dinamica - a new model to simulate and study landscape dynamics
TRANSCRIPT
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DINAMICA - a new model to simulate and study landscape dynamicsBritaldo Silveira Soares-Filho a, Gustavo Coutinho Cerqueira b, Cássio Lopes
Pennachinc
a Department of Cartography - [email protected], phone: 5531 34995417, fax: 553134995415, b Remote Sensing Center - [email protected], Federal University of MinasGerais, Av. Antônio Carlos, 6627 – 31270-900, Belo Horizonte, MG – Brazilc Intelligenesis do Brasil Ltda - [email protected], Av. Brasil 1438, 1505, BeloHorizonte, 30140-003 -Brazil______________________________________________________________________
Abstract
This paper reports the development of a new spatial simulation model of landscape
dynamics - DINAMICA. This model was initially conceived for the simulation of the
Amazon Landscape dynamics, particularly the landscapes evolved in areas occupied by
small farms. For testing its performance, the model was used to simulate the landscape
change patterns produced by the Amazonian colonists in clearing the forest, cultivating
the land, and eventually abandoning it to give birth to the ecological processes of
vegetation succession. The study area is located in the northern part of the state of Mato
Grosso, Brazil, which represents a typical Amazonian colonization frontier. To validate
the model, it was run for two sub-areas of the colonization region, using an eight year
time span, starting in 1986. The simulated landscape maps were compared with the
observed landscapes using the multiple resolution fitting procedure and a set of
landscape structure measures, including fractal dimension, contagion index and the
number of patches for each type of landscape element. The results from validation
methods for the two areas have shown a good performance of the model, indicating its
appropriateness for the study and simulation of spatial patterns created by landscape
dynamics in Amazonian colonization regions occupied by small farms. Possible
applications of the DINAMICA model include the evaluation of landscape
fragmentation produced by different architectures of colonization projects and the
prediction of a region’s spatial pattern evolution according to pre-defined transition
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rates. As contributions to the art of landscape dynamics modeling, the development of
DINAMICA model presents: 1) multi-scale vicinity-based transitional functions, 2)
incorporation of spatial feedback approach to a stochastic multi-step simulation
engineering, and 3) the application of logistic regression to calculate the spatial dynamic
transition probabilities.
Keywords: landscape change; cellular automata; simulation model; DINAMICA;
Amazonian landscape dynamics
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1. Introduction
A landscape can be considered a result of a succession of states evolving across time
(Forman and Godron, 1986). Consequently, its constant evolution can lead to
remarkable changes in the environment which may produce enormous ecological
impacts. Therefore, there is a need for characterization of the complex of mechanisms
and agents involved in this dynamics to better understand it and thenceforth to try to
predict the possible paths of a landscape evolution and its future ecological
implications.
To achieve this goal it is necessary to unveil the processes that rule landscape
changes at several spatial and temporal scales, considering that this set of processes is
comprised of local agents, regional forces, and international geographic patterns.
Furthermore, this study requires advanced analytical tools specially designed to analyze
and reproduce, in a computer environment, the spatial patterns resulting from landscape
changes (Turner, 1989; Turner and Gardner, 1990; Sklar and Costanza, 1990). These
methods will in turn enable the formulation of reliable models capable of simulating the
evolution of spatial phenomena, such as landscape changes.
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Simulation models have become a top research topic in many areas of study. The
importance of simulation models, as a new investigation device, is attributed to its
capacity of multiplying our individual imagination and thereby permitting groups of
individuals to share, interchange, and improve mental models of a certain common
reality, independently of its complexity (Lévy, 1998). Thus, simulation models can be
envisaged as a heuristic device useful to test hypothesis about landscape evolution under
several scenarios, which can be translated as different regional social-economic,
political, and environmental frameworks. In light of the outcome of the model, a better
conservation strategy or management plan can be selected by confronting the alternative
results produced by different simulation inputs. Hence, simulation models, which
realistically reproduce spatial land change patterns, are required today to quantitatively
evaluate complex environmental issues at local, regional, and global scales (Steyaert,
1993).
In a historical view, it can be said that the spatial framework of Landscape Ecology
studies provided the bases for the development of a new class of models that attempts to
describe and predict the evolution of ecological attributes in sub-units of area with
distinct localization and configuration (Baker, 1989). These models aim at integrating
diverse temporal and spatial scales to represent ecological system dynamics at the
landscape level (Sklar and Costanza, 1990). As a result, several researchers have
dedicated themselves to the development of landscape dynamics simulation models,
thus contributing to a diversity of approaches, which can be found in works such as
Turner (1987, 1988), Wilkie and Finn (1988), Parks (1990), Southworth et al. (1991),
Dale et. al. (1993, 1994), Flamm and Turner (1994), Gilruth et al. (1995), and Wear et
al. (1996).
