FUTURES: urban growth as patches

Towns and suburbs grow in patches: a housing estate, a shopping centre, a row of houses along a new road. Their growth is driven by what is already built nearby, it varies from one county to the next, and its patches have sizes and shapes that differ from region to region. FUTURES was built to simulate exactly this. In this part you will fit its three pieces on the Plum Island Ecosystems between 1985 and 1991 (a development pressure, a potential in each of two subregions, a library of the sizes of the patches developed), calibrate the compactness of its patches, simulate the eight years from 1991 to 1999, compare them with the map of 1999, and move the setting that decides between infill and sprawl. At the end you will know when FUTURES is the right model for a question about urban growth and how to read what it gives.

The idea#

Where it comes from#

FUTURES (FUTure Urban-Regional Environment Simulation) was developed by Meentemeyer et al. (2013) at North Carolina State University to project the urbanisation of the fast-growing region around Charlotte, North Carolina, and its consequences for forests and farmland. Its authors wanted three things models of the time did not combine: a demand for developed land that comes from the population of each county, a probability of development that differs between counties, and patches of development whose sizes and shapes are learned from the patches that really appeared. It has since been rewritten as the open-source GRASS GIS modules r.futures (Petrasova et al., 2016) and used, for example, to compare conservation strategies under urban growth (Dorning et al., 2015). Terraflux runs r.futures's algorithms through the landschaft library.

FUTURES models one transition, from undeveloped to developed, in three submodels:

• DEMAND: how many cells each subregion (a county, a municipality) develops each year;
• POTENTIAL: the probability that each undeveloped cell is developed, from a regression per subregion;
• PGA (patch-growing algorithm): where the demand goes, as patches grown from seeds drawn on the potential.

The development pressure#

The best predictor of development is development nearby. FUTURES measures it as a development pressure: for every cell, a sum over the developed cells around it, weighted by their distance.

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The potential#

The probability that an undeveloped cell becomes developed is a logistic function of the pressure and other drivers, with coefficients that may differ between subregions:

The patch-growing algorithm#

Every year, each subregion must develop its demand. The PGA draws a seed among the undeveloped cells in proportion to their potential; a seed is accepted with a probability equal to its potential; from it, a patch grows to a size drawn from the patch library (the sizes of the patches developed in the calibration period), adding cells next to the patch with a preference for compact shapes set by the compactness (an exponent of the distance to the seed, drawn around compactness_mean within compactness_range). The pressure is updated around every new cell, and the next seed is drawn. The incentive power raises the potential to a power before the seeds are drawn: above 1 it favours the cells of highest potential, mostly near existing development (infill); below 1 it flattens the differences and lets development jump into the countryside (sprawl).

When to use it#

FUTURES suits questions about urban and suburban growth: how much farmland or forest a region loses to development, where, and in what patches; how planning policies (infill, zoning, protected land) change it; how population scenarios translate into land. Its strengths are the demand by subregion, the pressure that makes growth self-reinforcing, and patches whose sizes come from data.

It is not a model of several land uses competing (CLUE-S, PLUS), nor of forest transitions (Dinamica EGO): it knows developed and undeveloped only. Choosing a land change model compares the families.

What you need#

• Maps of developed land (1 developed, 0 not, no value outside the study area) at two dates at least: here the built class (2) of Plum Island's land use maps of 1985, 1991 and 1999, made with raster.calc (lu == 2).
• Drivers of the potential (here slope and elevation of pie_drivers.tif).
• Subregions: a map of ids (counties). Plum Island has no counties in its data, so the map is cut into a western (1) and an eastern half (2) to show how subregions work.
• A demand per subregion and year: here the yearly trend of 1985-1991 (281 cells in the west, 257 in the east), kept for 1992-1999.

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Build the model#

Step 1: the pressure, the potential and the patch library#

format = "terraflux/1"

[[nodes]]
id = "lu1985"
op = "raster.read"
params = { path = "pie_landuse.tif", band = "1985" }

[[nodes]]
id = "lu1991"
op = "raster.read"
params = { path = "pie_landuse.tif", band = "1991" }

[[nodes]]
id = "drivers"
op = "raster.read"
params = { path = "pie_drivers.tif" }

[[nodes]]
id = "counties"
op = "raster.read"
params = { path = "subregions.tif" }

[[nodes]]
id = "dev1985"
op = "raster.calc"
inputs = { lu = "lu1985.raster" }
params = { expression = "lu == 2" }

