Calibration and Assessment of Multitemporal Image-based Cellular Automata Urban Growth Modeling

This paper focuses on the calibration and assessment of cellular automata model forurban growth modeling.. cellular automata has been used by Batty andXie 1994a to model urban growth of

Calibration and Assessment of Multitemporal Image-based

Cellular Automata Urban Growth Modeling

Sharaf Alkheder and Jie Shan *

Geomatics Engineering, School of Civil Engineering Purdue University, 550 Stadium Mall Drive, West Lafayette, IN 47907, USA

Phone: (765) 494-2168, Fax: (765) 496-1105

* Corresponding author Email: jshan@ecn.purdue.edu

This paper focuses on the calibration and assessment of cellular automata model forurban growth modeling The basic model design is a function of multitemporalsatellite imagery and population density A number of transition rules considering themost influential urbanization variables are introduced in the cellular automata Suchvariables include land use, road networks, and population density The cellular modeltransition rules are calibrated both spatially and temporally to determine the optimalrule values to ensure the modeling accuracy Spatially, the model is calibrated pertownship (about 3x3 square miles each) such that spatial variability of the urbangrowth process can be taken into account Temporal calibration is performed by using

a sequence of remote sensing images from which land use information at differentyears is extracted For assessment purpose, the multitemporal imagery is divided intotwo sets: training and testing data Training data are used for model calibration andtesting data for evaluating the prediction results The proposed evaluation measuresinclude fitness (for urban level match) and two types of modeling errors (for urbanpattern match), based on which the optimal rules with closest modeling results toreality are selected The study shows that the use of images reduces the need for alarge number of input data and hence the modeling uncertainty either from input data

or through propagation Evaluation on the rule variogram reveals that the transitionrule values are correlated spatially and vary with the urbanization level The paperreports the study outcome over city Indianapolis, Indiana for the past three decadesusing Landsat TM images and population data Modeling results, on the calibrationand prediction sides, show close match with reality for both urban level and patternquality measures This close match is a result of spatial calibration that takes intoaccount the specific urbanization nature of each spatial unit in the modeling process

Modeling results as compared to reality show more connectivity and smoothness aswell

Keywords: Cellular Automata; Thematic Imagery; Calibration; Urban Modeling

1. INTRODUCTION

Excessive progress has been achieved in urban dynamic modeling to understand theurban growth process (Meaille and Wald 1990; Batty and Xie 1994a and 1994b).Some urbanization models focus more on the physical aspects of the urban growthprocess (Wilson, 1978), while others on social factors (Jacobs, 1961) An example ofthe physical models is the land use transition model of Alonso and Muth in landscapeeconomics (Wilson, 1978) Social models simulate the urbanization process according

to the difference between individuals' intentions and their behavior (Clarke et al, 1997; Portugali et al, 1997) According to Clarke et al (1997), urban growth models

can be designed either for a specific geographical location such as BASS II whichmodels the urbanization process for the San Francisco Bay area only (Landis, 1992)

or as general models such as HILT (human-induced land transformations) where itsgrowth rules are designed to be general enough to fit different city structures

Yang and Lo (2003) classify urban dynamic models into three categories: Cellularautomata (CA) based models as in Clarke et al (1997), probability based models such

as Veldkamp and Fresco (1996) model, and GIS weighted models like Pijanowski et

al (1997) model The cellular automata-based models are becoming popular in recentliterature mainly because of its ability to model and visualise spatial complexphenomena (Takeyama and Couclelis, 1997) Urban cellular automata modelsperform better as compared to conventional mathematical models (Batty and Xie,1994a) and simplify the simulation of complex systems (Wolfram, 1986; Waldrop,1992) The fact that urban process is entirely local in nature also makes cellularautomata a preferred choice (Clarke and Gaydos, 1998)