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In spite of the contribution of the aforementioned works, environmental dynamics
modeling is still a very difficult task, considering the fact that it requires the knowledge
of several ecological mechanisms and processes which are often unknown (Nyerges,
1993) or too complex to be represented by means of the present technology. Thus, the
development of landscape dynamics simulation models still poses a great challenge to
science.
The aim of this paper is to contribute to the development of landscape dynamics
modeling, describing the conception and operation of a new spatial simulation model –
DINAMICA - which is designed to simulate the genesis and development of land
change spatial patterns. This model was initially conceived for the simulation of the
Amazon Landscape dynamics, particularly the landscapes evolved in areas occupied by
colonists - farm properties smaller than 100 hectares. For testing its performance, the
model was used to simulate the landscape change patterns produced by the Amazonian
colonists in clearing the forest, cultivating the land, and eventually abandoning it to give
birth to the ecological processes of succession. The study area is located in the northern
part of the state of Mato Grosso, Brazil, which represents a typical Amazonian
colonization frontier.
DINAMICA belongs to a new class of models which incorporates many of the
concepts embedded in previous researches, as aforementioned, and introduces
cybernetic properties of a typical cellular automaton model, based on the ideas of works
such as White and Engelen (1993, 1994, 1997), Wagner (1997), Wu (1998), and White
et al. (2000 a).
With new propositions, the DINAMICA model involves a multiple time step
stochastic simulation with dynamic spatial transition probabilities calculated within a
cartographic neighborhood. Its engineering employs special functions designed to
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reproduce cartographically the dimensions and forms of landscape patches. For the
model parameterization, logistic regression is applied to indicate the areas most
favorable for each type of transition. The data used in this analysis were obtained
mainly from satellite imagery.
2. Methods
2.1. Model structure
As mentioned by White et al. (2000 b), a cellular automaton can be considered as a
dynamical type of spatial model in which the state of each cell in an array depends on
the previous state of the cells within a cartographic neighborhood, according to a set of
state transition rules. According to White et al. (2000 b), a conventional cellular
automaton consists of:
1) a Euclidean space divided into an array of identical cells;
2) a cell neighborhood of a defined size and shape;
3) a set of discrete cell states;
4) a set of transition rules, which determine the state of a cell as a function of the states
of cells in the neighborhood;
5) discrete time steps with all cell states updated simultaneously.
Generally designed to simulate landscape changes, DINAMICA embodies the above
concepts by using a sequence of specially developed algorithms, which are described in
following text (Fig.1).
The model data
The software uses as its main input a landscape map (land-use and cover map), e.g.,
a thematic map obtained from satellite imagery. Yet, as input, the model employs
selected spatial variables which are structured in two cartographic subsets according to
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their dynamic or static nature. It generates, as output, simulated landscape maps (one for
each time step), the spatial transition probability maps, which depict the probability of a
cell at a position (x,y) to change from state i to state j, and the dynamic spatial variable
maps. In other words, the model receives a set of raster data cubes and outputs new
ones.
For running the simulations, the model works in phases, each one having its own
parameters:
1) the number of time steps;
2) the transition matrix, being the rates fixed within the phase;
3) an eventual saturation value for each type of landscape element;
4) the minimum sojourn time for each type of transition before a cell changes its state;
5) the coefficients of the logistic equations applied to calculate each spatial Pij;
6) the percentage of transitions executed by each transitional function together with its
options.
Each phase has fixed parameters; consequently the software can be run using
multiple phases, each one consisting of several time steps.
The calculation of the dynamic variables
As aforementioned, a subset of variables used in the simulation is static. These
variables need to be calculated only once before the simulation starts; then they are
stored in a raster data cube. They can comprise any type of environmental data, e.g., the
spatial variables used in this work (Fig. 1).
Another subset of variables is dynamic, which means that it needs to be calculated
before each iteration of the software program. These variables are represented by maps
showing the distances from a cell, in a particular state, to the nearest cell in a different
state of the landscape elements presented in the main input (the landscape map) and the
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landscape element's sojourn time. As a consequence, there will be one map of distances
and one of sojourn time for each landscape element.
The distance calculation procedure consists of computing, in a first step, the frontier
cells for each landscape element patch, whose locations are stored in a 2D tree. In a
second step, the landscape map is again scanned to calculate the Euclidean distance
between the null cells (a different state cell) to the nearest frontier cell stored in the 2D
tree. The sojourn time maps are calculated by adding one time unit for the non-changed
cells after each iteration.
The sojourn time can be used in two ways. First, it can be applied to restrict a
change only after a cell has remained in a given state for a specified period of time.
Second, it can also be used to derive transitions which are considered deterministic,
based only on the sojourn time, such as the evolvement of secondary forests from the
regeneration areas which have reached a certain age.
