CHAPTER
XV
REVIEW OF
SPATIAL DYNAMIC MODELS
15
.1.
Spatial/Land Use Change Modelling
Model
ing of land use changes as a function of its biophysical and socio

economic driving forces
provides insights into the extent and location of land use changes and its effects. The
modeling
framework is a methodology to model near future land use changes bas
ed upon actual and past
land use conditions. This chapter describes how changes in land use are allocated in the model.
15
.2.
Probabilistic Model
The Probabilistic Model is a statistical analysis of the quantitative relationships between the
actual land
use distribution and (potential) driving forces underlies the allocation procedure.
Based upon thus derived multiple regression equations, areas with potential for increase or
decrease in cover percentage of a certain land use type are identified. Actual
allocation is
modified by autonomous developments and competition between land use types. A multi

scale
approach is followed to account for the scale dependencies of driving factors of land use change.
This approach provides a balance between bottom

up eff
ects as result of local conditions and
top

down effects as result of changes at national and regional scales.
15
.
2.1
General structure of the
Model
The model consists of four main modules: a demand module; a population
growth
module; a
yield module; and an allocation module (Fig.
15.2.1.
1).
Fig.
15.2.1.
1. General structure of the
probabilistic
modelling framework
.
The demand module calculates, at
the national level, changes in
demand for
land use
taking into
account population growth. The
c
alculations are based upon trends of the past in combination
with projections of future
land use
demand. The population module calculates changes in
p
opulation and associated demographic characteristics based u
pon different projections (e.g.
Lutz et al., 1994).
The calculated changes in demand and population will influence the spatial distribution and
importance of different land use types and associated productions. The yield module calculates,
in a spatial exp
licit way, changes in yield level and yield distribution over the country. The
changes in the distribution of different land use types are calculated in the allocation module. In
principal it is assumed that all changes in demand are satisfied by changes i
n land use following
the theoretical framework of Boserup (1965)
.
The allocation module simulates the pattern of land
use change, not the total quantity of change,
which is calculated at the national level in the
demand module. All calculations are done on
a yearly basis.
15.2.2
. Multi

scale approach
The model includes two spatially explicit scales at which land use is allocated in addition to the
aggregated national scale
on which demands are calculated
.
These allocation scales are artificial
aggregation
levels consisting of grid based data at two different resolutions. A relatively coarse
scale is used to calculate the general trends of the changes in land use pattern and to capture the
influence of land use drivers that act over considerable distance. B
ased upon the general pattern
of land use change calculated at this coarse allocation scale, but taking local constraints into
account, the land use pattern is calculated at a finer scale level. Depending on the
application,
area studied and data availabil
ity the resolution of analysis will vary.
15.2.3
. Calculation of changes in land use
The allocation module is based upon a spatial analysis of the complex interaction between land
use, socio

economic conditions and biophysical constraints. This interacti
on is captured by an
empirical analysis of historic and present land use. This empirical analysis is used to identify the
most important biophysical and socio

economic drivers of land use, as well as the quantitative
relationships between these drivers and
the surface area of the different land use types.
The
Logit
regression models are assumed to give a good description of the land use distribution
under the
actual biophysical and socio

economic conditions in a country. The regression models
are used to c
alculate the cover percentage of the different land use types under the biophysical
and socio economic conditions in a certain grid
.
The results (further on referred to as ‘regression
cover’) represent the average cover of the different land use types
expected on basis of the
present biophysical and socio

economic conditions at the coarse allocation scale.
15
.3.
Cellular Automata Model
The urban growth dynamic implemented in the Cellular Automata Model is defined through four
steps:
(i) Spontaneous Gr
owth
Spontaneous growth defines the occurrence of random urbanization of land. In the cellular
automaton framework this means that any non

urbanized cell on the lattice has a certain (small)
probability of becoming urbanized in any time step.
(ii) New
Spreading Center Growth
The next urban growth step is defined though the dynamics of new spreading centers. As the
name indicates, this step determines whether any of the new, spontaneously urbanized cells will
become new urban spreading centers.
(iii) Ed
ge
Growth
E
dge

growth dynamics define the part of the growth that stems from existing spreading centers.
This growth propagates both the new centers generated in th
e present step ii
and the more
established centers from earlier times. Thus, if a non

urban cell has at least three urbanized
neighboring cells, it has a certain global probability to also become urbanized
(iv) Road