Many urban cellular automata models are reported The model of White and Engelen(1992a; 1992b) involves reduction of space to square grids, based on which a set ofinitial conditions is defined Transition rules are implemented recursively untilmatching the reference historical data cellular automata has been used by Batty andXie (1994a) to model urban growth of Cardiff, Wales, and Savannah, Georgia Later,

Batty et al (1999) develop a model that tests many hypothetical urban simulations to

evaluate different model structures Based on the work of von Neumann (1966),

Hagerstrand (1967), Tobler (1979), and Wolfram (1994), Clarke et al (1997) propose

the SLEUTH model, which is able to modify parameter settings when the growth rateexceeds or drops below a critical value Clarke and Gydos (1998) use SLEUTH tomodel the urban growth in San Francisco Bay region and Washington DC/Baltimorecorridor Yang and Lo (2003) use SLEUTH model to simulate future urban growth inAtlanta, Georgia with different growth scenarios Wu (2002) develops a stochasticcellular automata model to simulate rural-to-urban land conversions in the city ofGuangzhou, China

Calibration of cellular automata models is essential to achieve accurate modelingoutcome, however, it has been ignored until recent efforts to develop cellularautomata as a reliable procedure for urban development simulation (Wu 2002).Calibration is meant to determine the optimal values for parameters in the transitionrules so that the modeled urban growth closely matches real urban growth Thedifficulty of calibration is partially due to the complexity of urban developmentprocess (Batty et al 1999) Clarke et al (1997) use visual tests to establish parameterranges, to provide initial parameter values and to check if urban pattern matches realdata (Clarke et al, 1997) About a dozen of statistical measures are calculated forcertain features to check the match between real data and modeling results Such

visual and statistical tests are repeated for each parameter set Wu and Webster (1998)use multicriteria evaluation (MCE) to identify the parameter values for their cellularautomata model, while neural networks (NN) are used by Li and Yeh (2001) The factthat most urban cellular automata models need large number of data input variables isnot free of risk Many uncertainties show up in the simulation output These can resultfrom the uncertainty in the input data, uncertainty propagation through the model, andthe uncertainty of the model itself in term of what degree the model represents thereality Previous research shows that real city modeling is very sensitive to data errors(Li and Yeh, 2003) Therefore, it is beneficial to minimize the need for large inputdata to reduce modeling uncertainty and redundancy of input variables

This paper is focused on two important aspects in urban cellular automata modeling:cellular automata model calibration and assessment First, our cellular automatamodel is designed to reduce the amount of input data For this purpose a historical set

of satellite imagery is used as an alternative to cadastral maps as being used inliterature We believe that building the model over the imagery directly is morerealistic as compared to cadastral maps The imagery is a rich source of informationincluding land cover, urban extent and growth constrains (e.g water resources) Thiswill reduce the need of having different sets of input data layers In addition,uncertainty of urban modeling that usually rises from having multiple input datalayers (and hence variable precisions) will be reduced Other data that is not included

in the imagery (such as population density) can be used as extra input layers.Secondly, most cellular automata models assume that one set of transition rules willfit the whole study area As a matter of fact, some regions in a study area may havedifferent urbanization behavior than others Based on this understanding, we arguethat calibration should be carried out both spatially and temporally Spatial calibration

takes into account the spatial variability in urban process In this study, the study area

is divided into townships, each of which forms a calibration unit Transition rules arecalibrated to find the best values that fit the urban dynamics for each township.Temporal calibration is based on multitemporal imagery and allows transition rulevalues to change over time to meet the variable urban pattern in time Finally,modeling results are assessed quantitatively and qualitatively Three measures (one forurban count and two for modeling errors) are introduced for this purpose.Quantitatively, calibrated rules should be able to reproduce the same urban count asreality, and qualitatively they should be able to reproduce the same urban pattern Therules that produce urban count close to real imagery with minimum modeling errorsare selected The system is implemented first on synthetic city to study the effect ofgrowth factors on urban process and then expanded to model the historical urbangrowth of Indianapolis, Indiana over the last three decades