The calculation of the transition rates and quantities
The transition rates are passed on to the program as a fixed parameter within a given
phase (Fig. 2). As their values represent a percentage of transition, it is necessary to
calculate the amount of cells to be changed by each specified transition during an
iteration, which is done by multiplying the number of cells of each landscape element
occurring in a time step by the transition rate. To be used for projection purposes, the
software also incorporates a parameter, called Saturation Value, which forces the
stopping of the transition i to j, when the number of cells in state i reaches a minimum
value. This effect takes into account the asymptotic shape of the diffusion curve (Fig.
3), and it is calculated by using the following equation:
Rate’ (ij) = Rate(ij)*(M-v)/(M+v) (1)
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where M is a landscape element percentage area remaining at a time and v is the
minimum area that a landscape element will remain.
The calculation of the spatial transition probabilities
The spatial transition probabilities are calculated for each cell in the landscape raster
map and each specified transition. The method selected employs a general politomic
logistic equation in the form of:
)()()(
)(
...)/),((
VnGiVGiVjGi
VjGi
eee
eVyxjiP ⇒⇒⇒
⇒
+++=⇒ 21
(2)
where V is a vector of the k spatial variables, measured at location x,y and represented
by V1xy, V2xy, ..., Vkxy, and each Gij(V) is calculated by the following equation:
xykijkxyijij VVVGij ,,,,,)( βββ +++= K110 (3)
An approach to estimate the coefficients of the equations Gij(V), and consequently
the influence of each selected variable on the transitions, is by using logistic regression.
With respect to the application of this method, it is worth mentioning that the transitions
are what must be modeled, instead of the state frequencies. Thus, these regressions
model the chances of a cell to change to one of the states, chosen along the line of the
transition matrix, or to remain unchangeable (Fig. 2).
The result of this procedure is a set of maps; each depicting the probability of a cell
to change to another state. According to the current cell state, one of the spatial
transition probability maps stored in the output cube data will be picked up to be used
by the transitional functions (Expander and Patcher).
The transitional functions
A relevant issue to be considered with regard to landscape simulation models,
especially the mosaic structured models, refers to the neighborhood influence in the
transition probabilities and, consequently, in the landscape patch dynamics. One way in
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which the DINAMICA software deals with this matter consists in splitting the cell
selection mechanism into two processes. The first process is dedicated only to the
expansion or contraction of previous landscape patches, and it is called Expander
Function. In turn, the second process is designed to generate or form new patches
through a seedling mechanism, and this function is called Patcher Function.
The combination of the two processes is shown in the equation bellow:
Qij = r*(Expander Function) + s*(Patcher Function) (4)
where Qij stands for the total amount of transitions of type ij specified per
simulation step and r and s represent respectively the percentage of transitions executed
by each function, being that r + s = 1.
Varying the proportions of transitions executed by the two functions, the simulation
outputs can be made to approximate the structure of a landscape.
Both of the previously mentioned functions employ a stochastic selecting
mechanism. The applied algorithm consists in reading firstly the landscape map to pick
up the cells with higher probabilities and arranged them orderly in a data array. In
sequence, the selection of cells takes place randomly from top to bottom of the data
array (the internal stochastic choosing mechanism can be loosened or tightened
depending on the degree of randomization desired), storing at the end the locations of
the selected cells. In a second step, the landscape map is again scanned to execute the
selected transitions. In this way, it is guaranteed that there will be no bias from the
reading of the landscape map, which it is always done sequentially from the top left of
the raster.
The previously described procedures are used in both transitional functions. For
each function, the software controls the iteration number needed to accomplish the
amount of specified transitions. In the event that the Expander function does not realize
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the amount of desired transitions, after a limited number of iterations, it passes on to the
Patcher Function a residual number of transitions, so that the total number of transitions
always reaches an expected value. As a result, the final landscape map will only need to
be evaluated with regard to its spatial configuration, since the distributional model will
be coincident to that of the reference landscape.
The Expander function
The Expander algorithm is expressed in the following equation:
If nj > 3 then P'(ij)(xy) = P(ij)(xy) else P'(ij)(xy) = P(ij)(xy)* (nj)/4 (5)
where nj corresponds to the amount of cells of type j occurring in a window 3*3. This
method guarantees that the maximum Pij' will be the original Pij, whenever a cell type i
is surrounded by at least 50% of type j neighboring cells (Fig. 4).
Despite the fact that DINAMICA is a mosaic structured model, it can work using
this function with the concept of landscape patches, which are either contracted or
expanded depending on the type of transition.
The Patcher function
The Patcher function aims at reproducing the actual landscape structure, impeding
the formation of tiny single cell patches, which would probably occur if only a simple
cell allocation process were used.