Influenced
Growth
The final growth step, road

influenced
growth, is determined by the existing transportation
infrastructure as well as the most recent urbanization done under steps
i
,
ii and iii
.
If a road is
found within a given maximal radius of the selected cell, a temporary urban cell is placed at the
poin
t on the road that is closest to the selected cell. Next, this temporary urban cell conducts a
random walk along the road (or roads connected to the original road). The final location of this
temporary
urbanized cell is then considered as a new urban sprea
ding nucleus.
A
growth cycle
the basic unit of model growth consists of these four steps and, if modeling land
cover, the deltatron change dynamics.
15.3
.
1
General structure of the Model
Growth Cycle
A
growth cycle
is the basic unit of
model
growth
execution. I
t begins by setting each of the
c
oefficients to a unique value. Each of the growth rules
is
then applied. Finally, the resulting
growth rate is evaluated. If the growth rate exceeds or falls short of the CRITICAL_HIGH or
CRITICAL_LOW values, mo
del self

modification is applied. Self

modification will slightly
alter the coefficient values to simulate accelerated or depressed growth that is related with
boom and bust conditions in urban development.
Growth Coefficients
Five coefficient
s:
dispersion, b
reed
, s
pread
, s
lope
and r
oad gravity coefficient
s
values affect
how the growth rules are applied. These values are calibrated by comparing simulated land
cover change to a study area's historical data. The descriptions below outline the five
c
oefficient values, which types of growth they affect, and how the applied values are derived
from the coefficients.
1) Set coefficient values
( _ _ _ _ _ )
2) Apply Growth Rules
3) Self

Modification
1.
UGM
2.
DLM
Calculate growth rate (GR)
If (GR > CRITICAL_HIGH)
modify coefficients for
BOOM
state
If (GR < CRITICAL_LOW)
modify coefficients for
BUST state
Growth rules
The model begins with a set of initial
conditions which is the input data configuration of the
landscape. A set of decision, or growth, rules is then applied to the data to simulate urban driven
land cover change.
(i) Spontaneous Growth
Spontaneous growth defines the occurrence of random urbanization of land. In the cellular
automaton framework this means that any non

urbanized cell on the lattice has a certain (small)
probability of becoming urbanized in any time step. Thus, whether a gi
ven cell U(i,j,t) at
coordinate (i,j) at time t will be urbanized at time t+1 can be expressed by
(1)
U(i,j,t+1) = f1[ dispersion_coefficient , slope_coefficient , U(i,j,t), random ],
where the parameter
dispersion_coefficient
( diffusion_coefficient in previous literature (Clarke,
Hoppen, Gaydos 1996)) determines the (small) spontaneous, global urbanization probability, and
the
slope_coefficient
parameter determines the weighted probability of the local slope. The
stochasticity of the process is indicated by random. If the cell is already urbanized or excluded
from urbanization, i
t will not change, and therefore the ability to transition also depends on the
cells own current value.
Spontaneous Growth:
F(
dispersion_coefficient
,
slope_coefficient
)
{
for (p < dispersion_value)
{
select pixel location
(i,j)
at random
if (
(i,j
) is available for urbanization)
{
(i,j)
= urban
New S
preading Center Growth
}
}
} end spontaneous growth
Spontaneous growth example and pseudo code.
(ii) New Spreading Center Growth
The next urban growth step is defined though the dynamics of new spreading centers. As the name
indicates, this step determines whether any of the new, spontaneously urbanized cells will become
new urban spreading centers. The global parameter,
breed_coefficient
, defines the probability for
each new urbanized cell
U(i,j,t+1)
to become a new spreading center
U'(i,j,t+1)
, given two
neighboring cells also are available for ur
banization
(2)
U'(i,j,t+1) = f2[ breed

coefficient, U(i,j,t+1), random ],
where
(k,l)
are nearest neighbors to
(i,j)
. If the cell is allowed to become a spreading center, two
additional cells adjacent to the new spreading center cell also have to be urbanized. Thus an urban
spreading center is defined as a location with three or more adjacent urbanized cells. The
actual
ization of this step is dependent upon the
slope_coefficient

weighted topography and the
availability of neighborhood cells to make the transition.
New Spreading Center Growth:
F(
breed_coefficient
,
slope_coefficient
)
{
if (random_
integer <
breed_coefficient
)
if (two neighborhood pixels are available
for urbanization)
(i,j)
neighbors = urban
} end new spreading center growth
New spreading center growth example and pseudo code.
(iii) Edge Growth
Edge

growth dynamics define the part of the growth that stems from existing spreading centers.
This growth propagates both the new centers generated in step ii in this time step, time
(t+1)
, and
the more established centers from earlier times. Thus, if a n
on

urban cell has at least three urbanized
neighboring cells, it has a certain global probability to also become urbanized defined by the
spread_coefficient
, given it is
possible to build on the cell (
slope_coefficient
). Thus this edge
growth can be expressed by
(3)
U(i,j,t+1) = F3[ spread_coefficient, slope_coefficient, U(i,j,t), U(k,
l), random ],
where
(k,l)
belongs to the nearest neighborhood of
(i,j)
.
Edge Growth:
F(
spread_coefficient
,
slope_coefficient
)
{
for (all non

edge pixels
(i,j)
)
if (
(i,j)
is urban) and (random_integer
<
spread_coefficient
)
if (at least two urban neighbors exist)
if (a randomly chosen, non

urban
neighbor is available for urbanization)
(i,j)
neighbor = urban
} end edge growth
Edge growth example and pseudo code.
(iv) Road