[[nodes]]
id = "dev1991"
op = "raster.calc"
inputs = { lu = "lu1991.raster" }
params = { expression = "lu == 2" }

[[nodes]]
id = "pressure"
op = "futures.devpressure"
inputs = { developed = "dev1985.map" }
params = { size = 10, method = "gravity", gamma = 1.5, scaling_factor = 1.0 }

[[nodes]]
id = "predictors"
op = "stack.set"
inputs = { stack = "drivers.raster", map = "pressure.map" }
params = { name = "devpressure" }

[[nodes]]
id = "new_west"
op = "raster.calc"
inputs = { a = "dev1985.map", b = "dev1991.map", c = "counties.raster" }
params = { expression = "None if a == 1 or c != 1 else b" }

[[nodes]]
id = "potential_west"
op = "glm.fit"
inputs = { landscape = "new_west.map", drivers = "predictors.stack" }
params = { formulas = { 1 = ["devpressure", "slope", "elevation"] } }

[[nodes]]
id = "library"
op = "futures.patch_library"
inputs = { start = "dev1985.map", end = "dev1991.map", subregions = "counties.raster" }
params = { per_subregion = true }

The fitted coefficients:

In both halves the pressure raises the potential and the slope lowers it, but the east responds twice as strongly to the pressure: its development clings more closely to existing development. This is why subregions matter.

Step 2: calibrate the compactness#

The library gives the sizes; the shape is set by the compactness. futures.calibrate (r.futures.calib) simulates the calibration period with every combination of compactness_means, compactness_ranges and discount_factors, repeat times each, and compares the simulated patches with the observed ones: the histograms of their areas and of their compactness (perimeter / (2√(π area))), by a chi-square distance. It ranks the combinations by the mean of the two errors. Tried here: means 0.1, 0.3, 0.5, 0.7 and 0.9, ranges 0.05 and 0.3, four repeats each:

(the errors are relative to the largest of each; the full table is calibration.csv). The best, a mean of 0.7 and a range of 0.3, is used below. The ranking is not sharp: with four repeats, the first two are close. More repeats narrow it.

Step 3: simulate 1991 to 1999#

format = "terraflux/1"

[[nodes]]
id = "land"
op = "raster.read"
params = { path = "pie_landuse.tif", band = "1991" }

[[nodes]]
id = "end"
op = "raster.read"
params = { path = "pie_landuse.tif", band = "1999" }

[[nodes]]
id = "drivers"
op = "raster.read"
params = { path = "pie_drivers.tif" }

[[nodes]]
id = "counties"
op = "raster.read"
params = { path = "subregions.tif" }

[[nodes]]
id = "developed"
op = "raster.calc"
inputs = { lu = "land.raster" }
params = { expression = "lu == 2" }

[[nodes]]
id = "pressure"
op = "futures.devpressure"
inputs = { developed = "developed.map" }
params = { size = 10, method = "gravity", gamma = 1.5, scaling_factor = 1.0 }

[[nodes]]
id = "futures"
op = "futures.simulation"
inputs = { landscape = "developed.map", pressure = "pressure.map", drivers = "drivers.raster", subregions = "counties.raster", seed = 1 }
params = { potential = { ids = [1, 2], intercept = [-3.234651, -4.223066], devpressure = [0.06569, 0.132427], predictors = ["slope", "elevation"], coefficients = [[-0.035282, 0.000257], [-0.028677, 0.004273]] }, demand = { subregions = [1, 2], rows = [[281, 257], [281, 257], [281, 257], [281, 257], [281, 257], [281, 257], [281, 257], [281, 257]], years = [1992, 1993, 1994, 1995, 1996, 1997, 1998, 1999] }, patch_sizes = { ids = [1, 2], lists = [[1, 1, 1, 2, 3, 5, 9], [1, 1, 2, 2, 4, 7, 12]] }, neighborhood = 10, approach = "gravity", gamma = 1.5, scaling_factor = 1.0, discount_factor = 1.0, compactness_mean = 0.7, compactness_range = 0.3, num_neighbors = 4, seed_search = "probability", incentive_power = 1.0 }

futures.simulation (r.futures.simulation) is one node that runs every year itself; it is not built from a loop, because its pieces (the pressure updated around every new cell, the seeds, the patches) are intertwined within a year. Its inputs: landscape (developed or not at the start), pressure (at the start, with the same settings as the potential was fitted with), drivers (the predictors by name), subregions and seed. Its settings:

• potential: the coefficients, { ids, intercept, devpressure, predictors, coefficients }, a value per subregion (copied from the two regressions above);
• demand: { subregions, rows, years }, a row per year with the cells of each subregion;
• patch_sizes: the library (futures.patch_library's output, { ids, lists });
• the pressure's neighborhood, approach, gamma and scaling_factor (the same as node 6);
• the PGA's compactness_mean, compactness_range, discount_factor (multiplies the patch sizes), num_neighbors (4 or 8) and seed_search;
• incentive_power (1: neither infill nor sprawl favoured).