This function employs a device which searches for cells around a chosen location
for a combined transition. This is done firstly by electing the core cell of the new patch
and then selecting a specific number of cells around the core cell, according to their Pij
transition probabilities, for the combined transition (Fig. 5). As an example of the
application of this function, the new deforested patches, produced by the simulations in
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this work, are set to be equal to an area, about 5 hectares, equivalent to the yearly forest
cleared by a typical Amazonian colonist.
For each phase of the simulation, the percentage of the amount of transitions is set
for each of the aforementioned functions. Yet to avoid infinite loops, a maximum
number of iterations for the Expander Function is specified. The number of successful
transited cells is subtracted from the total number desired, and the remaining number is
passed on to the following function.
Both described functions also incorporate an allocation device which is responsible
for locating the cells with the highest transition probabilities for each ij desired
transition. This function stores these cells and organizes them for subsequent selection.
In this process, each newly selected cell will form a core for a new patch or an
expansion fringe, which is still needed to be developed by the use of the transitional
functions.
The size of the new patches and the expansion fringes are set up according to a
lognormal probability distribution; this implies that as input it is necessary to specify the
parameters of this distribution represented by the average size and the variance of each
type of new patch and expansion fringe to be formed.
The software interface
The DINAMICA software was written in object oriented C++ language and its
present version runs on 32 bits Windows system. All the DINAMICA model
parameters are set by using its graphical interface before they are saved to a script file.
This makes easier for users to understand and operate the software as well as to check
for input inconsistencies. Besides outputting the raster cubes, the software also
produces a log file which reports the numbers per types of transitions achieved after
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each step of its execution. The DINAMICA software, including the manuals and the
database of the test areas, can be accessed via internet on: www.csr.ufmg.br.
2.2. Test performance
The test site and its land change conceptual model
For testing its performance, the software was applied to simulate the landscape
changes of a typical Amazonian frontier colonization region, located in the Northern
state of Mato Grosso, Brazil (Fig. 6). The occupation of this region began to evolve at a
large scale at the end of the seventies, with the installation of several colonization
projects, consisting mainly of colonist from rural cooperatives of Southern Brazil.
Within this region, two sub-areas were selected for testing the simulations (Fig. 6).
Those areas, known as Terra Nova (140,870 hectares) and Guarantã (127,480 hectares)
correspond respectively to subsets of colonization projects, which are occupied by
colonists: farm properties smaller than 100 hectares.
The landscape changes produced by the Amazonian colonists in clearing the forest,
cultivating the land, and eventually abandoning it to give birth to the ecological
processes of succession, can be simplified for the study region in four landscape
elements (states) and six transitions according to the conceptual model illustrated in Fig.
7.
The simplification of the conceptual model can be justified by the fact that most of
the deforested areas in this region are occupied by pastures in several stages of use and
abandon. Furthermore, it can se said that the crops, despite a few exceptions found in
the region, are seen only as an initial stage of the forest-pasture conversion process.
With respect to this model, if we consider that the transitions regeneration to
secondary forest and secondary forest to mature forest are deterministic, based solely
on the sojourn time of the vegetation succession, we can reduce the model’s transition
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matrix to 9 elements, of which only six occur in reality. Notice, in this case, that the
transition matrix will reduce to a vector, since the other transitions are complementary.
Thus, the simulations will deal only with the transitions: forest-deforestation,
deforestation-regeneration, and regeneration-deforestation.
In effect, this taxonomic simplification does not compromise the model, considering
that it was not initially conceived to focus on the current land uses of these regions, but
the spatial configuration arising from the forest fragmentation and the forest recovery of
the abandoned areas.
The time span for running the model
The time span chosen for running the model encompassed eight years and was
divided into two periods: 1986 to 1991 and 1991 to 1994. These periods correspond to
distinct phases of the regional landscape dynamics. The first phase shows a slower
annual deforestation rate as a consequence of the gold rush, which took place in the
region and attracted colonists to work in the alluvial mines (known in Brazil as
garimpos) instead of cultivating their lands. The second phase corresponds to the
decline of the gold rush and is characterized by more intense land changes provoked by
the return of the colonists to cultivate theirs lands as well as by the increase in pressure
for land holding due to internal migration from the mining areas.
In this way, three TM/Landsat images were selected to characterize the distinct
phases of the landscape dynamics: the first one acquired in June, 1986, corresponding
to a short time after the beginning of the Landsat-5 operation; the second in June,
1991, the most intensive phase of the garimpo, and the third in May, 1994, concomitant
to the field work. The satellite images were classified in order to generate the
multitemporal maps, which depict the distribution of the landscape elements: forest,
deforestation, and regeneration. The three resulting landscape maps were cross-
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tabulated to produce the transition matrices for the two periods of analysis (Table 1).
Subsequently, these matrices were derived by means of the Markovian chain property
(Equation 6) to give the annual transition rates, because the simulations were set to run
in time steps of one year.
1−= HHVP tt (6)
where P is the original transition matrix, H and V are its eigenvector and eigenvalue
matrices, and t is a fraction or a multiple of its time span.