Influenced Growth
The final growth step, road

influenced growth, is determined by the existing transportation
infrastructure as well as the most recent urbanization done under steps
i
,
ii
and
iii
. With a
probability defined by
breed_coefficient
, newly urbanized cells (at time
t+1
) are selected, and the
existence of a road is sought in their neighborhoods. If a road is found within a given maximal
radius (determined by
road_gravity_coefficient
) of the selected cell, a temporary urban cell is
placed at the point on the road that is closest to the selected cell. Next, this temporary urban cell
conducts a random walk along th
e road (or roads connected to the original road) where the number
of steps is determined by the parameter
dispersion_coefficient
. The final location of this temporar
y
urbanized cell is then considered as a new urban spreading nucleus. If a neighboring cell to the
temporary urbanized cell (on the road) is available for urbanization, it will happen (randomly picked
among possible candidates). If two adjacent cells to th
is newly urbanized cell are also available for
urbanization it will happen (randomly picked among candidates). Thus the creation of the
temporary urbanized cell on the road is defined by
(4.1)
U'(k,l,t+1) = f4.1[ U(i,j,t+1), road_gravity_coefficient, R(
m,n), random ]
where
i,j,k,l,m
, and
n
are cell coordinates, and
R(m,n)
defines a road cell. The random walk on the
road may be expressed by
(4.2)
U''(i,j,t+1) = f4.2[ U'(k,l,t+1), dispersion_coefficient, R(m,n), random ].
where
(i,j)
are road cells nei
ghboring
(k,l)
. If we define the location of the temporary urbanized cell
at the end of the random walk by
(p,q)
, the new adjacent urban spreading center will be defined by
(4.3)
U'''(i,j,t+1) = f4.3[ U''(p,q,t+1), R(m,n), slope_coefficient, random ],
a
nd two additional adjacent urbanized cells may be added using
(4.4)
U''''(i,j,t+1) = f4.4[ U'''(p,q,t+1), slope_coefficient, random ],
where
(i,j)
and
(k,l)
belong to the nearest neighborhood of
(p,q)
. The four steps above are
collectively referred to a
s a
road trip
. Each attempt to select a newly urbanized pixel to move to a
road is a new road trip. The number of attempted road trips in any given growth cycle is determined
by the
breed coefficient
breed_coefficient
.
Road

Influenced Growth:
F(
breed_coefficient
,
road_gravity_coefficient
,
dispersion_coefficient
,
slope_coefficient
)
{
for (p <=
breed_coefficient
)
{
road_gravity = value which is a function of
image size and
road_gravity_coefficient
max_search = maximum distance, determined by
road_gravity, for which a road pixel is searched
(i,j)
= randomly selected pixel, urbanized within the
current growth cycle
road_found = search outward from
(i,j)
, up to
max_search, for a road pixel
if (road_found)
{
walk along the road, in randomly selected
directions, for a number of steps determined
by the road_value and the
dispersion_coefficient
if (a neighboring pixel is available for urbanization)
(i,j)
neighbor = urban
if (two ne
ighbors of the newly urban pixel
are available for urbanization)
two urban pixel neighbors = urban
} } } end road

influenced growth
Road

influenced growth example and pseudo code.
Basic Simulation
A
simulation is a step up in complexity from a
growth cycle
.
A simulation is a series of growth
cycles that begins at a start date and completes at a stop date.
15.2.3
. Calculation of changes in land use
Initial Conditions
A simulation must be initialized
with a set of conditions. These initial conditions are a) an
integer value, or seed, that initializes the random number generator b) a value for each of the five
growth coefficients and c)
Model
input images most closely representing the start date.
With
these initial conditions growth cycles are generated. It is assumed that one growth cycle
represents a year of growth. Following this assumption:
number of growth cycles in a simulation = stop_date