It also takes zoning, restrictions and FUTURES' flood submodel (climate, with catchments, flood maps, damage and adaptation), not used here. Its outputs: maps (developed or not after each year), development (the year each cell was developed) and info (per year and subregion, the cells developed and the demand carried over when it could not be met: overflow).

Read the results#

Eight years#

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PlayYear

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Each subregion meets its demand every year (with a few cells carried to the next year when a patch overshoots, the overflow columns of info).

Against 1999#

Observed 1999

Simulated 1999

validate.futures (r.futures.validation) compares the simulated map of 1999 with the observed one (both 0/1), from the map of 1991: a figure of merit of 0.034, kappa simulation of 0.020, a quantity disagreement of 0.006 and an allocation disagreement of 0.053. The figure of merit is as low as Dinamica EGO's on the same landscape (Validating a simulation): Plum Island's new development is scattered in small patches that no model with these drivers places cell by cell. The quantity disagreement has a clear cause: the trend of 1985-1991 asked for 4,304 new cells, while 3,105 were developed between 1991 and 1999. Growth slowed in the 1990s; a demand from population projections, as FUTURES intends, would have done better than a trend.

What to report#

The demand of each subregion and its source; the pressure's settings; the potential's coefficients per subregion; the patch library (its period, the number and sizes of patches); the calibrated compactness and how it was chosen; the incentive power; the validation against an observed map; replicates (several seeds).

Settings that change the answer#

The incentive power: infill or sprawl#

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PlayIncentive power

From 0.25 to 1 the share of new development that touches the development of 1991 rises from 53% to 77%: the incentive power is FUTURES' lever for policies of compact growth. At 2, and more so at 4, something else happens: the potentials, raised to the power, become so small that few seeds are accepted, and the simulation fails to meet the demand (1,166 of 4,304 cells at 2; 16 at 4), carrying the rest over year after year (negative overflow in info). Read info after every run with an incentive power above 1, and if the demand is not met, raise the potential (a larger scaling factor of the pressure) or keep the power lower.

Other settings#

• The pressure's radius, decay and scaling: they must be the same in futures.devpressure (the potential's fit) and in the simulation, or the coefficients no longer apply.
• The patch library: a library from a period of large developments makes large patches; discount_factor scales the sizes (below 1, smaller patches).
• The demand: the most consequential choice, and the one validation exposed here.

Common pitfalls#

• Fitting the potential on all cells. It must be fitted on the cells undeveloped at the start, with their state at the end; developed cells would teach the model where development is, not where it appears.
• A pressure computed differently in the fit and in the simulation.
• Subregions with no development in the calibration period: their regression cannot be fitted; merge them.
• Ignoring overflow: a demand not met is a silent failure of the scenario.
• One run. The PGA is stochastic: run replicates (Calibration and many runs).

Going further#

• MOLUSCE, the next part, and PLUS, the other patch-generating models.
• A demand per region: subregions in the other models.
• Validating a simulation.
• The operators: futures.devpressure, futures.patch_library, futures.calibrate, futures.simulation, validate.futures.

References#

• Dorning, M. A., Koch, J., Shoemaker, D. A., & Meentemeyer, R. K. (2015). Simulating urbanization scenarios reveals tradeoffs between conservation planning strategies. Landscape and Urban Planning, 136, 28–39. https://doi.org/10.1016/j.landurbplan.2014.11.011
• Meentemeyer, R. K., Tang, W., Dorning, M. A., Vogler, J. B., Cunniffe, N. J., & Shoemaker, D. A. (2013). FUTURES: multilevel simulations of emerging urban–rural landscape structure using a stochastic patch-growing algorithm. Annals of the Association of American Geographers, 103(4), 785–807. https://doi.org/10.1080/00045608.2012.707591
• Petrasova, A., Petras, V., Van Berkel, D., Harmon, B. A., Mitasova, H., & Meentemeyer, R. K. (2016). Open source approach to urban growth simulation. The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, XLI-B7, 953–959. https://doi.org/10.5194/isprs-archives-XLI-B7-953-2016