The input variables
The multitemporal landscape maps were used with cartographic ancillary data to
point out the most important spatial variables and their weights of influence on the
analyzed transitions. The logistic regression method was used in this procedure; its
outcome consists in the spatial transition probability maps. As aforementioned, these
maps represent the chances of a cell, with coordinates x and y, and in a state i, to change
to another state. In synthesis, these variables are used to incorporate, in the simulation
analysis, the colonists’ decision rules about where they cultivate their lands and
eventually abandon them; as well as the ecological processes and natural barriers to the
recovery of the forest landscape.
The spatial variables used in this work to calculate the spatial transition probabilities
are soil, vegetation, distance to rivers, distance to main roads, distance to secondary
roads, altitude, slope, distance to forest, distance to deforested areas, distance to
regeneration patches, and the urban attraction factor. This last variable is evaluated
through a gravitational model, which involves the sum of the inverse of distances, raised
to the second power, from a rural area cell to all cells classified as urban. A detailed
description of the method used to calculate the spatial transition probabilities is reported
by Soares-Filho et al. (2000).
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The cell map resolution was chosen to be equal to 100 meters, taking into account
the average size of the regeneration patches – the smallest occurring patches - which
correspond to 3.76 hectares for the selected areas. This resolution results in data arrays
consisting of 360 per 400 cells for Guarantã and 664 per 484 cells for Terra Nova.
2. Model calibration
As the DINAMICA transitional functions present many calibration possibilities, the
software needed to be run several times in order to adjust the parameters of these
functions. For this purpose, a set of simulation series was run adopting the empirical
transitions rates, as shown in Table 1, and varying the mixture of the transitional
functions – Patcher and Expander, as well as the expected mean patch size of each
landscape element. In this manner, the effect produced on the simulated landscape
structures by these varying parameters was tested. To evaluate the results, the simulated
landscapes were measured by using a set of landscape quantitative indices, including
fractal dimension (double fractal dimension), contagion index and the number of
patches for each type of landscape element (McGarigal and Marks, 1995).
From the graphs in Fig. 8, it can observed that as the proportion of transitions
executed by the Patcher function grows at the expense of the Expander function, the
contagion index decreases while the fractal dimension and the number of patches of the
landscape elements increase. When the patcher process prevails over the Expander
function, the patches become smaller, more fragmented, and consequently showing a
higher fractal dimension and a less connectivity, as indicated by the lower contagion
index. In turn, as the expected mean patch sizes are set larger, the contagion index
increases and the fractal dimension and the number of patches decrease, as a result of
forming more core areas.
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As main observation attained from these graphics, we can assert that the mixtures of
these transitional functions show a predictable behavior with regard to the simulated
landscape structures, thus leading us to conclude that an optimized solution for a
particular landscape can be achieved by means of defining a certain combination of
these functions together with their parameters.
Subsequently, a new set of simulations was performed to approximate the simulated
landscapes to the ones of Terra Nova and Guarantã. Still, in this calibration phase, some
coefficients of the logistic equation had to be slightly modified to account for
interactions resulted by the effect of using multiple time steps and transitional functions,
which were not considered by the logistic regression analysis. As a consequence, the
urban attraction factor coefficient had to be diminished, since it was added to the effect
of the Expander function to provoke an extreme concentration of the deforestation
around the urban centers. The simulation input data for the best achieved results for
both areas are presented in Table 2 and Table 3.
3. Results and discussion
Since DINAMICA has a stochastic structure, the final obtained models were run
twenty times for each area, using the best adjustment achieved (Tables 2 and 3 and Fig.
9). To validate the simulations, the simulated landscape maps were compared with the
observed landscapes using the set of landscape indices previously mentioned - fractal
dimension, contagion index and the number of patches for each type of landscape
element - plus the multiple resolution fitting procedure (Turner et al., 1989).
The results from the validation methods for the two areas are presented in Table 4.
They show that the simulations were able to reproduce the contagion indices with an
average deviation of 10.7% for Terra Nova and 6.9% for Guarantã.
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In turn, the fractal dimension of the landscape elements were reproduced with a
maximum average deviation of 2.8% for Terra Nova and 10.8% for Guarantã.
For Terra Nova, the simulation model matches from 93.9% to 95.6% the numbers of
landscape element patches. For Guarantã, the numbers of deforestation and regeneration
patches attained, respectively, an average approximation of 97.1 and 94%. On the other
hand, the simulation did not approximate the number of forest patches for the Guarantã
sub-area, where the average deviation exceeded 183%.