2. PRINCIPLES OF CELLULAR AUTOMATA

Cellular automata is originally introduced by Ulam and von Neumann in 1940s as aframework to study the behaviour of complex systems (von Neumann, 1966) It iscommonly defined as a dynamical discrete system in space and time that operates on auniform grid under certain rules It consists of four components – pixels, their states(such as land use classes), neighborhood (square, circle etc) and transition rules.cellular automata computation is iterative, with the future state of a pixel beingdetermined based on the current pixel’s state, neighborhood, and transition rules.Based on the work of Codd (1968), Sipper (1997) provides a formal definition of two

dimensional (2-D) cellular automata Let I represents a set of integers, a cellular space

associated with the set I I can be defined The neighborhood function for pixel  is

g( )      1,  2, ,  n (1)

where;  (i = 1…n) represents the index of the neighborhood pixels Figure 1 shows i

an example of a 2-D cellular automata grid system, where I=5 represents the total

space of pixels in a grid of 5 5= 25 pixels As an example, the state of pixel  isurban and it is surrounded by a 3  3 square neighborhood This means that there areeight neighbors for with , i = 1…8 The neighborhood of pixel  can be i

presented as a city-block metric :

   ( , ) x x + y y (2)given that  ( ,x y ) and ( ,x y ) The function ( , )   defines the set of pixels

 around pixel  such that I Ix 

Figure 1 An example of 2-D cellular automata

In Figure 1, a 3 3 neighborhood is selected for The metric  represents the

dimension of the square neighborhood region of x  = ±1 in the x-direction andx

y = ±1 in the y-direction with a total of 9 pixels The neighborhood statey

function h t() is defined as:

t( ) ( ( ), (t t 1), , (t ))

n

h   v  v    v    (3)where( ( ), (t t 1), , (t ))

n

v  v   v   are the states of pixel  and its

neighborhood pixels at time t The selected neighborhood kernel in Figure 1 for the

center pixel has h t()= [water, road, urban, water, urban, urban, road, road, urban](in row-first order)

Finally, the relationship between the state of pixel  at time (t+1) and its neighborhood states at time t can be expressed as:

vt1( )   f h ( ( ))t  (4)

where ( ( ))f h t  is the transition function that represents the cellular automatatransition rules defined on  and its neighborhood states Typically, the transitionfunction ( ( ))t

f h  uses IF THEN rules over ( )t

h  to identify the future state of  at

time t+1

3. MODELING OF SYNTHETIC CITY

This section implements the cellular automata principle to a synthetic city to study theeffect of modeling parameters on the urban growth process It mimics the realitythrough introducing complex structures for an urban system Figure 2 presents theimage of 200 200 pixels used as an input to the cellular automata algorithm Sixclasses are defined: Road, River, Lake, Pollution Source, Urban, and Non-urban.The design of cellular automata model needs to reflect the effect of land use on theurban growth process Transportation system encourages and drives the urban

development For example, commercial centres should have access to road networkfor customer’s visit and goods delivery Therefore, cellular automata rules related toroad should encourage urban development for pixels near roads River and lake pixelsshould be constrained such that no urban growth is allowed on these locations toconserve water resources On the other hand, lakes are considered as one of theattractive factors for urban development especially residential and recreational types,

so the corresponding rule needs to show this effect on urban development Thepollution sources are included as one of the constraints for urban development due totheir effect on the degradation of ecological system The designed cellular automatarules should discourage urban growth in such locations Based on the aboveconsiderations, the following rules are used

 IF a test pixel is urban, river, road, lake or pollution source, THEN no change

 IF a test pixel is non-urban AND there is no pollution pixel in its neighborhood,then four cases are defined:

1 IF three or more of the neighborhood pixels are urban, THEN change the testpixel to urban

2 IF one or more of the neighborhood pixels are road AND one or more areurban, THEN change the test pixel to urban

3 IF one or more of the neighborhood pixels are lake AND one or more areurban, THEN change the text pixel to urban

4 ELSE keep non-urban

The above cellular automata rules first check the growth constraint to preserve certainland cover classes (e.g water) then test the possibility of urban development for non-urban pixels based on the urbanization level in the neighborhood Figure 2 shows thesimulated urban growth results after 0, 25, 50 and 60 growth steps with the 3  3

neighborhood The effect of road on driving the urban development is clear where thegrowth rate is higher near the road and its pattern follows the road’s direction Highergrowth rate towards the lakes is also noticeable The restriction on growth in locationsclose to pollution sources succeeds in decreasing the urban development rate andcreates “buffer” zones around such places Finally, the growth constraint on waterpixels succeeds as well in conserving the water resources in future urban growth.

Step 50 Step 60

Figure 2 cellular automata urban growth modeling for synthetic city

Lake

Lake

Road Pollution

sources

River

Urban

Non-urban

4. MODELING OF CITY INDIANAPOLIS

This section applies the cellular automata urban modeling approach to a real city Thestudy area, transition rules, and evaluation criteria will be discussed

4.1 Study area and data

Indianapolis, Indiana, USA is selected for the study Indianapolis is located in MarionCounty at latitude 39° 44' N and longitude of 86° 17' W as shown in Figure 3 Itencounters recognizable accelerated growth in population and urban infrastructureover the last few decades It grows from a small part of Marion County in early 1970s

to cover the entire county and parts of the neighboring counties in 2003 The necessityarises to model the urban growth over time for sustainable planning and distribution

of infrastructure services

a Indianapolis (US Census Bureau) b Township map

Figure 3 City of Indianapolis and township map, Indiana, USA

0 2 4 8 12 16

Kilometers

Two types of data are used as input to the cellular automata model: land use data(thematic imagery) and population density Historical satellite images of the years

1973 (MSS 4 bands), 1982, 1987, 1992 and 2003 (TM 7 bands) in UTM NAD 1983projection are collected over the study area Images are classified using Anderson et al[1976] classification system to produce 5 land use maps containing 7 classes namely;water, road, residential, commercial, forest, pasture and row crops Commercial andresidential classes represent urban class of interest in this study All classified imagesare resampled to 60-m resolution as input to the cellular automata

Besides the images, year 1990 and 2000 population census tracts maps (see Figure 4,population per tract) are also used To prepare the population density grids as the input

to cellular automata, the following procedure is used for both years (1990 and 2000).The area of each census tract is computed and used to produce the tract populationdensity by dividing its population by its area The centroids for all census tractsbesides the overall city centroid for the study area are computed The distance fromeach census tract centroid to the city centroid is identified Population densities forcensus tracts within specified distance range are averaged to reduce the variability indata For example, an average population density for all census tracts within 2 km iscalculated, then another average density is calculated for tracts within 2-4 km and soforth An exponential function is fitted representing population density as a function

of distance from the city center for both year 1990 and 2000 separately:

POPULATION DENSITY A e B DISTANCE

from city centroid using Equation 5 These population density grids are used asanother input to cellular automata.

Figure 4 Population and census tracts for year 1990 (left) and 2000 (right)

4.2 Transition rules

The design of the cellular automata consists of defining the transition rules,calibrating them, and evaluating the modeling results for prediction purpose Cellularautomata transition rules are designed as a function of land use, growth constraints,and population density A 3x3 neighborhood is used to minimize the number of inputvariables to the model The rules identify the urban level in the neighborhood neededfor a test pixel to urbanize and take into account no growth constraints for certain landcover classes The effect of closeness to urban area and infrastructure is alsoconsidered in the rules definition The following rules are identified:

1 IF a test pixel is water, road or urban (residential or commercial), THEN nochange

2 IF a test pixel is non urban (forest, pasture or row crop), THEN:

- IF its population density is equal or greater than a threshold (P i) ANDthe number of neighborhood residential pixels is equal or greater than a

threshold (R i), THEN change the test pixel to residential

- IF its population density is equal or greater than a threshold (P i) ANDthe number of neighborhood commercial pixels is equal or greater than

a threshold (C i), THEN change to commercial

- ELSE keep non urban

4.3 Evaluation criteria and rule calibration

Evaluation and calibration are performed township by township A township map is asemi-grid as shown atop the image in Figure 3 There are a total of 24 townships in

the area Dividing the study area into townships will take into consideration the effect

of site specific features (spatial calibration) in each township on urban growth Thesame cellular automata transition rules are defined for all townships; however,different townships may have different rule values Spatial calibration is to find the

optimal cellular automata parameters values (R,C,P) i for each township For this

purpose, cellular automata is run for all possible combinations (R,C,P) i in the search

space The search space for both R i and C i is respectively from 0 to 8 (the possible

neighborhood size of 3x3 kernel) with integer increment of 1 The search space for P i

ranges from 0 to 3 with increment of 0.1 Cellular automata runs for a total of 2511(8x8x31) combinations

An evaluation scheme with three measures is designed for each township:

1 Fitness measure: this is a quantitative measure representing the ratio ofsimulated urban pixel count to the ground truth count:

_





count urban

truth Ground

count urban

Simulated Fitness (6)

2 Type I modeling error: this is the first qualitative measure used to identify theurban class modeling mistakes It counts the pixels that are urban in theground truth image but non-urban in the simulated image.

% 100 _

_

count Urban count TypeI

3 Type II modeling error: this is the second qualitative measure used to identifythe non-urban class modeling mistakes Type II error counts the pixels that arenon-urban in real but urban in the simulated image

% 100 _

_

_

count Urban No

count TypeII



count Total

count TypeII count TypeI

Fitness measure is introduced to indicate how a specific (R,C,P) i combinationsucceeds in reproducing the same real urban level within a township A rulecombination is said to overestimate the township urbanization level if the fitness ismore than 100%, while a fitness less than 100% means an underestimation of theurbanization level Type I and Type II errors represent the pixel by pixel differencebetween the simulation results and ground truth It also provides a strict measure forthe mismatch between the simulated and real urban patterns Such errors need to beminimized for accurate modeling Among all the rule combinations, the one withminimum total error and with fitness value closest to 100% (within ±10%) is selected

as the best

The cellular automata modeling starts running from 1973 till 1982, which is the firstground truth image used for calibration The best rule combination is selected based

on the above evaluation criteria In the next step, temporal calibration is implementedthrough recalibrating the cellular automata rules with 1987 ground truth image Thesame procedure above is repeated at 1987 to find the best set of rule values for eachtownship to reproduce the growth pattern at 1987 The objective of recalibration at

1987 is to take the temporal urban dynamics change into consideration in thecalibration process By doing this, the transition rules can be exposed to the changes

in urban growth pattern over time and hence can be adapted to such dynamics The next step is to evaluate the prediction capability of cellular automata modelingwithout calibration at the destination year The set of calibrated rules for all townshipsthat best simulate the ground truth at 1987 are used to predict the future growth at

1992 Prediction at 1992 represents short term prediction of 5 years Predicted image

at 1992 is evaluated on a township basis for the three evaluation measures The nextprediction is performed at 2003 for a long term period of 11 years starting from 1992using the best rules after calibration at 1992 Table 1 shows modeling results at thecalibrated years (1982 and 1987), while Table 2 shows those for 1992 and 2003prediction results Simulated images (1982 and 1987) and predicted images (1992 and2003) are shown in Figures 5 to 8, respectively

Table 1 Year 1982 and 1987 calibrated numerical results

Type I Error

%

Type II Error

Type II Error (%)

Table 2 Year 1992 and 2003 prediction numerical results

Township#

Fitness(%)

TypeIError(%)

TypeIIError(%)

E

 (%) Fitness

(%)

TypeIError(%)

TypeIIError(%)

REAL: 1982 CALIBRATED: 1982

Figure 5 CA calibrated image at 1982