start_date
Concluded
Simulation
of growth cycles.
When the required number of growth cycles has been generated, the simulation concludes.
Input
The model requires several basic data layers exported from a geographic data base in grayscale
GIF image format. For all layers, 0 is a nonexistent or null value, while 0 < n < 255 is a "live", or
existing, value. The model requires all input layers to hav
e a consistent number of rows and
columns. For statistical calibration of the model, at least four urban time periods must be used.
Also, for purposes of calibration, the roads must be represented in two or more time periods. The
model requires two land us
e layers for deltatron land use modeling. All layers should be checked
for agreement; urban areas should not be present locations defined as undevelopable in the
excluded layer.
Format standards for all data types
grayscale GIF images
images are derived f
rom grids in the same projection
images are derived from grids of the same map extent
images the same resolution (row x column count is consistent)
images follow the required
naming format
The following images were created as part of a "fictional" data set to demonstrate format,
calibration and implementation of SLEUTH. Their purpose is to illustrate the requirements and
functions of the model rather than represe
nt processes of a specific city or region. Some images'
values on this page have been altered in order to illustrate their content and should not be
confused with the actual input image data which may be accessed from our download page.
Slope
The slope is commonly derived from a digital
elevation model (DEM), but other elevation
source data may be used. Cell values must be in
percent
slope, not degree, which is a common
default in some GIS software.
%slope equation:
Pixel value range: 0
–
0
ia湤⁵ne
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畳u a来猠獨潵s搠牥灲ese湴⁵湩煵n慮搠 la獳⸠
c潲xa浰meⰠ楦⁴桥⁁湤e牳潮rie癥氠fc桥浥⁷a猠
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睨w牥 (
R,G,B)
represents the red, green and blue
color bands in the image, and
class
is the land
cover type associated with the (R,G,B) value.
This information is entered in the
land cover
colorable
section of the scenario file where
pix
is
the (R,G,B) value and
name
is the class land
cover type.
Pixel value range: 0
–
㈵5
(R,G,B)
class
(1,1,1)
urban
(2,2,2)
agriculture
(3,3,3)
range land
(4,4,4)
forest
Excluded
The excluded image defines all locations that are
resistant to urbanization. Areas where urban
development is considered impossible, open water
bodies or national parks for example, are given a
value of 100 or greater. Locations that are
available for urban development have a value of
zero (0).
Pixels may contain any value between (0

100) if
the representation of partial exclusion of an area is
desired

unprotected wetlands could be an
example: Development is not likely, but there is
no
zoning to prevent it.
Pixel value range: 0

255 (values > 100, are read
as 100)
Urban
The urban extent for the start year, or
seed
, is used
to initialize the model and is the basis for the CA
driven urban growth. For calibration, the earliest
urban year is used as the seed, and subsequent
urban layers, or
control years
, are used to measure
several statistical best fit values. For thi
s reason,
at least four urban layers are needed for
calibration: one for initialization and three
additional for a least

squares calculation.
The definition of "urban extent" is up to the
creators of the data set. The model simply
requires a binary classi
fication of urban/nonurban.
Methods used in the past include digitizing city
maps and aerial photographs, thresholding
remotely sensed images or block densities from
census data.
Pixel value range: 0 = nonurban
0 < n < 256 = urban
Transportation
The road influenced growth dynamic included in
SLEUTH simulates the tendency of urban
development to be attracted to locations of
increased accessibility. A transportation network
can have major influence upon how a region
develops. To inc
lude this effect in calibration
several road layers, that change with the city's
growth over time, are desirable. SLEUTH will be
initialized with the earliest road layer. As growth
cycles, or "time", pass and the date for a more
recent road layer is reache
d, the new layer will be
read in and development will proceed from there.
Road network images may be binary (road/non

road) or have relative values:
weighting 1
weighting 2
pixel values
pixel values
accessibility
4
100
high
2
50
medium
1
25
low
0
0
none
note that the relative weighting of the two
schemes above are equivalent and would have an
identical effect if applied to the same data. For
more information see
road
weighting
.
Pixel value range:
binary: 0 = non

road, 0 < n < 256 = road
relative: (see above)
Hillshade
In order to give spatial context to the urban extent
data, a background image is incorporated into
image output. This must be a grayscale image,
and a hillshaded DEM (pictured here) is
commonly used.
To give further definition to a region, bodies of
water
may also be represented. This occurs by any
pixels in the background image whose values are
zero (0) being filled with the
water
color defined
in the scenario file. *Note: this will also mean that
any heavily shaded locations that have a zero
value will a
lso be filled with the WATER color.
This can be avoided by remapping any zero
values in the hillshade image to one (1) before
adding the water mask.
If
water
is defined as black (R,G,B = 0,0,0) zero
value pixels will remain black in the output
images.
Gro
wth coefficients d
o not necessarily remain static throughout an application. In response to
rapid or depressed growth rates, the coefficients may be increased or decreased to further
encourage system wide growth rate trends.
A second level of growth rules,
termed self