The multiple resolution fitting procedure evaluates the model spatial adherence to
the reference landscape with regard to the equal number of occurring equivalent cells
within a specified cell neighborhood, which is defined by a window size that is
considered a spatial resolution (Turner et al., 1989). The idea of using this procedure
consists in not discarding models which do not exactly match considering only a cell by
cell comparison, but show a good spatial approximation within a certain cell vicinity. In
this manner, the multiple resolution fitting procedure showed that the average simulated
landscape of Terra Nova started with a spatial fitting of 82.4% for the highest resolution
of 100 meters, and reached a fitting of 90% for a window size of 1800 meters. In turn,
the average simulated landscape of Guarantã showed a fitting of 63.6% at its highest
resolution, which reached 80% for a window size of 1400 meters (Fig. 10).
With respect to the achieved spatial adjustments, it is worth mentioning that
deforestation spreads throughout the region, generally from the road network, thus
being strongly influenced by the colonization project architecture. In this way, the lower
spatial fitting achieved for Guarantã sub-area can be explained by the fact that many
land invasions occurred in the central part of this region, deforesting an area originally
designed to be a collective forest reserve. Consequently, this process could not be
modeled by this specific simulation due to the lack of information about where the
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squatters would open the roads. This situation did not take place in the Terra Nova area,
where the simulation achieved a much better spatial adjustment.
Finally, we can mention that the low adjustment obtained for the number of forest
patches for the Guarantã sub-area could be, in part, explained by the high sensitivity of
this index, which can be drastically affected by any small change in the spatial
arrangement of the cells (Fig. 11).
3. Conclusions
The simulations were performed by using spatial data which could be obtained
mainly from satellite imagery. In view of this fact, it can be said that the achieved
results are encouraging, especially considering the fallible character of environmental
prediction. Moreover, the convergent results of both areas indicate that application of
the DINAMICA model can be transposed to investigate the landscape dynamics of other
Amazonian areas occupied by colonists, as long as some initial conditions are known.
Thus, this model can be applied to these Amazonian areas to describe the spatial
configuration arising from the forest fragmentation and the forest regeneration of the
abandoned areas. In this manner, DINAMICA deals with the effects of the deforestation
and regeneration processes on the fragmentation and recovery of the original forest
habitats. Thenceforth, the following issues can be addressed to the DINAMICA model:
− Which deforested areas will become secondary forests?
− What will be the spatial configuration of the landscape elements in a given future
time?
− What will be the ecological consequences of that landscape structure?
− Where will the forest fragments remain?
− How can the architecture of a colonization project influence the development of a
landscape, especially forest fragmentation?
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As DINAMICA focuses on the development of spatial patterns produced by land
change, this makes the software useful to investigate many other types of environmental
dynamic phenomena, considering the fact that its transitional functions can be adapted
to replicate different landscape structures and to work with any spatial resolution or at
diverse cartographic scales. Futhermore, DINAMICA can model any type and any
number of transitions as well as embrace any span of time, divided into any number of
time steps and phases with pre-defined transition rates. It can also incorporate, by means
of the logistic equations, any kind and number of data to calculate the spatial transition
probabilities. Still, the DINAMICA software can handle substantive large data arrays
with a good performance. The eight-step simulations executed for testing the model
took only six to seven minutes using a standard 400 MHz PC.
Therefore, the main contributions of the DINAMICA software to the art of
landscape change modeling can be mentioned as:
1) The development of multi-scale vicinity-based transitional functions;
2) the incorporation of spatial feedback approach to a stochastic multi-step simulation
engineering;
3) the application of logistic regression to calculate the spatial dynamic transition
probabilities.
In light of this development, DINAMICA presents us with new application
possibilities which will permit us to test hypothesis on how the many facts involved in
landscape evolution interact. Hence, this software can be envisaged as an investigation
device useful to decision makers to foresee the evolution of the spatial pattern of a
region, according to a particular environmental planning or caused by some type of
human occupation.
4. Acknowledgements
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The authors thank the project “Land use and Health in Amazon”, carried out by
CEDEPLAR-UFMG and coordinated by Dr. Diana Sawyer, for the financial support of
this work, and John Di Fiore for the proficient review of this paper.
5. References cited
Baker, W.L., 1989. A review of models of landscape change. Landscape Ecology, 2:
111-133.
Dale, V.H., O’Neill, R.V., Pedlowski, M., Southworth, F., 1993. Causes and effects of
land-use change in central Rondônia, Brazil. Photogrammetric Engineering and
Remote Sensing, 59: 997-1005.
