aggregate map. The neighbourhood (grey) of a focal cell (black) is highlighted in each virtual landscape.
Figure 2.
Maximum Distance Accessible in 100 Steps
Maximum distance accessible in 100 cell-to-cell steps from the origin (star) in five virtual landscapes. The geometry of the regular grids is immediately apparent from the accessible regions of the von Neumann (yellow), Moore (red), and hexagonal (green) landscape models. Accessibility in all the irregular geometries was similar, and lay between that of the Dirichlet (circular blue line) and CGD4 (dashed circular blue line) virtual landscapes. London (shaded grey) is not accessible with some geometries, but is completely within reach of others.
Table 1.
Description of Area and Neighbourhood for Parcels in the Interior of All the Virtual Landscapes
Figure 3.
The distribution of step lengths possible in five virtual landscapes. The von Neumann (orange, at 1.0) and hexagonal (green, at 1.074 km) landscapes only allow a single step length, whereas the Moore geometry allows two steps (red, at 1.0 and 1.41). A single instance of a Dirichlet landscape (blue, mean 1.095 km, gamma distributed with shape = 1.98, rate = 1.8 × 104) allows a distribution of step lengths that vary from parcel to parcel but have a mean similar to the hexagonal geometry. Other irregular landscapes have step length distributed similarly (CGD4 shown, blue dashed line, mean 1.15 km).
Figure 4.
Angular Variation in Accessibility
The minimum number of steps required to travel a range of distances was measured every 10° across a 90° angle in four virtual landscapes. The standard deviation increased linearly with increased distance (d) in the regular grids; von Neumann (dashed line) has trend 0.14d (R2 = 0.9995), Moore (dash dot) 0.09d (R2 = 0.9994), and hexagons (dash dot dot) 0.05d (R2 = 0.9952). Standard deviation in the Dirichlet virtual landscape (solid line) increased with trend 0.46 ln(d) (R2 = 0.968).
Figure 5.
Population Distributions after Random Movement in Different Virtual Landscapes
A matrix showing population distributions after a number of random movement scenarios. Rows (from top to bottom): (1) von Neumann; (2) Moore; (3) hexagon; (4) Dirichlet; and (5) vector landscapes. Columns (from left to right): (A) random movement with t = 5 (time steps) and p = 1 (probability of movement in a time step); (B) random movement, t = 10, p = 0.5; and (C) random movement, t = 100, p = 1. Colours represent population density in each cell on a common scale, ranging from yellow (low density) through orange, red, and purple to blue (high density). In row 5, the vector points are only represented in one colour.
Figure 6.
Population Distributions after Directed Random Movement in Different Virtual Landscapes
A matrix showing population distributions after a number of directed random movement scenarios. Rows (from top to bottom): (1) von Neumann; (2) Moore; (3) hexagon; (4) Dirichlet; and (5) vector landscapes. Columns (from left to right): (A) directed random movement, t = 50, p = 1, b = 0.0 (probability of backward movement); (B) directed random movement, t = 100, p = 0.5, b = 0.0; and (C) directed random movement, t = 50, p = 0.5, b = 0.1. Colours represent population density in each cell on a common scale, ranging from yellow (low density) through orange, red, and purple to blue (high density). In row (5), the vector points are only represented in one colour.
Figure 7.
Random Movement in a Single Dirichlet Landscape
Population distributions in a Dirichlet landscape are shown after: (top) random movement with t = 100 (time steps), p = 1 (probability of movement in a time step); (centre) semi-directed movement with t = 50, p = 1, b = 0.1 (probability of choosing a neighbour closer to the origin); and (bottom) directed movement with t = 50, p = 1, b = 0. In the left column, individuals move into a neighbouring parcel with probability 1/(number of neighbours). In the right column, individuals move into a neighbouring parcel with probability proportional to the length of the shared boundary. There is no significant difference in the population distributions despite the difference in neighbour choice.
Figure 8.
Qualitative Differences in Raster Representation of a Real Extent
Three alternative raster representations of the UK at 10 km resolution, formed against the BNG. In (A), the origin is at BNG (0,0) and the raster is aligned with the BNG. In (B), the origin has been shifted to BNG (−5000,−5000) but the orientation is unchanged. In (C), the origin is at BNG (0,0), but the raster has been rotated by 45°. The arrows refer to the orientation of the raster grid. Observe the variation in shape and size of the Orkney and Shetland isles (the two groups of islands north of the mainland).
Figure 9.
Variation in the Area Measurement of a Real Extent
The area of the UK was measured from raster datasets at resolution 1, 10, 50, and 100 km, as a percentage of a vector polygon area (245,660 km2), and the mean calculated. The y-error bars denote coefficient of variation. The mean is always an underestimate, and worsens at lower resolutions.
Figure 10.
Space and Direction from a Fixed Point in Multiple Irregular Landscapes
(A) A single Dirichlet landscape showing three fixed points (+). One (top left) occupies a cell inland, while another (top right) occupies a coastal cell that is restricted by the edge of the extent. The third (centre) is shown with the available directions for movement in that landscape instance.
(B) A second random instance of a Dirichlet tessellation in the same extent. The three fixed points are highlighted, with their respective cells and directions of movement.
(C) The sum of the observations of space and direction around the three points after only five Dirichlet landscapes. Light grey lines indicate the boundaries of cells in all five landscapes. The kernel associated with inland and coastal points is shown in shades of grey, with the lightest shade showing an area only associated with the point in one landscape instance and black being the area common to all five. Although clearly not circular, given enough iterations, all points approach a circular kernel of influence unless restricted by the extent. Available directions for movement across all five landscapes are shown from the third point, demonstrating that movement in any direction is equally possible.
Figure 11.
Creation of a Coarse-Grain Dirichlet Landscape from a Raster
To create a Dirichlet landscape, starting points are chosen randomly (top left). The vector Dirichlet landscape (top right) is shown for comparison. Random starting points are translated into a raster grid (bottom left). All other raster squares are then assigned to the nearest coloured square (measured between centroids) and boundaries dissolved to produce the CGD landscape (bottom right).