modification rules, is prompted by an unusually high or low
growth rate. The growth rate is the sum of the four different
types of growth
defined by the model for
each model
growth cycle
, or "year." The limits CRITICAL_HIGH and CRITICAL_LOW (defined in the
scenario_file
) kick off an increase or decrease in three of the growth control parameters: dispersion,
breed, and spread. If the growth rate exceeds the CRITICAL_HIGH, the coefficients are increased by a
multiplier great
er than one: BOOM. This increase imitates the tendency of an expanding system to grow
ever more rapidly. If the growth rate falls below the CRITICAL_LOW, the coefficients are decreased by
a multiplier less than one, BUST, causing growth to taper off as it
does in a depressed or saturated
system.
15
.4.
Compar
ative
Analysis
Mode process flow
Each mode has variations on how simulations are executed.
Prediction process flow
Prediction mode is a collection of Monte Carlo simulations. The coefficient set and ini
tial images
for a prediction run are identical for every simulation, but the initializing seed value is altered a
the iteration
number of times, with each simulation evolving slightly differently due to the
modified random number series.
Growth Cycles
It
is assumed that one growth cycle represents a year of growth. Following this assumption:
number of growth cycles in a simulation = stop_date

start_date.
Conclude Simulation
When the required number of growth cycles has been generated, the simulation concludes.
Testing process flow
Test mode was created as a way to generate a set of historical data simulations, such as those
described in calibration mode process flow, for
a single coefficient set without requiring the
START_* and STOP* coefficients to be set to identical values. Only the START_* values are
used when testing. The START* value must be greater than the STOP_* value.
Test mode can produce the same
statistic and
output
files that may be generated in calibrate
mode. Additionally, on the final Monte Carlo run, test mode will generate annual images of land
cover change. In this way, a visual and further statistical evaluation of calibration coefficient
performance m
ay be made.
Initial Conditions
Each simulation in a test mode is initialized with the START_* values of each coefficient type
and
model
images as described in a
basic simulation
. The seed value for the first simulation is
initialized with the
random seed
value
. After a simulation is completed, the initializing seed that
began that simulation is reset and a new simulation is run.
Growth Cycles
It is assumed that one growth cycle represents a year of growth. Following this assumption:
number of growth cycl
es in a simulation = stop_date

start_date.
Conclude Simulation
When the required number of growth cycles has been generated, the simulation concludes.
Calibration process flow
Calibration is the most complex of the different mode types. Each coefficient set combination
created by the coefficient START_, STOP_ and STEP_
values will initialize a run
. Each run will
be executed MONTE_CARLO_ITERATIONS number of times. The RANDOM_SE
ED
value
initializes the first Monte C
arlo simulation of every run.
The run initializing seed value is set in the scenario file with the
random seed value. The number
of Monte C
arlo iterations is set in the scenario
.
Several statistic and image files may b
e generated in calibrate mode by setting preferences in the
scenario file. However, due to the computational requirements of calibration, it is recommended
that these write flags are set to OFF. Instead, once a few top coefficient sets are identified,
stat
istics and image files for these runs may be generated in test mode.
Initial Conditions
Each run of a calibration job is initialized with a permutation of the coefficient ranges. Each run
will be executed MONTE_CARLO_ITERATIONS number of times. The first m
onte carlo of
each run is initialized with the RANDOM_SEED value. After a simulation is completed, the
initializing seed that began that
simulation is reset and a new Monte C
arlo simulation is run. This
process continues MC number of t
imes. When the number
of Monte C
arlo iterations for that run
has been completed, a coefficient value will be incremented and a new run initialized. This will
continue until all possible coefficient permutations have been completed.
Generate Simulations
It is assumed that one
growth cycle represents a year of growth. Following this assumption:
number of growth cycles in a simulation = stop_date

start_date.
As growth cycles (or years) complete, time passes. When a cycle completes that has a matching
date from the urban input
layers, a gif image of simulated data is produced and several metrics of
urban form are measured and stored in memor
y. When the required number of Monte C
arlo
simulations has been completed the measurements for each metric are averaged over the number
of
monte carlo iterations. These averaged values are then compared to the input urban data, and
Pearson regression scores are calculated for that run. These scores are written to the
control_stats.log
file and used to assess coefficient set performance.
Conclude Simulation
When the required number of growth cycles has been generated, the simulation concludes.
15
.5.
Evaluation and Selection
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