Dale, V.H., O’Neill, R.V., Southworth, F., Pedlowski, M., 1994. Modeling effects of
land management in the Brazilian Amazonian settlement of Rondônia. Conservation
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Table 1The transitions matrices of the sub-areas for the two periods analyzed
Guarantã Terra Nova86x91 86x91
Deforestation Regeneration Forest DeforestationRegeneration ForestDeforestation 78.86 0.00 22.78 77.41 0.00 21.27Regeneration 21.14 0.00 0.00 22.59 0.00 0.00Forest 0.00 0.00 77.22 0.00 0.00 78.73
91x94 91x94Deforestation Regeneration Forest DeforestationRegeneration Forest
Deforestation 61.67 26.42 16.07 69.94 35.50 19.40Regeneration 38.33 73.58 0.00 30.06 64.50 0.00Forest 0.00 0.00 83.93 0.00 0.00 80.60
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Table 2Logistic regression coefficients
Terra Nova sub-area - Phase 1 - - Phase 2 -DR FD DR RD FD
Constant 4.08 -1.9621 0.0246 2.788 0.043
Vegetation type 2 0 2.0793 0.3138 0 2.07
Vegetation type 3 0 2.0462 0.2 0 2.506
Soil type 2 -1.12 0 0 0 -0.7414
Soil type 3 -1.459 0 0 0 1.5237
Altitude 0 0 0.00551 -0.01304 0
Slope 0 0 0.06231 0 0
Urban Attraction -0.003 0 -7.50E-05 0 -0.00387
D. to main roads 6.78E-05 -7.32E-05 0 0 -3.24E-05
D. to secondary roads 0 -4.38E-03 0 -1.00E-03 -1.34E-04
D. to river 2.72E-04 -7.51E-05 -2.76E-04 3.99E-04 0
D. to deforestation 0 -4.39E-04 0 0 -6.07E-03
D. to regeneration 0 0 -4.51E-04 0 0
D. to forest -6.83E-03 0 -0.00271 0.00286 0
Guarantã sub-area- Phase 1 - - Phase 2 -DR FD DR RD FD
Constant 0.287 -4.083 -2.4181 -0.6259 -0.7143
Vegetation type 2 0 2.2334 0 0 1.7841
Vegetation type 3 0 1.5923 0 0 1.8828
Soil type 2 -1.4534 0 0 0 0
Soil type 3 -1.5089 0 0 0 0
Altitude 0.01638 0 0.00837 0 -0.01302
Slope 0 0 0.03744 0 0
Urban Attraction -0.00403 0.00018 0 0 0.00036
D. to main roads 3.50E-05 0 -1.23E-05 0 0
D. to secondary roads 0 -2.03E-04 0 -0.00087 0
D. to river 0 -3.44E-05 0 0 0
D. to deforestation 0 -4.32E-05 0 0 -7.72E-05
D. to regeneration -4.59E-06 0 0 0 -3.31E-04
D. to forest -4.59E-03 0 -0.00458 0 0
D. (distance). DR: deforestation-regeneration, FD: forest-deforestation, and RD:Regeneration-deforestation. d. means distance.
26
Table 3Input parameters for the best simulation results
Terra NovaDeforestation-Regeneration Regeneration-Deforestation Forest-Deforestation
Steps Expan. Mean Var. Rate S.T Expan. Mean Var. Rate S.T Expan. Mean Var. Rate S.T1 to 5 0.2 1 1 0.0391 5 0.6 3 1 0 0 0.6 5 2 0.0466 06 to 8 0.2 1 1 0.1199 3 0.6 3 1 0.1617 0 0.8 5 2 0.0696 0
GuarantãDeforestation-Regeneration Regeneration-Deforestation Forest-Deforestation
Steps Expan. Mean Var. Rate S.T Expan. Mean Var. Rate S.T Expan. Mean Var. Rate S.T1 to 5 0.8 1 1 0.0311 5 0 0 0 0 0 0.6 5 1 0.0503 06 to 8 0.6 1 1 0.1600 3 0.8 3 1 0.166 0 0.8 10 1 0.0553 0
Expan. represents the proportion of transitions executed by the Expander function,being the Patcher function complementary to the Expander. S.T represents sojourntime.
27
Table 4Landscape structures indices obtained for the average simulated landscapes* comparedwith the ones of the reference landscapes
TERRA NOVAForest Deforestation Regeneration
NP DLFD NP DLFD NP DLFD ContagionReference Landscape 791 1.45 671 1.49 2473 1.58 25.21
Simulated Landscape 764 1.41 712 1.49 2387 1.6 27.9% deviation 3.4 2.8 6.1 0.0 3.5 1.3 10.7
GUARANTÃForest Deforestation Regeneration
NP DLFD NP DLFD NP DLFD ContagionReference Landscape 507 1.39 823 1.48 2300 1.55 28.39
Reference Landscape 1438 1.48 847 1.64 2162 1.66 26.44% deviation 183.6 6.5 2.9 10.8 6.0 7.1 6.9
* n= 20, NP: number of patches; DLFD: Double fractal dimension.
28
Figure Captions
Fig. 1. Flowchart of the DINAMICA software
Fig. 2. The transition matrix for four landscape elements
Fig. 3. Hypothetical deforestation curve obtained by using the saturation function(Equation 1)
Fig. 4. Pij arrays before a) and after b) the convolution of the Expander operator
Fig. 5. The selection of cells around an allocated core cell by the Patcher function
Fig. 6. TM/Landsat of May, 1994, comprising the study region and its localization withrespect to Brazil and the state of Mato Grosso. The two selected sub-areas arecontoured in white
Fig. 7. Model of landscape changes in the study area. The indicated time spans for thetransitions “regeneration to secondary forest” and “secondary forest to forest” mayvary greatly
Fig. 8. Setting the percentage of transitions executed by each transitional function andthe mean size of the produced patches (m) to replicate the desired landscapestructure. The values of m are expressed in hectares. Graphics a to e were obtainedfor the forest-deforestation transition and graphic f for the deforestation-regenerationtransition
Fig. 9. Simulated landscape maps of Terra Nova, compared with the observed landscapemaps
Fig. 10. Results of the multiple resolution fitting for both areas. Average values for 20runs. Average fitting for Terra Nova = 0.88 and for Guarantã = 0.77
Fig. 11. Modification of the number of patches as a result of the rearrangement of asingle cell: a) two patches, b) one patch
29
landscape map static variables
Pdr
final landscape map
Pfd
Prd
Number of phases,Phase 1:• n° of time steps,• the transition matrixwithin the phase,• the saturation values,• the minimum sojourn timefor each type of transition,• the coefficients of thelogistic equations,• the percentage oftransitions executed by eachtransitional function and itsparameters:maximum number ofiterations for the expanderfunction, mean patch sizeand patch size variance.Phase 2: ..........,Phase 3: ..........,Phase n: ...........
sojourn time in state i
calculates thedynamic distances
calculates thespatial
probabilities
calculatesthe amount
of transitions
executes theExpanderfunction
executes thePatcherfunction
t = tn
END
changes the cells
distance to deforestationdistance to forest
distance to regeneration
soilvegetation
altitudeslope
distance to riversdistance to secondary roads
distance to main roadsurban attraction factor
iterates
calculates sojourn time
dynamic variables
n=N (N =maximum number of iterations)
distances to landscapeelements
transition probability mapsInput Parameters
30
44434241
34333231
24232221
14131211
31
0
10
20
30
40
50
60
70
80
90
1 5 9 13 17 21 25 29 33 37
years
% d
efor
esta
tion
deforestation
deforestation rate
saturation threshold
32
ji
i
j
33
allocated core cell
generated patch
34
TERRANOVA
GUARANTÃ
J-1Mato Grosso
BRAZIL
35
FORESTFOREST
DEFORESTEDDEFORESTEDAREASAREAS
REGENERATIONREGENERATION
SECONDARYFOREST
TIME > 90 YEARS
TIME > 20 YEARS
Dense Rain ForestOpen Rain Forest
Alluvial ForestDense SavannahOpen Savannah
Pastures in diversestages of use andagricultural areas
Young andintermediatesuccessions
36
Number of deforestation patches varying the forest-deforestation transition
600
1100
1600
2100
2600
3100
1 0.8 0.6 0.4 0.2 0%Expander to Patcher
NP
m=5
m=4
m=3
m=2
m=1
Double fractal dimension of deforestation patches varying the forest-deforestation transition
1.47
1.52
1.57
1.62
1.67
1 0.8 0.6 0.4 0.2 0%Expander to Patcher
DLF
D
m=5
m=4
m=3
m=2
m=1
Number of forest patches varying the forest-deforestation transition
200400600800
10001200140016001800
1 0.8 0.6 0.4 0.2 0%Expander to Patcher
NP
m=5
m=4
m=3
m=2
m=1
Double fractal dimension of forest patches varying the forest-deforestation transition
1.35
1.4
1.45
1.5
1.55
1.6
1.65
1.7
1 0.8 0.6 0.4 0.2 0%Expander to Patcher
DLF
D
m=5
m=4
m=3
m=2
m=1
Contagion index varying the forest-deforestation transition
162126313641465156
1 0.8 0.6 0.4 0.2 0%Expander to Patcher
% C
onta
gion
m=5
m=4
m=3
m=2
m=1
Number of regeneration patches varying the deforestation-regeneration transition
1200
1700
2200
2700
3200
3700
1 0.8 0.6 0.4 0.2 0%Expander to Patcher
NP
m=5
m=4
m=3
m=2
m=1
a) b)
c)
e) f)
d)
37
deforestationregeneration
forest
TERRA NOVA 86
deforestationregeneration
forest
SIMULATED 91
deforestationregeneration
forest
SIMULATED 94
deforestationregeneration
forest
TERRA NOVA 86
deforestationregeneration
forest
TERRA NOVA 91
deforestationregeneration
forest
TERRA NOVA 94
10 5 0 5 10 KM
38
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1 2 7 12 18 25 33
window size in cells
% fi
tting
Guarantã
TerraNova
39
a) b)